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abstract: 'To lowest order in the coupling strength, the spin-orbit coupling in quantum dots results in a spin-dependent Aharonov-Bohm flux. This flux decouples the spin-up and -down random matrix theory ensembles of the quantum dot. We employ this ensemble and find significant changes in the distribution of the Coulomb blockade peak height, in particular a decrease of the width of the distribution. The puzzling disagreement between standard random matrix theory and the experimental distributions by Patel [*et al.*]{} might possibly be attributed to these spin-orbit effects.'
author:
- 'K. Held$^*$, E. Eisenberg, and B. L. Altshuler'
title: 'Random matrix theory for closed quantum dots with weak spin-orbit coupling'
---
The spin-orbit coupling in a two-dimensional semiconductor quantum well mainly contributes through the Rashba [@Rashba] and Dresselhaus [@Dress] terms, arising from the asymmetry of the confining potential and the lattice structure, respectively. It is much weaker than in three-dimensional semiconductors where it is induced mainly by impurities, which are absent in a high-mobility two-dimensional electron gas. The spin-orbit scattering is further suppressed if the two-dimensional system is confined to a quantum dot; estimates of the spin-flip rates were given in Ref. . This fact is of great importance for future applications of quantum dots as spintronics devices. However, it was shown that spin-orbit scattering has a significant effect in the presence of an in-plane magnetic field [@Halperin; @Oreg; @Khaetskii], which explains [@Halperin] recent experiments [@Folk].
In this paper, we discuss another manifestation of the spin-orbit coupling in confined structures, which takes place even in the absence of appreciable spin-flip scattering. Aleiner and Fal’ko recently showed [@Aleiner01a] that a weak spin-orbit coupling creates a spin-dependent Aharonov-Bohm flux. While this flux does not flip spins, it can change the random matrix ensemble of the quantum dot. For broken time reversal symmetry, the spin-up and spin-down parts of the spectrum are completely uncorrelated and described by independent Gaussian unitary ensembles (GUE) [@Aleiner01a]. The possibility of such an ensemble was raised by Alhassid [@Alhassid01], while the relation to the spin-orbit coupling was already suggested by Lyanda-Geller and Mirlin [@Lyanda]. In the present paper, we study the statistical distribution of the Coulomb blockade peak height in this ensemble, and find the distribution to be narrowed. This might explain the discrepancy between a recent experiment by Patel [*et al.*]{} [@Patel98] and standard random matrix theory (RMT) [@Alhassid98; @Alhassid01] at low temperatures.
In Ref. , the free-electron Hamiltonian with Rashba and Dresselhaus spin-orbit terms was expanded to second order in the coordinates, under the assumption that $L_{1,2}/\lambda_{1,2}\ll 1$ ($L_{1,2}$: lateral dimensions of the two-dimensional quantum dot; $\lambda_{1,2}$: characteristic length scale of the spin-orbit coupling which is proportional to the inverse spin-orbit coupling strength). One obtains $$\begin{aligned}
\tilde{H} &=&\frac{1}{2m}\left( \vec{p}-e\vec{A}-\vec{a}_{\bot }\frac{\sigma
_{z}}{2}-\vec{a}_{\Vert }\right) ^{2}+u(\vec{r}). \label{Ham}\end{aligned}$$ Here, $u(\vec{r})$ is the (disordered) confining potential; $\vec{p}=\vec{P}-e\vec{A}$ is the kinetic momentum with the canonical momentum $\vec{P}$ and the vector potential $$\begin{aligned}
\vec{A} &=&B_{z}[\vec{r}{\bf \times }\vec{n}_{z}]/2c;\hspace{0.12in}\vec{a}%
_{\bot }=[\vec{r}{\bf \times }\vec{n}_{z}]/(2\lambda _{1}
\lambda_{2});\hspace{%
0.12in} \label{aperp} \\
\vec{a}_{\Vert } &=&\frac{1}{6}\frac{[\vec{r}{\bf \times }\vec{n}_{z}]}{%
\lambda _{1}\lambda _{2}}\left( \frac{x_{1}\sigma _{1}}{\lambda _{1}}+\frac{%
x_{2}\sigma _{2}}{\lambda _{2}}\right) \label{apar};\end{aligned}$$ $\sigma_i$ denote the Pauli matrices and $B$ is the magnetic field in the direction \[001\] perpendicular to the lateral quantum dot. The coordinates $x_1$ and $x_2$ are along the directions $[110]$ and $[1\bar{1}0]$ and we neglected the Zeeman splitting term as we are interested in the behavior at relatively low magnetic fields. The term $\vec{a}_{\Vert }$ is responsible for spin-flips but it is of higher order in the spin-orbit coupling strength than $\vec{a}_{\bot}$. Thus, it will be neglected in the following as we assume the spin-orbit coupling to be weak such that $\vec{a}_{\bot}$ dominates. The $\vec{a}_{\bot}$ term has exactly the same form as the vector potential $\vec{A}$ except for its spin-dependence. As an electron collects an Aharonov-Bohm flux on a close path due to the vector potential $\vec{A}$, it also collects a spin-dependent flux due to $\vec{a}_{\bot}$. This spin dependent flux translates to a spin-dependent effective magnetic field, so that the electrons feel a total magnetic field of strength $B^{\rm eff}_{\sigma}= B+ \frac{c}{e}\frac{1}{\lambda_1 \lambda_2}
\frac {\sigma}{2}
$ with $\sigma=\pm\hbar$ for up- and down-spin, respectively. An increase of the flux changes the matrix elements, and scrambles the eigenenergies and eigenvectors. In the absence of spin-orbit coupling, the flux is exactly the same for spin-up and spin-down electrons such that their eigenenergies and eigenvectors are degenerate. If the spin-orbit terms are present, but no external magnetic field is applied, the time-reversal symmetry is preserved, and the states are still Kramers degenerate (up-spin and down-spin see the same magnitude of magnetic field with opposite signs). However, when spin-orbit coupling and external magnetic field are present, electrons with different spin see different magnetic fields, and their eigenenergies and eigenvectors decorrelate. If the spin-dependent flux is large enough spin-up and spin-down eigenenergies and eigenvectors are distributed according to two [*independent*]{} GUEs [@Aleiner01a].
Before we analyze this weak spin-orbit RMT ensemble, we study the decorrelation of the eigenenergies and eigenvectors due to a change in the magnetic field, to determine how much flux is needed in order to have two uncorrelated ensembles. The additional flux translates to a change of the random matrix described by [@Aleiner01] $$H= \frac{H_1+x H_2}{\sqrt{1+x^2}}
\label{RMTmodel}$$ with RMT matrices $H_1$ and $H_2$ in the unitary ensemble. As the perturbation $x$ increases from zero the eigenenergies $E_i(x)$ and eigenfunctions $\Psi_i(x)$ of $H$ change. We analyze the decorrelations of the energies via the level diffusion correlator $C_E \!=\! \langle\!\langle\sqrt{(E_i(x)-E_i(0))^2 }\rangle\!\rangle /
\Delta$, where $ \langle\!\langle\cdots\rangle\!\rangle $ means averaging over different realizations and different levels $i$. This correlator has been shown to have a universal form [@Attias95]. The decorrelation of the eigenfunctions is measured by $C_{\Gamma}\! =\! \langle\!\langle |\, \langle \psi_i(x)|\psi_i(0)\rangle
\,|^2 \rangle\!\rangle$. It can be shown that this correlator also measures the correlations of the level tunneling rates and has a universal form as well [@Alhassid96]. The results are presented in Fig. \[figcorr\] and show that the correlations in both quantities disappear at about the same value of $x\sqrt{N}\! \approx\! 1$, where $N$ is the size of the random matrix. Hence we conclude that the decorrelation of the eigenvalues and the eigenfunctions (dot-lead coupling) occur together. Thus, the spin-orbit coupling leads to a crossover from two degenerate GUE spectra, to an ensemble of two uncorrelated GUE spectra.
![Correlations of the level tunneling rates $C_\Gamma$ and the rescaled spectral diffusion correlator $C_E$ for the RMT model (\[RMTmodel\]), as a function of $x\sqrt{N}$ for different matrix sizes $N$. The data collapse for different $N$ indicates the universality. []{data-label="figcorr"}](rmt2.eps){width="2.4in"}
For the above RMT model, the crossover to two independent GUE ensembles occurs at $x\sqrt{N}\! \approx\! 1$. The corresponding flux difference needed to decorrelate the spectrum is given by the following relation [@Aleiner01] $${x}\sqrt{N}=\chi\sqrt{g_T}\frac{\delta\Phi_{\rm eff}}{\Phi_0}
\label{estPhi}$$ where $\delta\Phi_{\rm eff}$ is the flux difference, $\Phi_0$ is the quantum unit of flux, $g_T$ denotes the Thouless conductance, and $\chi$ is a non-universal sample-dependent constant of order unity. We thus realize that one needs about $1/\sqrt{g_T}$ flux quanta to crossover to two uncorrelated GUE ensembles.
Let us now estimate the strength of spin-orbit interaction required to create this amount of flux difference. As mentioned above, the difference in effective flux between the two spin sectors is $\delta\Phi_{\rm eff}/\Phi_0=
L_1L_2/(\lambda_1\lambda_2)$. The $\lambda$’s are connected to the Rashba and Dresselhaus spin orbit parameters $\gamma$ and $\eta$ via $1/\lambda_1\lambda_2 =4(\gamma^2-\eta^2)$. Independent estimates of $\gamma$ and $\eta$ are not available, but, in principle, can be obtained [@Aleiner01a]. One can get an approximate value via the better known parameter $Q_{\rm SO}^2=({\hbar v_F}/{E_F} )^2 (\gamma^2+\eta^2)$: $$\begin{aligned}
\left | \frac{1}{\lambda_1\lambda_2} \right | & \leq &
Q_{\rm SO}^2 \left (\frac{2E_F}{\hbar v_F}\right )^2 = Q_{\rm SO}^2 k_F^2 \\
\frac{\delta\Phi_{\rm eff}}{\Phi_0}& = & Q_{\rm SO}^2 \, k_F L_1 \, k_F L_2\end{aligned}$$ Typical experimental values are $k_F L_{1,2}\sim 50$, $g_T\sim 10-100$, and estimates for $Q_{\rm SO}$ are in the range $4-16 \times 10^{-3}$ [@Halperin]. Thus, we estimate $\sqrt{g_T}\; \delta\Phi_{\rm eff}/\Phi_0=0.1-6.4$, i.e., the right hand side of Eq. (\[estPhi\]) can be expected to be of order unity [@Zumbuehl02]. Hence, the spin-orbit effect is strong enough to decorrelate (or to start to decorrelate) the spin-up and spin-down sector, while being weak enough not to yield significant spin-scattering. A strong spin-scattering, which can be generated by the application of an in-plane magnetic field, would mix the two spin species and result in a single GUE.
We will now analyze this situation where the weak spin-orbit coupling results in two uncorrelated GUE ensembles for spin-up and spin-down electrons and the results do not depend on the spin-orbit strength. Under the assumption of a constant Coulomb interaction [@Aleiner01] and applying the Master equation for sequential tunneling through the quantum dot, the conductance of a closed quantum dot is given by [@Beenakker91] (for a review see Ref. ) $$\!G\!=\!\frac{e^2}{{\rm k_B} T}
%\!\!\sum_{\sigma=\uparrow,\downarrow}\! \sum_{i=1}^\infty
\!\! \sum_{i\sigma}
\!
\frac{\Gamma^L_{i \sigma}
\Gamma^R_{i \sigma}}{\Gamma^L_{i \sigma}\!+\!\Gamma^R_{i \sigma}}
P_{\rm eq}(N)P(E_{i \sigma}|N)[1\!-\!f(E_{i \sigma}\!-\!\mu)]. \label{Eq:G}$$ Here $\Gamma^{L(R)}_{i \sigma}$ is the tunneling rate between the $i$th one-particle eigenlevel of the dot with spin $\sigma$ and the left (right) lead, $E_{i \sigma}$ is the one-particle eigenenergy of this level, $P_{\rm eq}(N)$ denotes the equilibrium probability to find $N$ electrons in the dot (we assume the typical experimental situation where the Coulomb blockade only allows $N$ and $N+1$ electrons in the quantum dot), $P(E_{i \sigma}|N)$ is the canonical probability to have the $i$th level of the spin-$\sigma$ sector occupied given the presence of $N$ electrons in the dot, and $f(E-\mu)$ is the Fermi function at the effective chemical potential $\mu$ which includes the charging energy. In Eq. (\[Eq:G\]), $\Gamma^{L(R)}_{i \sigma}$ is distributed according to the Porter-Thomas distribution for the GUE $P_{2}(\Gamma) = \frac{1}{\overline{\Gamma}} \exp(-\Gamma/
\overline{\Gamma})$, which only depends on the mean value $\overline{\Gamma}$ of the distribution (we assume this mean value to be the same for the coupling to the left and right lead in the following). At zero temperature, only one level ($i_1$,$\sigma_1$) contributes in Eq. (\[Eq:G\]) such that $\mu=E_{i_1 \sigma_1}$, $P(E_{i \sigma}|N)=1$, and $P_{\rm eq}(N)=1/2$. Thus, the zero temperature average conductance is given by $\langle G\rangle = \frac{1}{12} \frac{\hbar \bar{\Gamma}}{k_B T} \frac{e^2}{\hbar}$ and the ratio of standard deviation to mean-value becomes $\sigma(G)/\langle G\rangle=2/\sqrt{5}$. Here, we have used $\langle{\Gamma^R_{i \sigma}}/{(\Gamma^L_{i \sigma}\!+\!\Gamma^R_{i \sigma})} \rangle={1}/{3}$ and $\langle{\Gamma^R_{i \sigma}}^2/({\Gamma^L_{i \sigma}\!+\!\Gamma^R_{i \sigma}})^2 \rangle={1}/{5}$ for the GUE distribution. At low temperatures, there are a few realizations of the RMT eigenlevel distribution where a second level ($i_2$,$\sigma_2$) is within an interval of order $k_BT$ around the first level at the Fermi energy. Then, the second level also contributes to the conductance through the quantum dot. Neglecting the shift of the chemical potential due to the second level (i.e., keeping $\mu=E_{i_1 \sigma_1}$), we calculated this two level situation. This gives the leading behavior in $k_B T/\Delta$ for Eq. (\[Eq:G\]): $$\begin{aligned}
%\!\!\!\!\!\!\!\langle G\rangle &\!\!=\!\!& \frac{1}{12} \left(1+[9 \ln2-5\ln3]
%\frac{k_B T}{\Delta}\right)\frac{\hbar \bar{\Gamma}}{k_B T} \frac{e^2}{\hbar},\\
\!\!\!\!\!\!\!\frac{\sigma(G)}{\langle G\rangle}&\!\!=\!\!&\frac{2}{\sqrt{5}}
\left(\!1\!+\!\left[\frac{781}{9} \ln 2\! -\!\frac{127}{3} \ln3\!-\!\frac{409}{27}\right]\frac{k_B
T}{\Delta}\!\right)\!.
\label{Eq:lowT}\end{aligned}$$
For general temperatures, we maximize numerically the conductance Eq. (\[Eq:G\]) w.r.t. $\mu$, and averaged over 100000 RMT realizations of the eigenenergies and the dot-lead couplings. The results are shown in Fig. \[fig:Sigma\], in comparison to the experiment of Patel [*et al.*]{}. In contrast to the standard RMT result [@Patel98], the RMT ensemble for weak spin-orbit coupling describes the width of the conductance distribution and its change with temperature reasonably well at low temperatures, without any adjustable parameter. Compared to the standard GUE, the width of the distribution is reduced at low temperatures because of the absence of level repulsion for levels with opposite spin. This results in higher probability to find a close-by level (with opposite spin and independent tunneling rate), and leads to more RMT realizations in which two or more levels contribute to the low-temperature conductance. Having more independent channels for the conductance makes the probability distribution more Gaussian and decreases its width. At higher temperatures, the experimental results are not adequately described by spin-orbit effects alone. In the regime $k_B T \! \gtrsim \!\Delta$ however [@Eisenberg01], one has to account for inelastic scattering $\Gamma_{\rm in}$. Taking the limit $\Gamma_{\rm in}\! \rightarrow\! \infty$, we obtain the high temperature asymptotic behavior $\frac{\sigma(G)}{\langle G\rangle} \!=\!\sqrt{\frac{1}{24}\frac{\Delta}{k_BT}}$ which gives reasonable results except for the quantum dot with diamond-symbols. Note that upon reducing $k_B T/\Delta$, $\Gamma_{\rm in}$ will decrease, resulting in a crossover from the dot-dashed to the dashed line in Fig. \[fig:Sigma\]. The inelastic scattering rates of [@Eisenberg01] would imply that the dashed $\Gamma_{\rm in}\!=\!\infty$-line is approached in the range $k_B T$=1.5-4 $\Delta$.
![Width of the conductance distribution $\sigma(G)/\langle G\rangle$ vs. temperature. At low temperatures, the RMT ensemble for weak spin-orbit interaction \[dashed line; dotted line: low temperature behavior Eq. (\[Eq:lowT\])\] well describes the experiment [@Patel98] (symbols correspond to slightly different quantum dots), in contrast to standard RMT (solid line) [@Patel98]. At higher temperatures, a further suppression is due to inelastic scattering processes (dot-dashed line: $\Gamma_{\rm in}\!= \!\infty$ high-$T$ asymptote). []{data-label="fig:Sigma"}](figSigma.ps){width="2.8in"}
In Fig. \[fig:Stat\], we compare the full probability distribution with the experimental one [@Patel98] at $k_B T=0.1\Delta$ and $k_B T=0.5\Delta$. Within the experimental statistical fluctuation, a good agreement is achieved without any free parameters, much better than for the standard RMT [@Patel98]. This suggests that the spin-orbit strength is sufficient to fully decorrelated the spin-up and spin-down ensembles. With an estimate of the experimental Thouless conductance $g_T \approx 20$ obtained from $g_T\approx \sqrt{N}$, this means that a spin-orbit coupling strength $Q_{\rm SO} \gtrsim 10^{-2}$ is required in the quantum dot of Ref. (where we set $\chi=1$ in Eq. (\[estPhi\])). In general, the crossover to the weak spin-orbit regime occurs at $Q_{\rm SO}^2 (k_F L)^{5/2}\gtrsim 1$. Thus, the size of the dot and the parameter $Q_{\rm SO}$ which depends on the dot’s specific asymmetry of the confining potential determine whether this quantum dot is in the weak spin-orbit regime. The size dependence might explain why earlier measurements by Chang [*et al.*]{} [@Chang96] using very small quantum dots showed agreement with the standard RMT without spin-orbit interaction. A similar agreement was found by Folk [*et al.*]{} [@Folk96], despite using similar large quantum dots as in Ref. . The contradictory results of [@Folk96] and [@Patel98] might be due to the better statistics of the latter experiment, or could be explained within the framework presented here, as following from differences in the confining potential (which might be, e.g., caused by differences in the realization of the two dimensional electron gas and the gate voltage), translating into differences in $Q_{\rm SO}$. Alternatively, it is possible that the spin-orbit effect in both samples is weak, and the deviations from RMT in [@Patel98] should be explained by another mechanism (e.g. exchange [@UsajPC]).
In order to validate that the quantum dot is indeed in the weak spin-coupling regime described here, we suggest to repeat the experiment with an in-plane magnetic field. A strong in-plane magnetic field should drive the system towards the strong spin-orbit scattering limit. In general, one would expect the spin-orbit scattering to suppress $\sigma(G)/\langle G\rangle$. However, in the case of weak spin-orbit coupling the in-plane magnetic field, which drives the system towards a single GUE, regenerates the level repulsion. Therefore, we predict $\sigma(G)/\langle G\rangle$ to increase upon applying an in-plane magnetic field at low temperatures. Another crucial test to the the weak-spin orbit scenario is the behavior in the absence of a magnetic field. In this case, the degeneracy is preserved but the spin-orbit coupling drives the system from the Gaussian orthogonal to the unitary ensemble [@Aleiner01a]. One implication is a strong suppression of the magnetoconductance.
![RMT predictions with weak spin-orbit coupling (dashed line) for the probability distribution of the Coulomb blockade peak conductance for a quantum dot at $k_B T=0.1\Delta$ (left figure) and $k_B T=0.5\Delta$ (right figure), compared with the Patel [*et al.*]{} experiment [@Patel98] (histograms) and standard RMT theory [@Patel98] (solid line). There are no free parameters in these distributions.[]{data-label="fig:Stat"}](figStat01.ps "fig:"){width="1.8in"} ![RMT predictions with weak spin-orbit coupling (dashed line) for the probability distribution of the Coulomb blockade peak conductance for a quantum dot at $k_B T=0.1\Delta$ (left figure) and $k_B T=0.5\Delta$ (right figure), compared with the Patel [*et al.*]{} experiment [@Patel98] (histograms) and standard RMT theory [@Patel98] (solid line). There are no free parameters in these distributions.[]{data-label="fig:Stat"}](figStat05.ps "fig:"){width="1.83in"}
Finally we note that the disagreements between RMT predictions and the results of [@Patel98] can not be attributed to dephasing. Had this been the case, this experiment would indicate an appreciable dephasing even at low temperatures, in contradiction with theoretical predictions [@Altshuler97]. However, recent measurements of the low-temperatures dephasing rates [@Folk00] are consistent with theory [@Eisenberg01], and furthermore, it has been shown by Rupp and Alhassid [@Rupp02] that dephasing alone can not explain the results of [@Patel98]. Our calculation shows that the spin-orbit coupling without dephasing can describe the low-temperature part of [@Patel98], and that the inclusion of strong dephasing gives reasonable agreement for the high-temperature part.
In conclusion, we analyzed the effect of weak spin-orbit coupling on closed quantum dots in the presence of a perpendicular magnetic field which breaks the time-reversal symmetry. In this regime which can be realized for (some) quantum dots, the spin-orbit coupling does not lead to one non-degenerate GUE ensemble but to two independent GUEs for spin-up and -down electrons. This has important consequences, in particular, at low temperatures, as there is no level-repulsion for levels with opposite spins. The statistical distribution of the conductance peak maximum shows a good agreement with recent experimental distributions by Patel [*et al.*]{} [@Patel98], but disagrees with experiments for similar sized quantum dots [@Folk96]. The exchange interaction might yield similar changes in the statistical distribution [@UsajPC], and it is unclear at present whether the complete explanation for the peak heights statistics behavior is given by the weak spin-orbit RMT. More experiments are needed to clarify the relative importance of the two effects and to explain the experimental contradiction mentioned above. If the spin-orbit effect is dominant, we predict an increase of the width of the distribution upon applying a strong in-plane magnetic field and a very low magnetoconductance. We further note that without spin-degeneracy there will also be [*no*]{} $\delta$-function-like contribution in the level-spacing distribution, in contrast to standard RMT.
We acknowledge helpful discussions with I.L. Aleiner, Y. Alhassid, H.U. Baranger, C.M. Marcus, T. Rupp, and G. Usaj. This work has been supported by ARO, DARPA, and the Alexander von Humboldt foundation.
[10]{}
E. I. Rashba, Sov. Phys. Solid State 2, 1109 (1960). Yu. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984); G. Lommer, F. Malcher, and U. Rössler, Phys. Rev. Lett. [**60**]{}, 728 (1988).
G. Dresselhaus, Phys. Rev. [**100**]{}, 580 (1955).
A. V. Khaetskii and Yu. V. Nazarov, Phys. Rev. B [**61**]{}, 12639 (2000).
B. I. Halperin [*et al.*]{}, Phys. Rev. Lett. [**86**]{}, 2106 (2001).
For a review see, Y. Oreg [*et al.*]{}, in [*Nano-Physics and Bio-Electronics*]{}, T. Chakraborty, F. Peeters, and U. Sivan (eds.), Elsevier (2002).
J. A. Folk [*et al.*]{}, Phys. Rev. Lett. 86, 2102 (2001).
I. L. Aleiner and V. I. Fal’ko, Phys. Rev. Lett. [**87**]{}, 256801 (2002).
Y. Alhassid, Rev. Mod. Phys. [**72**]{}, 895 (2000).
Y. B. Lyanda-Geller and A. D. Mirlin, Phys. Rev. Lett. [**72**]{}, 1894 (1994).
S. R. Patel,[*et al.*]{}, Phys. Rev. Lett. [**81**]{}, 5900 (1998).
Y. Alhassid, M. Gökçedağ, and A. D. Stone, Phys. Rev. B [**58**]{}, R7524 (1998).
I. L. Aleiner, P. W. Brouwer and L. I. Glazman, Physics Reports [**358**]{}, 309 (2002).
H. Attias and Y. Alhassid, Phys. Rev. [**E 52**]{}, 4776 (1995).
Y. Alhassid and H. Attias, Phys. Rev. Lett. [**76**]{}, 1711 (1996).
Zumbühl [*et al.*]{}, Phys. Rev. Lett. [**89**]{}, 276803 (2002), experimentally estimated $\lambda_{1,2}$=4-5$\,\mu$m for a $5.8\,\mu$m$^2$ open dot. If transferred directly to the smaller dot of [@Patel98] this would correspond to the lower limit of our estimate.
C. W. J. Beenakker, Phys. Rev. B [**44**]{}, 1646 (1991).
E. Eisenberg, K. Held, and B. L. Altshuler, Phys. Rev. Lett. [**88**]{}, 136801 (2002).
A. M. Chang [*et al.*]{}, Phys. Rev. Lett. [**76**]{}, 1695 (1996). J. A. Folk [*et al.*]{}, Phys. Rev. Lett. [**76**]{}, 1699 (1996).
B. L. Altshuler [*et al.*]{}, Phys. Rev. Lett. [**78**]{}, 2803 (1997).
J. A. Folk, [*et al.*]{}, Phys. Rev. Lett. [**87**]{}, 206802 (2001).
T. Rupp, Y. Alhassid, and S. Malhotra, Phys. Rev. [**B 65**]{}, 193304 (2002).
G. Usaj and H. U. Baranger, cond-mat/0211649; T. Rupp and Y. Alhassid, cond-mat/0212126.
| ArXiv |
---
abstract: 'We consider two systems of wave equations whose wave-packet solutions have trajectories that are altered by the “anomalous velocity” effect of a Berry curvature. The first is the matrix Weyl equation describing cyclotron motion of a charged massless fermion. The second is Maxwell equations for the whispering-gallery modes of light in a cylindrical waveguide. In the case of the massless fermion, the anomalous velocity is obscured by the contribution from the magnetic moment. In the whispering gallery modes the anomalous velocity causes the circumferential light ray to creep up the cylinder at the rate of one wavelength per orbit, and can be identified as a continuous version of the Imbert-Federov effect.'
author:
- MICHAEL STONE
title: Berry phase and anomalous velocity of Weyl fermions and Maxwell photons
---
Introduction
============
In many quantum systems the motion of a wave-packet is governed by semiclassical equations of the form [@blount; @sundaram-niu; @niu; @horvathy] &=& -+ e(), \[EQ:lorentz\]\
&=& + . \[EQ:chang-niu\] In the absence of the last term in the second equation these would just be Hamilton’s equations for a particle with hamiltonian ${\mathcal H}({\bf x},{\bf k})=\varepsilon({\bf k})+V({\bf x})$ moving in a magnetic field. The additional $\dot {\bf k}\times {\bm \Omega}$ term in (\[EQ:chang-niu\]) is the [*anomalous velocity*]{} correction to the na[ï]{}ve group velocity $\partial \varepsilon/\partial{\bf k}$.
The vector ${\bf \Omega}$ is a function of the kinetic momentum ${\bf k}$ only, and is a Berry curvature which has different origins in different systems. For a Bloch electron in an energy band in a solid the curvature accounts for the effects of all other bands. In particle-physics applications the curvature arises from the intrinsic angular momentum of the particle. In all cases it affects the velocity because different momentum components of a localized wave-packet accumulate different geometric phases when both ${\bf k}$ is changing and the Berry curvature is non-zero [@chong]. These ${\bf k}$-dependent geometric phases are just as significant in determining the wave-packet position as the ${\bf k}$-dependent dynamical phases arising from the dispersion equation $\omega=\varepsilon({\bf k})$.
A particularly simple example occurs in in the dynamics of massless relativistic fermions and in Weyl semimetals where bands touch at a point. In both systems the wavepackets are solutions of a Weyl “half-Dirac” equation and the Berry curvature arises because the spin (or pseudo-spin) vector is locked to the direction of the momentum ${\bf k}$. The forced precession of a spin-$S$ vector is the paradigm in the original Berry-phase paper [@berry1] and the corresponding curvature is simply ([**k**]{})= S . \[EQ:berry-curvature\] Even this basic example gives rise to much physics — the axial and gauge anomalies [@stephanov; @stone-dwivedi; @dwivedi-stone] and the chiral magnetic and vortical effect [@vilenkin; @fukushima; @son3].
The semiclassical analyses that reveal the anomalous velcity in (\[EQ:chang-niu\]) are quite intricate as they have to go beyond the leading WKB ray tracing equation. It is the aim of this paper to consider two simple systems in which the predicted anomalous velocity effect can be sought directly in the stationary eigenfunctions of underlying wave equation. In both cases the curvature is given by (\[EQ:berry-curvature\]). The first (in section \[SEC:cyclotron\]) is the circular motion of a massless charged spin-1/2 particle in a magnetic field. The second (in section \[SEC:whisper\]) is the circular motion of a spin-1 photon in an optical fibre waveguide. In the first case the presence of the anomalous velocity is obscured by the coupling of the magnetic field to the particle’s magnetic moment. The second unambiguously displays the expected anomalous velocity drift.
Cyclotron orbits {#SEC:cyclotron}
================
We start by considering the cyclotron motion of a massless Weyl fermion with positive charge $e$ in a magnetic field ${\bf B}= -B \hat{\bf z}$. The field is derived from a vector potential ${\bf A}= B(y,-x)/2$ and its downward direction has been chosen so that the particle orbits in an anti-clockwise direction about the $z$ axis.
The Weyl Hamiltonian for a right-handed spin-1/2 particle is H= -i(-ie[**A**]{}), where ${\bm \sigma}=(\sigma_1,\sigma_2,\sigma_3)$ denotes the Pauli matrices. We are using natural units in which $\hbar =c=1$, although we will occasionally insert these symbols when it helps to illuminate the discussion.
Acting on functions proportional to $e^{ik_z z}$ we have H\^2 = [I]{}(- \^2 + r\^2 +eBL\_3 +k\_z\^2) + eB\_3, \[EQ:weyl-squared\] where $L_z= -i(x\partial_y-y\partial x)=-i\partial_\phi$ is the canonical (as opposed to kinetic) angular momentum and ${\mathbb I}$ denotes the 2-by-2 identity matrix. The eigenvalues of the scalar Shrödinger operator in parenthesis in (\[EQ:weyl-squared\]) are E\^2\_[n,l,k\_z]{} = eB{2n+|l|+l+ 1}+k\_z\^2, and the corresponding eigenfunctions are \_[n,l,k\_z]{}(r,) =()\^[(|l|+1)/2]{} r\^[|l|]{}(-) L\^[|l|]{}\_n() e\^[il]{}e\^[ik\_z z]{}. Both $n$ and $l$ are integers and $L^{|l|}_n$ is the associated Laguerre polynomial. When $n=0$, $k_z=0$, and $l>0$, the wavefunction $\varphi_{0,l,0}(r,\phi)$ corresponds to a particle describing a circular cyclotron orbit with the origin as its centre and radius R\_l=. If we decrease $l$ while staying in the same Landau level ([*i.e.*]{} by increasing $n$ so as to keep $E^2_{n,l}$ fixed) the classical circular orbit keeps the same radius but its centre moves away from the origin and is smeared-out in $\theta$ over the full $2\pi$. When $l=0$ the circle passes through the origin. For $l$ negative, the energy no longer depends on $l$ and the Landau level keeps $n$ fixed while $l$ continues to decrease. The classical orbit still has the original radius, but no longer encloses the origin. In particular the case $n=k_z=0$ and $l<0$ corresponds particles in the lowest Landau level but with different orbit centres.
By applying the projection operator $P= (E+H)/2E$ to the Schrödinger eigenfunction we find that the cyclotron-motion eigenfunctions of the Weyl hamiltonian $H$ with $n=0$, $l>0$, and longitudinal momentum $k_z$ are \_[0,l,k\_z]{}(r,,z)= e\^[ikz]{}e\^[il]{}r\^l (-). These states have energy $E_{l, k_z} = \sqrt{2leB+k_z^2}$ and the orbit radius is still R\_l= . At $k_z=0$, the angular velocity of a wave-packet is = .|\_[k\_z=0]{}= ensuring that $v_\phi= R_l\dot \phi=c=1$.
There is a special case where $l=n=0$ and \_[0,0, k\_z]{}=e\^[ik\_z z]{} (
01
) (-) with $E=-k_z$. This mode only exists as a positive energy mode for $k_z<0$. It is this unbalanced mode, with a density of $eB/2\pi$ per unit area in the $x$, $y$ plane that is the source of the chiral-magnetic-effect current \_[CME]{}= \_0\^= , of a gas of zero-temperature Weyl fermions with chemical potential $\mu$ [@vilenkin; @fukushima].
Consider the $l>0$, $k_z=0$ orbits. Even though these orbits possess no component of momentum in the $z$ direction, plugging the time dependence of the classical orbital momentum ${\bf k}$ into the anomalous-velocity formula (\[EQ:chang-niu\]) suggests that they should creep down the $z$ axis. To compute the predicted creep-rate we observe that a particle of helicity $S$ whose spin direction is forced to describe a circle of co-latitude $\theta$ on a sphere with polar co-ordinates $\theta$, $\phi$ accumulates Berry phase at the rate [@berry1] \_[Berry]{} =-S(1-). For $S=+1/2$, and using our expression for $\dot \phi$, this becomes \_[Berry]{}= 12(-1)= 12 (-1). where = \~. In an energy eigenstate this accumulating geometric phase should be indistinguishable from the accumulating $-Et$ dynamical phase. In other words $\dot \gamma_{\rm Berry}$ should appear as a contribution to the energy of E\_[Berry]{}&=& [const]{}. - 12\
&\~& [const]{}. -12 . This energy adds z= = -12 \[EQ:drift1\] to the group velocity over above that expected from the $E=c|{\bf k}|$ energy-momentum relation. The velocity (\[EQ:drift1\]) corresponds to drift rate of one-half of a de Broglie wavelength $\lambda_{\rm Broglie}$ per orbit — the de Broglie wavelength being here defined as the wavelength associated with the kinetic momentum ${\bf k}$ so that $
E = {2\pi \hbar c }/{\lambda_{\rm Broglie}}
$.
Unfortunately, except for the special case $l= 0$, there is no sign of any contribution to the energy linear in $k_z$ in the exact solution of the eigenvalue problem! Instead we have E== + 12 k\_z\^2/ +. The reason for the absence is that there is another linear-in-$k_z$ contribution to the energy coming from the Weyl particle’s magnetic moment [@son3].
One might question whether a massless particle can have a magnetic moment as no time ever passes on a null-vector world-line. Nonetheless, in the laboratory frame, a particle obeying the Weyl equation possesses an energy-dependent effective moment of $${\bm \mu}_{\rm Weyl}=\pm \frac{e\hat{\bf k}}{2E}. \quad \hbox{($\pm $ for positive/negative helicity)}.$$ This moment is precisely what is required for the Larmor precession frequency $$\Omega_{\rm Larmor}= -B \left|\frac{\bm \mu}{\bf S}\right|$$ to coincide with the orbital frequency and so ensure that the spin remains aligned with the momentum. The appendix contains a derivation of ${\bm \mu}_{\rm Weyl}$ from the Weyl equation and explains how the Dirac gyromagnetic ratio $g=2$ continues to to be valid even for massless particles. For a positive-helicity particle the interaction of this moment with the magnetic field provides an energy shift of $$\delta E_{\rm magnetic}=-{\bm\mu}_{\rm Weyl}\cdot {\bf B}= -\frac{e{\bf B}\cdot {\hat {\bf k}}}{2E}= \frac{eB\cos\theta}{2\sqrt{2leB}}=+\frac 12 \frac{k_z}{2l} .$$ that precisely cancels the Berry phase contribution. This cancellation does not apply to all eigenstates and does not affect the total chiral magnetic effect, but as explained in [@son3] it does complicate its simple semiclassical derivation in [@stephanov].
Photons in whispering-gallery modes {#SEC:whisper}
===================================
We were thwarted in our attempt to observe an anomalous drift velocity in an exact solution of the Weyl equation because the accumulating geometric Berry phase was obscured by a dynamical phase arising from the particle’s magnetic moment. We therefore seek the effect in the motion of a moment-free massless spinning particle. An obvious candidate is the spin-1 photon. To make it clear that a photon should obey similar semiclassical equations to a massless fermion it helps to rewrite Maxwell’s equations in a form that affords a direct comparison with the Weyl equation. The idea for a such a rewriting is apparently due to Riemann. The complete set of Maxwell’s equations equations were first published in 1865 and Riemann died in 1866. The rewriting is nonetheless ascribed to him by Heinrich Weber in his edition of Riemann’s lectures published in 1901 [@riemann]. The idea was independently discovered by Silberstein [@silberstein] in 1907, and has been recently extensively championed by the Bia[ł]{}ynicki-Birula’s [@birula]. The Riemann-Silberstein equations make use the complex-valued fields \^&=& [**E**]{}i c[**B**]{}\
&=& [**E**]{}i Z [**H**]{}, where Z= is the wave impedance. Provided that $Z$ does not vary with position or time, we can combine the two Maxwell “curl" equations as i \_t \^\_i = c\_[local]{} \_[ijk]{} \_j \^\_k . \[EQ:RS1\] Here $c_{\rm local}\equiv 1/\sqrt{\mu\epsilon}$ may vary with position. At non-zero frequency, the two Maxwell “divergence” equations follow from curl equations and do not need to be separately imposed. Once we define the spin-1 generator ${\bm \Sigma}_i$ to be the matrix with entries $- i \epsilon_{ijk}$, eq. (\[EQ:RS1\]) becomes a pair of Weyl equations i\_t \^= c\_[local]{} () \^, one for the left-helicity chiral field and one for the right-helicity field. to make the analogy with the Dirac-Weyl equation as close as possible, we have inserted an $\hbar$ on both sides of (\[EQ:RS1\]) so that we can exhibit the equation in terms of the quantum mechanical momentum operator $\hat {\bf p}= -i\hbar \nabla$. We conclude that as the direction of the wave-momentum vector precesses, the photon spin is forced to follow it. The photon wavefunction will then acquire a geometric Berry phase that is twice as a large as that of the Weyl fermion, and this phase must have a similarly-proportioned effect on the semi-classical particle trajectory.
The only significant difference between the spin-$1/2$ Weyl equation and Maxwell equations is that the Maxwell “wave-function” obeys a Majorana condition (\^+)\^\*= \^- that indicates that the photon is its own antiparticle.
To obtain an optical analogue of cyclotron motion, we consider the whispering-gallery modes of light in a step-index optical fibre. In whispering-gallery modes a light beam orbits the fibre circumferentially rather than propagating along its length. We may think of the orbit as a closely spaced sequence of total internal reflections off the step discontinuity in the refractive index. (See figure \[FIG:bliokh\]). A separation of the orbits of left and right circularly polarized beams was predicted from the anomalous velocity equations in [@bliokh-bliokh] and experimentally verified in [@bliokh-verification]. In this section we seek to derive the separation directly from the Maxwell-equation eigenmodes.
Recall that the refractive index $n$ is obtained from the material parameters as $n/c= \sqrt{\epsilon\mu}$ and, in natural units where $c=1$, the local speed of light is $1/n$. We take the axis of the fibre as the $z$ axis and its core to have refractive index $n=n_1$ for $r<R$. The cladding will have index $n=n_2$ for $r>R$. All fields will have a tacit factor of $e^{ik_z z-i\omega t}$ so our differential operators act only on functions of the transverse co-ordinates $x$, $y$. We define parameters $\gamma$ by \^2= (\^2-k\_z\^2) where the sign is chosen so that $\gamma$ is real.
The transverse field components ${\bf E}_\perp$ and ${\bf H}_\perp$ are expressed in terms of the longitudinal components $E_z$ and $H_z$ by \_&=&{k\_z \_E\_z - [**e**]{}\_z \_H\_z},\
[**H**]{}\_&=&{k\_z \_H\_z + [**e**]{}\_z \_E\_z}, where, for our step fibre (no gradients of $\epsilon$ or $\mu$ away from the discontinuity) -\_\^2 H\_z +(k\_z\^2 -\^2) H\_z&=&0,\
-\_\^2 E\_z +(k\_z\^2 -\^2)E\_z&=&0. \[EQ:wave-equations\] To be guided we need $k_z^2-\epsilon\mu \omega^2$ to be negative in the core, and positive in the cladding. The solutions to (\[EQ:wave-equations\]) are then E\_z=
A[J]{}\_l(\_1 r)e\^[il]{} ,& r<R,\
B [K]{}\_l(\_2 r) e\^[il]{}, & r>R,
H\_z=
C[J]{}\_l(\_i r)e\^[il]{} ,& r<R,\
D [K]{}\_l(\_2 r) e\^[il]{}, & r>R.
Here ${\rm J}_l$ and ${\rm K}_l$ are the Bessel function and modified Bessel function respectively.
Below the fibre cutoff frequency the quantity $\gamma^2=\epsilon\mu \omega^2 -k_z^2$ is always positive. Light is no longer completely confined, so the eigen-frequencies have a negative imaginary that implies an exponential decay in time. The corresponding eigenfunctions must have outgoing waves at infinity, and so will be of the form E\_z=
A[J]{}\_l(\_i r)e\^[il]{} ,& r<R,\
B [H]{}\^[(1)]{}\_l(\_2 r) e\^[il]{}, & r>R,
H\_z=
C[J]{}\_l(\_i r)e\^[il]{} ,& r<R,\
D [H]{}\^[(1)]{}\_l(\_2 r) e\^[il]{}. & r>R,
Here ${\rm H}^{(1)}_l$ is a Hankel function of the first kind.
Whispering-gallery modes have small $k_z$ and large azimuthal quantum number $l$. These modes are always below the fibre cutoff, and so $\gamma^2$ does not change sign at $r=R$. If we consider the waves near the point $(x,y,z)=(R,0,0)$ then we have E\_x&=& ( k\_z - )\
E\_y&=& ( + )\
H\_x&=& ( k\_z + )\
H\_y&=& ( - ) On substituting the functional form of the solutions for $r<R$ these become E\_x&=& ( k\_z \_1 [J’]{}\_l(\_1r)A-\_l(\_1 r) C)\
E\_y&=& ( \_l (\_1 r)A +\_1 [J’]{}\_l(\_1 r) C)\
H\_x&=& ( k\_z \_1 J’(\_1 r) C +\_l (\_1 r)A )\
H\_y&=& ( \_l(\_1 r) C -[\_1 ]{} [J’]{}\_l(\_1 r) A), and for $r>R$ E\_x&=& ( k\_z \_2 [H’]{}\^[(1)]{}\_l(\_2r)B-\^[(1)]{}\_l(\_2 r) D)\
E\_y&=& ( \^[(1)]{}\_l(\_2 r)B +[\_2 ]{}\_2 [H’]{}\^[(1)]{}\_l(\_2r) D)\
H\_x&=& ( k\_z \_2 [H’]{}\^[(1)]{}\_l(\_2r) D +\^[(1)]{}\_l(\_2 r)B )\
H\_y&=& ( \^[(1)]{}\_l(\_2 r)D -[\_2 \_2 ]{} [H’]{}\^[(1)]{}\_l(\_2r) B). The boundary conditions are that the tangential components $ E_z$, $E_y$ and $H_z$, $H_y$ be continuous. The continuity of the normal components of ${\bf D}$ and ${\bf B}$ is then ensured by the Maxwell “curl” equations.
If $k_z =0$, then $\gamma= n \omega= \sqrt{\mu\epsilon} \,\omega $ and the boundary condition equations break into two blocks. The $E_z$ and $H_y$ continuity equations for one block are, respectively, \_l(n\_1R)A &=& [H]{}\^[(1)]{}\_l(n\_2 R)B,\
J’(n\_1R)A &=& \^[(1)]{}\_l (n\_2 R) B. The $H_z$ and $E_y$ continuity equations for the other are \_l(n\_1R)C &=& [H]{}\^[(1)]{}\_l(n\_2 R)D,\
J’(n\_1R)C &=& \^[(1)]{}\_l (n\_2 R) D. There are therefore two families of whispering-gallery modes. The first is comprises TM modes (transverse when looking along the fibre) that have $H_z= 0$ (and therefore non-zero $A$, $B$) with frequencies determined by the eigenvalue equation = . \[EQ:TM\] The second comprises TE modes with $E_z=0$ (and therefore non-zero $C$, $D$), with = . \[EQ:TE\] Equivalently the eigenfrequencies are the zeros of (R) && \_l(n\_1 R)[H]{}\_l\^[(1)]{}(n\_2 R) - \^[(1)]{}(n\_2 R)[J]{}\_l(n\_1 R)\
[Det\_[TE]{}]{}(R) && \_l(n\_1 R)[H]{}\_l\^[(1)]{}(n\_2 R) - \^[(1)]{}(n\_2 R)[J]{}\_l(n\_1 R). In general TE and TM modes with the same $l$ are not degenerate.
Consider first the case of $n_1<n_2$. This is not the situation in a practical optical fibre where the core always has a higher refractive index, but there are still low-$Q$ resonant modes whose partial confinement arises because grazing-angle incidence provides strong reflection. In this case, as $r$ increases, the Hankel functions are approaching their asymptotic oscillating region \^[(1)]{}\_l(n\_2 r ) \~ e\^[in\_2 r ]{}+… before ${\rm J}_l(n_1 \omega r)$ starts to oscillate. For $l$ large and $n_1/n_2 \approx 1$ the RHS of equations (\[EQ:TM\]) and (\[EQ:TE\]) are slowly varying functions of $x=n_1\omega r$. They have a very small real part and a negative imaginary part. Consequently the eigenfrequencies of both TE and TM modes are close to zeros of ${\rm J}_l(x)$ but we must give $x$ a negative imaginary part to make the determinants vanish. The eigenfrequencies are therefore given by $$n_1\omega R= z_{l,n}= \xi_{n,l}- i \eta_{l,n}$$ where $\xi_{l,n}$ is the $n$-th zero of ${\rm J}_l(x)$ and $\eta_{l,n}$ is a positive quantity that differs for TE and TM modes.
Now we introduce a small $k_z$. The pairs of equations are now coupled, but we see that the term with $k_z$ on the LHS of the $E_y$ and $H_y$ continuity equations multiplies ${\rm J}_l(x)$, and this quantity is small at resonance. We therefore neglect these terms. If we look at the $E_x$ and $H_x$ continuity equations (whose validity was previously enforced by the others) and neglect the RHS radiation fields, we find that the eigenvalue equations simplify to k\_z n\_1 \_l(n\_1R) A -\_l(n\_1R)C&=&0,\
k\_z n\_1 \_l(n\_1 R) C+\_l(n\_1R)A&=&0. \[EQ:reduced-eqs-1\] As we are very close to zeros of the Bessel function, we can set ${\rm J}_l(z_{l,n}+n_1\delta \omega R)\approx n_1 R \,{\rm J'}_l(z_{l,n})\,\delta \omega$ and (\[EQ:reduced-eqs-1\]) reduce to k\_z A- il C&=&0,\
k\_z C +il n\_1\^2 A&=&0. \[EQ:reduced-eqs-2\] These two equations coincide once we set $(A,C)=(1,\pm i n_1)$, which corresponds to right and left circularly polarized light in a medium of refractive index $n_1$. They give a frequency shift of = k\_z. The longitudinal group velocity of a wave packet centered around $k_z=0$ s therefore z = . Since $n_ 1\omega R\sim l$ for these modes and $n_1 \omega = k_\phi$ is the wavenumber for the light in the core, we can write this equation as z = = where $\lambda_{\rm glass}$ is the wavelength of the light in the fibre core. As the time for one orbit is $2\pi R n_1$, we find that the rate of drift is one (in glass) wavelength per orbit. This is exactly what we expect from the berry phase argument in section \[SEC:cyclotron\]. The photons can only make a few orbits, however, before they escape or become depolarized due to the TE mode (which, looking along the circumferential ray, is a linearly polarized beam with the ${\bf E}$ field in the radial direction) having a shorter lifetime that the TM (which is a linearly polarized beam with the ${\bf E}$ field in the $z$ direction).
In an actual fibre we have $n_1>n_2$. In this regime the Bessel function ${\rm J}_l(n_1\omega r)$ begins to oscillate while the Hankel function ${\rm H}^{(1)}_l(n_2 \omega r )$ is still almost real and exponentially decreasing. In the region of $\omega R$ corresponding to total internal reflection, the fields outside the glass decay almost to zero as they would for total internal reflection off a flat interface — but eventually the Hankel function begins to oscillate and the fields become outgoing radiation.
We assume that $k_z$ is small, so that we can ignore the $k_z$’s in $\gamma_1$ and $\gamma_2$. The boundary condition equations are then ( [J]{}\_l(\_1R ) C - \^2\_l(\_1 R) A )&=& ( [H]{}\_l\^[(1)]{}(\_2R ) D - \^2\_l\^[(1)]{}(\_2 R) B )\
[J]{}\_l(\_1R)A &=& [H]{}\^[(1)]{}\_l(\_2 R )B\
( [J]{}\_l(\_1R ) A - \^2\_l(\_1 R) C )&=& ( [H]{}\_l\^[(1)]{}(\_2R ) B - \^2\_l\^[(1)]{}(\_2 R) D ).\
[J]{}\_l(\_1R)C &=& [H]{}\^[(1)]{}\_l(\_2 R )D
From now on we make the assumption that the impedance does not change from core to cladding— [*I.e.*]{} that $\sqrt{{\mu_1}/{\epsilon_1}}= \sqrt{{\mu_2}/{\epsilon_2}}$. This impedence matching condition might be hard to engineer, but given that we are seeking a mathematical illustration of the anomalous-velocity equation rather than proposing experimental verification it is not unreasonable. The matching means that the right and left handed Riemann Silberstein fields do not mix. It also ensures the degeneracy of the $k_z=0$ TE and TM modes, both eigenfrequences being determined by the same vanishing condition $${\rm Det}(\omega R) \equiv {\rm J'}_l(n_1 \omega R){\rm H}_l^{(1)}(n_2 \omega R) - {\rm H'}^{(1)}(n_2 \omega R){\rm J}_l(n_1 \omega R)=0.$$ See figure \[FIG:symmetric\] for a plot that locates the zeros of ${\rm Det}(\omega R)$.
For non-zero $k_z$ the TE and TM pairs of equations are again coupled, but, accepting the impedance-matching condition we can decouple the four equations into a different two pairs of equations — one for each helicity. We set (A,C)&=& (1, i )X\_+\
(B,D)&=&(1,i ) Y\_+ and similarly, with the sign before the $i$ changed, for $X_-$, $Y_-$. Using the notation $c_{1,2}=1/n_{1,2}$, the equations for the “+” pair become ( c\_1\^2 [J]{}\_l(R/c\_1) +\^2 [J’]{}\_l(R/c\_1)) X\_+ &=& (\^2 [H]{}\_l\^[(1)]{}(R/c\_2) +\^2 [[H]{}’]{}\_l\^[(1)]{}(R/c\_2))Y\_+\
[J]{}\_l (R/c\_1)X\_+&=& [H]{}\_l\^[(1)]{}(R/c\_2)Y\_+ Dividing the first by the second equation and rearranging gives (c\_1\^2-[c\_2]{}\^2) =\^2( - ). When $k_z=0$ the vanishing of the RHS is the eigenvalue condition. To find $\partial\omega/\partial k_z$ at $k_z=0$, we therefore need to compute the derivative of the RHS at the points at which it vanishes. To do this we make use of the asymptotic formula
\~-1\
which is accurate for large $x$ provided that we stay away from places where ${\rm J}_l(x)$ vanishes \[see figure \[FIG:bessel\]\]. This condition is satisfied at the points of interest \[see figure \[FIG:symmetric\]\]. We may also use the formulæ &\~& -1\
&\~& - ,x<l. These last two approximations \[see figure \[FIG:hankel\]\] imply that ()’ = -()\^2\~0, which is not quite right, but the derivative on the LHS is $O(1/l)$ in the region of interest and can be neglected. We also note that we can evaluate ()’ = -()\^2 at the unperturbed eigenvalue by exploiting the fact that ${\rm J}_l'/{\rm J}_l={\rm H}'_l/{\rm H}_l$ at that point. We find (c\_1\^2-[c\_2]{}\^2) k\_z &=&\
&=& . Thus = k\_z= k\_z. The opposite circularly-polarized have an opposite frequency shift. We have therefore recovered the same drift equation z= = that we found for the $n_1<n_2$ fibre. Recall that this drift is at a rate of one wavelength per orbit.
Discussion
==========
We can compare the mode-expansion rate of drift with that expected from angular momentum conservation about an axis perpendicular to the reflection interface. For a particle of momentum ${\bf p}$ and helicity $S$, and treating the orbit as a series of grazing-angle reflections, each deflection through an angle $\delta \phi $ causes a change in the perpendicular spin component of $2S \sin(\delta \phi/2)$ that must be compensated for by a change in the orbital angular momentum of $-|{\bf p}| \delta z$. Thus $|{\bf p}| \dot z= - S\dot \phi$. Since $\dot\phi R= c/n_{\rm glass}$ we have z = - S . In this picture the drift may be understood as a continuous version of the Imbert-Fedorov effect [@imbert; @federov].
Agreement with the results of the previous section requires us to identify the magnitude of the photon momentum $|{\bf p}|$ with $2\pi \hbar/\lambda_{\rm glass} =n_{\rm glass} \hbar \omega/c$, This is the Minkowski expression for the momentum of a photon in a medium — as opposed to Abraham’s expression for the momentum which places the $n_{\rm glass}$ in the denominator. (For a review the Abraham-Minkowski momentum controversy see [@abraham-minkowski].) Minkowski’s momentum is today understood to be the [*pseudo-momentum*]{} which is conserved as a result of the homogeneity of the medium [@blount-pseudomomentum]. It is the rotational symmetry of the medium about the normal to the core-cladding interface that is responsible for the angular momentum conservation, so the appearance of the Minkowski momentum is not surprising.
That a continuous process of reflection and rotation can transport a circularly polarized beam of light parallel to itself through an arbitrary distance is related to the fact that a continuous sequence of Lorentz boosts and rotations in free space can perform an arbitrary “Wigner translation" of a finite polarized beam [@stone-wigner].
Acknowledgements
================
This project was supported by the National Science Foundation under grant NSF DMR 13-06011. I would like to thank Konstantin Bliokh and Misha Stephanov for e-mail discussions.
Appendix: Gordon decomposition, the Weyl magnetic moment, and $g=2$.
====================================================================
The orginal Gordon decomposition of the Dirac 4-current [@gordon] shows that for any solution $\psi$ of the massive Dirac equation (i\^(\_-m)=0, \[EQ:Dirac-massive\] the four-current can be expressed as |\^= (|\^-(\^|) )+ \_(|\^), where \^ = \[\^,\^\] is the Lorentz generator.
Gordon’s decomposition of the current into a particle number-flux and bound spin contribution clearly requires $m\ne 0$. There is, however, a version that is valid in both massive and massless cases: assume that $\psi({\bf r},t)=\psi({\bf r})\exp\{-iEt\}$ and make use of the Dirac equation in the Hamiltonian form \_t&=& --im,\
\_t |&=& +| +i m|, with $\beta=\gamma^0$, $\alpha^i=\gamma^0\gamma^i$ and so find e| = (- ()) + (). Here ${\bm \gamma}= (\gamma^1,\gamma^2,\gamma^3)$, and = with (S\_x,S\_y,S\_z)= (\^[23]{},\^[31]{},\^[12]{}) so that =12 . With the particle-number density identified with $\rho= \psid\psi$, we can again interpret the first term in the decomposition as the current ${\bf j}_{\rm free}= e\rho {\bf k}/E= e\rho {\bf v}$ due to particles moving at speed ${\bf v}={\bf k}/E$. The second term, ${\bf j}_{\rm bound}= (e/E)\nabla\times {\bf S}$ is the current due to the gradients in the intrinsic magnetic moment density. The magnetic moment itself is found by integrating by parts to show that \_[bound]{}d\^3x =(e E)d\^3 x = d\^3 x. For a single massless particle whose spin-1/2 is locked to the direction $\hat {\bf k}$ of its kinetic momentum this is \_[Weyl]{}= , as claimed in section \[SEC:cyclotron\].
For the both massive and massless case we also have an expression for the momentum density as part of the symmetric Belinfante-Rosenfeld energy-momentum tensor T\^\_[BR]{}= (|\^\^- (\^|) \^+|\^\^-(\^|) \^). Using the Dirac equation we evaluate $T^{0\mu}_{\rm BR}=({\mathcal E},{\bf P})$ to find ${\mathcal E}=E\psid \psi$, and = 1[2i]{}(()- ()) +12 . (If we used the non-symmetric canonical energy-momentum tensor T\^\_[canonical]{}= (|\^\^- (\^|) \^), we do would not find the bound spin-momentum contribution.)
Again integrating by parts, we recover the spin contribution to the total angular momentum density as (12 )d\^3x = d\^3x, so the division by 2 in the spin contribution to the momentum density is correct. The absence of a division by 2 in the formula for the current reflects the $g=2$ gyromagnetic ratio of the electron. In other words a spin-density gradient is twice as effective at making an electric current as it is at contributing to the momentum.
[99]{}
E. N. Adams, E. I. Blount, [*Energy bands in the presence of an external force field II: Anomalous velocities*]{}, J. Phys. Chem. Solids, [**10**]{} 286-303 (1959); E. I. Blount in [*Solid State Physics*]{}, edited by F. Seitz and D. Turnbull (Academic Press New York, 1962) Vol 13, p305.
G. Sundaram, Q. Niu, [*Wave Packet Dynamics in Slowly Perturbed Crystals – Energy Gradient Correction and Berry Phase Effects*]{}, Phys. Rev. [**B59**]{} 14915-14925 (1999).
D. Xiao, J. Shi, Q. Niu, [*Berry Phase Correction to Electron Density of States in Solids*]{}, Phys. Rev. Lett. [**95**]{} 137204 1-4 (2005).
C. Duval, Z. Horváth, P. A. Horváthy, L. Martina, P. C. Stichel, [*Berry Phase Correction to Electron Density in Solids*]{}, Modern Physics Letters [**B 20**]{} 373-378 (2006).
Y. D. Chong, [*Berry’s phase and the anomalous velocity of Bloch wavepackets*]{}, Phys. Rev. B [**81**]{} 052303 1-2 (2010).
M. V. Berry, [*Quantal Phase Factors Accompanying Adiabatic Changes*]{}, Proc. Roy. Soc. Lond. [**A392**]{}, 45-57 (1984).
M. A. Stephanov, Y. Yin, [*Chiral Kinetic Theory*]{}, Phys. Rev. Lett. [**109**]{} 162001 1-5 (2012).
M. Stone, V. Dwivedi, [*A Classical Version of the Non-Abelian Gauge Anomaly*]{} Phys. Rev. D [**88**]{} 045012 1-8 (2013).
V. Dwivedi, M. Stone, [*Classical chiral kinetic theory and anomalies in even space-time dimensions*]{} J. Phys. A 47 025401 1-20 (2014).
A. Vilenkin, [*Microscopic partity violating effects: neutrino fluxes from rotating black holes amd rotating thermal radiation*]{}, Phys Rev D [bf 20]{} 1807-1812; [*Equilibrium parity-violating current in a magnetic field*]{} Phys. Rev. D [**22**]{} 3080-3084 (1980).
K. Fukushima, D. E. Kharzeev, H. J. Warringa, [*Chiral magnetic effect*]{}, Phys. Rev. D 78, 074033 (2008).
J-Y. Chen, D. T. Son, M. A. Stephanov, H-U. Yee, Y. Yin, [*Lorentz Invariance in Chiral Kinetic Theory*]{}, Phys. Rev. Lett. [**113**]{} 182302 1-5 (2014).
H. M. Weber, [*Die partiellen Differential-Gleichungen der mathematischen Physik nach Riemann’s Vorlesungen*]{}, p348, (Friedrich Vieweg und Sohn, Braunshweig 1901).
L. Silberstein, [*Elektromagnetische Grundgleichungen in bivectorieller Behandlung*]{}, Annalen der Phys. [**22**]{} 579-586 (1907); [*Nachtrag zur Abhandlung über ÓElektromagnetische Grundgleichungen in bivectorieller Behandlung*]{}, Annalen der Phys. [**24**]{} 783-784 (1907).
I. Bia[ł]{}ynicki-Birula, Z. Bia[ł]{}ynicki-Birula, [*The role of the Riemann-Silberstein vector in classical and quantum theories of electromagnetism*]{}, J. Phys. A [**46**]{} 053001-32 (2013). K. Yu. Bliokh, Yu. P. Bliokh, [*Modified geometrical optics of a smoothly inhomogeneous isotropic medium: The anisotropy, Berry phase, and the optical Magnus effect*]{}, Phys. Rev. E 70, 026605-9 (2004).
K. Y. Bliokh, A. Niv, V. Kleiner, E. Hasman [*Geometrodynamics of spinning light*]{}, Nature Photon. 2, 748-753 (2008).
F. I. Fedorov, [*To the theory of total reflection*]{}, Doklady Akademii Nauk SSSR 105, 465-468 (1955) \[translated and reprinted in J. Opt. [**15**]{} 014002-3 (2013)\].
C. Imbert, [*Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam*]{}, Phys. Rev. D [**5**]{} 787-796 (1972).
E. I. Blount, Bell Telephone Laboratories technical memorandum 38139-9. (1971).
R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, H. Rubinsztein-Dunlop, [*Colloquium: Momentum of an electromagnetic wave in dielectric media*]{}, Rev. Mod. Phys. [**79**]{} 1197-1216 (2007); Erratum: Rev. Mod. Phys. [**81**]{}, 443 (2009).
M. Stone, V. Dwivedi, T. Zhou, [*Wigner Translations and the Observer Dependence of the Position of Massless Spinning Particles*]{}, Phys. Rev. Lett. [ **114**]{} 210402 1-4 (2015)
W. Gordon, [*Der Strom der Diracschen Elektronentheorie*]{}, Z. Phys. [**50**]{} 630-632 (1928).
| ArXiv |
---
abstract: 'We propose the cascade attribute learning network (CALNet), which can learn attributes in a control task separately and assemble them together. Our contribution is twofold: first we propose attribute learning in reinforcement learning (RL). Attributes used to be modeled using constraint functions or terms in the objective function, making it hard to transfer. Attribute learning, on the other hand, models these task properties as modules in the policy network. We also propose using novel cascading compensative networks in the CALNet to learn and assemble attributes. Using the CALNet, one can zero shoot an unseen task by separately learning all its attributes, and assembling the attribute modules. We have validated the capacity of our model on a wide variety of control problems with attributes in time, position, velocity and acceleration phases.'
author:
- 'Zhuo Xu\*$^{1}$, Haonan Chang\*$^{2}$, and Masayoshi Tomizuka$^{1}, \emph{Fellow, IEEE}$[^1][^2][^3]'
title: '**Cascade Attribute Learning Network** '
---
INTRODUCTION
============
Reinforcement learning (RL) [@sutton_book] has been successful in solving many control problems rooted in fixed Markov Decision Processes (MDPs) environments [@nn_policy][@gae][@visuomotor]. However, the extremely close interactions between the RL algorithms and the MDPs leads to the difficulty to reuse the knowledge learned from one task in new tasks. This difficulty further impedes RL policies from being adept in solving high dimensional complicated tasks. For example, it is easy to train an autonomous vehicle to travel from an origin position to a target position. However, if one takes into consideration a bunch of vehicle and pedestrian obstacles, the difficulty of the problem could grow overwhelming for shallow policy models. In order to avoid all the obstacles, one would have to train a deep policy network with very sparse reward input. Therefore, the training process usually requires an unbearably large amount of computation. What makes it worse is that such policies are hardly reusable in other scenarios, even if the new task is very similar to the previous one. Suppose a speed limit requirement is added to the autonomous driving task, although the input of the policy network is already tediously high dimensional, given there is no entrance for the speed limit information to enter the policy network, it is not possible that the pretrained policy accomplishes the new task, no matter how the policy network is tuned. Therefore, RL frameworks with fixed policy models can hardly address such high dimensional and complicated tasks in environments of great variance.
We propose to address this problem from a new perspective: modularizing complicated and high dimensional problems using a series of attributes. The attributes refer especially to global characteristics or requirements that take effect throughout the task. An example of attribute learning is shown in Fig. 1. Concretely, to solve the complicated driving problem, one first decompose the requirements of the task into a target reaching attribute, an obstacle avoidance attribute and a speed limit attribute, then train the modular network for each of the attributes, and finally assemble the attribute networks together to produce the overall policy. Modularizing a task using a series of attributes has three main intriguing advantages:
![Autonomous driving as an example of modularizing a complicated task into multiple attributes using the cascade attribute learning network (CALNet).[]{data-label="figurelabel"}](figure1.jpg)
1. Decomposing a high dimensional complicated task into low dimensional attributes makes the training process much easier and faster.
2. Trained attribute modules can be reused in new tasks, making it possible to build up versatile policies that can adjust to changes in tasks by assembling attribute modules.
3. In attribute learning, specific state information is provided only to its corresponding attribute modules. This decoupling formulation makes it possible to dynamically manage state space in high dimensional environments.
In order to modularize the attributes, we propose a simple but efficient RL framework called the cascade attribute learning network (CALNet). The brief idea of the CALNet is shown in Fig. 1. In CALNet, the attribute modules are connected in cascade series. Each attribute module receives both the output of its preceding module and its corresponding states, and returns the action that satisfies all the attributes ahead of it. The details of the CALNet architecture and the training methods are described in Section III. Using the CALNet, one can zero shoot an unseen task by separately learning all the attributes in the task and assemble the attribute modules in series together. The reminder of this paper is organized as follows: the related works and the background of RL are introduced in Section II. In Section III, the architecture of the CALNet and the implementation details are described. In Section IV, we show simulation results to validate the proposed model using a variety of robots and attributes and give discussions on the experiments. The conclusions are given in Section V.
Related Work and Background
===========================
Related Work
------------
There have been lots of attempts to create versatile intelligence that can not only solve complicated tasks, but adjust to changes in the tasks as well. Transfer learning [@reinforcement_transfer][@transfer] is a key tool that makes the use of previously learned knowledges for the better or faster learning of new knowledges. Rusu et al [@progressive1][@progressive2] designed a multi-column (network) framework, referred to as progressive network, in which newly added columns are laterally connected to previously learned columns for knowledge transfer. Drafty et al [@transferable_policy] and Braylan et al [@reuse_module] also designed interesting network architectures for knowledge transfer in MAV control and video game playing. For the combinations of transfer learning and imitation learning, Ammar et al [@unsupervised_transfer] uses unsupervised learning to map states for transfer, assuming the existence of distance function between different state spaces. Gupta et al [@invariant_feature] learns an invariant feature between different dimensional states and use demonstrations to increase the density of the rewards. Our work differs from those works mainly in that we put emphasis on the modularization of attributes, which are concrete and meaningful modules that can be conveniently assembled into various combinations.
There are other methods seeking to learn a globally general policy: Meta learning [@meta] attempts to build self-adaptive learners that improve their bias through accumulating experience. One shot imitation learning [@one_shot], for example, is a meta learning framework which is trained using a number of different tasks so that new skills could be learned from a single expert demonstration. Curriculum learning (CL)[@curriculum] trains a model on a sequence of cognate tasks that get more and more challenging gradually, so as to solve hard tasks that could not be learned from scratching. Florensa et al [@reverse_curriculum] applied reverse curriculum generation (RCL) in RL. In the early stage of the training process, the RCL initializes the agent state to be very close to the target state, making the policy very easy to train. They then gradually increase the random level of the initial state as the RL model performs better and better. Our policy training strategy is inspired by the idea of CL and achieved satisfying robustness for the policies. There are also researches in training modular neural networks, [@modular_robot_task] investigates the combinations of multiple robots and tasks, while [@modular_subtask] investigates the combinations of multiple sequential subtasks. Our work, different from those works, looks into modularization in a different dimension. We investigate the modularization of attributes, the characteristics or requirements that take effect throughout the whole task.
Deep Reinforcement Learning Background
--------------------------------------
The objective of RL is to maximize the expected sum of the discounted rewards $R_t = \mathbb{E} \sum_{k=0}^\infty \gamma^{k} \cdot r_{t+k}$ in an agent-environment-interacting MDP. The agent observes state $s_t$ at time $t$, and selects an action $a_t$ according to its policy $\pi_\theta$ parameterized by $\theta$. The environment receives $s_t$ and $a_t$, and returns the next state $s_{t+1}$ and the reward in this step $r_t$. The $\gamma$ in the objective function is a discounting coefficient. The main approaches for reinforcement learning include deep Q-learning (DQN) [@dqn], asynchronous advantage actor critic (A3C) [@a3c], trust region policy optimization (TRPO) [@trpo], and proximal policy optimization (PPO) [@ppo]. Approaches used in continuous control are mostly policy gradient methods, i.e. A3C, TRPO, and PPO. Vanilla policy gradient method updates the parameters $\theta$ by ascending the log probability of action $a_t$ with higher advantage $\hat{A_t}$. The surrogate objective function is $$L(\theta) = \hat{\mathbb{E}}_t \left[ \log \pi_{\theta}(a_t \mid s_t) \cdot \hat{A_t} \right] \eqno{(1)}$$ Although A3C uses the unbiased estimator of policy gradient, large updates can prevent the policy from converging. TRPO introduces a constraint to restrict the updated policy from being too far in Kullback-Leibler (KL) distance [@kl] from the old policy. Usually, TRPO solves an unconstrained optimization with a penalty punishing the KL distance between $\pi_{\theta}$ and $\pi_{\theta_{old}}$, specifically, $$L(\theta) = \hat{\mathbb{E}}_t \left[ \frac{ \pi_{\theta}(a_t \mid s_t)}{\pi_{\theta_{old}}(a_t \mid s_t)} \cdot \hat{A_t} -\beta \cdot \textrm{KL} \left( \pi_{\theta} , \pi_{\theta_{old}} \right) \right] \eqno{(2)}$$ However, the choice of the penalty coefficient $\beta$ has been a problem [@ppo]. Therefore, PPO modifies TRPO by using a simple clip function parameterized by $\epsilon$ to limit the policy update. Specifically, $$L(\theta) = \hat{\mathbb{E}}_t \left[ \min \left( \frac{ \pi_{\theta}}{\pi_{\theta_{old}}}, \textrm{clip} \left(\frac{ \pi_{\theta}}{\pi_{\theta_{old}}}, 1+\epsilon, 1-\epsilon \right) \right)\hat{A_t} \right] \eqno{(3)}$$ This simple objective turns out to perform well while enjoying better sample complexity, thus we are using PPO as the default RL algorithm in our policy training. We are also inspired by [@dppo] to build a distributed framework with multiple threads to speed up the training process.
The advantage function $\hat{A_t}$ describes how better a policy is compared to a baseline. Traditionally the difference between the estimated Q value and value functions is applied as the advantage [@a3c]. Recently Schulman et al [@gae] proposed using generalized advantage estimation (GAE) to leverage the bias and variance of the advantage estimator.
The CALNet
==========
Problem Formulation
-------------------
We consider an agent performing a complicated task with multiple attributes. Since the agent is fixed, its action space is a fixed space, which we call $A$. We decompose the task into a series of attributes, denoted $\{ 0,1,2,\ldots \}$. We refer to the $0^{th}$ attribute as the base attribute, which usually corresponds to the most fundamental goal of the task, such as the target reaching attribute in the autonomous driving task. We define the state space of each attribute to be the minimum state space that fully characterizes the attribute, denoted $S=\{S_0,S_1,S_2,S_3\ldots \}$. For example, let the base attribute be the target reaching attribute, and the $1^{st}$ attribute be the obstacle avoidance attribute. Then $S_0$ consists of the states of the agent and the target, while $S_1$ consists of the states of the agent and the obstacle, and yet does not include the states of the target.
Each attribute has an unique reward function as well, denoted $R=\{R_0,R_1,R_2,R_3\ldots\}$. Each $R_i$ is a function mapping a state action pair to a real number reward, i.e. $R_i: S_i \times A \rightarrow \mathbb{R}$. Similarly, there is a specific transition probability distribution for each attribute, denoted: $P=\{P_0, P_1, P_2, P_3 \ldots \}$. And for each attribute, its transition function takes in the state action pairs and outputs the states for the next timestep, that is, $P_i : S_i \times A \rightarrow S_i$.
A key characteristic in our problem formulation is that the state spaces for different attributes can be different. This formulation enables the attribute learning network to dynamically manage the state space of the task. Specifically, the states of the $i^{th}$ attribute, $s_i$, is fed to the module of the $i^{th}$ attribute in the network.
Network architecture
--------------------
The architecture of the CALNet is shown in Fig. 2. and Fig. 3. Both the training phase (Fig. 2.) and the testing phase (Fig. 3.) of the CALNet are implemented in cascade orders. In the training phase, first a RL policy $\pi_0$ is trained to accomplish the goal of the base attribute. The base attribute network takes in $s_0\in S_0$ and outputs $a_0 \in A$, the reward and transition functions of the MDP are given by $R_0$ and $P_0$. This process is a default RL training process.
Then the $1^{st}$ attribute module is trained in series of the base attribute module. The $1^{st}$ attribute module consists of a compensate network and a weighted sum operator. The compensate network is fed with state $s_1 \in S_1$, and action $a_0$ chosen by $\pi_0$. The output of the compensate network is the compensate action $a^{c}_1$, which is used to compensate $a_0$ to produce the overall action $a_1$. The reward for the MDP is given by $R_0+R_1$ so that the requirements for both attributes are satisfied. The new transition function may not be directly calculated using $P_0$ and $P_1$, but it can be easily obtained from the environment. Since the parameters of the base attribute network are pretrained, the cascading attribute network would extract the features of the attribute by exploring the new MDP under the guidance of the base policy.
It is noted that in the weighted sum operator, the weight of the compensative action $a^{c}_1$ is initiated to be small and increased over the training time. That is, at the early stage of the training process, mainly $a_0$ takes effect, while $a^{c}_1$ gradually gets to influence the overall $a_1$ as the training goes on. For other attributes, the training method is the same with that of the $1^{st}$ attribute.
In the testing phase, the designated attribute modules are connected in series following the base attribute, as shown in Fig. 3. In the CALNet, the $i^{th}$ attribute module takes in $s_i$ and $a_{i-1}$, and outputs $a_i$ that satisfies all the attributes before the $i^{th}$ module. The final output $a_j$ is the overall output that satisfies all the attributes in the attribute array.
 \[figurelabel\]
 \[figurelabel\]
Training Method
---------------
To guarantee the capacity of the CALNet, the policies need to meet two requirements:
1. The attribute policies should be robust over the state space, rather than being effective only at the states that are close to the optimal trajectory. This requirement guarantees the attribute policies to be instructive when more compensate actions are added on the top of them.
2. The compensate action for a certain attribute should be close to zero if the agent is in a state where this attribute is not active. This property increases the capability of multi-attribute structures.
For the sake of the robustness of the attribute policies, we apply CL to learn a general policy that can accomplish the task starting from any initial state. The CL algorithm first trains a policy with fixed initial state. As the training goes on, the random level of the initial state is smoothly increased, until the initial state is randomly sampled from the whole state space. The random level is increased only if the policy is capable enough for the current random level.
For example, consider the task of moving a ball to reach a target point in a 2 dimensional space. In each episode, the initial position of the ball is randomly sampled in a circular area. The random level in this case is the radius of the circle. In the early training stage, the radius is set to be very small, and the initial position is almost fixed. As the policy gains more and more generality, the reward in each episode increases. Once the reward reaches a threshold, the random level is increased, and the initial position of the ball is sampled from a larger area. The terminal random level corresponds to the circumstance where the circular sampling area fully covers the working zone. If the policy performs well under the terminal random level, the policy is considered successfully trained. The pseudocode for this process is shown in Algorithm 1.
$RandomLevel = $ Initial Random Level $\lambda = 1 + $ Random Level Increase Rate $N = $ Batch Number $LongTermR = Queue()$ Update the policy using PPO $Rewards \gets RunEpisode(N)$ $LongTermR.append(Rewards)$ $RandomLevel = RandomLevel \times \lambda$ $Clear(LongTermR)$
To guarantee the second requirement, an extra loss term that punishes the magnitude of the compensative action, $l^{c}_i \propto -\|a^{c}_i\|^2$, is added to the reward function so as to reduce $\|a^{c}_i\|$ when attribute $i$ is not active.
Experiments
===========
Our experiments aim to validate the capability and advantage of the CALNet. In this Section, first we introduce the experiment setup, we then show the capability of the CALNet to modularize and assemble attributes in multi-attribute tasks. In the last part of this section, we compare the CALNet with the baseline RL algorithm, and show that the CALNet can adjust to complicated tasks more easily.
Setup
-----
The experiments are powered by the MuJoCo physics simulator [@mujoco]. The policy functions in our experiments are Gaussian distributed obtained using fully connected neural networks, built using TensorFlow. The baseline RL algorithm we use is the PPO [@ppo] method with GAE [@gae] as the advantage estimator.
We design three robots as agents in our experiments. They are a robot arm in 2 dimensional space, a moving ball in 2 dimensional space, and a robot arm in 3 dimensional space. For all three robots we have enabled both position control and force control modes.
For each agent we have designed 5 attributes:
 \[figurelabel\]
### reaching (base attribute)
The reaching task is a natural selection for the base attribute. For the ball agent, the goal is to collide the target object. For the robot arm agents, the goal is to touch the target object.
### obstacle (position phase)
The obstacle attribute is to add an rigid obstacle ball in the space. Negative rewards are given if the robot collides the obstacle. Therefore, in baseline RL training, the agent can be dissuaded from exploring the right direction.
### automated door (time phase)
The automated door attribute is purely time controlled. The door blocking the target is opened only at some certain time. This attribute is harder than an obstacle, since it punishes the agent even if it goes to the right direction at a wrong time.
 \[figurelabel\]
 \[figurelabel\]
### speed limit (velocity phase)
The speed limit attribute adds a time-variant speed limit on the agent. The agent gets punished if it surpasses the speed limit. But if the robot’s speed is too slow, it may not be able to finish the task in one episode.
### force disturbance (acceleration phase)
The force disturbance attribute adds a time-variant force disturbance to the agent (or each joint for the arm).
CALNet Performance
------------------
The first set of experiments test the capability of the CALNet to learn attributes and assemble learned attributes. We first train the base attribute module using the baseline RL algorithm with CL, and then use the cascading modules to modularize the different attributes based on the pretrained base module. The results show that all the attributes can be successfully added to the base attribute using the CALNet. Fig. 5. shows some of the examples of the agent performing different attributes combinations.
We also test the transferability of the cascading modules and the capability of the CALNet of modeling tasks with multiple attributes. Concretely, we first train two attribute modules in parallel based on the pretrained base module. Then we connect the two attribute modules in series following the base attribute module. The CALNet structure is the same as the one shown in Fig. 3. The policy derived using the assembled network can zero shoot most of the tasks satisfying requirements of both attributes.
Fig. 6. shows two examples of the CALNet zero shooting a task where the moving ball reaches the target while avoiding two obstacles simultaneously. It is emphasized that this task is never trained before. We achieve zero shooting simply by connecting two pretrained obstacle attribute modules in series following the base module. Undeniably as the attributes grow more complicated and the number of attributes gets larger, it would require a certain amount of finetune. However, the advantage of modularizing and assembling attributes is remarkable, since the finetuning process is much easier and faster compared to training a new policy from scratch (as discussed in Section IV-C).
Comparison with Baseline RL Methods
-----------------------------------
We compare the capability of the CALNet and the baseline RL by comparing their training processes on a same task. We consider the MDP in which the ball agent gets to the target while avoiding an obstacle. The CALNet is trained with CL. For baseline RL trained with CL, in many cases it is to too hard for the agent to reach the target. Therefore, we have also implemented RCL, which let the initial state be very close to the target in the early stage of the training phase. Using RCL, the RL could gain positive reward very fast. The challenge would be whether the RL algorithm could maintain high reward level as the random level increases.
For CALNet, the base attribute has been trained, and we train the obstacle avoidance attribute module based on the base module. For the baseline RL, the task is trained from scratching. The focus of the comparison is placed on the responding reward and random level in CL versus the training iterations.
The reward and the random level curves are shown in Fig. 7, with the horizontal axis representing the training iterations. It is shown that the baseline RL using CL barely learns anything. This is because the reward is too sparse and the agent is consistently receiving punish from the obstacle, and fells into some local minimum. For the baseline RL using RCL, in the early stage, the average discounted reward in an episode is high as expected. But as the random level rises, the performance of the baseline RL with RCL drops. Therefore, the random level increases slowly as the training goes on.
The CALNet, on the other hand, is able to overcome the misleading punishments from the obstacle, thanks to the guidance of the instructive base attribute policy. As a result, the random level of the CALNet rises rapidly, and the CALNet achieves terminal random level more than 10 times faster than the baseline. These results indicate that the attribute module learns substantial knowledge of the attribute as the CL based training goes on.
 \[comparision\]
Conclusions
===========
In this paper, we propose the attribute learning and present the advantages of using this novel method to modularize complicated tasks. The RL framework we propose, the CALNet, uses cascading attribute modules to model the characteristics of the attributes. The attribute modules are trained with the guidance of the pretrained base attribute module. We validated the effectiveness of the CALNet of modularizing and assembling attributes, and showed the advantages of the CALNet in solving complicated tasks compared to the baseline RL. Our future work includes transferring attributes between different base attributes and even different agents. Another potential direction is to investigate the attribute learning models that can assemble lots of attributes. We believe that attribute learning can help human build versatile controllers more easily.
[99]{}
Sutton, Richard S., and Andrew G. Barto. Reinforcement learning: An introduction. Vol. 1. No. 1. Cambridge: MIT press, 1998. Levine, Sergey, and Pieter Abbeel. “Learning neural network policies with guided policy search under unknown dynamics.” Advances in Neural Information Processing Systems. 2014. Schulman, John, et al. “High-dimensional continuous control using generalized advantage estimation.” arXiv preprint arXiv:1506.02438 (2015). Levine, Sergey, et al. “End-to-end training of deep visuomotor policies.” Journal of Machine Learning Research 17.39 (2016): 1-40. Taylor, Matthew E., and Peter Stone. “Transfer learning for reinforcement learning domains: A survey.” Journal of Machine Learning Research 10.Jul (2009): 1633-1685. Pan, Sinno Jialin, and Qiang Yang. “A survey on transfer learning.” IEEE Transactions on knowledge and data engineering 22.10 (2010): 1345-1359. Rusu, Andrei A., et al. “Progressive neural networks.” arXiv preprint arXiv:1606.04671 (2016). Rusu, Andrei A., et al. “Sim-to-real robot learning from pixels with progressive nets.” arXiv preprint arXiv:1610.04286 (2016). Daftry, Shreyansh, J. Andrew Bagnell, and Martial Hebert. “Learning transferable policies for monocular reactive MAV control.” International Symposium on Experimental Robotics. Springer, Cham, 2016. AlexanderBraylan, MarkHollenbeck, and RistoMiikkulainen ElliotMeyerson. “Reuse of neural modules for general video game playing.” (2016). Ammar, Haitham Bou, et al. “Unsupervised cross-domain transfer in policy gradient reinforcement learning via manifold alignment.” Proc. of AAAI. 2015. Gupta, Abhishek, et al. “Learning invariant feature spaces to transfer skills with reinforcement learning.” arXiv preprint arXiv:1703.02949 (2017). Vilalta, Ricardo, and Youssef Drissi. “A perspective view and survey of meta-learning.” Artificial Intelligence Review 18.2 (2002): 77-95. Duan, Yan, et al. “One-Shot Imitation Learning.” arXiv preprint arXiv:1703.07326 (2017). Bengio, Yoshua, et al. “Curriculum learning.” Proceedings of the 26th annual international conference on machine learning. ACM, 2009. Florensa, Carlos, et al. “Reverse Curriculum Generation for Reinforcement Learning.” arXiv preprint arXiv:1707.05300 (2017). Devin, Coline, et al. “Learning modular neural network policies for multi-task and multi-robot transfer.” Robotics and Automation (ICRA), 2017 IEEE International Conference on. IEEE, 2017. Andreas, Jacob, Dan Klein, and Sergey Levine. “Modular multitask reinforcement learning with policy sketches.” arXiv preprint arXiv:1611.01796 (2016). Mnih, Volodymyr, et al. “Human-level control through deep reinforcement learning.” Nature 518.7540 (2015): 529-533. Mnih, Volodymyr, et al. “Asynchronous methods for deep reinforcement learning.” International Conference on Machine Learning. 2016. Schulman, John, et al. “Trust region policy optimization.” Proceedings of the 32nd International Conference on Machine Learning (ICML-15). 2015. Schulman, John, et al. “Proximal Policy Optimization Algorithms.” arXiv preprint arXiv:1707.06347 (2017). Kullback, Solomon, and Richard A. Leibler. “On information and sufficiency.” The annals of mathematical statistics 22.1 (1951): 79-86. Heess, Nicolas, et al. “Emergence of Locomotion Behaviours in Rich Environments.” arXiv preprint arXiv:1707.02286 (2017). Todorov, Emanuel, Tom Erez, and Yuval Tassa. “MuJoCo: A physics engine for model-based control.” Intelligent Robots and Systems (IROS), 2012 IEEE/RSJ International Conference on. IEEE, 2012.
[^1]: \* Both authors contributed equally to this work
[^2]: $^{1}$Zhuo Xu and Masayoshi Tomizuka are with the Dept. of Mechanical Engineering, University of California, Berkeley CA 94720, USA
[^3]: $^{2}$Haonan Chang is with the Dept. of Mechanical Engineering, Tsinghua University, Beijing 100084, China
| ArXiv |
---
abstract: 'We analyse the response of a spatially extended direction-dependent local quantum system, a detector, moving on the Rindler trajectory of uniform linear acceleration in Minkowski spacetime, and coupled linearly to a quantum scalar field. We consider two spatial profiles: (i) a profile defined in the Fermi-Walker frame of an arbitrarily-accelerating trajectory, generalising the isotropic Lorentz-function profile introduced by Schlicht to include directional dependence; and (ii) a profile defined only for a Rindler trajectory, utilising the associated frame, and confined to a Rindler wedge, but again allowing arbitrary directional dependence. For (i), we find that the transition rate on a Rindler trajectory is non-thermal, and dependent on the direction, but thermality is restored in the low and high frequency regimes, with a direction-dependent temperature, and also in the regime of high acceleration compared with the detector’s inverse size. For (ii), the transition rate is isotropic, and thermal in the usual Unruh temperature. We attribute the non-thermality and anisotropy found in (i) to the *leaking* of the Lorentz-function profile past the Rindler horizon.'
author:
- 'Sanved Kolekar[^1]'
date: November 2019
title: |
Directional dependence of the Unruh effect\
for spatially extended detectors
---
Introduction
============
The Unruh effect [@Fulling:1972md; @Davies:1974th; @Unruh:1976db] states that an observer of negligible spatial size on a worldline of uniform linear acceleration in Minkowski spacetime reacts to the Minkowski vacuum of a relativistic quantum field by thermal excitations and de-excitations, in the Unruh temperature $g/(2\pi)$, where $g$ is the observer’s proper acceleration. The acceleration singles out a distinguished direction in space, and an observer with direction-sensitive equipment will in general see a direction-dependent response; however, for the Lorentz-invariant notion of direction-sensitivity introduced in [@Takagi:1985tf], the associated temperature still turns out to be equal to $g/(2\pi)$, independently of the direction. For textbooks and reviews, see [@Birrell:1982ix; @Crispino:2007eb; @Fulling:2014wzx].
In this paper we address the response of uniformly linearly accelerated observers in Minkowski spacetime, operating direction-sensitive equipment of nonzero spatial size. We ask whether the temperature seen by these observers is still independent of the direction. The question is nontrivial: while a spatially pointlike detector with a monopole coupling is known to be a good approximation for the interaction between the quantum electromagnetic field and electrons on atomic orbitals in processes where the angular momentum interchange is insignificant [@MartinMartinez:2012th; @Alhambra:2013uja], finite size effects can be expected to have a significant role in more general situations [@DeBievre:2006pys; @Hummer:2015xaa; @Pozas-Kerstjens:2015gta; @Pozas-Kerstjens:2016rsh; @Pozas-Kerstjens:2017xjr; @Simidzija:2018ddw]. Also, the notion of a finite size accelerating body has significant subtlety: while a rigid body undergoing uniform linear acceleration in Minkowski spacetime can be defined in terms of the boost Killing vector, different points on the body have differing values of the proper acceleration, and the body as a whole does not have an unambiguous value of ‘acceleration’. It would be interesting to ask whether the resultant transition rate would be thermal at all. And if yes, then at what temperature. The observer’s response could hence well be expected to depend explicitly on the body’s shape as well as the size.
A related issue is the following: An analysis of a direction dependent point-like detector re-affirms that the Unruh bath is isotropic even though there is a preferred direction in the Rindler frame, the spatial direction along the direction of acceleration [@Takagi:1985tf]. However, analysing drag forces on drifting particles in the Unruh bath reveals through the Fluctuation Dissipation Theorem that the quantum fluctuations in the Unruh bath are not isotropic [@Kolekar:2012sf]. These anisotropies could be relevant for direction dependent spatially extended detectors whose length scales are of the order or greater than the correlation scales associated with the quantum fluctuations.
We consider the response of spatially extended direction-dependent detectors in uniform linear acceleration in two models of such a detector: (i) a spatial sensitivity profile that generalises the isotropic Lorentz-function considered by Schlicht [@schlicht] to include spatial anisotropy, and (ii) a spatial sensitivity profile defined in terms of the geometry of the Rindler wedge, and explicitly confined to this wedge, following De Bievre and Merkli [@DeBievre:2006pys].
We begin in Section \[schlichtsection\] by briefly reviewing a detector with an isotropic Lorentz-function spatial profile [@schlicht], highlighting the role of a spatial profile as the regulator of the quantum field’s Wightman function, and recalling how the Unruh effect thermality arises for this detector.
In section \[directiondetsection\] we generalise the Lorentz-function spatial profile to include spatial anisotropy, initially for an arbitrarily-accelerated worldline, relying on the Fermi-Walker frame along the trajectory. We then specialise to a Rindler trajectory of uniform linear acceleration. We find that the transition rate is non-thermal, and angle dependent. Thermality is however restored in the low and high frequency regimes, and also in the regime of high acceleration compared with the inverse of the detector’s spatial extent.
In section \[rindlerframesection\] we analyse a profile defined in the Rindler frame of a Rindler trajectory, and confined to the Rindler wedge, following De Bievre and Merkli [@DeBievre:2006pys]. We find that the transition rate is isotropic and thermal at the usual Unruh temperature.
In section \[discsection\] we discuss and resolve the discrepancy of these two outcomes. The key property responsible for the non-thermality and anisotropy for the Lorentz-function profile is that this profile leaks outside the Rindler wedge, past the Rindler horizon. The leaking is an unphysical side effect of a detector model with a noncompact spatial profile, and it is unlikely to have a counterpart in spatially extended detectors with a more fundamental microphysical description. We leave the development of such spatially extended detector models subject to future work.
Spatially isotropic Lorentz-function profile\[schlichtsection\]
===============================================================
In this section we briefly review Schlicht’s generalisation [@schlicht] of a two-level Unruh-DeWitt detector [@Unruh:1976db; @DeWitt:1979] to a nonzero spatial size.
We consider a massless scalar field $\phi$ in four-dimensional Minkowski spacetime, and a two-level quantum system, a detector, localised around a timelike worldline $x(\tau)$, parametrised in terms of the proper time $\tau$. The interaction Hamiltonian reads $H_{int} = c \, m(\tau) \,\chi(\tau) \phi(\tau)$, where $c$ is a coupling constant, $m(\tau)$ is the detector’s monopole moment operator, $\chi(\tau)$ is the switching function that specifies how the interaction is turned on and off, and $\phi(\tau)$ is the spatially smeared field operator. The formula for $\phi(\tau)$ is $$\phi(\tau) = \int d^3\xi \; f_{\epsilon} ({\bm \xi}) \, \phi(x(\tau, {\bm \xi}))
\ ,
\label{smearedoperator}$$ where ${\bm \xi} = (\xi^1, \xi^2, \xi^3)$ stands for the spatial coordinates associated with the local Fermi-Walker transported frame and $x(\tau, {\bm \xi})$ is a spacetime point written in terms of the Fermi-Walker coordinates. The smearing profile function $f_{\epsilon} ({\bm \xi})$ specifies the spatial size and shape of the detector in its instantaneous rest frame. In linear order perturbation theory, the detector’s transition probability is then proportional to the response function, $${\cal F}(\omega) = \int_{-\infty}^{\infty} du \, \chi(u) \int_{-\infty}^{\infty} ds \, \chi(u -s) e^{- i \omega s} \, W(u,u-s)
\ ,
\label{transprobability}$$ where $\omega$ is the transition energy, $W(\tau,\tau^\prime) = \langle \Psi | \phi(\tau) \phi(\tau^\prime) | \Psi \rangle$ and $|\Psi \rangle$ is the initial state of the scalar field. The choice for the smearing profile function $f_{\epsilon}$ made in [@schlicht] was the three-dimensional isotropic Lorentz-function, $$f_{\epsilon}({\bm \xi})= \frac{1}{\pi^2} \frac{\epsilon}{{(\xi^{2}+\epsilon^{2})}^2}
\ ,
\label{schlichtprofile}$$ where the positive parameter $\epsilon$ of dimension length characterises the effective size.
The selling point of the profile function is that it allows the switch-on and switch-off to be made instantaneous; for a strictly pointlike detector, by contrast, instantaneous switchings would produce infinities and ambiguities [@schlicht]. In particular, for a detector that is switched off at proper time $\tau$, the derivative of ${\cal F}$ with respect to $\tau$ can be understood as a transition rate, in the ‘ensemble of ensembles’ sense discussed in [@Langlois:2005if; @Louko:2007mu]. If the switch-on takes place in the infinite past, the transition rate formula becomes $${\dot {\cal F}}(\omega) = 2 \operatorname{Re}\int_0^\infty ds \,e^{-i \omega s} \, W(\tau,\tau - s)
\ .
\label{eq:transrate-schlicht}$$
When the trajectory is the Rindler trajectory of uniform linear acceleration of magnitude $g>0$, and $|\Psi\rangle$ is the Minkowski vacuum, the transition rate becomes [@schlicht] $${\dot {\cal F}}(\omega) = \frac{1}{2 \pi} \; \frac{(\omega /g)}{1+ \epsilon^2} \; \frac { e^{\frac{2\omega }{g} \tan^{-1}\left( g \epsilon \right)}}
{
e^{\frac{2 \pi \omega}{g}} -1
}
\ .
\label{schlichttrans}$$ In the limit $\epsilon \rightarrow 0$, ${\dot {\cal F}}$ reduces to the Planckian formula in the Unruh temperature $g/(2\pi)$, consistently with other ways of obtaining the response of a pointlike detector in the long time limit [@Unruh:1976db; @DeWitt:1979; @letaw; @Takagi:1986kn; @Fewster:2016ewy].
For $\epsilon$ strictly positive, ${\dot {\cal F}}$ is no longer Planckian. However, we wish to observe here that ${\dot {\cal F}}$ is still thermal, in the sense that it satisfies the detailed balance condition, $$\begin{aligned}
{\dot {\cal F}}(-\omega) = e^{\beta\omega}{\dot {\cal F}}(\omega)
\ , \end{aligned}$$ where the inverse temperature now reads $\beta = \bigl(2\pi - 4 \tan^{-1}( g \epsilon) \bigr) /g$. The temperature is thus higher than the usual Unruh temperature. This feature has to our knowledge not received attention in the literature, and we shall discuss its geometric origins in section \[discsection\].
Spatially anisotropic Lorentz-function profile {#directiondetsection}
==============================================
In this section we generalise the isotropic Lorentz-function profile to include spatial anisotropy.
General trajectory
------------------
Let $x(\tau)$ again be a timelike worldline parametrised by its proper time $\tau$, so that the four-velocity $u^a := \frac{dx^a}{d\tau}$ is a unit timelike vector. The four-acceleration vector $a^a := u^b \nabla_b u^a$ is orthogonal to $u^a$, and its direction is Fermi-Walker transported along the trajectory only when the trajectory stays in a timelike plane, as seen by considering the torsion and hypertorsion of the trajectory in the Letaw-Frenet equations [@letaw; @kolekar].
We define the direction dependence by writing the expression for $\phi(\tau)$ in Eq.(\[smearedoperator\]) as $$\frac{d \phi(\tau)}{d \Omega} = \int d\xi \; f_{\epsilon} ({\bm \xi}) \, \phi(x(\tau, {\bm \xi})) = \phi_\Omega(\tau)
\ ,
\label{smearedoperatorang}$$ such that the ${\bm \xi}$ points in the direction of $\Theta_0$ and $\phi_0$. Integrating $\phi_\Omega(\tau)$ all over the solid angle then reproduces the smeared field operator $\phi(\tau)$ in the Schlicht case. Assuming the two level quantum system to couple linearly to $\phi_\Omega$, one can then proceed to calculate the transition rate as per formula in Eq.(\[transprobability\]) with the corresponding $W_\Omega(\tau,\tau^\prime)$ equal to $ \langle \Psi | \phi_\Omega(\tau) \phi_\Omega(\tau^\prime) | \Psi \rangle$. Equivalently, one can consider a detector whose spatial profile has the radial dependence of and depends on the angles $\Theta_0$ and $\phi_0$ through $$\begin{aligned}
f_{\epsilon}({\bm \xi},\Theta_{0})= \frac{1}{2\pi^3} \frac{\epsilon}{{(\xi^{2}+\epsilon^{2})}^2} \frac{\delta (\theta - \Theta_{0})}{\sin\Theta_{0}} \delta(\phi - \phi_0)
\ ,
\label{angprof}\end{aligned}$$ where $\theta$ and $\phi$ are measured in the ${\bm \xi} = (\xi^1, \xi^2, \xi^3)$ space. One can once again note that integrating over the solid angle $d\Omega_0 = \sin\Theta_{0} d\Theta_{0} d\phi_{0}$ yields the isotropic profile .
Following the steps in section \[schlichtsection\], the transition rate formula becomes $$\begin{aligned}
{\dot {\cal F}}_{\Theta_0}(\omega) = 2 \operatorname{Re}\int_0^\infty ds \,e^{-i \omega s} \, W_{\Theta_0}(\tau,\tau - s)
\ ,
\label{angtransitionrate}\end{aligned}$$ where $$\begin{aligned}
W_{\Theta_0}(\tau,\tau^\prime) = \langle \Psi | \phi(\tau, \Theta_0) \phi(\tau^\prime,\Theta_0) | \Psi \rangle
\ ,
\label{angwhitmannfunction}\end{aligned}$$ and the smeared field operator reads $$\begin{aligned}
\phi(\tau, \Theta_0) = \int d^3\xi \; f_{\epsilon} ({\bm \xi}, \Theta_0) \, \phi(x(\tau, {\bm \xi}))
\ ,
\label{smearedRARF}\end{aligned}$$ provided these expressions are well defined. We shall now show that the expressions are well defined provided $\Theta_0 \ne \pi/2$, and we give a more explicit formula for ${\dot {\cal F}}_{\Theta_0}$.
Suppose hence from now on that $\Theta_0 \ne \pi/2$. We work in global Minkowski coordinates in which points on Minkowski spacetime are represented by their position vectors, following the notation in [@schlicht]. The trajectory is written as $x^b(\tau)$. At each point on the trajectory, we introduce three spacelike unit vectors $e^b_{(\alpha)}(\tau)$, $\alpha = 1,2,3$, which are orthogonal to each other and to $u^b(\tau) = \frac{dx^b(\tau)}{d\tau}$, and are Fermi-Walker transported along the trajectory. We coordinatise the hyperplane orthogonal to $u^b(\tau)$ by ${\bm \xi} = (\xi^1, \xi^2, \xi^3)$ by $$\begin{aligned}
x^b(\tau, {\bf \xi}) = x^b(\tau) + \xi^\alpha e^b_{(\alpha)}(\tau)
\ . \end{aligned}$$
Using (\[angwhitmannfunction\]) and (\[smearedRARF\]), we obtain $$W_{\Theta_0}(\tau,\tau^\prime) = \frac{1}{(2 \pi)^3} \int \frac{d^3 {\bf k}}{2 \omega({\bf k})} \;
g_{\Theta_0} ( {\bf k}, \tau) \, g^*_{\Theta_0} ( {\bf k}, \tau^\prime)
\ ,
\label{gwhitmann}$$ where $\omega({\bf k}) = \sqrt{{\bf k}^2}$ and $$g_{\Theta_0} ( {\bf k}, \tau) = \int d^3\xi \; f_{\epsilon}({\bm \xi},\Theta_{0}) e^{i k_b x^b(\tau, \, {\bm \xi} )}
\ .
\label{gdef}$$ The index $\alpha$ refers to values $(1,2,3)$ and $e^b_{(\alpha)}(\tau)$ are the orthogonal Fermi unit-basis vectors in the spatial direction orthogonal to $u^b(\tau)$. Then defining $3$ - vector $\tilde{{\bf k}}$ having components $(\tilde{{\bf k}})_\alpha = k_b {\bf e}^b_{(\alpha)}(\tau)$ and working in spherical co-ordinates in the ${\bf \xi}$- space, we can recast Eq. (\[gdef\]) to get $$\begin{aligned}
g_{\Theta_0}( {\bf k}, \tau) &=& \frac{1}{\pi} e^{i k_b x^b(\tau)} \int_0^\infty d\xi \, \frac{\xi^2\epsilon}{(\xi^2+\epsilon^2)^2} e^{i \xi \cos\Theta_0 |\tilde{{\bf k}}|}
\notag
\\
& = & e^{i k_b x^b(\tau)} \left( I_R + I_M \right)
\ , \end{aligned}$$ where $I_R$ and $I_M$ are the real and imaginary parts of the integral. Here, $\tilde{{\bf k}}$ is oriented along the $z$ direction in the ${\bf \xi}$- space and the angle $\Theta_0$ is measured from the $z$ direction. The real part can be evaluated by contour integration, with the result $$\begin{aligned}
I_R &=& \frac{1}{\pi} \int_{0}^\infty d\xi \frac{\xi^2\epsilon}{(\xi^2+\epsilon^2)^2} \cos(\xi |\tilde{{\bf k}}| \cos\Theta_0) \nonumber \\
&=& \frac{1}{4} \frac{\partial}{\partial \epsilon} \left( \epsilon e^ {- \epsilon |\tilde{{\bf k}}| |\cos{\Theta_0}| }\right) \nonumber \\
&=& \frac{1}{4} \left( 1-\epsilon |\tilde{{\bf k}}| |\cos\Theta_0| \right) e^{-\epsilon |\tilde{{\bf k}}| |\cos \Theta_0|}
\ .
\label{IR}\end{aligned}$$ The imaginary part can be reduced to the exponential integral $E_1$ [@dlmf], with the result $$\begin{aligned}
I_M = \frac{i \operatorname{sgn}(a)}{4\pi}
\Bigl[
(|a|+1) e^{|a|} E_1(|a|) + (|a|-1) e^{-|a|} \operatorname{Re}\bigl( E_1(-|a|) \bigr)
\Bigr]
\ , \end{aligned}$$ where $a = \epsilon |\tilde{{\bf k}}| \cos \Theta_0$. Note that the replacement $\Theta_0 \to \pi - \Theta_0$ leaves $I_R$ invariant but gives $I_M$ a minus sign.
To proceed further, we assume that the profile function is invariant under $\Theta_0 \rightarrow \pi - \Theta_0$, that is, under $\cos(\Theta_0) \rightarrow - \cos(\Theta_0)$. Since $I_M$ is an odd function of $\cos(\Theta_0)$, it does not contribute to $g_{\Theta_0} \left( {\bf k}, \tau \right)$ under such an invariance whereas $I_R$ being even in $\cos(\Theta_0)$ contributes. Physically, this would mean that the direction sensitive detector reads off the average of two transition rates from the $\Theta_0$ and $\pi - \Theta_0$ directions respectively. Eq.(\[gdef\]) then becomes $$g_{\Theta_0} \left( {\bf k}, \tau \right) = \frac{1}{4} \frac{\partial}{\partial \epsilon} \left( \epsilon \; e^ {- \epsilon |\tilde{{\bf k}}| |\cos{\Theta_0}| } \; e^{i k_b x^b(\tau)}\right)$$ Using the fact that $k_b$ is a null vector, it is straightforward to show that $|\tilde{{\bf k}}| = - [k_b u^b(\tau)]$. Substituting the above expression in Eq.(\[gwhitmann\]) and upon performing the straightforward ${\bf k}$ integral, we can write a compact expression for $W_{\Theta_0}$ of the following form $$\begin{aligned}
W_{\Theta_0}(\tau,\tau^\prime) &=& \frac{-1}{16} \frac{\partial}{\partial \epsilon^\prime} \frac{\partial}{\partial \epsilon^{\prime \prime}} \Biggl( \frac{4\pi\epsilon^\prime \epsilon^{\prime \prime} }{ \left[ T\left(\epsilon^\prime, \epsilon^{\prime \prime}, \tau, \tau^\prime \right) \right]^2 - \left[ X \left( \epsilon^\prime, \epsilon^{\prime \prime}, \tau, \tau^\prime \right) \right]^2 } \Biggr)_{\epsilon^\prime = \epsilon, \epsilon^{\prime \prime} = \epsilon}
\label{wfinalcompact}\end{aligned}$$ where the functions $T\left(\epsilon^\prime, \epsilon^{\prime \prime}, \tau, \tau^\prime \right)$ and $X\left(\epsilon^\prime, \epsilon^{\prime \prime}, \tau, \tau^\prime \right)$ are found to be $$\begin{aligned}
T\left(\epsilon^\prime, \epsilon^{\prime \prime}, \tau, \tau^\prime \right) &=& \left(t(\tau) -t(\tau^\prime) \right) - i |\cos\Theta_{0}| \left( \epsilon^{\prime \prime} {\dot{t}}(\tau) + \epsilon^{\prime} {\dot{t}}(\tau^\prime) \right) \label{Tdef} \\
X\left(\epsilon^\prime, \epsilon^{\prime \prime}, \tau, \tau^\prime \right) &=& \left({\bf x}(\tau) -{\bf x}(\tau^\prime) \right) - i |\cos\Theta_{0}| \left( \epsilon^{\prime \prime} {\dot{{\bf x}}}(\tau) + \epsilon^{\prime} {\dot{{\bf x}}}(\tau^\prime) \right) \label{Xdef}\end{aligned}$$ The overdot refers to the derivative with respect to the proper time $\tau$ or $\tau^\prime$. Expanding the above expression, $W_{\Theta_0}(\tau,\tau^\prime) $ can also be written as $$\begin{aligned}
W_{\Theta_0}(\tau,\tau^\prime) = \frac{1}{16} & \bigg\{& \frac{4 \pi}{-T^2 + X^2} + \frac{i 8 \pi \epsilon^{\prime \prime} |\cos\Theta_{0}| \left[ -T\dot{t} + X \dot{{\bf x}} \right] }{\left( -T^2+X^2 \right)^2} \nonumber \\
&& + \frac{i 8 \pi \epsilon^{\prime} |\cos\Theta_{0}| \left[-T \dot{t}^{\prime} + X \dot{{\bf x}}^{\prime} \right]}{\left(-T^2+X^2 \right)^2} \nonumber \\
&& - \frac{32 \pi \epsilon^{\prime} \epsilon^{\prime\prime} |\cos\Theta_{0}| \left[ -T \dot{t}^\prime + X \dot{{\bf x}}^{\prime} \right] \left[ -T \dot{t}+X\dot{{\bf x}} \right] }{\left( -T^2+X^2 \right)^3} \nonumber \\
&& + \frac{8 \pi \epsilon^{\prime} \epsilon^{\prime\prime} |\cos\Theta_{0}|^{2} \left[- \dot{t} \dot{t}^{\prime} + \dot{{\bf x}}\dot{{\bf x}}^{\prime} \right] }{ \left( -T^2+X^2 \right)^2} \; \bigg\}_{\epsilon^\prime = \epsilon, \epsilon^{\prime \prime} = \epsilon}
\label{Wfinal}\end{aligned}$$ The first term is of the familiar form one gets for the total transition rate, however one must note that both the functions $T\left(\epsilon^\prime, \epsilon^{\prime \prime}, \tau, \tau^\prime \right)$ and $X\left(\epsilon^\prime, \epsilon^{\prime \prime}, \tau, \tau^\prime \right)$ are dependent on the angle $\Theta$.
Another feature of Eq.(\[Wfinal\]), is that, $\epsilon$ and $|\cos\Theta_0|$ always appear as a product in the expression. Given that $|\cos\Theta_0|$ is always non-negative, one can formally absorb it in the definition of $\epsilon$ itself. Then, in the point-like limit of the detector, that is, when taking the $\epsilon \rightarrow 0$, one will arrive at an expression which is independent of the angular direction. Thus to have a direction dependence in the transition rate, one needs to have the spatial extension of the detector modelled using a finite positive $\epsilon$ parameter in the present model.
We have thus finished our construction of the direction dependent spatially extended detector. Substituting Eq.(\[Wfinal\]) in Eq.(\[angtransitionrate\]) gives us the angular transition rate of the detector. The expression is general and will hold for any accelerating trajectory in a flat spacetime.
Rindler trajectory
------------------
We shall now analyse the direction dependent transition rate for the special case of the Rindler trajectory.
Substituting for the trajectory $t(\tau) = (1/g)\sinh (g\tau) $, $x(\tau) =(1/g) \cosh (g\tau) $ and $y = z = 0$ in Eqs.(\[Tdef\]) and (\[Xdef\]), we have $$\begin{aligned}
-T^2+X^2
& = \frac{2}{g^2} \bigg\{ 1 - \sqrt{1-{c^{\prime}}^2} \sqrt{1-{c^{\prime\prime}}^2} \cosh \bigl[ g(\tau-\tau^{\prime})-i(\alpha_{c^{\prime\prime}}+\alpha_{c^{\prime}} ) \bigr]
\notag
\\
& \hspace{8ex}
- \left( \frac{{c^\prime}^2 + {c^{\prime\prime}}^2 }{2} \right)
\bigg\}
\ ,
\label{TXRindler}\end{aligned}$$ where $c^{\prime\prime}=i|\cos\Theta_{0}| g \epsilon^{\prime\prime}$, $c^{\prime}=i|\cos\Theta_{0}| g \epsilon^{\prime}$, $\cos \alpha_{c^{\prime\prime}} = 1/\sqrt{1-{c^{\prime\prime}}^2}$ and $\cos \alpha_{c^{\prime}} = 1/\sqrt{1-{c^{\prime}}^2}$. As is expected for a stationary trajectory, the above expression depends on the proper time $\tau$ and $\tau^\prime$ through their difference $\tau - \tau^\prime$ only. Further substituting Eq.(\[TXRindler\]) in Eqs.(\[wfinalcompact\]) and (\[angtransitionrate\]), we get $${\dot {\cal F}}_{\Theta_0}(\omega) =\frac{1}{16} \frac{\partial}{\partial \epsilon^{\prime}} \frac{\partial}{\partial \epsilon^{\prime\prime}} \bigg\{ 2 \operatorname{Re}\int_0^{\infty} ds \, e^{-i\omega s} \frac{4\pi\epsilon^{\prime\prime} \epsilon^{\prime}}{-T^{2}+X^{2}} \bigg\}_{\epsilon^\prime = \epsilon, \epsilon^{\prime \prime} = \epsilon}
\label{angFinter}$$ Identifying the symmetry in the integrand under the simultaneous exchange of $s \rightarrow -s$ and $i \rightarrow -i$ using Eq.(\[TXRindler\]), we can express the integral as a contour integral over the full real line $s \rightarrow (-\infty, \infty)$. Further, $(- T^2 + X^2)$ has the periodicity in $s \rightarrow s + 2\pi i /g$. One can then close the contour at $s + 2\pi i /g$ and evaluate the residue at the poles, to get $$\begin{aligned}
{\dot {\cal F}}_{\Theta_0}(\omega)
&=
\frac{\partial}{\partial \epsilon^{\prime}} \frac{\partial}{\partial \epsilon^{\prime\prime}} \bigg\{\frac{\pi^{2}g\epsilon^{\prime} \epsilon^{\prime\prime}e^{\frac{\omega}{g} \left[ \tan^{-1} \left( |\cos\Theta_{0}|g\epsilon^{\prime} \right)+ \tan^{-1}\left( |\cos\Theta_{0}|g\epsilon^{\prime \prime} \right)\right]} }{4 \left( e^{\frac{2 \pi \omega}{g}}-1\right)}
\notag
\\[1ex]
& \hspace{13ex}
\times \frac{\sin \left( \frac{\omega}{g} \cosh^{-1} (c) \right)}{\sqrt{1-{c^{\prime}}^2} \sqrt{1-{c^{\prime \prime}}^2} \sinh \left[ \cosh^{-1} (c) \right] } \; \; \bigg\}_{\epsilon^\prime = \epsilon, \epsilon^{\prime \prime} = \epsilon}\end{aligned}$$ where $c = [1 - ({c^\prime}^2 + {c^{\prime \prime}}^2)/2]/\sqrt{1 - {c^\prime}^2}\sqrt{1- {c^{\prime \prime}}^2}$. Note that $c\ge1$. Considering only the factor inside the braces (without the partial derivatives), it does seem to satisfy the KMS condition with the inverse of the temperature being $2 \pi/ g - (2/g)\tan^{-1} \left( |\cos\Theta_{0}|g\epsilon^{\prime} \right)+ (2/g)\tan^{-1}\left( |\cos\Theta_{0}|g\epsilon^{\prime \prime} \right)$. One gets such a result in the Schlicht case for the total transition rate with $\epsilon^{\prime} = \epsilon^{\prime \prime}$ and without the $\cos\Theta_0$ dependence. However, in the present case, the additional partial derivatives break the KMS property for ${\dot {\cal F}}_{\Theta_0}(\omega) $.
Another way to approach at the final expression is to first differentiate the integrand in Eq.(\[angFinter\]) and then perform the contour integration. This leads to the following $${\dot {\cal F}}_{\Theta_0}(\omega) = 2 \operatorname{Re}\int_0^{\infty} ds \, e^{-i\omega s} \, \frac{1}{D_\epsilon(s)}$$ where $$\begin{aligned}
\frac{1}{D_\epsilon(s)} = \frac{ g^{2} \pi\bigg\{ 3 b^2 \epsilon^2 + b^4 \epsilon^4 - 2(1- b^2 \epsilon^2)\sinh^{2}\left[ \frac{gs}{2}-i\alpha \right] - 2 i b \epsilon \sinh\left[ gs - i2\alpha \right] \bigg\}}{32 \left( 1+b^2\epsilon^2 \right)^3 \sinh^{4}\left[ \frac{gs}{2}-i\alpha \right]}
\label{denominator}\end{aligned}$$ and $b = g |\cos\Theta_0|$. This contour integral can be calculated using the similar procedure outlined for integral in Eq.(\[angFinter\]) for each of the three terms. One finally gets the angular transition rate to be $$\begin{aligned}
{\dot {\cal F}}_{\Theta_0}(\omega) &=& \frac{\pi g^{2}}{32 {\left(1+b^2 \epsilon^2 \right)}^3} \; \; \frac { e^{\frac{2\omega }{g} \tan^{-1}\left( g|\cos\Theta_{0}|\epsilon \right)}}{\left( e^{\frac{2 \pi \omega}{g}} -1\right) } \nonumber \\
&& \times \bigg\{ \; \; \frac{16 \pi}{3} \left(3 b^2 \epsilon^2 + b^4 \epsilon^4 \right) \frac{\omega}{g} \left(4+\frac{\omega^2}{g^2} \right) \nonumber \\
&& \; \; \; \; \; \; + 16 \pi \left(1-b^2\epsilon^2 \right) \frac{\omega}{g} + 32 \pi b \epsilon \frac{\omega^2}{g^2} \; \; \bigg\}\end{aligned}$$ Thus ${\dot {\cal F}}_{\Theta_0}(\omega)$ is not KMS thermal, in general, except when $\Theta_0 = \pi/2$. In the case $\Theta_0 = \pi/2$, $b$ vanishes and one recovers the usual Unruh temperature. Interestingly, even though the regularization in Eq.(\[Wfinal\]) does not hold in the $\Theta_0 = \pi/2$ case, we find that the final expression is indeed finite for the case.
For $\Theta_0 \neq \pi/2$, the quadratic term in the polynomial of $(\omega/g)$ breaks the thermality of the whole expression by just a sign. One can check that the polynomial in the braces does not possess a real root and hence is positive for all real values of $\omega$. Thus the transition rate ${\dot {\cal F}}_{\Theta_0}(\omega)$ is always positive as expected.
In the low frequency regime $|\omega/g| \ll 1$, the terms linear in $(\omega/g)$ dominate compared to the quadratic and cubic terms. Whereas, in the high frequency regime $|\omega/g| \gg 1$, the term cubic in $(\omega/g)$ dominate compared to the linear and quadratic terms. Hence, in both these limits, ${\dot {\cal F}}_{\Theta_0}(\omega)$ is KMS with the inverse of the temperature being equal to $2 \pi/ g - (4/g)\tan^{-1} \left( |\cos\Theta_{0}|g\epsilon \right)$, that is one observes a angle dependent temperature. In the $\Theta_0 = \pi/2$ direction, the temperature is same as the usual Unruh temperature while it increases as $\Theta_0$ decreases in the domain $0 \leq \Theta_0 \leq \pi/2$. Along the direction of acceleration, it is the maximum.
Let us look at the combination $\epsilon |\cos\Theta_0|$. As mentioned earlier, $\epsilon$ and $|\cos\Theta_0|$ always appear as a product in the expression. Let us assume that for $\Theta_0 \neq \pi/2$, the product $\epsilon |\cos\Theta_0| \gg 1$ by assuming $\epsilon \gg 1$ for a finite $|\cos\theta_0|$. In this case, the term with pre-factor $b^4 \epsilon^4$ dominates over rest of the terms in the braces and ${\dot {\cal F}}_{\Theta_0}(\omega)$ is KMS thermal with the inverse of the temperature being equal to $2 \pi/ g - (4/g)\tan^{-1} \left( |\cos\Theta_{0}|g\epsilon \right)$. The $\epsilon$ parameter represents the length scale of the spatial extension of the detector. A large $\epsilon$ would signify a sufficiently extended detector. Whereas, as mentioned earlier, in the point-like limit of the detector $\epsilon \rightarrow 0$, one recovers the usual isotropic Unruh temperature. This suggests that the features mentioned above are especially due to the spatial extension of the detector.
We comment on this spurious result in the discussion section after analysing the spatially extended detector from the Rindler co-moving frame of reference in the next section.
One might suspect that for the direction dependence feature of the spatially extended detector to vanish in the point-like limit $\epsilon \rightarrow 0^+$, could possibly be a feature of the transition rate of the detector wherein one has formally subtracted an infinite constant term by taking the difference of the transition probability of the detector at $\tau$ and $\tau + d\tau$ to arrive at the transition rate expression. The constant infinite term may, perhaps, contain the information about the direction dependence $\Theta_0$ even in the point-like limit $\epsilon \rightarrow 0^+$.
One can check for the above suspicion by explicitly computing the transition probability for the extended detector when the detector is switched ON and switch OFF smoothly *around* the finite proper time $\tau_0$ and $\tau_f$ respectively. The general expression for the transition probability is given as $${\cal F}(\omega) = \int_{-\infty}^{\infty} du \, \chi(u) \int_{-\infty}^{\infty} ds \, \chi(u -s) e^{- i \omega s} \, W_{\Theta_0, \epsilon}(u,u-s)
\label{transitionprobability}$$ where $\chi(\tau)$ is the smooth switching function which vanishes for $\tau < \tau_0$ and $\tau > \tau_f$ while it is unity for $\tau_0 < \tau < \tau_f$ and is smooth in its transition. For the stationary Rindler trajectory, we have $W_{\Theta_0, \epsilon}(u,u-s) = W_{\Theta_0, \epsilon}(s)$ as shown in Eqs.(\[TXRindler\]) and (\[wfinalcompact\]). Hence in the above equation Eq.(\[transitionprobability\]) for transition probability, we can interchange the sequence of integration and perform the $u$ integral first to get $${\cal F}(\omega) = \int_{-\infty}^{\infty} ds \, e^{- i \omega s} \, W_{\Theta_0, \epsilon}(s) \, Q(s)
\label{transitionprobability2}$$ where $Q(s) = \int_{-\infty}^{\infty} du \, \chi(u) \chi(u -s)$ is also a smooth analytic function in complex $s$ plane. One can then perform the contour integral in Eq.(\[transitionprobability2\]), by choosing an appropriate contour and evaluating the residues of the expression at the poles of the Wightman function $W_{\Theta_0, \epsilon}(s)$ to obtain a finite result. However, from the expression in Eq.(\[denominator\]), one can see that the combination of $\epsilon$ and $|\cos{\Theta_0}|$ always appears as product, hence evaluating the contour integral in Eq.(\[transitionprobability2\]) would preserve the product structure which would imply that taking the point-like limit $\epsilon \rightarrow 0^+$ would make the $\Theta_0$ dependence to go away. In-fact, one can also verify that even in the non-stationary case, the product feature of the $\epsilon$ and $|\cos{\Theta_0}|$ would still hold and the direction dependence would vanish again in the the point-like limit $\epsilon \rightarrow 0^+$.
A spatially extended detector in the Rindler frame {#rindlerframesection}
==================================================
Our aim in this section is to investigate the response of an spatially extended detector, working in its co-moving frame and coupled to the Minkowski vacuum state of the scalar field, following De Bievre and Merkli [@DeBievre:2006pys]. We consider the corresponding centre of mass, with co-ordinates $(x_0(\tau))$, of the detector to follow the Rindler trajectory with uniform acceleration $g$. We work in Rindler co-ordinates with the following form of the metric $$ds^2 = \exp{(2 g z)} \left( - dt^2 + dz^2 \right) + d x^2_{\perp}
\label{rindlermetric}$$
We further assume the usual monopole interaction Hamiltonian term proportional to the value of the field on the trajectory, but now the field is replaced by the smeared field $\phi(\tau)$ obtained through $$\phi(\tau) = \int dz d^2 x_{\perp} e^{g z} f \left(z, x_{\perp}, z_0(\tau), x_{\perp 0}(\tau) \right) \phi(x)
\label{smeared}$$ where $dz d^2 x_{\perp} e^{g z}$ is the 3-spatial volume of the $t = $ constant hypersurface and $f \left(z, x_{\perp}, z_0(\tau), x_{\perp 0}(\tau) \right)$ is the profile function which encodes the spatial geometry of the extended detector itself. For the particular detector considered, $z_0(\tau) =0 = x_{\perp 0}(\tau) $, the detector is centred, with its centre of mass at the origin.
The pullback of the Wightman function relevant for calculating the detector response function is $$\begin{aligned}
W(\tau,\tau^\prime) = \langle 0_M | \phi(\tau) \phi(\tau^\prime) | 0_M \rangle
\label{whitmannfunction}\end{aligned}$$ with the transition rate being $${\dot {\cal F}}(E) = 2 \operatorname{Re}\int_0^\infty ds \,e^{-i E s} \, W(\tau,\tau - s)$$
The quantised scalar field in terms of the mode solutions for the metric in Eq.(\[rindlermetric\]) is $$\phi(x) = \int d\omega \int d^2 k_{\perp} \left[ {\hat a}_{\omega, k_{\perp}} v_{\omega, k_{\perp}}(x) + {\hat a}^{\dagger}_{\omega, k_{\perp}} v^{\star}_{\omega, k_{\perp}}(x) \right]
\label{field}$$ where the mode solutions are given in terms of the modified Bessel function as $$v_{\omega, k_{\perp}}(x) = \sinh \left[\frac{\pi\omega /g}{4\pi^{4}g} \right]^{1/2} K_{i\omega /g} \left[ \frac{\sqrt{k_{\bot}^{2} + m^{2}}}{g e^{-g z}} \right] e^{ik_{\perp} \cdot x_{\perp} - i \omega t}
\label{modsol}$$ The smeared field operator defined in Eq.(\[smeared\]) can then be expressed as $$\phi(\tau) = \int d\omega \int d^2 k_{\perp} \left[ {\hat a}_{\omega, k_{\perp}} h_{\omega, k_{\perp}}(\tau) + {\hat a}^{\dagger}_{\omega, k_{\perp}} h^{\star}_{\omega, k_{\perp}}(\tau) \right]$$ with the corresponding smeared field modes to be $$\begin{aligned}
h_{\omega, k_{\perp}}(\tau) &=& \sinh \left[\frac{\pi\omega /g}{4\pi^{4}g} \right]^{1/2} e^{ - i \omega t} \; u_{\omega, k_{\perp}}\left(z_0(\tau), x_{\perp 0}(\tau) \right) \end{aligned}$$ and $$\begin{aligned}
u_{\omega, k_{\perp}}\left(z_0(\tau), x_{\perp 0}(\tau) \right) &=& \int dz \, d^2x_{\perp} e^{g z} f \left(z, x_{\perp}, z_0(\tau), x_{\perp 0}(\tau) \right) \nonumber \\
&& \times K_{i\omega /g} \left[ \frac{\sqrt{k_{\bot}^{2} + m^{2}}}{g e^{-g z}} \right] e^{ik_{\perp} \cdot x_{\perp}} \end{aligned}$$ The pullback of the Wightman function given in Eq.(\[whitmannfunction\]) is then expressed in terms of the smeared field modes to become $$\begin{aligned}
W(\tau,\tau^\prime) &=& \int d\omega \int d^{2}k_{\perp} \left[ \left( \eta_{\omega}+1 \right) h_{\omega, k_{\perp}}(\tau) h^{\star}_{\omega, k_{\perp}}(\tau^\prime) + \eta_{\omega} \, h^{\star}_{\omega, k_{\perp}}(\tau) h_{\omega, k_{\perp}}(\tau^\prime) \right] \nonumber \\
&=& \int d\omega \int d^{2}k_{\perp} \sinh \left[\frac{\pi\omega /g}{4\pi^{4}g} \right] \bigg[ \left( \eta_{\omega}+1 \right) u_{\omega, k_{\perp}}(\tau) u^{\star}_{\omega, k_{\perp}}(\tau^\prime) e^{- i \omega (\tau - \tau^\prime)} \nonumber \\
\; \; \; & & + \; \eta_{\omega} \, u^{\star}_{\omega, k_{\perp}}(\tau) u_{\omega, k_{\perp}}(\tau^\prime) \, e^{ i \omega (\tau - \tau^\prime)} \bigg] \end{aligned}$$ where $\eta_{\omega} = 1/(\exp{(\beta \omega)} - 1)$ being the Planckian factor with the usual Unruh temperature. Since for the Rindler trajectory, we have $z_0(\tau) =0 = x_{\perp 0}(\tau) $, hence the $u_{\omega, k_{\perp}}$ are just constants. Then $W(\tau, \tau^\prime) = W(\tau - \tau^\prime) = W(s)$ as expected for a Killing trajectory. The transition rate is straightforward to obtain and we get, $$\begin{aligned}
{\dot {\cal F}}(E)
&=& \int d\omega \int d^{2}k_{\perp} \sinh \left[\frac{\pi\omega /g}{4\pi^{4}g} \right] \bigg[ \Theta(\omega + E) \left( \eta_{\omega}+1 \right) u_{\omega, k_{\perp}} u^{\star}_{\omega, k_{\perp}} \nonumber \\
\; \; \; & & + \; \Theta(\omega - E)\; \eta_{\omega} \, u^{\star}_{\omega, k_{\perp}} u_{\omega, k_{\perp}} \bigg] \end{aligned}$$ Thus, ${\dot {\cal F}}(E) $, satisfies the KMS condition for an arbitrary profile function, $$\frac{{\dot {\cal F}}(E)}{{\dot {\cal F}}(-E)} = \frac{\eta_{E}}{\eta_{E}+1} = e^{- \beta E}$$ with the usual Unruh temperature.
The above result regarding the thermality is quite general and holds for any arbitrary smooth profile function which falls off to Rindler’s spatial infinity. One could even have included a direction dependent angle as in the case of Eq.(\[angprof\]). However, the result would still be the same, since the spatial part does not contribute to the $\tau$ integral in the co-moving frame.
Discussion {#discsection}
==========
We have analysed two models for a spatially extended detectors having direction dependence on the Rindler trajectory. The first model is based on Schlicht type construction with a direction-sensitive Lorentz-function profile for the smeared field operator with a characteristic length $\epsilon$ and defined in the Fermi co-ordinates attached to the uniformly accelerated trajectory. Whereas the second model has a very general direction sensitive profile for the smeared field operator but defined in the Rindler wedge corresponding to the trajectory. The transition rate for the two models were found to differ significantly when evaluated on the Rindler trajectory. In the first model, the spectrum was obtained to be anisotropic and non-KMS in general. Only in the two limits for the frequency $\omega/g \ll 1$ and $\omega/g \gg 1$ was the spectrum KMS thermal with a direction weighted temperature $2 \pi/ g - (4/g)\tan^{-1} \left( |\cos\Theta_{0}|g\epsilon \right)$. Further, for an arbitrary frequency but for $\epsilon /g \gg 1$ the spectrum if KMS thermal again with the same direction dependent temperature. In contrast, for the second model, the transition rate was found to be KMS thermal and isotropic for an arbitrary direction sensitive profile for the detector.
The reason for the discrepancy in results of the two models can be understood by analysing the tails of the profiles chosen relative to the Rindler horizon. In the first model with the Schlicht type profile, the constant time slices chosen in the Fermi co-ordinates extend all the way throughout the Rindler horizon since these slices form a subset of the Cauchy surfaces foliating the global flat spacetime. Hence the Lorentz-function profile defined on such a Cauchy surface has a tail extending much beyond the Rindler horizon which implies that the spatially extended detector is made up of constituents which leak outside the Rindler wedge. In such a case, when the proper time increases, the points at constant spatial coordinates on the orthogonal spatial hypersurfaces in the other Rindler wedge move to the past, and this casts doubt on the transition probability formula that involves the response function, since the formula is derived from time-dependent perturbation theory and involves time-ordered evolution. It is then a pure coincidence that Schlicht’s derivation of the spectrum in Eq.(\[schlichttrans\]) is KMS thermal with a higher temperature than the usual Unruh temperature, since the validity of the quantum description of the formula itself is suspect other than in the case when $\epsilon$ is set to zero wherein the detector model is restricted to the Rindler wedge with the usual Unruh temperature. However one can expect the transition rate expression derived in Eq.(\[angtransitionrate\]) and (\[Wfinal\]) is to be valid for detector trajectories not involving casual horizons.
On the other hand, If the support of the detector’s profile is contained in the Rindler wedge, then the corresponding transition rate is KMS in the usual Unruh temperature of the Rindler trajectory as is evident from the results of the second model of the detector defined in the Rindler wedge. This is regardless whether the peak of the spatial profile chosen coincides with the reference trajectory of the detector. The underlying reason is that the monopole interaction defines the energy gap of the detector with respect to the proper time of the reference trajectory, even when the proper time at the peak of the profile may be quite different from the proper time at the reference trajectory, which is the Rindler trajectory in the present case. One could question whether this way to define the interaction is a reasonable model of the microphysics of an extended body. When the profile has a peak, perhaps a more reasonable model would be to choose the reference trajectory to coincide with the peak of the profile. Thus, based on the second model defined in the Rindler wedge, we conclude that the Unruh effect is directionally isotropic with the usual Unruh temperature for spatially extended direction dependent detectors.
Nevertheless, there could be some interest in getting a more quantitative control of what happens for the Schlicht type detector model when the profile leaks outside the Rindler wedge, beyond Schlicht’s Lorentz-function profile. Schlicht’s profile gives KMS spectrum for the case discussed in the section \[schlichtsection\] but at a different temperature; greater than the usual Unruh temperature. One could question whether other leaking profiles still give KMS (at some temperature) or does any deviation from Schlicht’s profile necessarily break KMS.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Jorma Louko for helpful discussions and useful comments on the draft. SK thanks the Department of Science and Technology, India for financial support and the University of Nottingham for hospitality where part of the work was completed.
[99]{}
S. A. Fulling, Phys. Rev. D [**7**]{}, 2850 (1973). P. C. W. Davies, J. Phys. A [**8**]{}, 609 (1975). W. G. Unruh, Phys. Rev. D [**14**]{}, 870 (1976).
S. Takagi, Phys. Lett. [**148B**]{}, 116 (1984). N. D. Birrell and P. C. W. Davies, *Quantum Fields in Curved Space* (Cambridge University Press, Cambridge, 1982).
L. C. B. Crispino, A. Higuchi and G. E. A. Matsas, Rev. Mod. Phys. [**80**]{}, 787 (2008) \[arXiv:0710.5373 \[gr-qc\]\].
S. Fulling and G. Matsas, Scholarpedia [**9**]{}, no. 10, 31789 (2014). E. Martín-Martínez, M. Montero and M. del Rey, Phys. Rev. D [**87**]{}, 064038 (2013) \[arXiv:1207.3248 \[quant-ph\]\].
Á. M. Alhambra, A. Kempf and E. Martín-Martínez, Phys. Rev. A [**89**]{}, 033835 (2014) \[arXiv:1311.7619 \[quant-ph\]\].
S. De Bievre and M. Merkli, Class. Quant. Grav. [**23**]{}, 6525 (2006) \[arXiv:math-ph/0604023\].
D. Hümmer, E. Martín-Martínez and A. Kempf, Phys. Rev. D [**93**]{}, 024019 (2016) \[arXiv:1506.02046 \[quant-ph\]\].
A. Pozas-Kerstjens and E. Martín-Martínez, Phys. Rev. D [**92**]{}, 064042 (2015) \[arXiv:1506.03081 \[quant-ph\]\].
A. Pozas-Kerstjens and E. Martín-Martínez, Phys. Rev. D [**94**]{}, 064074 (2016) \[arXiv:1605.07180 \[quant-ph\]\].
A. Pozas-Kerstjens, J. Louko and E. Martín-Martínez, Phys. Rev. D [**95**]{}, 105009 (2017) \[arXiv:1703.02982 \[quant-ph\]\].
P. Simidzija and E. Martín-Martínez, Phys. Rev. D [**98**]{}, 085007 (2018) \[arXiv:1809.05547 \[quant-ph\]\].
S. Kolekar and T. Padmanabhan, Phys. Rev. D [**86**]{}, 104057 (2012) \[arXiv:1205.0258 \[gr-qc\]\].
S. Schlicht, Class. Quant. Grav. **21**, 4647 (2004) \[arXiv:gr-qc/0306022\].
B. S. DeWitt, in [*General Relativity: an Einstein centenary survey*]{}, edited by S. W. Hawking and W. Israel (Cambridge University Press, Cambridge, 1979).
P. Langlois, “Imprints of spacetime topology in the Hawking-Unruh effect,” PhD Thesis, University of Nottingham, arXiv:gr-qc/0510127.
J. Louko and A. Satz, Class. Quant. Grav. [**25**]{}, 055012 (2008) \[arXiv:0710.5671 \[gr-qc\]\].
J. Letaw, Phys. Rev. D **23**, 1709 (1981).
S. Takagi, Prog. Theor. Phys. Suppl. [**88**]{}, 1 (1986). C. J. Fewster, B. A. Juárez-Aubry and J. Louko, Class. Quant. Grav. [**33**]{}, 165003 (2016) \[arXiv:1605.01316 \[gr-qc\]\].
S. Kolekar and J. Louko, Phys. Rev. D [**96**]{}, 024054 (2017) \[arXiv:1703.10619 \[hep-th\]\].
*NIST Digital Library of Mathematical Functions.* `http://dlmf.nist.gov/`, Release 1.0.19 of 2018-06-22. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, and B. V. Saunders, eds.
[^1]: [email protected]
| ArXiv |
---
abstract: 'We present a comparison of Doppler-shifted H$\alpha$ line emission observed by the Global Jet Watch from freshly-launched jet ejecta at the nucleus of the Galactic microquasar SS433 with subsequent ALMA imaging at mm-wavelengths of [*the same*]{} jet ejecta. There is a remarkable similarity between the transversely-resolved synchrotron emission and the prediction of the jet trace from optical spectroscopy: this is an a priori prediction not an a posteriori fit, confirming the ballistic nature of the jet propagation. The mm-wavelength of the ALMA polarimetry is sufficiently short that the Faraday rotation is negligible and therefore that the observed ${\bf E}$-vector directions are accurately orthogonal to the projected local magnetic field. Close to the nucleus the ${\bf B}$-field vectors are perpendicular to the direction of propagation. Further out from the nucleus, the ${\bf B}$-field vectors that are coincident with the jet instead become parallel to the ridge line; this occurs at a distance where the jet bolides are expected to expand into one another. X-ray variability has also been observed at this location; this has a natural explanation if shocks from the expanding and colliding bolides cause particle acceleration. In regions distinctly separate from the jet ridge line, the fractional polarisation approaches the theoretical maximum for synchrotron emission.'
author:
- 'Katherine M. Blundell , Robert Laing, Steven Lee and Anita Richards,'
title: |
SS433’s jet trace from ALMA imaging and Global Jet Watch spectroscopy:\
evidence for post-launch particle acceleration
---
Introduction
============
Since shortly after its discovery four decades ago the prototypical Galactic microquasar SS433 has been known to eject oppositely-directed jets whose launch axis precesses with a cone angle of about 19 degrees approximately every 162 days and whose speeds average to about a quarter of the speed of light. Striking images of the emission at radio (cm) wavelengths reveal a zigzag/corkscrew structure that arises because of the above properties modulated by light-travel time effects arising from its orientation with respect to our line-of-sight [@Hjellming1981; @Stirling2002; @Blundell04; @Roberts2008; @Miller-Jones2008]. The optical spectra of this object are characterised by a strong Balmer H$\alpha$ emission line complex close to the rest-wavelength of this line, and also blue-shifted and red-shifted lines whose observed wavelengths change successively on a daily basis according to the instantaneous speed and angle of travel with respect to our line of sight. Fitted parameters to the kinematic model developed from the first few years of optical spectroscopy were presented by e.g. @Margon84 and @Eikenberry01. Hitherto the timing of optical spectroscopy and spatially resolved radio imaging has not permitted the observation of the same ejecta both at launch and after propagation. We present the first mm-wave image of SS433 from ALMA in combination with optical spectroscopy (Sec\[sec:SpeedsFromZeds\]) of the same ejecta observed during the year prior to the ALMA observations (Sec\[sec:ALMA\]). This allows us to distinguish ballistic motion post-launch from deceleration [e.g. @Stirling2004].
A long standing question is why SS433’s jet ejecta are primarily line emitting at launch yet synchrotron emitting at largest distances from the nucleus; the polarisation changes explored in Sec.\[sec:polar\] shed light on this. Inference of the magnetic-field structure in the jets is complicated by the combined effects of Faraday rotation and time-variable structure. Previous studies [@Stirling2004; @Roberts2008; @Miller-Jones2008] have been hampered by lack of resolution and frequency coverage as well as the uncertain effects of spatial- and temporal-variations in Faraday rotation. The dependence of Faraday rotation on the square of the wavelength ($\lambda$), means that wide-band observations at mm wavelengths allow the projected field direction to be determined accurately in a single observation even close to the core, where Faraday rotation measures may be large [@Roberts2008]. We present our polarimetric 230GHz results in Sec.\[sec:polar\].
Optical spectroscopy and inference from Doppler shifts {#sec:SpeedsFromZeds}
======================================================
Spectra of SS433 spanning a wavelength range of approximately 5800 to 8500 Angstroms, and having a spectral resolution of $\sim4000$ were observed in the year prior to the ALMA observations whenever this target was a nighttime object. These were carried out with the multi-longitude Global Jet Watch telescopes each of which is equipped with an Aquila spectrograph; the design and testing of these high-throughput spectrographs are described by @Lee2018. The observatories, astronomical operations, processing and calibration of the spectroscopic data streams are described in @Blundell2018. Almost all of these spectra contain a pair of so-called “moving lines” arising from the most recently launched jet bolides in SS433. The wavelengths corresponding to the centroids of the blue-shifted and red-shifted H$\alpha$ emission were converted into redshift pairs with respect to H$\alpha$ in the rest frame of SS433 according to its systemic velocity with respect to Earth [@lockman2007]. From these redshift and blueshift pairs from a given spectrum were derived the launch speed of each pair of bolides [@Blundell05 equation 2]. This avoids the approximation of constant ejection speed, which has been shown to be inaccurate from archival spectroscopy [@Blundell05; @Blundell11]. Assuming that the subsequent motion is ballistic (this assumption is discussed in Sec\[sec:ballistic\]), and adopting the standard kinematic model [@Hjellming1981], the locations they attain by the Julian Date of the mid-point of the ALMA observations (2457294.4836) are calculated, and plotted in Fig\[fig:combine\]. The assumed parameters of the kinematic model using the notation of @Eikenberry01 and @Hjellming1981 are: cone angle $\theta = 19^\circ$ (Hjellming et al. use $\psi$), inclination $i = 79^\circ$, rotation on the sky $\chi = 10^\circ$ (position angle $+100^\circ$), period $P = 162.34$day (Blundell et al., in preparation) and distance $d =
5.5$kpc [@Blundell04; @lockman2007]. The ejection phase was determined by fitting to the observed redshift pairs from JD 2457000 to JD 2457293.5. The phase $\phi = (2\pi/P)(t - t_{\rm ref}) + \phi_0$ with $\phi_0 =
-0.241$rad for a reference Julian date of $t_{\rm ref} =
2456000$. $\phi$ is used as in equation 1 of @Eikenberry01; @Hjellming1981 denote the same quantity by $\Omega(t_0-t_{\rm ref})$.
Millimetre polarimetric imaging {#sec:ALMA}
===============================
SS433 was observed using 27 ALMA antennas between 2015 September 28 21:26 and September 29 01:46 UT. Three execution blocks were run almost in sequence and under similar conditions. The precipitable water vapour column was around 1.4mm. The correlator was set up in Time Division Multiplex mode with a total bandwidth of 7.5GHz, in four 1.75-GHz spectral windows (spw) centred at 224, 226, 240 and 242GHz. Each spw was divided into 64 spectral channels and XX, YY, XY and YX correlations were recorded. The longest and shortest baselines were 2270 and 43m, sensitive to angular scales $\la 3.7$arcsec.
The quasar J1751+0939 was used as a bandpass, polarization and flux scale calibrator and J1832+0731 was used as the phase reference source on an approximately 8 min cycle. The total integration time on SS433 was $\approx$2hr. Initial data reduction followed standard ALMA scripts, executed in CASA [@Schnee2014]. The flux density of J1751+0939 during these observations was taken to be 3.7275Jy at 232.86GHz with a spectral index $\alpha = -$0.441 (defined in the sense $S(\nu) \propto \nu^{-\alpha}$) and the total flux scale uncertainty is about 10%. Polarization leakage was calibrated as described by @Nagai16. Several iterations of [clean]{} in multi-frequency synthesis mode [@Rau] followed by self-calibration were used to improve the imaging of SS433. The final iteration of amplitude and phase self-calibration was made by combining all four spectral windows using a model with two Taylor series terms. We show the zero-order Taylor series $I$ image after self-calibration, together with polarised intensity images derived from $Q$ and $U$ for the entire band (we demonstrate below that Faraday rotation is negligible for our frequency range). The off-source rms levels are 13, 11 and 12$\mu$Jybeam$^{-1}$ in $I$, $Q$ and $U$, respectively, consistent with the expectations for thermal noise alone. The restoring beam has FWHM $0.19 \times 0.16$arcsec$^2$.
The $I$ image (Fig\[fig:combine\], central panel greyscale) shows the familiar zigzag/corkscrew shape of SS433. The peak flux density at 233GHz is 86.0 mJy/beam. The in-band spectral index of the core is $-0.29 \pm 0.14$.
Comparison of time-extrapolated spectroscopy with ALMA imaging {#sec:ballistic}
==============================================================
![[*Central panel:*]{} mm-wave image made from ALMA observations in 2015 September described in Sec\[sec:ALMA\] overlaid with symbols representing the positions attained by the bolides whose speeds are measured via optical spectroscopy and assumed to move ballistically after launch. [*Upper two and lower two panels:*]{} Representative spectra from four different dates prior to the ALMA observations are shown, revealing a pair of Doppler shifted lines corresponding to emission from the oppositely moving jet bolides, recently launched and still optically radiant. These spectra are from each of four different observatories, from bottom to top, namely eastern Australia (GJW-OZ), Western Australia (GJW-WA), Chile (GJW-CL) and South Africa (GJW-SA). []{data-label="fig:combine"}](combine_redblue_axis_labels.eps){width="53.00000%"}
Fig\[fig:combine\] shows excellent agreement between the jet trace predicted by the redshift and blueshift pairs from the optical spectroscopy and the brightness distribution subsequently measured by ALMA at mm-wavelengths. There are only three free parameters in the superposition: two positional offsets to align the ejection centre with the peak of the radio emission and the rotation of the precession axis on the sky (which was taken from @Hjellming1981, not determined independently). This is the first time that it has been possible to demonstrate the superposition directly with optical spectroscopy covering the appropriate time period. We find no evidence for significant deceleration of the jet post-launch (which would be evinced by radio emission systematically lagging the optical bolides). In particular, we can rule out a deceleration of 0.02$c$ just outside the launch location as suggested[^1] by @Stirling2004: this would lead to a systematic offset of $\approx$0.3arcsec between the predicted jet trace and the ALMA brightness distribution at projected distances $\ga$1arcsec, most obviously on the East side of the source; this is not seen. More stringent constraints on deceleration can be obtained from a comparison of the jet trace predicted by optical spectroscopy with VLA radio imaging between 8GHz and 12GHz, in which the trace is detectable out to much greater distances from the nucleus than is possible in our current 230-GHz observations. We will address this comparison in a future paper, together with possible correlations of speed with launch angle.
$B$-field structure {#sec:polar}
===================
The apparently conflicting results in the literature for the relation between the projected magnetic-field direction and the underlying jet flow in SS433 can be understood by consideration of the different distances from the nucleus probed by these studies. @Stirling2004 and @Roberts2008 found a preferential alignment between the magnetic field and the jet ridge line from $\approx$0.4 – 2arcsec from the nucleus, whereas figure 8 of @Miller-Jones2008 suggests that the field is instead parallel to the ballistic velocity of the jet knots at distances larger than 2arcsec.
Our measurements of the magnetic-field direction are much less affected by Faraday rotation than those in earlier work. @Stirling2004 found a mean rotation measure (RM) of 119radm$^{-2}$ (excluding the core), which implies a position angle rotation of 0.01$^\circ$ at 230GHz. Even for the RM’s of $\approx$600radm$^{-2}$ estimated for distances within 0.4arcsec of the core [@Stirling2004; @Roberts2008], the inferred rotation is still only 0.06$^\circ$ at this ALMA band. We also find no evidence for any wavelength-dependent rotation across our observing band. In particular, the position angles measured for the individual spws at the location of the core (where we might expect maximum Faraday rotation) are consistent with the mean value for the band with an rms scatter of 1.0$^\circ$ and Fig. \[fig:paplot\] shows no systematic trend. We therefore conclude that the position angles plotted in Fig. \[fig:fracpolvectors\] are not significantly affected by Faraday rotation.
![The direction of the ${\bf B}$-field at four different wavelengths plotted as a function of $\lambda^2$. All measurements are consistent with $-17$ degrees. The solid bars represent errors due to thermal noise alone while the dotted bars include systematic errors. \[fig:paplot\]](paplot.eps){width="6.5cm"}
While the fractional polarisation of the nucleus of SS433 at 230GHz is low ($p = 0.011$), the position angle is still securely determined. The inset to Fig\[fig:fracpolvectors\] shows that within $\approx$0.35arcsec of the nucleus the orientation of the $B$-field vectors is consistent with being perpendicular to the jet ridge line (and also to line of radial ejection, which is indistinguishable from it at this distance), consistent with the tentative suggestion by @Roberts2008. At a distance of $\approx$0.35arcsec, the degree of polarisation increases to $p \approx 0.1$. Here, the field directions in both jets become parallel to the ridge line and clearly inconsistent with the direction of ballistic motion. This relative orientation persists out to at least 0.7arcsec, beyond which we cannot measure accurate position angles. Our result is consistent with that of @Stirling2004, but with higher angular resolution and lower uncertainties from Faraday rotation. For distances from the launch-point exceeding $\approx$2arcsec, Miller-Jones et al (2008) report the magnetic field of the jet as being parallel to the local velocity vector (using the value for rotation measure reported by Stirling et al 2004).
The transition in field direction at 0.35arcsec ($3 \times
10^{14}$m) may bear on the oft-debated question of whether the outflow in SS433 is best described as a succession of independent bolides or a continuous jet. It is interesting to compare this distance with the point at which the expanding bow shocks surrounding neighbouring bolides first intersect. If we assume that one bolide is ejected per day at a speed of 0.26$c$, their radial separation is $\approx$6.7 $\times 10^{12}$m. If the shock expansion speed is comparable with the expansion rate of the radio knots measured with VLBI ($\approx 0.015c$; @Jeffrey2016), then the shock fronts will indeed expand into each other and interact when the bolides have travelled $\approx 3 \times 10^{14}$m from the nucleus, roughly where the change in field direction occurs. Bolides will coalesce to form larger structures which will then cease to interact with one another, when the paths of successive bolide conglomerates are too angularly divergent. Thereafter, the magnetic field observed to be associated with the jet trace will no longer reflect the details of the bolides as launched but rather their interactions with the (magnetised) medium through which they flow. This magnetic field behaviour appears to dominate for distances from the launch-point exceeding $\approx$2arcsec.
Very highly polarised emission is observed away from the jet trace on the East side of the source (labelled A in Fig \[fig:fracpolvectors\]). Both @Roberts2008 and @Miller-Jones2008 have drawn attention to these off-ridgeline regions being significantly more polarised than the jet ridgeline itself in VLA data at 15GHz and 8GHz respectively. Our 230GHz data show significantly higher fractional polarisation values (0.6 – 0.7) in these regions. Comparison of all these data suggest the degree and direction of polarisation may change with precession period and possibly also distance from the nucleus.
The interaction of bolides into larger coalescences has been reasoned above to occur where there is a change in polarisation behaviour namely at approximately 0.35arcsec from the nucleus. We note that this coincides with the region reported by @Migliari2002 [@Migliari2005] to show distinct X-ray variability which we suggest arises from shocks formed by the coalescence. Such a mechanism would naturally give rise to the stochastic nature of the X-ray variations reported by @Migliari2002 [@Migliari2005].
![Panel (a) shows a colour scale depicting the fractional polarisation, $p$, of SS433 averaged over the ALMA observing band. Points are shown blanked (grey) wherever the total intensity $I < 5\sigma_I$. Panel (b) shows vectors whose lengths are proportional to $p$ (with the scale indicated by the labelled bar) and whose directions are along the apparent ${\bf B}$-field direction (i.e. rotated by 90$^\circ$ from the ${\bf E}$-vector direction with no correction for Faraday rotation: see text). The vectors are plotted where $I > 5\sigma_I$ and $P > 3\sigma_I$ and are superimposed on a grey-scale of total intensity, as indicated by the wedge labelled in mJy/beam. The inset shows the core with vectors plotted on an expanded scale for $I > 1$mJy/beam and $P > 3\sigma_I$. Panel (c): as (b), but with vectors superimposed on the inferred locations reached by pairs of plasma bolides (green crosses, as in Fig\[fig:combine\]). The inset again shows the core region.[]{data-label="fig:fracpolvectors"}](pol.eps){width="8.5cm"}
Conclusions
===========
Over one precession period of the jets in the Galactic microquasar SS433 is shown to be traced out at mm-wavelengths in our ALMA imaging. This shows remarkable concordance with the predicted trace of the same ejecta from Global Jet Watch spectroscopy at earlier epochs.
At mm-wavelengths the Faraday Rotation towards SS433 is negligible. By 0.35arcsec from the nucleus the ${\bf B}$-field direction has changed from being perpendicular to parallel to the jet ridge line. This occurs where bolides are expected to have expanded into one another, and where X-ray variability has been reported, consistent with the onset of particle acceleration and the change from line-emission at launch to dominant synchrotron emission further out.
This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2013.1.01369.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. A great many organisations and individuals have contributed to the success of the Global Jet Watch observatories and these are listed on [www.GlobalJetWatch.net]{} but we particularly thank the University of Oxford and the Australian Astronomical Observatory.
Blundell, K.M. & Bowler, M.G., 2004, , 616, L159
Blundell, K.M. & Bowler, M.G. 2005, , 622, L129
Blundell, K.M., Schmidtobreick, L., & Trushkin S., 2011, MNRAS, 417, 2401
Blundell, K.M., Lee., S., Porter, A., et al., in prep.
Eikenberry, S., et al, 2001, , 561, 1027
Hjellming, R.M. & Johnston, K.J. 1981, , 246, L141
Jeffrey, R. M., Blundell, K. M., Trushkin, S. A., & Mioduszewski, A. J. 2016, , 461, 312
Lee, S., Blundell, K., Gillingham, P.R., in prep
Lockman, F.J., Blundell, K.M., Goss, W.M. 2007 MNRAS 381 881
Margon, B., 1984, , 22, 507
Migliari, S., Fender, R., & M[é]{}ndez, M. 2002, Science, 297, 1673
Migliari, S., Fender, R. P., Blundell, K. M., M[é]{}ndez, M., & van der Klis, M. 2005, , 358, 860
Miller-Jones, J. C. A., Migliari, S., Fender, R. P., et al. 2008, , 682, 1141-1151
Nagai, H., Nakanishi, K., Paladino, R., et al, 2016, , 824, 132
Rau, U. & Cornwell, T.J., 2011, , 532, 71
Roberts, D. H., Wardle, J. F. C., Lipnick, S. L., Selesnick, P. L., & Slutsky, S. 2008, , 676, 584
Schnee, S.L., Brogan, C., Espada, D., et al 2014 in Observatory Operations: Strategies, Processes and Systems, Proc. SPIE, 9149, 91490
Stirling, A.M., Jowett, F. H., Spencer, R. E., et al, 2002, , 337, 657
Stirling, A. M., Spencer, R. E., Cawthorne, T. V., & Paragi, Z. 2004, , 354, 1239
[^1]: The lower speeds suggested by @Stirling2004 are a direct consequence of their adoption of a smaller distance $d = 4.8$kpc for SS433.
| ArXiv |
---
abstract: 'The Jahn-Teller problems of C$_{60}$ anions involving $t_{1g}$ next lowest unoccupied molecular orbital (NLUMO) were theoretically investigated. The orbital vibronic coupling parameters for the $t_{1g}$ orbitals were derived from the Kohn-Sham orbital levels with hybrid B3LYP functional by the frozen phonon approach. With the use of these coupling parameters, the vibronic states of the first excited C$_{60}^-$ were calculated, and were analyzed. The dynamical Jahn-Teller stabilization energy of the first excited C$_{60}^-$ is stronger than that of the ground electronic states, resulting in two times larger splitting of vibronic levels than those of the ground state C$_{60}^-$. The present coupling parameters prompt us to understand more about the excited C$_{60}$.'
author:
- Zhishuo Huang
- Dan Liu
title: 'Dynamical Jahn-Teller effect in the first excited C$_{60}^-$'
---
Introduction
============
Highly symmetric C$_{60}$ exhibits complex Jahn-Teller dynamics characterized by orbital-vibration entanglement in various charged and excited states [@Chancey1997; @Bersuker2006; @Dunn2015]. Among these states, negatively charged C$_{60}$ is one of the most interesting cases because it often serves as a building brick of materials [@Gunnarsson2004; @Capone; @Alloul; @Kamaras; @Takabayashi; @Nomura2016; @Otsuka]. In order to comprehend thoroughly the role of the building brick, many properties of negatively charged C$_{60}$ should be understood clearly, especially about JT effect involved properties. Though JT effect, including dynamic JT effect, of C$_{60}$ anions have been intensively investigated [@Auerbach1994; @Manini1994; @Yu1994; @Dunn1995; @Gunnarsson1995; @OBrien1996; @Tosatti1996; @Yu1997; @Manini1998; @Sookhun2003; @Dunn2005; @Tomita2005; @Hands2008; @Frederiksen2008; @Iwahara2010; @Dunn2012; @Klupp2012; @Stochkel2013; @Ponzellini; @Kundu2015; @Iwahara2018; @Liu2018a; @Liu2018b; @Matsuda2018], it is only last years that the actual situation in the ground electronic states of C$_{60}^{n-}$ molecule $(n = 1-5)$ has been established with accurate coupling parameters, which showed the importance of dynamic JT effect [@Liu2018a; @Liu2018b].
So far, the works about the dynamic JT effect in negatively charged C$_{60}$ have been almost always in the ground electronic configuration populating only the lowest unoccupied molecular orbitals, which is the $t_{1u}$ orbital. However, to our knowledge, neither the vibronic coupling parameters for excited electronic configuration, say $t_{1g}$ next lowest unoccupied molecular orbial (NLUMO), nor the relevant JT effect has been theoretically investigated much. While it is believed that the nature of excited C$_{60}$ anions involving the next lowest unoccupied molecular orbital is of fundamental importance to interpret absorption spectra of isolated C$_{60}^-$ [@Kato1991; @Kato1993; @Kodama1994; @Kondo1995; @Kwon2001; @Kwon2002; @Tomita2005; @Stochkel2013; @Watariguchi2016], electron transfer process of fullerene [@ET1; @ET2], and excitation spectra of alkali-doped fullerides [@Knupfer1997; @Chibotaru1999; @Chibotaru2000], and the JT effect involving the NLUMO must be significant in highly alkali doped [@Knupfer1997] and alkali-earth/rare-earth doped fullerides [@Chen1999; @Margadonna2000; @Iwasa2003; @Li2003; @He2005; @Akada2006; @Heguri2010]. Furthermore, it might also be important [@Nava2018] in recently reported light induced superconductivity of alkali-doped fullerides [@Mitrano2016; @Cantaluppi2017]. Recently, bound excited states of C$_{60}^-$ have been theoretically investigated [@EX1; @EX2; @EX3; @EX4; @EX5], and the stability of the first excited ${}^2T_{1g}$ states of C$_{60}^-$ has been confirmed, nevertheless, the vibronic problem has not been investigated.
In this work, we address the dynamical JT effect of first excited C$_{60}^-$ anion populating the $t_{1g}$ NLUMO. The vibronic coupling parameters are derived from the data obtained by density functional theory (DFT) calculations with hybrid B3LYP exchange-correlation functional. Using these coupling parameters, the vibronic states are obtained by numerically diagonalizing the dynamical JT Hamiltonian matrix, and are analyzed.
Jahn-Teller Effect
==================
Model Hamiltonian {#Sec:H}
-----------------
The $t_{1g}$ next LUMO of neutral C$_{60}$ with $I_h$ symmetry is triply degenerate and separated from the other orbital levels [@Chancey1997]. According to the selection rule, the $t_{1g}$ orbitals couple to totally symmetric $a_g$ and five-fold degenerate $h_g$ representation as in the case of $t_{1u}$ orbitals [@Jahn1937]: $$\begin{aligned}
[t_{1g} \otimes t_{1g}] = a_g \oplus h_g.
\label{Eq:selection}\end{aligned}$$ In this work, we take the equilibrium structure of C$_{60}$ as the reference. Therefore, besides the $h_g$ modes, the vibronic couplings to the $a_g$ modes are nonzero. The linear vibronic Hamiltonian of C$_{60}^-$ in the first excited electronic $(t_{1g}^1)$ configuration resembles to that for the ground $t_{1u}^1$ electronic configuration [@OBrien1969; @Auerbach1994; @OBrien1996; @Chancey1997]: $$\begin{aligned}
H &=& H_a + H_h,
\label{Eq:H}
\\
H_a &=&
\frac{1}{2}
\left(
p_a^2 + \omega_a^2 q_{a}^2
\right) + V_a q_{a},
\label{Eq:Ha}
\\
H_h &=&
\sum_{\gamma = \theta, \epsilon, \xi, \eta, \zeta} \frac{1}{2}\left(p_{h\gamma}^2 + \omega_h^2 q_{h \gamma}^2\right)
\nonumber\\
&&+
V_h
\begin{pmatrix}
\frac{1}{2} q_{h\theta} - \frac{\sqrt{3}}{2} q_{h\epsilon} & \frac{\sqrt{3}}{2} q_{h\zeta} & \frac{\sqrt{3}}{2} q_{h\eta} \\
\frac{\sqrt{3}}{2} q_{h\zeta} & \frac{1}{2} q_{h\theta} + \frac{\sqrt{3}}{2} q_{h\epsilon} & \frac{\sqrt{3}}{2} q_{h\xi} \\
\frac{\sqrt{3}}{2} q_{h\eta} & \frac{\sqrt{3}}{2} q_{h\xi} & -q_{h\theta} \\
\end{pmatrix}.
\label{Eq:Hh}\end{aligned}$$ Here, $q_{\Gamma\gamma}$ and $p_{\Gamma\gamma}$ ($\gamma = \theta, \epsilon, \xi, \eta, \zeta$ for $\Gamma = h$) are mass-weighted normal coordinates and conjugate momenta, respectively, $\omega_\Gamma$ is frequency, and $V_\Gamma$ the vibronic coupling parameters. The basis of the marix is in the order of $|T_{1g}x\rangle$, $|T_{1g}y\rangle$, $|T_{1g}z\rangle$. The representation for normal coordinates and conjugate momenta possess the symmetry of real $d$-type ($(2z^2-x^2-y^2)/\sqrt{6}$, $(x^2-y^2)/\sqrt{2}$, $\sqrt{2}yz$, $\sqrt{2}zx$, $\sqrt{2}xy$), as they are in consistent with the original and most used representation [@OBrien1969; @Auerbach1994; @Manini1994; @Obrien1996; @Chancey1997]. The bases are different from those ($Q$) of some previous work [@Dunn1995]. The relation between them are $$\begin{aligned}
\begin{split}
q_\theta =\sqrt{\frac{3}{8}} Q_\theta + \sqrt{\frac{5}{8}} Q_\epsilon,\\
q_\epsilon=\sqrt{\frac{3}{8}} Q_\theta - \sqrt{\frac{5}{8}} Q_\epsilon.
\end{split}\end{aligned}$$ In the above equation, the indices $g$ or $u$ indicating the parity and the indices $\mu$ distinguishing the frequencies are omitted for simplicity. They are added when necessary for the discussion.
Adiabatic potential energy surface
----------------------------------
The model Hamiltonians for the ground electronic configuration and the first excited configuration are the same. Therefore, many electronic properties of the ground and the first excited electronic configurations are common too. The depth of the adiabatic potential energy surface (APES) with respect to the reference structure is given by [@OBrien1969] $$\begin{aligned}
U_\text{min} &=& -E_a - E_\text{JT}
\nonumber\\
&=& -\frac{V_a^2}{2\omega_a^2} - \frac{V_h^2}{2\omega_h^2},
\label{Eq:Umin}\end{aligned}$$ with $$\begin{aligned}
q_{a,0} = -\frac{V_a}{\omega_a^2}, \quad \left|\bm{q}_{h,0} \right| = \frac{V_h}{\omega_h^2},\end{aligned}$$ where $E_a$ and $E_\text{JT}$ are the first and the second terms in the last expression in Eq. (\[Eq:Umin\]), respectively, and $\bm{q}_{h}$ is the list of $q_{h\gamma}$. The APES has two-dimensional continuous trough [@OBrien1969], suggesting the presence of SO(3) symmetry [@OBrien1971; @Pooler1980].
Vibronic states
---------------
As in the case of the JT problem for the ground electronic configuration [@OBrien1971; @Romestain1971; @Pooler1980], the vibronic angular momenta $\hat{\bm{J}}$ also exist in the first excited state [@Chancey1997]: $$\begin{aligned}
[\hat{H}_h, \hat{\bm{J}}^2] = [\hat{H}_h, \hat{J}_z] = 0.
\label{Eq:symmetry}\end{aligned}$$ Therefore, the eignestates of $\hat{H}$ (vibronic states) are expressed by $J$, $M_J$, and principal quantum number $\alpha$, $$\begin{aligned}
\hat{H}_h|\alpha JM_J\rangle &=& E_{\alpha J} |\alpha JM_J\rangle.
\label{Eq:vibronicproblem}\end{aligned}$$ The analytical treatments of the vibronic states in the strong limit of vibronic coupling [@OBrien1969; @OBrien1971; @Auerbach1994; @OBrien1996; @Iwahara2018] and weak coupling limit [@Manini1994] have been discussed much. Nevertheless, for the quantitative description of C$_{60}$ ions, only numerical approach can provide accurate description.
For numerical calculations, it is convenient to expand the vibornic states as $$\begin{aligned}
|\alpha J M_J \rangle &=& \sum_\gamma \sum_{\bm{n}_h} |T_{1g} \gamma\rangle \otimes |\bm{n} \rangle C_{\gamma \bm{n}; \alpha J M_J}.\end{aligned}$$ Here, $\bm{n}_h = (n_{h\theta}, n_{h\epsilon}, n_{h\xi}, n_{h\eta}, n_{h\zeta})$ is the set of vibrational quantum numbers of the Harmonic oscillation part of Eq. (\[Eq:Hh\]). Such an expansion using the direct products of the electronic states and the eigenstates of harmonic oscillator has been developed long time ago [@Longuet-Higgins1958] and has been routinely used to study dynamical JT problems including fullerene anion [@OBrien1971; @Auerbach1994; @Gunnarsson1995; @OBrien1996; @Iwahara2010; @Iwahara2013; @Ponzellini; @Liu2018a].
In the present calculations, the vibrational basis is truncated as $$\begin{aligned}
0 \le n_{h(\mu)\gamma}, \quad \sum_{\mu\gamma} n_{h(\mu)\gamma} \le 7,\end{aligned}$$ because the dimension of the Hamiltonian matrix rapidly increases. To take account of the eight sets of $h_g$ modes in real C$_{60}$, $\mu$ is added in the condition. For the diagonalization of the vibronic Hamiltonian (\[Eq:Hh\]), Lanczos algorithm was applied [@Pooler1984].
Results
=======
Orbital vibronic coupling parameters
------------------------------------
![ The JT splitting of the NLUMO levels with respect to $q_{h_g(8)\epsilon}$ deformation (in atomic unit). The black points and gray lines indicate the DFT values and model energy, respectively. []{data-label="Fig:V"}](Vh8.eps){width="8cm"}
---------- ------- ------------------- ------------ ------------ ------------ ------------ ------------ ------------
$J$
$\Gamma$ $\mu$ $\omega_{\Gamma}$ $V_\Gamma$ $g_\Gamma$ $E_\Gamma$ $V_\Gamma$ $g_\Gamma$ $E_\Gamma$
$a_g$ 1 496 $-0.449$ $-0.418$ 5.38 $-0.264$ $-0.245$ 1.849
2 1470 $-2.480$ $-0.452$ 18.66 $-2.380$ $-0.422$ 16.543
$h_g$ 1 273 $-0.406$ $-0.926$ 14.50 0.192 0.455 3.415
2 437 $-0.476$ $-0.536$ 7.78 0.450 0.503 6.886
3 710 $-1.061$ $-0.577$ 14.64 0.754 0.396 7.069
4 774 $-0.594$ $-0.284$ 3.86 0.554 0.259 3.256
5 1099 $-0.498$ $-0.141$ 1.35 0.766 0.209 3.038
6 1250 $-1.664$ $-0.387$ 11.61 0.578 0.132 1.360
7 1428 0.125 0.024 0.05 2.099 0.394 13.867
8 1575 $-2.113$ $-0.348$ 11.79 2.043 0.326 10.592
---------- ------- ------------------- ------------ ------------ ------------ ------------ ------------ ------------
The orbital vibronic coupling parameters are defined by the gradients of the $t_{1g}$ NLUMO level: $$\begin{aligned}
v_a = \left.\frac{\partial \epsilon_{t_{1g}z}}{\partial q_a}\right|_{\bm{q} = \bm{0}},
\quad v_h = -\left.\frac{\partial \epsilon_{t_{1g}z}}{\partial q_{h\theta}}\right|_{\bm{q} = \bm{0}},\end{aligned}$$ where $\bm{q}$ is the set of all normal coordinates. In the present case, the vibronic coupling parameters $V_\Gamma$ correspond to the orbital vibronic coupling parameters $v_\Gamma$: $$\begin{aligned}
V_\Gamma = v_\Gamma,\end{aligned}$$ in a good approximation because of the very small mixing of the orbitals under JT deformation.
The vibronic coupling parameters are derived by fitting the model potential to the gradients of NLUMO levels calculated in Ref. . The calculations were done using DFT calculations with hybrid B3LYP functional, because, indicated by the previous studies, B3LYP could give closer parameters to the experimental data[@Iwahara2013; @Matsuda2018], and has a good agreement wit GW approximation calculations [@Faber2011]. The derived coupling parameters are listed in Table \[Table:V\], and one of the fittings are shown in Fig. \[Fig:V\] (see for the others Supplemental Materials).
The stabilization energies in the first excited electron configuration are $E_a = 24.04$ and $E_h = 65.58$ meV, which are by 30.7 % and 32.5% larger than the stabilization energies of $a_{g}$ and $h_{g}$ mode for the ground configuration, respectively. Moreover, almost all the vibronic coupling parameters for the $h$ modes in the $T_{1g}$ state are opposite compared with the case for the $T_{1u}$ state. The difference in sign indicates that the relative displacements in the ground and excited electronic states are large, and hence, the vibronic progression under the transition ${}^2T_{1g} \leftarrow {}^2T_{1u}$ tends to be stronger than that under the photoelectron spectra of C$_{60}^-$. This tendency is seen in experimental absorption spectra of C$_{60}^-$ [@Kondo1995; @Tomita2005] and photoelectron spectra of C$_{60}^-$ [@Wang2005; @Huang2014].
Vibronic states
---------------
The ground vibronic levels (termed by $J$=1) for the $T_{1u}$ (LUMO) and $T_{1g}$ (NLUMO) electornic states are $-96.5$ and $-113.8$ meV, respectively. Previous study shows that for LUMO, the contributions from the static and the dynamic JT effect to the ground energy is almost the same[@Liu2018a], but this is the not the same situation for NLUMO, as is shown in Table. \[Table:Energy\_contrib\]. The ratio of the dynamical contribution to the static contribution is smaller in NLUMO case than in LUMO case, which is consistent [@Auerbach1994; @OBrien1996] with the stronger vibronic coupling in NLUMO than in LUMO.
Orbital E$_{total}$ E$_{static}$ E$_{dynamic}$ Ratio
----------------- ------------- -------------- --------------- -------
NLUMO $-113.8$ $-65.6$ $-48.2$ 0.74
LUMO[@Liu2018a] $-96.46$ $-50.3$ $-46.2$ 0.92
: Contributions to the ground vibronic energy (E$_{total}$) of NLUMO and LUMO of C$_{60}^{-}$. E$_{static}$, and E$_{dynamic}$ represent the static JT and dynamic JT stabilization energies. Ratio refers the ratio between E$_{static}$ and E$_{dynamic}$ (E$_{dynamic}$/E$_{static}$).[]{data-label="Table:Energy_contrib"}
The low-energy vibronic levels are shown in Fig. \[Fig:Vibronic\_E\_level\] (see also Table. \[Table:Energy\_level\]) for NLUMO, and LUMO, and are compared with the vibrational levels of neutral C$_{60}$. For easy comparision, the energies of ground states of LUMO, NLUMO and the lowest vibrational state were shifted to a same level (0 meV). The group of the first excited vibronic levels ($J = 3,2,1$) split more in NLUMO than in LUMO, as expected from the stronger vibronic coupling in the former: The splitting of the former, 13.3 meV, is about two times larger than that of LUMO (4.4 meV). Such splitting may be observed as fine structure in e.g. high-resolution absorption spectra of C$_{60}^-$.
![Low-lying vibronic levels with respect to NLUMO and LUMO of C$_{60}^{-}$. The numbers next to the energy levels are $J$ with the numbers in the parenthesis are the degeneracy.[]{data-label="Fig:Vibronic_E_level"}](Vibronic_E_level.eps){width="8cm"}
$J$ NLUMO LUMO
----- --- ------------ -----------
1 $-113.815$ $-96.469$
2 $-74.589$ $-60.753$
3 $-57.370$ $-38.126$
4 $-47.520$ $-29.757$
5 $-40.736$ $-26.841$
6 $-31.168$ $-11.395$
7 $-25.493$ $-8.099$
8 - $-5.411$
9 - $-4.123$
1 $-78.132$ $-61.918$
2 $-58.955$ $-40.873$
3 $-46.127$ $-29.004$
4 $-31.122$ $-12.071$
5 $-26.748$ $-8.264$
6 - $-6.155$
7 - $-4.265$
1 $-87.854$ $-65.135$
2 $-63.252$ $-46.786$
3 $-55.282$ $-31.703$
4 $-40.633$ $-25.806$
5 $-33.815$ $-14.620$
6 $-32.593$ $-12.575$
7 $-25.581$ $-8.417$
8 - $-3.843$
9 - $-1.607$
1 $-46.772$ $-28.549$
2 $-34.314$ $-14.108$
3 $-27.326$ $-7.724$
1 $-60.505$ $-33.554$
2 $-36.593$ $-14.985$
3 $-27.281$ $ -$
: The vibronic energy levels with respect to NLUMO and LUMO of C$_{60}^{-}$ (meV). The data for LUMO are taken Ref. , and the number in the parentheses indicate $J$.[]{data-label="Table:Energy_level"}
Conclusion
==========
In this work, the vibronic states of the first excited C$_{60}^-$ were calculated based on the orbital vibronic coupling parameters in the $t_{1g}$ orbital, which is compared with that in $t_{1u}$. The results for the $t_{1g}^1$ configuration showed much stronger dynamic JT stabilization than that for the $t_{1u}^-$ configuration, which induces greater splitting group of the first excited vibronic states too. Combining the present vibronic coupling constant and those from Ref. [\[1.3\]]{}, it is possible to investigate the luminescence spectra[@Akimoto2002] involving $t_{1g}$ NLUMO.
The authors thank Naoya Iwahara for his help with numerical calculations. They also gratefully acknowledge funding by the China Scholarship Council.
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| ArXiv |
---
abstract: 'This presents an overview of relativistic hydrodynamic modeling in heavy-ion collisions prepared for Hot Quarks 2016, at South Padre Island, TX, USA. The influence of the initial state and viscosity on various experimental observables are discussed. Specific problems that arise in the hydrodynamical modeling at the Beam Energy Scan are briefly discussed.'
author:
- 'Jacquelyn Noronha-Hostler'
title: Hydrodynamic Overview at Hot Quarks 2016
---
Introduction
============
The Quark-Gluon Plasma (QGP) existed microseconds after the Big Bang where its size was significantly larger and its expansion significantly slower than the QGP experimentally measured in heavy ion collisions. The QGP produced in the laboratory is the hottest, smallest, and densest fluid known to mankind so the techniques used to describe its properties are still being developed. With the discovery that the QGP formed at RHIC and the LHC was a nearly perfect liquid, a “standard model" of heavy-ion collisions has emerged that includes fluctuating initial conditions, event-by-event relativistic viscous hydrodynamics, and a hadronic afterburner.
The main signatures of nearly perfect fluidity are the flow harmonics (Fourier coefficients of the particle spectra) that can be reproduced within relativistic hydrodynamical calculations with an extremely small shear viscosity to entropy density ratio, $\eta/s\sim 0.08$. Originally, it was understood that there should be a lower bound for $\eta/s\sim 1/4\pi$ by combining a quasiparticle description with the uncertainty principle [@Danielewicz:1984ww]. The derivation of the KSS limit from strong coupling holography [@Kovtun:2003wp] initially gave support to such a bound, although, now it is known that there are holographic examples where $\eta/s$ can be even smaller, e.g., [@Kats:2007mq; @Brigante:2007nu; @Brigante:2008gz; @Buchel:2008vz; @Critelli:2014kra; @Finazzo:2016mhm]. It is expected that a minimum exists close to the crossover temperature [@Aoki:2006we] as one goes from a strongly interacting QGP phase into an eventually weakly interacting hadron gas phase [@NoronhaHostler:2008ju; @NoronhaHostler:2012ug]. Further references for the temperature dependence of viscosity can be found in [@Noronha-Hostler:2015qmd].
While a reasonable estimate for the range of values of $\eta/s$ can be made using sophisticated Bayesian techniques [@Bernhard:2015hxa; @Bernhard:2016tnd], its exact value is extremely dependent on the initial state formed immediately after the two heavy-ions collide. Over the years, a plethora of initial state models have been developed, each of which has a corresponding range of valid transport coefficients that allow for reasonable fit to experimental data. One of the most pressing issues remaining in heavy ions is finding observables either sensitive to only the initial state or only to transport coefficients, which will be discussed in detail in this proceedings.
Collective Flow
===============
When two heavy ions are smashed together, clear geometrical effects occur depending on if they hit head-on (central collisions), have a grazing collision (peripheral), or hit somewhere in between (mid-central). Experimentally, heavy ions are collided billions of times (each collision is an event) and each event produces a different number of particles that participated in the collision, known as multiplicity. The more central the collisions, the larger multiplicities are produced so the events are then sorted by their multiplicities into central classes where $0\%$ centrality have the highest multiplicities and $100\%$ centrality indicates the lowest possible multiplicities. Central collisions produce on average a circular shape in the transverse plane to the beam axis whereas as an approximate almond shape is produced for mid-central collisions and beyond.
Due to quantum fluctuations in the initial position of protons and neutrons within each ion, a multitude of shapes can be produced [@Alver:2010gr]. Each highly inhomogeneous initial condition runs separately through hydrodynamics on an event-by-event basis. Experimentally the initial state cannot be measured directly, rather pressure gradients convert the initial geometrical shapes into a corresponding momentum space anisotropy, measured via flow harmonics. To obtain the flow harmonics, one calculates the Fourier coefficients of the particle spectra (with special care to reproduce the exact way experimentalists measure flow harmonics [@Luzum:2012da] where multiplicity weighing and centrality rebinning should not be ignored [@Gardim:2016nrr; @Betz:2016ayq]).
A number of parameters that go into hydrodynamical modeling such as the initial time after which one assumes the system admits a hydrodynamic description, and also the switching temperature below which a hadronic transport is used. The initial time, $\tau_0$, depends on the collisional energy as well as the initial condition type and these issues were discussed in more detail in the Beam Energy Scan and anisotropic hydrodynamics talks at this conference. The maximum switching temperature, $T_{SW}$, is constrained by the hadronization temperature indicated from Lattice QCD [@Borsanyi:2014ewa] to ensure that one switches to the correct degrees of freedom.
![\[fig:IC\]Initial conditions used in heavy ion collisions organized by their basic assumptions.](profs.pdf){width="20pc"}
The flow harmonics themselves are most sensitive to the choice in the initial conditions as well as the transport coefficients ($\eta/s$, bulk viscosity to entropy density, $\zeta/s$, and their corresponding relaxation times- $\tau_{\pi}$ and $\tau_{\Pi}$, respectively). In the next two sections experimental observables that constrain the initial state and transport coefficients are discussed.
Separating the Initial State from Transport Coefficients
========================================================
At LHC energies a number of initial conditions such as IP-Glasma [@Gale:2012rq], EKRT [@Niemi:2015qia], and Trento (tuned to IP-Glasma) [@Moreland:2014oya] manage to fit well the two and four particle flow cumulants as well as the distribution of $v_n$’s. In Fig. \[fig:IC\] a schematic cartoon of the most well-known initial condition models categorized by their basic properties is shown. Additionally, extremely accurate predictions for the flow harmonics of LHC run 2 to the order of $\sim 5\%$ [@Niemi:2015voa; @Noronha-Hostler:2015uye] were made and later experimentally confirmed in [@Adam:2016izf]. However, the different models used in these predictions varied parameters like viscosity, freeze-out, and the inclusion of initial flow.
It has been well-established that an approximately linear relationship exists between the initial energy/entropy density eccentricities, $\varepsilon_n$, and the experimentally measured flow harmonics $v_n$’s [@Teaney:2010vd; @Gardim:2011xv; @Teaney:2012ke; @Niemi:2012aj; @Gardim:2014tya] and that sub-nucleon fluctuations do not appear to play a significant role in the calculation of the lowest order harmonics in large collision systems [@Noronha-Hostler:2015coa]. That being said, higher order flow harmonics are significantly more complicated and depend on a variety of eccentricities [@Gardim:2011xv; @Gardim:2014tya]. Additionally, $v_1$ is especially complicated [@Gardim:2014tya] and may depend on the full $T^{\mu\nu}$ initialization [@Gardim:2011qn].
![\[fig:mom\]First moment (mean), second moment (variance), third moment (skewness), and fourth moment (kurtosis) of a distribution.](momentsdis.pdf){width="20pc"}
In order to more easily quantify the distribution of flow harmonics, multiparticle cumulants are used. Cumulants of the $v_2$ distribution are directly connected to the moments of the distribution via $v_2\{4\}/v_2\{2\})^4=2-\langle v_2^4\rangle/\langle v_2^2\rangle^2$, which can indicate the degree of which the system is fluctuating (see Fig. \[fig:mom\]). If there are no fluctuations in the system the $p_T$-integrated $v_2\{4\}/v_2\{2\}\rightarrow 1$, whereas $v_2\{4\}/v_2\{2\}< 1$ and $v_2\{4\}\sim v_2\{6\} \sim \dots$ is a sign of the collective behavior measured in heavy ion collisions. Note, however, that the higher order cumulants are not exactly identical, small deviations can exist due to the skewness of the initial conditions [@Giacalone:2016eyu]. Finally, complications do exist for more peripheral collisions where deviations are seen between the linear mapping of the initial eccentricities and the final elliptical flow [@Niemi:2015qia], which can be explained due to cubic response [@Noronha-Hostler:2015dbi].
Recently, symmetric cumulants [@ALICE:2016kpq] that measure the correlation of different order flow harmonics on an event-by-event basis have been measured in PbPb collisions and $SC(3,2)$ (which involves elliptic and triangular flow) appears to be almost entirely driven by the initial eccentricities. While it was thought that $SC(4,2)$ was sensitive to the choice in viscosity, much of that disappears after a proper treatment of multiplicity weighing and centrality binning [@Gardim:2016nrr].
Another possibility that exists for extracting the properties of the initial state is via the correlations between soft and hard sectors of heavy ion collisions, i.e., soft-hard event engineering (SHEE) as described in [@Noronha-Hostler:2016eow] where it was found that the high $p_T$ flow harmonics are directly linked to fluctuating initial eccentricities (due to the path length dependence of jet quenching). Moreover, it was recently found that viscosity plays little role in the high $p_T$ flow harmonics [@Betz:2016ayq] so it’s especially interesting to exploit this relationship. Further studies may also be possible in the heavy flavor sector as well [@Nahrgang:2014vza], as done for instance in [@Prado:2016xbq].
Interestingly enough, it appears that there is a direct link between the symmetric cumulants and the event plane correlations [@Giacalone:2016afq], though the latter are significantly more sensitive to viscous effects [@Niemi:2015qia]. In fact, event plane correlations appear to be very sensitive to the temperature dependence of $\eta/s$ where a small viscosity in the hadron resonance gas phase seems to be preferred (such a small $\eta/s$ can occur due to a large number of massive, degenerate states close to the crossover transition [@NoronhaHostler:2008ju; @NoronhaHostler:2012ug]). Additionally, $v_4\{4\}^4$ may indicate that a larger $\eta/s$ is needed but further testing is still needed [@Giacalone:2016mdr].
The extraction of transport coefficients is significantly more intricate due to the interplay between shear and bulk viscosity [@Noronha-Hostler:2013gga; @Noronha-Hostler:2014dqa; @Ryu:2015vwa]. This is further complicated due to the large influence of the bulk viscous corrections at hadronization. In fact, different implementations of the bulk viscosity correction to the particle distribution at freeze-out can either increase the shear viscosity [@Bernhard:2016tnd; @Noronha-Hostler:2013gga; @Noronha-Hostler:2014dqa] or decrease it, as in [@Ryu:2015vwa]. Furthermore, second order transport coefficients may also become relevant [@Molnar:2013lta; @Finazzo:2014cna] especially in small collision systems, and the effects of many of these coefficients have not been systematically studied within the context of event-by-event hydrodynamics.
Additionally, $\langle p_T\rangle$ was recently used in [@Ryu:2015vwa] (and later in [@Bernhard:2016tnd]) to help constrain the temperature dependence of the bulk viscosity. There is a very intricate connection between the temperature dependence of $\zeta/s$ and the equation of state (bulk viscosity is only nonzero in a non-conformal system, which indicates that there should be a peak in bulk viscosity at the crossover [@Karsch:2007jc]) so this quantity is expected to play an even more significant role in the Beam Energy Scan.
Beam Energy Scan
================
So far in this paper we focused on LHC energies where the baryon chemical potential, $\mu_B$, is approximately zero. With the Beam Energy Scan at RHIC one is able to probe matter that would correspond to other regions in the QCD Phase Diagram and, hopefully, also approach the critical end point. However, a significant number of questions remain at almost all level of hydrodynamical modeling. Initial calculations within kinetic theory/hadron resonance gas [@Demir:2008tr; @Denicol:2013nua; @Kadam:2014cua], which do not contain a critical point, show a decrease in $\eta/s$ as one increases $\mu_B$. However, if the dynamic universality class of the QCD critical point is that of model H [@Son:2004iv], then one would expect the transport coefficients to diverge at the critical point. Assuming that the dynamic universality class is H and there is a diverging viscosity, then this would be a nice explanation for the turning off of triangular flow observed at low energies [@Adamczyk:2016exq].
However, there may be a more simple explanation for the decrease in $v_3$ at low energies that connects to the decreased amount of time spent in the hydrodynamic phase as the beam energy decreases [@Auvinen:2013sba]. This would be more consistent either with no critical point (or perhaps one placed at much larger $\mu_B$) or perhaps with a different dynamic universality class such as that of model B [@Rougemont:2015ona]. Significantly more research needs to be done to better understand both the location of the critical point as well as the dynamic properties of the expanding QGP in this regime and, hopefully, the results from the upcoming Beam Energy Scan II run at STAR will lead to unambiguous experimental verification of QCD critical behavior.
Outlook
=======
Relativistic viscous hydrodynamics has become a powerful tool to explore the strongly interacting QCD matter formed in heavy ion collisions. Significant progress has been made in finding “orthogonal" measurements that can independently aid in extracting either the initial state of the system or its transport properties. A number of open-ended questions still remain such as if there is still perfect fluidity close to the QCD critical point as well as what special considerations need to be taken into account at finite baryon density. While not covered here, other important questions discussed at this conference remain, e.g., the need for anisotropic hydrodynamics [@Nopoush:2014pfa] and also pre-equilibrium flow [@Heinz:2015arc; @Keegan:2016cpi]. Additionally, the origin of flow harmonics (and their corresponding cumulants) in small systems is still up for debate [@prit].
Acknowledgements {#acknowledgements .unnumbered}
================
JNH thanks the HotQuarks2016 organizers for inviting her to give the Hydrodynamic Overview of Heavy-Ion Collisions and she was supported by the National Science Foundation under grant no. PHY-1513864.
References {#references .unnumbered}
==========
[9]{}
P. Danielewicz and M. Gyulassy, Phys. Rev. D [**31**]{}, 53 (1985). doi:10.1103/PhysRevD.31.53 P. Kovtun, D. T. Son and A. O. Starinets, JHEP [**0310**]{}, 064 (2003) doi:10.1088/1126-6708/2003/10/064 \[hep-th/0309213\]. Y. Kats and P. Petrov, JHEP [**0901**]{}, 044 (2009) doi:10.1088/1126-6708/2009/01/044 \[arXiv:0712.0743 \[hep-th\]\]. M. Brigante, H. Liu, R. C. Myers, S. Shenker and S. Yaida, Phys. Rev. D [**77**]{}, 126006 (2008) doi:10.1103/PhysRevD.77.126006 \[arXiv:0712.0805 \[hep-th\]\]. M. Brigante, H. Liu, R. C. Myers, S. Shenker and S. Yaida, Phys. Rev. Lett. [**100**]{}, 191601 (2008) doi:10.1103/PhysRevLett.100.191601 \[arXiv:0802.3318 \[hep-th\]\]. A. Buchel, R. C. Myers and A. Sinha, JHEP [**0903**]{}, 084 (2009) doi:10.1088/1126-6708/2009/03/084 \[arXiv:0812.2521 \[hep-th\]\]. R. Critelli, S. I. Finazzo, M. Zaniboni and J. Noronha, Phys. Rev. D [**90**]{}, no. 6, 066006 (2014) doi:10.1103/PhysRevD.90.066006 \[arXiv:1406.6019 \[hep-th\]\].
S. I. Finazzo, R. Critelli, R. Rougemont and J. Noronha, arXiv:1605.06061 \[hep-ph\]. Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz and K. K. Szabo, Nature [**443**]{}, 675 (2006) doi:10.1038/nature05120 \[hep-lat/0611014\].
J. Noronha-Hostler, J. Noronha and C. Greiner, Phys. Rev. Lett. [**103**]{}, 172302 (2009) doi:10.1103/PhysRevLett.103.172302 \[arXiv:0811.1571 \[nucl-th\]\]. J. Noronha-Hostler, J. Noronha and C. Greiner, Phys. Rev. C [**86**]{}, 024913 (2012) doi:10.1103/PhysRevC.86.024913 \[arXiv:1206.5138 \[nucl-th\]\]. J. Noronha-Hostler, arXiv:1512.06315 \[nucl-th\].
J. E. Bernhard, P. W. Marcy, C. E. Coleman-Smith, S. Huzurbazar, R. L. Wolpert and S. A. Bass, Phys. Rev. C [**91**]{}, no. 5, 054910 (2015) doi:10.1103/PhysRevC.91.054910 \[arXiv:1502.00339 \[nucl-th\]\]. J. E. Bernhard, J. S. Moreland, S. A. Bass, J. Liu and U. Heinz, Phys. Rev. C [**94**]{}, no. 2, 024907 (2016) doi:10.1103/PhysRevC.94.024907 \[arXiv:1605.03954 \[nucl-th\]\]. B. Alver and G. Roland, Phys. Rev. C [**81**]{}, 054905 (2010) Erratum: \[Phys. Rev. C [**82**]{}, 039903 (2010)\] doi:10.1103/PhysRevC.82.039903, 10.1103/PhysRevC.81.054905 \[arXiv:1003.0194 \[nucl-th\]\]. M. Luzum and J. Y. Ollitrault, Phys. Rev. C [**87**]{}, no. 4, 044907 (2013) doi:10.1103/PhysRevC.87.044907 \[arXiv:1209.2323 \[nucl-ex\]\]. F. G. Gardim, F. Grassi, M. Luzum and J. Noronha-Hostler, arXiv:1608.02982 \[nucl-th\].
B. Betz, M. Gyulassy, M. Luzum, J. Noronha, J. Noronha-Hostler, I. Portillo and C. Ratti, arXiv:1609.05171 \[nucl-th\]. S. Borsanyi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti and K. K. Szabo, Phys. Rev. Lett. [**113**]{}, 052301 (2014) doi:10.1103/PhysRevLett.113.052301 \[arXiv:1403.4576 \[hep-lat\]\]. C. Gale, S. Jeon, B. Schenke, P. Tribedy and R. Venugopalan, Phys. Rev. Lett. [**110**]{}, no. 1, 012302 (2013) doi:10.1103/PhysRevLett.110.012302 \[arXiv:1209.6330 \[nucl-th\]\]. H. Niemi, K. J. Eskola and R. Paatelainen, Phys. Rev. C [**93**]{}, no. 2, 024907 (2016) doi:10.1103/PhysRevC.93.024907 \[arXiv:1505.02677 \[hep-ph\]\]. J. S. Moreland, J. E. Bernhard and S. A. Bass, Phys. Rev. C [**92**]{}, no. 1, 011901 (2015) doi:10.1103/PhysRevC.92.011901 \[arXiv:1412.4708 \[nucl-th\]\]. H. Niemi, K. J. Eskola, R. Paatelainen and K. Tuominen, Phys. Rev. C [**93**]{}, no. 1, 014912 (2016) doi:10.1103/PhysRevC.93.014912 \[arXiv:1511.04296 \[hep-ph\]\]. J. Noronha-Hostler, M. Luzum and J. Y. Ollitrault, Phys. Rev. C [**93**]{}, no. 3, 034912 (2016) doi:10.1103/PhysRevC.93.034912 \[arXiv:1511.06289 \[nucl-th\]\]. J. Adam [*et al.*]{} \[ALICE Collaboration\], Phys. Rev. Lett. [**116**]{}, no. 13, 132302 (2016) doi:10.1103/PhysRevLett.116.132302 \[arXiv:1602.01119 \[nucl-ex\]\]. D. Teaney and L. Yan, Phys. Rev. C [**83**]{}, 064904 (2011) doi:10.1103/PhysRevC.83.064904 \[arXiv:1010.1876 \[nucl-th\]\].
F. G. Gardim, F. Grassi, M. Luzum and J. Y. Ollitrault, Phys. Rev. C [**85**]{}, 024908 (2012) doi:10.1103/PhysRevC.85.024908 \[arXiv:1111.6538 \[nucl-th\]\]. D. Teaney and L. Yan, Phys. Rev. C [**86**]{}, 044908 (2012) doi:10.1103/PhysRevC.86.044908 \[arXiv:1206.1905 \[nucl-th\]\]. H. Niemi, G. S. Denicol, H. Holopainen and P. Huovinen, Phys. Rev. C [**87**]{}, no. 5, 054901 (2013) doi:10.1103/PhysRevC.87.054901 \[arXiv:1212.1008 \[nucl-th\]\]. F. G. Gardim, J. Noronha-Hostler, M. Luzum and F. Grassi, Phys. Rev. C [**91**]{}, no. 3, 034902 (2015) doi:10.1103/PhysRevC.91.034902 \[arXiv:1411.2574 \[nucl-th\]\].
J. Noronha-Hostler, J. Noronha and M. Gyulassy, Phys. Rev. C [**93**]{}, no. 2, 024909 (2016) doi:10.1103/PhysRevC.93.024909 \[arXiv:1508.02455 \[nucl-th\]\]. F. G. Gardim, F. Grassi, Y. Hama, M. Luzum and J. Y. Ollitrault, Phys. Rev. C [**83**]{}, 064901 (2011) doi:10.1103/PhysRevC.83.064901 \[arXiv:1103.4605 \[nucl-th\]\]. G. Giacalone, L. Yan, J. Noronha-Hostler and J. Y. Ollitrault, arXiv:1608.01823 \[nucl-th\]. J. Noronha-Hostler, L. Yan, F. G. Gardim and J. Y. Ollitrault, Phys. Rev. C [**93**]{}, no. 1, 014909 (2016) doi:10.1103/PhysRevC.93.014909 \[arXiv:1511.03896 \[nucl-th\]\]. J. Adam [*et al.*]{} \[ALICE Collaboration\], arXiv:1604.07663 \[nucl-ex\].
G. Giacalone, L. Yan, J. Noronha-Hostler and J. Y. Ollitrault, Phys. Rev. C [**94**]{}, no. 1, 014906 (2016) doi:10.1103/PhysRevC.94.014906 \[arXiv:1605.08303 \[nucl-th\]\]. J. Noronha-Hostler, B. Betz, J. Noronha and M. Gyulassy, Phys. Rev. Lett. [**116**]{}, no. 25, 252301 (2016) doi:10.1103/PhysRevLett.116.252301 \[arXiv:1602.03788 \[nucl-th\]\].
M. Nahrgang, J. Aichelin, S. Bass, P. B. Gossiaux and K. Werner, Phys. Rev. C [**91**]{}, no. 1, 014904 (2015) doi:10.1103/PhysRevC.91.014904 \[arXiv:1410.5396 \[hep-ph\]\]. C. A. G. Prado, J. Noronha-Hostler, M. R. Cosentino, M. G. Munhoz, J. Noronha and A. A. P. Suaide, arXiv:1609.06093 \[nucl-th\]. G. Giacalone, L. Yan, J. Noronha-Hostler and J. Y. Ollitrault, arXiv:1608.06022 \[nucl-th\]. J. Noronha-Hostler, G. S. Denicol, J. Noronha, R. P. G. Andrade and F. Grassi, Phys. Rev. C [**88**]{}, 044916 (2013) doi:10.1103/PhysRevC.88.044916 \[arXiv:1305.1981 \[nucl-th\]\]. J. Noronha-Hostler, J. Noronha and F. Grassi, Phys. Rev. C [**90**]{}, no. 3, 034907 (2014) doi:10.1103/PhysRevC.90.034907 \[arXiv:1406.3333 \[nucl-th\]\].
S. Ryu, J.-F. Paquet, C. Shen, G. S. Denicol, B. Schenke, S. Jeon and C. Gale, Phys. Rev. Lett. [**115**]{}, no. 13, 132301 (2015) doi:10.1103/PhysRevLett.115.132301 \[arXiv:1502.01675 \[nucl-th\]\].
E. Molnar, H. Niemi, G. S. Denicol and D. H. Rischke, Phys. Rev. D [**89**]{}, no. 7, 074010 (2014) doi:10.1103/PhysRevD.89.074010 \[arXiv:1308.0785 \[nucl-th\]\]. S. I. Finazzo, R. Rougemont, H. Marrochio and J. Noronha, JHEP [**1502**]{}, 051 (2015) doi:10.1007/JHEP02(2015)051 \[arXiv:1412.2968 \[hep-ph\]\].
F. Karsch, D. Kharzeev and K. Tuchin, Phys. Lett. B [**663**]{}, 217 (2008) doi:10.1016/j.physletb.2008.01.080 \[arXiv:0711.0914 \[hep-ph\]\]. N. Demir and S. A. Bass, Phys. Rev. Lett. [**102**]{}, 172302 (2009) doi:10.1103/PhysRevLett.102.172302 \[arXiv:0812.2422 \[nucl-th\]\]. G. S. Denicol, C. Gale, S. Jeon and J. Noronha, Phys. Rev. C [**88**]{}, no. 6, 064901 (2013) doi:10.1103/PhysRevC.88.064901 \[arXiv:1308.1923 \[nucl-th\]\].
G. P. Kadam and H. Mishra, Nucl. Phys. A [**934**]{}, 133 (2014) doi:10.1016/j.nuclphysa.2014.12.004 \[arXiv:1408.6329 \[hep-ph\]\].
D. T. Son and M. A. Stephanov, Phys. Rev. D [**70**]{}, 056001 (2004) doi:10.1103/PhysRevD.70.056001 \[hep-ph/0401052\]. L. Adamczyk [*et al.*]{} \[STAR Collaboration\], Phys. Rev. Lett. [**116**]{}, no. 11, 112302 (2016) doi:10.1103/PhysRevLett.116.112302 \[arXiv:1601.01999 \[nucl-ex\]\]. J. Auvinen and H. Petersen, Phys. Rev. C [**88**]{}, no. 6, 064908 (2013) doi:10.1103/PhysRevC.88.064908 \[arXiv:1310.1764 \[nucl-th\]\]. R. Rougemont, J. Noronha and J. Noronha-Hostler, Phys. Rev. Lett. [**115**]{}, no. 20, 202301 (2015) doi:10.1103/PhysRevLett.115.202301 \[arXiv:1507.06972 \[hep-ph\]\].
M. Nopoush, R. Ryblewski and M. Strickland, Phys. Rev. C [**90**]{}, no. 1, 014908 (2014) doi:10.1103/PhysRevC.90.014908 \[arXiv:1405.1355 \[hep-ph\]\]. U. W. Heinz and J. Liu, Nucl. Phys. A [**956**]{}, 549 (2016) doi:10.1016/j.nuclphysa.2016.01.065 \[arXiv:1512.08276 \[nucl-th\]\]. L. Keegan, A. Kurkela, A. Mazeliauskas and D. Teaney, JHEP [**1608**]{}, 171 (2016) doi:10.1007/JHEP08(2016)171 \[arXiv:1605.04287 \[hep-ph\]\]. See P. Tribedy, HotQuarks 2016 proceedings to appear.
| ArXiv |
---
abstract: 'This paper addresses the question of whether a “rigid molecule” (one which does not deform in an external field) used as the conducting channel in a standard three-terminal MOSFET configuration can offer any performance advantage relative to a standard silicon MOSFET. A self-consistent solution of coupled quantum transport and Poisson’s equations shows that even for extremely small channel lengths (about $1~nm$), a “well-tempered” molecular FET demands much the same electrostatic considerations as a “well-tempered” conventional MOSFET. In other words, we show that just as in a conventional MOSFET, the gate oxide thickness needs to be much smaller than the channel length (length of the molecule) for the gate control to be effective. Furthermore, we show that a rigid molecule with metallic source and drain contacts has a temperature independent subthreshold slope much larger than $60~mV/decade$, because the metal-induced gap states in the channel prevent it from turning off abruptly. However, this disadvantage can be overcome by using semiconductor contacts because of their band-limited nature.'
author:
- |
Prashant Damle, Titash Rakshit, Magnus Paulsson and Supriyo Datta [^1] [^2] [^3] [^4] [^5]\
School of Electrical and Computer Engineering\
Purdue University\
West Lafayette, IN 47907
title: 'Current-voltage characteristics of molecular conductors: two versus three terminal'
---
Molecular electronics, MOSFETs, electrostatic analysis, quantum transport, Non-equilibrium Green’s function (NEGF) formalism.
Introduction {#sec:intro}
============
are promising candidates as future electronic devices because of their small size, chemical tunability and self-assembly features. Several experimental molecular devices have recently been demonstrated (for a review of the experimental work see [@reed_review]). These include two terminal devices where the conductance of a molecule coupled to two contacts shows interesting features such as a conductance gap [@reed_expt], asymmetry [@reichert_asymm_iv] and switching [@chen]. Molecular devices where a third terminal produces a negative differential resistance [@lang_fet], or suppresses the two terminal current [@emberly_fet] have been theoretically studied, but most of the work on modeling the current-voltage (IV) characteristics of molecular conductors has focused on two-terminal devices (see, for example, [@datta_expt; @emberly_two_pdt; @diVentra; @rdamle; @taylor_siesta; @palacios_fullerene] and references therein).
The purpose of this paper is to analyze a three-terminal molecular device assuming that the molecule behaves essentially like a rigid solid. Unlike solids, molecules are capable of deforming in an external field and it may be possible to take advantage of such conformational effects to design transistors with superior characteristics. However, in this paper we do not consider this possibility and simply address the question of whether a “rigid molecule” used as the conducting channel in a standard three-terminal MOSFET configuration can offer any performance advantage relative to a standard silicon MOSFET.
Although rigorous ab initio models are available in the literature [@diVentra; @rdamle; @taylor_siesta; @palacios_fullerene], they normally do not account for the three-terminal electrostatics that is central to the operation of transistors. For this reason we have used a simple model Hamiltonian whose parameters have been calibrated by comparing with ab initio models. We believe that a simple model Hamiltonian with rigorous electrostatics is preferable to an ab initio Hamiltonain with simplified electrostatics since the essential physics of a rigid molecular FET lies in its electrostatics.
The role of electrostatic considerations in the design of conventional silicon MOSFETs (with channel lengths ranging from $10~nm$ and above) is well understood. For the gate to have good control of the channel conductivity, the gate insulator thickness has to be much smaller than the channel length. Also, for a given channel length and gate insulator thickness, a double gated structure is superior to a single gated one, simply by virtue of having two gates as opposed to one. If a molecule is used as the channel in a standard three-terminal MOSFET configuration, the effective channel length is very small - about $1~nm$. Would similar electrostatic considerations apply for such small channel lengths? In this paper we answer this question in the affirmative. Specifically we will show that:
- [ The only advantage gained by using a molecular conductor for an FET channel is due to the reduced dielectric constant of the molecular environment. To get good gate control with a single gate the gate oxide thickness needs to be less than 10% of the channel (molecule) length, whereas in conventional MOSFETs the gate oxide thickness needs to be less than 3% of the channel length [@taur_ning]. With a double gated structure, the respective percentages are 60% and 20% [@zhibin_ballistic].]{}
- [ Relatively poor subthreshold characteristics (a [*temperature independent*]{} subthreshold slope much larger than $60~mV/decade$) are obtained even with good gate control, if metallic contacts (like gold) are used, because the metal-induced gap states in the channel preclude it from turning off abruptly. Preliminary results with a molecule coupled to doped silicon source and drain contacts, however, show a temperature dependent subthreshold slope ($\sim k_BT/q$). We believe this is due to the band-limited nature of the silicon contacts, and we are currently investigating this effect.]{}
Overall this study suggests that superior saturation and subthreshold characteristics in a molecular FET can only arise from novel physics beyond that included in our model. Further work on molecular transistors should try to capitalize on the additional degrees of freedom afforded by the “soft” nature of molecular conductors [@titash] - a feature that is not included in this study.
Although there has been no experimental report of a moleculer FET to date [^6] , judging from the historical development of the conventional silicon MOSFET, it is reasonable to expect that a single gated structure would be easier to fabricate than a double gated one. With this in mind, in this paper we mainly focus on a single gated molecular FET geometry (see Fig. \[fig:scheme\]). Few key results with a double gated geometry will be shown wherever appropriate to emphasize the differences between the single and double gated structures. The paper is organized as follows: Section \[sec:theory\] contains a brief description of the theoretical formulation and the simulation procedure. Section \[sec:results\] presents the simulation results along with an explanation of the underlying physics. Section \[sec:conclusion\] summarizes this paper.
Theory {#sec:theory}
======
A schematic figure of a molecule coupled to gold contacts (source and drain) is shown in Fig. \[fig:scheme\]a. As an example we use the Phenyl Dithiol (PDT) molecule which consists of a phenyl ring with thiol (-SH) end groups. A gate terminal modulates the conductance of the molecule. We use a simple model Hamiltonian $H$ to describe the molecule (Fig. \[fig:scheme\]b). The effect of the source and drain contacts is taken into account using self-energy functions $\Sigma_1$ and $\Sigma_2$ [@datta_book]. Scattering processes may be described using another self-energy matrix $\Sigma_p$. However, in this paper we focus on coherent or ballistic transport ($\Sigma_p=0$). The source and drain contacts are identified with their respective Fermi levels $\mu_1$ and $\mu_2$. Our simulation consists of iteratively solving a set of coupled equations (Fig. \[fig:scheme\]c) - the Non-Equilibrium Green’s Function (NEGF) formalism [@datta_book; @datta_tut] equations for the density matrix $\rho$ and the Poisson’s equation for the self-consistent potential $U_{SC}$. Given $H$, $U_{SC}$, $\Sigma_1$, $\Sigma_2$, $\mu_1$ and $\mu_2$ the NEGF formalism has clear prescriptions to obtain the density matrix $\rho$ from which the electron density and the current may be calculated. Once the electron density is calculated we solve the Poisson’s equation to obtain the self-consistent potential $U_{SC}$. The solution procedure thus consists of two iterative steps:
- [ [**Step 1**]{}: calculate $\rho$ given $U_{SC}$ using NEGF]{}
- [ [**Step 2**]{}: calculate $U_{SC}$ given $\rho$ using Poisson’s equation]{}
The above two steps are repeated till neither $U_{SC}$ nor $\rho$ changes from iteration to iteration. It is worth noting that the self-consistent potential obtained by solving Poisson’s equation (Eq. \[eq:poisson\]) may be augmented by an appropriate exchange-correlation potential that accounts for many electron effects using schemes like Hartree-Fock (HF) or Density Functional Theory (DFT) [@szabo]. In this paper we do not consider the exchange-correlation effects.
Step 1: To obtain $\rho$ from $U_{SC}$
--------------------------------------
The central issue in non-equilibrium statistical mechanics is to determine the density matrix $\rho$; once it is known, all quantities of interest (electron density, current etc.) can be calculated. A good introductory discussion of the concept of density matrix may be found in [@datta_tut]. To obtain the density matrix $\rho$ from the self-consistent potential $U_{SC}$ using the NEGF formalism, we need to know the Hamiltonian $H$, the contact self-energy matrices $\Sigma_{1,2}$ and the contact Fermi levels $\mu_{1,2}$. In this section we describe how we obtain these quantities, and then present a brief outline of the NEGF equations.
[*Hamiltonian*]{}: We use a simple basis consisting of one $p_z$ (or $\pi$) orbital on each carbon and sulfur atom. It is well known that the PDT molecule has $\pi$ conjugation - a cloud of $\pi$ electrons above and below the plane of the molecule that dictate the transport properties of the molecule [@magnus_paper]. The on-site energies of our $p_z$ orbitals correspond to the energies of valence atomic $p_z$ orbitals of sulfur and carbon (apart from a constant shift of all levels which is allowed as it does not affect the wavefunctions). The carbon-carbon interaction energy is $2.5~eV$ which is widely used to describe carbon nanotubes [@saito_cnt_book]. The sulfur-carbon coupling of $1.5~eV$ is empirically determined to obtain a good fit to the ab initio energy levels obtained using the commercially available quantum chemistry software Gaussian ’98 [@gaussian] (Fig. \[fig:pi\]).
Our model is very similar to the well established $p_z$ orbital based Hückel theory used by many quantum chemists. Although we use a simple model Hamiltonian to describe the molecule, we believe that the essential qualitative physics and chemistry of the molecule is captured. This is because both the molecular energy levels and the wavefunctions closely resemble those calculated from a much more sophisticated ab initio theory (Fig. \[fig:pi\]).
[*Self-energy*]{}: Self-energy formally arises out of partitioning the molecule-contact system into a molecular subsystem and a contact subsystem. The contact self-energy $\Sigma$ is calculated knowing the contact surface Green’s function $g$ and the coupling between the molecule and contact $\tau$. For a molecule coupled to two contacts (source and drain) the molecular Green’s function at an energy $E$ is then written as [@datta_book] ($I$: identity matrix, $H$: molecular Hamiltonian, $U_{SC}$: self-consistent potential):
$$G=[EI-H-U_{SC}-\Sigma_1-\Sigma_2]^{-1}
\label{eq:G}$$
where the contact self-energy matrices are
$$\Sigma_{1,2}=\tau_{1,2}g_{1,2}\tau_{1,2}^\dagger
\label{eq:Sig}$$
We model the gold FCC (111) contacts using one $s$-type orbital on each gold atom. The coupling matrix element between neighboring $s$ orbitals is taken equal to $-4.3~eV$ - this gives correct surface density of states (DOS) of $0.07~/(eV-atom)$ for the gold (111) surface [@papaconst]. The site energy for each $s$ orbital is assumed to be $-8.74~eV$ in order to get the correct gold Fermi level of $\sim -5.1~eV$. The surface Green’s function $g$ is calculated using a recursive technique explained in detail in [@manoj_thesis]. The contact-molecule coupling $\tau$ is determined by the geometry of the contact-molecule bond. It is believed [@larsen] that when a thiol-terminated molecule like Phenyl Dithiol is brought close to a gold substrate, the sulfur bonds with three gold atoms arranged in an equilateral triangle. For a good contact extended Hückel theory predicts a coupling matrix element of about $2~eV$ between the sulfur $p_z$ orbital and the three gold $s$ orbitals. However to simulate the bad contacts typically observed in experiments [@reed_expt; @diVentra] we reduce the coupling by a factor of five (this factor is also treated as a parameter, and our results do not change qualitatively for a range of values of this parameter).
Unlike the Hamiltonian, the self-energy matrices are non-Hermitian. The anti-Hermitian part of the self-energy, also known as the broadening function:
$$\Gamma_{1,2}=i(\Sigma_{1,2}-\Sigma_{1,2}^\dagger)
\label{eq:Gam}$$
is related to the lifetime of an electron in a molecular eigenstate. Thus upon coupling to contacts, the molecular density of states (Fig. \[fig:dos\]) looks like a set of broadened peaks.
[*Where is the Fermi energy?*]{}: When a molecule is coupled to contacts there is some charge transfer between the molecule and the contacts, and the contact-molecule-contact system attains equilibrium with one Fermi level $E_f$. A good question to ask is where $E_f$ lies relative to the molecular energy levels. The answer is not clear yet, the position of $E_f$ seems to depend on what contact model one uses. A jellium model [@diVentra] for the contacts predicts that $E_f$ is closer to the LUMO level for PDT whereas an extended Hückel theory based model [@tian] predicts that $E_f$ is closer to the HOMO level (see Fig. \[fig:pi\] and the related caption for a description of HOMO and LUMO levels). Our ab initio model [@rdamle] seems to suggest that for gold contacts, $E_f$ ($\sim -5.1~eV$) lies a few hundred millivolts above the PDT HOMO. In this paper we will use $E_f=-5.1~eV$ and set the molecular HOMO level (obtained from the $\pi$ model) equal to the ab initio HOMO level ($\sim
-5.4~eV$) (see Figs. \[fig:pi\], \[fig:dos\]). Once the location of the equilibrium Fermi energy $E_f$ is known, we can obtain the source and drain Fermi levels $\mu_1$ and $\mu_2$ under non-equilibrium conditions (non-zero $V_{DS}$): $\mu_1=E_f$ and $\mu_2=E_f-qV_{DS}$.
[*NEGF equations*]{}: Given $H$, $\Sigma_{1,2}$, contact Fermi energies $\mu_{1,2}$ and the self-consistent potential $U_{SC}$, NEGF has clear prescriptions [@datta_book] to obtain the density matrix $\rho$. The density matrix can be expressed as an energy integral over the correlation function $-iG^<(E)$, which can be viewed as an energy-resolved density matrix:
$$\rho = \int dE[-iG^<(E)/2\pi] \label{rho}$$
The correlation function is obtained from the Green’s function $G$ (eq. \[eq:G\]) and the broadening functions $\Gamma_{1,2}$ (eq. \[eq:Gam\]): $$-i{G}^< = G\left({f_1\Gamma_1 + f_2\Gamma_2}\right)G^\dagger$$ where $f_{1,2}(E)$ are the Fermi functions with electrochemical potentials $\mu_{1,2}$: $$f_{1,2}(E) = \left( 1 + \exp{\left[ {{E-\mu_{1,2}}\over{k_BT}}\right]}\right)^{-1}$$
The density matrix so obtained can be used to calculate the electron density $n{(\vec{r})}$ in real space using the eigenvectors of the Hamiltonian $\Psi_\alpha {(\vec{r})}$ expressed in real space:
$$n{(\vec{r})}=\sum_{\alpha,\beta} \Psi_\alpha {(\vec{r})}\Psi_\beta ^* {(\vec{r})}\rho_{\alpha \beta}
\label{eq:nofr}$$
The total number of electrons $N$ may be obtained from the density matrix as:
$$N={\rm trace}(\rho)
\label{eq:N}$$
The density matrix may also be used to obtain the terminal current [@datta_book]. For coherent transport, we can simplify the calculation of the current by using the transmission formalism where the transmission function [@datta_book]: $$T(E) = {\rm trace} \left[ \Gamma_1 G \Gamma_2 G^\dagger \right]$$ is used to calculate the terminal current $$I = (2q/h)\int_{-\infty}^\infty dE~T(E)~
\left(f_1(E)-f_2(E) \right)$$
Step 2: To obtain $U_{SC}$ from $\rho$
--------------------------------------
The Poisson’s equation relates the real space potential distribution $U {(\vec{r})}$ in a system to the charge density $n {(\vec{r})}$. We assume a nominal charge density $n_0 {(\vec{r})}$ obtained by solving the NEGF equations with $U {(\vec{r})}=0$ (at $V_{GS}=V_{DS}=0$). The Poisson’s equation is then solved for the [*change*]{} in the charge density ($n-n_0$) from the nominal value $n_0$ [^7] :
$$\vec{\nabla}\cdot\left(\epsilon\vec{\nabla}U {(\vec{r})}\right) = -q^2(n {(\vec{r})}-n_0 {(\vec{r})})
\label{eq:poisson}$$
The Poisson (or Hartree) potential $U$ may be augmented by an appropriate exchange-correlation potential $U_{xc}$. In this paper, we do not take into account the exchange-correlation effects ($U_{xc}=0$). We have two schemes to solve the Poisson’s equation: (1) simple Capacitance Model and (2) rigorous numerical solution over a 2D grid in real space.
[*Capacitance Model*]{}: We use a simplified picture of the molecule as a quantum dot with some nominal [*total*]{} charge $N_0$ (at $V_{GS}=V_{DS}=0$) and some average potential $U$ arising because of the [*change*]{} $N-N_0$ in this nominal charge due to the applied bias. Thus $U$, $N_0$ and $N$ are numbers and not matrices. The total charge $N$ can be calculated from the NEGF density matrix using Eq. \[eq:N\]. $U$ is the solution to the Poisson’s equation, and may be written as the sum of two terms: (1) A Laplace (or homogeneous) solution $U_L$ with zero charge on the molecule but with applied bias and (2) A particular (or inhomogeneous) solution $U_P$ with zero bias but with charge present on the molecule. Thus $U=U_L+U_P$. $U_L$ is easily written down in terms of the capacitative couplings $C_{MS}$, $C_{MD}$ and $C_{MG}$ of the molecule (Fig. \[fig:RC\]) with the source, drain and gate respectively:
$$U_L=\beta (-qV_{GS}) + \frac{(1-\beta)}{2} (-qV_{DS})
\label{eq:UL}$$
where
$$\beta=\frac{C_{MG}}{C_{MS}+C_{MD}+C_{MG}}
\label{eq:beta}$$
is a parameter ($0 < \beta < 1$) and is a measure of how good the gate control is. Gate control is ideal when $C_{MG}$ is very large as compared to $C_{MS}$ and $C_{MD}$ [^8] . In this case, $\beta=1$ and the Laplace solution $U_L=-qV_{GS}$ is essentially tied to the gate. An estimate of gate control may be obtained from the numerical grid solution explained below by plotting $\beta$ as a function of gate oxide thickness (Fig. \[fig:beta\]).
The particular solution $U_P$ may be written in terms of a charging energy $U_0$ as:
$$U_P=U_0(N-N_0)
\label{eq:UP}$$
The charging energy is treated as a parameter, and may be estimated as follows. The capacitance of a sphere of radius $R$ is $4\pi \epsilon R$. If we distribute a charge of one electron on this sphere, the potential of the sphere is $q/4\pi \epsilon R$. For $R=1~nm$ the value of this potential is about $1.4~eV$. Thus we use a charging energy $U_0 \sim 1~eV$. $U_0$ is the charging energy per electron per molecule and may also be estimated from the numerical grid solution by finding the average potential in the region occupied by the molecule and carrying one electronic charge distributed equally. This numerical procedure also yields $U_0 \sim 1~eV$ and is used to estimate the charging energy while comparing the capacitance model with the numerical grid solution (see Fig. \[fig:compare\] and the related caption).
With the simple capacitance model just described, the Poisson’s solution $U$ is just a number. The self-consistent potential that adds to the $p_z$ Hamiltonian (see Eq. \[eq:G\]) is then calculated as $U_{SC}=UI$, where $I$ is the identity matrix of the same size as that of the Hamiltonian.
[*Numerical solution*]{}: We use a 2D real space grid to solve the discretized Poisson’s equation for the geometry shown in Fig. \[fig:scheme\]a. The applied gate, source and drain voltages provide the boundary conditions. We use a dielectric constant of 3.9 for silicon dioxide and 2 for the self-assembled monolayer (SAM) [@sam_dielectric].
The correct procedure to obtain the real space charge density $n {(\vec{r})}$ (see Eq. \[eq:poisson\]) from the $p_z$ orbital space density matrix $\rho$ is to make use of Eq. \[eq:nofr\]. However, we simplify the calculation of $n {(\vec{r})}$ by observing that a carbon or sulfur $p_z$ orbital has a spread of about five to six Bohr radii (1 Bohr radius $a_B=0.529~\AA$). So for each atomic site $\alpha$ we distribute a charge equal to $\rho_{\alpha \alpha}$ equally in a cube with side $\sim~10a_B$ centered at site $\alpha$.
The solution to Poisson’s equation yields the real space potential distribution. However, the self-consistent potential $U_{SC}$ that needs to be added to the Hamiltonian (Eq. \[eq:G\]) is in the $p_z$ orbital space. We assume that $U_{SC}$ is a diagonal matrix with each diagonal entry as the value of the real space solution $U$ at the appropriate atomic position. For example, the diagonal entry in $U_{SC}$ corresponding to the sulfur based $p_z$ orbital would be equal to $U(\vec{r}_S)$ where $\vec{r}_S$ is the position vector of the sulfur atom.
Results {#sec:results}
=======
The self-consistent procedure (Fig. \[fig:scheme\]c) is done with the two types of Poisson solutions discussed above. The simple capacitance model is fast while the 2D numerical solution is slow but more accurate. The capacitance model has two parameters, namely $\beta$ which is a measure of the gate control, and $U_0$ which is the charging energy. These parameters can be extracted using the 2D numerical solution. We will first present results with the capacitance model by assuming ideal gate control, or $\beta=1$. This ideal case is useful to explain the current saturation mechanism. We will then compare the results obtained from the capacitance model with those obtained from the numerical solution, and show that the two match reasonably well.
Ideal gate control, on state
----------------------------
Fig. \[fig:ideal\_iv\] shows the molecular IV characteristic obtained by self-consistently solving the coupled NEGF - capacitance model Poisson’s equations. We contrast the IV for ideal gate control ($\beta=1$, Fig. \[fig:ideal\_iv\]a,b) with that for no gate control ($\beta=0$, Fig. \[fig:ideal\_iv\]c,d). For each case, we have shown the IV for positive as well as negative drain voltage. We observe the following:
- [With ideal gate control the IV is asymmetric with respect to the drain bias. For positive drain bias, we see very little gate modulation of the current. For negative drain bias we see current saturation and good gate modulation - the IV looks like that of a MOSFET.]{}
- [With no gate control the IV is symmetric with respect to the drain bias. There is no gate modulation.]{}
These features of the IV characteristic may be understood as follows (Fig. \[fig:mechanism\]). Let us first consider the ideal gate case. Since the gate is held at a fixed potential [*with respect to the source*]{}, the molecular DOS does not shift relative to the source Fermi level $\mu_1$ as the drain bias is changed [^9] . For negative drain bias (Fig. \[fig:mechanism\]a), the drain Fermi level $\mu_2$ moves up (towards the LUMO) with respect to the molecular DOS. Since the drain current depends on the DOS lying between the source and drain Fermi levels, the current saturates for increasing negative drain bias because the tail of the DOS dies out as the drain Fermi level moves towards the LUMO. If the gate bias is now made more negative, the molecular levels shift up relative to the source Fermi level, thereby bringing in more DOS in the energy range between $\mu_1$ and $\mu_2$ (referred to as the $\mu_1$-$\mu_2$ window from now on) , and the current increases. Thus we get current saturation and gate modulation.
For positive drain bias (Fig. \[fig:mechanism\]b), $\mu_2$ moves down (towards the HOMO) with respect to the molecular DOS. The current increases with positive drain bias because more and more DOS is coming inside the $\mu_1$-$\mu_2$ window as $\mu_2$ moves towards the HOMO peak. Once $\mu_2$ crosses the HOMO peak, the current levels off. This is the resonant tunneling mechanism. If the gate bias is now made more negative, no appreciable change is made in the DOS inside the $\mu_1$-$\mu_2$ window, and the maximum current remains almost independent of the gate bias.
Now let us contrast this with the case where no gate is present. In this case, due to the applied drain bias $V_{DS}$, the molecular DOS floats up by roughly $-qV_{DS}/2$ with respect to the source Fermi level. For either negative (Fig. \[fig:mechanism\]c) or positive (Fig. \[fig:mechanism\]d) drain bias, the current flow mechanism is resonant tunneling. Since the equilibrium Fermi energy lies closer to the HOMO, for negative drain bias $\mu_1$ crosses HOMO while for positive drain bias $\mu_2$ crosses HOMO [@datta_expt; @toymodel]. No gate modulation is seen as expected, and the IV is symmetric with respect to $V_{DS}$.
Ideal gate control - off state
------------------------------
Fig. \[fig:subth\] shows the log scale drain current as a function of gate bias at high drain bias. We note that despite assuming ideal gate control, the subthreshold slope of this molecular FET is about $300~mV/decade$ which is much worse than the ideal room temperature $k_BT/q=60~mV/decade$ that a good MOSFET can come close to achieving. It is also worth noting here that our simulation is done at low temperature - the subthreshold slope of the molecular FET is [*temperature independent*]{} and only depends on the molecular DOS as explained below.
The poor subthreshold slope may be understood as follows. As the gate voltage is made more positive, the molecular DOS shifts down with respect to the $\mu_1$-$\mu_2$ window. The HOMO peak thus moves away from the $\mu_1$-$\mu_2$ window, and fewer states are available to carry the current. The rate at which the current decreases with increasing positive gate bias thus depends on the rate at which the tail of the DOS in the HOMO-LUMO gap dies away with increasing energy (Fig. \[fig:dos\]). Typically we find that the tail of the DOS dies away at the rate of several hundred milli electron-volts of energy per decade, and this slow fall in the DOS determines the subthreshold slope. The slow fall in the molecular DOS may be attributed to the metal-induced gap (MIG) states - the gold source and drain contacts have a sizeable DOS near the Fermi energy, and are separated only by a few angstroms [^10]. Since the molecule is assumed to be rigid, the molecular DOS has no temperature dependence and hence neither does the subthreshold slope. Thus the molecular FET with a rigid molecule acting as the channel is a very poor switching device even with ideal gate control.
Estimate of Gate Control
------------------------
The 2D numerical Poisson’s solution may be used to estimate the gate control as follows. From Eq. \[eq:UL\] we see that
$$\beta=\left. -\frac{1}{q} ~ \frac{\partial U_L}{\partial V_{GS}}\right | _{V_{DS}}
\label{eq:beta_num}$$
Thus $\beta$ may be estimated from the numerical solution by slightly changing $V_{GS}$ (keeping $V_{DS}$ constant) and calculating how much the Laplace’s solution changes over the region occupied by the molecule. A plot of $\beta$ calculated using this method as a function of the gate oxide thickness $T_{ox}$ is shown in Fig. \[fig:beta\].
Knowing that the channel length (length of the PDT molecule) is about $1~nm$, It is evident from Fig. \[fig:beta\] that in order to get good gate control ($\beta > 0.8$) the gate oxide thickness ($T_{ox}$) needs to be about one tenth of the channel length ($L_{ch}$), or about $1~\AA$! Thus we need $L_{ch}/T_{ox} \sim 10$ to get a good molecular FET. In a well-designed conventional bulk MOSFET, $L_{ch}/T_{ox} \sim 40$ [@taur_ning]. This difference between a molecular FET and a conventional FET may be understood by noting that the dielectric constant of the molecular environment (=2) is about 6 times smaller than that of silicon (=11.7) [@lundstrom_private].
Fig. \[fig:beta\] also shows $\beta$ as a function of $T_{ox}$ calculated using the 2D numerical Laplace’s solution over a double gated molecular FET structure. In this case, we find that to get good gate control, we need $L_{ch}/T_{ox} \sim 1.6$. Thus a double gated structure is superior to a single gated one for a given $L_{ch}$ and $T_{ox}$, as is also expected for conventional silicon MOSFETs. The reason for this is simply that two gates can better control the channel than one.
Comparison: Capacitance model vs. Numerical Poisson’s solution
--------------------------------------------------------------
Fig. \[fig:compare\] compares the IV characteristic obtained by solving the self-consistent NEGF-Poisson’s equations with the numerical Poisson’s solution and the capacitance model. The parameters $\beta$ and $U_0$ for the capacitance model were extracted from the numerical solution. We see a reasonable agreement between the two solutions despite the simplifications made in the capacitance model (the capacitance model assumes a flat potential profile in the region occupied by the molecule, which may not be true, especially at high bias) . For $t_{ox}=1.5~nm$ (Fig. \[fig:compare\]a) there is very little gate modulation and no saturation as expected. In this case $\beta=0.28$ (Fig. \[fig:beta\]) and the IV resembles that shown in Fig. \[fig:ideal\_iv\]c more than the one in Fig. \[fig:ideal\_iv\]a. Also seen in Fig. \[fig:compare\] are the results for $t_{ox}=1~\AA$. For this case $\beta=0.82$ and we observe reasonable saturation and gate control. For realistic oxide thicknesses, however, we expect to observe an IV like the one shown in Fig. \[fig:compare\]a. We have also calculated the IV characteristics with a double gated geometry (not shown here), and as expected from Fig. \[fig:beta\], saturating IVs can be obtained for more realistic oxide thicknesses ($\sim ~ 7~\AA$).
Conclusion {#sec:conclusion}
==========
We have presented simulation results for a three terminal molecular device with a rigid molecule acting as the channel in a standard MOSFET configuration. The NEGF equations for quantum transport are self-consistently solved with the Poisson’s equation. We conclude the following:
1. [The current-voltage (IV) characteristics of molecular conductors are strongly influenced by the electrostatics, just like conventional semiconductors. With good gate control, the IV characteristics will saturate for one polarity of the drain bias and increase monotonically if the polarity is reversed. By contrast two-terminal symmetric molecules typically show symmetric IV characteristics.]{}
2. [The only advantage gained by using a molecular conductor for an FET channel is due to the reduced dielectric constant of the molecular environment. To get good gate control with a single gate the gate oxide thickness needs to be less than 10% of the channel (molecule) length, whereas in conventional MOSFETs the gate oxide thickness needs to be less than 3% of the channel length. With a double gated structure, the respective percentages are 60% and 20%.]{}
3. [Relatively poor subthreshold characteristics (a [*temperature independent*]{} subthreshold slope much larger than $60~mV/decade$) are obtained even with good gate control, if metallic contacts (like gold) are used, because the metal-induced gap states in the channel preclude it from turning off abruptly. Preliminary results with a molecule coupled to doped silicon source and drain contacts, however, show a temperature dependent subthreshold slope ($\sim k_BT/q$). We believe this is due to the band-limited nature of the silicon contacts, and we are currently investigating this effect.]{}
4. [Overall this study suggests that superior saturation and subthreshold characteristics in a molecular FET can only arise from novel physics beyond that included in our model. Further work on molecular transistors should try to exploit the additional degrees of freedom afforded by the “soft” (as opposed to rigid) nature of molecular conductors [@titash].]{}
[*Acknowledgments*]{}: We would like to thank M. Samanta, A. Ghosh, R. Venugopal and M. Lundstrom for useful discussions. This work was supported by the NSF under grants number 9809520-ECS and 0085516-EEC and by the Semiconductor Technology Focus Center on Materials, Structures and Devices under contract number 1720012625.
[^1]: Corresponding author: Prashant Damle
[^2]: Telephone: (765) 494 3383
[^3]: Fax: (765) 494 2706
[^4]: email: [email protected]
[^5]: ©2002 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.
[^6]: The authors are aware of one experimental claim (J.H. Schön et al., Nature 413, page 713, 2001) reporting superior molecular FET with a single gated geometry. This claim, however, has been strongly questioned (see article by R.F. Service in Science 298, page 31, 2002).
[^7]: The potential distribution corresponding to the nominal charge density (when no drain or gate bias is applied) is included in the calculation of the molecular Hamiltonian [@datta_expt].
[^8]: We have assumed that $C_{MS}=C_{MD}$ in eq. \[eq:UL\], which is reasonable because the center of the molecule is equidistant from the source and drain contacts in our model (see Fig. \[fig:scheme\]). In general, if the source (drain) is closer to the molecule, then $C_{MS}$ ($C_{MD}$) will be bigger [@datta_expt]. With $C_{MS}=C_{MD}$, the molecular Laplace potential is $V_{DS}/2$ in the absence of a gate ($\beta=0$), as is evident from eq. \[eq:UL\] (also see Fig. \[fig:mechanism\]c,d and the related caption).
[^9]: This is true as long as the charging energy $U_0 \sim 1~eV$, which is typically the case. For high charging energies the particular solution $U_P$ can dominate the Laplace solution $U_L$ (see eqs. \[eq:UL\],\[eq:UP\] and related discussion), thereby reducing gate control.
[^10]: For ballistic silicon MOSFETs, due to the band-limited nature of the doped silicon source/drain contacts, the MIG DOS is negligible. The subthreshold slope at a finite temperature is thus determined by the rate at which the difference in the source and drain [*Fermi function tails*]{} falls as a function of energy. This rate depends on the temperature, and the subthreshold slope is thus proportional to $k_BT/q$ ($\approx 60~mV$ at room temperature) for ballistic Si MOSFETs [@zhibin_ballistic]. Preliminary results for a molecular FET with doped silicon source and drain contacts do show a subthreshold slope proportional to $k_BT/q$; we are currently investigating this effect.
| ArXiv |
TIT/HEP–499\
[hep-th/0307206]{}\
July, 2003\
[\
]{}\
[ **Minoru Eto $^{a}$**]{} [^1], [ **Nobuhito Maru $^{b}$**]{} [^2] and [ **Norisuke Sakai $^{a}$**]{} [^3]
1.5em
[ *$^{a}$Department of Physics, Tokyo Institute of Technology\
Tokyo 152-8551, JAPAN\
and\
$^{b}$Theoretical Physics Laboratory\
RIKEN (The Institute of Physical and Chemical Research)\
2-1 Hirosawa, Wako, Saitama 351-0198, JAPAN* ]{} [**Abstract**]{}\
Introduction
============
In the brane-world scenario [@LED; @RS1; @RS2], our four-dimensional world is to be realized on topological defects such as walls. To obtain realistic unified theories beyond the standard model, supersymmetry (SUSY) has been most useful [@DGSW]. Moreover, SUSY helps to construct topological defects like walls as BPS states [@WittenOlive] that preserve part of SUSY. For a realistic model, understanding SUSY breaking has been an important problem, which is addressed in the SUSY brane-world scenario extensively [@BULK]–[@MSSS]. Models have been constructed that realize one such idea : coexistence of BPS and anti-BPS walls produces SUSY breaking automatically [@MSSS]. In particular, the SUSY breaking effects are suppressed exponentially as a function of distance between walls. On the other hand, non-BPS multi-wall configurations are not protected by SUSY and need not be stable. Such non-BPS wall configurations was successfully stabilized by introducing topological quantum numbers, such as a winding number [@MSSS2; @SakaiSugisaka]. The physical reason behind the stability is simple : a BPS wall and an anti-BPS wall with winding numbers generally exert repulsion, which then pushes each other at anti-podal points of the compactified dimension.
One of the most attractive models in the brane-world scenario is the model with the warped metric [@RS1; @RS2]. A possible solution of the gauge hierarchy problem was proposed in the two brane-model [@RS1], and a localization of graviton on a single brane was found even in a noncompact space [@RS2] at the cost of fine-tuning between bulk cosmological constant and boundary cosmological constant at orbifold fixed points. Supersymmetrization of the thin-wall model has also been constructed in five dimensions [@ABN]–[@FLP]. It is natural to ask if the infinitely thin branes in these models can be replaced by physical smooth wall configurations made out of scalar fields [@CGR]–[@SkTo]. We have succeeded in constructing BPS as well as non-BPS solutions in the ${\cal N}=1$ supergravity coupled with a chiral scalar multiplet in four dimensions [@EMSS]. A similar BPS solution has also been constructed in five-dimensional supergravity [@AFNS; @Eto:2003ut]. In the limit of vanishing gravitational coupling $\kappa \rightarrow 0$, our model reduces to the model having the exact solution of non-BPS multi-walls [@MSSS2]. Therefore the model is likely to be stable thanks to the winding number near the weak gravity limit. However, we need to address the issue of stability in the presence of gravity, since the radius of the extra dimension is now a dynamical variable which might introduce instability into the model. There have been a number of works to analyze the stability of the infinitely thin wall [@GiLa]–[@CsabaCsaki], especially in the presence of a stabilizing mechanism due to Goldberger and Wise [@GoWi].
The purpose of our paper is to study the stability of the model with winding number in the presence of gravity and to analyze the mass spectrum of fluctuations on the BPS and non-BPS solutions. We find that there are zero modes of transverse traceless fluctuations localized on the wall which play the role of the graviton in our world on the wall. The BPS solution has also gravitino zero mode which is localized on the wall and forms a supermultiplet with the graviton under the surviving supergravity transformation with the Killing spinor of the BPS solution. We obtain that the BPS solution has no other zero modes, and no tachyonic fluctuations. For instance, we find that possible additional massless tensor and scalar modes are either gauge degrees of freedom or unphysical (the mode function is not normalizable). As for the non-BPS solution, we find that another possible zero modes of the transverse traceless fluctuations of metric can be gauged away and that there exists no zero mode other than the graviton localized on the wall. To obtain a concrete estimate of the mass spectrum, we need to use approximations. We use small width approximation where the width $\Lambda^{-1}$ of the wall is small compared to the radius $R$ of compactified extra dimension. We find that the non-BPS solution has no tachyonic fluctuations in spite of the dynamical role played by the radius of the compactified dimension. Tensor as well as scalar fluctuations have massive modes, without any tachyons. This result shows that our non-BPS solution is stable without introducing an additional stabilizing mechanism such as the Goldberger-Wise mechanism [@GoWi].
The lightest massive scalar mode is usually called radion. We can evaluate the mass of the radion on our non-BPS background at least for $R \gg \Lambda^{-1}$, where $R$ is the radius of the compactified dimension and $\Lambda^{-1}$ is the width of the wall. We find that the mass squared of the radion is given by $$m^2_0 \propto \Lambda^2
e^{-\pi R \Lambda}
\label{eq:radion-mass1}$$ It is interesting to note that the mass scale is given by the inverse wall width $\Lambda$, and that it becomes exponentially light as a function of the distance $\pi R$ between the two walls. This behavior is precisely the same as the previous model in the global SUSY case [@MSSS2].
Modes of fermions including gravitino are also analyzed. We find that the Nambu-Goldstone modes can be reproduced in the limit of vanishing gravitational coupling both for bosonic and fermionic modes.
Our BPS solution has a smooth limit of thin walls where it reproduces the Randall-Sundrum model [@EMSS]. In the original Randall-Sundrum model, the fine-tuning was necessary between the boundary and the bulk cosmological constants. However, the necessary relation between bulk and boundary cosmological constants is now an automatic consequence of the equation of motion of scalar fields and Einstein equation in our model. We no longer need to impose a fine-tuning on input parameters of the model.
Sec.2 summarizes our model and solutions briefly. Sec.3 separates various bosonic modes with respect to the surviving Lorentz symmetry (tensor and scalar modes) and addresses the question of stability of the BPS solution. Sec.4 discusses the stability of non-BPS solution and evaluates the mass of the radion. Sec.5 deals with the fermionic modes. The gauge fixing to the Newton gauge is justified in Appendix A, and some illustrative cases of potential in the conformal coordinate are worked out in Appendix B.
Brief review of BPS domain wall in SUGRA
========================================
Lagrangian and BPS equations
-----------------------------
We consider a chiral multiplet containing scalar $\phi$ and fermion $\chi$ with the minimal kinetic term and the superpotential $P$, and the gravity multiplet containing vielbein $e_m{^{\underline{a}}}$ and gravitino $\psi_m{^{\alpha}}$. The local Lorentz vector indices are denoted by letters with the underline as $\underline{a}$, and the vector indices transforming under general coordinate transformations are denoted by Latin letters as $m, n=0, \dots, 3$. The left(right)-handed spinor indices[^4] are denoted by undotted (dotted) indices as ${\alpha} ({\dot \alpha})$. Then the $\mathcal{N}=1$ supergravity Lagrangian is given in four-dimensional spacetime as [@WessBagger] $$\begin{aligned}
e^{-1}\mathcal{L} &=&
- \frac{1}{2\kappa^2}R
+ \varepsilon^{klmn}\bar\psi_k\bar\sigma_l
\tilde{\mathcal{D}}_m\psi_n
\nonumber\\
&&- g^{mn}\partial_m\phi^*\partial_n\phi
- {\rm e}^{\kappa^2\phi^*\phi}
\left(|D_\phi P|^2 - 3\kappa^2|P|^2\right)
-i \bar\chi\bar\sigma^m\mathcal{D}_m\chi
\nonumber\\
&& - \frac{\sqrt{2}}{2}\kappa
\left(\partial_n\phi^*\chi\sigma^m\bar\sigma^n\psi_m
+ \partial_n\phi\bar\chi\bar\sigma^m\sigma^n
\bar\psi_m\right)
\nonumber\\
&&+ \frac{\kappa^2}{4}
\left(i\varepsilon^{klmn}\psi_k\sigma_l\bar\psi_m
+ \psi_m\sigma^n\bar\psi^m\right)\chi\sigma_n\bar\chi
- \frac{\kappa^2}{8}\chi\chi\bar\chi\bar\chi
\nonumber\\
&&- {\rm e}^{\frac{\kappa^2}{2}\phi^*\phi}\bigg[
\kappa^2\left(P^*\psi_m\sigma^{mn}\psi_n
+ P\bar\psi_m\bar\sigma^{mn}\bar\psi_n\right)\nonumber\\
&&+\frac{i\kappa}{\sqrt{2}}
\left(D_\phi P\chi\sigma^m\bar\psi_m
+ D_{\phi^*}P^*\bar\chi\bar\sigma^m\psi_m\right)
\nonumber\\
&& + \frac{1}{2}
\left(\mathcal{D}_\phi D_\phi P \chi\chi
+ \mathcal{D}_{\phi^*}D_{\phi^*}P^*\bar\chi\bar\chi\right)
\bigg],
\label{SUGRA_Lag}\end{aligned}$$ where the gravitational coupling $\kappa$ is the inverse of the four-dimensional Planck mass $M_{\rm Pl}$, $g_{mn}$ is the metric of the spacetime and $e$ is the determinant of the vierbein $e_m{^{\underline{a}}}$. The generalized supergravity covariant derivatives are defined as follows : $$\begin{aligned}
\label{covder}
\begin{array}{rll}
\mathcal{D}_m\chi &= &\partial_m\chi + \chi\omega_m
- \dfrac{i\kappa^2}{2}{\rm Im}\!
\left[\phi^*\partial_m\phi\right]\chi,\\
\tilde{\mathcal{D}}_m\psi_n
&= &\partial_m\psi_n + \psi_n\omega_m
+ \dfrac{i\kappa^2}{2}{\rm Im}\!
\left[\phi^*\partial_m\phi\right]\psi_n,\\
D_\phi P &= &\partial_\phi P + \kappa^2\phi^*P,\\
\mathcal{D}_\phi D_\phi P
&= &\partial_\phi^2P + 2\kappa^2\phi^*D_\phi P
- \kappa^4\phi^{*2}P,
\end{array}\end{aligned}$$ where $\omega_m$ is the spin connection and we use the notation ${\rm Im}\!\left[X\right]
\equiv \dfrac{X-X^*}{2i}$ in what follows. The scalar potential in the supergravity Lagrangian (\[SUGRA\_Lag\]) is given by $$\begin{aligned}
V(\phi,\phi^*) = {\rm e}^{\kappa^2\phi^*\phi}
\left(|D_\phi P|^2 - 3\kappa^2|P|^2\right).
\label{eq:scalar-potential}\end{aligned}$$
The above Lagrangian (\[SUGRA\_Lag\]) is invariant under the supergravity transformation : $$\begin{aligned}
\begin{array}{rll}
\delta_\zeta e_m{^{\underline{a}}}
&= &i\kappa\left(\zeta\sigma^{\underline{a}}\bar\psi_m
+ \bar\zeta\bar\sigma^{\underline{a}}\psi_m\right),\\
\delta_\zeta \psi_m
&= &2\kappa^{-1}\mathcal{D}_m\zeta
+ i\kappa{\rm e}^{\frac{\kappa^2}{2}\phi^*\phi}
P\sigma_m\bar\zeta
- \dfrac{i\kappa}{2}\sigma_{mn}\zeta\chi\sigma^n\bar\chi
- \dfrac{i\kappa^2}{2}{\rm Im}\!
\left[\phi^*\delta_\zeta\phi\right]\psi_m,\\
\delta_\zeta \phi &= &\sqrt{2}\ \zeta\chi,\\
\delta_\zeta \chi &= &i\sqrt{2}\ \sigma^m\bar\zeta
\hat{\mathcal{D}}_m\phi
- \sqrt{2}\ {\rm e}^{\frac{\kappa^2}{2}\phi^*\phi}
D_{\phi^*}P^*\zeta
+ \dfrac{i\kappa^2}{2}{\rm Im}\!
\left[\phi^*\delta_\zeta\phi\right]\chi,
\end{array}
\label{eq:SUGRAtransf}\end{aligned}$$ where $\zeta$ is a local SUSY transformation parameter and the covariant derivatives are given by $$\begin{aligned}
\begin{array}{rll}
\hat{\mathcal{D}}_m\phi
&= &\partial_m\phi - \dfrac{\sqrt{2}}{2}
\kappa\bar\psi_m\bar\chi,\\
\mathcal{D}_m\zeta &= &\partial_m\zeta + \zeta\omega_m
+ \dfrac{i\kappa^2}{2}{\rm Im}\!
\left[\phi^*\partial_m\phi\right]\zeta.
\end{array}\end{aligned}$$
Next we turn to derive the equations of motion for solutions which depend on only one “extra” coordinate $x^2=y$ under the warped metric Ansatz $$\begin{aligned}
ds^2 = g_{mn}dx^mdx^n
= {\rm e}^{2A(y)}\eta_{\mu\nu}dx^\mu dx^\nu + dy^2
\quad (\mu,\nu = 0,1,3),\label{warped_metric}\end{aligned}$$ where Greek indices $\mu=0, 1, 3$ denote three-dimensional vector transforming under general coordinate transformations, and $\eta_{\mu\nu} = {\rm diag}(-,+,+)$ denotes three dimensional flat spacetime metric. All the geometrical quantities can be written in terms of the function $A(y)$ in the warp factor and its derivatives with respect to the extra coordinate $y$. For later convenience, we write formulas in general $D$ space-time dimensions in the following :
1. vierbein $$\begin{aligned}
e_m{^{\underline{a}}}
= {\rm diag}\left({\rm e}^A,\ {\rm e}^A,\ 1,\
{\rm e}^A\right),\quad
e_{\underline{a}}{^m}
= {\rm diag}\left({\rm e}^{-A},\
{\rm e}^{-A},\ 1,\ {\rm e}^{-A}\right),\end{aligned}$$
2. spin connection $$\begin{aligned}
(\chi \omega_m)_\alpha = {1 \over 2}
\omega_{m\underline{ab}}
\left(\sigma^{\underline{ab}}\right)_\alpha{}^\beta
\chi_\beta,
\qquad
\omega_{m\underline{ab}} =
\dot{A}\left(\delta_{\underline{a}}{^2}e_{\underline{b}m}
- \delta_{\underline{b}}{^2}e_{\underline{a}m}\right),
\label{eq:spin-connec}\end{aligned}$$
3. Ricci tensor $$\begin{aligned}
R_{mn} = {\rm e}^{2A}\left(
\ddot{A} + (D-1)
\dot{A}^2\right)
\eta_{\mu\nu}\delta_m{^\mu}\delta_n{^\nu}
+ (D-1)
\left(
\ddot{A} +
\dot{A}^2\right)\delta_m{^2}\delta_n{^2},\end{aligned}$$
where a dot denotes a derivative with respect to $y$, $\dot A\equiv dA/dy$, and we turn off all the fermionic fields as a tree level solution. The energy momentum tensor is given in terms of the scalar potential $V(\phi, \phi^*)$ in (\[eq:scalar-potential\]) $$\begin{aligned}
T_{mn} = \partial_m\phi^*\partial_n\phi
+ \partial_m\phi\partial_n\phi^*
- g_{mn}\left(g^{kl}\partial_k\phi^*\partial_l\phi
+ V(\phi,\phi^*)\right).\end{aligned}$$
Plugging these into the Einstein equation[^5] $R_{mn} = - \kappa^2\tilde{T}_{mn}$, we obtain $$\begin{aligned}
\ddot{A} = -{2\over D-2}\kappa^2
\dot\phi^*\dot\phi,\quad
\dot{A}^2 = \frac{2\kappa^2}{(D-1)(D-2)}\left(
\dot\phi^*\dot\phi-V(\phi,\phi^*)\right).
\label{Einstein_eq}\end{aligned}$$ The field equation for the scalar $\phi$ in the chiral multiplet takes the form : $$\begin{aligned}
\ddot\phi
+ (D-1) \dot{A}\dot\phi
= \frac{\partial V}{\partial\phi^*}.
\label{field_eq}\end{aligned}$$ Notice that only two out of the three equations in Eqs.(\[Einstein\_eq\]) and (\[field\_eq\]) are independent (assuming only one real component, say the real part of the scalar field $\phi$ is nontrivial in the solution). Any one of three equations are automatically satisfied if others are satisfied.
It is well known that special type of solutions for these nonlinear second order differential equations are obtained as solutions of a set of the first order differential equations, the so-called BPS equations which guarantees the partial conservation of SUSY. Similarly to the global SUSY case, the BPS equations can be derived from the half SUSY condition where we parametrize the conserved SUSY parameter as $$\begin{aligned}
\zeta(y) = {\rm e}^{i\theta(y)}
\sigma^{\underline{2}}\bar\zeta(y).
\label{eq:half-susy}\end{aligned}$$ That is, we demand that the bosonic configuration should satisfy $\delta_\zeta\chi =
\delta_\zeta\psi_m=0$ for the parameter $\zeta(y)$ in Eq.(\[eq:half-susy\]). The BPS equations for the metric are derived from the condition for the gravitino. From $m=\mu=0, 1, 3$ components the first order equation for the warp factor $A$ is derived : $$\begin{aligned}
0 = \delta_\zeta\psi_\mu
= \kappa^{-1}{\rm e}^A
\left[{\rm e}^{i\theta}\dot{A}
+ i\kappa^2{\rm e}^{\frac{\kappa^2}{2}\phi^*\phi}
P\right]\sigma_{\underline{\mu}}\bar\zeta ,\end{aligned}$$ $$\begin{aligned}
\dot{A}
= - i\kappa^2{\rm e}^{-i\theta}
{\rm e}^{\frac{\kappa^2}{2}\phi^*\phi}P.\label{BPS_A}\end{aligned}$$ From $m=2$ component we find the first order equation for the Killing spinor $\zeta$ : $$\begin{aligned}
0 = \delta_\zeta\psi_2 = 2\kappa^{-1}
\left[\dot\zeta
+ \frac{i\kappa^2}{2}{\rm Im}\!
\left[\phi^*\dot\phi
\right]\zeta
+ \frac{i\kappa^2}{2}{\rm e}^{-i\theta}
{\rm e}^{\frac{\kappa^2}{2}\phi^*\phi}P\zeta\right].\end{aligned}$$ Rewriting the half SUSY condition as $\zeta_{\underline{\alpha}} ={\rm
e}^{\frac{i}{2}\left(\theta + \frac{\pi}{2}\right)}
|\zeta_{\underline{\alpha}}|$ and substituting it into the above equation, we find [@CGR] $$\begin{aligned}
\dot{|\zeta_{\underline{\alpha}}|}
= \frac{\dot{A}}{2}
|\zeta_{\underline{\alpha}}|,\quad
\dot\theta
= - \kappa^2{\rm Im}\!\left[\phi^*\dot\phi
\right].\label{BPS_kill}\end{aligned}$$ On the other hand, the first order equation for the matter field $\phi$ is derived from the half SUSY condition for the matter fermion $\chi$ : $$\begin{aligned}
\dot\phi
= -i{\rm e}^{i\theta}{\rm e}^{\frac{\kappa^2}{2}
\phi^*\phi}D_{\phi^*}P^*.\label{BPS_phi}\end{aligned}$$ Eq.(\[BPS\_A\]), (\[BPS\_kill\]) and (\[BPS\_phi\]) are collectively called BPS equations. One can easily show that solutions of the BPS equations satisfy the equations of motion (\[Einstein\_eq\]) and (\[field\_eq\]). Notice that the Eq.(\[BPS\_phi\]) and the second equation of Eq.(\[BPS\_kill\]) do not contain the metric, so we can solve this as if the scalar field decouples from gravity. Once the configuration of the scalar field $\phi$ and the phase $\theta$ are determined, the warp factor $A$ is obtained from Eq.(\[BPS\_A\]). Finally, the Killing spinor $\zeta$ is also determined from the first equation of Eq.(\[BPS\_kill\]).
Exact BPS solution
------------------
Recently, we found the exact BPS solutions for the periodic model in SUGRA [@EMSS], by allowing the gravitational correction for the superpotential as follows $$\begin{aligned}
P(\phi)
= {\rm e}^{-\frac{\kappa^2}{2}\phi^2}
\times \frac{\Lambda^3}{g^2}
\sin\frac{g}{\Lambda}\phi,\label{P_mod}\end{aligned}$$ where $\Lambda$ is a coupling with unit mass dimension and $g$ is a dimensionless coupling. We introduced this modification for the superpotential in SUGRA to maintain the periodicity of the model with the aid of the Kähler transformation. This modification for the superpotential gives SUSY vacua which do not depend on the gravitational coupling $\kappa$. This was crucial for us to obtain the exact BPS solutions in SUGRA. The superpotential (\[P\_mod\]) yields the following scalar potential : $$\begin{aligned}
V = \frac{\Lambda^4}{g^2}
{\rm e}^{2\kappa^2({\rm Im}\left[\phi\right])^2}
\left[
\left|\cos\frac{g}{\Lambda}\phi
- \frac{2i\kappa^2\Lambda}{g}{\rm Im}\!\left[\phi\right]
\ \sin\frac{g}{\Lambda}\phi\right|^2
- \frac{3\kappa^2\Lambda^2}{g^2}
\left|\sin\frac{g}{\Lambda}\phi\right|^2
\right].\end{aligned}$$ The SUSY vacua are determined from the condition $D_\phi P = 0$. For the above modified superpotential we find that the SUSY vacua are periodically distributed at $\phi =
\dfrac{\Lambda}{g}\left(\dfrac{\pi}{2}
+ n\pi\right),\ (n\in\mathbb{Z})$ on the real axis in the complex $\phi$ plane.
In order to determine the scalar field configuration, we need to solve the second equation of Eq.(\[BPS\_kill\]) for $\theta$ together with the equation for scalar field : $$\begin{aligned}
\dot\phi
= - i{\rm e}^{i\theta}{\rm e}^{i\kappa^2\phi^*{\rm Im}[\phi]}
\frac{\Lambda^2}{g}
\left[\cos\frac{g}{\Lambda}\phi^*
+ \frac{2i\Lambda\kappa^2}{g}{\rm Im}[\phi]
\sin\frac{g}{\Lambda}\phi^*\right].
\label{eq:matterBPSeq}\end{aligned}$$ To solve Eqs.(\[BPS\_kill\]) and (\[eq:matterBPSeq\]), we choose $\phi_{\rm I} \equiv {\rm Im}[\phi] = 0$ and $\theta = \pm\dfrac{\pi}{2}$ at a point, say $y=y_i$ as an initial condition for the imaginary part $\phi_{\rm I}(y)$ of the scalar field and the phase $\theta(y)$. Then these equations tell that $\dot\phi_{\rm I}
= \dot\theta
= 0$ at $y=y_i$. Therefore we find $\phi_{\rm I} = 0$ and $\theta = \pm\dfrac{\pi}{2}$ at any $y$. At this stage, only the real part of $\phi$ has a nontrivial configuration in the extra dimension $y$. We shall call those scalar fields that have nontrivial configurations as a function of the coordinate of extra dimension, as “active” scalar fields. The scalar potential along $\phi_{\rm I}=0$ surface is given by the following potential $V_{\rm R}$ for the real part $\phi_{\rm R} \equiv\dfrac{\phi+\phi^*}{2}$ of the scalar field : $$\begin{aligned}
V_{\rm R}(\phi_{\rm R})
= \frac{\Lambda^4}{g^2}
\left[\cos^2\frac{g}{\Lambda}\phi_{\rm R}
- \frac{3\kappa^2\Lambda^2}{g^2}
\sin^2\frac{g}{\Lambda}\phi_{\rm R}\right].
\label{eq:real-scalar-pot}\end{aligned}$$ It has been shown that the following form of scalar potential with a real “superpotential” $\hat P(\phi_{\rm R})$ of a real scalar field $\phi_{\rm R}$ ensures the existence of a stable AdS vacuum in gravity theories in $D$ dimensions [@Boucher; @PKT] : $$\begin{aligned}
V_{\rm R} = {D-2 \over 2}\left[{D-2 \over 2}
\left(\frac{d\hat P}{d\phi_{\rm R}}\right)^2
-(D-1)\kappa^2\hat P^2\right],\label{potential_ads}\end{aligned}$$ if there is a critical point in $\hat P(\phi_R)$, even though supersymmetry is not required in this form. Let us note that our scalar potential is compatible with the above form of the scalar potential. In our case, the “superpotential” is given by $$\begin{aligned}
\hat P(\phi_{\rm R}) = \frac{\Lambda^3}{g^2}\sin
\frac{g}{\Lambda}\phi_{\rm R} .\end{aligned}$$ Since this $\hat P$ has critical points at $\phi_R=\dfrac{\Lambda}{g}
\left(\dfrac{\pi}{2}+n\pi\right)$, our scalar potential $V_{\rm R}(\phi_{\rm R})$ in Eq.(\[eq:real-scalar-pot\]) has these critical points as stable AdS vacua.
The remaining BPS equations for the active scalar field and the warp factor are of the form : $$\begin{aligned}
\dot\phi_{\rm R}
= \pm \frac{d\hat P}{d\phi_{\rm R}}
= \pm \frac{\Lambda^2}{g}
\cos\frac{g}{\Lambda}\phi_{\rm R},\quad
\dot{A}
= \mp \kappa^2\hat P
= \mp \frac{\kappa^2\Lambda^3}{g^2}
\sin\frac{g}{\Lambda}\phi_{\rm R}.
\label{BPS_eq_R}\end{aligned}$$ Let us solve these BPS equations by choosing a SUSY vacuum $\phi_{\rm R}
=\dfrac{\Lambda}{g}\left(\mp(-1)^n\dfrac{\pi}{2}+n\pi\right)$ as an initial condition at $y=-\infty$. We shall consider the solution for the BPS equations (\[BPS\_eq\_R\]) with the sign correlated to the sign of the initial condition at $y=-\infty$. The exact BPS solutions are found to be of the form : $$\begin{aligned}
\phi_{\rm R}
= \frac{\Lambda}{g}\left[(-1)^n
\left\{2\tan^{-1}{\rm e}^{\pm\Lambda(y-y_0)}
- \frac{\pi}{2}\right\} + n\pi\right],\quad
{\rm e}^{A}
= \left[\cosh\Lambda(y-y_0)
\right]^{-\frac{k}{\Lambda}},
\label{BPS_sol}\end{aligned}$$ where $k\equiv \dfrac{\kappa^2\Lambda^3}{g^2}$ is the inverse of the curvature radius of the AdS spacetime at infinity. These solutions interpolate between the two SUSY vacua, from $\phi_{\rm R} =
\dfrac{\Lambda}{g}\left(\mp(-1)^n \dfrac{\pi}{2}
+ n\pi\right)$ at $y =
-\infty$ to $\phi_{\rm R} = \dfrac{\Lambda}{g}\left(\pm(-1)^n\dfrac{\pi}{2} +
n\pi\right)$ at $y = +\infty$. We denote $y_0$ the modulus parameter of these solutions and we suppress an integration constant for $A$ which amounts to an irrelevant normalization constant of metric. Eq.(\[BPS\_kill\]) determines the Killing spinors which has two real Grassmann parameters $\epsilon_1, \epsilon_2$ corresponding to the two conserved SUSY directions on the BPS solution[^6] : $$\begin{aligned}
\zeta
= {\rm e}^{\frac{i}{2}\left(\theta
+ \frac{\pi}{2}\right)}
{\rm e}^{\frac{A}{2}}\times
\left(\begin{array}{c}
\epsilon_1 \\
\epsilon_2
\end{array}\right),
\label{eq:Killingspinor}\end{aligned}$$ $$\begin{aligned}
{\rm e}^{\frac{i}{2}\left(\theta
+ \frac{\pi}{2}\right)}
{\rm e}^{\frac{A}{2}}
=
\left\{
\begin{array}{ll}
i\left[\cosh\Lambda(y-y_0)\right]^{
-\frac{k}{2\Lambda}},
\quad&{\rm for}\ \theta = \dfrac{\pi}{2},\\
\left[\cosh\Lambda(y-y_0)\right]^{
-\frac{k}{2\Lambda}},
\quad&{\rm for}\ \theta = -\dfrac{\pi}{2}.
\end{array}
\right.\end{aligned}$$
Our model has a smooth limit of thin walls where it reproduces the Randall-Sundrum model [@EMSS]. Notice that we do not need any fine-tuning of input parameters of the model, in contrast to the original Randall-Sundrum model. The necessary fine-tuning between bulk and boundary cosmological constants is now an automatic consequence of the equation of motion of scalar fields and Einstein equation in our model.
non-BPS solution
----------------
Assuming that only single real scalar field $\phi_{\rm R}$ has nontrivial classical configuration, the equations (\[Einstein\_eq\]) and (\[field\_eq\]) reduce to $$\begin{aligned}
\ddot{A}
=-{2\kappa^2 \over D-2}\dot\phi_{\rm R}^2
,\qquad
\dot{A}^2
= \frac{2\kappa^2}{(D-1)(D-2)}
\left(\dot\phi_{\rm R}^2
-V_{\rm R}\right),
\qquad
\ddot\phi_{\rm R}
+ (D-1) \dot{A}\dot\phi_{\rm R}
= \frac{1}{2}\frac{dV_{\rm R}}{d\phi_{\rm R}}.
\label{eom_real_phi}\end{aligned}$$ It has been shown that the above set of coupled second order differential equations is equivalent to the following set of nonlinear differential equations [@DFGK; @SkTo]. Given the scalar potential $V_{\rm R}(\phi_{\rm R})$, we should find a real function $W(\phi_{\rm R})$ by solving the following first order nonlinear differential equation $${d W(\phi_{\rm R}) \over d\phi_{\rm R}}
= \pm {2 \over D-2}
\sqrt{V_{\rm R}(\phi_{\rm R})+
{(D-1)(D-2) \over 2}\kappa^2W^2(\phi_{\rm R})} .
\label{eq:nonlinear-eq}$$ Then $\phi_{\rm R}(y)$ and $A(y)$ are obtained by solving the following two first order differential equations $$\dot\phi_{\rm R}(y)=
{D-2 \over 2}\dfrac{d W(\phi_{\rm R})}{d \phi_{\rm R}},
\qquad
\dot A(y) = - \kappa^2W(\phi_{\rm R}) .
\label{phi-A}$$ If we choose the “superpotential” $\hat P$ as a real function $W$, (\[eq:nonlinear-eq\]) and (\[phi-A\]) are satisfied by the scalar potential (\[potential\_ads\]) and the BPS equations (\[BPS\_eq\_R\]). Therefore these set of first order nonlinear differential equations includes all the BPS solutions as part of the solutions. However, it is important to realize that (\[eq:nonlinear-eq\]) and (\[phi-A\]) are equivalent to the set of Einstein equation and the scalar filed equation, and hence give all the non-BPS solutions as well.
We have been able to construct non-BPS multi-wall solutions to the Einstein equation (\[Einstein\_eq\]) and the field equation (\[field\_eq\]) using the above method of nonlinear equations [@EMSS]. We have also found that BPS solutions are the only solution that do not encounter singularities at any finite $y$. To obtain any other regular solution, especially non-BPS solutions, negative cosmological constant has to be introduced at some boundary. Since we are interested in periodic array of walls where extra dimension can be identified as a torus $S^1$ with possible division by discrete groups (orbifolds), we introduced the cosmological constant and obtained a number of interesting non-BPS solutions [@EMSS].
The above nonlinear differential equation (\[eq:nonlinear-eq\]) gives a set of solution curves which fill once and only once the entire $(\phi_{\rm R}, W)$ plane except forbidden regions defined by $V_{\rm R}+(D-1)(D-2)\kappa^2W^2/2 \le 0$. Let us denote the solution curve starting from an initial condition $W_0$ at $\phi_{\rm R} = \phi_{{\rm R},0}$ as $W(\phi_{\rm R}; (\phi_{{\rm R},0}, W_0))$. A boundary cosmological constant $\lambda_i$ at $y_i$ gives a jump of derivative of the function $A(y)$ in the warp factor. Let us denote the value of the scalar field at the boundary $y_i$ as $\phi_{{\rm R},i}$. Eq.(\[phi-A\]) shows that this jump of $A(y)$ is satisfied by cutting the solution curve and jump to another solution curve at $\phi_{{\rm R},i}$ with the constraint $$\lambda_i
=2 \left(W (\phi_{{\rm R},i}+\epsilon)-W (\phi_{{\rm R},i}-\epsilon)\right) .
\label{eq:cosm_const_W}$$ Since we are interested in minimum amount of inputs at boundaries, we wish to implement only the boundary cosmological constant without any boundary potential for scalar fields $\phi_{\rm R}$, contrary to many other approaches characteristic of the Goldberger-Wise type of the stabilization mechanism [@GoWi], [@DFGK], [@SkTo]. Therefore we need to maintain the derivative $dW/d\phi_{\rm R}$ to be smoothly connected at the boundary.
Since Eq.(\[eq:nonlinear-eq\]) gives the same value of derivative $dW/d\phi_{\rm R}$ for $\pm W$, we can connect the solution curve at any value of $\phi_{{\rm R},i}$ if we switch from a solution curve going through $W, \phi_{{\rm R},i}$ to another one going through $-W, \phi_{{\rm R},i}$. Eq.(\[eq:cosm\_const\_W\]) gives the necessary cosmological constant at this boundary as $\lambda=4W(\phi_{{\rm R},i})$. There may be other possibilities to connect the solution curves, but this is the simplest possibility that covers many interesting situations.
To be definite, we shall consider walls that have simple symmetry property under the parity $Z_2$ : $\phi_{\rm R} \rightarrow -\phi_{\rm R}$. Let us start a solution curve going through $\phi_{\rm R}=0, W_0>0$. Then the solution curve goes above the forbidden region. To obtain a non-BPS solution which is odd under the $Z_2$ transformation, we place a boundary at $\phi_{\rm R}=0$ with a positive cosmological constant by an amount $\lambda_0=4W_0>0$. On the other hand, we can place a boundary at any $\phi_{\rm R}>0$ with a negative cosmological constant $\lambda_1=-4W(\phi_{{\rm R},1}, (\phi_{\rm R}=0, W_0))$. However, we can obtain a multi-wall solution that have simple transformation property under the $Z_2$ by placing another boundary at integer multiple of $\phi_{\rm R}=\Lambda \pi/(2g)$.
If we place the first boundary at the vacuum point $\phi_{\rm R}=\Lambda \pi/(2g)$, we obtain a simplest model in the sense that the energy density at the second boundary at $\phi_{\rm R}=\Lambda \pi/(2g)$ is purely made of negative cosmological constant $$\lambda =
-4W(\phi_{\rm R}=\Lambda \pi/(2g)).
\label{eq:negaive-cosm-pi/2}$$ The magnitude of this negative cosmological constant becomes the same as the total energy of the wall centered at $\phi_{\rm R}=0$ in the limit of large separation of two boundaries. Since the solution admits $S^1/(Z_2\times Z_2)$ symmetry, we call the coordinate at the second boundary $y=\pi R/2$. The behavior of this non-BPS solutions in the $W, \phi$ plane is illustrated in Fig.\[fig:half\_wind\_sol\](a). The corresponding function $A(y)$ in the warp factor is illustrated in Fig.\[fig:half\_wind\_sol\](b), where one should note that $A(y)$ is linear near the second boundary at $y=\pi R/2$, showing that only the boundary cosmological constant exists apart from the bulk cosmological constant there.
As another solution, we can place the second boundary at $\phi_{\rm R}=\Lambda \pi/g$, where the active scalar field $\phi_{\rm R}$ develops another wall configuration. In this case, the negative cosmological constant $-4W(\phi_{\rm R}=\Lambda \pi/g)$ placed at the second boundary has magnitude which becomes twice the total energy of the wall centered at $\phi_{\rm R}=0$ in the limit of large separation of two boundaries. The behavior of this non-BPS solutions in the $W, \phi$ plane is illustrated in Fig.\[fig:wind\_sol\](a). The corresponding function $A(y)$ in the warp factor is illustrated in Fig.\[fig:wind\_sol\](b), where one should note that the function $A(y)$ has additional kink behavior deviating from the linear exponent near the second boundary at $y=\pi R$, showing that there is an additional smooth positive energy density centered around the boundary besides the negative boundary cosmological constant in contrast to the previous $S^1/(Z_2\times Z_2)$ example in Fig.\[fig:half\_wind\_sol\].
Bosonic Fluctuation and the BPS Solution
========================================
A Bogomolo’nyi bound has been derived for the energy density of the BPS domain walls in $\mathcal{N}=1$ SUGRA in four-dimensional spacetime [@CGR]. They used the generalized Israel-Nester-Witten tensor, which was originally applied to a simple proof of the positive ADM mass conjecture in general relativity. However, the ADM mass may not be well-defined for domain walls, since they are extended to infinity. Therefore it is presumably still useful to check that there is really stability of the fluctuation on our wall configuration even in the case of BPS solutions. We shall present a general formalism to analyze the modes and their stability, and then apply it to the fluctuations around the BPS background configurations in this section. The equations and procedures obtained in this section can also be used to the non-BPS background solutions with appropriate additional inputs, which is dealt with in Sec.\[sc:stability=nonBPS\].
Mode equations for the bosonic sector
-------------------------------------
We start with the metric perturbation in the Newton gauge [@TanakaMontes], [@CsabaCsaki] : $$\begin{aligned}
ds^2 = {\rm e}^{2A}\left(\eta_{\mu\nu}+h^{\rm TT}_{\mu\nu}
+ 2B\eta_{\mu\nu}\right)dx^\mu dx^\nu
+ (1-2(D-3)B)dy^2,
\label{eq:newton-gauge}\end{aligned}$$ where $h^{\rm TT}_{\mu\nu}$ is transverse traceless $\eta^{\mu\nu}h^{\rm TT}_{\mu\nu}=0, \;
\partial^\mu h^{\rm TT}_{\mu\nu}=0$. Some details for the procedure of this gauge fixing are given in Appendix A. This gauge is useful since the linearized equations become very simple. The linearized Einstein equations in $D$ space-time dimensions ($D=4$ in our specific model) read : $$\begin{aligned}
{2}
&\left({\rm e}^{-2A}\square_{D-1} +
\partial_y^2
+ (D-1)\dot{A}\partial_y
\right)h^{\rm TT}_{\mu\nu}
= 0,\label{Newton_1}\\
&\left({\rm e}^{-2A}\square_{D-1} + \partial_y^2
+ (3D-5)\dot{A}\partial_y
+ 2(D-3)\left((D-1)\dot{A}^2
+
\ddot{A}\right)
\right)B
= -{2\kappa^2\over D-2}\frac{dV_{\rm R}}{d\phi_{\rm R}}
\varphi_{\rm R},
\label{Newton_2}\\
&\left(\partial_y + (D-3)\dot{A}
\right)B = - {2\kappa^2\over D-2}\kappa^2\dot\phi_{\rm R}
\varphi_{\rm R},\label{Newton_3}\end{aligned}$$ where the first line comes from the traceless part of $(\mu,\nu)$ component of the linearized Einstein equations, the second line from the trace part of $(\mu,\nu)$ and the last from $(\mu,2)$ component. The $(2,2)$ component of the linearized equation is not shown, since it can be derived from Eqs.(\[Newton\_1\])-(\[Newton\_3\]). The linearized field equations give : $$\begin{aligned}
{2}
&\left({\rm e}^{-2A}\square_{D-1} + \partial_y^2
+ (D-1)\dot{A}\partial_y
- \frac{1}{2}\frac{d^2V_{\rm R}}{d\phi_{\rm R}^2}
\right)\varphi_{\rm R}
= -2(D-2) \dot\phi_{\rm R}\partial_yB
- (D-3)\frac{dV_{\rm R}}{d\phi_{\rm R}}B,\label{active}\\
&\left({\rm e}^{-2A}\square_{D-1} + \partial_y^2
+ (D-1)\dot{A}\partial_y
- \frac{1}{2}\frac{\partial^2V}{\partial \phi_{\rm I}^2}
\bigg|_{\rm background}\right)\varphi_{\rm I}
= 0,\label{inert}\end{aligned}$$ where $\varphi_{\rm R(I)}$ denotes the real (imaginary) part of the fluctuation of the scalar field $\phi=\phi_{\rm R}+\varphi_{\rm R}+i\varphi_{\rm I}$ around the background field configuration $\phi_{\rm R}$. Notice that the solutions of the linearized Einstein equation automatically satisfy the linearized field equations for the active scalar field $\varphi_{\rm R}$. Therefore, the Eqs.(\[Newton\_1\])– (\[inert\]) constitute the full set of independent linearized equations for the fields $h^{\rm TT}_{\mu\nu},\ B,\ \varphi_{\rm R}$ and $\varphi_{\rm I}$.
Tensor perturbation : localized massless graviton {#sc:tensot-perturb}
-------------------------------------------------
First we show that the linearized equation for the transverse traceless mode (graviton) given in Eq.(\[Newton\_1\]) can be brought into a Schrödinger form. It can again be rewritten into a form of the supersymmetric quantum mechanics (SQM) which ensures the stability of the system. For that purpose we change the coordinate $y$ into the conformally flat coordinate $z$ defined as $$\begin{aligned}
dz\equiv{\rm e}^{-A(y)}dy, \qquad
ds^2={\rm e}^{2A(y)}\left(
\eta_{\mu\nu}dx^\mu dx^\nu + dz^2
\right). \end{aligned}$$ We also redefine the field as $\tilde{h}^{\rm TT}_{\mu\nu} \equiv
{\rm e}^{{D-2\over 2}A}h^{\rm TT}_{\mu\nu}$. In the following we use prime to denote a derivative in terms of $z$. Then the linearized equation (\[Newton\_1\]) becomes $$\begin{aligned}
\square_{D-1}\tilde{h}^{\rm TT}_{\mu\nu}(x,z)
= \left[-\partial_z^2 + \mathcal{V}_t(z)\right]
\tilde{h}^{\rm TT}_{\mu\nu}(x,z),
\qquad
\mathcal{V}_t(z) = \left({D-2\over 2}\right)^2A'{^2}
+ {D-2\over 2}A'',
\label{eq:schrod-tt}\end{aligned}$$ where $\mathcal{V}_t(z)$ is the potential in this “Schrödinger” type equation. For our BPS background solution (\[BPS\_sol\]) the Schrödinger potential takes the form : $$\begin{aligned}
\mathcal{V}_t(y) =
\big[\cosh\Lambda(y-y_0)\big]^{-\frac{2k}{\Lambda}}
\left[-\frac{k\Lambda}{\cosh^2\Lambda(y-y_0)}
+ 2k^2\tanh^2\Lambda(y-y_0)\right],
\label{sch_tensor}\end{aligned}$$ where $4T^3\equiv 4g^{-2}\Lambda^3$ is the tension of the wall and $k=\kappa^2T^3$. Although our model contains three parameters $\Lambda,\ g$ and $\kappa$, this potential depends on only two parameters $k$ and $\Lambda$. If we take the thin wall limit where $\Lambda\rightarrow\infty$ fixing $4T^3$, we obtain (putting $y_0=0$) $$\begin{aligned}
\frac{\Lambda}{\cosh^2\Lambda y(z)}
\rightarrow 2\delta(z),\quad
\tanh^2\Lambda y(z) \rightarrow 1,\quad
\left[\cosh\Lambda y(z)
\right]^{-\frac{k}{\Lambda}}
\rightarrow
\frac{1}{\left(k|z|+1\right)^2},\end{aligned}$$ with $kz={\rm sgn}(y){\rm e}^{k|y|}-1$. Thus the Schrödinger potential (\[sch\_tensor\]) becomes precisely the potential of the Randall-Sundrum model : $$\begin{aligned}
\mathcal{V}_t(z) \rightarrow
\frac{2k^2}{\left(k|z| + 1\right)^2}
- 2k\delta(z).\end{aligned}$$
We find that the part of action quadratic in $\tilde{h}^{\rm TT}_{\mu\nu}$ has no $z$ dependent weight $$\begin{aligned}
S \sim \int dzd^{D-1}x\ \eta^{\mu\rho}
\eta^{\nu\lambda}\tilde{h}^{\rm TT}_{\mu\nu}
\left(\square_{D-1}+\partial_z^2
-\frac{1}{2}\mathcal{V}_t\right)
\tilde{h}^{\rm TT}_{\rho\lambda},
\label{eq:quadratic-h-action}\end{aligned}$$ in conformity with the absence of the linear term [@CEHS] in $\partial_z$ in the Shrödinger type equation (\[eq:schrod-tt\]). We stress that this is written in terms of the conformal coordinate $z$ and the redefined field $\tilde{h}^{\rm TT}_{\mu\nu}$.
Defining mode equations by eigenvalue equations $\mathcal{H}_t
\psi_n(z)=
\left[-\partial_z^2 + \mathcal{V}_t(z)\right]
\psi_n(z)=m_n^2\psi_n(z)$ with mass squared eigenvalues $m_n^2$, and assuming mode functions $\psi_{n}(z)$ to form a complete set, the transverse traceless fields can be expanded into a set of effective fields $\hat{h}^{{\rm TT} (n)}_{\mu\nu}(x)$ $$\begin{aligned}
\tilde{h}^{\rm TT}_{\mu\nu}(x,z)
= \sum_n \hat{h}^{{\rm TT} (n)}_{\mu\nu}(x)\psi_{n}(z). \end{aligned}$$ Then the above quadratic action (\[eq:quadratic-h-action\]) becomes $$\begin{aligned}
S \sim \sum_{n, k}\int dz\ \psi_n(z) \psi_k(z) \ \cdot
\int d^{D-1}x\ \eta^{\mu\rho}
\eta^{\nu\lambda}
\hat{h}^{\rm TT(n)}_{\mu\nu}
\left[\square_{D-1}-m_k^2\right]
\hat{h}^{\rm TT(k)}_{\rho\lambda} . \end{aligned}$$ Therefore the inner product for the mode function $\psi(z)$ should be defined as as $$\begin{aligned}
\langle\psi_1|\psi_2\rangle = \int dz\ \psi_1(z)\psi_2(z),
\label{eq:inner-prod}\end{aligned}$$ for which the usual intuition of quantum mechanics works.
The Hamiltonian $\mathcal{H}_t$ can now be expressed in a SQM form as follows $$\begin{aligned}
\mathcal{H}_t = Q_t^\dagger Q_t,
\qquad
Q_t\equiv - \partial_z + {D-2 \over 2}A',
\qquad
Q_t^\dagger
\equiv \partial_z + {D-2 \over 2}A',
\label{eq:SQM-TT}\end{aligned}$$ where the “supercharge” $Q_t$ and $Q_t^\dagger$ are adjoint of each other at least for BPS background where no boundary condition has to be imposed. Therefore the Hamiltonian $\mathcal{H}_t$ is a nonnegative definite Hermitian operator[^7], and its eigenvalues are nonnegative definite. Therefore we can conclude that the tensor perturbation has no tachyonic modes which destabilize the background field configurations at least for BPS solutions.
There are two possible zero modes in the tensor perturbation. One is the state which is annihilated by $Q_t|\tilde{h}^{\rm TT(+)}_{\mu\nu}\rangle=0$, and another is the state defined as $Q_t^\dagger\left(Q_t|\tilde{h}^{\rm TT(-)}_{\mu\nu}
\rangle\right)=0$ where $Q_t|\tilde{h}^{\rm TT(-)}_{\mu\nu}\rangle\neq0$. Then zero modes are of the form : $$\begin{aligned}
{2}
&\tilde{h}^{\rm TT(0)}_{\mu\nu}(x,z)
= \hat{h}^{\rm TT(+)}_{\mu\nu} (x)\
{\rm e}^{{D-2 \over 2}A(z)}
+ \hat{h}^{\rm TT(-)}_{\mu\nu} (x)\
{\rm e}^{{D-2 \over 2}A(z)}\int dz\ {\rm
e}^{-(D-2)A(z)},
\label{eq:tensor-zero-mode}\end{aligned}$$ where $A(z) = A(y(z))$. Notice that we must verify the normalizability of the wave-function to obtain a physical massless effective field in the case of noncompact space such as our BPS background. In the case of non-BPS background, the boundary condition has to be verified, which we shall consider in Sec.\[sc:stability=nonBPS\]. The first mode $\hat{h}^{\rm TT(+)}_{\mu\nu}$ in Eq.(\[eq:tensor-zero-mode\]) is normalizable if $\int dz\ {\rm e}^{(D-2)A(z)} < \infty$, corresponds to the graviton which is localized at the wall with a positive energy density. namely, if ${\rm e}^{(D-2)A}$ falls off faster than $|z|^{-1}$ [@CEHS]. For our BPS solution (\[BPS\_sol\]) the asymptotic behavior of the warp factor ${\rm e}^A$ is of order $|z|^{-1}$. Therefore we obtain a normalizable massless transverse traceless mode $\hat{h}^{\rm TT(+)}_{\mu\nu}$ which gives the physical graviton localized on the wall.
On the other hand, the second term $\hat{h}^{\rm TT(-)}_{\mu\nu}$ in Eq.(\[eq:tensor-zero-mode\]) is not normalizable and is unphysical since $\left({\rm e}^{D-2\over 2}A\int dz\
{\rm e}^{-(D-2)A}\right)^2\sim |z|^4$ at $|z| \rightarrow \infty$ for our BPS solution. If there exists a regulator brane with a negative tension at some $y$, this mode can become normalizable and localizes at the negative tension brane in contrast to the graviton. If there are no bulk scalar fields (contrary to our model) as in the original Randall-Sundrum model of single wall, this zero mode corresponds to the physical massless field which was called radion in Ref.[@Charmousis:1999rg].
Our specific four-dimensional model of non-BPS wall gives a three-dimensional effective theory on the wall. Transverse traceless mode of graviton in three dimensions has no dynamical degree of freedom except possible topological modes. However, our formalism and analysis can be applied at each step to general $D$-dimensional theories, once we obtain the relevant non-BPS solutions in such theories. In that respect, we believe that our findings should still be useful.
![[]{data-label="V_tensor"}](V_tensor.eps){width="6.5cm"}
The Schödinger potential can always be expressed in terms of $y$, but is difficult in terms of $z$ explicitly[^8], since it is generally difficult to solve $dz = {\rm e}^{-A}dy$ explicitly. If we express the potential in terms of $y$, we obtain a volcano type potential as shown in Fig.\[V\_tensor\]. The width of the well is $\sim 2\Lambda^{-1}$ and the depth is $\sim k\Lambda$.
Next we turn to analysis of the massive Kaluza-Klein (KK) mode. There are no modes with negative mass squared in the tensor perturbation, as we have already shown. Since the Schrödinger potential (\[sch\_tensor\]) vanishes asymptotically ($z=\pm\infty$), all the massive KK modes are continuum scattering states with eigenvalues $m^2>0$. In order to examine the mode functions of the massive KK modes, we look into the region far from the wall, namely $\Lambda|y|\gg1$. Since ${\rm e}^{A} \simeq {\rm e}^{-k|y|}$, $kz\simeq {\rm sgn}(y){\rm e}^{k|y|}-1$, we find that the Schrödinger potential becomes $$\begin{aligned}
\mathcal{V}_t(z) \simeq
\frac{2k^2}{\left(k|z| + 1\right)^2}
\qquad \left(\Lambda|y|\gg1\right).
\label{large_y_tensor}\end{aligned}$$ This happens to be the same potential as that in the Randall-Sundrum single wall model [@RS2], in spite of different spacetime dimensions. The wave functions of the continuum massive modes for this potential are known to be expressed as linear combinations of Bessel functions at the region far from the wall [@RS2].
The active scalar perturbation {#sc:active-scalar}
-------------------------------
Next we study the perturbation of the active scalar field $\varphi_{\rm R}$. Notice that the fluctuation $\varphi_{\rm R}$ around the active scalar field background $\phi_{\rm R}$ can be reduced to the trace part $B$ of the metric perturbation through Eq.(\[Newton\_3\]). Therefore we mainly concentrate on the trace (scalar) part of the metric perturbation $B$ in what follows. The linearized equation which contains only $B$ can be derived by combining Eq.(\[Newton\_2\]) and (\[Newton\_3\]) and using the background field equation : $$\begin{aligned}
\left[{\rm e}^{-2A}\square_{D-1} + \partial_y^2
+ \left((D-3)\dot{A}-2\frac{\ddot\phi_{\rm R}}
{\dot\phi_{\rm R}}
\right)\partial_y
+ 2(D-3)\left(\ddot{A}-\dot{A}
\frac{\ddot\phi_{\rm R}}{\dot\phi_{\rm R}}\right)
\right]B=0.\label{B_eq}\end{aligned}$$ In order to transform this into the Schrödinger form, we change the coordinate from $y$ to $z$ and redefine the field as $\tilde{B}\equiv{\rm e}^{{D-2 \over 2}A}
\phi_{\rm R}'{^{-1}}B$. Substituting this into Eq.(\[B\_eq\]), we find the Schödinger type equation for the scalar perturbation; $$\begin{aligned}
\mathcal{H}_e \tilde{B} \equiv \left[-\partial_z^2
+ \mathcal{V}_e(z)\right]\tilde{B}
= \square_{D-1}\tilde{B},
\label{Ham_scalar}\end{aligned}$$ where the Schrödinger potential $\mathcal{V}_e(z)$ is defined by $$\begin{aligned}
\mathcal{V}_e(z) \equiv
- \frac{\phi_{\rm R}'''}{\phi_{\rm R}'}
+ 2 \left(\frac{\phi_{\rm R}''}{\phi_{\rm R}'}\right)^2
+(D-4){A''\phi_R'' \over \phi_R'}
- {3D-10 \over 2}A'' + \left({D-2 \over 2}\right)^2A'{^2}.
\label{eq:scalar-fluc-pot}\end{aligned}$$ Similarly to the tensor perturbation, the inner-product for the scalar perturbation $B$ should be defined in terms of the conformal coordinate $z$ and the redefined field $\tilde{B}$.
Plugging our solution (\[BPS\_sol\]) into this, we find $$\begin{aligned}
\mathcal{V}_e =
\left[\cosh\Lambda (y-y_0)\right]^{-\frac{2k}{\Lambda}}
\left[
\Lambda^2 + k\Lambda
\left(1+\frac{1}{\cosh^2\Lambda (y-y_0)}\right)
\right].
\label{Sch_scalar}\end{aligned}$$
![[]{data-label="V_scalar"}](V_scalar.eps){width="6.5cm"}
We stress that $\mathcal{V}_e$ can be expressed in terms of $y$, but not in terms of $z$, since it is generally difficult to solve $dz={\rm e}^{-A}dy$. This potential $\mathcal{V}_e$ has the following properties : i) it is positive definite, ii) it vanishes asymptotically at infinity, and iii) the height of $\mathcal{V}_e$ is of order $\Lambda^2$ as shown in Fig.\[V\_scalar\]. From i) , it follows that there are no tachyonic modes since the wave function of such modes will necessarily diverge either at $y=\infty$ or $y=-\infty$. Therefore we can conclude that the background configuration (\[BPS\_sol\]) is stable under the active scalar perturbation. From ii), it follows that the spectrum of the massive modes is continuous starting from zero. From iii), the potential diverges at any finite point $y$ in the thin wall limit $(\Lambda\rightarrow\infty)$.
Though we can not find the exact solutions for the massive KK modes, zero modes can be found by rewriting the Hamiltonian (\[Ham\_scalar\]) into SQM form as follows : $$\begin{aligned}
\mathcal{H}_e = Q_e^\dagger Q_e,
\quad\quad
Q_e \equiv
-\partial_z +
\left[\log\left({\rm e}^{-{D-2 \over 2}A}
\dfrac{A'}{\phi_{\rm R}'}\right)\right]',
\quad
Q_e^\dagger \equiv
\partial_z +
\left[\log\left({\rm e}^{-{D-2 \over 2}A}
\dfrac{A'}{\phi_{\rm R}'}\right)\right]'.
\label{eq:SQM-scalar}\end{aligned}$$ To show this, we use the identity $\phi_{\rm R}'\left(A'''-2A'A''\right) =
2\phi_{\rm R}''\left(A''-A'{^2}\right)$. Similarly to the tensor perturbation there are two zero modes of $\mathcal{H}_e$ : $$\begin{aligned}
{2}
&\tilde{B}^{(0)}(x,z)
= \hat{B}^{(+)}(x)\ \frac{A'}{\phi_{\rm R}'}
{\rm e}^{-{D-2 \over 2}A}
+ \hat{B}^{(-)}(x)\ \frac{A'}{\phi_{\rm R}'}
{\rm e}^{-{D-2 \over 2}A}
\int dz\ \frac{\phi_{\rm R}'{^2}}{A'{^2}}
{\rm e}^{(D-2)A} .
\label{zero_scalar}\end{aligned}$$ Both zero modes are unphysical by the following reasons. The first term is unphysical, in the sense that this is eliminated by a gauge transformation preserving the Newton gauge (\[eq:newton-gauge\]) : $$\begin{aligned}
\xi_2 = \hat{B}^{(+)}(x){{\rm e}^{-(D-3)A} \over D-3},
\quad
\xi_\mu
= - \hat{B}^{(+)}_{,\mu}(x)
{{\rm e}^{2A} \over D-3}\int dy\ {\rm e}^{-(D-1)A},
\label{gauge_transf_newton}\end{aligned}$$ where $\xi_m$ is an infinitesimal coordinate transformation parameters. The transformation law is given in Appendix A. The second term is unphysical since it diverges at infinity and is not normalizable as illustrated in Fig.\[V\_scalar\_z\].
Next we turn to analysis for the massive KK modes. As we have mentioned above, massive modes are continuous from zero. Similarly to the tensor perturbation, properties of mode functions can be examined by analyzing the behavior of the potential in the region far from the wall. In the region where $|y|\Lambda\gg1$, the Schrödinger potential (\[Sch\_scalar\]) becomes : $$\begin{aligned}
\mathcal{V}_e(z) \simeq
\frac{\Lambda^2 + k\Lambda}{\left(k|z| +1\right)^2}.
\label{large_y_scalar}\end{aligned}$$ This potential is very similar to the Schrödinger potential (\[large\_y\_tensor\]) for the tensor perturbation. Therefore all the massive modes are given by a linear combination of Bessel functions asymptotically at $|z|\rightarrow\infty$. Although these two Schrödinger potentials (\[large\_y\_tensor\]) and (\[large\_y\_scalar\]) have the same $z$ dependence asymptotically $|y|\Lambda\gg1$, their behaviors in the thin wall limit are very different. The potential (\[large\_y\_tensor\]) depends only on $k$ (fixed in the thin-wall limit), but not on $\Lambda$. On the other hand, the potential (\[large\_y\_scalar\]) is proportional to polynomials in $\Lambda$. Therefore, the latter diverges in thin wall limit whereas the former is finite. This can be understood as follows. The perturbation of the trace part of the metric $B$ is related to the active scalar field perturbation $\varphi_{\rm R}$ through Eq.(\[Newton\_3\]). Since all the massive KK modes associated with the active scalar field become infinitely heavy in the thin wall limit, the massive KK modes for the perturbation of the trace part of the metric freeze simultaneously. In this limit only the tensor perturbations remain which correspond to the known modes of the RS model[^9] .
The zero modes $\hat{B}^{(+)}(x)\dfrac{A'}{\phi_{\rm R}'}
{\rm e}^{-{D-2 \over 2}A}$ of the fluctuation of the trace part of the metric $B$ in Eq.(\[zero\_scalar\]) can be translated into the perturbation of the active scalar field $\varphi_{\rm R}$ by means of Eq.(\[Newton\_3\]) : $$\begin{aligned}
\varphi_{\rm R}^{(0)}(x,y)
= \hat\varphi_{\rm R}^{(+)}(x)\ \dot\phi_{\rm R}
{\rm e}^{-(D-3)A}
\rightarrow
\hat\varphi_{\rm R}^{(+)}(x)\ \dot\phi_{\rm R},\quad
\left(\kappa\rightarrow0\right).\end{aligned}$$ where $\hat\varphi_{\rm R}^{(+)}(x)\equiv
(D-2)\hat{B}_{\rm R}^{(+)}(x)/2$. In weak gravity limit $(\kappa\rightarrow0)$, ${\rm e}^{A}$ reduces to a constant. Then we find that this zero mode is localized on the wall and that it corresponds to the Nambu-Goldstone boson corresponding to the spontaneously broken translational invariance.
Analysis for the perturbation about $\phi_{\rm I}$
---------------------------------------------------
In our tree level solution the imaginary part of the scalar field $\phi$ vanishes identically and does not contribute to the energy momentum tensor. Therefore it does not affect the spacetime geometry. We shall call scalar fields with no nontrivial field configuration as inert field. In the linear order of perturbations we found that the fluctuation $\varphi_{\rm I}$ of this inert field decouples from any other fluctuations, as shown in Eq.(\[inert\]).
In order to find the spectrum of $\varphi_{\rm I}$, we first bring Eq.(\[inert\]) into a Schrödinger form by changing the coordinate from $y$ to $z$ and redefining the field $\tilde\varphi_{\rm I}\equiv
{\rm e}^{{D-2 \over 2}A}\varphi_{\rm I}$. Then we obtain $$\begin{aligned}
\mathcal{H}_{\rm I}\tilde\varphi_I
\equiv \left[-\partial_z^2
+ \mathcal{V}_{\rm I}(z)\right]\tilde\varphi_{\rm I}
= \square_{D-1}\tilde\varphi_{\rm I},
\quad
\mathcal{V_{\rm I}}(z)
\equiv \mathcal{V}_t(z)
+ {\rm e}^{2A}\dfrac{1}{2}
\dfrac{\partial^2V}{\partial\phi_{\rm I}^2}\bigg|.
\label{eq:inert-hamilton}\end{aligned}$$ where ${\cal V}_t(z)$ is the potential for transverse traceless part of the metric defined in Eq.(\[eq:schrod-tt\]). To obtain more concrete informations on the spectrum, we need to examine properties of each model. For our model we find $$\begin{aligned}
\frac{1}{2}\frac{\partial^2V}{\partial\phi_{\rm I}^2}\bigg|
&=& \Lambda^2
+ \frac{\kappa^2\Lambda^4}{g^2}
\left(1+2\cos^2\frac{g}{\Lambda}\phi_{\rm R}\right)
- \frac{2\kappa^4\Lambda^6}{g^4}
\sin^2\frac{g}{\Lambda}\phi_{\rm R}
\label{eq:inert-potential}\end{aligned}$$ We shall discuss generic property of this inert scalar for the non-BPS background in Sec.\[sc:stability=nonBPS\].
If we choose the BPS solution as our background, we can rewrite the potential by using the BPS equations (\[BPS\_eq\_R\]) $$\begin{aligned}
\frac{1}{2}\frac{\partial^2V}{\partial\phi_{\rm I}^2}\bigg|
&=& \Lambda^2 + 2k\Lambda - 2\ddot{A} - 2\dot{A}^2,\end{aligned}$$ Then the Schrödinger potential $\mathcal{V}_{\rm I}$ takes the form : $$\begin{aligned}
\mathcal{V}_{\rm I} = A'{^2} - A''
+ {\rm e}^{2A}\left(\Lambda^2 + 2k\Lambda\right).\end{aligned}$$ We illustrate $\mathcal{V}_I$ in terms of $y$ in Fig.\[V\_inert\]. For vanishing gravitational coupling $\kappa \rightarrow 0$, $\mathcal{V}_{\rm I}$ reduces to a constant $\Lambda^2$, which agrees with the model of global SUSY in Ref.[@EMSS]. On the other hand, the potential $\mathcal{V}_{\rm I}$ acquires regions of negative values when $\kappa$ becomes large.
![[]{data-label="V_inert"}](V_inert.eps){width="6.5cm"}
In the case of the BPS background, we can show that there are no tachyonic modes in this inert scalar sector with the aid of the SQM. Let us introduce a supercharge as $Q_{\rm I}
= -\partial_z - A'$ and $Q_{\rm I}^\dagger=\partial_z - A'$. Then the Hamiltonian $\mathcal{H}_{\rm I}$ can be rewritten as $$\begin{aligned}
\mathcal{H}_{\rm I} = Q_{\rm I}^\dagger Q_{\rm I}
+ {\rm e}^{2A}\left(\Lambda^2 + 2k\Lambda\right).\end{aligned}$$ The first term is a nonnegative definite Hermitian operator and the second term is never negative. Therefore, we can conclude that eigenvalues of $\mathcal{H}_{\rm I}$ are always nonnegative and there are no tachyonic mode.
Stability of Non-BPS multi-Walls {#sc:stability=nonBPS}
=================================
For non-BPS solutions, the positivity of the energy of the fluctuation and the associated stability is entirely nontrivial. In the limit of vanishing gravitational coupling, however, our supergravity model reduces to a global SUSY model that has been shown to be stable [@EMSS]. Since the mass gap in the global SUSY model should not disappear even if we switch on the gravitational coupling infinitesimally, the massive scalar fluctuations in the global SUSY model should remain massive at least for small gravitational coupling. On the other hand, we need to watch out a possible new tachyonic instability associated with the metric fluctuations.
As for the transverse traceless part of the metric, we have already shown that there are two possible zero mode candidates $ \hat{h}^{\rm TT(+)}_{\mu\nu} (x)\ {\rm e}^{A(z)}$ and $\hat{h}^{\rm TT(-)}_{\mu\nu} (x)\
{\rm e}^{A(z)}\int dz\ {\rm e}^{-2A(z)}$ in Eq.(\[eq:tensor-zero-mode\]). In the non-BPS solution, we no longer need to worry about the normalizability of the wave function. Instead, we need to satisfy the boundary condition imposed by the presence of the boundary cosmological constants. To impose the boundary condition, we have to use coordinate system which is more more appropriate to specify the position of the boundary. This is achieved by going to the Gaussian normal coordinates [@TanakaMontes]. We have been using the Newton gauge to study the mass spectrum in the Shrödinger type equation. We can follow the argument in Ref.[@TanakaMontes] to obtain general coordinate transformations $\xi_m$ from the Newton gauge to the Gaussian normal gauge : $$\xi_2(x,y)={1 \over 2}\int_{0}^{y} dy' h_{22}
+ \bar \xi_2^{(\pm)}(x),$$ $$\xi_\mu(x,y)=
-{1 \over 2}\int_{y^{(\pm)}}^ydy'e^{-2A}
\int_{y^{(\pm)}}^{y'}dy'' h_{22, \mu}
-\bar \xi_{2,\mu}^{(\pm)}\int_{y^{(\pm)}}^ydy'e^{-2A}
+ \bar \xi_\mu^{(\pm)}(x),$$ where $\bar \xi_2^{(\pm)}, \bar \xi_\mu^{(\pm)}$ depend on $x$ only. Boundary conditions in the Newton gauge are found to be [@TanakaMontes] $$\left[\partial_y h_{\mu\nu}^{TT}
+2e^{-2A}\bar \xi_{2,\mu,\nu}^{(\pm)}
\right]_{y^{(\pm)}-0}^{y^{(\pm)}+0}
=0
, \label{const_tensor}$$ $$\left[\left(\partial_y+\dot A\right) h_{22}
+ 2\ddot{A} \bar \xi_2^{(\pm)}
\right]_{y^{(\pm)}-0}^{y^{(\pm)}+0}
=0. \label{const_scalar}$$ We find that the former mode $ \hat{h}^{\rm TT(+)}_{\mu\nu} (x)\ {\rm e}^{A(z)}$ receives no constraint from the boundary condition, and is still a physical massless mode localized on the wall which should be regarded as the graviton in the effective theory. The latter mode $\hat{h}^{\rm TT(-)}_{\mu\nu} (x)\
{\rm e}^{A(z)}\int dz\ {\rm e}^{-2A(z)}$ is constrained by the boundary condition (\[const\_tensor\]). Since the constraint (\[const\_tensor\]) and (\[const\_scalar\]) relate $\bar \xi_2^{(\pm)}$ to $\hat h^{(-)}_{\mu\nu}$ and $\hat h_{22}(x)$, this mode should be classified as a scalar type perturbation. On the other hand, $\hat h^{(-)}_{\mu\nu}$ can be gauged away through the gauge transformation (\[gauge\_transf\_newton\]), which preserves the Newton gauge[^10]. Therefore this mode becomes unphysical in the presence of the bulk scalar field like in our model. Since $\bar \xi_2$ corresponds to the physical distance between the two branes [@Charmousis:1999rg], our system automatically incorporates the stabilization mechanism without an additional bulk scalar fields. In the thin wall limit, the wall scalar field freezes out and the fluctuation of the scalar field ceases to be related to the scalar perturbation of the metric, so that $\hat h_{\mu\nu}^{(-)}$ can no longer be gauged away. As a result of restoration of $\hat h_{\mu\nu}^{(-)}$, any distance between two walls is admitted as classical solutions and the model becomes meta-stable.
To evaluate the mass spectrum of massive modes, we use the small width approximation. Then the asymptotic behavior of the potential gives the wave functions expressed by means of the Bessel functions as in the Randall-Sundrum model. These eigenvalue spectrum is approximately equally spaced just like the plane wave solutions. These (almost) continuum modes should give the corrections to the effects of localized graviton, similarly to the Randall-Sundrum model. Therefore we obtain a massless graviton localized on the wall and a tower of massive KK modes for transverse traceless part of the metric.
Since the active scalar field $\varphi_{\rm R}$ exhibits a mass gap without any tachyon, we expect that there should be no tachyonic instability at least for small enough gravitational coupling. We have found in Eq.(\[Newton\_3\]) that the active scalar field $\varphi_{\rm R}$ can be reduced to the trace part of the metric $B$. This implies that there should be no tachyon in the transverse traceless mode as well at least for small gravitational coupling, since both degrees of freedom represent one and the same dynamical degree of freedom. In fact, we have observed that the potential ${\cal V}_{e}$ defined in Eq.(\[eq:scalar-fluc-pot\]) is everywhere positive and has no tachyon.
There are two possible zero mode candidates as given in Eq.(\[zero\_scalar\]). However, both of them are unphysical by the following reason. The first one can be eliminated by a gauge transformation. The second one has now no problem of normalization, since the extra dimension is now a finite interval, but it cannot satisfy the boundary conditions [@TanakaMontes].
To evaluate the mass of the lightest scalar particle, which is usually called radion, we use the thin-wall approximation where the wall width is assumed to be small compared to the radius of compactification $R$. To make this approximation, we separate the potential (\[eq:scalar-fluc-pot\]) for the trace part of the metric fluctuation into unperturbed and perturbation as $$\mathcal{V}_e(z) \equiv
\mathcal{V}_e^{(0)}(z) +
\mathcal{V}_e^{(1)}(z) ,$$ $$\mathcal{V}_e^{(0)}(z) =
\phi_{\rm R}' e^{{D-4 \over 2}A}\left({ e^{-{D-4 \over 2}A}
\over \phi_{\rm R}'}\right)'',
\qquad
\mathcal{V}_e^{(1)}(z) =
{D-3 \over D-2} 2 \kappa^2 \left(\phi_{\rm R}'\right)^2,$$ where we considered in $D$-dimensions instead of $4$ dimensions. The zero-th order eigenfunction for the lowest eigenvalue is found to be $$\tilde B^{(0)}(z)=
{{\cal N} \over {\rm e}^{{D-2 \over 2}A}\phi_{\rm R}'}
,$$ with the vanishing eigenvalue $(m_0^{(0)})^2=0$ and a normalization factor ${\cal N}$. The first order eigenfunction is given by $$\tilde B^{(1)}(z)=
{{\cal N} \over {\rm e}^{{D-4 \over 2}A}\phi_{\rm R}^{'}}
\int^z dz' {\rm e}^{(D-4)A} \phi_{\rm R}^{'2}
\int^{z'} dz'' {\rm e}^{-(D-4)A} \phi_{\rm R}^{'-2}
\left(\mathcal{V}_e^{(1)}(z)
-(m_0^{(1)})^2\right)
,$$ where the first correction to the mass squared eigenvalue is denoted as $(m_0^{(1)})^2$. By using Eq.(\[Newton\_3\]), the trace part of the metric can be transformed into the active scalar fluctuation $\varphi_{\rm R}$ as $$\varphi_0^{(1)}(y)=-{D-2 \over \kappa^2}
{\cal N} \dot\phi_{0}
\int_{0}^{y_{(+)}} dy'
\left({D-3 \over D-2}\kappa^2 {\rm e}^{-(D-3)A}
-{(m_0^{(1)})^2{\rm e}^{-(D-1)A} \over 2 \dot\phi_{\rm R}^{2}}
\right)
,$$ where the position of the wall is at $y=0$ and the boundary with the negative cosmological constant is $y_{(+)}$. To satisfy the correct boundary conditions, we have to require that this first order eigenfunction $\varphi_0^{(1)}$ should vanish at the boundary [@TanakaMontes]. This determines the first order mass squared as $$(m_0^{(1)})^2=2\kappa^2
\frac{D-3}{D-2}
\frac{\displaystyle \int_0^{y_{(+)}} dy\ {\rm e}^{-A}}{
\displaystyle \int_{0}^{y_{(+)}} dy\
\dfrac{{\rm e}^{-3A}}{\dot \phi_{0}^2}},$$
Taking $D=4$ as in our model, and applying to the case of the first example of non-BPS background in Eq.(\[eq:negaive-cosm-pi/2\]) with the symmetry $Z_2\times Z_2$, we should identify $y_{(+)}=\pi R/2$ and obtain $$m^2_0 \approx 8\Lambda^2
e^{-(1+\alpha^2)\pi \Lambda R}
\left(1+{3 \over 2}\alpha^2\right)
2^{\alpha^2},
\qquad
\alpha^2 \equiv
{\kappa^2 T^3 \over \Lambda },
\label{eq:radion-mass}$$ where $4T^3$ is the tension (energy density) of the wall. For the other background solution with the $Z_2\times Z_2$ symmetry, we should identify the boundary with the negative cosmological constant as $y_{(+)}=\pi R$, and obtain the same result. It is interesting to note that the mass scale is given by the inverse wall width $\Lambda$, and that it becomes exponentially light as a function of the distance between the walls, even though the radion mass receives a complicated gravitational corrections. It is appropriate to fix the wall tension and the gravitational coupling in taking the small width limit $\Lambda R \rightarrow \infty$. Then we obtain a simple mass formula in the limit [^11] $$m^2_0 \approx 8\Lambda^2
e^{-\pi \Lambda R} \rightarrow 0 .
\label{eq:radion-mass2}$$ This characteristic feature of lightest massive scalar fluctuation is precisely the same as the global SUSY case [@MSSS2]. The lightest massive mode in that case results from the fact that two walls have no communication when they are far apart, and the translation zero modes of each wall becomes massless as the separation between walls goes to infinity.
The mass spectrum of the inert scalar fluctuations $\varphi_{\rm I}$ is determined by the Schrördinger form of the eigenvalue problem (\[eq:inert-hamilton\]) with the potential $\mathcal{V_{\rm I}}(z)$. The potential has the same term as the transverse traceless mode $\mathcal{V_{\rm t}}(z)$ with an additional term $\frac{1}{2}\frac{\partial^2V}{\partial\phi_{\rm I}^2}\bigg|$ in Eq.(\[eq:inert-potential\]) which is nonnegative definite provided the gravitational coupling is not too strong $\kappa \le {g \over \Lambda}$ $$\begin{aligned}
\frac{1}{2}\frac{\partial^2V}{\partial\phi_{\rm I}^2}\bigg|
&=& \Lambda^2\left(1+3\frac{\kappa^2\Lambda^2}{g^2}\right)
- 2\frac{\kappa^4\Lambda^4}{g^2}
\left(1+ \frac{\kappa^2\Lambda^2}{g^2}\right)
\sin^2\frac{g}{\Lambda}\phi_{\rm R}
\nonumber \\
&\ge &
\Lambda^2\left(1-\frac{\kappa^2\Lambda^2}{g^2}\right)
\left(1+2\frac{\kappa^2\Lambda^2}{g^2}\right) \ge 0
.
\label{eq:inert-positivity}\end{aligned}$$ Therefore inert scalar does not produce any additional tachyonic instability.
Fermions
========
In the previous two sections, we focused on the stability of BPS and Non-BPS wall configurations and studied its fluctuations. In this section, we turn to the fermionic part of the model and study its fluctuation. We shall consider only the BPS solutions for simplicity, since it allows massless gravitino, whereas the non-BPS solutions do not. The part of the Lagrangian (\[SUGRA\_Lag\]) quadratic in fermion fields (with arbitrary powers of bosons) can be rewritten as $$\begin{aligned}
\label{fermionlag}
e^{-1}{\cal L}_{{\rm fermion}}^{{\rm quadratic}} &=&
-i {\rm e}^{-A} {\bar{\chi}}{\bar{\sigma}}^{\underline{\mu}}
{\cal D}_\mu \chi
-i {\bar{\chi}}{\bar{\sigma}}^{\underline{2}} {\cal D}_2 \chi
+ \varepsilon^{\underline{\kappa 2 \mu \nu}}
{\rm e}^{-3A}
{\bar{\psi}}_{\kappa} {\bar{\sigma}}_{\underline{2}}
\tilde{{\cal D}}_\mu \psi_{\nu}
+ \vep^{\underline{\kappa \lambda 2 \nu}}
{\rm e}^{-2A} {\bar{\psi}}_{\kappa} {\bar{\sigma}}_{\underline{\lambda}}
\tilde{{\cal D}}_2 \psi_{\nu}
\nonumber \\
&&
+ \vep^{\underline{2 \lambda \mu \nu}}
{\rm e}^{-2A} {\bar{\psi}}_{2} {\bar{\sigma}}_{\underline{\lambda}}
\tilde{{\cal D}}_{\mu} \psi_{\nu}
+ \vep^{\underline{\kappa \lambda \mu 2}}
{\rm e}^{-2A} {\bar{\psi}}_{\kappa} {\bar{\sigma}}_{\underline{\lambda}}
\tilde{{\cal D}}_{\mu} \psi_{2}
\nonumber \\
&&
-\frac{\kappa}{\sqrt{2}} {\rm e}^{-A} \dot{\phi}^* \chi
\sig^{\underline{\mu}}
{\bar{\sigma}}^{\underline{2}} \psi_\mu
- \frac{\kappa}{\sqrt{2}} {\rm e}^{-A} \dot{\phi} {\bar{\chi}}{\bar{\sigma}}^{\underline{\mu}}
\sig^{\underline{2}} {\bar{\psi}}_\mu
+ \frac{\kappa}{\sqrt{2}} \dot{\phi}^* \chi \psi_2
+ \frac{\kappa}{\sqrt{2}} \dot{\phi} {\bar{\chi}}{\bar{\psi}}_2
\nonumber \\
&& - \kappa^2 {\rm e}^{{\kappa^2 \over 2}\phi^*\phi}
\biggl[
P^* {\rm e}^{-2A} \psi_\mu \sig^{\underline{\mu \nu}}
\psi_\nu
+ P {\rm e}^{-2A} {\bar{\psi}}_\mu {\bar{\sigma}}^{\underline{\mu \nu}}
{\bar{\psi}}_\nu
+ 2 {\rm e}^{-A}
(P^* \psi_\mu \sig^{\underline{\mu 2}} \psi_2
+ P {\bar{\psi}}_\mu \sig^{\underline{\mu 2}} {\bar{\psi}}_2)
\nonumber \\
&& +
{\rm e}^{-A} \frac{i\kappa}{\sqrt{2}} ( D_\phi P \chi
\sig^{\underline{\mu}} \bar{\psi}_\mu
+ D_{\phi^*} P^* {\bar{\chi}}{\bar{\sigma}}^{\underline{\mu}} \psi_\mu)
+ \frac{i\kappa}{\sqrt{2}}
(D_\phi P \chi \sig^{\underline{2}} {\bar{\psi}}_2
+ D_{\phi^*} P^* {\bar{\chi}}{\bar{\sigma}}^{\underline{2}} \psi_2)
\nonumber \\
&&
+ {\frac{1}{2}}( {\cal D}_\phi D_\phi P \chi^2
+ {\cal D}_{\phi^*} D_{\phi^*} P^* {\bar{\chi}}^2)
\biggr].
\label{eq:fermion-lag}\end{aligned}$$ The terms in the fourth line is quadratic in gravitino without any derivatives, which can be regarded as mass terms for gravitino. We find that they are $Z_2$ odd under $\phi \rightarrow -\phi$. In this respect, our model provides an explicit realization of the condition to have a smooth limit of vanishing width of the wall [@BCY] and in agreement with one version of the five-dimensional supergravity on the orbifold [@FLP]. For our modified superpotential $P$ given in Eq.(\[P\_mod\]), $D_\phi P$ and ${\cal D}_\phi D_\phi P$ are of the form: $$\begin{aligned}
D_\phi P
&=& {\rm e}^{-\frac{\kappa^2}{2}\phi^2}
\left[\kappa^2 (\phi^* - \phi)
\frac{\Lambda^3}{g^2} {\rm sin} \frac{g}{\Lambda} \phi
+
\frac{\Lambda^2}{g} {\rm cos}\frac{g}{\Lambda}\phi
\right],\\
{\cal D}_\phi D_\phi P
&=& 2 \kappa^2 (\phi^*-\phi) \partial_\phi P
- \left(\kappa^2 + \kappa^4 (\phi^2 - \phi^{*2})
+ \frac{g^2}{\Lambda^2}\right)P, \\
&=&{\rm e}^{-\frac{\kappa^2}{2}\phi^2}
\left[ -\left(\kappa^2 + \frac{g^2}{\Lambda^2}
- \kappa^4 (\phi - \phi^*)^2\right)
\frac{\Lambda^3}{g^2}
{\rm sin}\frac{g}{\Lambda} \phi
+ 2 \kappa^2 (\phi^* - \phi)
\frac{\Lambda^2}{g} {\rm cos}\frac{g}{\Lambda} \phi
\right].
\label{eq:DDP}\end{aligned}$$
Gravitino
---------
In this subsection, we will explore a massless gravitino which is a superpartner of the massless localized graviton under the SUGRA transformation with the conserved Killing spinor (\[eq:Killingspinor\]). Before studying equations of motion for gravitino, we will supertransform the wave function of the localized massless graviton to find conditions that the physical gravitino should satisfy. Let us focus on SUGRA transformation law for vierbein in Eq.(\[eq:SUGRAtransf\]), $$\delta_\zeta e_m{^{\underline{a}}}
= i \kappa \left( \zeta \sigma^{\underline{a}} \bar{\psi}_m
+ \bar{\zeta} \bar{\sigma}^{\underline{a}} \psi_m \right).
\label{vtrf}$$ The preserved SUSY along the Killing spinor $\zeta{(K)}$ in Eq.(\[eq:Killingspinor\]) with $\theta=\pi/2$ is given by $$\begin{aligned}
\zeta^\alpha{(K)}
&=& -i \bar{\zeta}_{\dot{\alpha}}(K)
\bar{\sigma}^{2 \dot{\alpha} \alpha}
= i {\rm e}^{A/2} [\epsilon_2, -\epsilon_1], \\
\bar{\zeta}_{\dot{\alpha}}(K)
&=& -i \zeta^\alpha{(K)} \sigma^2_{\alpha \dot{\alpha}}
= i{\rm e}^{A/2} [-\epsilon_1, -\epsilon_2].
\label{eq:Killing-spinor}\end{aligned}$$ Denoting the fluctuations $h_{mn}$ of the metric around the background spacetime metric $g_{mn}^{\rm background}\equiv diag({\rm e}^{2A}\eta_{\mu\nu}, 1)$ as $g_{mn}=g_{mn}^{\rm background} + h_{mn}$, the following linearized 3D SUGRA transformations with the Killing spinor $\zeta(K)$ are obtained for the metric fluctuations $\delta h_{mn}
=\delta(e_m{}^{\underline{a}}
e_{n \underline{a}})
=\delta e_m{}^{\underline{a}}
e_{n \underline{a}}
+ e_{m\underline{a}}
\delta e_{n}{}^{\underline{a}}$ $$\begin{aligned}
\label{3dmntrf}
\delta_{\zeta(K)} h_{\mu \nu} &=& i \kappa {\rm e}^A \zeta(K)
(\sigma_{\underline{\mu}} \bar{\psi}_\nu -i \sigma^{\underline{2}}
\bar{\sigma}_{\underline{\mu}} \psi_\nu + \sigma_{\underline{\nu}}
\bar{\psi}_\mu -i \sigma^{\underline{2}} \bar{\sigma}_{\underline{\nu}} \psi_\mu), \\
\label{3d2mtrf}
\delta_{\zeta(K)} h_{2 \mu} &=& i \kappa \zeta(K)
\left\{ \sigma_{\underline{2}} \bar{\psi}_\mu + i \psi_\mu
+ (\sigma_{\underline{\mu}} \bar{\psi}_2 -i\sigma^{\underline{2}}
\bar{\sigma}_{\underline{\mu}} \psi_2) {\rm e}^A \right\}, \\
\label{3d22trf}
\delta_{\zeta(K)} h_{22} &=& 2i \kappa \zeta(K) (\sigma_2 \bar{\psi}_2
+i \psi_2). \end{aligned}$$
In sect.\[sc:tensot-perturb\], we have imposed the gauge fixing condition (Newton Gauge) for graviton $$h_{22}= - \frac{1}{3} {\rm e}^{-2A}\eta^{\mu\nu}h_{\mu\nu} \equiv - \frac{1}{3}h,
\label{eq:gauge-fix-graviton1}$$ $$h_{2\mu}=0.
\label{eq:gauge-fix-graviton2}$$ We can algebraically decompose $h_{\mu\nu}$ into traceless part ${\rm e}^{2A}h_{\mu\nu}^{TT}$ and trace part $h$. We have found that the localized graviton zero mode is contained in the traceless part $$\begin{aligned}
\eta^{\mu \nu} h_{\mu \nu} = 0.
\label{eq:traceless}\end{aligned}$$ Equation of motion shows that the localized graviton zero mode also satisfies the transverse condition: $$\begin{aligned}
\eta^{\lambda \mu} \partial_\lambda h_{\mu \nu} = 0.
\label{eq:transverse}\end{aligned}$$ The matter fermion of course do not have the graviton zero mode : $\varphi=0$.
It is useful to decompose Weyl spinors in four dimensions into two 2-component Majorana spinors in three dimensions. For instance gravitinos $\psi_m$ are decomposed into two 2-component Majorana spinor-vectors $\psi_m^{(1)}$ and $\psi_m^{(2)}$ (real and imaginary part of the Weyl spinor-vector) $$\begin{aligned}
\label{majo1}
\psi_{m\alpha}^{(1)} \equiv \psi_{m\alpha}
- i \sigma^2_{\alpha\dot\alpha} \bar{\psi}_m^{\dot\alpha}
=-i\sigma^2_{\alpha\dot\alpha}\bar\psi_m^{(1)\dot\alpha},\end{aligned}$$ $$\begin{aligned}
\label{majo2}
\psi^{(2)}_{m\alpha} \equiv \psi_{m\alpha}
+ i \sigma^2_{\alpha\dot\alpha} \bar{\psi}_m^{\dot\alpha}
=i\sigma^2_{\alpha\dot\alpha}\bar\psi_m^{(2)\dot\alpha}. \end{aligned}$$ Similarly to the traceless and trace part decomposition of graviton (symmetric tensor), gravitino (vector-spinor) can also be algebraically decomposed into its traceless part $\psi_\mu^T$ and trace part $\bar \psi$ as $$\psi_\mu=\psi_\mu^T
- {1 \over 3}\sigma_{\underline{\mu}}\bar\psi,
\qquad
\bar \sigma^{\underline{\mu}}\psi_\mu^T=0,
\qquad
\bar \sigma^{\underline{\mu}}\psi_\mu=\bar\psi.$$
Let us make a SUGRA transformations of the physical state conditions (\[eq:gauge-fix-graviton1\])–(\[eq:transverse\]) for gravitons with the conserved Killing spinor $\zeta(K)$ in Eq.(\[eq:Killing-spinor\]). The SUGRA transformations with $\zeta(K)$ of Eqs.(\[eq:gauge-fix-graviton1\]), (\[eq:gauge-fix-graviton2\]) (\[eq:traceless\]) give $$\begin{aligned}
0 = \delta_{\zeta(K)} \left( h_{22} + \frac{1}{3} h\right)
= 2i \kappa \zeta(K) \left(
\sigma_{\underline{2}}
\bar{\psi}_2^{(1)}
+ \frac{1}{3} {\rm e}^{-A}\sigma^{\underline{\mu}}
\bar\psi_\mu^{(2)}
\right),
\label{eq:h22gauge-trans}\end{aligned}$$ $$\begin{aligned}
0 &=& \delta_{\zeta(K)} h_{2\mu}
= i \kappa \zeta
\left(
\sigma_{\underline{2}} \bar{\psi}_\mu^{(1)}
+ {\rm e}^A
\sigma_{\underline{\mu}} \bar{\psi}_2^{(2)}
\right).
\label{eq:gauge-fix-graviton2(2)}\end{aligned}$$ $$\begin{aligned}
0 &=& \delta_{\zeta(K)} \eta^{\mu \nu} h_{\mu \nu}
=
2i\kappa {\rm e}^{-A} \zeta
\sigma^{\underline{\mu}} \bar{\psi}_{\mu}^{(2)}.
\label{eq:traceless-gauge-tr}\end{aligned}$$ These result suggest the most natural gauge fixing condition for local gauge SUGRA transformations $$\begin{aligned}
\psi_2 =0,
\label{eq:psi2-gauge}\end{aligned}$$ which can always be chosen. Then, the above gauge fixing conditions (\[eq:h22gauge-trans\])–(\[eq:traceless-gauge-tr\]) are translated as $\bar\psi_\mu^{(1)} = 0$ and the traceless condition for $\psi^{(2)}$ $$\begin{aligned}
\sigma^{\underline{\mu}}\bar\psi_\mu = 0.
\label{gravitino3}\end{aligned}$$ Therefore we expect[^12] that the localized massless gravitino should be contained in the traceless part of $\psi_\mu^{(2)}$. The SUGRA transformation of the remaining condition (\[eq:transverse\]) gives the transverse condition for the $\psi_\mu^{(2)}$. Similarly to the graviton case, the localized gravitino should not have matter component $$\chi=0.
\label{eq:no-matter-fermion}$$
Let us now examine the equations of motion for gravitino $\psi_\mu$ coupled with the matter fermion $\chi$, which are obtained by varying the action (\[eq:fermion-lag\]). If we impose the conditions (\[eq:psi2-gauge\]), (\[gravitino3\]), and (\[eq:no-matter-fermion\]) on the gravitino equations of motion, we obtain $$\begin{aligned}
0 &=&
{\rm e}^{-3A}
\vep^{\underline{\mu 2 \rho \nu}} {\bar{\sigma}}_{\underline{2}}
\partial_\rho \psi_\nu
+{\rm e}^{-2A}
\left(
- {\frac{1}{2}}\dot{A} \vep^{\underline{\mu 2 \rho \nu}}
{\bar{\sigma}}_{\underline{\rho}} \psi_\nu
+ \vep^{\underline{\mu \rho 2 \nu}}
{\bar{\sigma}}_{\underline{\rho}} \partial_2 \psi_\nu
- \kappa^2{\rm e}^{{\kappa^2 \over 2}\phi^*\phi}
P \bar\sigma^{\mu\nu}{\bar{\psi}}_\nu
\right) \nonumber \\
&=&
{\rm e}^{-3A}
\vep^{\underline{\mu 2 \rho \nu}} {\bar{\sigma}}_{\underline{2}}
\partial_\rho \psi_\nu
+{\rm e}^{-2A}
\left[
- \vep^{\underline{\mu 2 \rho \nu}}
{\bar{\sigma}}_{\underline{\rho}}
\left(\partial_2 +{\frac{1}{2}}\dot{A}\right) \psi_\nu
+ \dot{A} \eta^{\mu\nu}{\bar{\psi}}_\nu
\right], \end{aligned}$$ where we have used the BPS equation (\[BPS\_A\]) for background fields. Possible zero mode should give a vanishing eigenvalue for the operator in the parenthesis : $$\begin{aligned}
0 &=&
- \vep^{\underline{\mu 2 \rho \nu}}
{\bar{\sigma}}_{\underline{\rho}}
\left(\partial_2 +{\frac{1}{2}}\dot{A}\right) \psi_\nu
+ \dot{A} \eta^{\mu\nu}{\bar{\psi}}_\nu
=-i \eta^{\mu\nu} \bar{\sigma}^{\underline{2}}
\left(\partial_2 + {\dot{A}\over 2}\right)
\psi_\nu
+ \dot{A} \eta^{\mu\nu}{\bar{\psi}}_\nu
\label{eom1}\end{aligned}$$ where $\vep^{\underline{\mu \rho 2 \nu}}
{\bar{\sigma}}_{\underline{\rho}}= i
({\bar{\sigma}}^{\underline{\mu}}
\sig^{\underline{2}} {\bar{\sigma}}^{\underline{\nu}}
- \eta^{\underline{\mu \nu}} {\bar{\sigma}}^{\underline{2}})$ is used in the second equality. In terms of the 2-component Majorana spinors (\[majo1\]) and (\[majo2\]), we obtain $$\begin{aligned}
\left(\partial_2 + \frac{3}{2}\dot A\right)\bar\psi^{(1)TT}_\mu
+ \left(-\partial_2 + \frac{1}{2}\dot A\right)\bar\psi^{(2)TT}_\mu = 0.\end{aligned}$$ Since $\psi^{(1)}_\mu = 0$, we obtain $$\begin{aligned}
\left(-\partial_2 + \frac{1}{2}\dot A\right)\bar\psi^{(2)TT}_\mu = 0.\end{aligned}$$ Therefore, we find the gravitino zero mode in the transverse traceless part of the 2-component Majorana vector-spinor $\bar\psi^{(2)TT}_\mu$ with the wave function $$\begin{aligned}
\bar\psi^{(2)TT}_\mu(y) = {\rm e}^{\frac{A(y)}{2}}.\end{aligned}$$
Now we see that the localized massless gravitino wave function is in precise agreement with that expected from the preserved SUGRA transformation with the Killing spinor $\zeta(K)$ : $$\begin{aligned}
e_{\mu}{^{\underline{a}}} \sim \zeta(K) \sigma^{\underline{a}}\psi_{\mu}.\end{aligned}$$ Since the wave function of the graviton and the Killing spinor are $e_\mu{^{\underline{a}}} \sim {\rm e}^{A}$ and $\zeta(K) = {\rm e}^{\frac{A}{2}}$, we find $\psi_\mu \sim {\rm e}^{\frac{A}{2}}$.
Matter Fermion
--------------
In this subsection, we study the fluctuation of matter fermion. By varying the Lagrangian (\[eq:fermion-lag\]) with respect to $\chi$, we obtain the equation of motion for matter fermion $\chi$. Using the gauge choice $\psi_2 = {\bar{\psi}}_2 = 0$ to the equation of motion, we find $$\begin{aligned}
0 &=&
-i{\rm e}^{A} \bar\sig^{\underline{\mu}}
\partial_\mu {\chi}
-\frac{3}{2}i \dot{A} {\bar{\sigma}}_2 {\chi}
-i {\bar{\sigma}}^{\underline{2}} \partial_2 {\chi}
-{\rm e}^{\frac{\kappa^2}{2}\phi^* \phi}
{\cal D}_{\phi^*} D_{\phi^*} P^* \bar{{\chi}}
\nonumber \\
&&- \frac{\kappa}{\sqrt{2}} \dot{\phi}
{\bar{\sigma}}^{\underline{\mu}} \sig^{\underline{2}} {\bar{\psi}}_\mu
-{\rm e}^{\frac{\kappa^2}{2}\phi^* \phi}
\frac{i\kappa}{\sqrt{2}} D_{\phi^*} P^*
{\bar{\sigma}}^{\underline{\mu}} \psi_\mu
. \end{aligned}$$ The second line gives the mixing term between the trace part of gravitino $\bar\psi=\bar\sigma^\mu\psi_\mu$ and the matter fermion $\chi$. Using the BPS equation (\[BPS\_phi\]), the mixing term can be rewritten as $$\begin{aligned}
- \frac{\kappa}{\sqrt{2}} \dot{\phi}
{\bar{\sigma}}^{\underline{\mu}} \sig^{\underline{2}} {\bar{\psi}}_\mu
-\frac{i\kappa}{\sqrt{2}} \dot{\phi}
{\bar{\sigma}}^{\underline{\mu}} \psi_\mu
=
-\frac{i\kappa}{\sqrt{2}} \dot{\phi}
{\bar{\sigma}}^{\underline{\mu}}\left(\psi_\mu
- i\sig^{\underline{2}} \bar \psi_\mu\right)
=
-\frac{i\kappa}{\sqrt{2}} \dot{\phi}
{\bar{\sigma}}^{\underline{\mu}}\psi_\mu^{(1)} ,\end{aligned}$$ where we used the 2-component Majorana spinor notation defined in Eqs.(\[majo1\]), (\[majo2\]). Since the mixing occurs only with $\psi_\mu^{(1)}$, it is also useful to decompose matter fermions into 2-component Majorana spinors, similarly to Eqs.(\[majo1\]), (\[majo2\]). Then the matter equation of motion is decomposed into two parts with opposite transformation property under the charge conjugation $\sigma^2$ $$-i{\rm e}^{A} \bar\sig^{\underline{\mu}}
\partial_\mu {\chi}^{(2)}
= i {\bar{\sigma}}^{\underline{2}} \left[
\partial_2 +\frac{3}{2} \dot{A}
-{\rm e}^{\frac{\kappa^2}{2}\phi^* \phi}
{\cal D}_{\phi^*} D_{\phi^*} P^* \right]
{\chi}^{(1)}
,$$ $$-i{\rm e}^{A} \bar\sig^{\underline{\mu}}
\partial_\mu {\chi}^{(1)}
=i {\bar{\sigma}}^{\underline{2}} \left[
\partial_2 +\frac{3}{2} \dot{A}
+{\rm e}^{\frac{\kappa^2}{2}\phi^* \phi}
{\cal D}_{\phi^*} D_{\phi^*} P^* \right]
{\chi}^{(2)}
-i\sqrt2 \kappa \dot{\phi} \bar \sigma^{\underline{\mu}}
\psi_\mu^{(1)}
.$$ It is now clear that we have a zero mode consisting of purely $\chi^{(1)}$ : $$-i{\rm e}^{A} \bar\sig^{\underline{\mu}}
\partial_\mu {\chi}^{(1)}
=0
,
\qquad \chi^{(2)}=0,
\qquad \psi_\mu=0$$ The zero mode wave function for matter fermion is given by $$\left[
\partial_2 +\frac{3}{2} \dot{A}
-{\rm e}^{\frac{\kappa^2}{2}\phi^* \phi}
{\cal D}_{\phi^*} D_{\phi^*} P^* \right]
{\chi}^{(1)} =0 ,$$ whose solution is given by $${\chi}^{(1)}_0 \sim {\rm e}^{-3A/2} {\rm exp}
\left[ \int dy \ {\rm e}^{\frac{\kappa^2}{2}\phi^* \phi}
{\cal D}_\phi
D_\phi P \right],
\label{matterzero}$$ Using (\[eq:DDP\]), the integral in (\[matterzero\]) in BPS case reads $$\begin{aligned}
\label{intm32}
\int dy \ {\rm e}^{\frac{\kappa^2}{2}\phi^* \phi}
{\cal D}_\phi D_\phi P
&=&
-(\kappa^2 + \frac{g^2}{\Lambda^2}) \frac{\Lambda^3}{g^2}
\int dy \ {\rm sin} \frac{g}{\Lambda} \phi
=-(1+\kappa^2 \frac{\Lambda^2}{g^2} ) \Lambda
\int dy \ {\rm tanh}(\Lambda y) \nonumber \\
&=&
-(1+\kappa^2 \frac{\Lambda^2}{g^2} )
{\rm log}({\rm cosh}(\Lambda y)) .\end{aligned}$$ In the second equality, $P={\rm e}^{-\kappa^2 \phi^2/2}
\frac{\Lambda^3}{g^2}{\rm sin}(g\phi/\Lambda)$ in Eq.(\[P\_mod\]) is substituted and $\phi = \phi^*$ is taken into account. In the last equality, the BPS solution $\phi = \frac{\Lambda}{g}
\left(
2{\rm tan}^{-1}e^{\Lambda(y-y_0)} - \frac{\pi}{2}
\right)$ in Eq.(\[BPS\_sol\]) with $n=0$ is considered. Then the zero mode wave function of matter fermion $\chi^{(1)}$ is given by $$\begin{aligned}
\label{mf01}
\chi^{(1)}_0 &\sim& {\rm e}^{-3A/2}
\left[{\rm cosh}(\Lambda y)
\right]^{-\left[1 + \kappa^2 \frac{\Lambda^2}{g^2} \right]}. \end{aligned}$$
In the weak gravity limit, the zero mode of matter fermion reduces to the Nambu-Goldstone fermion associated with the spontaneously broken SUSY [@MSSS2] $$\begin{aligned}
\label{mf01-1}
\chi^{(1)}_0
\to
\frac{1}{{\rm cosh}(\Lambda y)}, \qquad
\kappa \to 0 . \end{aligned}$$ As expected in the global SUSY limit, the wave function is localized at the wall where two out of four SUSY are broken. Let us note, however, that this zero mode of matter fermion should be unphysical except at $\kappa =0$ limit. For any finite values of $\kappa$, it should be possible to gauge away this zero mode, precisely analogously to the zero modes $\hat{B}^{(+)}$ in Eq.(\[zero\_scalar\]) in the matter scalar sector in sect.\[sc:active-scalar\]. In fact we can see that the $A$ dependence (warp factor) of the zero modes of active scalar $\hat{B}^{(+)}(x)$ and the matter fermion $\chi^{(1)}_0(x)$ agrees with the surviving SUSY transformation generated by the Killing spinor $\zeta(K)$, and will form a supermultiplet under the surviving SUGRA, since $\phi_{{\rm R}}^{(0)}(y) \sim e^{-A}, \chi^{(1)}_0(y)
\sim {\rm e}^{-3A/2}$ and $\zeta(y) \sim {\rm e}^{A/2}$ $$\begin{aligned}
\delta_\zeta \phi(x,y)_{{\rm R}}
= \sqrt{2} \zeta(x,y) \chi^{(1)}(x,y). \end{aligned}$$
On the other hand, ${\chi}^{(2)}$ should contain another Nambu-Goldstone fermion corresponding to the SUSY charges broken by the negative tension brane, if we consider non-BPS multi-wall configurations. Noting that the mixing term is suppressed by the Planck scale $M_P$, the zero mode equation of motion for ${\chi}^{(2)}$ in weak gravity limit $\kappa \to 0$ is given by $$\begin{aligned}
0 &=& \partial_2 {\chi}^{(2)}
+ {\rm e}^{\frac{\kappa^2}{2}\phi^* \phi}
{\cal D}_\phi D_\phi P {\chi}^{(2)}, \\
&\to& \partial_2 {\chi}^{(2)}
- \Lambda {\rm sin}\frac{g}{\Lambda}
\phi~{\chi}^{(2)}
= \partial_2 {\chi}^{(2)} - \Lambda {\rm tanh}(\Lambda y)
{\chi}^{(2)},
\quad (\kappa \to 0) \end{aligned}$$ where BPS solution $\phi = \frac{\Lambda}{g}
\left(
2{\rm tan}^{-1}e^{\Lambda(y-y_0)} - \frac{\pi}{2}
\right)$ is substituted in the last equality, thus the zero mode wave function becomes $${\chi}^{(2)}_0 \to {\rm cosh}(\Lambda y)~(\kappa \to 0),$$ which is not normalizable and hence unphysical even in the limit of $\kappa\rightarrow 0$. We know from the exact solution of the non-BPS two-wall solution[@MSSS2], that this wave function results when taking the limit of large radius to obtain the BPS solution. In that limit, SUSY broken on the second wall at $y=\pi R$ is restored and the corresponding Nambu-Goldstone fermion, which was localized on the second brane, becomes non-normalizable and unphysical. This is precisely our zero mode wave function $\chi^{(2)}_0$.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Daisuke Ida, Kazuya Koyama, Tetsuya Shiromizu, and Takahiro Tanaka for useful discussions in several occasions. One of the authors (M.E.) gratefully acknowledges support from the Iwanami Fujukai Foundation. This work is supported in part by Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology, Japan No.13640269 (NS) and by Special Postdoctoral Researchers Program at RIKEN (NM).
Appendix A
==========
In this appendix we show the gauge fixing to the Newton gauge. The most general fluctuation around the background metric (\[warped\_metric\]) takes the form : $$\begin{aligned}
ds^2 &=& {\rm e}^{2A}\left(\eta_{\mu\nu} + h^{\rm T}_{\mu\nu}
+ 2\eta_{\mu\nu}B\right)dx^\mu dx^\nu
+ 2 f_{\mu}\ dx^\mu dy + \left(1 - 2C\right)dy^2,\end{aligned}$$ where the trace part of the $(\mu,\nu)$ component of fluctuations is denoted as $2\eta_{\mu\nu}B$ and the traceless part is denoted as $h^{\rm T}_{\mu\nu}$. $f_\mu$ denotes the fluctuation of $(\mu,2)$ component and $-2C$ denotes the fluctuation of $(2,2)$ component.
After a tedious calculation we find the linearized Ricci tensors : $$\begin{aligned}
R^{(1)}_{\mu\nu}
&\!\!\!=&\!\!\!
\frac{1}{2}{\rm e}^{2A}\left({\rm e}^{-2A}\square_{D-1}
+ \partial_y^2
+ (D-1)\dot{A}\partial_y
+ 2(D-1)\dot{A}^2
+ 2\ddot{A}
\right)
h^{\rm T}_{\mu\nu}
- h^{\rm T}_{(\mu\rho,\nu)}{^{,\rho}}\nonumber\\
&\!\!\!&\!\!\!
+ \eta_{\mu\nu}{\rm e}^{2A}\left({\rm e}^{-2A}\square_{D-1}
+ \partial_y^2
+ 2(D-1)\left(\dot{A}\partial_y
+ \dot{A}^2 \right)
+ 2\ddot{A}
\right)B
+ (D-3)B_{,\mu,\nu}\nonumber\\
&\!\!\!&\!\!\!
- \eta_{\mu\nu}\dot{A}
f_\rho{^{,\rho}}
- \left(\partial_y
+ (D-3)\dot{A}
\right)f_{(\mu,\nu)}
+ \eta_{\mu\nu}{\rm e}^{2A}\left(\dot{A}\partial_y
+ 2\ddot{A}
+ 2(D-1)\dot{A}^2
\right)C
- C_{,\mu,\nu},
\\\ \nonumber\\
R^{(1)}_{\mu2}
&\!\!\!=&\!\!\!
- \frac{1}{2}\left(-{\rm e}^{-2A}\square_{D-1}
- 2(D-1)\dot{A}^2
- 2\ddot{A}
\right)f_{\mu}
- \frac{1}{2}{\rm e}^{-2A}f_{\rho,\mu}{^{,\rho}}\nonumber\\
&\!\!\!&\!\!\!
+ (D-2)\dot{A}
C_{,\mu}
+ (D-2)\partial_y
B_{,\mu}
- \frac{1}{2}\partial_y
h^{\rm T}_{\mu\rho}{^{,\rho}},
\\\ \nonumber\\
R^{(1)}_{22}
&\!\!\!=&\!\!\!
(D-1)\left(\partial_y^2
+ 2\dot{A}\partial_y
\right)B
- {\rm e}^{-2A}\partial_y
f_{\rho}{^{,\rho}}
- \left({\rm e}^{-2A}\square_{D-1} - (D-1)\dot{A}\partial_y
\right)C,\end{aligned}$$ where we define $B_{,\mu} = \partial_\mu B$, $h^{\rm T}_{(\mu\rho,\nu)} = \dfrac{1}{2}\left(h^{\rm T}_{\mu\rho,\nu} +
h^{\rm T}_{\nu\rho,\mu}\right)$, $f_\rho{^{,\rho}} =
\eta^{\rho\lambda}\partial_\lambda f_\rho$ and $\square_{D-1} =
\eta^{\rho\lambda}\partial_\rho\partial_\lambda$. We also find the linearized energy momentum tensor as follows : $$\begin{aligned}
\tilde{T}^{(1)}_{\mu\nu}
&\!\!\!=&\!\!\! {2 \over D-2}{\rm e}^{2A}\left[
V_{\rm R}h^{\rm T}_{\mu\nu} + \eta_{\mu\nu}\left(2V_{\rm R}B
+ \frac{dV_{\rm R}}{d\phi_{\rm R}}\varphi_{\rm R}\right)\right],\\
\tilde{T}^{(1)}_{\mu2}
&\!\!\!=&\!\!\! 2\dot\phi_{\rm R}
\varphi_{{\rm R},\mu} + {2 \over D-2}V_{\rm R}f_\mu,\\
\tilde{T}^{(1)}_{22}
&\!\!\!=&\!\!\!
4 \dot\phi_{\rm R}\partial_y
\varphi_{\rm R}
+ {2 \over D-2}\frac{dV_{\rm R}}{d\phi_{\rm R}}\varphi_{\rm R}
- {4 \over D-2}V_{\rm R}C,\end{aligned}$$ where $\varphi_{\rm R}$ is the fluctuation around the background active scalar field $\phi_{\rm R}$. Notice that the fluctuation $\varphi_{\rm I}$ about the background configuration for the imaginary part $\phi_{\rm I}$ decouples from any other fields in linear order of the fluctuations. We can obtain the linearized Einstein equations by plugging these into $R^{(1)}_{mn} = -\kappa^2\tilde{T}^{(1)}_{mn}$.
The above results are the most general in the sense that we do not fix any gauge for the fluctuations. As a next step, we wish to fix the gauge that simplifies the linearized equations. The “Newton” gauge is known as a candidate of such a gauge [@TanakaMontes; @CsabaCsaki]. The gauge transformation laws for the fluctuations are of the form : $$\begin{aligned}
\delta h^{\rm T}_{\mu\nu}
=
- \hat\xi_{(\mu,\nu)}
+ \dfrac{2}{D-1}\eta_{\mu\nu}\hat\xi_{\rho}{^{,\rho}},\quad
\delta B
=
- \dot{A}
\xi_2 - \dfrac{1}{D-1}\hat\xi_{\rho}{^{,\rho}},
\nonumber\\
\delta f_{\mu}
=
- {\rm e}^{2A}\partial_y
\hat{\xi}_{\mu} - \xi_{2,\mu},\quad
\delta C
=
\partial_y
\xi_{2},\quad
\delta\varphi_{\rm R} = - \dot\phi_{\rm R}
\xi_2,\end{aligned}$$ where $\xi_m$ is an infinitesimal coordinate transformation parameter $\delta x_m \equiv \xi_m$ and $\hat\xi_\mu \equiv {\rm e}^{-2A}\xi_\mu$. Using these four gauge freedom, we fix $f_\mu = 0$ and $(D-3)B = C$. The residual gauge transformation should satisfy $$\begin{aligned}
\partial_y
\hat\xi_\mu + {\rm e}^{-2A}\xi_{2,\mu}=0,\quad
\left(\partial_y
+(D-3)\dot{A}
\right)\xi_2 =
-\dfrac{D-3}{D-1}\hat\xi_\rho{^{,\rho}}.
\label{residual}\end{aligned}$$ In this gauge the linearized Einstein equations take the form : $$\begin{aligned}
{2}
\!\!\!&\frac{1}{2}{\rm e}^{2A}
\left({\rm e}^{-2A}\square_{D-1} +
\partial_y^2
+ (D-1)\dot{A}\partial_y
+ 2(D-1)\dot{A}^2
+ 2\ddot{A}
\right)h^{\rm T}_{\mu\nu}
- h^{\rm T}_{(\mu\rho,\nu)}{^{,\rho}}
+ \frac{1}{D-1}\eta_{\mu\nu}h^{\rm T}_{\rho\lambda}
{^{,\rho,\lambda}}
\nonumber \\
\!\!\!&
= - {2 \over D-2}\kappa^2 {\rm e}^{2A}
V_{\rm R}h^{\rm T}_{\mu\nu},
\label{1st_gauge_1}
\\
\!\!\!&{\rm e}^{2A}\left({\rm e}^{-2A}\square_{D-1}
+ \partial_y^2
+ (3D-5)\dot{A}\partial_y
+ 2(D-2)\left((D-1)\dot{A}^2
+ \ddot{A}\right)
\right)B
- \frac{1}{D-1}h^{\rm T}_{\rho\lambda}{^{,\rho,\lambda}}
\nonumber \\
\!\!\!&= -{2 \over D-2}\kappa^2{\rm e}^{2A}
\left(2V_{\rm R}B + \frac{dV_{\rm R}}
{d\phi_{\rm R}}\varphi_{\rm R}\right),\label{1st_gauge_2}\\
\!\!\!&(D-2)\left(\partial_y + (D-3)\dot{A}
\right)B_{,\mu}
- \frac{1}{2}\partial_y
h^{\rm T}_{\mu\rho}{^{,\rho}}
= -2\kappa^2\dot\phi_{\rm R}
\varphi_{{\rm R},\mu},\label{1st_gauge_3'}
\\
\!\!\!&\left(-(D-3){\rm e}^{-2A}\square_{D-1} +
(D-1)\partial_y^2
+ (D-1)^2\dot{A}\partial_y
\right)B
\nonumber \\
\!\!\!&
= -2\kappa^2\left(
2\dot\phi_{\rm R}\partial_y
\varphi_{\rm R}
+ \frac{1}{D-2}\frac{dV_{\rm R}}{d\phi_{\rm R}}
\varphi_{\rm R} - {2(D-3) \over D-2}V_{\rm R}B
\right),\label{1st_gauge_4'}\end{aligned}$$ where the Eq.(\[1st\_gauge\_1\]) is the traceless part of the $(\mu,\nu)$ component whereas the Eq.(\[1st\_gauge\_2\]) is the trace part of it. Denoting $D(x,y) = (D-2)\left(\partial_y+(D-3)\dot{A}\right)B
+ 2\kappa^2\dot\phi_{\rm R}
\varphi_{\rm R}$, the Eq.(\[1st\_gauge\_3’\]) is rewritten as $$\begin{aligned}
D_{,\mu}(x,y) =
\dfrac{1}{2}\partial_y
h^{\rm T}_{\mu\rho}{^{,\rho}}.\label{1st_gauge_3}\end{aligned}$$ Summing the background Einstein equation Eq.(\[1st\_gauge\_2\]) multiplied by $(D-3){\rm e}^{-2A}$ and Eq.(\[1st\_gauge\_4’\]) gives $\left(\partial_y + (D-1)\dot{A}\right)D(x,y)
= \dfrac{D-3}{2(D-1)}{\rm e}^{-2A}
h^{\rm T}_{\rho\lambda}{^{,\rho,\lambda}}$. Then we find $$\begin{aligned}
\displaystyle D(x,y) = E(x)\ {\rm e}^{-(D-1)A(y)} +
\dfrac{D-3}{2(D-1)}{\rm e}^{-(D-1)A}\int dy\
{\rm e}^{(D-3)A}
h^{\rm T}_{\rho\lambda}{^{,\rho,\lambda}}
\label{1st_gauge_4}\end{aligned}$$ where $E(x)$ is an arbitrary function of $x$. At this stage, the linearized Einstein equations are Eq.(\[1st\_gauge\_1\]), (\[1st\_gauge\_2\]), (\[1st\_gauge\_3\]) and (\[1st\_gauge\_4\]).
Next we attempt to eliminate the longitudinal mode of $h^{\rm T}_{\mu\nu}$ by using the residual gauge freedom. That is, we wish to set $v_\mu \equiv h^{\rm T}_{\mu\rho}{^{,\rho}} = 0$. For that purpose we first derive the equations of motion for $v_\mu$. Taking a divergence of Eq.(\[1st\_gauge\_1\]), we find $$\begin{aligned}
\left(\partial_y
+ (D-1)\dot{A}
\right)\partial_yv_\mu
= \frac{D-3}{D-1}
{\rm e}^{-2A}v_\rho{^{,\rho}}{_{,\mu}}.\label{eq_v}\end{aligned}$$ This equation can be solved as follows : i) taking divergence, we can determine $v_\rho{^{,\rho}}$, ii) regarding the solution $v_\rho{^{,\rho}}$ as a source, we can determine $v_\mu$. The gauge transformation law of $v_\mu$ takes the form $$\begin{aligned}
\delta v_\mu
= - \square_{D-1} \hat\xi_\mu -
\frac{D-3}{D-1}\hat\xi_{\rho,\mu}{^{,\rho}}.\label{delta_v}\end{aligned}$$ We want to set $0 = v_\mu + \delta v_\mu$ by using the gauge transformation (\[delta\_v\]) whose $\hat\xi_\mu$ satisfies the condition (\[residual\]) for the residual gauge transformation. Notice that the equations for $v_\mu$ and $\delta v_\mu$ are identical second order differential equations since the gauge transformation law consistent with the gauge condition (\[residual\]) does not change the form of the equation (\[eq\_v\]). We can also verify this from the condition (\[residual\]) straightforwardly as follows. From Eq.(\[residual\]) we find the identity ${\rm e}^{-2A}\square_{D-1}\xi_2 =
{D-1 \over D-3}\partial_y\left(\partial_y+
(D-3)\dot{A}\right)\xi_2$. Combining this and Eq.(\[delta\_v\]), we find $\delta v_\rho{^{,\rho}} = {2(D-2)(D-1) \over (D-3)^2}
{\rm e}^{2A}\left(\partial_y+(D-1)\dot{A}\right)
\partial_y
\left(\partial_y+(D-3)\dot{A}\right) \xi_2$ and $\partial_y\delta v_\mu =
{2(D-2) \over D-3}\partial_y
\left(\partial_y + (D-3)\dot{A}\right)\xi_{2,\mu}$. Hence, $\delta v_\mu$ satisfies just the same equation as Eq.(\[eq\_v\]) : $$\begin{aligned}
\left(\partial_y + (D-1)\dot{A}\right)
\partial_y\delta v_\mu
= \frac{D-3}{D-1}{\rm e}^{-2A}
\delta v_\rho{^{,\rho}}{_{,\mu}}.\end{aligned}$$ Therefore, $v_\mu$ can be eliminated in the gauge, if we can set at a given $y=y_0$ surface $$\begin{aligned}
v_\mu = - \delta v_\mu,\quad
\partial_y
v_\mu = - \partial_y
\delta v_\mu.\label{tt_condition}\end{aligned}$$ To clear matters, we introduce new functions $\Lambda_\mu(x) \equiv \hat\xi_\mu(x,y_0)$, $\Xi_\mu(x) \equiv \left(\partial_y
\hat\xi_\mu\right)(x,y_0)$, $\Gamma(x) \equiv \xi_2(x,y_0)$, $\Delta(x) \equiv \left(\partial_y
\xi_2\right)(x,y_0)$ which are defined at $y=y_0$ surface. In terms of these functions Eq.(\[tt\_condition\]) can be rewritten as $$\begin{aligned}
\square_{D-1}\Lambda_\mu
+ \frac{D-3}{D-2}\Lambda_{\rho,\mu}{^{,\rho}}
= \mathcal{A}_\mu,\quad
\square_{D-1}\Xi_\mu
+ \frac{D-3}{D-2}\Xi_{\rho,\mu}{^{,\rho}}
= \mathcal{B}_\mu,\label{const_surface}\end{aligned}$$ where $\mathcal{A}_\mu(x) \equiv v_\mu(x,y_0)$ and $\mathcal{B}_\mu(x) \equiv \left(\partial_y
v_\mu\right)(x,y_0)$. Similarly, the gauge condition (\[residual\]) at $y=y_0$ surface can be rewritten as $$\begin{aligned}
\Xi_{\mu} + {\rm e}^{-2A}\Gamma_{,\mu} = 0,\quad
\Delta + (D-3)\dot{A}\Gamma
= -\frac{D-3}{D-1}\Lambda_\rho{^{,\rho}}.\label{residual2}\end{aligned}$$ $\Lambda_\mu$ and $\Xi_\mu$ can be determined similarly to the Eq.(\[eq\_v\]). Next, we determine $\Gamma$ from the first equation of Eq.(\[residual2\]). However, this equation does not necessarily have a solution for a general function $\Xi_\mu$. To see this in detail, plug this into the second equation of Eq.(\[const\_surface\]) and we obtain $\square_{D-1}\Gamma_{,\mu} = -
\dfrac{D-2}{2D-5}{\rm e}^{2A}\mathcal{B}_\mu$. This equation can be solved if and only if $\mathcal{B}_\mu$ is expressed as a gradient of some function. In our case we obtain $\mathcal{B}_\mu = 2\partial_\mu D$ from Eq.(\[1st\_gauge\_3\]). Hence, a solution $\Gamma$ of the first equation of Eq.(\[residual2\]) exists. At the end, $\Delta$ is determined from the second equation of Eq.(\[residual2\]). In this gauge, we obtain $D(x,y) = E{\rm e}^{-(D-1)A}$ where $E$ is a constant from Eq.(\[1st\_gauge\_3\]) and (\[1st\_gauge\_4\]). We set $E=0$ since we require that the fluctuations $B$ and $\varphi_{\rm R}$ should vanish at infinity $|x|\rightarrow\infty$ on the wall. Thus we established our gauge choice (Newton gauge) and the constraints for the residual gauge transformations are (\[residual\]) and $$\begin{aligned}
\square_{D-1}\hat\xi_\mu + \frac{D-3}{D-1}
\hat\xi_\rho{^{,\rho}}{_{,\mu}}=0.\end{aligned}$$
Appendix B
==========
For a special case where $k\Lambda^{-1}$ is an integer[^13], we can express the Schrödinger potential in terms of $z$ explicitly. As an illustrative example, let us take $k=\Lambda$, where we find (putting $y_0=0$) ${\rm e}^{A}=(\cosh ky)^{-1}$, $z=k^{-1}\sinh ky$, $A(z)=-\dfrac{1}{2}\log\left(1+k^2z^2\right)$. The Schrödinger potential for the tensor perturbation takes the form : $$\begin{aligned}
\mathcal{V}_t(z) = - \frac{k^2(1-2k^2z^2)}{(1+k^2z^2)^2}.\end{aligned}$$ There remains only one parameter controlling both the width of the wall and the magnitude of the gravitational coupling, similarly to Ref.[@Gremm:1999pj]. Zero mode wave functions can also be expressed in terms of the $z$ coordinate explicitly and are shown in Fig.\[V\_tensor\_z\] : $$\begin{aligned}
\tilde{h}^{\rm TT(0)}_{\mu\nu}(x,z)
= \hat{h}^{\rm TT(+)}_{\mu\nu}(x)\ \frac{1}{\sqrt{1+k^2z^2}}
+ \hat{h}^{\rm TT(-)}_{\mu\nu}(x)\
\frac{z+k^2z^3/3}{\sqrt{1+k^2z^2}}.\end{aligned}$$
![[]{data-label="V_tensor_z"}](V_tensor_z.eps){width="6.5cm"}
The Schrödinger potential $\mathcal{V}_e$ for the active scalar perturbation can be also expressed in terms of $z$ : $$\begin{aligned}
\mathcal{V}_e =
\frac{k^2\left(3+2k^2z^2\right)}{\left(1+k^2z^2\right)^2},\end{aligned}$$ and zero modes are of the form : $$\begin{aligned}
\hat{B}^{(0)}(x,z) = \hat{B}^{(+)}(x)\ z\sqrt{1+k^2z^2}
+ \hat{B}^{(-)}(x)\ \left(1+z\tan^{-1}kz\right)\sqrt{1+k^2z^2}.\end{aligned}$$ These are shown in Fig.\[V\_scalar\_z\].
![[]{data-label="V_scalar_z"}](V_scalar_z.eps){width="6.5cm"}
[100]{} N. Arkani-Hamed, S. Dimopoulos and G. Dvali, [*Phys. Lett.*]{} [**B429**]{} (1998) 263 \[hep-ph/9803315\]; I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. Dvali, [*Phys. Lett.*]{} [**B436**]{} (1998) 257 \[hep-ph/9804398\]. L. Randall and R. Sundrum, [*Phys. Rev. Lett.*]{} [**83**]{} (1999) 3370 \[hep-ph/9905221\]. L. Randall and R. Sundrum, [*Phys. Rev. Lett.*]{} [**83**]{} (1999) 4690 \[hep-th/9906064\]. S. Dimopoulos and H. Georgi, [*Nucl. Phys. *]{} [**B193**]{} (1981) 150; N. Sakai, Z. f. Phys. [**C11**]{} (1981) 153; E. Witten, [*Nucl. Phys. *]{} [**B188**]{} (1981) 513; S. Dimopoulos, S. Raby, and F. Wilczek, [*Phys. Rev. *]{} [**D24**]{} (1981) 1681. E. Witten and D. Olive, [*Phys. Lett. *]{} [**B78**]{} (1978) 97. E.A. Mirabelli and M.E. Peskin, [*Phys. Rev.*]{} [**D58**]{} (1998) 065002 \[hep-th/9712214\]. T. Gherghetta and A. Riotto, [*Nucl. Phys.*]{} [**B623**]{} (2002) 97 \[hep-th/0110022\]. I.L. Buchbinder, S. James Gates, Jr., Hock-Seng Goh, W.D. Linch III, M.A. Luty, Siew-Phang Ng, and J. Phillips, “Supergravity loop contributions to brane world supersymmetry breaking” \[hep-th/0305169\]; R. Rattazzi, C.A. Scrucca and A. Strumia, “Brane to brane gravity mediation of supersymmetry breaking” \[hep-th/0305184\]. L. Randall and R. Sundrum, [*Nucl. Phys.*]{} [**B557**]{}, 79 (1999) \[hep-th/9810155\]; G. F. Giudice, M. A. Luty, H. Murayama and R. Rattazzi, JHEP [**9812**]{}, 027 (1998) \[hep-ph/9810442\]. D.E. Kaplan, G.D. Kribs and M. Schmaltz, [*Phys. Rev.*]{} [**D62**]{} (2000) 035010 \[hep-ph/9911293\]; Z. Chacko, M.A. Luty, A.E. Nelson and E. Ponton, [*JHEP*]{} [**0001**]{} (2000) 003 \[hep-ph/9911323\]. T. Kobayashi and K. Yoshioka, [*Phys. Rev. Lett.*]{} [**85**]{} (2000) 5527 \[hep-ph/0008069\]; Z. Chacko and M.A. Luty, [JHEP]{} [**0105**]{} (2001) 067 \[hep-ph/0008103\]. N. Arkani-Hamed, L.J. Hall, D. Smith and N. Weiner, [*Phys. Rev.*]{} [**D63**]{} (2000) 056003 \[hep-ph/9911421\]. N. Maru, N. Sakai, Y. Sakamura, and R. Sugisaka, [*Phys. Lett.*]{} [**B496**]{} (2000) 98, \[hep-th/0009023\]. N. Maru, N. Sakai, Y. Sakamura, and R. Sugisaka, [*Nucl. Phys.*]{} [**B616**]{} (2001) 47 \[hep-th/0107204\] ; N. Maru, N. Sakai, Y. Sakamura, and R. Sugisaka, the Proceedings of the 10th Tohwa international symposium on string theory, American Institute of Physics, 607, pages 209-215, (2002) \[hep-th/0109087\]; “SUSY Breaking by stable non-BPS configurations”, to appear in the Proceedings of the Corfu Summer Institute on Elementary particle Physics, Corfu, September 2001 \[hep-th/0112244\]. N. Sakai and R. Sugisaka, [*Int. J. Mod. Phys.*]{} [**A17**]{} (2002) 4697 \[hep-th/0204214\]. R. Altendorfer, J. Bagger and D. Nemeschansky, [*Phys. Rev.*]{} [**D63**]{} (2001) 125025, \[hep-th/0003117\]. T. Gherghetta and A. Pomarol, [*Nucl. Phys.*]{} [**B586**]{} (2000) 141, \[hep-ph/0003129\]; A. Falkowski, Z. Lalak and S. Pokorski, [*Phys. Lett.*]{} [**B491**]{} (2000) 172, \[hep-th/0004093\]; E. Bergshoeff, R. Kallosh and A. Van Proeyen, [*JHEP*]{} [**0010**]{} (2000) 033, \[hep-th/0007044\]. M. Cvetic, F. Quevedo, and S. Rey, [*Phys. Rev. Lett.*]{} [**67**]{} (1991) 1836; M. Cvetic, S. Griffies, and S. Rey, [*Nucl. Phys.*]{} [**B381**]{} (1992) 301 \[hep-th/9201007\]; M. Cvetic, and H.H. Soleng, [*Phys. Rep.*]{} [**B282**]{} (1997) 159 \[hep-ph/9804398\]. F.A. Brito, M. Cvetic, and S.C. Yoon, [*Phys. Rev.*]{} [**D64**]{} (2001) 064021, \[hep-ph/0105010\]. O. DeWolfe, D.Z. Freedman, S.S. Gubser, and A. Karch, [*Phys. Rev. *]{} [**D62**]{} (2000) 046008, \[hep-th/9909134\]. K. Skenderis, and P.K. Townsend, [*Phys. Lett.*]{} [**B468**]{} (1999) 46, \[hep-th/9909070\]; G.W.Gibbons and N.D. Lambert, [*Phys. Lett.*]{} [**B488**]{} (2000) 90, \[hep-th/0003197\]. M. Eto, N. Maru, N. Sakai, and T. Sakata, [*Phys. Lett. *]{}[**B553**]{} (2003) 87 \[hep-th/0208127\]. M. Arai, S. Fujita, M. Naganuma, and N. Sakai, [*Phys. Lett. *]{}[**B556**]{} (2003) 192, \[hep-th/0212175\]. M. Eto, S. Fujita, M. Naganuma and N. Sakai, “BPS multi-walls in five-dimensional supergravity,” \[hep-th/0306198\]. C. Charmousis, R. Gregory and V. A. Rubakov, Phys. Rev. D [**62**]{}, 067505 (2000) \[arXiv:hep-th/9912160\]. J. Garriga, and T. Tanaka, [*Phys. Rev. Lett.*]{} [**84**]{} (2000) 2778, \[hep-th/9911055\]. C. Csaki, J. Erlich, T.J. Hollowood, and Y. Shirman, Nucl.Phys. [**B581**]{} (2000) 309, \[hep-th/0001033\]. T. Tanaka and X. Montes, Nucl.Phys. [**B582**]{} (2000) 259, \[hep-th/0001092\]. M. Gremm, Phys. Lett. B [**478**]{}, 434 (2000) \[arXiv:hep-th/9912060\]. C. Csaki, M.L. Graesser, and G.D. Kribs, Phys.Rev. [**D63**]{} (2001) 065002, \[hep-th/0008151\]. W.D. Goldberger and M.B. Wise, [*Phys. Rev. Lett. *]{} [**83**]{} (1999) 4922 \[hep-th/9907447\].
J. Wess and J. Bagger, “Supersymmetry and Supergravity”, 1991, Princeton University Press.
W. Boucher, Nucl. Phys. [**B242**]{} (1984) 282.
P.K. Townsend, Phys. Lett. [**148B**]{} (1984) 55.
[^1]: *e-mail address: [email protected]*
[^2]: *e-mail address: [email protected], Special Postdoctoral Researcher*
[^3]: *e-mail address: [email protected]*
[^4]: We follow conventions of Ref.[@WessBagger] for the spinor and other notations.
[^5]: We define $\tilde{T}_{mn}\equiv
T_{mn} - \frac{1}{D-2}g_{mn}T^k{_k}$.
[^6]: These Killing spinors are the corrected results of those in our previous work [@EMSS].
[^7]: Adjoint relation between $Q_t$ and $Q_t^\dagger$ and the Hermiticity of $\mathcal{H}_t$ are assured by the inner product defined in Eq.(\[eq:inner-prod\]) without $z$ dependent weight.
[^8]: For a special case where $k\Lambda^{-1}$ is an integer, we can express the Schrödinger potential in terms of $z$ explicitly. We show this in Appendix B
[^9]: Generically speaking, the fluctuations of the inert scalar field $\varphi_{\rm I}$ can be an exception depending on the potential, although the inert scalar $\varphi_{\rm I}$ in our model is also frozen in the thin wall limit.
[^10]: For simplicity, we have assumed $Z_2$ parity of the metric perturbation to be even.
[^11]: This mass squared is factor two larger compared to the value of lightest massive scalar in the global SUSY model[@MSSS2]. We have not understood this discrepancy.
[^12]: One should have in mind that it is desirable to choose a gauge fixing condition for SUGRA transformations to become a supertransformation under $\zeta(K)$ of the gauge fixing condition for general coordinate transformations. However, it may not be logically mandatory.
[^13]: Then we can no longer take the thin wall limit of $\Lambda\rightarrow \infty$ with $k$ fixed.
| ArXiv |
---
abstract: 'In this paper, we collect various structural results to determine when an integral homology $3$–sphere bounds an acyclic smooth $4$–manifold, and when this can be upgraded to a Stein embedding. In a different direction we study whether smooth embedding of connected sums of lens spaces in $\mathbb{C}^2$ can be upgraded to a Stein embedding, and determined that this never happens.'
address:
- |
Department of Mathematics\
Georgia Institute of Technology\
Atlanta\
Georgia
- |
Department of Mathematics\
University of Alabama\
Tuscaloosa\
Alabama
author:
- 'John B. Etnyre'
- Bülent Tosun
title: Homology spheres bounding acyclic smooth manifolds and symplectic fillings
---
Introduction
============
The problem of embedding one manifold into another has a long, rich history, and proved to be tremendously important for answering various geometric and topological problems. The starting point is the Whitney Embedding Theorem: every compact $n$–dimensional manifold can be smoothly embedded in $\mathbb {R}^{2n}$. In this paper we will focus on smooth embeddings of 3–manifolds into $\R^4$ and embeddings that bound a convex symplectic domain in $(\R^4, \omega_{std})$. One easily sees that given such an embedding of a (rational) homology sphere, it must bound a (rational) homology ball. Thus much of the paper is focused on constructing or obstructing such homology balls.
Smooth embeddings
-----------------
In this setting, an improvement on the Whitney Embedding Theorem, due to Hirsch [@Hirsch:embedding] (also see Rokhlin [@Rohlin:3manembedding] and Wall [@Wall:embedding]), proves that every $3$–manifold embeds in $\mathbb{R}^5$ smoothly. In the smooth category this is the optimal result that works for all $3$–manifolds; for example, it follows from a work of Rokhlin that the Poincaré homology sphere $P$ cannot be embeded in $\mathbb{R}^4$ smoothly. On the other hand in the topological category one can always find embeddings into $\R^4$ for any integral homology sphere by Freedman’s work [@Freedman:4manifolds]. Combining the works of Rokhlin and Freedman for $P$ yields an important phenomena in $4$–manifold topology: there exists a closed oriented non-smoothable $4$–manifold — the so called $E_8$ manifold. In other words, the question of [*when does a $3$–manifold embeds in $\mathbb{R}^4$ smoothly*]{} is an important question from the point of smooth $4$–manifold topology. This is indeed one of the question in the Kirby’s problem list (Problem $3.20$) [@Kirby:problemlist]. Since the seminal work of Rokhlin in 1952, there has been a great deal of progress towards understanding this question. On the constructive side, Casson-Harrer [@CH], Stern, and Fickle [@Fickle] have found many infinite families of integral homology spheres that embeds in $\mathbb{R}^4$. On the other hand techniques and invariants, mainly springing from Floer and gauge theories, and symplectic geometry [@FS:R; @Manolescu:T; @OS:grading], have been developed to obstruct smooth embeddings of $3$–manifolds into $\mathbb{R}^4$. It is fair to say that despite these advances and lots of work done in the last seven decades, it is still unclear, for example, which Brieskorn homology spheres embed in $\mathbb{R}^4$ smoothly and which do not.
A weaker question is whether an integral homology sphere can arise as the boundary of an acyclic $4$–manifold? Note that a homology sphere that embeds in $\mathbb{R}^4$ necessarily bounds an acyclic manifold, and hence is homology cobordant to the $3$–sphere. Thus a homology cobordism invariant could help to find restrictions, and plenty of such powerful invariants has been developed. For example, for odd $n$, $\Sigma(2,3,6n-1)$ and $\Sigma(2,3,6n+1)$ have non-vanishing Rokhlin invariant. For even $n$, $\Sigma(2,3,6n-1)$ has $R=1$, where $R$ is the invariant of Fintushel and Stern, [@FS:R]. Hence none of these families of homology spheres can arise as the boundary of an acyclic manifold. On the other hand, for $\Sigma(2,3,12k+1)$ all the known homology cobordism invariants vanish. Indeed, it is known that $\Sigma(2,3,13)$ [@AKi] and $\Sigma(2,3,25)$ [@Fickle] bound contractible manifolds of Mazur type. Motivated by the questions and progress mentioned above and view towards their symplectic analogue, we would like to consider some particular constructions of three manifolds bounding acyclic manifolds.
Our first result is the following, which follows by adapting a method of Fickle.
\[main1\] Let $K$ be a knot in the boundary of an acyclic, respectively rationally acyclic, $4$–manifold $W$ which has a genus one Seifert surface $F$ with primitive element $[b]\in H_1(F)$ such that the curve $b$ is slice in $W.$ If $b$ has $F$–framing $s$, then the homology sphere obtained by $\frac{1}{(s\pm 1)}$ Dehn surgery on $K$ bounds an acyclic, respectively rationally acyclic, $4$–manifold.
Fickle [@Fickle] proved this theorem under the assumption that $\partial W$ was $S^3$ and $b$ was an unknot, but under these stronger hypothesis he was able to conclude that the homology sphere bounds a contractible manifold.
Fintushel and Stern conjectured, see [@Fickle], the above theorem for $\frac{1}{k(s\pm 1)}$ Dehn surgery on $K$, for any $k\geq 0$. So the above theorem can be seen to verify their conjecture in the $k=1$ case.
As noted by Fickle, if the conjecture of Fintushel and Stern is true then all the $\Sigma(2,3,12k+1)$ will bound acyclic manifolds since they can be realized by $-1/2k$ surgery on the right handed trefoil knot that bounds a Seifert surface containing an unknot for which the surface gives framing $-1$.
Notice that if $b$ is as in the theorem, then the Seifert surface $F$ can be thought of as obtained by taking a disk around a point on $b$, attaching a 1–handle along $b$ (twisting $s$ times) and then attaching another 1–handle $h$ along some other curve. The proof of Theorem \[main1\] will clearly show that $F$ does not have to be embedded, but just ribbon immersed so that cutting $h$ along a co-core to the handle will result in a surface that is “ribbon isotopic" to an annulus. By ribbon isotopic, we mean there is an 1-parameter family of ribbon immersions between the two surfaces, where we also allow a ribbon immersion to have isolated tangencies between the boundary of the surface and an interior point of the surface.
Consider the (zero twisted) $\pm$ Whitehead double $W_\pm(K_p)$ of $K_p$ from Figure \[exampleKp\]. In [@Cha07], Cha showed that $K_p$ is rationally slice. That is $K_p$ bounds a slice disk in some rational homology $B^4$ with boundary $S^3$. (Notice that $K_1$ is the figure eight knot originally shown to be rationally slice by Fintushel and Stern [@FintushelStern84].) Thus Theorem \[main1\] shows that $\pm 1$ surgery on $W_\pm(K_p)$ bounds a rationally acyclic 4–manifolds. This is easy to see as a Seifert surface for $W_\pm(K_p)$ can be made by taking a zero twisting ribbon along $K_p$ and plumbing a $\pm$ Hopf band to it.
[exampleKp]{} (44.5, 51)[$-p$]{} (48,6)[$p$]{}
Moreover, from Fickle’s original version of the theorem, $\pm\frac12$ surgery on $W_\pm(K_p)$ bounds a contractible manifold.
We can generalize this example as follows. Given a knot $K$, we denote by $R_m(K)$ the $m$-twisted ribbon of $K$. That is take an annulus with core $K$ such that its boundary components link $m$ times. Now denote by $P(K_1, K_2, m_1, m_2)$ the plumbing of $R_{m_1}(K_1)$ and $R_{m_2}(K_2)$. If the $K_i$ are rationally slice then $\frac{1}{m_i\pm 1}$ surgery on $P(K_1, K_2, m_1, m_2)$ yields a manifold bounding a rationally acyclic manifolds; moreover, if the $K_i$ are slice in some acyclic manifold, then the result of these surgeries will bound an acyclic manifold.
[**Symplectic embeddings.**]{} Another way to build examples of integral homology spheres that bound contractible manifolds is via the following construction. Let $K$ be a slice knot in the boundary of a contractible manifold $W$ (e.g. $W=B^4$), then $\frac{1}{m}$ Dehn surgery along $K$ bounds a contractible manifold. This is easily seen by removing a neighborhood of the slice disk from W (yielding a manifold with boundary 0 surgery on $K$) and attaching a 2–handle to a meridian of $K$ with framing $-m$. With this construction one can find examples of three manifolds modeled on not just Seifert geometry, for example $\Sigma(2,3,13)$ is the result of $1$ surgery on Stevedore’s knot $6_1$ but also hyperbolic geometry, for example the boundary of the Mazur cork is the result of $1$ surgery on the pretzel knot $P(-3,3,-3)$, which is also known as $\overline {9}_{46}$. See Figure \[fig:smooth\].
![On the left is the 3-manifold $Y_{m,n}$ described as a smooth $\frac{1}{m}$ surgery on the slice knot $P(3,-3,-n)$ for $n\geq 3$. On the right is the contractible Mazur-type manifold $W_{m,n}$ with $\partial W_{m,n}\cong Y_{m,n}$. Note the $m=1,~n=3$ case yields the original Mazur manifolds (with reversed orientation).[]{data-label="fig:smooth"}](RegSlicetoContractible.pdf){width="16cm"}
We ask the question of when $\frac{1}{m}$ surgery on a slice knot produces a [*Stein contractible*]{} manifold. Here there is an interesting asymmetry not seen in the smooth case.
\[regslice\] Let $L$ be a Legendrian knot in $(S^3, \xi_{std})$ that bounds a regular Lagrangian disc in $(B^4, w_{std})$. Contact $(1+\frac{1}{m})$ surgery on $L$ (so this is smooth $\frac{1}{m}$ surgery) is the boundary of a contractible Stein manifolds if and only if $m>0$.
This result points out an interesting angle on a relevant question in low dimensional contact and symplectic geometry: which compact contractible 4-manifolds admit a Stein structure? In [@MT:pseudoconvex] the second author and Mark found the first example of a contractible manifold without Stein structures with either orientation. This manifold is a Mazur-type manifold with boundary the Brieskorn homology sphere $\Sigma(2,3,13)$. A recent conjecture of Gompf remarkably predicts that Brieskorn homology sphere $\Sigma(p,q,r)$ can never bound acyclic Stein manifolds. It is an easy observation that $\Sigma(2,3,13)$ is the result of smooth $1$ surgery along the stevedore’s knot $6_1$. The knot $6_1$ is not Lagrangian slice, and indeed if Gompf conjecture is true, then by Theorem \[regslice\] $\Sigma(2,3,13)$ can never be obtained as a smooth $\frac{1}{n}$ surgery on a Lagrangian slice knot for any natural number $n$. Motivated by this example, Theorem \[regslice\], and Gompf’s conjecture we make the following weaker conjecture.
No non-trivial Brieskorn homology sphere $\Sigma(p,q,r)$ can be obtained as smooth $\frac{1}{n}$ surgery on a regular Lagrangian slice knot.
On the other hand as in Figure \[fig:smooth\] we list a family of slice knots, that are regular Lagrangian slice because they bound decomposable Lagrangian discs and by [@CET] decomposable Lagrangian cobordisms/fillings are regular. We explicitly draw the contractible Stein manifolds these surgeries bound in Figure \[fig:stein\].
![Stein contractible manifold with $\partial X_{m,n}\cong Y_{m,n}$.[]{data-label="fig:stein"}](RegSlicetoContractibleStein.pdf){width="12cm"}
A related embedding question is the following: when does a lens space L(p,q) embeds in $\mathbb{R}^4$ or $S^4$? Two trivial lens spaces, $S^3$ and $S^1\times S^2$ obviously have such embeddings. On the other hand, Hantzsche in $1938$ [@Hantzsche] proved, by using some elementary algebraic topology that if a $3$–manifold $Y$ embeds in $S^4$, then the torsion part of $H_1(Y)$ must be of the form $G \oplus G$ for some finite abelian group $G$. Therefore a lens space $L(p,q)$ for $|p|>1$ never embeds in $S^4$ or $\mathbb{R}^4$. For punctured lens spaces, however the situation is different. By combining the works of Epstein [@Epstein] and Zeeman [@Zeeman], we know that, a punctured lens space $L(p,q)\setminus B^3$ embeds in $\mathbb{R}^4$ if and only if $p>1$ is odd. Note that given such an embedding a neighborhood of $L(p,q)\setminus B^3$ in $\mathbb{R}^4$ is simply $(L(p,q)\setminus B^3)\times [-1,1]$ a rational homology ball with boundary $L(p,q)\# L(p,p-q)$ (recall $-L(p,q)$ is the same manifold as $L(p,p-q)$).
One way to see an embedding of $L(p,q)\# L(p,p-q)$ into $S^4$ is as follows: First, it is an easy observation that if $K$ is a doubly slice knot (that is there exists a smooth unknotted sphere $S\subset S^4$ such that $S\cap S^3=K$), then its double branched cover $\Sigma_2(K)$ embeds in $S^4$ smoothly. Moreover by a known result of Zeeman $K \# m(K)$ is a doubly slice knot for any knot $K$ (here $m(K)$ is the mirror of $K$). It is a classic fact that $L(p,q)$ is a double branched cover over the the 2-bridge knot $K(p,q)$ (this is exactly where we need $p$ to be odd, as otherwise $K(p,q)$ is a link). In particular $L(p,q)\#L(p,p-q)$, being double branched cover of doubly slice knot $K(p,q)\#m(K(p,q))$, embeds in $S^4$ smoothly. On the other hand, Fintushel-Stern [@FS:lensspace] and independently Gilmer-Livingston [@GilmerLivingston] showed this is all that could happen. That is they proved that $L(p,q)\#L(p,q')$ embeds in $S^4$ if and only if $L(p,q')=L(p,p-q)$ and $p$ is odd. In particular for $p$ odd, $L(p,q)\#L(p,p-q)$ bounds a rational homology ball in $\mathbb{R}^4$. A natural question in this case is to ask whether any of this smooth rational homology balls can be upgraded to be Symplectic or Stein submanifold of $\mathbb{C}^2$. We prove that this is impossible.
\[main2\] No contact structure on $L(p,q)\#L(p,p-q)$ has a symplectic filling by a rational homology ball. In particular, $L(p,q)\#L(p,p-q)$ cannot embed in $\mathbb{C}^2$ as the boundary of exact symplectic submanifold in $\mathbb{C}^2$.
Donald [@Donald] generalized Fintushel-Stern and Gilmer-Livingston’s construction further to show that for $L=\#_{i=1}^h L(p_i,q_i)$, the manifold $L$ embeds smoothly in $\mathbb{R}^4$ if and only if there exists $Y$ such that $L\cong Y\#-Y$. Our proof of Theorem \[main2\] applies to this generalization to prove none of the sums of lens spaces which embed in $\mathbb{R}^4$ smoothly can bound an exact symplectic manifold in $\mathbb{C}^2.$
To prove this theorem we need a preliminary result of independent interest.
\[prop\] If a symplectic filling $X$ of a lens space $L(p,q)$ is a rational homology ball, then the induce contact structure on $L(p,q)$ is a universally tight contact structure $\xi_{std}$.
Recall that every lens space admits a unique contact structure $\xi_{std}$ that is tight when pulled back the covering space $S^3$. Here we are not considering an orientation on $\xi_{std}$ when we say it is unique. On some lens spaces the two orientations on $\xi_{std}$ give the same oriented contact structure and on some they are different.
After completing a draft of this paper, the authors discovered that this result was previously proven by Fossati [@Fossati19pre] and Golla and Starkston [@GollaStarkston19pre]. As the proof we had is considerably different we decided to present it here.
[**Acknowledgements:**]{} We are grateful to Agniva Roy for pointing out the work of Fossati and of Golla and Starkston. The first author was partially supported by NSF grant DMS-1906414. Part of the article was written during the second author’s research stay in Montreal in Fall 2019. This research visit was supported in part by funding from the Simons Foundation and the Centre de Recherches Mathmatiques, through the Simons-CRM scholar-in-residence program. The second author is grateful to CRM and CIRGET, and in particular to Steve Boyer for their wonderful hospitality. The second author was also supported in part by a grant from the Simons Foundation (636841, BT)
Bounding acyclic manifolds
==========================
We now prove Theorem \[main1\]. The proof largely follows Fickle argument from [@Fickle], but we repeat it here for the readers convince (and to popularize Fickle’s beautiful argument) and to note where changes can be made to prove our theorem.
Suppose the manifold $\partial W$ is given by a surgery diagram $D$. Then the knot $K$ can be represented as in Figure \[TheKnot\]. There we see in grey the ribbon surface $F$ with boundary $K$ and the curve $b$ on the surface.
[TheKnot]{} (48, 68)[$D$]{} (103, 15)[$K$]{} (62, 15)[$b$]{} (11, 10)[$-s+1$]{} (90, 33)[$0$]{}
The result of $\frac{1}{s-1}$ surgery on $K$ is obtained by doing $0$ surgery on $K$ and $(-s+1)$ surgery on a meridian as shown in Figure \[TheKnot\]. (The argument for $\frac{1}{s+1}$ surgery is analogous and left to the reader.) Now part of $b$ is the core of one of the 1–handles making up $F$. So we can handle slide $b$ and the associated 1–handle over the $(-s+1)$ framed unknot to arrive at the left hand picture in Figure \[Step1\]. Then one may isotope the resulting diagram to get to the right hand side of Figure \[Step1\].
[Step1]{} (30, 26)[$D$]{} (84, 26)[$D$]{} (48, 11)[$0$]{} (100, 11)[$0$]{} (11, 6)[$-s+1$]{} (63, 6)[$-s+1$]{} (14.5, 17.5)[$1$]{}
We now claim the left hand picture in Figure \[Step2\] is the same manifold as the right hand side of Figure \[Step1\]. To see this notice that the green part of the left hand side of Figure \[Step2\] consists of two 0-framed knots. Sliding one over the other and using the new 0-framed unknot to cancel the non-slid component results in the right hand side of Figure \[Step1\].
Before moving forward we discuss the strategy of the remainder of the proof. The left hand side of Figure \[Step2\] represents the 3-manifold $M$ obtained from $\partial W$ by doing $\frac{1}{s-1}$ surgery on $K$. We will take $[0,1]\times M$ and attach a 2–handle to $\{1\}\times M$ to get a 4-manifold $X$ with upper boundary $M'$ so that $M'$ is obtained from $W$ by removing a slice disk $D$ for $b$. Since $W$ is acyclic, the complement of $D$ will be a homology $S^1\times D^3$. Let $W'$ denote this manifold. Attaching $X$ upside down to $W'$ (that is attaching a 2–handle to $W'$) to get a 4–manifold $W''$ with boundary $-M$. Since $-M$ is a homology sphere, we can easily see that $W''$ is acyclic. Thus $-W''$ is an acyclic filling of $M$.
Now to see we can attach the 2–handle to $[0,1]\times M$ as described above, we just add a 0-framed meridian to the new knot unknot on the left hand side of Figure \[Step2\]. This will result in the diagram on the right hand side of Figure \[Step2\].
[Step2]{} (32, 29)[$D$]{} (87, 29)[$D$]{} (50, 11)[$0$]{} (97, 11)[$0$]{} (45.5, 19)[$0$]{} (26.5, 17.5)[$0$]{} (11, 6.5)[$-s+1$]{} (70, 6)[$-s+1$]{}
We are left to see that the right hand side of Figure \[Step2\] is the boundary of $W$ with the slice disk for $b$ removed. To see this notice that the two green curves in Figure \[Step2\] co-bound an embedded annulus with zero twisting (the grey in the figure) and one boundary component links the $(-s+1)$ framed unknot and the other does not. Sliding the former over the latter results in the left hand diagram in Figure \[Step3\]. Cancelling the two unknots from the diagram results in the right hand side of Figure \[Step3\] which is clearly equivalent to removing the slice disk $D$ for $b$ from $W$.
[Step3]{} (34, 31)[$D$]{} (78, 31)[$D$]{} (26, 3.5)[$0$]{} (26.5, 17.5)[$0$]{} (11, 3)[$-s+1$]{} (92, 6)[$0$]{}
Stein fillings
==============
We begin this section by proving Theorem \[regslice\] concerning smooth $\frac 1 m$ surgery on a Lagrangian slice knot.
We begin by recalling a result from [@CET] that says contact $(r)$ surgery on a Legendrian knot $L$ for $r\in(0,1]$ is strongly symplectically fillable if and only if $L$ is Lagrangian slice and $r=1$. Thus $(1+1/m)$ contact surgery for $m<0$ will never be fillable, much less fillable by a contractible Stein manifold.
We now turn to the $m>0$ case and start by a particularly helpful visualization of the knot $L$ (here and forward $L$ stands both for the knot type and Legendrain knot that realizing the knot type that bounds the regular Lagrangian disk). By [@CET Theorem $1.9$, Theorem $1.10$], we can find a handle presentation of the 4-ball $B^4$ made of one 0–handle, and $n$ cancelling Weinstein $1$– and $2$–handle pairs, and a maximum Thurston-Bennequin unknot in the boundary of the 0–handle that is disjoint from $1$– and $2$–handles such that when the $1$– and $2$–handle cancellations are done the unknot becomes $L$. See Figure \[Handle\].
[Handle2]{} (56, 2)[$L$]{}
Now smooth $1/m$ surgery on $L$ can also be achieved by smooth $0$ surgery (which corresponds to taking the complement of the slice disk) on $L$ followed by smooth $-m$ surgery on its meridian.
As the proof of Theorem 1.1 in [@CET] shows, removing a neighborhood of the Lagrangian disk $L$ bounds from $B^4$ gives a Stein manifold with boundary $(+1)$ contact surgery on $L$ (that is smooth $0$ surgery on $L$). Now since the meridian to $L$ can clearly be realized by an unknot with Thurston-Bennequin invariant $-1$, we can stabilize it as necessary and attach a Stein $2$–handle to it to get a contractible Stein manifold bounding $(1+1/m)$ contact surgery on $L$ for any $m>1$.
For the $m=1$ case we must argue differently. One may use Legendrian Reidemeister moves to show that in any diagram for $L$ as described above the $2$–handles pass through $L$ as shown on the left hand side of Figure \[normalform\].
[NormalForm]{} (74, 7.75)[$-1$]{}
Smoothly doing contact $(1+1/1)$–surgery on $L$ (that is smooth $1$ surgery) is smoothly equivalent to replacing the left hand side of Figure \[normalform\] with the right hand side and changing the framings on the strands by subtracting their linking squared with $L$.
Now notice that if we realize the right hand side of Figure \[normalform\] by concatenating $n$ copies of either diagram in Figures \[sandz\] (where $n$ is the number of red strands in Figure \[normalform\]) then the Thurston-Bennequin invariant of each knot involved in Figure \[normalform\] is reduced by the linking squared with $L$. Thus we obtain a Stein diagram for the result of $(2)$ contact surgery on $L$.
[sandz]{}
Notice that the diagram clearly describes an acyclic 4–manifolds and moreover the presentation for its fundamental group is the same as for the presentation for the fundamental group of $B^4$ given by the original diagram. Thus the 4–manifolds is contractible.
We now turn to the proof that connected sums of lens spaces can never have acyclic symplectic fillings, but first prove Proposition \[prop\] that says any contact structure on a lens space that is symplectically filled by a rational homology ball must be universally tight.
Let $X$ be a rational homology ball symplectic filling of $L(p,q)$. We show the induces contact structure must be the universally tight contact structure $\xi_{std}$. This will follow from unpacking recent work of Menke [@menke2018jsjtype] where he studies exact symplectic fillings of a contact $3$–manifold that contains a [*mixed torus*]{}.
We start with the set-up. Honda [@Honda:classification1] and Giroux [@Giroux:classification] have classified tight contact structures on lens spaces. We review the statement of Honda in terms of the Farey tessellation. We use notation and terminology that is now standard, but see see [@Honda:classification1] for details. Consider a minimal path in the Farey graph that starts at $-p/q$ and moves counterclockwise to $0$. To each edge in this path, except for the first and last edge, assign a sign. Each such assignment gives a tight contact structure on $L(p,q)$ and each tight contact structures comes from such an assignment. If one assigns only $+$’s or only $-$’s to the edges then the contact structure is universally tight, and these two contact structures have the same underlying plane field, but with opposite orientations. We call this plane field (with either orientation) the the universally tight structure $\xi_{std}$ on $L(p,q)$. All the other contact structures are virtually overtwisted, that is they are tight structures on $L(p,q)$ but become overtwisted when pulled to some finite cover. The fact that at some point in the path describing a virtually overtwisted contact structure the sign must change is exactly the same as saying a Heegaard torus for $L(p,q)$ satisfies Menke’s mixed torus condition.
\[mixed\] Let $(Y, \xi)$ denote closed, co-oriented contact $3$–manifold and let $(W, \omega)$ be its strong (resp. exact) symplectic filling. If $(Y,\xi)$ contains a mixed torus $T$, then there exists a (possibly diconnected) symplectic manifold $(W', \omega')$ such that:
- $(W', \omega')$ is a strong (rep. exact) symplectic filling of its boundary $(Y',\xi')$.
- $\partial W'$ is obtained from $\partial W$ by cutting along $T$ and gluing in two solid tori.
- $W$ can be recovered from $W'$ by symplectic round $1$–handle attachment.
In our case we have $X$ filling $L(p,q)$. Suppose the contact structure on $L(p,q)$ is virtually overtwisted. The theorem above now gives a symplectic manifold $X'$ two which a round 1–handle can be attached to recover $X$; moreover, $\partial X'$ is a union of two lens spaces or $S^1\times S^2$. However, Menke’s more detailed description of $\partial X'$ shows that $S^1\times S^2$ is not possible. We digress for a moment to see why this last statement is true. When one attaches a round 1–handle, on the level of the boundary, one cuts along the torus $T$ and then glues in two solid tori. Menke gives the following algorithm to determine the meridional slope for these tori. That $T$ is a mixed torus means there is a path in the Farey graph with three vertices having slope $r_1, r_2,$ and $r_3$, each is counterclockwise of the pervious one and there is an edge from $r_i$ to $r_{i+1}$ for $i=1,2$. The torus $T$ has slope $r_2$ and the signs on the edges are opposite. Now let $(r_3,r_1)$ denote slopes on the Farey graph that are (strictly) counterclockwise of $r_3$ and (strictly) clockwise of $r_1$. Any slope in $(r_3,r_1)$ with an edge to $r_2$ is a possible meridional slope for the glued in tori, and these are the only possible slopes. Now since our $r_i$ are between $-p/q$ and $0$ we note that if there was an edge from $r_2$ to $-p/q$ or $0$ then $r_2$ could not be part of a minimal path form $-p/q$ to $0$ that changed sign at $r_2$. Thus when we glue in the solid tori corresponding to the round 1–handle attachment, they will not have meridional slope $0$ or $-p/q$ and thus we cannot get $S^1\times S^2$ factors.
The manifold $X'$ is either connected or disconnected. We notice that it cannot be connected because it is know that any contact structure on a lens space is planar [@Schoenenberger05], and Theorem 1.2 from [@Etnyre:planar] says any filling of a contact structure supported by a planar open book must have connected boundary. Thus we know that $X'$ is, in fact, disconnected. So $X'=X'_1\cup X'_2$ with $\partial X'_i$ a lens space. The Mayer–Vietoris sequence for the the decomposition of $X'$ into $X'_1\cup X'_2$ (glued along an $S^1\times D^2$ in their boundaries) shows that $H_1$ of $X_1'$ or $X_2'$ has rank 1, while both of their higher Betti numbers are 0. But now the long exact sequence for the pair $(X'_i,\partial X'_i)$ implies that $b_1$ must be 0 for both the $X_i'$. This contradiction shows that a symplectic manifold which is rational homology ball and with boundary $L(p,q)$ must necessarily induce the universally tight contact structure on the boundary.
The statement about embeddings follows directly from the statement about symplectic fillings. To prove that result let $X$ be an exact symplectic filling of $L(p,q)\# L(p,p-q)$ that is also a rational homology ball. Observe that there is an embedded sphere in $\partial X$ as it is reducible. Eliashberg’s result in [@CE Theorem $16.7$] says that $X$ is obtained from another symplectic manifold with convex boundary by attaching a 1–handle. Thus $X\cong X_1\natural X_2$ where $X_1$ and $X_2$ are exact symplectic manifolds with $\partial X_1=L(p,q)$ and $\partial X_2=L(p,p-q)$ or $X\cong X'\cup (\text{1--handle})$ where $X'$ is symplectic 4-manifold with the disconnected boundary $\partial X'\cong L(p,q)\sqcup L(p,p-q)$.
As argued above in the proof of Proposition \[prop\] it is not possible to have $X'$ with disconnected boundary being lens spaces and we must be in the case $X\cong X_1\natural X_2$; moreover, since $X$ is a rational homology balls, so are the $X_i$. Moreover, since $X_1$ and $X_2$ are symplectic filling of their boundaries, they induce tight contact structures on $L(p,q)$ and $L(p,p-q)$), respectively.
Proposition \[prop\] says that these tight contact structures must be, the unique up to changing orientation, universally tight contact structures $\xi_{std}$ on $L(p,q)$ and $\xi'_{std}$ on $L(p,p-q)$. Thus we have that $X_1$ and $X_2$ are rational homology balls, and are exact symplectic fillings of $(L(p,q), \xi_{std})$, and $(L(p,p-q), \xi'_{std})$, respectively. In [@Lisca08 Corollary 1.2(d)] Lisca classified all such fillings. According to Lisca’s classification, symplectic rational homology ball fillings of $(L(p,q), \xi_{std})$ are possible exactly when $(p,q)=(m^2, mh-1)$ for some $m$ and $h$ co-prime natural numbers, and similarly for $(L(p,p-q), \xi'_{std})$ exactly when $(p,p-q)=(m^2, mk-1)$ for $m$ and $k$ co-prime natural numbers. Now simple calculation shows that, the only possible value for $m$ satisfying these equations is $m=2$. In particular, we get that $p=4$, but then we must have $\{q, p-q\}=\{1,3\}$, and 3 cannot be written as $2k-1$, for $k$ co-prime to $2$. Thus there is no such $X$.
[10]{}
Selman Akbulut and Robion Kirby, *Mazur manifolds.*, The Michigan Mathematical Journal **26** (1979), no. 3, 259–284.
Andrew Casson and John Harer, *Some homology lens spaces which bound rational homology balls*, Pacific Journal of Mathematics **96** (1981), no. 1, 23–36.
Jae Choon Cha, *The structure of the rational concordance group of knots*, Mem. Amer. Math. Soc. **189** (2007), no. 885, x+95. [MR ]{}[2343079]{}
K. Cieliebak and Y. Eliashberg, *From [S]{}tein to [W]{}einstein and [B]{}ack – [S]{}ymplectic [G]{}eometry of [A]{}ffine [C]{}omplex [M]{}anifolds*, Colloquium Publications, vol. 59, Amer. Math. Soc., 2012.
James Conway, John B Etnyre, and B[ü]{}lent Tosun, *[Symplectic Fillings, Contact Surgeries, and Lagrangian Disks]{}*, International Mathematics Research Notices (2019), rny291.
Andrew Donald, *Embedding [S]{}eifert manifolds in [$S^4$]{}*, Trans. Amer. Math. Soc. **367** (2015), no. 1, 559–595. [MR ]{}[3271270]{}
D. B. A. Epstein, *Embedding punctured manifolds*, Proc. Amer. Math. Soc. **16** (1965), 175–176. [MR ]{}[208606]{}
John B. Etnyre, *Planar [O]{}pen [B]{}ook [D]{}ecompositions and [C]{}ontact [S]{}tructures*, IMRN **79** (2004), 4255–4267.
Henry Clay Fickle, *Knots, [${\bf Z}$]{}-homology [$3$]{}-spheres and contractible [$4$]{}-manifolds*, Houston J. Math. **10** (1984), no. 4, 467–493. [MR ]{}[774711]{}
Ronald Fintushel and Ronald Stern, *Rational homology cobordisms of spherical space forms*, Topology **26** (1987), no. 3, 385–393. [MR ]{}[899056]{}
Ronald Fintushel and Ronald J. Stern, *A [$\mu$]{}-invariant one homology [$3$]{}-sphere that bounds an orientable rational ball*, Four-manifold theory ([D]{}urham, [N]{}.[H]{}., 1982), Contemp. Math., vol. 35, Amer. Math. Soc., Providence, RI, 1984, pp. 265–268. [MR ]{}[780582]{}
[to3em]{}, *Pseudofree orbifolds*, Ann. of Math. (2) **122** (1985), no. 2, 335–364. [MR ]{}[808222]{}
Edoardo Fossati, *Topological constraints for stein fillings of tight structures on lens spaces*, 2019.
Michael Hartley Freedman, *The topology of four-dimensional manifolds*, J. Differential Geometry **17** (1982), no. 3, 357–453. [MR ]{}[679066]{}
Patrick M. Gilmer and Charles Livingston, *On embedding [$3$]{}-manifolds in [$4$]{}-space*, Topology **22** (1983), no. 3, 241–252. [MR ]{}[710099]{}
Emmanuel Giroux, *Structures de contact en dimension trois et bifurcations des feuilletages de surfaces*, Invent. Math. **141** (2000), no. 3, 615–689. [MR ]{}[1779622]{}
Marco Golla and Laura Starkston, *The symplectic isotopy problem for rational cuspidal curves*, 2019.
W. Hantzsche, *Einlagerung von [M]{}annigfaltigkeiten in euklidische [R]{}äume*, Math. Z. **43** (1938), no. 1, 38–58. [MR ]{}[1545714]{}
Morris W. Hirsch, *On imbedding differentiable manifolds in euclidean space*, Ann. of Math. (2) **73** (1961), 566–571. [MR ]{}[124915]{}
Ko Honda, *On the [C]{}lassification of [T]{}ight [C]{}ontact [S]{}tructures [I]{}*, Geom. Topol. **4** (2000), 309–368.
Rob Kirby, *Problems in low dimensional manifold theory*, Algebraic and geometric topology ([P]{}roc. [S]{}ympos. [P]{}ure [M]{}ath., [S]{}tanford [U]{}niv., [S]{}tanford, [C]{}alif., 1976), [P]{}art 2, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 273–312. [MR ]{}[520548]{}
Paolo Lisca, *On symplectic fillings of lens spaces*, Trans. Amer. Math. Soc. **360** (2008), no. 2, 765–799. [MR ]{}[2346471]{}
Ciprian Manolescu, *Pin(2)-equivariant [S]{}eiberg-[W]{}itten [F]{}loer homology and the triangulation conjecture*, J. Amer. Math. Soc. **29** (2016), no. 1, 147–176. [MR ]{}[3402697]{}
Thomas E. Mark and Bülent Tosun, *Obstructing pseudoconvex embeddings and contractible [S]{}tein fillings for [B]{}rieskorn spheres*, Adv. Math. **335** (2018), 878–895. [MR ]{}[3836681]{}
Michael Menke, *A jsj-type decomposition theorem for symplectic fillings*, 2018.
Peter Ozsv[á]{}th and Zolt[á]{}n Szab[ó]{}, *Absolutely graded floer homologies and intersection forms for four-manifolds with boundary*.
V. A. Rohlin, *The embedding of non-orientable three-manifolds into five-dimensional [E]{}uclidean space*, Dokl. Akad. Nauk SSSR **160** (1965), 549–551. [MR ]{}[0184246]{}
Stephan Schönenberger, *Planar open books and symplectic fillings*, Ph.D. thesis, University of Pennsylvania, 2005.
C. T. C. Wall, *All [$3$]{}-manifolds imbed in [$5$]{}-space*, Bull. Amer. Math. Soc. **71** (1965), 564–567. [MR ]{}[175139]{}
E. C. Zeeman, *Twisting spun knots*, Trans. Amer. Math. Soc. **115** (1965), 471–495. [MR ]{}[195085]{}
| ArXiv |
---
abstract: 'We study the $\beta$ analogue of the nonintersecting Poisson random walks. We derive a stochastic differential equation of the Stieltjes transform of the empirical measure process, which can be viewed as a dynamical version of the Nekrasov’s equation in [@MR3668648 Section 4]. We find that the empirical measure process converges weakly in the space of c[á]{}dl[á]{}g measure-valued processes to a deterministic process, characterized by the quantized free convolution, as introduced in [@MR3361772]. For suitable initial data, we prove that the rescaled empirical measure process converges weakly in the space of distributions acting on analytic test functions to a Gaussian process. The means and the covariances are universal, and coincide with those of $\beta$-Dyson Brownian motions with the initial data constructed by the Markov-Krein correspondence. Our proof relies on integrable features of the generators of the $\beta$-nonintersecting Poisson random walks, the method of characteristics, and a coupling technique for Poisson random walks.'
address: |
Harvard University\
E-mail: [email protected]
author:
- Jiaoyang Huang
bibliography:
- 'References.bib'
title: '$\beta$-Nonintersecting Poisson Random Walks: Law of Large Numbers and Central Limit Theorems'
---
Introduction
============
$\beta$-nonintersecting Poisson random walks
--------------------------------------------
Let $\tilde{{\bm{x}}}(t)=(\tilde x_1(t),\tilde x_2(t),\cdots, \tilde x_n(t))$ be the continuous-time *Poisson random walk* on ${\mathbb{Z}}_{{\geqslant}0}^n$, i.e. particles independently jump to the neighboring right site with rate $n$. The generator of ${{\bm{x}}}(t)$ is given by $$\begin{aligned}
\tilde {{{\mathcal}L}}^n f(\tilde {{\bm{x}}})=\sum_{i=1}^nn\left(f(\tilde{{\bm{x}}}+{\bm{e}}_i)-f(\tilde {{\bm{x}}})\right),\end{aligned}$$ where $\{{\bm{e}}_i\}_{1{\leqslant}i{\leqslant}n}$ is the standard vector basis of ${{\mathbb R}}^n$. $\tilde {{\bm{x}}}(t)$ conditioned never to collide with each other is the *nonintersecting Poisson random walk*, denoted by $\tilde {{\bm{x}}}(t)=(x_1(t),x_2(t),\cdots,x_n(t))$. The nonintersecting condition has probability zero, and therefore, needs to be defined through a limit procedure which is performed in [@MR1887625]. The nonintersecting Poisson random walk is a continuous time Markov process on $$\begin{aligned}
{\mathbb{W}}^n_1=\{({\lambda}_1+(n-1),{\lambda}_2+(n-2),\cdots,{\lambda}_n): ({\lambda}_1,{\lambda}_2,\cdots,{\lambda}_n)\in {\mathbb{Z}}_{{\geqslant}0}^n, {\lambda}_1{\geqslant}{\lambda}_2{\geqslant}\cdots{\geqslant}{\lambda}_n{\geqslant}0\},\end{aligned}$$ with generator $$\begin{aligned}
{{{\mathcal}L}}_1^n f({{\bm{x}}})=n\sum_{i=1}^n\frac{V({{\bm{x}}}+{\bm{e}}_i)}{V({{\bm{x}}})}\left(f({{\bm{x}}}+{\bm{e}}_i)-f({{\bm{x}}})\right)=n\sum_{i=1}^{n}\left(\prod_{j:j\neq i}\frac{x_i-x_j+1}{x_i-x_j}\right)\left(f({{\bm{x}}}+{\bm{e}}_i)-f({{\bm{x}}})\right),\end{aligned}$$ where $V({{\bm{x}}})=\prod_{1{\leqslant}i<j{\leqslant}n}(x_i-x_j)$ is the Vandermond determinant in variables $x_1,x_2,\cdots,x_n$.
If instead of the Poisson random walk, we start from $n$ independent Brownian motions with mean $0$ and variance $t/n$, then the same conditioning leads to the celebrated *Dyson Brownian motion* with $\beta=2$, which describes the stochastic evolution of eigenvalues of a Hermitian matrix under independent Brownian motion of its entries. For general $\beta>0$, the *$\beta$-Dyson Brownian motion* ${{\bm{y}}}(t)=(y_1(t), y_2(t),\cdots, y_n(t))$ is a diffusion process solving $$\begin{aligned}
\label{e:DBM1}
{{\rm d}}y_i(t)=\sqrt{\frac{2}{\beta n}}{{\rm d}}{{\mathcal B}}_i(t)+\frac{1}{n}\sum_{j\neq i}\frac{1}{y_i(t)-y_j(t)}{{\rm d}}t,\quad i=1,2,\cdots, n,\end{aligned}$$ where $\{({{\mathcal B}}_1(t), {{\mathcal B}}_2(t),\cdots, {{\mathcal B}}_n(t))\}_{t{\geqslant}0}$ are independent standard Brownian motions, and $\{{{\bm{y}}}(t)\}_{t>0}$ lives on the Weyl chamber ${\mathbb{W}}^n=\{({\lambda}_1,{\lambda}_2,\cdots,{\lambda}_n): {\lambda}_1>{\lambda}_2>\cdots>{\lambda}_n\}$.
The nonintersecting Poisson random walk can be viewed as a discrete version of the Dyson Brownian motion with $\beta=2$. For general $\beta>0$, we fix $\theta=\beta/2$ and define the *$\beta$-nonintersecting Poisson random walk*, denoted by ${{\bm{x}}}(t)=(x_1(t),x_2(t),\cdots,x_n(t))$, as a continuous time Markov process on $$\begin{aligned}
\label{e:defWtheta}
{\mathbb{W}}^n_\theta=\{({\lambda}_1+(n-1)\theta,{\lambda}_2+(n-2)\theta,\cdots,{\lambda}_n): ({\lambda}_1,{\lambda}_2,\cdots,{\lambda}_n)\in {\mathbb{Z}}_{{\geqslant}0}^n, {\lambda}_1{\geqslant}{\lambda}_2{\geqslant}\cdots{\geqslant}{\lambda}_n{\geqslant}0\},\end{aligned}$$ with generator $$\begin{aligned}
\label{e:generator}
{{{\mathcal}L}}^n_\theta f({{\bm{x}}})=\theta n\sum_{i=1}^n\frac{V({{\bm{x}}}+{\bm{e}}_i)}{V({{\bm{x}}})}\left(f({{\bm{x}}}+{\bm{e}}_i)-f({{\bm{x}}})\right)=\theta n\sum_{i=1}^{n}\left(\prod_{j:j\neq i}\frac{x_i-x_j+\theta}{x_i-x_j}\right)\left(f({{\bm{x}}}+{\bm{e}}_i)-f({{\bm{x}}})\right).\end{aligned}$$
In the beautiful article [@MR3418747], Gorin and Shkolnikov constructed certain multilevel discrete Markov chains whose top level dynamics coincide with the $\beta$-nonintersecting Poisson random walks. However, we use slightly different notations, and speed up time by $n$. In [@MR3418747], the $\beta$-nonintersecting Poisson random walks are constructed as stochastic dynamics on Young diagrams. We recall that a Young diagram $\bm\lambda$, is a non-increasing sequence of integers $$\begin{aligned}
\bm\lambda=({\lambda}_1,{\lambda}_2,{\lambda}_3, \cdots), \quad \lambda_1{\geqslant}\lambda_2{\geqslant}{\lambda}_3{\geqslant}\cdots{\geqslant}0.\end{aligned}$$ We denote $\ell_{{\bm{\lambda}}}$ the number of non-empty rows in ${\bm{\lambda}}$, i.e. ${\lambda}_{\ell_{{\bm{\lambda}}}}>0, {\lambda}_{\ell_{{\bm{\lambda}}}+1}={\lambda}_{\ell_{{\bm{\lambda}}}+2}=\cdots =0$, and $|{\bm{\lambda}}|=\sum_{i=1}^{\ell_{{\bm{\lambda}}}}{\lambda}_i$ the number of boxes in ${\bm{\lambda}}$. Let ${\mathbb{Y}}^n$ denote the set of all Young diagrams with at most $n$ rows, i.e. $\ell_{{\bm{\lambda}}}{\leqslant}n$. A box $\Box\in {\bm{\lambda}}$ is a pair of integers, $$\begin{aligned}
\Box=(i,j)\in {\bm\lambda}, \text{ if and only if } 1{\leqslant}i{\leqslant}\ell_\lambda, 1{\leqslant}j{\leqslant}\lambda_i.\end{aligned}$$ We denote ${\bm{\lambda}}'$ the transposed diagram of $\bm\lambda$, defined by $$\begin{aligned}
\lambda_j'=|\{i: 1{\leqslant}j{\leqslant}\lambda_i\}|, \quad 1{\leqslant}j{\leqslant}{\lambda}_1.\end{aligned}$$ For a box $\Box=(i,j)\in {\bm{\lambda}}$, its arm $a_\Box$, leg $l_\Box$, co-arm $a_\Box'$ and co-leg $l_\Box'$ are $$\begin{aligned}
a_\Box=\lambda_i-j,\quad l_\Box=\lambda_j'-i,\quad a_\Box'=j-1,\quad l_\Box'=i-1.\end{aligned}$$
Given a $\beta$-nonintersecting Poisson random walk ${{\bm{x}}}(t)$, we can view it as a growth process on ${\mathbb{Y}}^n$, by defining ${\bm{\lambda}}(t)$ by $$\begin{aligned}
\label{e:defla}
{\lambda}_i(t)=x_i(t)-(n-i)\theta, \quad 1{\leqslant}i{\leqslant}n.\end{aligned}$$ Since ${{\bm{x}}}(t)\in {\mathbb{W}}_\theta^n$, we have ${\lambda}_1(t){\geqslant}{\lambda}_2(t){\geqslant}\cdots{\geqslant}{\lambda}_n(t){\geqslant}0$, and thus ${\bm{\lambda}}(t)$ is a continuous time Markov process on ${\mathbb{Y}}^n$. Its jump rate is given in [@MR3418747 Proposition 2.25] rescaled by $n$, which, after simplification, is the same as . There is a simple formula for the transition probability of ${\bm{\lambda}}(t)$ with zero initial data [@MR3418747 Proposition 2.9, 2.28]. However, there are no simple formulas for the transition probabilities of ${\bm{\lambda}}(t)$ with general initial data.
\[t:density\] Suppose the initial data of ${\bm{\lambda}}(t)$ is the empty Young diagram. Then for any fixed $t>0$, the law of ${\bm{\lambda}}(t)$ is given by $$\begin{aligned}
\label{e:density}
{\mathbb{P}}_t({\lambda}_1,{\lambda}_2,\cdots, {\lambda}_n)=
e^{-\theta t n^2}(\theta tn)^{|{\bm{\lambda}}|}\prod_{\Box\in {\bm{\lambda}}}\frac{\theta n+a_\Box'-\theta l_\Box'}{(a_\Box+\theta l_\Box+\theta)(a_\Box+\theta l_\Box +1)}.\end{aligned}$$
It is proven in [[@MR3418747 Theorem 3.2]]{} that the Markov process $\bm{\lambda}(t)$ converges in the diffusive scaling limit to the $\beta$-Dyson Brownian motion.
Fix $\theta=\beta/2{\geqslant}1/2$ and let $\varepsilon>0$ be a small parameter. Let ${{\bm{x}}}(t)$ be the $\beta$-nonintersecting Poisson random walk starting at ${{\bm{x}}}(0)\in {\mathbb{W}}^n_\theta$. We define $\bm{\lambda}(t)$ as in and the rescaled stochastic process $\bm{\lambda}^\varepsilon(t)=({\lambda}_1^{{\varepsilon}}(t),{\lambda}_2^{{\varepsilon}}(t),\cdots, {\lambda}_n^{{\varepsilon}}(t))$ be defined through, $$\begin{aligned}
{\lambda}_i^{{\varepsilon}}(t){\mathrel{\mathop:}=}\varepsilon ^{1/2}\left(\frac{{\lambda}_i(t/\varepsilon)}{\theta n}-\frac{t}{\varepsilon }\right),\quad i=1,2,\cdots, n.
$$ Suppose that as ${{\varepsilon}}\rightarrow 0$, the initial data $\bm{\lambda}^{{\varepsilon}}(0)$ converges to a point ${{\bm{y}}}(0)\in {\mathbb{W}}^n$. Then the process $\bm{\lambda}^{{\varepsilon}}(t)$ converges in the limit ${{\varepsilon}}\rightarrow 0$ weakly in the Skorokhod topology towards the $\beta$-Dyson Brownian motion ${{\bm{y}}}(t)=(y_1(t), y_2(t),\cdots, y_n(t))$ as in .
Notations
---------
Throughout this paper, we use the following notations:
We denote ${{\mathbb R}}$ the set of real numbers, ${{\mathbb C}}$ the set of complex numbers, ${{\mathbb C}}_+$ the set of complex numbers with positive imaginary parts, ${{\mathbb C}}_-$ the set of complex numbers with negative imaginary parts, ${\mathbb{Z}}$ the set of integers, and ${\mathbb{Z}}_{{\geqslant}0}$ the set of non-negative integers.
We denote $M_1({{\mathbb R}})$ the space of probability measures on ${{\mathbb R}}$ equipped with the weak topology. A metric compatible with the weak topology is the *L[é]{}vy metric* defined by $$\begin{aligned}
{\rm{dist}}(\mu,\nu){\mathrel{\mathop:}=}\inf_{\varepsilon}\{\mu(-\infty, x-\varepsilon)-\varepsilon{\leqslant}\nu(-\infty, x){\leqslant}\mu(-\infty, x+\varepsilon)+\varepsilon \text{ for all $x$}\}.\end{aligned}$$
Let $({\mathbb{M}},\operatorname{dist}(\cdot,\cdot))$ be a metric space, either ${{\mathbb C}}^m$ or ${{\mathbb R}}^m$ with the Euclidean metric or $M_1({{\mathbb R}})$ with the L[é]{}vy metric. The set of c[á]{}dl[á]{}g functions, i.e. functions which are right continuous with left limits, from $[0,T]$ to ${\mathbb{M}}$ is denoted by $D([0,T],{\mathbb{M}})$ and is called the *Skorokhod space*. Let $\Lambda$ denote the set of all strictly increasing, continuous bijections from $[0,T]$ to $[0,T]$. The *Skorokhod metric* on $D([0,T],{\mathbb{M}})$ is defined by $$\begin{aligned}
\operatorname{dist}(f,g)=\inf_{{\lambda}\in \Lambda}\max\left\{\sup_{0{\leqslant}t{\leqslant}T}|{\lambda}(t)-t|, \sup_{0{\leqslant}t{\leqslant}T}\operatorname{dist}(f(t), g({\lambda}(t)))\right\}.\end{aligned}$$ Let $Z^n$, $Z$ be random variables taking value in the Skorokhod space $D([0,T],{\mathbb{M}})$. We say $Z_n$ converges *almost surely* towards $Z$, if $Z_n\rightarrow Z$ in $D([0,T], {\mathbb{M}})$ for the Skorokhod metric almost surely. We say $Z_n$ *weakly converges* towards $Z$, denoted by $Z^n\Rightarrow Z$, if for all bounded Skorokhod continuous functions $f$, $$\begin{aligned}
\lim_{n\rightarrow \infty}{\mathbb{E}}[f(Z^n)]\rightarrow {\mathbb{E}}[f(Z)].\end{aligned}$$ We refer to [@MR1431297 Chapter 1] and [@MR1876437 Chapter 3] for nice presentations on weak convergence of stochastic processes in the Skorokhod space.
A *random field* is a collection of random variables indexed by elements in a topological space. If the collection of random variables are jointly Gaussian, we call it a *Gaussian random field*. Let $(g^n(z))_{z\in \Omega}$, $(g(z))_{z\in \Omega}$ be ${{\mathbb C}}$-valued random fields indexed by an open subset $\Omega\subset {{\mathbb C}}\setminus{{\mathbb R}}$. We say $(g^n(z))_{z\in \Omega}$ weakly converges towards $(g(z))_{z\in \Omega}$ in the sense of finite dimensional distributions, if for any $z_1,z_2,\cdots, z_m\in \Omega$ the random vector $(g^n(z_j))_{1{\leqslant}j{\leqslant}m}$ weakly converges to $(g(z_j))_{1{\leqslant}j{\leqslant}m}$. Let $\{(g_t^n(z))_{z\in \Omega}\}_{0{\leqslant}t{\leqslant}T}$, $\{(g_t(z))_{z\in \Omega}\}_{0{\leqslant}t{\leqslant}T}$ be random field valued random processes. We say $\{(g_t^n(z))_{z\in \Omega}\}_{0{\leqslant}t{\leqslant}T}$ weakly converges towards $\{(g_t(z))_{z\in \Omega}\}_{0{\leqslant}t{\leqslant}T}$ in the sense of finite dimensional processes, if for any $z_1,z_2,\cdots, z_m\in \Omega$ the random process $\{(g_t^n(z_j))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$ weakly converges to $\{(g_t(z_j))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$ in the Skorokhod space $D([0,T], {{\mathbb C}}^m)$.
Main results
------------
In this paper, we study the asymptotic behavior of the $\beta$-nonintersecting Poisson random walks, as the number of particles $n$ goes to infinity.
We consider $\beta$-nonintersecting Poisson random walks ${{\bm{x}}}(t)=(x_1(t), x_2(t),\cdots, x_n(t))$, with initial data ${{\bm{x}}}(0)=(x_1(0), x_2(0),\cdots, x_n(0))$. We define the empirical measure process $$\begin{aligned}
\label{e:defemp}
\mu^n_t=\frac{1}{n}\sum_{i=1}^n \delta_{x_i(t)/\theta n},\end{aligned}$$ which can be viewed as a random element in $D([0,T], M_1({{\mathbb R}}))$, the space of right-continuous with left limits processes from $[0,T]$ into the space $M_1({{\mathbb R}})$ of probability measures on ${{\mathbb R}}$.
The law of large numbers theorem states that the empirical measure process $\{\mu_t^n\}_{0{\leqslant}t{\leqslant}T}$ converges in $D([0,T], M_1({{\mathbb R}}))$. We need to assume that the initial empirical measure $\mu_0^n$ converges in the L[é]{}vy metric as $n$ goes to infinity in $M_1({{\mathbb R}})$.
We denote the Stieltjes transform of the empirical measure at time $t$ as $$\begin{aligned}
\label{e:defmt}
m^n_t(z)=\frac{1}{n}\sum_{i=1}^{n}\frac{1}{x_i(t)/\theta n-z}=\int \frac{{{\rm d}}\mu^n_t(x)}{x-z},\end{aligned}$$ where $z\in {{\mathbb C}}\setminus {{\mathbb R}}$.
\[t:LLN\] Fix $\theta>0$. We assume that there exists a measure $\mu_0\in M_1({{\mathbb R}})$, such that the initial empirical measure $\mu_0^n$ converges in the L[é]{}vy metric as $n$ goes to infinity towards $\mu_0$ almost surely (in probability). Then, for any fixed time $T>0$, $\{\mu^n_t\}_{0{\leqslant}t{\leqslant}T}$ converges as $n$ goes to infinity in $D([0,T], M_1({{\mathbb R}}))$ almost surely (in probability). Its limit is the unique measure-valued process $\{\mu_t\}_{0{\leqslant}t{\leqslant}T}$, so that the density satisfies $0{\leqslant}{{\rm d}}\mu_t(x)/{{\rm d}}x{\leqslant}1$, and its Stieltjes transform $$\begin{aligned}
\label{e:defmt}
m_t(z)=\int \frac{{{\rm d}}\mu_t(x)}{x-z},\end{aligned}$$ satisfies the equation $$\begin{aligned}
\label{e:limitMST}
m_t(z)=m_0(z)-\int_0^t e^{- m_s(z)}{\partial}_z m_s(z){{\rm d}}s,\end{aligned}$$ for $z\in {{\mathbb C}}\setminus {{\mathbb R}}$.
Assumption in Theorem \[t:LLN\] is equivalent to that the Stieltjes transform of the initial empirical measure $$\begin{aligned}
\lim_{n\rightarrow \infty} m^n_0(z)=\int \frac{{{\rm d}}\mu_0(x)}{x-z}{=\mathrel{\mathop:}}m_0(z),\end{aligned}$$ almost surely (in probability), for any $z\in {{\mathbb C}}\setminus {{\mathbb R}}$.
The Stieltjes transform of $ \mu_t$ is characterized by , $$\begin{aligned}
\label{e:dif1}
{\partial}_t m_t(z)=-e^{-m_t(z)}{\partial}_z m_t(z)={\partial}_z (e^{- m_t(z)}).\end{aligned}$$ This is a complex Burgers type equation, and can be solved by the method of characteristics. We define the characteristic lines, $$\begin{aligned}
\label{e:dif2}
{\partial}_t z_t(z)=e^{- m_t( z_t(z))}, \quad z_0(z)=z.\end{aligned}$$ If the context is clear, we omit the parameter $z$, i.e. we simply write $z_t$ instead of $z_t(z)$. Plugging into , and applying the chain rule we obtain $
{\partial}_t m_t(z_t)=0.
$ It implies that $ m_t(z)$ is a constant along the characteristic lines, i.e. $m_t(z_t(z))=m_0(z_0(z))=m_0(z)$. And the solution of the differential equation is given by $$\begin{aligned}
\label{e:deft}
z_t(z)=z+te^{-m_0(z)}, \quad 0{\leqslant}t< -\frac{{\mathop{\mathrm{Im}}}[z]}{{\mathop{\mathrm{Im}}}[e^{-m_0(z)}]}{=\mathrel{\mathop:}}{{\frak t}}(z).\end{aligned}$$ We conclude that the Stieltjes transform $m_t(z)$ is given by $$\begin{aligned}
\label{e:tmtrelation}
m_t(z+te^{-m_0(z)})=m_0(z).\end{aligned}$$ Later we will prove that for any time $t{\geqslant}0$, there exists an open set $\Omega_t\subset{{\mathbb C}}\setminus {{\mathbb R}}$ defined in , such that $z_t(z)=z+te^{-m_0(z)}$ is conformal from $\Omega_t$ to ${{\mathbb C}}\setminus {{\mathbb R}}$, and is a homeomorphism from the closure of $\Omega_t\cap {{\mathbb C}}_+$ to ${{\mathbb C}}_+\cup {{\mathbb R}}$, and from the closure of $\Omega_t\cap {{\mathbb C}}_-$ to ${{\mathbb C}}_-\cup {{\mathbb R}}$.
The central limit theorem states that the rescaled empirical measure process $\{n(\mu_t^n-\mu_t)\}_{0{\leqslant}t{\leqslant}T}$ weakly converges in the space of distributions acting on analytic test functions to a Gaussian process. We need to assume that the rescaled initial empirical measure $n(\mu_0^n-\mu_0)$ weakly converges to a measure.
We define the rescaled fluctuation process $$\begin{aligned}
\label{e:defgt}
g_t^n(z)=n(m_t^n(z)- m_t(z))=n\int \frac{{{\rm d}}(\mu_t^n(x)- \mu_t(x))}{x-z},\end{aligned}$$ which characterizes the behaviors of the rescaled empirical measure process $\{n(\mu_t^n-\mu_t)\}_{0{\leqslant}t{\leqslant}T}$.
\[a:initiallaw\] We assume there exists a constant ${{\frak a}}$, such that that for any $z\in {{\mathbb C}}\setminus{{\mathbb R}}$, $$\begin{aligned}
{\mathbb{E}}\left[|g_0^n(z)|^2\right]{\leqslant}{{\frak a}}({\mathop{\mathrm{Im}}}[z])^{-2},\end{aligned}$$ and the random field $(g_0^n(z))_{z\in {{\mathbb C}}\setminus{{\mathbb R}}}$ weakly converges to a deterministic field $(g_0(z))_{z\in {{\mathbb C}}\setminus {{\mathbb R}}}$, in the sense of finite dimensional distributions.
Assumption \[a:initiallaw\] implies that the initial empirical measure $\mu_0^n$ converges in the L[é]{}vy metric as $n$ goes to infinity towards $\mu_0$ in probability.
\[t:CLT\] Fix $\theta>0$. We assume Assumption \[a:initiallaw\]. Then for any fixed time $T>0$, the process $\{(g^n_t(z_t(z)))_{z\in\Omega_T}\}_{0{\leqslant}t{\leqslant}T}$ converges weakly towards a Gaussian process $\{(g_t(z_t(z)))_{z\in \Omega_T}\}_{0{\leqslant}t{\leqslant}T}$, in the sense of finite dimensional processes, with initial data $(g_0(z))_{z\in {{\mathbb C}}\setminus{{\mathbb R}}}$ given in Assumption \[a:initiallaw\], means $$\begin{aligned}
\label{e:mean0}
{\mathbb{E}}[g_t(z_t(z))]&=\mu(t, z){\mathrel{\mathop:}=}\frac{g_0(z)}{1-t{\partial}_zm_0(z)e^{-m_0(z)}}
+\left(\frac{1}{2}-\frac{1}{2\theta}\right)\frac{t(({\partial}_z m_0(z))^2-{\partial}_z^2 m_0(z))e^{-m_0(z)}}{(1-t{\partial}_z m_0(z)e^{-m_0(z)})^2},\end{aligned}$$ and covariances $$\begin{aligned}
\begin{split}\label{e:variance0}
{{\rm{cov}}}[g_s(z_s(z)), g_t(z_t(z'))]&=\sigma(s, z, t, z'){\mathrel{\mathop:}=}\frac{1}{\theta}\frac{1}{(1-s{\partial}_zm_0(z)e^{-m_0(z)})(1-t{\partial}_zm_0(z')e^{-m_0(z')})}\\
&\times\left(\frac{1}{(z-z')^2}-\frac{(1-(s\wedge t){\partial}_zm_0(z)e^{-m_0(z)})(1-(s\wedge t){\partial}_zm_0(z')e^{-m_0(z')})}{(z-z'+(s\wedge t)(e^{-m_0(z)}-e^{-m_0(z')}))^2}\right)\\
{{\rm{cov}}}[g_s(z_s(z)), \overline{g_t(z_t(z'))}]&=\sigma(s, z,t,\bar z'),
\end{split}\end{aligned}$$ where $$\begin{aligned}
\sigma(s, z,t,z){\mathrel{\mathop:}=}\lim_{z'\rightarrow z}\sigma(s,z,t,z')
&=\frac{(s\wedge t)e^{-m_0(z)}(2({\partial}_z m_0(z))^3-6{\partial}_z m_0(z){\partial}_z^2 m_0(z)+2{\partial}_z^3m_0(z))}{12\theta(1-(s\wedge t){\partial}_z m_0(z)e^{-m_0(z)})^3(1-(s\vee t){\partial}_z m_0(z)e^{-m_0(z)})}\\
&+\frac{(s\wedge t)^2e^{-2m_0(z)}(({\partial}_z m_0(z))^4+3({\partial}_z^2 m_0(z))^2-2{\partial}_z m_0(z){\partial}_z^3m_0(z))}{12\theta(1-(s\wedge t){\partial}_z m_0(z)e^{-m_0(z)})^3(1-(s\vee t){\partial}_z m_0(z)e^{-m_0(z)})}.\end{aligned}$$
We can rewrite the means and covariances in terms of the characteristic lines $z_t(z)$: $$\begin{aligned}
\begin{split}\label{e:meanandvar}
\mu(t,z)&=\frac{g_0(z)}{{\partial}_z z_t(z)}+\left(\frac{1}{2}-\frac{1}{2\theta}\right)\frac{{\partial}_z^2 z_t(z)}{({\partial}_z z_t(z))^2},\\
\sigma(s, z, t,z')&=\frac{1}{\theta}\frac{1}{{\partial}_zz_s(z){\partial}_zz_t(z')}\left(\frac{1}{(z-z')^2}-\frac{{\partial}_z z_{s\wedge t}(z){\partial}_z z_{s\wedge t}(z')}{(z_{s\wedge t}(z)-z_{s\wedge t}(z'))^2}\right).
\end{split}\end{aligned}$$ We will prove in Section \[s:compareDBM\], the means and the covariances are universal, and coincide with those of $\beta$-Dyson Brownian motions with initial data constructed by the Markov-Krein correspondence.
To study the fluctuation of the rescaled empirical measure process $\{n(\mu_t^n-\mu_t)\}_{0{\leqslant}t{\leqslant}T}$ with analytic functions as test functions, we need to assume that the extreme particles are bounded.
\[a:ibound\] We assume there exists a large number ${{\frak b}}$, such that $$\begin{aligned}
\label{e:ibound}
{{\frak b}}n{\geqslant}x_1(0){\geqslant}x_2(0){\geqslant}\cdots {\geqslant}x_n(0).\end{aligned}$$
\[t:CLT2\] Fix $\theta>0$. We assume Assumptions \[a:initiallaw\] and \[a:ibound\]. Then for any fixed time $T>0$ and real analytic functions $f_1, f_2, \cdots, f_m$ on ${{\mathbb R}}$, the random process $$\begin{aligned}
\left\{\left(n\int f_j(x){{\rm d}}(\mu_{t}^{n}(x)-\mu_{t}(x))\right)_{1{\leqslant}j{\leqslant}m}\right\}_{0{\leqslant}t{\leqslant}T},\end{aligned}$$ converges as $n$ goes to infinity in $D([0,T], {{\mathbb R}}^m)$ weakly towards a Gaussian process $\{({{{\mathcal}F}}_j(t))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$, with means and covariances $$\begin{aligned}
{\mathbb{E}}[{{{\mathcal}F}}_j(t)]&=\frac{1}{2\pi{\mathrm{i}}}\oint_{{{\mathcal}C}}\mu(t, z_t^{-1}(w))f_j(w){{\rm d}}w,\\
{{\rm{cov}}}[{{{\mathcal}F}}_j(s), {{{\mathcal}F}}_k(t)]&=-\frac{1}{4\pi^2}\oint_{{{\mathcal}C}}\oint_{{{\mathcal}C}}\sigma(s, z_s^{-1}(w),t, z_t^{-1}(w'))f_j(w)f_k(w'){{\rm d}}w{{\rm d}}w',\end{aligned}$$ where the contours are sufficiently large depending on ${{\frak b}}$.
We prove in Proposition \[p:extremePbound\] that with exponentially high probability all particles $x_i(t)$ are inside an interval $[0, {{\frak c}}n]$. The contours in Theorem \[t:CLT2\] encloses a neighborhood of $[0,{{\frak c}}/\theta]$. Further, it is enough to assume in Theorem \[t:CLT2\] that $f_j$ are analytic only in a neighborhood of $[0,{{\frak c}}/\theta]$.
Related results
---------------
For the $\beta$-Dyson Brownian motion , the asymptotic behavior of the empirical measure process was studied in [@MR1217451; @MR1176727]. They found that the empirical measure process $$\begin{aligned}
\label{e:empirical}
\tilde \mu_t^n=\frac{1}{n}\sum_{i=1}^n \delta_{y_i(t)},\quad 1{\leqslant}i{\leqslant}n,\end{aligned}$$ converges weakly in the space of continuous measure-valued processes to a deterministic process $\tilde \mu_t$, characterized by the free convolution with semi-circle distributions. It was proven in [@MR1819483], that the rescaled empirical measure process $n(\tilde \mu_t^n-\tilde \mu_t)$ converges weakly in the space of distributions acting on a class of $C^6$ test functions to a Gaussian process, provided that the initial distributions $n(\tilde\mu_0^n-\tilde\mu_0)$ converge. The explicit formulas of the means and the covariances of the limit Gaussian process was derived in [@MR2418256].
More generally, the $\beta$-Dyson Brownian motion ${{\bm{y}}}(t)=(y_1(t), y_2(t),\cdots, y_n(t))$ with potential $V$ is given by $$\begin{aligned}
{{\rm d}}y_i(t)=\sqrt{\frac{2}{\beta n}}{{\rm d}}{{\mathcal B}}_i(t)+\frac{1}{n}\sum_{j\neq i}\frac{1}{y_i(t)-y_j(t)}{{\rm d}}t -\frac{1}{2}V'(y_i(t)){{\rm d}}t,\quad i=1,2,\cdots, n,\end{aligned}$$ where $\{({{\mathcal B}}_1(t), {{\mathcal B}}_2(t), \cdots, {{\mathcal B}}_n(t))\}_{t{\geqslant}0}$ are independent standard Brownian motions. It was proven in [@GDBM1; @GDBM2], that under mild conditions on $V$, the empirical measure process converges to a $V$-dependent measure-valued process, which can be realized as the gradient flow of the Voiculescu free entropy on the Wasserstein space over ${{\mathbb R}}$. The central limit theorem of the rescaled empirical measure process was proven in [@Un] for $\beta>1$ and sufficiently regular convex potential $V$.
The Wigner-Dyson-Mehta conjecture stated that the eigenvalue correlation functions of a general class of random matrices converge to the corresponding ones of Gaussian matrices. The Dyson Brownian motion plays a central role in the three-step approach to the universality conjecture in a series of works [@Landon2016; @fix; @MR2919197; @MR3429490; @MR3372074; @MR2810797; @kevin3], developed by Erd[ő]{}s, Yau and their collaborators. Parallel results were established in certain cases in [@MR2669449; @MR2784665], with a four moment comparison theorem.
The transition probability of the $\beta$-nonintersecting Poisson random walks with the fully-packed initial data $x_i(0)=(n-i)\theta$, $1{\leqslant}i{\leqslant}n$, is a discrete $\beta$ ensemble with Charlier weight. The discrete $\beta$ ensembles with general weights were introduced in [@MR3668648], which is a probability distribution $$\begin{aligned}
\label{e:disc}
{\mathbb{P}}_n(\ell_1,\ell_2,\cdots, \ell_n)=\frac{1}{Z_n}\prod_{1{\leqslant}i<j{\leqslant}n}\frac{\Gamma(\ell_i-\ell_j+1)\Gamma(\ell_i-\ell_j+\theta)}{\Gamma(\ell_i-\ell_j)\Gamma(\ell_i-\ell_j+1-\theta)}\prod_{i=1}^nw(\ell_i; N),\end{aligned}$$ on ordered $n$-tuples $\ell_1>\ell_2>\cdots \ell_n$ such that $\ell_i={\lambda}_i+(n-i)\theta$ and ${\lambda}_1{\geqslant}{\lambda}_2{\geqslant}\cdots{\geqslant}{\lambda}_N$ are integers. The discrete $\beta$ ensembles are discretizations for the $\beta$ ensembles of random matrix theory, which are probability distributions on $n$ tuples of reals $y_1>y_2>\cdots>y_n$, $$\begin{aligned}
\label{e:betaensemble}
{\mathbb{P}}_n(y_1,y_2,\cdots, y_n)=\frac{1}{Z_n}\prod_{1{\leqslant}i<j{\leqslant}n}|y_i-y_j|^\beta\prod_{i=1}^ne^{-nV(y_i)},\end{aligned}$$ where the potential $V$ is a continuous function. Under mild assumptions on the potential $V$, the $\beta$ ensembles exhibit a law of large number, i.e., the empirical measure $$\begin{aligned}
\mu^n=\frac{1}{n}\sum_{i=1}^n \delta_{y_i},\end{aligned}$$ converges to a non-random equilibrium measure $\mu$. For $\beta=1,2,4$ and $V(y)=y^2$, this statement dates back to the original work of Wigner [@MR0077805; @MR0083848]. We refer to [@MR2760897 Chapter 2.6] for the study of the $\beta$ ensembles with general $V$. In the breakthrough paper [@MR1487983], Johansson introduced the loop (or Dyson-Schwinger) equations to the mathematical community, and proved that the rescaled empirical measure satisfies a central limit theorem, i.e., for sufficiently smooth functions $f(y)$ the random variable $$\begin{aligned}
n\int f(x) ({{\rm d}}\mu^n(x)-{{\rm d}}\mu(x)).\end{aligned}$$ converges to a Gaussian random variable. We refer to [@MR3010191; @borot-guionnet2; @KrSh] for further development. The law of large numbers and the central limit theorems of the discrete $\beta$ ensemble were proven in [@MR3668648], using a discrete version of the loop equations [@Nekrasov].
In the special case when $\beta=2$, the central limit theorem for the global fluctuations of the nonintersecting Poisson random walk were obtained by various methods. For the fully-packed initial data, the central limit theorem was established in [@MR3148098] by the technique of determinantal point processes, in [@MR3552537; @MR3263029] by computations in the universal enveloping algebra of $U(N)$ and in [@Dui; @MR3556288] by employing finite term recurrence relations of orthogonal polynomials. For general initial data, the law of large numbers and the central limit theorems were proven in [@MR3361772; @BuGo], where the Schur generating functions were introduced to study random discrete models. Our results give a new proof of these results based on the dynamical approach.
Organization of the paper
-------------------------
In Section \[s:qfc\], we recall the quantized free convolution as introduced in [@MR3361772]. We show that the limit measure-valued process $\mu_t$ is characterized by the quantized free convolution. In Section \[s:compareDBM\], we compare the central limit theorems of the $\beta$-nonintersecting Poisson random walks with those of the $\beta$-Dyson Brownian motions. It turns out that the means and covariances of the limit fluctuation process coincide under Markov-Krein correspondence. In Section \[s:sde\], we collect some properties of the generator ${{{\mathcal}L}}_\theta^n$ of the $\beta$-nonintersecting Poisson random walks, and derive a stochastic differential equation of the Stieltjes transform of the empirical measure process, which relies on the integrable features of the generator. The stochastic differential equation can be viewed as a dynamical version of the Nekrasov’s equation in [@MR3668648 Section 4], which is crucial for the proof of central limit theorems of the discrete $\beta$ ensembles. In Section \[s:LLN\] and \[s:CLT\], we prove the law of large numbers and central limit theorem of the $\beta$-nonintersecting Poisson random walks. We directly analyze the stochastic differential equation satisfied by the Stieltjes transform using the method of characteristics as in [@HL], where the method of characteristics was used to derive the rigidity of the Dyson Brownian motion. Since the $\beta$-nonintersecting Poisson random walks are jump processes, the analysis is more sophisticated than that of the Dyson Brownian motion. In Section \[s:extremep\] we derive an estimate of the locations of extreme particles, by a coupling technique, and prove the central limit theorem with analytic test functions.
Finally we remark that by analyzing the stochastic differential equation of the Stieltjes transform of the empirical measure process as in [@HL], one can prove the optimal rigidity estimates and a mesoscopic central limit theorem for the $\beta$-nonintersecting Poisson random walks.
**Acknowledgement.** The author heartily thanks Vadim Gorin for constructive comments on the draft of this paper.
Law of large numbers for the empirical measure process
======================================================
Quantized free convolution {#s:qfc}
--------------------------
In this section we study the limit measure-valued process $ \mu_t$. To describe it, we need the concept of quantized free convolution as introduced in [@MR3361772]. The quantized free convolution is a quantized version of the free convolution originally defined by Voiculescu [@MR799593; @MR839105] in the setting of operator algebras. Given a probability measure $\mu$, we denote its Stieltjes transform by $m_\mu(z)=\int {{\rm d}}\mu(x)/(x-z)$, for any $z\in {{\mathbb C}}\setminus{{\mathbb R}}$. The $R$-transform is defined as $$\begin{aligned}
\label{e:defRf}
R_\mu(z){\mathrel{\mathop:}=}m_\mu^{-1}(-z)-\frac{1}{z},\end{aligned}$$ where $m_\mu^{-1}(z)$ is the functional inverse of $m_\mu(z)$, i.e. $m_\mu(m_{\mu}^{-1}(z))=m_{\mu}^{-1}(m_\mu(z))=z$. The free convolution is a unique operation on probability measures $(\mu, \nu)\mapsto \mu\boxplus\nu$, which agrees with the addition of the $R$-transforms: $$\begin{aligned}
R_\mu(z)+R_{\nu}(z)=R_{\mu\boxplus\nu}(z)\end{aligned}$$ It was proven in [@MR1094052] that the asymptotic distribution of eigenvalues of sums of independent random matrices is given by the free convolution.
The quantized free convolution is an operation on probability measures which have bounded by $1$ density with respect to the Lebesgue measure. One gets the quantized free convolution by replacing the $R$-transform in with the quantized $R$-transform $$\begin{aligned}
\label{e:defR}
R_\mu^{\text{quant}}(z){\mathrel{\mathop:}=}m_\mu^{-1}(-z)-\frac{1}{1-e^{-z}}.\end{aligned}$$ The quantized free convolution is a unique operation on probability measures $(\mu, \nu)\mapsto \mu\otimes\nu$, which agrees with the addition of the quantized $R$-transforms: $$\begin{aligned}
R_\mu^{\text{quant}}(z)+R_{\nu}^{\text{quant}}(z)=R_{\mu\otimes\nu}^{\text{quant}}(z)\end{aligned}$$ It was proven in [@MR3361772 Theorem 1.1] that the quantized free convolution characterizes the tensor product of two irreducible representation of unitary group.
The Markov-Krein correspondence [@MR1618739; @MR0167806] gives an exact relationship between the free convolution and the quantized free convolution.
\[t:MKcor\] For every probability measure $\mu$ on ${{\mathbb R}}$ which has bounded by $1$ density with respect to the Lebesgue measure, there exists a probability measure $Q(\mu)$ such that $$\begin{aligned}
\label{e:defQ}
m_{Q(\mu)}(z)=1-e^{-m_{\mu}(z)},\end{aligned}$$ where $m_\mu(z)$ and $m_{Q(\mu)}(z)$ are Stieltjes transforms of $\mu$ and $Q(\mu)$ respectively. We denote the operator $$\begin{aligned}
\tilde Q(\mu)=r\circ Q \circ r(\mu),\end{aligned}$$ where $r$ is the reflection of a measure with respect to the origin. The operator $\tilde Q$ intertwines the free convolution and the quantized free convolution, i.e. for any two probability measures $\mu_1, \mu_2$ as above, we have $$\begin{aligned}
\tilde Q(\mu_1\otimes\mu_2)=\tilde Q(\mu_1)\boxplus \tilde Q(\mu_2).\end{aligned}$$
Theorem \[t:MKcor\] essentially reduces the quantized free convolution to the free convolution. Properties of the quantized free convolution, e.g., existence and uniqueness, follow from their counterparts of the free convolution. We sketch the construction of the operator $Q$ in Remark \[r:proofMKcor\] in Section \[s:sde\].
The limit measure-valued process $ \mu_t$ can be described by the quantized free convolution. We denote $ R_t^{\text{quant}}(z)$ the quantized $R$-transform of the measure $ \mu_t$. From , we have $$\begin{aligned}
\left(m_t\right)^{-1}(z)=\left(m_0\right)^{-1}(z)+te^{-z},\end{aligned}$$ and $$\begin{aligned}
R_t^{\text{quant}}(z)= R_0^{\text{quant}}(z)+te^{z}.\end{aligned}$$ There exists a family of measures $ \nu_t$ such that the quantized $R$-transform of $ \nu_t$ is given by $te^{z}$. The Stieltjes transform $m_{\nu_t}(z)$ of $\nu_t$ is given by $$\begin{aligned}
z e^{2m_{\nu_t}(z)}+(1-t-z)e^{m_{\nu_t}(z)}+t=0.\end{aligned}$$ We can solve for $m_{\nu_t}(z)$, and the density of $ \nu_t$ is given by for $t{\leqslant}1$, $$\begin{aligned}
\label{e:defnu1}
{{\rm d}}\nu_t(x)/{{\rm d}}x=
\left\{
\begin{array}{cc}
\frac{1}{\pi} \text{arccot} \left(\frac{x+t-1}{\sqrt{4xt-(x+t-1)^2}}\right), &(1-\sqrt{t})^2{\leqslant}x{\leqslant}(\sqrt{t}+1)^2, \\
1,& x< (1-\sqrt{t})^2,\\
0, & x> (\sqrt{t}+1)^2,
\end{array}
\right.\end{aligned}$$ for $t>1$, $$\begin{aligned}
\label{e:defnu2}
{{\rm d}}\nu_t(x)/{{\rm d}}x=
\left\{
\begin{array}{cc}
\frac{1}{\pi} \text{arccot} \left(\frac{x+t-1}{\sqrt{4xt-(x+t-1)^2}}\right), &(1-\sqrt{t})^2{\leqslant}x{\leqslant}(\sqrt{t}+1)^2, \\
0, & x< (\sqrt{t}-1)^2 \text{ or } x> (\sqrt{t}+1)^2.
\end{array}
\right.\end{aligned}$$ We can conclude from the discussion above,
The limit measure $ \mu_t$ is the quantized free convolution of the initial measure $ \mu_0$ with the measure $ \nu_t$ as defined in and : $$\begin{aligned}
\mu_t= \mu_0\otimes \nu_t.\end{aligned}$$
In the rest of this section, we collect some properties of the Stieltjes transform $m_t$, the characteristic lines $z_t$ and the logarithmic potential $h_t$ of the measure $\mu_t$, $$\begin{aligned}
\label{e:defht}
h_t(z)=\int\log(x-z){{\rm d}}\mu_t(x), \quad z\in {{\mathbb C}}\setminus {{\mathbb R}}.\end{aligned}$$ We remark that ${\partial}_z h_t(z)=-m_t(z)$.
\[p:ztp\] For any time $t{\geqslant}0$, we define an open set $\Omega_t\subset{{\mathbb C}}\setminus {{\mathbb R}}$ $$\begin{aligned}
\label{e:defOmega}
\Omega_t{\mathrel{\mathop:}=}\left\{z\in {{\mathbb C}}\setminus {{\mathbb R}}: \int \frac{{{\rm d}}Q(\mu_0)(x)}{|x-z|^2}<\frac{1}{t}\right\},\end{aligned}$$ where the operator $Q$ is defined in Theorem \[t:MKcor\]. Then, $z_t(z)=z+te^{-m_0(z)}$ is conformal from $\Omega_t$ to ${{\mathbb C}}\setminus {{\mathbb R}}$, and is a homeomorphism from the closure of $\Omega_t\cap {{\mathbb C}}_+$ to ${{\mathbb C}}_+\cup {{\mathbb R}}$, and from the closure of $\Omega_t\cap {{\mathbb C}}_-$ to ${{\mathbb C}}_-\cup {{\mathbb R}}$. Moreover for any $z\in \Omega_t$, $|{\mathop{\mathrm{Im}}}[z_s]|$ is monotonically decreasing for $0{\leqslant}s{\leqslant}t$, i.e., $|{\mathop{\mathrm{Im}}}[z_s]|{\geqslant}|{\mathop{\mathrm{Im}}}[z_t]|$.
Thanks to Theorem \[t:MKcor\], we have $$\begin{aligned}
z_t(z)=z+te^{-m_0(z)}=z+t-tm_{Q(\mu_0)}(z),\end{aligned}$$ and the proposition follows from [@MR1488333 Lemma 4].
\[p:estimatemt\] Fix $T>0$. For any $0{\leqslant}t{\leqslant}T$ and $z\in \Omega_T$ as defined in , we have $$\begin{aligned}
\begin{split}\label{e:mtestimate}
&({\partial}_z m_t)(z_t(z))=\frac{{\partial}_zm_0(z)}{1-t{\partial}_z m_0(z)e^{-m_0(z)}},\quad
({\partial}^2_z m_t)(z_t(z))=\frac{{\partial}_z^2 m_0(z)-t({\partial}_zm_0(z))^3e^{-m_0(z)}}{(1-t{\partial}_z m_0(z)e^{-m_0(z)})^3},\\
&({\partial}_t m_t)(z_t(z))=-\frac{{\partial}_z m_0(z)e^{-m_0(z)}}{1-t{\partial}_z m_0(z)e^{-m_0(z)}},\quad ({\partial}_t h_t)(z_t(z))=-e^{-m_0(z)}.
\end{split}\end{aligned}$$
The first three relations follow directly by taking derivative of . For the last relation, we have $$\begin{aligned}
{\partial}_z(({\partial}_t h_t)(z_t(z)))
=-({\partial}_t m_t)(z_t){\partial}_z z_t(z)={\partial}_z(-e^{-m_0(z)}).\end{aligned}$$ The last relation follows by noticing that $\lim_{z\rightarrow\infty}({\partial}_t h_t)(z_t(z))=0$.
Comparing with $\beta$-Dyson Brownian motion {#s:compareDBM}
--------------------------------------------
In this section, we compare the central limit theorems of the $\beta$-nonintersecting random walks with those of the $\beta$-Dyson Brownian motion. For general $\beta>0$, we recall the $\beta$-Dyson Brownian motion ${{\bm{y}}}(t)=(y_1(t), y_2(t),\cdots, y_n(t))$ is a diffusion process solving $$\begin{aligned}
\label{e:DBM3}
{{\rm d}}y_i(t)=\sqrt{\frac{2}{\beta n}}{{\rm d}}{{\mathcal B}}_i(t)+\frac{1}{n}\sum_{j\neq i}\frac{1}{y_i(t)-y_j(t)}{{\rm d}}t +{{\rm d}}t,\quad i=1,2,\cdots, n,\end{aligned}$$ where $\{({{\mathcal B}}_1(t), {{\mathcal B}}_2(t),\cdots, {{\mathcal B}}_n(t))\}_{t{\geqslant}0}$ are independent standard Brownian motions, and $\{{{\bm{y}}}(t)\}_{t>0}$ lives on the Weyl chamber ${\mathbb{W}}^n=\{({\lambda}_1,{\lambda}_2,\cdots,{\lambda}_n): {\lambda}_1>{\lambda}_2>\cdots>{\lambda}_n\}$.
The expression is slightly different from . We add a constant drift term in , so that it matches with the dynamics of the $\beta$-nonintersecting Poisson random walks.
We denote the empirical measure process of , $$\begin{aligned}
\tilde \mu_t^n {\mathrel{\mathop:}=}\frac{1}{n}\sum_{i=1}^n \delta_{y_i(t)}.\end{aligned}$$ It follows from [@MR1217451; @MR1176727], if the initial empirical measure $\tilde \mu_0^n$ converges in the L[é]{}vy metric as $n$ goes to infinity towards a probability measure $\tilde \mu_0$ almost surely (in probability), then, for any fixed time $T>0$, $\{\tilde \mu^n_t\}_{0{\leqslant}t{\leqslant}T}$ converges as $n$ goes to infinity in $D([0,T], M_1({{\mathbb R}}))$ almost surely (in probability). The Stieltjes transform of the limit measure-valued process $\{\tilde\mu_t\}_{0{\leqslant}t{\leqslant}T}$ $$\begin{aligned}
\tilde m_t(z)=\int \frac{{{\rm d}}\tilde \mu_t(x)}{x-z},\quad z\in {{\mathbb C}}\setminus {{\mathbb R}},\end{aligned}$$ is characterized by $$\begin{aligned}
\begin{split}\label{e:stDBM}
& \tilde m_t(\tilde z_t(z))=\tilde m_0(z),\\
& \tilde z_t(z)=z+t-t\tilde m_0(z),
\end{split}\end{aligned}$$ where $\tilde z_t(z)$ is well-defined on the domain $$\begin{aligned}
\tilde \Omega_t{\mathrel{\mathop:}=}\left\{z\in {{\mathbb C}}\setminus {{\mathbb R}}: \int \frac{{{\rm d}}\tilde \mu_0(x)}{|x-z|^2}<\frac{1}{t}\right\}.\end{aligned}$$ We recall the limit empirical measure process $\mu_t$ of the $\beta$-nonintersecting Poisson random walks from Theorem \[t:LLN\], its Stieltjes transform $m_t(z)$ in , and the key relations and . We also recall the Markov-Krein correspondence operator $Q$ from Theorem \[t:MKcor\]. If we take the $\tilde \mu_0=Q(\mu_0)$, by the defining relation of $Q$, we have $$\begin{aligned}
\tilde m_0(z)=1-e^{-m_0(z)}.\end{aligned}$$ Therefore, the characteristic lines for $m_t(z)$ and $\tilde m_t(z)$ are the same: $$\begin{aligned}
\label{e:cline}
\tilde z_t(z)=z+t-t\tilde m_0(z)=z+te^{-m_0(z)}=z_t(z).\end{aligned}$$ The Stieltjes transforms $m_t(z)$ and $\tilde m_t(z)$ satisfy $$\begin{aligned}
\tilde m_t(\tilde z_t(z))
=\tilde m_0(z)
=1-e^{-m_0(z)}
=1-e^{-m_t(z_t(z))}
=1-e^{-m_t(\tilde z_t(z))}.\end{aligned}$$ Since $\tilde z_t(z)$ is a surjection onto ${{\mathbb C}}\setminus {{\mathbb R}}$, we get $\tilde \mu_t=Q(\mu_t)$.
We denote the rescaled fluctuation process $$\begin{aligned}
\tilde g_t^n(z)=n(\tilde m_t^n(z)- \tilde m_t(z)).\end{aligned}$$ It follows from [@MR1819483; @MR2418256], if there exists a constant ${{\frak a}}$, such that for any $z\in {{\mathbb C}}\setminus{{\mathbb R}}$, $$\begin{aligned}
{\mathbb{E}}\left[|\tilde g_0^n(z)|^2\right]{\leqslant}{{\frak a}}({\mathop{\mathrm{Im}}}[z])^{-2},\end{aligned}$$ and the random field $(\tilde g_0^n(z))_{z\in {{\mathbb C}}\setminus{{\mathbb R}}}$ weakly converges to a deterministic field $(\tilde g_0(z))_{z\in {{\mathbb C}}\setminus {{\mathbb R}}}$, in the sense of finite dimensional distributions, then, for any fixed time $T>0$, the process $\{(\tilde g^n_t(\tilde z_t(z)))_{z\in\tilde \Omega_T}\}_{0{\leqslant}t{\leqslant}T}$ converges weakly towards a Gaussian process $\{(\tilde g_t(\tilde z_t(z)))_{z\in \tilde \Omega_T}\}_{0{\leqslant}t{\leqslant}T}$, in the sense of finite dimensional processes, with initial data $(\tilde g_0(z))_{z\in {{\mathbb C}}\setminus{{\mathbb R}}}$, means and covariances $$\begin{aligned}
\begin{split}\label{e:meanandvar2}
{\mathbb{E}}[\tilde g_t(\tilde z_t(z))]&=\tilde \mu(t, z)=\frac{\tilde g_0(z)}{ {\partial}_z\tilde z_t(z)}+\left(\frac{1}{2}-\frac{1}{2\theta}\right)\frac{{\partial}_z^2\tilde z_t(z)}{({\partial}_z \tilde z_t(z))^2},\\
{{\rm{cov}}}[\tilde g_s(\tilde z_s(z)), \tilde g_t(\tilde z_t(z'))]&=\tilde \sigma(s, z, t, z')=\frac{1}{\theta}\frac{1}{{\partial}_z\tilde z_s(z){\partial}_z\tilde z_t(z')}\left(\frac{1}{(z-z')^2}-\frac{{\partial}_z \tilde z_{s\wedge t}(z){\partial}_z \tilde z_{s\wedge t}(z')}{(\tilde z_{s\wedge t}(z)-\tilde z_{s\wedge t}(z'))^2}\right),\\
{{\rm{cov}}}[\tilde g_s(\tilde z_s(z)), \overline{\tilde g_t(\tilde z_t(z'))}]&=\tilde \sigma(s, z,t,\bar z').
\end{split}\end{aligned}$$
The statements in [@MR1819483; @MR2418256] are for $\beta$-Dyson Brownian motions with quadratic potential, which differ from by a rescaling of time and space. follows from [@MR2418256 Theorem 2.3] by a change of variable.
By comparing with , if we replace the characteristic lines $\tilde z_t(z)$ in the expressions of the means and covariances of the random field $\{(\tilde g_t(\tilde z_t(z)))_{z\in\tilde \Omega_T}\}_{0{\leqslant}t{\leqslant}T}$ by $z_t(z)$, we get the means and variances of the random field $\{(g_t( z_t(z)))_{z\in \Omega_T}\}_{0{\leqslant}t{\leqslant}T}$. If we take the $\tilde \mu_0=Q(\mu_0)$ and $\tilde g_0(z)=g_0(z)$, then from the discussion above, we have $\tilde \mu_t=Q(\mu_t)$ and $\tilde z_t(z)=z_t(z)$ from . Thus, $$\begin{aligned}
\{(\tilde g_t(\tilde z_t(z)))_{z\in\tilde \Omega_T}\}_{0{\leqslant}t{\leqslant}T}\stackrel{d}{=}\{(g_t( z_t(z)))_{z\in \Omega_T}\}_{0{\leqslant}t{\leqslant}T},\end{aligned}$$ in the sense of finite dimensional processes. From the discussion above, we have that the following diagram commutes
\[e:commute\] (\_0, g\_0(z)) & & & & & &&(\_t, g\_t(z\_t(z)))\
(\_0, g\_0(z)) & & & & & &&(\_t, g\_t(z\_t(z))),
where $Q$ is the Markov-Krein correspondence from Theorem \[t:MKcor\], and $I$ is the identity map.
Stochastic differential equation for the Stieltjes transform {#s:sde}
------------------------------------------------------------
In this section we derive a stochastic differential equation for the Stieltjes transform of the empirical measure process $\mu_t^n$.
\[p:msde\] The Stieltjes transform of the empirical measure process satisfies the following stochastic differential equation $$\begin{aligned}
\begin{split}\label{e:msde}
m^n_t(z_t)=m^n_0(z)
+\int_0^t {\partial}_z m^n_s(z_s) e^{-m_s(z_s)}{{\rm d}}s&+\theta n\int_{0}^t\left(\prod_{j=1}^n\left(1+\frac{1}{n}\frac{1}{z_s-x_j(s)/\theta n}\right)\right.\\
&-\left.\prod_{j=1}^n\left(1+\frac{1}{n}\frac{1}{(z_s-1/\theta n)-x_j(s)/\theta n}\right)\right){{\rm d}}s+M^n_t(z),
\end{split}\end{aligned}$$ where $z_t$ is defined in with $z_0=z$ and $M^n_t(z)$ is a Martingale starting at $0$, with quadratic variations, $$\begin{aligned}
\begin{split}\label{e:qv}
[ M^n(z), M^n(z)]_t=\sum_{0{\leqslant}s{\leqslant}t}(m^n_s(z_s)-m^n_{s-}(z_s))^2, \\ [ M^n(z), \overline{M^n(z)}]_t=\sum_{0{\leqslant}s{\leqslant}t}|m^n_s(z_s)-m^n_{s-}(z_s)|^2.
\end{split}\end{aligned}$$
We remark that the integrand in also appears in the Nekrasov’s equation in [@MR3668648 Section 4], which is crucial for the proof of the central limit theorems of the discrete $\beta$ ensembles. We can view as a dynamical version of the Nekrasov’s equation.
The $\beta$-nonintersecting random walk ${{\bm{x}}}(t)$ is a continuous time Markov jump process. We recall its generator ${{{\mathcal}L}}^n_\theta $ from . By It[ó]{}’s formula, $$\begin{aligned}
\label{e:ito}
M^n_t(z){\mathrel{\mathop:}=}m^n_t(z_t)-m^n_0(z_t)-\int_0^t {\partial}_z m^n_s(z_s){\partial}_t z_s{{\rm d}}s-\int_{0}^t {{{\mathcal}L}}^n_\theta m^n_s(z_s){{\rm d}}s,\end{aligned}$$ is a martingale with quadratic variations given by . To estimate the integrand ${{{\mathcal}L}}_\theta^n m_s^n(z_s)$ in , we need some algebraic facts about the generator ${{{\mathcal}L}}_\theta^n$.
\[l:average\] For any $\theta\in {{\mathbb R}}$, we have $$\begin{aligned}
\label{e:average}
\sum_{i=1}^{n} \prod_{j:j\neq i}\frac{x_i-x_j+\theta}{x_i-x_j}=n.\end{aligned}$$
We can rewrite the left hand side of in terms of the Vandermond determinant in variables $x_1,x_2,\cdots, x_n$, $$\begin{aligned}
\sum_{i=1}^{n} \prod_{j:j\neq i}\frac{x_i-x_j+\theta}{x_i-x_j}
=\sum_{i=1}^n \frac{V({{\bm{x}}}+ \theta {\bm{e}}_i)}{V({{\bm{x}}})}.\end{aligned}$$ We notice that $\sum_{i=1}^n V({{\bm{x}}}+\theta {\bm{e}}_i)$ is a degree $n(n-1)/2$ polynomial in variables $x_1,x_2,\cdots,x_n$. More importantly, it is antisymmetric. Therefore, there exists a constant $C(\theta, n)$ depending on $\theta$ and $n$ such that $$\begin{aligned}
\label{e:average2}
\sum_{i=1}^n V({{\bm{x}}}+\theta {\bm{e}}_i)=C(\theta, n)V({{\bm{x}}}).\end{aligned}$$ We conclude from by comparing the coefficient of the term $x_1^{n-1}x_2^{n-2}\cdots x_{n-1}$.
The following identity will be crucial for the derivation of the stochastic differential equation of the Stieltjes transforms of the empirical measure process $\mu_t$.
\[c:average3\] For any $\theta\in {{\mathbb R}}$, we have $$\begin{aligned}
\label{e:average3}
\sum_{i=1}^{n} \left(\prod_{j:j\neq i}\frac{x_i-x_j+\theta}{x_i-x_j}\right)\frac{1}{n}\frac{1}{x_i/\theta n-z}=1-\prod_{j=1}^n\left(1+\frac{1}{n}\frac{1}{z-x_j/\theta n}\right).\end{aligned}$$
We use Lemma \[l:average\] for the vector $(x_1,x_2,\cdots, x_n, \theta n z)$, $$\begin{aligned}
\begin{split}
n+1&=\sum_{i=1}^{n} \left(\prod_{j:j\neq i}\frac{x_i-x_j+\theta}{x_i-x_j}\right)\frac{x_i-\theta n z+\theta}{x_i-\theta n z}
+\prod_{j=1}^n\frac{\theta n z-x_j+\theta}{\theta n z-x_j}\\
&=\sum_{i=1}^{n} \left(\prod_{j:j\neq i}\frac{x_i-x_j+\theta}{x_i-x_j}\right)+\sum_{i=1}^{n} \left(\prod_{j:j\neq i}\frac{x_i-x_j+\theta}{x_i-x_j}\right)\frac{\theta}{x_i-\theta n z}
+\prod_{j=1}^n\left(1+\frac{\theta}{\theta n z-x_j}\right)\\
&=n+\sum_{i=1}^{n} \left(\prod_{j:j\neq i}\frac{x_i-x_j+\theta}{x_i-x_j}\right)\frac{\theta}{x_i-\theta n z}
+\prod_{j=1}^n\left(1+\frac{\theta}{\theta n z-x_j}\right).
\end{split}\end{aligned}$$ The claim follows by rearranging.
\[r:proofMKcor\] We can use Corollary \[c:average3\] to give a construction of the operator $Q$ in Theorem \[t:MKcor\]. Let $\mu$ be a measure as in Theorem \[t:MKcor\]. For any large integer $m>0$, we discretize $\mu$ on the scale $1/m$ and define $$\begin{aligned}
\mu^m{\mathrel{\mathop:}=}\frac{1}{m}\sum_{i=1}^m \delta_{y_i^m},\quad
\frac{i-1/2}{m}=\int_{y_i^m}^{\infty}{{\rm d}}\mu(x), \quad 1{\leqslant}i{\leqslant}m.\end{aligned}$$ As $m$ goes to infinity, $\mu^m$ weakly converges to $\mu$. Since the density of $\mu$ is bounded by $1$, we have $y_{i}^m-y_{i+1}^m{\geqslant}1/m$ for all $1{\leqslant}i{\leqslant}m-1$. The Perelomov-Popov measure is defined as $$\begin{aligned}
Q^m(\mu^m){\mathrel{\mathop:}=}\frac{1}{m}\sum_{i=1}^m \prod_{j:j\neq i}\frac{y_i^m-y_{j}^m+1/m}{y_i^m-y_j^m}\delta_{y_i^m}.\end{aligned}$$ Since $y_{i}^m-y_{i+1}^m{\geqslant}1/m$, $Q^m(\mu^m)$ is a positive measure. Moreover, thanks to Lemma \[l:average\], $Q^m(\mu^m)$ is a probability measure. We denote the Stieltjes transform of $\mu$, $\mu^m$ and $Q^m(\mu^m)$ by $m_{\mu}(z)$, $m_{\mu^m}(z)$ and $m_{Q^m(\mu^m)}(z)$ respectively. Since $\mu^m$ weakly converges to $\mu$ as $m$ goes to infinity, $$\begin{aligned}
\lim_{m\rightarrow \infty}m_{\mu^m}(z)=m_\mu(z),\quad z\in {{\mathbb C}}\setminus{{\mathbb R}}.\end{aligned}$$ For the Stieltjes transform $m_{Q^m(\mu^m)}(z)$, we use Corollary \[c:average3\], $$\begin{aligned}
m_{Q^m(\mu^m)}(z)
&=\frac{1}{m}\sum_{i=1}^m \prod_{j:j\neq i}\frac{y_i^m-y_{j}^m+1/m}{y_i^m-y_j^m}\frac{1}{y_i^m-z}\\
&=1-\prod_{j=1}^m\left(1-\frac{1}{m}\frac{1}{y_j^m-z}\right)
=1-\exp\left\{-\frac{1}{m}\sum_{j=1}^m\frac{1}{y_j^m-z}+\operatorname{O}\left(\frac{1}{m}\right)\right\}\\
&=1-\exp\left\{-m_{\mu^m}(z)+\operatorname{O}\left(\frac{1}{m}\right)\right\}\rightarrow 1-e^{-m_\mu(z)},\end{aligned}$$ as $m$ goes to infinity. Since $$\begin{aligned}
\lim_{y\rightarrow \infty}{\mathrm{i}}y\left(1-e^{-m_\mu({\mathrm{i}}y)}\right)=-1,\end{aligned}$$ by [@MR1962995 Theorem 1], there exists a probability measure $Q(\mu)$ with Stieltjes transform $1-e^{-m_\mu(z)}$, and $Q^m(\mu^m)$ weakly converges to $Q(\mu)$.
For the integrand ${{{\mathcal}L}}^n_\theta m^n_s(z)$, we have $$\begin{aligned}
\begin{split}
{{{\mathcal}L}}^n_\theta m^n_s(z)
&=\theta n\sum_{i=1}^{n}\left(\prod_{j:j\neq i}\frac{x_i(s)-x_j(s)+\theta}{x_i(s)-x_j(s)}\right)\left(\frac{1}{n}\frac{1}{(x_i(s)+1)/\theta n-z_s}-\frac{1}{n}\frac{1}{x_i(s)/\theta n-z_s}\right)\\
&=\theta n\left(\prod_{j=1}^n\left(1+\frac{1}{n}\frac{1}{z_s-x_j(s)/\theta n}\right)
-\prod_{j=1}^n\left(1+\frac{1}{n}\frac{1}{(z_s-1/\theta n)-x_j(s)/\theta n}\right)\right).
\end{split}\end{aligned}$$ where we used Corollary \[c:average3\]. Combining with , this finishes the proof Proposition \[p:msde\]
Law of large numbers {#s:LLN}
--------------------
In this section we analyze , and prove the Law of large numbers for the $\beta$-nonintersecting Poisson random walk.
We define an auxiliary process $N^n_t$, which counts the number of jumps for the $\beta$-nonintersecting Poisson random walk ${{\bm{x}}}(t)$, $$\begin{aligned}
\label{e:defNt}
N^n_t=\sum_{i=1}^n \left(x_i(t)-x_i(0)\right).\end{aligned}$$ The Poisson process $N^n_t$ will be used later to control the martingale term $M^n_t(z)$ in .
$N^n_t$ is a Poisson process, starting at $0$, with jump rate $\theta n^2$.
According to the generator of the $\beta$-nonintersecting Poisson random walk, the process $N^n_t$ increases $1$ with rate $$\begin{aligned}
\theta n\sum_{i=1}^n\sum_{j:j\neq i}\frac{x_i-x_j+\theta}{x_i-x_j}=\theta n^2,\end{aligned}$$ where we used Proposition \[e:average\].
In the following we simplify the stochastic differential equation of $m_t(z_t)$, i.e. the second integrand in . By Proposition \[p:ztp\], for any $z\in \Omega_T$, and $0{\leqslant}t{\leqslant}T$, we have $|{\mathop{\mathrm{Im}}}[z_t]|{\geqslant}|{\mathop{\mathrm{Im}}}[z_T]|>0$. Therefore, we have the trivial bound $1/|x_j(s)/\theta n-z_t|=\operatorname{O}(1)$, where the implicit constant depends on ${\mathop{\mathrm{Im}}}[z_T]$.
For the first term in the second integrand in , by the Tylor expansion $$\begin{aligned}
\begin{split}\label{e:t1}
&\phantom{{}={}}\prod_{j=1}^n\left(1+\frac{1}{n}\frac{1}{z_s-x_j(s)/\theta n}\right)
=\exp\left(\sum_{j=1}^n \ln\left(1+\frac{1}{n}\frac{1}{z_s-x_j(s)/\theta n}\right)\right)\\
&=\exp\left(\sum_{j=1}^n \frac{1}{n}\frac{1}{z_s-x_j(s)/\theta n}-\frac{1}{2n^2}\frac{1}{(z_s-x_j(s)/\theta n)^2}+\frac{1}{3n^3}\frac{1}{(z_s-x_j(s)/\theta n)^3}+\operatorname{O}\left(\frac{1}{n^4}\right)\right)\\
&=e^{-m^n_s(z_s)}\left(1-\frac{1}{2}\frac{{\partial}_z m^n_s(z_s)}{n}+\frac{1}{8}\frac{({\partial}_z m^n_s(z_s))^2}{n^2}-\frac{1}{6}\frac{{\partial}_z^2 m^n_s(z_s)}{n^2}+\operatorname{O}\left(\frac{1}{n^3}\right)\right).
\end{split}\end{aligned}$$ Similarly, for the second term in the integrand, $$\begin{aligned}
\begin{split}\label{e:t2}
&\phantom{{}={}}\prod_{j=1}^n\left(1+\frac{1}{n}\frac{1}{(z_s-1/\theta n)-x_j(s)/\theta n}\right)
=\exp\left(\sum_{j=1}^n \ln\left(1+\frac{1}{n}\frac{1}{(z_s-1/\theta n)-x_j(s)/\theta n}\right)\right)\\
&=e^{-m^n_s(z)}\left(1-\left(\frac{1}{2}-\frac{1}{\theta}\right)\frac{{\partial}_z m^n_s(z)}{n}+\frac{1}{2}\left(\frac{1}{\theta}-\frac{1}{2}\right)^2\frac{({\partial}_z m^n_s(z_s))^2}{n^2}-\right.\\
&\phantom{{}=e^{-m^n_s(z_s)}\left(1-\left(\frac{1}{2}-\frac{1}{\theta}\right)\frac{{\partial}_z m^n_s(z_s)}{n}\right.}\left.-\frac{1}{2}\left(\frac{1}{\theta^2}-\frac{1}{\theta}+\frac{1}{3}\right)\frac{{\partial}_z^2 m^n_s(z_s)}{n^2}+\operatorname{O}\left(\frac{1}{n^3}\right)\right).
\end{split}\end{aligned}$$ The difference of and is $$\begin{aligned}
\begin{split}\label{e:t3}
&\phantom{{}={}}\prod_{j=1}^n\left(1+\frac{1}{n}\frac{1}{(z_s-1/\theta n)-x_j(s)/\theta n}\right)-\prod_{j=1}^n\left(1+\frac{1}{n}\frac{1}{(z_s-1/\theta n)-x_j(s)/\theta n}\right)\\
&=e^{-m^n_s(z_s)}\left(-\frac{{\partial}_z m^n_s(z_s)}{\theta n}+\left(\frac{1}{2}-\frac{1}{2\theta}\right)\frac{({\partial}_z m^n_s(z_s))^2-{\partial}_z^2 m^n_s(z_s)}{\theta n^2}+\operatorname{O}\left(\frac{1}{n^3}\right)\right).
\end{split}\end{aligned}$$ We can use to simplify the stochastic differential equation , $$\begin{aligned}
\begin{split}\label{e:newsde}
m^n_t(z_t)=m^n_0(z)
&+\int_0^t {\partial}_z m^n_s(z_s) \left(e^{-m_s(z_s)}-e^{-m^n_s(z_s)}\right){{\rm d}}s+\\
&+\left(\frac{1}{2}-\frac{1}{2\theta}\right)\int_{0}^t\frac{(({\partial}_z m^n_s(z_s))^2-{\partial}_z^2 m^n_s(z_s))e^{-m_s^n(z_s)}}{n}{{\rm d}}s+M^n_t(z)+\operatorname{O}\left(\frac{t}{n^2}\right),
\end{split}\end{aligned}$$ where the implicit constant depends on $\theta$ and ${\mathop{\mathrm{Im}}}[z_T]$.
In the following we estimate the martingale $M^n_t(z)$ using the Burkholder-Davis-Gundy inequality. For the quadratic variation of $M^n_t(z)$, we have $$\begin{aligned}
\begin{split}\label{e:quadvar}
&\phantom{{}={}}[ M^n(z), \overline{M^n(z)}]_t
=\sum_{0< s{\leqslant}t} |m^n_s(z_s)-m^n_{s-}(z_s)|^2\\
&=\frac{1}{n^2}\sum_{0< s{\leqslant}t\atop \Delta x_i(s)>0}\left|\frac{1}{x_i(s)/\theta n-z_s}-\frac{1}{x_i(s-)/\theta n-z_s}\right|^2
= \operatorname{O}\left(\frac{N^n_t}{ n^4}\right),
\end{split}\end{aligned}$$ where the implicit constant depends on $\theta$ and ${\mathop{\mathrm{Im}}}[z_T]$. It follows from the Burkholder-Davis-Gundy inequality, for any $p{\geqslant}1$, we have $$\begin{aligned}
\begin{split}\label{e:BDG}
{\mathbb{E}}\left[\left(\sup_{0{\leqslant}t{\leqslant}T}|M^n_t(z)|\right)^p\right]^{1/p}
&{\leqslant}Cp {\mathbb{E}}\left[[ M^n(z), \overline{M^n(z) }]_T^{p/2}\right]^{1/p}
\\
&=\operatorname{O}\left(\frac{p}{ n^2}{\mathbb{E}}\left[\left(N^n_T\right)^{p/2}\right]^{1/p} \right)=\operatorname{O}\left(\frac{T^{1/2}p^{3/2}}{n}\right).
\end{split}\end{aligned}$$ By the Markov’s inequality, we have $$\begin{aligned}
{\mathbb{P}}\left(\sup_{0{\leqslant}t{\leqslant}T}|M^n_t(z)|{\geqslant}\varepsilon\right){\leqslant}\left(\frac{CT^{1/2}p^{3/2}}{\varepsilon n}\right)^p.\end{aligned}$$ Therefore it follows by taking $p>1$, $\sup_{0{\leqslant}t{\leqslant}T}|M^n_t(z)|$ converges to zero almost surely as $n$ goes to infinity.
For any $0{\leqslant}t{\leqslant}T$, we can rewrite as $$\begin{aligned}
\begin{split}
|m^n_t(z_t)-m^n_0(z)|
&{\leqslant}|m^n_0(z)-m_0(z)|+C\left(\int_0^t |m_s(z_s)-m_0(z)|{{\rm d}}s+\frac{t}{n}\right)+\sup_{0{\leqslant}t{\leqslant}T}|M_t^n|\\
&=C\left(\int_0^t |m_s(z_s)-m_0(z)|{{\rm d}}s\right)+|m^n_0(z)-m_0(z)|+\operatorname{o}(1)
\end{split}\end{aligned}$$ where the constant $C$ depends on $\theta$, $T$ and ${\mathop{\mathrm{Im}}}[z_T]$, and the term $\operatorname{o}(1)$ converges to zero almost surely and is uniform for $0{\leqslant}t{\leqslant}T$. Thus, it follows from Gronwell’s inequality, $$\begin{aligned}
\sup_{0{\leqslant}t{\leqslant}T}|m_t(z_t)-m_0(z)|{\leqslant}C|m_0^n(z)-m_0(z)|+o(1),\end{aligned}$$ which converges to zero almost surely (in probability), if $|m_0^n(z)-m_0(z)|$ converges to zero almost surely (in probability). This finishes the proof of Theorem \[t:LLN\].
Central limit theorems for the empirical measure process
========================================================
In this section, we prove the central limit theorems for the rescaled empirical measure process $\{n(\mu^n_t-\mu_t)\}_{0{\leqslant}t {\leqslant}T}$ with analytic test functions.
Central limit theorems {#s:CLT}
----------------------
Theorem \[t:CLT\] follows from the following proposition.
\[p:CLT\] We assume Assumption \[a:initiallaw\]. Then for any values $z_1,z_2,\cdots, z_m\in {{\mathbb C}}\setminus{{\mathbb R}}$ and time $T<\min\{{{\frak t}}(z_1),{{\frak t}}(z_2),\cdots, {{\frak t}}(z_m)\}$ as defined in , the random processes $\{(g^n_t(z_t(z_j)))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$ converge weakly in the Skorokhod space $D([0,T], {{\mathbb C}}^m)$ towards a Gaussian process $\{(\mathcal{G}_j(t))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$, which is the unique solution of the system of stochastic differential equations $$\begin{aligned}
\begin{split}\label{e:limitprocess}
{{{\mathcal}G}}_j(t)&={{{\mathcal}G}}_j(0)+ \int_0^t \frac{{\partial}_zm_0(z_j)e^{-m_0(z_j)}}{1-s{\partial}_zm_0(z_j)e^{-m_0(z_j)}} {{{\mathcal}G}}_j(s){{\rm d}}s\\
&+\left(\frac{1}{2}-\frac{1}{2\theta}\right)\int_0^t\frac{(({\partial}_z m_0(z_j))^2-{\partial}_z^2 m_0(z_j))e^{-m_0(z_j)}}{(1-s{\partial}_z m_0(z_j)e^{-m_0(z_j)})^3}{{\rm d}}s+{{\mathcal W}}_j(t),\quad 1{\leqslant}j{\leqslant}m,
\end{split}\end{aligned}$$ with initial data $(g_0(z_j))_{1{\leqslant}j{\leqslant}m}$ given in Assumption \[a:initiallaw\], and $\{({{\mathcal W}}_j(t))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$ is a centered Gaussian process independent of $({{{\mathcal}G}}_j(0))_{1{\leqslant}j{\leqslant}m}$, and $$\begin{aligned}
\begin{split}\label{e:covW1}
\langle {{\mathcal W}}_j, {{\mathcal W}}_k\rangle_t=-\frac{1}{\theta}\int_0^t\frac{({\partial}_s m_s)(z_s(z_j))+({\partial}_sm_s)(z_s(z_k))}{(z_s(z_j)-z_s(z_k))^2}-\frac{2(e^{-m_0(z_j)}-e^{-m_0(z_k)})}{(z_s(z_j)-z_s(z_k))^3}{{\rm d}}s,\\
\langle{{\mathcal W}}_j, \bar{{{\mathcal W}}}_k\rangle_t=-\frac{1}{\theta}\int_0^t\frac{({\partial}_s m_s)(z_s(z_j))+({\partial}_sm_s)(z_s(\bar{z}_k))}{(z_s(z_j)-z_s(\bar{z}_k))^2}-\frac{2(e^{-m_0(z_j)}-e^{-m_0(\bar{z}_k)})}{(z_s(z_j)-z_s(\bar{z}_k))^3}{{\rm d}}s,
\end{split}\end{aligned}$$ where $$\begin{aligned}
\begin{split}\label{e:covW2}
\langle {{\mathcal W}}_j, {{\mathcal W}}_j\rangle_t
&=\lim_{z_k\rightarrow z_j}-\frac{1}{\theta}\int_0^t\frac{({\partial}_s m_s)(z_s(z_j))+({\partial}_sm_s)(z_s(z_k))}{(z_s(z_j)-z_s(z_k))^2}-\frac{2(e^{-m_0(z_j)}-e^{-m_0(z_k)})}{(z_s(z_j)-z_s(z_k))^3}{{\rm d}}s\\
&=-\frac{1}{6\theta }\int_0^t ({\partial}_s{\partial}^2_z m_s)(z_s(z_j)){{\rm d}}s.
\end{split}\end{aligned}$$
We can solve for $\{({{{\mathcal}G}}_j(t))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$ explicitly. From , we have $$\begin{aligned}
\begin{split}\label{e:rearrange}
{{\rm d}}(1-t{\partial}_zm_0(z_j)e^{-m_0(z_j)}){{{\mathcal}G}}_j(t)&=\left(\frac{1}{2}-\frac{1}{2\theta}\right)\frac{(({\partial}_z m_0(z_j))^2-{\partial}_z^2 m_0(z_j))e^{-m_0(z_j)}}{(1-t{\partial}_z m_0(z_j)e^{-m_0(z_j)})^2}{{\rm d}}t\\
&+(1-t{\partial}_zm_0(z_j)e^{-m_0(z_j)}){{\rm d}}{{\mathcal W}}_j(t).
\end{split}\end{aligned}$$ We integrate both sides of , $$\begin{aligned}
{{{\mathcal}G}}_j(t)=\frac{{{{\mathcal}G}}_j(0)}{1-t{\partial}_zm_0(z_j)e^{-m_0(z_j)}}
+\left(\frac{1}{2}-\frac{1}{2\theta}\right)\frac{t(({\partial}_z m_0(z_j))^2-{\partial}_z^2 m_0(z_j))e^{-m_0(z_j)}}{(1-t{\partial}_z m_0(z_j)e^{-m_0(z_j)})^2}
+ {{\mathcal B}}_j(t),\end{aligned}$$ where $$\begin{aligned}
{{\mathcal B}}_j(t)=\frac{1}{(1-t{\partial}_zm_0(z_j)e^{-m_0(z_j)})}\int_0^t (1-s{\partial}_zm_0(z_j)e^{-m_0(z_j)}){{\rm d}}{{\mathcal W}}_j(s).\end{aligned}$$ By a straightforward (but tedious and lengthy) calculation, using , and , we get the covariances of $\{({{\mathcal B}}_j(t))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$, $$\begin{aligned}
{{\rm{cov}}}[ {{\mathcal B}}_j(s), {{\mathcal B}}_k(t)]
&=\frac{\int_0^{s\wedge t} (1-u{\partial}_zm_0(z_j)e^{-m_0(z_j)})(1-u{\partial}_zm_0(z_k)e^{-m_0(z_k)}){{\rm d}}\langle{{\mathcal W}}_j, {{\mathcal W}}_k\rangle_u}{(1-s{\partial}_zm_0(z_j)e^{-m_0(z_j)})(1-t{\partial}_zm_0(z_k)e^{-m_0(z_k)})}\\
&=\frac{1}{\theta}\frac{1}{ (1-s{\partial}_zm_0(z_j)e^{-m_0(z_j)})(1-t{\partial}_zm_0(z_k)e^{-m_0(z_k)})}\\
&\times\left(\frac{1}{(z_j-z_k)^2}-\frac{(1-(s\wedge t){\partial}_zm_0(z_j)e^{-m_0(z_j)})(1-(s\wedge t){\partial}_zm_0(z_k)e^{-m_0(z_k)})}{(z_j-z_k+(s\wedge t)(e^{-m_0(z_j)}-e^{-m_0(z_k)}))^2}\right)=\sigma(s, z_j,t,z_k)\\
{{\rm{cov}}}[ {{\mathcal B}}_j(s), \overline{{{\mathcal B}}_k(t)}]&=\sigma(s,z_j,t, \bar z_k),\end{aligned}$$ and $$\begin{aligned}
&\phantom{{}={}}{{\rm{cov}}}[{{\mathcal B}}_j(s), {{\mathcal B}}_j(t)]=\frac{\int_0^{s\wedge t} (1-u{\partial}_zm_0(z_j)e^{-m_0(z_j)})^2{{\rm d}}\langle{{\mathcal W}}_j, {{\mathcal W}}_j\rangle_u}{(1-s{\partial}_zm_0(z_j)e^{-m_0(z_j)})(1-t{\partial}_zm_0(z_j)e^{-m_0(z_j)})}\\
&=\frac{(s\wedge t)e^{-m_0(z_j)}(2({\partial}_z m_0(z_j))^3-6{\partial}_z m_0(z_j){\partial}_z^2 m_0(z_j)+2{\partial}_z^3m_0(z_j))}{12\theta(1-(s\wedge t){\partial}_z m_0(z_j)e^{-m_0(z_j)})^3(1-(s\vee t){\partial}_z m_0(z_j)e^{-m_0(z_j)})}\\
&+\frac{(s\wedge t)^2e^{-2m_0(z_j)}(({\partial}_z m_0(z_j))^4+3({\partial}_z^2 m_0(z_j))^2-2{\partial}_z m_0(z_j){\partial}_z^3m_0(z_j))}{12\theta(1-(s\wedge t){\partial}_z m_0(z_j)e^{-m_0(z_j)})^3(1-(s\vee t){\partial}_z m_0(z_j)e^{-m_0(z_j)})}=\sigma(s,z_j,t, z_j)=\lim_{z_k\rightarrow z_j}\sigma(s, z_j,t, z_k).
$$ This finishes the proof of Theorem \[t:CLT\].
We divide the proof of Proposition \[p:CLT\] into three steps. In Step one we prove the tightness of the processes $\{(g^n_t(z_t(z_j)))_{1{\leqslant}j{\leqslant}m})\}_{0{\leqslant}t{\leqslant}T}$ as $n$ goes to infinity. In Step two, we prove that the martingale term $\{(nM_t^n(z_j))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$ converges weakly to a centered complex Gaussian process. In Step three, we prove that the subsequential limits of $\{(g^n_t(z_t(z_j)))_{1{\leqslant}j{\leqslant}m})\}_{0{\leqslant}t{\leqslant}T}$ solve the stochastic differential equation . Proposition \[p:CLT\] follows from this fact and the uniqueness of the solution to .
[*Step one: tightness.*]{}
We first prove the tightness of the martingale term.
We assume the assumptions of Proposition \[p:CLT\]. Then as $n$ goes to infinity, the random processes $\{(nM^n_t(z_j))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$, and $\{(n^2[ M^n(z_j), M^n(z_k)]_t)_{1{\leqslant}j,k{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$ are tight.
We apply the sufficient condition for tightness of [@MR1943877 Chapter 6, Proposition 3.26]. We need to check the modulus conditions: for any $\varepsilon>0$ there exists a $\delta>0$ such that $$\begin{aligned}
\label{e:modulus1}
&{\mathbb{P}}\left(\sup_{1{\leqslant}j{\leqslant}m}\sup_{0{\leqslant}t{\leqslant}t'{\leqslant}T, t'-t{\leqslant}\delta}|n(M^n_{t'}(z_j)-M^n_t(z_j))|{\geqslant}\varepsilon\right){\leqslant}\varepsilon,\\
\label{e:modulus2}&{\mathbb{P}}\left(\sup_{1{\leqslant}j,k{\leqslant}m}\sup_{0{\leqslant}t{\leqslant}t'{\leqslant}T, t'-t{\leqslant}\delta}\left|n^2([ M^n(z_j), M^n(z_k)]_{t'}-[ M^n(z_j), M^n(z_k)]_t)\right|{\geqslant}\varepsilon\right){\leqslant}\varepsilon.\end{aligned}$$
For , since $\{M^n_{t'}(z_j)-M^n_t(z_j)\}_{t{\leqslant}t'{\leqslant}T\vee t+\delta}$ is a martingale, it follows from the Burkholder-Davis-Gundy inequality, for any $p{\geqslant}1$, we have $$\begin{aligned}
\begin{split}
&\phantom{{}={}}{\mathbb{E}}\left[\left(\sup_{t{\leqslant}t'{\leqslant}T\vee t+\delta}|n(M^n_{t'}(z_j)-M^n_t(z_j))|\right)^p\right]^{1/p}\\
&{\leqslant}Cpn {\mathbb{E}}\left[[ M^n(z_j)-M^n_t(z_j), \overline{M^n(z_j)-M_t^n(z_j) }]_{T\vee t+\delta}^{p/2}\right]^{1/p}
\\
&=\operatorname{O}\left(\frac{p}{ n}{\mathbb{E}}\left[\left(N^n_{T\vee t+\delta}-N^n_t\right)^{p/2}\right]^{1/p} \right)=\operatorname{O}\left(\delta^{1/2}p^{3/2}\right).
\end{split}\end{aligned}$$ where the implicit constant depends on $\theta$ and $\min_{1{\leqslant}j{\leqslant}m}|{\mathop{\mathrm{Im}}}[z_T(z_j)]|$. By the Markov’s inequality, we have $$\begin{aligned}
{\mathbb{P}}\left(\sup_{t{\leqslant}t'{\leqslant}T\vee t+\delta}|n(M^n_{t'}(z_j)-M^n_t(z_j))|{\geqslant}\varepsilon\right){\leqslant}\left(\frac{C\delta^{1/2}p^{3/2}}{\varepsilon }\right)^p.\end{aligned}$$ Let $t_k=(k-1)\delta\vee T$ for $1{\leqslant}k{\leqslant}\lfloor1/\delta\rfloor$. By a union bound $$\begin{aligned}
\begin{split}
&\phantom{{}={}}{\mathbb{P}}\left(\sup_{1{\leqslant}j{\leqslant}m}\sup_{0{\leqslant}t{\leqslant}t'{\leqslant}T, t'-t{\leqslant}\delta}|n(M^n_{t'}(z_j)-M^n_t(z_j))|{\geqslant}\varepsilon\right)\\
&{\leqslant}\frac{m}{\delta} \sup_{1{\leqslant}j{\leqslant}m}\sup_{1{\leqslant}k{\leqslant}\lfloor 1/\delta \rfloor}{\mathbb{P}}\left(\sup_{t_k{\leqslant}t{\leqslant}T\vee(t_k+\delta)}|n(M^n_{t}(z_j)-M^n_{t_k}(z_j))|{\geqslant}\varepsilon/2\right)\\
&{\leqslant}\frac{m}{\delta}\left(\frac{2C\delta^{1/2}p^{3/2}}{\varepsilon}\right)^p{\leqslant}\varepsilon.
\end{split}\end{aligned}$$ if we take $p>2$ and $\delta$ small enough. This finishes the proof of .
The modulus of the process $n^2[M^n(z_j), M^n(z_k)]_{t}$ is dominated by the Poisson process $n^{-2}N_t^n$ in the following sense. For any $0{\leqslant}t{\leqslant}T$ and $t{\leqslant}t'{\leqslant}T\vee(t+\delta)$, $$\begin{aligned}
\begin{split}\label{e:control}
&\phantom{{}={}}\left|n^2([ M^n(z_j), M^n(z_k)]_{t'}-[ M^n(z_j), M^n(z_k)]_t)\right|\\
&{\leqslant}n^2\sum_{t< s{\leqslant}t'} |m^n_s(z_s(z_j))-m^n_{s-}(z_s(z_j))||m^n_s(z_s(z_k))-m^n_{s-}(z_s(z_k))|\\
&=\sum_{t< s{\leqslant}t'\atop \Delta x_i(s)>0}\left|\frac{1}{x_i(s)/\theta n-z_s(z_j)}-\frac{1}{x_i(s-)/\theta n-z_s(z_j)}\right|\left|\frac{1}{x_i(s)/\theta n-z_s(z_k)}-\frac{1}{x_i(s-)/\theta n-z_s(z_k)}\right|\\
&= \operatorname{O}\left(\frac{N^n_{t'}-N^n_t}{ n^2}\right),
\end{split}\end{aligned}$$ where the implicit constant depends on $\theta$ and $\min_{1{\leqslant}j{\leqslant}m}|{\mathop{\mathrm{Im}}}[z_T(z_j)]|$. By the same argument as for , we have that $n^{-2}N_t^n$ satisfies the modulus condition: $$\begin{aligned}
\label{e:modulus3}{\mathbb{P}}\left(\sup_{0{\leqslant}t{\leqslant}t'{\leqslant}T, t'-t{\leqslant}\delta}\left|\frac{N_{t'}^n-N_t^n}{n^2}\right|{\geqslant}\varepsilon\right){\leqslant}\varepsilon.\end{aligned}$$ The claim follows from combining and .
We assume the assumptions of Proposition \[p:CLT\]. Then as $n$ goes to infinity, the random processes $\{(g^n_t(z_t(z_j)))_{1{\leqslant}j{\leqslant}m})\}_{0{\leqslant}t{\leqslant}T}$ are tight.
We apply the sufficient condition for tightness of [@MR1943877 Chapter 6, Proposition 3.26], and check the modulus condition: for any $\varepsilon>0$ there exists a $\delta>0$ such that $$\begin{aligned}
\label{e:modulus}
\sup_{n{\geqslant}1}{\mathbb{P}}\left(\sup_{1{\leqslant}i{\leqslant}m}\sup_{0{\leqslant}t{\leqslant}t'{\leqslant}T, t'-t{\leqslant}\delta}|g^n_{t'}(z_{t'}(z_j))-g^n_t(z_t(z_j))|{\geqslant}\varepsilon\right){\leqslant}\varepsilon.\end{aligned}$$
Before we prove the modulus condition , we first prove that as $n$ goes to infinity, the random processes $\{(g_t^n(z_t(z_j)))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$ are stochastically bounded, i.e. for any $\varepsilon>0$, there exists $M>0$ such that $$\begin{aligned}
\label{e:stBound}
\sup_{n{\geqslant}1}{\mathbb{P}}\left(\sup_{1{\leqslant}j{\leqslant}m}\sup_{0{\leqslant}t{\leqslant}T}|g^n_t(z_t(z_j))|{\geqslant}M\right){\leqslant}\varepsilon.\end{aligned}$$ By rearranging , for any $1{\leqslant}i {\leqslant}m$, we get $$\begin{aligned}
\begin{split}\label{e:gsde1}
g_t^n(z_t(z_j))&=g_0^n(z_j)+\int_0^t {\partial}_z m^n_s(z_s(z_j))e^{-m_0(z_j)} n\left(1-e^{-(m^n_s(z_s(z_j))-m_0(z_j))}\right){{\rm d}}s+\\
&+\left(\frac{1}{2}-\frac{1}{2\theta}\right)\int_{0}^t\left(({\partial}_z m^n_s(z_s(z_j)))^2-{\partial}_z^2 m^n_s(z_s(z_j))\right)e^{-m_s^n(z_s(z_j))}{{\rm d}}s+nM^n_t(z_j)+\operatorname{O}\left(\frac{t}{n}\right).
\end{split}\end{aligned}$$ For the second and third terms on the righthand side of , we have $$\begin{aligned}
\begin{split}\label{e:bound1}
\left|{\partial}_z m^n_s(z_s(z_j))e^{-m_0(z_j)} n\left(1-e^{-(m^n_s(z_s(z_j))-m_0(z_j))}\right)\right|&{\leqslant}C|g_s^n(z_s(z_j))|,\\
\left|({\partial}_z m^n_s(z_s(z_j)))^2-{\partial}_z^2 m^n_s(z_s(z_j))e^{-m_s^n(z_s(z_j))}\right|&{\leqslant}C,
\end{split}\end{aligned}$$ where the constants $C$ depends on $\min_{1{\leqslant}j{\leqslant}m}|{\mathop{\mathrm{Im}}}[z_T(z_j)]|$. From , we have $$\begin{aligned}
\begin{split}\label{e:bound2}
{\mathbb{E}}\left[\left(\sup_{0{\leqslant}t{\leqslant}T}|nM^n_t(z_j)|\right)^p\right]^{1/p}
&{\leqslant}CT^{1/2}p^{3/2},
\end{split}\end{aligned}$$ where the constant $C$ depends on $\theta$ and $\min_{1{\leqslant}j{\leqslant}m}|{\mathop{\mathrm{Im}}}[z_T(z_j)]|$. Combining , with the Gronwall’s inequality, we get that the process $\{(g_t^n(z_t(z_j))_{1{\leqslant}j{\leqslant}m}\}_{1{\leqslant}t{\leqslant}T}$ is stochastically bounded . On the event $\{\sup_{1{\leqslant}j{\leqslant}m}\sup_{0{\leqslant}t{\leqslant}T}|g^n_t(z_t(z_j))|{\leqslant}M\}$, for any $0{\leqslant}t{\leqslant}T$ and $t{\leqslant}t'{\leqslant}T\vee t+\delta$, we have $$\begin{aligned}
|g^n_{t'}(z_{t'}(z_j))-g^n_t(z_t(z_j))|{\leqslant}C(M+1)\delta+n(M_{t'}^n(z_j)-M_t^n(z_j))+\operatorname{O}\left(\frac{t}{n}\right).\end{aligned}$$ The claim follows from the tightness of $\{(nM_t^n(z_j))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$.
*Step two: weak convergence of the martingale term.*
We define a sequence of stopping times $\tau_0^n, \tau^n_1, \tau^n_2, \tau^n_3, \cdots$, where $\tau_0^n=0$ and for $l{\geqslant}1$, $\tau^n_l$ is the time of the $l$-th jump of the Poisson process $N_t^n$. The following estimate follows from the tail estimate of the exponential random variables.
\[c:waittime\] Fix $\theta>0$ and time $T>0$. For any $\varepsilon>0$, there exists a $M>0$ such that $$\begin{aligned}
\label{e:waittime}
\sup_{n{\geqslant}1}{\mathbb{P}}\left(\sup_{0<\tau^n_j{\leqslant}t} |\tau^n_j-\tau^n_{j-1}|{\geqslant}\frac{M\ln n}{n^2}\right){\leqslant}\varepsilon\end{aligned}$$
Since the waiting time of $N_t^n$ is an exponential random variable of rate $\theta n^2$, for any $j{\geqslant}1$, we have $$\begin{aligned}
\label{e:waittime2}
{\mathbb{P}}\left(|\tau^n_j-\tau^n_{j-1}|{\geqslant}\frac{M\ln n}{n^2}\right)=\exp\left\{-\theta n^2\frac{M\ln n}{n^2}\right\}=n^{-\theta M}.\end{aligned}$$ The claim follows from and a union bound.
\[c:Mconverge\] We assume the assumptions of Proposition \[p:CLT\]. Then as $n$ goes to infinity, the complex martingales $\{(nM_t^n(z_j))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$ converge weakly in $D([0,T], {{\mathbb C}}^m)$ towards a centered complex Gaussian process $\{({{\mathcal W}}_j(t))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$, with quadratic variation given by and .
We notice that $\overline{M_t^n(z_j)}=M_t^n(\bar{z}_j)$. Claim \[c:Mconverge\] follows from [@MR838085 Chapter 7, Theorem 1.4] and the weak convergence of the quadratic variations, $$\begin{aligned}
\label{e:var}n^2[ M^n(z_j), M^n(z_j) ]_t
&\Rightarrow -\frac{1}{6\theta }\int_0^t ({\partial}_s{\partial}^2_z m_s)(z_s(z_j)){{\rm d}}s, \\
\label{e:cov}n^2[ M^n(z_j), M^n(z_k) ]_t
&\Rightarrow-\frac{1}{\theta}\int_0^t\frac{({\partial}_s m_s)(z_s(z_j))+({\partial}_sm_s)(z_s(z_k))}{(z_s(z_j)-z_s(z_k))^2}-\frac{2(e^{-m_0(z_j)}-e^{-m_0(z_k)})}{(z_s(z_j)-z_s(z_k))^3}{{\rm d}}s.\end{aligned}$$ Thanks to , we know that the processes $\{(n^2[M^n_t(z_j), M^n_t(z_k)]_t)_{1{\leqslant}j,k{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$ are tight. For and , it remains to prove the weak convergence of any fixed time.
By definition, the quadratic variation $n^2[M^n_t(z_j), M^n_t(z_k)]_t$ is given by $$\begin{aligned}
\begin{split}\label{e:quadvar2}
n^2[ M^n(z_j), M^n(z_k) ]_t
&=
\sum_{0< s{\leqslant}t\atop \Delta x_i(s)>0}\left(\frac{1}{x_i(s)/\theta n-z_s(z_j)}-\frac{1}{x_i(s-)/\theta n-z_s(z_j)}\right)\\
&\phantom{{}=
\sum_{0< s{\leqslant}t\atop \Delta x_i(s)>0}}\left(\frac{1}{x_i(s)/\theta n-z_s(z_k)}-\frac{1}{x_i(s-)/\theta n-z_s(z_k)}\right)\\
&=\sum_{0< s{\leqslant}t\atop \Delta x_i(s)>0}\frac{1}{(\theta n)^2}\frac{1}{(x_i(s)/\theta n-z_s(z_j))^2(x_i(s)/\theta n-z_s(z_k))^2}+\operatorname{O}\left(\frac{N^n_t}{ n^3}\right),
\end{split}\end{aligned}$$ where the implicit constant depends on $\theta$ and $\min_{1{\leqslant}j{\leqslant}m}|{\mathop{\mathrm{Im}}}[z_T(z_j)]|$. We can further rewrite as a sum of differences. For , we have $$\begin{aligned}
\begin{split}\label{e:quadvarexp1}
&\phantom{{}={}}n^2[ M^n(z_j), M^n(z_j) ]_t
=\sum_{0< s{\leqslant}t\atop \Delta x_i(s)>0}\frac{1}{(\theta n)^2}\frac{1}{(x_i(s)/\theta n-z_s(z_j))^4}+\operatorname{O}\left(\frac{N^n_t}{ n^3}\right)\\
&=\sum_{0< s{\leqslant}t\atop \Delta x_i(s)>0}-\frac{1}{3\theta n}\left(\frac{1}{(x_i(s)/\theta n-z_s(z_j))^3}-\frac{1}{(x_i(s-)/\theta n-z_s(z_j))^3}\right)+\operatorname{O}\left(\frac{N^n_t}{ n^3}\right)\\
&=-\frac{1}{6\theta }\sum_{0< s{\leqslant}t\atop \Delta x_i(s)>0}\left({\partial}_z^2 m_{s}^n(z_s(z_j)) -{\partial}_z^2 m_{s-}^n(z_s(z_j))\right)+\operatorname{O}\left(\frac{N^n_t}{ n^3}\right)
\end{split}\end{aligned}$$ We recall the stopping times defined above Claim \[c:waittime\], and rewrite as $$\begin{aligned}
\begin{split}\label{e:quadvarexp2}
&\phantom{{}={}}n^2[ M^n(z_j), M^n(z_j) ]_t
=-\frac{1}{6\theta }\sum_{0<l{\leqslant}N_t^n}\left({\partial}_z^2 m_{\tau^n_l}^n(z_{\tau^n_l}(z_j)) -{\partial}_z^2 m_{\tau^n_{l-1}}^n(z_{\tau^n_l}(z_j))\right)+\operatorname{O}\left(\frac{N^n_t}{ n^3}\right)\\
&=-\frac{1}{6\theta }\sum_{0<l{\leqslant}N_t^n}\left({\partial}_z^2 m_{\tau^n_l}^n(z_{\tau^n_l}(z_j)) -{\partial}_z^2 m_{\tau^n_{l-1}}^n(z_{\tau^n_{l-1}}(z_j))-{\partial}^3_zm_{\tau^n_{l-1}}^n(z_{\tau^n_{l-1}}(z_j))(z_{\tau^n_l}(z_j)-z_{\tau^n_{l-1}}(z_j))\right)\\
&+\operatorname{O}\left(N_t^n\left(\frac{1}{ n^3}+\sup_{0<l{\leqslant}N_t^n} |z_{\tau^n_l}-z_{\tau^n_{l-1}}|^2\right)\right)\\
&=-\frac{1}{6\theta}\left({\partial}_z^2 m_{t}^n(z_{t}(z_j))-{\partial}_z^2 m_{0}^n(z_{0}(z_j))-\int_0^t{\partial}_z^3m_s^n(z_s(z_j)){{\rm d}}z_s(z_j)\right)\\
&+\operatorname{O}\left(N_t^n\left(\frac{1}{ n^3}+\sup_{0<l{\leqslant}N_t^n} |z_{\tau^n_l}-z_{\tau^n_{l-1}}|^2\right)+|z_t(z_j)-z_{\tau^n_{N_t^n}}(z_j)|\right)\\
&\Rightarrow -\frac{1}{6\theta }\int_0^t ({\partial}_s{\partial}^2_z m_s)(z_s(z_j)){{\rm d}}s,
\end{split}\end{aligned}$$ where in the last line we used Claim and that $z_t$ is Lipschitz with respect to $t$. This finishes the proof of . For , we have $$\begin{aligned}
&\phantom{{}={}}n^2[ M^n(z_j), M^n(z_k) ]_t
=\sum_{0< s{\leqslant}t\atop \Delta x_i(s)>0}\frac{1}{(\theta n)^2}\frac{1}{(x_i(s)/\theta n-z_s(z_j))^2(x_i(s)/\theta n -z_s(z_k))^2}+\operatorname{O}\left(\frac{N^n_t}{ n^3}\right)\\
&=\frac{1}{(\theta n)^2}\sum_{0< s{\leqslant}t\atop \Delta x_i(s)>0}\left(\frac{1}{(z_s(z_j)-z_s(z_k))^2}\left(\frac{1}{(x_i(s)/\theta n-z_s(z_j))^2}+\frac{1}{(x_i(s)/\theta n-z_s(z_k))^2}\right)\right.\\
&\left.-\frac{2}{(z_s(z_j)-z_s(z_k))^3}\left(\frac{1}{(x_i(s)/\theta n-z_s(z_j))}-\frac{1}{(x_i(s)/\theta n-z_s(z_k))}\right)\right)+\operatorname{O}\left(\frac{N^n_t}{ n^3}\right)\\
&=-\frac{1}{\theta}\sum_{0<s{\leqslant}t\atop \Delta x_i(s)>0}
\frac{(m^n_s(z_s(z_j))-m^n_{s-}(z_s(z_j)))+(m^n_s(z_s(z_k))-m^n_{s-}(z_s(z_k)))}{(z_s(z_j)-z_s(z_k))^2}\\
&-
\frac{2}{\theta}\sum_{0<s{\leqslant}t\atop \Delta x_i(s)>0}\frac{(h^n_s(z_s(z_j))-h^n_{s-}(z_s(z_j)))-(h^n_s(z_s(z_k))-h^n_{s-}(z_s(z_k)))}{ (z_s(z_j)-z_s(z_k))^3}+\operatorname{O}\left(\frac{N^n_t}{ n^3}\right),\end{aligned}$$ where $h_t^n(z)$ is the logarithmic potential of the empirical measure $\mu_t^n$, $$\begin{aligned}
h_t^n(z)=\int \ln(x-z){{\rm d}}\mu_t^n(x)=\frac{1}{n}\sum_{i=1}^n \ln (x_i(t)-z),\quad z\in {{\mathbb C}}\setminus{{\mathbb R}}.\end{aligned}$$ Thanks to Theorem , we have $$\begin{aligned}
h_t^n(z)\Rightarrow h_t(z)=\int \ln (x-z){{\rm d}}\mu_t,\quad z\in {{\mathbb C}}\setminus{{\mathbb R}},\end{aligned}$$ where the logarithmic potential $h_t(z)$ is defined in . By the same argument as in , we get $$\begin{aligned}
n^2\langle M^n(z_j), M^n(z_k) \rangle_t
&\Rightarrow
-\frac{1}{\theta}\int_0^t \frac{({\partial}_s m_s)(z_s(z_j))+({\partial}_sm_s)(z_s(z_k))}{(z_s(z_j)-z_s(z_k))^2}+\frac{2(({\partial}_s h_s)(z_s(z_j))-({\partial}_sh_s)(z_s(z_k)))}{(z_s(z_j)-z_s(z_k))^3}{{\rm d}}s\\
&=-\frac{1}{\theta}\int_0^t\frac{({\partial}_s m_s)(z_s(z_j))+({\partial}_sm_s)(z_s(z_k))}{(z_s(z_j)-z_s(z_k))^2}-\frac{2(e^{-m_0(z_j)}-e^{-m_0(z_k)})}{(z_s(z_j)-z_s(z_k))^3}{{\rm d}}s.\end{aligned}$$ This finishes the proof of .
*Step three: subsequential limit.*
In the first step, we have proven that as $n$ goes to infinity, the random processes $\{(g^n_t(z_t(z_j)))_{1{\leqslant}j{\leqslant}m})\}_{0{\leqslant}t{\leqslant}T}$ are tight. Without loss of generality, by passing to a subsequence, we assume that they weakly converge towards to a random process $\{({{{\mathcal}G}}_j(t))_{1{\leqslant}j{\leqslant}m})\}_{0{\leqslant}t{\leqslant}T}$. We check that the limit process satisfies the stochastic differential equation . The random process $\{(g^n_t(z_t(z_j)))_{1{\leqslant}j{\leqslant}m})\}_{0{\leqslant}t{\leqslant}T}$ satisfies the stochastic differential equation . For the first term on the righthand side of , by our assumption $(g_0^n(z_j))_{1{\leqslant}j{\leqslant}m}\Rightarrow (g_0(z_j))_{1{\leqslant}j{\leqslant}m}$. For the second term, by Theorem \[t:LLN\], we have $$\begin{aligned}
&\phantom{{}={}}\int_0^t {\partial}_z m^n_s(z_s(z_j))e^{-m_0(z_j)} n\left(1-e^{-(m^n_s(z_s(z_j))-m_0(z_j))}\right){{\rm d}}s\\
&=\int_0^t {\partial}_z m^n_s(z_s(z_j))e^{-m_0(z_j)} g^n_t(z_s(z_j)){{\rm d}}s
+\operatorname{O}\left(\int_0^tn|m_s^n(z_s(z_j))-m_0(z_j)|^2{{\rm d}}s\right)\\
&\Rightarrow \int_0^t {\partial}_z m_s(z_s(z_j))e^{-m_0(z_j)} {{{\mathcal}G}}_j(s){{\rm d}}s
= \int_0^t \frac{{\partial}_zm_0(z_j)e^{-m_0(z_j)}}{1-s{\partial}_zm_0(z_j)e^{-m_0(z_j)}} {{{\mathcal}G}}_j(s){{\rm d}}s\end{aligned}$$ where the last term vanishes, because the processes $\{(g^n_t(z_t(z_j)))_{1{\leqslant}j{\leqslant}m})\}_{0{\leqslant}t{\leqslant}T}$ are stochastically bounded, i.e. . For the third term, by Theorem \[t:LLN\] and Proposition \[p:estimatemt\], we have $$\begin{aligned}
\begin{split}
\int_{0}^t\left(({\partial}_z m^n_s(z_s(z_j)))^2-{\partial}_z^2 m^n_s(z_s(z_j))\right)e^{-m_s^n(z_s(z_j))}{{\rm d}}s&\Rightarrow
\int_{0}^t\left(({\partial}_z m_s(z_s(z_j)))^2-{\partial}_z^2 m_s(z_s(z_j))\right)e^{-m_0(z_j)}{{\rm d}}s\\
&=\int_0^t\frac{(({\partial}_z m_0(z_j))^2-{\partial}_z^2 m_0(z_j))e^{-m_0(z_j)}}{(1-s{\partial}_z m_0(z_j)e^{-m_0(z_j)})^3}{{\rm d}}s.
\end{split}\end{aligned}$$ For the fourth term, in Step two we have proven that $\{(nM_t^n(z_j))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$ converges weakly towards a centered complex Gaussian process $\{({{\mathcal W}}_j(t))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$, which is characterized by and . This finishes the proof of Theorem \[p:CLT\].
Extreme particles {#s:extremep}
-----------------
In the following we first derive a large deviation estimate of the extreme particles of the $\beta$-nonintersecting random walks. Then Theorem \[t:CLT2\] follows from Theorem \[t:CLT\] by a contour integral.
\[p:extremePbound\] Suppose the initial data ${{\bm{x}}}(0)$ satisfies Assumption \[a:ibound\]. For any time $t>0$, there exists a constant ${{\frak c}}$ depending on ${{\frak b}}$ and $t$, such that $$\begin{aligned}
\label{e:extremePbound}
{{\frak c}}n{\geqslant}x_1(t){\geqslant}x_2(t){\geqslant}\cdots {\geqslant}x_n(t), \end{aligned}$$ with probability at least $1-\exp(-cn)$.
We notice that the $\beta$-nonintersecting Poisson random walks are shift invariant. Suppose that the $\beta$-nonintersecting Poisson random walk ${{\bm{y}}}(t)$ starts from $(a+(n-1)\theta, a+(n-2)\theta, \cdots, a)$, where $a= a(n)\in {\mathbb{Z}}_{{\geqslant}0}$. Then it follows from Theorem \[t:density\] that for any fixed $t>0$, the law of ${{\bm{y}}}(t)$ is given by $$\begin{aligned}
\label{e:defPt}
{\mathbb{P}}_{t}(y_1,y_2,\cdots, y_n)=\frac{1}{Z_n}\prod_{1{\leqslant}i <j{\leqslant}n}\frac{\Gamma(y_i-y_j+1)\Gamma(y_i-y_j+\theta)}{\Gamma(y_i-y_j)\Gamma(y_i-y_j+1-\theta)}\prod_{i=1}^n\frac{(\theta t n)^{y_i-a}}{\Gamma(y_i-a+1)},\end{aligned}$$ where the partition function $Z_n$ is given by $$\begin{aligned}
Z_n=e^{\theta t n^2}(\theta t n)^{\theta(n-1)n/2}\prod_{i=1}^n\frac{\Gamma(i\theta)}{\Gamma(\theta)}.\end{aligned}$$
The measure ${\mathbb{P}}_t(y_1,y_2,\cdots, y_n)$ is a discrete $\beta$ ensemble studies in [@MR3668648]. The next proposition follows from [@MR3668648 Theorem 7.1].
\[p:0initialbound\] Take $a=\lceil {{\frak b}}n\rceil$ and $ t>0$. There exits a constant ${{\frak c}}$ depending ${{\frak b}}$ and $t$, such that the measure ${\mathbb{P}}_ t$ as in satisfies $$\begin{aligned}
{\mathbb{P}}_{ t}\left(y_1{\leqslant}{{\frak c}}n\right){\geqslant}1-\exp(-cn). \end{aligned}$$
Let ${{\bm{x}}}(t)$ be a $\beta$-nonintersecting Poisson random walk with initial data ${{\bm{x}}}(0)\in {\mathbb{W}}_\theta^n$ satisfying , and ${{\bm{y}}}(t)$ another independent $\beta$-nonintersecting Poisson random walk with initial data ${{\bm{y}}}(0)=(\lceil {{\frak b}}n\rceil+(n-1)\theta, \lceil{{\frak b}}n\rceil+(n-2)\theta, \cdots, \lceil {{\frak b}}n \rceil)$. Let ${{\frak c}}$ be as in Proposition \[p:0initialbound\], we prove by constructing a coupling of ${{\bm{x}}}(t)$ and ${{\bm{y}}}(t)$, that $$\begin{aligned}
\label{e:PBound}
{\mathbb{P}}(x_1( t){\leqslant}{{\frak c}}n){\geqslant}{\mathbb{P}}(y_1( t){\leqslant}{{\frak c}}n).\end{aligned}$$ Then the claim follows from combining Proposition \[p:0initialbound\] and .
We define the coupling $(\hat{{\bm{x}}}(t), \hat{{\bm{y}}}(t))$ as a Poisson random walk on ${\mathbb{W}}^n_\theta\times {\mathbb{W}}^n_\theta$, with initial data $(\hat{{\bm{x}}}(0), \hat{{\bm{y}}}(0))=({{\bm{x}}}(0), {{\bm{x}}}(0))$, and generator $$\begin{aligned}
\begin{split}
\hat{{{\mathcal}L}}^n_\theta f({{\bm{x}}}, {{\bm{y}}})&=\theta n\sum_{i=1}^n\left[\frac{V({{\bm{x}}}+\theta {\bm{e}}_i)}{V({{\bm{x}}})}-\frac{V({{\bm{y}}}+\theta {\bm{e}}_i)}{V({{\bm{y}}})}\right]_+\left(f({{\bm{x}}}+{\bm{e}}_i, {{\bm{y}}})-f({{\bm{x}}}, {{\bm{y}}})\right)\\
&+\theta n\sum_{i=1}^n\left[\frac{V({{\bm{y}}}+\theta {\bm{e}}_i)}{V({{\bm{y}}})}-\frac{V({{\bm{x}}}+\theta {\bm{e}}_i)}{V({{\bm{x}}})}\right]_+\left(f({{\bm{x}}}, {{\bm{y}}}+{\bm{e}}_i)-f({{\bm{x}}}, {{\bm{y}}})\right)\\
&+\theta n\sum_{i=1}^n\min\left\{\frac{V({{\bm{x}}}+\theta {\bm{e}}_i)}{V({{\bm{x}}})},\frac{V({{\bm{y}}}+\theta {\bm{e}}_i)}{V({{\bm{y}}})}\right\}\left(f({{\bm{x}}}+{\bm{e}}_i, {{\bm{y}}}+{\bm{e}}_i)-f({{\bm{x}}}, {{\bm{y}}})\right).
\end{split}\end{aligned}$$ where $[x]_+=\max\{x,0\}$. The marginal distributions of $\hat{{\bm{x}}}(t)$ and $\hat{{\bm{y}}}(t)$ coincide with those of ${{\bm{x}}}(t)$ and ${{\bm{y}}}(t)$ respectively, $$\begin{aligned}
\{\hat{{\bm{x}}}(s)\}_{0{\leqslant}s{\leqslant}t}\overset{d}{=}\{{{\bm{x}}}(s)\}_{0{\leqslant}s{\leqslant}t},\quad \{\hat{{\bm{y}}}(s)\}_{0{\leqslant}s{\leqslant}t}\overset{d}{=}\{{{\bm{y}}}(s)\}_{0{\leqslant}s{\leqslant}t}.\end{aligned}$$
For the initial data, we have $$\begin{aligned}
\hat x_i(0){\leqslant}{{\frak b}}n{\leqslant}\hat y_i(0), \quad 1{\leqslant}i{\leqslant}n.\end{aligned}$$ In the following we prove that the coupling process $(\hat {{\bm{x}}}(t), \hat{{\bm{y}}}(t))$ satisfies $$\begin{aligned}
\label{e:comparison1}
{\mathbb{P}}\left(\text{for all $t{\geqslant}0$ and $1{\leqslant}i{\leqslant}n$, }\hat x_i(t){\leqslant}\hat y_i(t)\right)=1.\end{aligned}$$ We define a sequence of stopping times, $\tau^n_1, \tau^n_2, \tau^n_3, \cdots$, where $\tau^n_k$ is the time of the $k$-th jump of the coupling process $(\hat {{\bm{x}}}(t), \hat {{\bm{y}}}(t))$. We prove by induction that $$\begin{aligned}
\label{e:comparison2}
{\mathbb{P}}\left(\text{for all $0{\leqslant}t{\leqslant}\tau^n_k$ and $1{\leqslant}i{\leqslant}n$, }\hat x_i(t){\leqslant}\hat y_i(t)\right)=1.\end{aligned}$$ Then follows by noticing that $\lim_{k\rightarrow\infty}\tau^n_k=\infty$. We assume that holds for $k$, we prove it for $k+1$. If $\hat x_i(\tau^n_k)<\hat y_i(\tau^n_k)$, then with probability one, $\hat x_i(\tau^n_{k+1}){\leqslant}\hat y_i(\tau^n_{k+1})$. If $\hat x_i(\tau^n_k)=\hat y_i(\tau^n_k)$, by our assumptions, $\hat{{\bm{x}}}(\tau^n_k), \hat {{\bm{y}}}(\tau^n_k)\in {\mathbb{W}}^n_\theta$ and $\hat x_j(\tau^n_k){\leqslant}\hat y_j(\tau^n_k)$ for all $1{\leqslant}j{\leqslant}n$, we have $$\begin{aligned}
0{\leqslant}\frac{\hat x_i(\tau^n_k)-\hat x_j(\tau^n_k)+\theta}{\hat x_i(\tau^n_k)-\hat x_j(\tau^n_k)}{\leqslant}\frac{\hat y_i(\tau^n_k)-\hat y_j(\tau^n_k)+\theta}{\hat y_i(\tau^n_k)-\hat y_j(\tau^n_k)},\quad j\neq i.\end{aligned}$$ Thus the jump rate from $(\hat {{\bm{x}}}(\tau^n_k), \hat{{\bm{y}}}(\tau^n_k))$ to $(\hat {{\bm{x}}}(\tau^n_k)+{\bm{e}}_i, \hat{{\bm{y}}}(\tau^n_k))$, $$\begin{aligned}
\begin{split}
&\phantom{{}={}}\theta n\left[\frac{V(\hat {{\bm{x}}}(\tau^n_k)+\theta {\bm{e}}_i)}{V(\hat {{\bm{x}}}(\tau^n_k))}-\frac{V(\hat{{\bm{y}}}(\tau^n_k)+\theta {\bm{e}}_i)}{V(\hat {{\bm{y}}}(\tau^n_k))}\right]_+\\
&=\theta n\left[\prod_{j:j\neq i}\frac{\hat x_i(\tau^n_k)-\hat x_j(\tau^n_k)+\theta}{\hat x_i(\tau^n_k)-\hat x_j(\tau^n_k)}-\prod_{j:j\neq i}\frac{\hat y_i(\tau^n_k)-\hat y_j(\tau^n_k)+\theta}{\hat y_i(\tau^n_k)-\hat y_j(\tau^n_k)}\right]_+= 0.
\end{split}\end{aligned}$$ vanishes. Therefore with probability one, $\hat x_i(\tau^n_{k+1}){\leqslant}\hat y_i(\tau^n_{k+1})$. This finishes the proof of and .
It follows from , $$\begin{aligned}
\label{e:PBound2}
{\mathbb{P}}(\hat x_1( t ){\leqslant}{{\frak c}}n){\geqslant}{\mathbb{P}}(\hat y_1( t ){\leqslant}{{\frak c}}n).\end{aligned}$$ Since the marginal distributions of $\hat{{\bm{x}}}(t)$ and $\hat{{\bm{y}}}(t)$ coincide with those of ${{\bm{x}}}(t)$ and ${{\bm{y}}}(t)$ respectively, follows from combining Proposition \[p:0initialbound\] and . This finishes the proof of Propostion \[p:extremePbound\].
We take a contour ${{{\mathcal}C}}$ which encloses a neighborhood of $[0,{{\frak c}}/\theta]$. Then with exponentially high probability we have $$\begin{aligned}
n\int f_j(x){{\rm d}}(\mu_t^n(x)-\mu_t(x))=\frac{1}{2\pi{\mathrm{i}}}\oint_{{{\mathcal}C}}g^n_t(w)f_j(w){{\rm d}}w, \quad 1{\leqslant}j{\leqslant}m.\end{aligned}$$ By Proposition \[e:defOmega\], $z_t(z)$ is a homeomorphism from the closure of $\Omega_t\cap {{\mathbb C}}_+$ to ${{\mathbb C}}_+\cup {{\mathbb R}}$, and from the closure of $\Omega_t\cap {{\mathbb C}}_-$ to ${{\mathbb C}}_-\cup {{\mathbb R}}$. By a change of variable, we have $$\begin{aligned}
\frac{1}{2\pi{\mathrm{i}}}\oint_{{{\mathcal}C}}g^n_t(w)f_j(w){{\rm d}}w=\frac{1}{2\pi{\mathrm{i}}}\oint_{z_t^{-1}({{{\mathcal}C}})} g^n_t(z_t(z))f_j(z_t(z)){{\rm d}}z_t(z), \quad 1{\leqslant}j{\leqslant}m.\end{aligned}$$ By the continuous mapping theorem of weak convergence, it follows from Theorem \[t:CLT\] $$\begin{aligned}
&\left\{\left(\frac{1}{2\pi{\mathrm{i}}}\oint_{z_t^{-1}({{{\mathcal}C}})} g^n_t(z_t(z))f_j(z_t(z)){{\rm d}}z_t(z)\right)_{1{\leqslant}j{\leqslant}m}\right\}_{0{\leqslant}t{\leqslant}T}\Rightarrow \{({{{\mathcal}F}}_j(t))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}\\
&{{{\mathcal}F}}_j(t){\mathrel{\mathop:}=}\frac{1}{2\pi{\mathrm{i}}}\oint_{z_t^{-1}({{{\mathcal}C}})} g_t(z_t(z))f_j(z_t(z)){{\rm d}}z_t(z),\quad 1{\leqslant}j{\leqslant}m,\end{aligned}$$ and the means and the covariances of the Gaussian process $\{({{{\mathcal}F}}_j(t))_{1{\leqslant}j{\leqslant}m}\}_{0{\leqslant}t{\leqslant}T}$ are given by $$\begin{aligned}
{\mathbb{E}}[{{{\mathcal}F}}_j(t)]&=\frac{1}{2\pi{\mathrm{i}}}\oint_{z_t^{-1}({{{\mathcal}C}})}\mu(t, z) f_j(z_t(z)){{\rm d}}z_t(z)=\frac{1}{2\pi{\mathrm{i}}}\oint_{{{{\mathcal}C}}}\mu(t, z_t^{-1}(w))f_j(w){{\rm d}}w\\
{{\rm{cov}}}[{{{\mathcal}F}}_j(s), {{{\mathcal}F}}_{k}(t)]&=-\frac{1}{4\pi^2}\oint_{z_s^{-1}({{{\mathcal}C}})}\oint_{z_t^{-1}({{{\mathcal}C}})}\sigma(s, z,t,z') f_j(z_s(z))f_k(z_t(z')){{\rm d}}z_s(z){{\rm d}}z_t(z')\\
&=-\frac{1}{4\pi^2}\oint_{{{{\mathcal}C}}}\oint_{{{{\mathcal}C}}}\sigma(s, z_s^{-1}(w),t,z_t^{-1}(w')) f_j(w)f_k(w'){{\rm d}}w{{\rm d}}w'.\end{aligned}$$ where $\mu(t,z)$ and $\sigma(s, z,t,z')$ are as defined in and . This finishes the proof of Theorem \[t:CLT2\].
| ArXiv |
---
abstract: 'We produce the family of Calabi-Yau hypersurfaces $X_{n}$ of $({\mathbb P}^{1})^{n+1}$ in higher dimension whose inertia group contains non commutative free groups. This is completely different from Takahashi’s result [@ta98] for Calabi-Yau hypersurfaces $M_{n}$ of ${\mathbb P}^{n+1}$.'
address:
- ' (Masakatsu Hayashi) Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyamacho 1-1, Toyonaka, Osaka 560-0043, Japan '
- ' (Taro Hayashi) Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyamacho 1-1, Toyonaka, Osaka 560-0043, Japan '
author:
- Masakatsu Hayashi and Taro Hayashi
title: 'Calabi-Yau hypersurfaces in the direct product of ${\mathbb P}^{1}$ and inertia groups'
---
Introduction
============
Throughout this paper, we work over ${\mathbb C}$. Given an algebraic variety $X$, it is natural to consider its birational automorphisms $\varphi {\colon}X \dashrightarrow X$. The set of these birational automorphisms forms a group ${\operatorname{Bir}}(X)$ with respect to the composition. When $X$ is a projective space ${\mathbb P}^{n}$ or equivalently an $n$-dimensional rational variety, this group is called the Cremona group. In higher dimensional case ($n \geq 3$), though many elements of the Cremona group have been described, its whole structure is little known.
Let $V$ be an $(n+1)$-dimensional smooth projective rational manifold. In this paper, we treat subgroups called the “inertia group" (defined below ) of some hypersurface $X \subset V$ originated in [@gi94]. It consists of those elements of the Cremona group that act on $X$ as identity.
In Section \[cyn\], we mention the result (Theorem \[tak\]) of Takahashi [@ta98] about the smooth Calabi-Yau hypersurfaces $M_{n}$ of ${\mathbb P}^{n+1}$ of degree $n+2$ (that is, $M_{n}$ is a hypersurface such that it is simply connected, there is no holomorphic $k$-form on $M_{n}$ for $0<k<n$, and there is a nowhere vanishing holomorphic $n$-form $\omega_{M_{n}}$). It turns out that the inertia group of $M_{n}$ is trivial (Theorem \[intro2\]). Takahashi’s result (Theorem \[tak\]) is proved by using the “Noether-Fano inequality". It is the useful result that tells us when two Mori fiber spaces are isomorphic. Theorem \[intro2\] is a direct consequence of Takahashi’s result.
In Section \[cy1n\], we consider Calabi-Yau hypersurfaces $$X_{n} = (2, 2, \ldots , 2) \subset ({\mathbb P}^{1})^{n+1}.$$ Let $${\operatorname{UC}}(N) {\coloneqq}\overbrace{{\mathbb Z}/2{\mathbb Z}* {\mathbb Z}/2{\mathbb Z}* \cdots * {\mathbb Z}/2{\mathbb Z}}^{N} = \operatorname*{\raisebox{-0.8ex}{\scalebox{2.5}{$\ast$}}}_{i=1}^{N}\langle t_{i}\rangle$$ be the *universal Coxeter group* of rank $N$ where ${\mathbb Z}/2{\mathbb Z}$ is the cyclic group of order 2. There is no non-trivial relation between its $N$ natural generators $t_{i}$. Let $$p_{i} {\colon}X_{n} \to ({\mathbb P}^{1})^{n}\ \ \ (i=1, \ldots , n+1)$$ be the natural projections which are obtained by forgetting the $i$-th factor of $({\mathbb P}^{1})^{n+1}$. Then, the $n+1$ projections $p_{i}$ are generically finite morphism of degree 2. Thus, for each index $i$, there is a birational transformation $$\iota_{i} {\colon}X_{n} \dashrightarrow X_{n}$$ that permutes the two points of general fibers of $p_{i}$ and this provides a group homomorphism $$\Phi {\colon}{\operatorname{UC}}(n+1) \to {\operatorname{Bir}}(X_{n}).$$
From now, we set $P(n+1) {\coloneqq}({\mathbb P}^{1})^{n+1}$. Cantat-Oguiso proved the following theorem in [@co11].
$($[@co11 Theorem 1.3 (2)]$)$\[iota\] Let $X_{n}$ be a generic hypersurface of multidegree $(2,2,\ldots,2)$ in $P(n+1)$ with $n \geq 3$. Then the morphism $\Phi$ that maps each generator $t_{j}$ of ${\operatorname{UC}}(n+1)$ to the involution $\iota_{j}$ of $X_{n}$ is an isomorphism from ${\operatorname{UC}}(n+1)$ to ${\operatorname{Bir}}(X_{n})$.
Here “generic” means $X_{n}$ belongs to the complement of some countable union of proper closed subvarieties of the complete linear system $\big| (2, 2, \ldots , 2)\big|$.
Let $X \subset V$ be a projective variety. The *decomposition group* of $X$ is the group $$\begin{aligned}
{\operatorname{Dec}}(V, X) {\coloneqq}\{f \in {\operatorname{Bir}}(V)\ |\ f(X) =X \text{ and } f|_{X} \in {\operatorname{Bir}}(X) \}.\end{aligned}$$ The *inertia group* of $X$ is the group $$\begin{aligned}
\label{inertia}
{\operatorname{Ine}}(V, X) {\coloneqq}\{f \in {\operatorname{Dec}}(V, X)\ |\ f|_{X} = {\operatorname{id}}_{X}\}.\end{aligned}$$
Then it is natural to consider the following question:
\[qu\] Is the sequence $$\begin{aligned}
\label{se}
1 \longrightarrow {\operatorname{Ine}}(V, X) \longrightarrow {\operatorname{Dec}}(V, X) \overset{\gamma}{\longrightarrow} {\operatorname{Bir}}(X) \longrightarrow 1\end{aligned}$$ exact, i.e., is $\gamma$ surjective?
Note that, in general, this sequence is not exact, i.e., $\gamma$ is not surjective (see Remark \[k3\]). When the sequence is exact, the group ${\operatorname{Ine}}(V, X)$ measures how many ways one can extend ${\operatorname{Bir}}(X)$ to the birational automorphisms of the ambient space $V$.
Our main result is following theorem, answering a question asked by Ludmil Katzarkov:
\[intro\] Let $X_{n} \subset P(n+1)$ be a smooth hypersurface of multidegree $(2, 2, \ldots, 2)$ and $n \geq 3$. Then:
- $\gamma {\colon}{\operatorname{Dec}}(P(n+1), X_{n}) \to {\operatorname{Bir}}(X_{n})$ is surjective, in particular Question $\ref{qu}$ is affirmative for $X_{n}$.
- If, in addition, $X_{n}$ is generic, there are $n+1$ elements $\rho_{i}$ $(1 \leq i \leq n+1)$ of ${\operatorname{Ine}}(P(n+1), X_{n})$ such that $$\langle \rho_{1}, \rho_{2}, \ldots , \rho_{n+1} \rangle \simeq \underbrace{{\mathbb Z}* {\mathbb Z}* \cdots * {\mathbb Z}}_{n+1} \subset {\operatorname{Ine}}(P(n+1), X_{n}).$$ In particular, ${\operatorname{Ine}}(P(n+1), X_{n})$ is an infinite non-commutative group.
Our proof of Theorem \[intro\] is based on an explicit computation of elementary flavour.
We also consider another type of Calabi-Yau manifolds, namely smooth hypersurfaces of degree $n+2$ in ${\mathbb P}^{n+1}$ and obtain the following result:
\[intro2\] Suppose $n \geq 3$. Let $M_{n} = (n+2) \subset {\mathbb P}^{n+1}$ be a smooth hypersurface of degree $n+2$. Then Question $\ref{qu}$ is also affirmative for $M_{n}$. More precisely:
- ${\operatorname{Dec}}({\mathbb P}^{n+1}, M_{n}) = \{ f \in {\operatorname{PGL}}(n+2, {\mathbb C}) = {\operatorname{Aut}}({\mathbb P}^{n+1})\ |\ f(M_{n}) = M_{n}\}$.
- ${\operatorname{Ine}}({\mathbb P}^{n+1}, M_{n}) = \{{\operatorname{id}}_{{\mathbb P}^{n+1}}\}$, and $\gamma {\colon}{\operatorname{Dec}}({\mathbb P}^{n+1}, M_{n}) \overset{\simeq}{\longrightarrow} {\operatorname{Bir}}(M_{n}) = {\operatorname{Aut}}(M_{n})$.
It is interesting that the inertia groups of $X_{n} \subset P(n+1) = ({\mathbb P}^{1})^{n+1}$ and $M_{n} \subset {\mathbb P}^{n+1}$ have completely different structures though both $X_{n}$ and $M_{n}$ are Calabi-Yau hypersurfaces in rational Fano manifolds.
\[k3\] There is a smooth quartic $K3$ surface $M_{2} \subset {\mathbb P}^{3}$ such that $\gamma$ is not surjective (see [@og13 Theorem 1.2 (2)]). In particular, Theorem \[intro2\] is not true for $n = 2$.
Preliminaries
=============
In this section, we prepare some definitions and properties of birational geometry and introduce the Cremona group.
Divisors and singularities
--------------------------
Let $X$ be a projective variety. A *prime divisor* on $X$ is an irreducible subvariety of codimension one, and a *divisor* (resp. *${\mathbb Q}$-divisor* or *${\mathbb R}$-divisor*) on $X$ is a formal linear combination $D = \sum d_{i}D_{i}$ of prime divisors where $d_{i} \in {\mathbb Z}$ (resp. ${\mathbb Q}$ or ${\mathbb R}$). A divisor $D$ is called *effective* if $d_{i}$ $\geq$ 0 for every $i$ and denote $D \geq 0$. The closed set $\bigcup_{i}D_{i}$ of the union of prime divisors is called the *support* of $D$ and denote Supp$(D)$. A ${\mathbb Q}$-divisor $D$ is called *${\mathbb Q}$-Cartier* if, for some $0 \neq m \in {\mathbb Z}$, $mD$ is a Cartier divisor (i.e. a divisor whose divisorial sheaf ${\mathcal O}_{X}(mD)$ is an invertible sheaf), and $X$ is called ${\mathbb Q}$-*factorial* if every divisor is ${\mathbb Q}$-Cartier.
Note that, since the regular local ring is the unique factorization domain, every divisor automatically becomes the Cartier divisor on the smooth variety.
Let $f {\colon}X \dashrightarrow Y$ be a birational map between normal projective varieties, $D$ a prime divisor, and $U$ the domain of definition of $f$; that is, the maximal subset of $X$ such that there exists a morphism $f {\colon}U \to Y$. Then ${\operatorname{codim}}(X\setminus U) \geq 2$ and $D \cap U \neq \emptyset$, the image $(f|_{U})(D \cap U)$ is a locally closed subvariety of $Y$. If the closure of that image is a prime divisor of $Y$, we call it the *strict transform* of $D$ (also called the *proper transform* or *birational transform*) and denote $f_{*}D$. We define $f_{*}D = 0$ if the codimension of the image $(f|_{U})(D \cap U)$ is $\geq$ 2 in $Y$.
We can also define the strict transform $f_{*}Z$ for subvariety $Z$ of large codimension; if $Z \cap U \neq \emptyset$ and dimension of the image $(f|_{U})(Z \cap U)$ is equal to $\dim Z$, then we define $f_{*}Z$ as the closure of that image, otherwise $f_{*}Z$ = 0.
Let $(X, D)$ is a *log pair* which is a pair of a normal projective variety $X$ and a ${\mathbb R}$-divisor $D \geq 0$. For a log pair $(X, D)$, it is more natural to consider a *log canonical divisor* $K_{X} + D$ instead of a canonical divisor $K_{X}$.
A projective birational morphism $g {\colon}Y \to X$ is a *log resolution* of the pair $(X, D)$ if $Y$ is smooth, ${\operatorname{Exc}}(g)$ is a divisor, and $g_{*}^{-1}(D) \cup {\operatorname{Exc}}(g)$ has simple normal crossing support (i.e. each components is a smooth divisor and all components meet transversely) where ${\operatorname{Exc}}(g)$ is an exceptional set of $g$, and a divisor $over$ $X$ is a divisor $E$ on some smooth variety $Y$ endowed with a proper birational morphism $g {\colon}Y \to X$.
If we write $$K_{Y} + \Gamma + \sum E_{i} = g^{*}(K_{X}+D) + \sum a_{E_{i}}(X, D)E_{i},$$ where $\Gamma$ is the strict transform of $D$ and $E_{i}$ runs through all prime exceptional divisors, then the numbers $a_{E_{i}}(X, D)$ is called the *discrepancies of $(X, D)$ along $E_{i}$*. The *discrepancy of* $(X, D)$ is given by $${\operatorname{discrep}}(X, D) {\coloneqq}\inf\{ a_{E_{i}}(X, D)\ |\ E_{i} \text{ is a prime exceptional divisor over } X\}.$$ The discrepancy $a_{E_{i}}(X, D)$ along $E_{i}$ is independent of the choice of birational maps $g$ and only depends on $E_{i}$.
Let us denote ${\operatorname{discrep}}(X, D) = a_{E}$. A pair $(X, D)$ is *log canonical* (resp. *Kawamata log terminal* ($klt$)) if $a_{E} \geq 0$ (resp. $a_{E} > 0$). A pair $(X, D)$ is *canonical* (resp. *terminal*) if $a_{E} \geq 1$ (resp. $a_{E} > 1$).
Cremona groups
--------------
Let $n$ be a positive integer. The *Cremona group* ${\operatorname{Cr}}(n)$ is the group of automorphisms of ${\mathbb C}(X_{1}, \ldots, X_{n})$, the ${\mathbb C}$-algebra of rational functions in $n$ independent variables.
Given $n$ rational functions $F_{i} \in {\mathbb C}(X_{1}, \ldots, X_{n})$, $1 \leq i \leq n$, there is a unique endomorphism of this algebra maps $X_{i}$ onto $F_{i}$ and this is an automorphism if and only if the rational transformation $f$ defined by $f(X_{1}, \ldots, X_{n}) = (F_{1}, \ldots, F_{n})$ is a birational transformation of the affine space ${\mathbb A}^{n}$. Compactifying ${\mathbb A}^{n}$, we get $${\operatorname{Cr}}(n) = {\operatorname{Bir}}({\mathbb A}^{n}) = {\operatorname{Bir}}({\mathbb P}^{n})$$ where Bir$(X)$ denotes the group of all birational transformations of $X$.
In the end of this section, we define two subgroups in ${\operatorname{Cr}}(n)$ introduced by Gizatullin [@gi94].
Let $V$ be an $(n+1)$-dimensional smooth projective rational manifold and $X \subset V$ a projective variety. The *decomposition group* of $X$ is the group $${\operatorname{Dec}}(V, X) {\coloneqq}\{f \in {\operatorname{Bir}}(V)\ |\ f(X) =X \text{ and } f|_{X} \in {\operatorname{Bir}}(X) \}.$$ The *inertia group* of $X$ is the group $${\operatorname{Ine}}(V, X) {\coloneqq}\{f \in {\operatorname{Dec}}(V, X)\ |\ f|_{X} = {\operatorname{id}}_{X}\}.$$
The decomposition group is also denoted by Bir$(V, X)$. By the definition, the correspondence $$\gamma {\colon}{\operatorname{Dec}}(V, X) \ni f \mapsto f|_{X} \in {\operatorname{Bir}}(X)$$ defines the exact sequence: $$\begin{aligned}
\label{seq}
1 \longrightarrow {\operatorname{Ine}}(V, X) = \ker \gamma \longrightarrow {\operatorname{Dec}}(V, X) \overset{\gamma}{\longrightarrow} {\operatorname{Bir}}(X).\end{aligned}$$
So, it is natural to consider the following question (which is same as Question \[qu\]) asked by Ludmil Katzarkov:
\[qexact\] Is the sequence $$\begin{aligned}
\label{exact}
1 \longrightarrow {\operatorname{Ine}}(V, X) \longrightarrow {\operatorname{Dec}}(V, X) \overset{\gamma}{\longrightarrow} {\operatorname{Bir}}(X) \longrightarrow 1\end{aligned}$$ exact, i.e., is $\gamma$ surjective?
In general, the above sequence is not exact, i.e., $\gamma$ is not surjective. In fact, there is a smooth quartic $K3$ surface $M_{2} \subset {\mathbb P}^{3}$ such that $\gamma$ is not surjective ([@og13 Theorem 1.2 (2)]).
Calabi-Yau hypersurface in ${\mathbb P}^{n+1}$ {#cyn}
==============================================
Our goal, in this section, is to prove Theorem \[intro2\] (i.e. Theorem \[ta\]). Before that, we introduce the result of Takahashi [@ta98].
Let $X$ be a normal ${\mathbb Q}$-factorial projective variety. The 1*-cycle* is a formal linear combination $C = \sum a_{i}C_{i}$ of proper curves $C_{i} \subset X$ which are irreducible and reduced. By the theorem of the base of Néron-Severi (see [@kl66]), the whole numerical equivalent class of 1-cycle with real coefficients becomes the finite dimensional ${\mathbb R}$-vector space and denotes $N_{1}(X)$. The dimension of $N_{1}(X)$ or its dual $N^{1}(X)$ with respect to the intersection form is called the *Picard number* and denote $\rho(X)$.
$($[@ta98 Theorem 2.3]$)$\[tak\] Let $X$ be a Fano manifold $($i.e. a manifold whose anti-canonical divisor $-K_{X}$ is ample,$)$ with $\dim X \geq 3$ and $\rho(X) = 1$, $S \in |-K_{X}|$ a smooth hypersurface with ${\operatorname{Pic}}(X) \to {\operatorname{Pic}}(S)$ surjective. Let $\Phi {\colon}X \dashrightarrow X'$ be a birational map to a ${\mathbb Q}$-factorial terminal variety $X'$ with $\rho(X') = 1$ which is not an isomorphism, and $S' = \Phi_{*}S$. Then $K_{X'} + S'$ is ample.
This theorem is proved by using the *Noether-Fano inequality* which is one of the most important tools in birational geometry, which gives a precise bound on the singularities of indeterminacies of a birational map and some conditions when it becomes isomorphism.
This inequality is essentially due to [@im71], and Corti proved the general case of an arbitrary Mori fiber space of dimension three [@co95]. It was extended in all dimensions in [@ta95], [@bm97], [@is01], and [@df02], (see also [@ma02]). In particular, a log generalized version obtained independently in [@bm97], [@ta95] is used for the proof of Theorem \[tak\].
After that, we consider $n$-dimensional *Calabi-Yau manifold* $X$ in this paper. It is a projective manifold which is simply connected, $$H^{0}(X, \Omega_{X}^{i}) = 0\ \ \ (0<i<\dim X = n),\ \ \textrm{and \ } H^{0}(X, \Omega_{X}^{n}) = {\mathbb C}\omega_{X},$$ where $\omega_{X}$ is a nowhere vanishing holomorphic $n$-form.
The following theorem is a consequence of the Theorem \[tak\], which is same as Theorem \[intro2\]. This provides an example of the Calabi-Yau hypersurface $M_{n}$ whose inertia group consists of only identity transformation.
\[ta\] Suppose $n \geq 3$. Let $M_{n} = (n+2) \subset {\mathbb P}^{n+1}$ be a smooth hypersurface of degree $n+2$. Then $M_{n}$ is a Calabi-Yau manifold of dimension $n$ and Question $\ref{qexact}$ is affirmative for $M_{n}$. More precisely:
- ${\operatorname{Dec}}({\mathbb P}^{n+1}, M_{n}) = \{ f \in {\operatorname{PGL}}(n+2, {\mathbb C}) = {\operatorname{Aut}}({\mathbb P}^{n+1})\ |\ f(M_{n}) = M_{n}\}$.
- ${\operatorname{Ine}}({\mathbb P}^{n+1}, M_{n}) = \{{\operatorname{id}}_{{\mathbb P}^{n+1}}\}$, and $\gamma {\colon}{\operatorname{Dec}}({\mathbb P}^{n+1}, M_{n}) \overset{\simeq}{\longrightarrow} {\operatorname{Bir}}(M_{n}) = {\operatorname{Aut}}(M_{n})$.
By Lefschetz hyperplane section theorem for $n \geq 3$, $\pi_{1}(M_{n}) \simeq \pi_{1}({\mathbb P}^{n+1}) = \{{\operatorname{id}}\}$, ${\operatorname{Pic}}(M_{n}) = {\mathbb Z}h$ where $h$ is the hyperplane class. By the adjunction formula, $$K_{M_{n}} = (K_{{\mathbb P}^{n+1}} + M_{n})|_{M_{n}} = -(n+2)h + (n+2)h = 0$$ in Pic$(M_{n})$.
By the exact sequence $$0 \longrightarrow {\mathcal O}_{{\mathbb P}^{n+1}}(-(n+2)) \longrightarrow {\mathcal O}_{{\mathbb P}^{n+1}} \longrightarrow {\mathcal O}_{M_{n}} \longrightarrow 0$$ and $$h^{k}({\mathcal O}_{{\mathbb P}^{n+1}}(-(n+2))) = 0\ \ \text{for}\ \ 1 \leq k \leq n,$$ $$H^{k}({\mathcal O}_{M_{n}}) \simeq H^{k}({\mathcal O}_{{\mathbb P}^{n+1}}) = 0\ \ \text{for}\ \ 1 \leq k \leq n-1.$$ Hence $H^{0}(\Omega^{k}_{M_{n}}) = 0$ for $1 \leq k \leq n-1$ by the Hodge symmetry. Hence $M_{n}$ is a Calabi-Yau manifold of dimension $n$.
By ${\operatorname{Pic}}(M_{n}) = {\mathbb Z}h$, there is no small projective contraction of $M_{n}$, in particular, $M_{n}$ has no flop. Thus by Kawamata [@ka08], we get ${\operatorname{Bir}}(M_{n}) = {\operatorname{Aut}}(M_{n})$, and $g^{*}h = h$ for $g \in {\operatorname{Aut}}(M_{n}) = {\operatorname{Bir}}(M_{n})$.
So we have $g = \tilde{g}|_{M_{n}}$ for some $\tilde{g} \in {\operatorname{PGL}}(n+1, {\mathbb C})$. Assume that $f \in {\operatorname{Dec}}({\mathbb P}^{n+1}, M_{n})$. Then $f_{*}(M_{n}) = M_{n}$ and $K_{{\mathbb P}^{n+1}} + M_{n} = 0$. Thus by Theorem \[tak\], $f \in {\operatorname{Aut}}({\mathbb P}^{n+1}) = {\operatorname{PGL}}(n+2, {\mathbb C})$. This proves (1) and the surjectivity of $\gamma$.
Let $f|_{M_{n}} = {\operatorname{id}}_{M_{n}}$ for $f \in {\operatorname{Dec}}({\mathbb P}^{n+1}, M_{n})$. Since $f \in {\operatorname{PGL}}(n+1, {\mathbb C})$ by (1) and $M_{n}$ generates ${\mathbb P}^{n+1}$, i.e., the projective hull of $M_{n}$ is ${\mathbb P}^{n+1}$, it follows that $f = {\operatorname{id}}_{{\mathbb P}^{n+1}}$ if $f|_{M_{n}} = {\operatorname{id}}_{M_{n}}$. Hence ${\operatorname{Ine}}({\mathbb P}^{n+1}, M_{n}) = \{{\operatorname{id}}_{{\mathbb P}^{n+1}}\}$, i.e., $\gamma$ is injective. So, $\gamma {\colon}{\operatorname{Dec}}({\mathbb P}^{n+1}, M_{n}) \overset{\simeq}{\longrightarrow} {\operatorname{Bir}}(M_{n}) = {\operatorname{Aut}}(M_{n})$.
Calabi-Yau hypersurface in $({\mathbb P}^{1})^{n+1}$ {#cy1n}
====================================================
As in above section, the Calabi-Yau hypersurface $M_{n}$ of ${\mathbb P}^{n+1}$ with $n \geq 3$ has only identical transformation as the element of its inertia group. However, there exist some Calabi-Yau hypersurfaces in the product of ${\mathbb P}^{1}$ which does not satisfy this property; as result (Theorem \[main\]) shows.
To simplify, we denote $$\begin{aligned}
P(n+1) &{\coloneqq}({\mathbb P}^{1})^{n+1} = {\mathbb P}^{1}_{1} \times {\mathbb P}^{1}_{2} \times \cdots \times {\mathbb P}^{1}_{n+1},\\
P(n+1)_{i} &{\coloneqq}{\mathbb P}^{1}_{1} \times \cdots \times {\mathbb P}^{1}_{i-1} \times {\mathbb P}^{1}_{i+1} \times \cdots \times {\mathbb P}^{1}_{n+1} \simeq P(n),\end{aligned}$$ and $$\begin{aligned}
p^{i} {\colon}P(n+1) &\to {\mathbb P}^{1}_{i} \simeq {\mathbb P}^{1},\\
p_{i} {\colon}P(n+1) &\to P(n+1)_{i}\end{aligned}$$ as the natural projection. Let $H_{i}$ be the divisor class of $(p^{i})^{*}({\mathcal O}_{{\mathbb P}^{1}}(1))$, then $P(n+1)$ is a Fano manifold of dimension $n+1$ and its canonical divisor has the form $\displaystyle{-K_{P(n+1)} = \sum^{n+1}_{i=1}2H_{i}}$. Therefore, by the adjunction formula, the generic hypersurface $X_{n} \subset P(n+1)$ has trivial canonical divisor if and only if it has multidegree $(2, 2, \ldots, 2)$. More strongly, for $n \geq 3$, $X_{n} = (2, 2, \ldots, 2)$ becomes a Calabi-Yau manifold of dimension $n$ and, for $n=2$, a $K3$ surface (i.e. 2-dimensional Calabi-Yau manifold). This is shown by the same method as in the proof of Theorem \[ta\].
From now, $X_{n}$ is a generic hypersurface of $P(n+1)$ of multidegree $(2, 2, \ldots , 2)$ with $n \geq 3$. Let us write $P(n+1) = {\mathbb P}^{1}_{i} \times P(n+1)_{i}$. Let $[x_{i1} : x_{i2}]$ be the homogeneous coordinates of ${\mathbb P}^{1}_{i}$. Hereafter, we consider the affine locus and denote by $\displaystyle x_{i} = \frac{x_{i2}}{x_{i1}}$ the affine coordinates of ${\mathbb P}^{1}_{i}$ and by ${\bf z}_{i}$ that of $P(n+1)_{i}$. When we pay attention to $x_{i}$, $X_{n}$ can be written by following equation $$\begin{aligned}
\label{xn}
X_{n} = \{ F_{i,0}({\bf z}_{i})x_{i}^{2} + F_{i,1}({\bf z}_{i})x_{i} + F_{i,2}({\bf z}_{i}) = 0 \}\end{aligned}$$ where each $F_{i,j}({\bf z}_{i})$ $(j = 0, 1, 2)$ is a quadratic polynomial of ${\bf z}_{i}$. Now, we consider the two involutions of $P(n+1)$: $$\begin{aligned}
\tau_{i} {\colon}(x_{i}, {\bf z}_{i}) &\to \left(-x_{i}- \frac{F_{i,1}({\bf z}_{i})}{F_{i,0}({\bf z}_{i})}, {\bf z}_{i} \right)\label{tau}\\
\sigma_{i} {\colon}(x_{i}, {\bf z}_{i}) &\to \left(\frac{F_{i,2}({\bf z}_{i})}{x_{i} \cdot F_{i,0}({\bf z}_{i})}, {\bf z}_{i} \right).\label{sigma}\end{aligned}$$ Then $\tau_{i}|_{X_{n}} = \sigma_{i}|_{X_{n}} = \iota_{i}$ by definition of $\iota_{i}$ (cf. Theorem \[iota\]).
We get two birational automorphisms of $X_{n}$ $$\begin{aligned}
\rho_{i} = \sigma_{i} \circ \tau_{i} {\colon}(x_{i}, {\bf z}_{i}) &\to \left( \frac{F_{i,2}({\bf z}_{i})}{-x_{i} \cdot F_{i,0}({\bf z}_{i}) - F_{i,1}({\bf z}_{i})}, \ {\bf z}_{i} \right)\\
\rho'_{i} = \tau_{i} \circ \sigma_{i} {\colon}(x_{i}, {\bf z}_{i}) &\to \left( -\frac{x_{i} \cdot F_{i,1}({\bf z}_{i}) + F_{i,2}({\bf z}_{i})}{x_{i}\cdot F_{i,0}({\bf z}_{i})}, \ {\bf z}_{i} \right).\end{aligned}$$ Obviously, both $\rho_{i}$ and $\rho'_{i}$ are in Ine$(P(n+1), X_{n})$, map points not in $X_{n}$ to other points also not in $X_{n}$, and $\rho_{i}^{-1} = \rho'_{i}$ by $\tau_{i}^{2} = \sigma_{i}^{2} = {\operatorname{id}}_{P(n+1)}$.
\[order\] Each $\rho_{i}$ has infinite order.
By the definiton of $\rho_{i}$ and $\rho'_{i} = \rho_{i}^{-1}$, it suffices to show $$\begin{aligned}
{\begin{pmatrix}
0 & F_{i,2}\\
-F_{i,0} & -F_{i,1}
\end{pmatrix}}^{k}
\neq \alpha I\end{aligned}$$ for any $k \in {\mathbb Z}\setminus \{0\}$ where $I$ is an identity matrix and $\alpha \in {\mathbb C}^{\times}$. Their eigenvalues are $$\frac{-F_{i,1} \pm \sqrt{F_{i,1}^{2} - 4F_{i,0}F_{i,2}}}{2}.$$ Here $F_{i,1}^{2} - 4F_{i,0}F_{i,2} \neq 0$ as $X_{n}$ is general (for all $i$).
If $\begin{pmatrix}
0 & F_{i,2}\\
-F_{i,0} & -F_{i,1}
\end{pmatrix}^{k}
= \alpha I$ for some $k \in {\mathbb Z}\setminus \{0\}$ and $\alpha \in {\mathbb C}^{\times}$, then $$\left(\frac{-F_{i,1} + \sqrt{F_{i,1}^{2} - 4F_{i,0}F_{i,2}}}{-F_{i,1} - \sqrt{F_{i,1}^{2} - 4F_{i,0}F_{i,2}}}\right)^{k} = 1,$$ a contradiction to the assumption that $X_{n}$ is generic.
We also remark that Proposition \[order\] is also implicitly proved in Theorem \[main\].
Our main result is the following (which is same as Theorem \[intro\]):
\[main\] Let $X_{n} \subset P(n+1)$ be a smooth hypersurface of multidegree $(2, 2, \ldots, 2)$ and $n \geq 3$. Then:
- $\gamma {\colon}{\operatorname{Dec}}(P(n+1), X_{n}) \to {\operatorname{Bir}}(X_{n})$ is surjective, in particular Question $\ref{qexact}$ is affirmative for $X_{n}$.
- If, in addition, $X_{n}$ is generic, $n+1$ elements $\rho_{i} \in {\operatorname{Ine}}(P(n+1), X_{n})$ $(1 \leq i \leq n+1)$ satisfy $$\langle \rho_{1}, \rho_{2}, \ldots , \rho_{n+1} \rangle \simeq \underbrace{{\mathbb Z}* {\mathbb Z}* \cdots * {\mathbb Z}}_{n+1} \subset {\operatorname{Ine}}(P(n+1), X_{n}).$$ In particular, ${\operatorname{Ine}}(P(n+1), X_{n})$ is an infinite non-commutative group.
Let ${\operatorname{Ind}}(\rho)$ be the union of the indeterminacy loci of each $\rho_{i}$ and $\rho^{-1}_{i}$; that is, $\displaystyle {\operatorname{Ind}}(\rho) = \bigcup_{i=1}^{n+1}\big({\operatorname{Ind}}(\rho_{i}) \cup {\operatorname{Ind}}(\rho^{-1}_{i})\big)$ where ${\operatorname{Ind}}(\rho_{i})$ is the indeterminacy locus of $\rho_{i}$. Clearly, ${\operatorname{Ind}}(\rho)$ has codimension $\geq 2$ in $P(n+1)$.
Let us show Theorem \[main\] (1). Suppose $X_{n}$ is generic. For a general point $x \in P(n+1)_{i}$, the set $p_{i}^{-1}(x)$ consists of two points. When we put these two points $y$ and $y'$, then the correspondence $y \leftrightarrow y'$ defines a natural birational involutions of $X_{n}$, and this is the involution $\iota_{j}$. Then, by Cantat-Oguiso’s result [@co11 Theorem 3.3 (4)], ${\operatorname{Bir}}(X_{n})$ $(n\geq 3)$ coincides with the group $\langle \iota_{1}, \iota_{2}, \ldots , \iota_{n+1} \rangle \simeq \underbrace{{\mathbb Z}/2{\mathbb Z}* {\mathbb Z}/2{\mathbb Z}* \cdots * {\mathbb Z}/2{\mathbb Z}}_{n+1}$.
Two involutions $\tau_{j}$ and $\sigma_{j}$ of $X_{n}$ which we construct in and are the extensions of the covering involutions $\iota_{j}$. Hence, $\tau_{j}|_{X_{n}} = \sigma_{j}|_{X_{n}} = \iota_{j}$. Thus $\gamma$ is surjective. Since automorphisms of $X_{n}$ come from that of total space $P(n+1)$, it holds the case that $X_{n}$ is not generic. This completes the proof of Theorem \[main\] (1).
Then, we show Theorem \[main\] (2). By Proposition \[order\], order of each $\rho_{i}$ is infinite. Thus it is sufficient to show that there is no non-trivial relation between its $n + 1$ elements $\rho_{i}$. We show by arguing by contradiction.
Suppose to the contrary that there is a non-trivial relation between $n+1$ elements $\rho_{i}$, that is, there exists some positive integer $N$ such that $$\begin{aligned}
\label{rho}
\rho_{i_{1}}^{n_{1}} \circ \rho_{i_{2}}^{n_{2}} \circ \cdots \circ \rho_{i_{l}}^{n_{l}} = {\operatorname{id}}_{P(n+1)}\end{aligned}$$ where $l$ is a positive integer, $n_{k} \in {\mathbb Z}\setminus\{0\}$ $(1\leq k \leq l)$, and each $\rho_{i_{k}}$ denotes one of the $n + 1$ elements $\rho_{i}$ $(1 \leq i \leq n+1)$ and satisfies $\rho_{i_{k}} \neq \rho_{i_{k+1}}$ $(0 \leq k \leq l-1)$. Put $N = |n_{1}| + \cdots + |n_{l}|$.
In the affine coordinates $(x_{i_{1}}, {\bf z}_{i_{1}})$ where $x_{i_{1}}$ is the affine coordinates of $i_{1}$-th factor ${\mathbb P}^{1}_{i_{1}}$, we can choose two distinct points $(\alpha_{1}, {\bf z}_{i_{1}})$ and $(\alpha_{2}, {\bf z}_{i_{1}})$, $\alpha_{1} \neq \alpha_{2}$, which are not included in both $X_{n}$ and ${\operatorname{Ind}}(\rho)$.
By a suitable projective linear coordinate change of ${\mathbb P}^{1}_{i_{1}}$, we can set $\alpha_{1} = 0$ and $\alpha_{2} = \infty$. When we pay attention to the $i_{1}$-th element $x_{i_{1}}$ of the new coordinates, we put same letters $F_{i_{1},j}({\bf z}_{i_{1}})$ for the definitional equation of $X_{n}$, that is, $X_{n}$ can be written by $$X_{n} = \{ F_{i_{1},0}({\bf z}_{i_{1}})x_{i_{1}}^{2} + F_{i_{1},1}({\bf z}_{i_{1}})x_{i_{1}} + F_{i_{1},2}({\bf z}_{i_{1}}) = 0 \}.$$ Here the two points $(0, {\bf z}_{i_{1}})$ and $(\infty, {\bf z}_{i_{1}})$ not included in $X_{n} \cup {\operatorname{Ind}}(\rho)$. From the assumption, both two equalities hold:
\_[i\_[1]{}]{}\^[n\_[1]{}]{} \_[i\_[l]{}]{}\^[n\_[l]{}]{}(0, [**z**]{}\_[i\_[1]{}]{}) = (0, [**z**]{}\_[i\_[1]{}]{}) &\
\_[i\_[1]{}]{}\^[n\_[1]{}]{} \_[i\_[l]{}]{}\^[n\_[l]{}]{}(, [**z**]{}\_[i\_[1]{}]{}) = (, [**z**]{}\_[i\_[1]{}]{}).\[infty\]
We proceed by dividing into the following two cases.
[(i). The case where $n_{1} > 0$. Write $\rho_{i_{1}} \circ \rho_{i_{1}}^{n_{1}-1} \circ \rho_{i_{2}}^{n_{2}} \circ \cdots \circ \rho_{i_{l}}^{n_{l}} = {\operatorname{id}}_{P(n+1)}$. ]{}
Let us denote $\rho_{i_{1}}^{n_{1}-1} \circ \cdots \circ \rho_{i_{l}}^{n_{l}}(0, {\bf z}_{i_{1}}) = (p, {\bf z}_{i_{1}}')$, then, by the definition of $\rho_{i_{1}}$, it maps $p$ to $0$. That is, the equation $F_{i_{1},2}({\bf z}'_{i_{1}}) = 0$ is satisfied. On the other hand, the intersection of $X_{n}$ and the hyperplane $(x_{i_{1}}=0)$ is written by $$X_{n} \cap (x_{i_{1}}=0) = \{F_{i_{1},2}({\bf z}_{i_{1}}) = 0\}.$$ This implies $(0, {\bf z}'_{i_{1}}) = \rho_{i_{1}}(p, {\bf z}'_{i_{1}}) = (0, {\bf z}_{i_{1}})$ is a point on $X_{n}$, a contradiction to the fact that $(0, {\bf z}_{i_{1}}) \notin X_{n}$.
[(ii). The case where $n_{1} < 0$. Write $\rho^{-1}_{i_{1}} \circ \rho_{i_{1}}^{n_{1}+1} \circ \rho_{i_{2}}^{n_{2}} \circ \cdots \circ \rho_{i_{l}}^{n_{l}} = {\operatorname{id}}_{P(n+1)}$. ]{}
By using the assumption , we lead the contradiction by the same way as in (i). Precisely, we argue as follows.
Let us write $\displaystyle x_{i_{1}} = \frac{1}{y_{i_{1}}}$, then $(x_{i_{1}} = \infty, {\bf z}_{i_{1}}) = (y_{i_{1}} = 0, {\bf z}_{i_{1}})$ and $X_{n}$ and $\rho^{-1}_{i_{1}}$ can be written by $$X_{n} {\coloneqq}\{F_{i_{1},0}({\bf z}_{i_{1}}) + F_{i_{1},1}({\bf z}_{i_{1}})y_{i_{1}} + F_{i_{1},2}({\bf z}_{i_{1}})y_{i_{1}}^{2} = 0\},$$ $$\rho^{-1}_{i_{1}} {\colon}(y_{i_{1}}, {\bf z}_{i_{1}}) \to \left(\ -\frac{F_{i_{1},0}({\bf z}_{i_{1}})}{F_{i_{1},1}({\bf z}_{i_{1}}) + y_{i_{1}}\cdot F_{i_{1},2}({\bf z}_{i_{1}})},\ {\bf z}_{i_{1}} \right).$$
Let us denote $ \rho_{i_{1}}^{n_{1}+1} \circ \rho_{i_{2}}^{n_{2}} \circ \cdots \circ \rho_{i_{l}}^{n_{l}} (y_{i_{1}} =0, {\bf z}_{i_{1}}) = (y_{i_{1}} = q, {\bf z}_{i_{1}}'')$, then $\rho^{-1}_{i_{1}}$ maps $q$ to $0$. That is, the equation $F_{i_{1},0}({\bf z}''_{i_{1}}) = 0$ is satisfied, but the intersection of $X_{n}$ and the hyperplane $(y_{i_{1}} = 0)$ is written by $$X_{n}\cap (y_{i_{1}} = 0) = \{F_{i_{1},0}({\bf z}_{i_{1}}) = 0\}.$$ This implies $(y_{i_{1}}=0, {\bf z}''_{i_{1}}) = \rho^{-1}_{i_{1}}(y_{i_{1}} = q, {\bf z}_{i_{1}}'') = (x_{i_{1}}=\infty, {\bf z}_{i_{1}})$ is a point on $X_{n}$; that is, $(x_{i_{1}}=\infty, {\bf z}_{i_{1}}) \in X_{n} \cap (x_{i_{1}}=\infty)$. This is contradiction.
From (i) and (ii), we can conclude that there does not exist such $N$. This completes the proof of Theorem \[main\] (2).
Note that, for the cases $n = 2$ and $1$, Theorem \[main\] (2) also holds though (1) does not hold.
[**Acknowledgements:** ]{} The authors would like to express their sincere gratitude to their supervisor Professor Keiji Oguiso who suggested this subject and has given much encouragement and invaluable and helpful advices.
[aaaaaa]{}
A. Bruno, K. Matsuki, *Log Sarkisov program*, Internat. J. Math. [**8**]{} no.4 (1997), 451-494.
S. Cantat, K. Oguiso, *Birational automorphism group and the movable cone theorem for Calabi-Yau manifolds of Wehler type via universal Coxeter groups*, preprint [arXiv:1107.5862](http://arxiv.org/abs/1107.5862), to appear in Amer. J. Math.
A. Corti, *Factoring birational maps of threefolds after Sarkisov*, J. Algebraic Geom. [**4**]{} no.2 (1995), 223-254.
T. de Fernex, *Birational transformations of varieties*, University of Illinois at Chicago Ph. D. Thesis (2002).
M. H. Gizatullin, *The decomposition, inertia and ramification groups in birational geometry*, Algebraic Geometry and its Applications, Aspects of Math. [**E25**]{} (1994), 39-45.
V. A. Iskovskikh, *Birational rigidity of Fano hypersurfaces in the framework of Mori theory*, Usp. Mat. Nauk [**56**]{} no.2 (2001), 3-86; English transl., Russ. Math. Surveys [**56**]{} no.2 (2001), 207-291.
V. A. Iskovskikh, Yu. I. Manin, *Three-dimensional quartics and counterexamples to the Lüroth problem*, Mat. Sb. [**86**]{} no.1 (1971), 140-166; English transl., Math. Sb. [**15**]{} no.1 (1971), 141-166.
Y. Kawamata, *Flops connect minimal models*, Publ. Res. Inst. Math. Sci. [**44**]{} no.2 (2008), 419-423.
S. Kleiman, *Toward a numerical theory of ampleness*, Ann. of Math. [**84**]{} no.3 (1966), 293-344.
K. Matsuki, *Introduction to the Mori Program*, Universitext, Springer, New York (2002).
K. Oguiso, *Quartic K3 surfaces and Cremona transformations*, Arithmetic and geometry of $K3$ surfaces and Calabi-Yau threefolds, Fields Inst. Commun. [**67**]{}, Springer, New York (2013), 455-460.
N. Takahashi, *Sarkisov program for log surfaces*, Tokyo University Master Thesis, 1995.
N. Takahashi, *An application of Noether-Fano inequalities*, Math. Z. [**228**]{} no.1 (1998), 1-9.
| ArXiv |
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abstract: 'We present optical candidates for 75 X-ray sources in a $\sim 1$ deg$^2$ overlapping region with the medium deep ROSAT survey (Molthagen et al. 1997). These candidates are selected using the multi-color CCD imaging observations made for the T329 field of the Beijing-Arizona-Taipei-Connecticut (BATC) Sky Survey, which utilizes the NAOC 0.6/0.9m Schmidt telescope with 15 intermediate-band filters covering the wavelength range 3360-9745 Å. These X-ray sources are relatively faint (CR $<< 0.2 s^{-1}$) and thus mostly are not included in the RBS catalog, they also remain as X-ray sources without optical candidates in a previous identification program carried out by the Hamburg Quasar Survey. Within their position-error circles, almost all the X-ray sources are observed to have one or more spatially associated optical candidates within them down to the magnitude $m_V \sim 23.1$. We have classified 149 of 156 detected optical candidates with 73 of the 75 X-ray sources with a new method which predicts a redshift for non-stellar objects, which we have termed the SED-based Object Classification Approach (SOCA). These optical candidates include: 31 QSOs, 39 stars, 37 starburst galaxies, 42 galaxies, and 7 “just” visible objects. Twenty-eight X-ray error circles have only one visible object in them: 9 QSOs, 3 normal galaxies, 8 starburst galaxies, 6 stars, and two of the “just” visible objects. We have also cross-correlated the positions of these optical objects with NED, the FIRST radio source catalog and the 2MASS catalog. Separately, we have also SED-classified the remaining 6011 objects in our field of view. Optical objects are found at the $6.5\sigma$ level above what one would expect from a random distribution, only QSOs are over-represented in these error circles at greater than 4$\sigma$ frequency. We estimate redshifts for all extragalactic objects, and find a good correspondence of our predicted redshift with the measured redshift (a mean error of 0.04 in $\Delta z$. There appears to be a supercluster at z $\sim$ 0.3-0.35 in this direction, including many of the galaxies in the X-ray error circles are found in this redshift range.'
author:
- |
Haotong Zhang, Suijian Xue, David Burstein,\
Xu Zhou, Zhaoji Jiang, Hong Wu, Jun Ma, Jiansheng Chen,\
and Zhenlong Zou
title: 'Multicolor Photometric Observations of Optical Candidates to Faint ROSAT X-ray Sources in a 1 deg$^2$ field of the BATC Survey'
---
1.5cm
[**keywords:**]{} X-rays: galaxies - galaxies: active - catalog: surveys
Introduction
============
Combined optical and X-ray data allow one to obtain information about the luminosity functions of various types of X-ray sources as well as their evolution with redshift. In turn, this information can be used to further constrain models for the production of the X-ray background at different flux levels. Much effort has so far been made on the optical identification of the X-ray sources in the ROSAT/Bright Source (RBS) catalog (e.g., Voges et al. 1999; Rutledge et al. 2000) as well as of X-ray sources in some individual ROSAT deep survey observations (e.g., Lehmann et al. 2001).
Yet, it is often unknown how much of the detections that occur in X-ray error circles are real associations of optical counterparts to these X-ray sources, and how much of these associations are due to random chance. To do this, one needs to have detected and identified all optical objects in a given image, and then see what percentages of these objects (QSOs, galaxies, stars) are then found near or within the areas covered by the X-ray error circles. This is precisely the kind of data we have for 75 X-ray sources detected with the ROSAT PSPC to a flux limit $S_x({\rm 0.1-2.4
keV}) \geq 5.3\times10^{-14}$ , in a 1 deg$^2$ field of view, as this field of view was also observed in our multicolor images for the Beijing-Arizona-Taipei-Connecticut Sky Survey (BATC survey). The relevant X-ray and optical data are presented in § 2. Details of the object classification procedure as well as selection of the X-ray candidates are given in § 3. Associated information that can be gleaned from these data are given in § 4. We summarize our results in § 5.
The data and analysis
=====================
The X-ray data
--------------
The X-ray data come from a catalog obtained from a medium deep ROSAT survey in the HQS field HS 47.5/22 (Molthagen, Wendker, & Briel, 1997). The survey consists of 48 overlapping ROSAT PSPC pointings which were added up to produce a final catalog containing 574 X-ray sources with broad band (0.1-2.4 keV) count rates between $\sim3\times10^{-3}\rm cts\ s^{-1}$ and $\sim0.2\rm
cts\ s^{-1}$, in a field of view (FOV) of $\sim 2.3\rm\ deg^2$. Molthagen et al. adopt an X-ray error circle of 2$\sigma$ + 10$''$ in radius, with the value of $\sigma$ coming from their observations. This is the X-ray error circle used in the present analysis.
There was a preliminary identification of these X-ray sources with the HQS plates (Molthagen et al, 1997). Only a few objects, all brighter than $m_B\approx18^m.5$, have recognizable spectra. At $m_{\rm B}>18^m.5$, many objects are generally classified as weak and extremely blue, blue or red. For many X-ray sources no spectral classification was possible, the optical object simply being visible or the field of view empty. 75 of the 574 HQS sources fall on one program field of the BATC survey, T329, centered at 09:56:24.46, +47:35:08.4 (J2000), forming a subsample of the ROSAT medium deep survey in a 1 deg$^2$ field. (One-third of the BATC fields are located with a known quasar in its center. For field T329, this quasar is PC0953+4749 with z = 4.46, originally discovered by Schneider, Schmidt & Gunn 1991. Ironically, this QSO is not an X-ray source in the HQS field.) The X-ray brightness distribution of these 75 sources is shown in Fig.\[f1\]. The distribution of these 75 sources in our field of view in shown in Fig.\[x\_dist\].
Molthagen et al. associate 25 optical candidates for these 75 X-ray sources, or a frequency of 1/3: 6 QSOs or active galaxies; 7 QSO/active galaxy candidates (classified as such or extremely blue); 1 star; 8 stellar candidates; 1 galaxy candidate; 2 faint red objects; 5 unidentified spectra (including overlaps); 39 visible on the HQS direct plate only; and 6 empty fields (i.e., no counterpart on the HQS plate).
The BATC optical data
---------------------
Optical observations of BATC field T329 were carried out from 1996-1999 as part of the BATC Survey. Our survey utilizes the 0.6/0.9m Schmidt telescope of the Xinglong Observing Station of the National Astronomical Observatory of China (NAOC), equipped with 15 intermediate-band filters covering the wavelength range 3360-9745Å. With this facility our survey is designed to do multi-color CCD ($2048\times2048$) imaging of 500 selected, $\sim
1$ deg$^2$ fields-of-view for multiple scientific purposes (cf. Fan et al. 1996; Shang et al. 1998; Zheng et al. 1999; Zhou et al. 1999; Yan et al. 2000; Kong et al. 2000; Ma et al. 2002; Wu et al. 2002).
The dataset for T329 consists of a number of individual direct CCD images in each of the 15 BATC passbands. These images are first treated individually (bias, dark and flat-fielding corrections) and then combined to comprise deep images. Information on the passbands used for the present study, including filter parameters, total exposure time, number of flux calibration images obtained, and the magnitude limit for that passband are given in Table \[table1\]. Details on the BATC flux calibration procedure are given in several previous papers (Fan et al. 1996; Zhou et al. 1999; Yan et al. 2000) and the reader is referred to those papers for this information. Further discussion of the observations made in field T329 that are separate from the X-ray identification issue are given in Zhou et al. (2003). The final product of the BATC observations of field T329 is a catalog of 6160 point-like optical objects in our 58 arcmin$^2$ field of view, with astrometry and photometry in 15 colors.
SED classification
------------------
We are in the process of developing a SED-based Object Classification Approach (termed SOCA) for the BATC photometric system (Zhang, et al., in preparation). The SED of each object in our field of view, observed through $n$ filters, is compared to the SED computed for a set of template spectra. The aim is to find the best fit between the observed photometry and the model photometry through a standard $\chi^2$ minimization procedure:
$$\chi^2=\sum^N_{i=1}(\frac{f^{obs}_i-A \cdot
f^{temp}_i}{{\sigma}^{obs}_i})^2$$
where $f^{obs}_i$ and $f^{temp}_i$ are the observed and the template fluxes in the $i$th band respectively, ${\sigma}^{obs}_i$ is the error on the observed flux in this band, $A$ is the normalization constant can be calculated by minimize the $\chi^2$. In order to apply this method to SED classification, we currently employ three sets of template spectra:
1. The stellar library of Gunn & Stryker (1983) is used including most spectral classes on the MK system (this will be updated when new data are available);
2. The observed spectra of nearby galaxies (Kinny et al. 1996),including normal galaxies (Elliptical ,S0, Sa, Sb and Sc) and starburst (SB) galaxies (SB1-6) with different internal extinctions are used. Normal galaxies are redshifted from 0 to 1.0 in step of 0.01, SB galaxies are redshifted from 0 to 1.5 in step of 0.01.
3. A QSO template set is composed of series of simulated quasar spectra. These spectra have been constructed by fixing the emission line intensity ratio (cf. Wilkes 1986), while varying the $ly\alpha$ equivalent width (65 $\pm$ 34 Å) and the continuum index $\alpha$ (-0.75$\pm$ 0.5). $Ly\alpha$ forest absorption has been modeled according to M[ø]{}ller (1990) and Madau (1995). Redshift estimates are set between 0.0 and 6.0 in steps of 0.01 in $z$.
Representative template spectra used in the present paper are given in Fig.\[f2\]. The template SEDs are obtained by convolving the template spectra with the measured passband of each filter. As the template SEDs are morphologically-classified, some templates may represent two or more morphologically-similar classes. For example, an SED classified as a starburst galaxy can also possibly be matched to that of a QSO.
A value of $\chi^2$ is calculated for the correspondence of every template to each object SED. The minimum $\chi^2$ for each kind of template (star/galaxy/QSO/starburst galaxy) is calculated. The template with the $\chi^2$ minimum fit is taken as the best fit. In this fitting process we include those objects with at least 5 filter observations (such as saturated stars). The redshift estimates found for non-stellar objects (galaxies, QSOs) by this template-fitting process are useful for statistical studies of this field of view.
Optical Candidates
==================
Optical Objects near or within the X-Ray Error Circles
------------------------------------------------------
The CCD limiting magnitudes range from 20.5 to 23.5 mag, tending to be fainter in the bluer filters, and brighter in the more sky-limited redder filters (cf. Table \[table1\]). Our deeper, direct CCD observations, combined with our ability to classify the SEDs of the objects detected, permit us both to detect more objects than the HQS survey, as well as to classify more of the objects detected.
The total area covered by the X-ray searches we have done for the 75 X-ray sources corresponds to 31.52 arcmin$^2$ or 0.00937 of the 3364 arcmin$^2$ sky area subtended by the BATC CCD. This area includes additional area searched beyond the nominal 2$\sigma$ error circle for 13 (17%) of the X-ray-detected objects, most of these within 1-2$''$ of the original error circle. The 6160 objects detected in the full image field were selected with the same criteria as those we use for the X-ray error circle. If the optical objects and the X-ray sources are randomly associated, we expect to detect $31.52/3364 \times 6160 = 58$ optical objects.
Our observations find optical candidates (stars, galaxies, galaxy groups, starburst galaxies, QSOs) in 73 of the 75 X-ray error circles. We detect a total of 156 optical objects in these 75 X-ray error circles. Of these we can definitely SOCA-identify 140, tentatively identify 9 more (7 galaxies and two stars), 7 objects are only “visible,” and one X-ray circle (RXJ0955.5+4735) which is blank in our image (but two stars are just outside this error circle). This makes a total of 149 optically-identified candidates found in the BATC catalog that can be SOCA-classified and are found in or near 73 of the 75 X-ray error circles in our field of view. One of the two remaining X-ray sources (RXJ0954.0+4756) has a “just” visible object within the X-ray error circle that has a position coincident with a known radio source, but is too faint to obtain a reliable SED. The other remaining source (RXJ0953.7+4722) also has one “just” visible source within its error circle.
We have a difference in objects detected in the X-ray circles to those randomly expected of $149 - 58 = 91\pm14$(assuming Gaussian errors). The difference between detected objects and random placement of objects in these X-ray error circles is significant at the 91/14 = 6.5$\sigma$ level. It would appear that the X-ray circles do tend to include more objects than randomly placed circles put on the rest of this field of view, when the data are sampled to faint magnitude levels.
Table \[table2\] gives the relevant information on the optical candidates that are associated with these X-ray sources. The first 4 columns in Table \[table2\] come from the original X-ray catalog: X-ray source name, brightness in the 0.1–2.4 keV passband, 2$\sigma$ error circle radius in units of arcseconds, and the original HQS identification. The label assigned to each candidate optical object, a,b,c,$\ldots$, plus the observed position of the optical candidate (in J2000 coordinates) are given in the next three columns. Columns 8-12 give the derived information for each optical candidate: $\Delta r$ is the offset of the optical candidate position from the center of the X-ray error circle; $m_V$ denotes the V magnitude of the optical candidate (an upper limit is given if the candidate is only visible, but not measurable), derived from the relation: $m_{\rm
V}=m_{\rm g} + 0.3233(m_{\rm f} - m_{\rm h}) + 0.0590$ (Zhou et al., 2003); $f_{xo}$ is the ratio of X-ray to optical flux, calculated from the 0.1-2.4 keV count-rate and optical V magnitude, vis. $f_{\rm xo} = {\rm log}(f_x/f_o) = {\rm log(PSPC\
counts/s}\times10^{-11}) +0.4m_V + 5.37$ (Maccacaro et al. 1988); Pred z is the redshift that the SOCA estimates for galaxies and QSOs; Where there are known candidates that are clearly identified on our images, the identity of these candidates are given. Finding charts of all the X-ray sources in our summed image at 7050Å(our j filter) are given in Fig.\[idt\].
The SEDs for all 149 SOCA-identified optical candidates in or near 73 of the 75 X-ray source error circles are given in Fig. \[sed\], in which also the best template fit is plotted for each optical candidate. The label on each SED gives the template plotted, the predicted redshift (if galaxy or QSO), and the value of the $\chi^2$ fit. In the case of two known objects (a nearby galaxy and a bright star; see next section), their previously-known identifications are given in place of the SOCA classification in Table \[table2\].
Candidate Associations
----------------------
Most X-ray sources contain more than one optical candidate within their error circles. Choosing which one is the probable X-ray source is educated guesswork at best. Rather than assign probabilities of the likelihood of each optical candidate’s association with these X-ray sources, we prefer to give the reader the statistics of how the candidates relate to the full data on the 6160 objects found in our field of view.
In the 75 X-ray error circles, to a magnitude limit of V $\approx$ 23, we have found: 31 QSOs, 39 stars, 37 starburst galaxies, 42 galaxies, and 7 “only visible” objects. If we take the 6160 objects we find in our field of view, excluding the objects within the X-ray error circles, the analogous counts for these objects are: 341 QSOs, 1912 stars, 1508 starburst galaxies, 2076 galaxies, and 174 objects unclassified for a variety of reasons (too few filters observed, in the halo of a bright star, etc.). If we assume these objects are randomly distributed in this field of view, we expect random associations within our X-ray error circles to be 3.2 QSOs, 17.9 stars, 14.1 starburst galaxies, and 19.5 galaxies (discarding the unclassified objects). Therefore, the objects found in the X-ray error circles rather than the random placement of those identified objects in the field of view are: QSOs: $31-3.2 =27.8\pm5.8$; stars: $39-17.9 = 21.1\pm7.9$ ; starburst galaxies: $37-14.1 =22.9\pm7.1$; and galaxies: $42-19.5
= 22.5\pm7.8$.
On the plus side, all objects are more represented within the X-ray error circles that what would be randomly there. On the minus side, only the QSOs have a highly significant overdensity ($4.7\sigma$) within the X-ray error circles, while stars, starburst galaxies and galaxies are only there at the 2.8-3.2$\sigma$ level. In Fig\[stardis\]-\[qsodis\] we show the distribution of each kind of object (QSO, star, starburst galaxy, galaxy) in our field of view, relative to the placement and size of the X-ray error circles. That QSOs are statistically most reliable detections comes as little surprise, as this was already well-known (e.g., Shanks et al. 1991, Georgantopoulos et al. 1996, McHardy et al. 1998).
We only find one optical candidate in 28 of these error circles, these being: 9 QSOs, 3 normal galaxies, 8 starburst galaxies, 6 Galactic stars, and 2 candidates that are “just” visible on our image. The single candidates in these X-ray error circles have an asterisk by their SOCA classification in Table\[table2\].
Supplemental Data and Redshift Distributions
============================================
Radio and Near-IR identifications
---------------------------------
We have cross-correlated the positions of all of the X-ray sources in our field-of-view with positions in the FIRST (Faint Images of Radio Sources at Twenty centimeters) radio survey, the infrared 2MASS (2 Micron All Sky Survey) catalog, and the NED catalog. We find seven of the X-ray error circles have FIRST radio sources within them that are coincident with optical sources, in addition to one associated with another radio source (HS0954+4815 as given by NED; see Table \[table2\]). Of these 8 associated radio sources, just two do not have SEDs in our data, and only three have another object in their X-ray error circle.
There are a total of 100 radio sources in our image. The probability of having one radio source within 2$''$ of any optical source in this image is 0.47 (taking the joint probability). This makes the probability of having all eight position matches for the radio sources with the optical sources at $2.3\times10^{-6}$ . Hence, most of the radio sources are likely associated with their optical counterparts.
In contrast, of the seven 2MASS object with position coincident with our optical candidates, only 2 are classified as galaxies \[one normal galaxy, RXJ0953.8+4740(a) and one starburst galaxy, RXJ0956.9+4731(a)\], and five are identified with late-type stars that are the dominant stellar candidates within these particular X-ray error circles.
Spectroscopic Observations and Objects of Special Note
------------------------------------------------------
We have obtained spectra of a subsample of our optically-detected candidates to test our classification method and SOCA redshift estimates. These spectra were taken with the slit spectrograph on NAOC’s 2.16 m telescope at its Xinglong Observing Station, and the Multiobject Fiber Spectrograph (MOFS) on the 6m telescope of the Special Astrophysical Observatory of the Russian Academy of Sciences. Nine of these spectra are shown in Fig.\[spec\], with the BATC fluxes overlayed. Included are 7 QSOs, 1 starburst galaxy and the HII galaxy associated with UGC 5354. In addition, a search of the SIMBAD and NASA Extragalactic Database (NED) catalogs comes up with redshifts for two additional QSOs associated with these X-ray sources (cf. Table \[table2\]).
The correspondence of spectroscopically-determined redshifts with our SOCA-determined redshifts is excellent for 10 of the objects (a mean error in z of 0.04 for these 10 objects), while it is off by 1.03 for one additional object (RXJ0958.5+4738(a); shown in Fig. \[spec\]). Examination in detail of the SOCA fitting procedure for this one object shows that the BATC filter system mistakes the emission line in the 9190[Å]{} filter for \[OII\] 3727, while the spectrum shows this must be H$\alpha$.
We have also made careful visual inspections for each of optical counterparts within the X-error circles, relating their visual appearance to the SEDs we obtain for them (something the reader can also do, using Fig. \[sed\] and Fig. \[idt\]). We give special note to the objects found in the following X-ray sources:
RXJ0953.8+4740: There is evidently a galaxy group in this error circle, comprised of objects b,c and d. This group has previously been identified as PDCS 36 (the 36th cluster/group found in the Palomar Distant Cluster Survey, Postman et al. 1996).
RXJ0954.0+4756: This object is too faint to identify optically, but its position is coincident with the radio source 7C0952+4814 = FIRST J095401.1+475644.
RXJ0954.8+4715: The second object in Table 2 listed for this source has a position coincident with the radio source CDS90-R307B = FIRST J095453.2+471533.
RXJ0955.1+4729: There are 7 objects found in the X-ray error circle. RXJ0955.1+4729(e) is a confirmed QSO at a redshift 2.15 and (d) is a late type star. What is interesting is that a,b,c,f and g are classified as normal galaxies at redshifts 0.33,0.36,0.36,0.39 and 0.33 respectively. Visual inspection also tends to put them at similar distances, making them a possible galaxy group.
RXJ0956.7+4729: There are 3 objects within the error circle. From the image we can see that all of them are within the diffraction spikes of the nearby bright star, thus their SED are suspect. As a result, we put question marks in their identification in Table \[table2\].
RXJ0958.8+4744: This is part of the nearby, interacting galaxy UGC 5354. Part of this galaxy system is a small, HII galaxy off to one side.
RXJ09058.9+4745: This is a bright, cataloged F8 star, BD+48-1823.
Redshift Distribution
---------------------
In assembling the estimated redshifts for our X-ray associated objects, we noticed that many of them tended to be clustered around a redshift of 0.30$\pm 0.05$). Given the accuracy of our redshifts for galaxies ($<$0.1 for most individual objects), this is significant. The top histogram in Fig \[redshift\] shows the redshift distribution for the 110 galaxies plus QSOs X-ray source candidates in our 58 arcmin$^2$ field of view. It is evident that there is a high overabundance of objects, mostly galaxies, in the redshift range 0.25-0.35. The bottom histogram in Fig \[redshift\] shows the redshift distribution for all 3663 SOCA-classified galaxies in our field of view.
While the peak at redshift 0.25-0.35 is still there in the full galaxy sample, the contrast of that peak appears to be more significant for those galaxy candidates found in or near the error circles for these X-ray sources. At this redshift, a degree-sized field of view corresponds to a $\sim 20$ Mpc region of space. This means that there is a collection of superclusters at this redshift interval. This is similar to what one would see if one looked back at the local universe via sighting down the angle through the Perseus cluster, the Local Supercluster, the Great Attractor and the Shapley concentration stretched out over a redshift range of nearly 20,000 km/sec. In other words, seeing an overdense region of galaxies over a redshift range of 0.1 is not that unusual in our universe.
Summary
=======
Based on the 15 color photometric observations, and SED-based object classification approach (SOCA), as well as the multiwavelength cross-correlations, we find 156 optical candidates, to a magnitude limit close to $V \approx 23$, within or near the error circles for all 75 X-ray sources in our field of view. Among them are 31 QSOs (nine of which have spectroscopic confirmation), 37 starburst galaxies (2 of which have spectroscopic confirmation), 42 normal galaxies (including 3 possible galaxy groups), 39 stars (one of which is a BD object), and 7 just “visible” objects with no classifications (two of which are coincident with known radio sources). Two of the X-ray error circles have only just “visible” objects in them. We find 8 radio sources (out of 100 in our image) that are coincident with an optical object within the X-ray error circles, making it likely that many of these are the optical counterparts to these radio sources.
Separately, we have also SED-classified 6011 additional objects in our 3364 arcmin$^2$ field of view to the same apparent magnitude limit (i.e., not including those found near or in the X-ray error circles). Of these 6011 objects, 341 are QSOs, 1912 are stars, 1508 are starburst galaxies, and 2076 are galaxies and 174 are unclassified for a number of objective reasons. The area of our X-ray circles subtends 31.52 arcmin$^2$, or 0.00937 of our full 3364 arcmin$^2$ field of view. Considering the objects detected to objects that could be randomly found in these error circles, we have: $6.5\sigma$ detections for all 149 classified objects, $4.7\sigma$ for the QSOs, and from $2.8-3.2\sigma$ for stars, starburst galaxies and galaxies. Twenty-eight error circles have only one object in or near them, including: 9 QSOs, 3 normal galaxies, 8 starburst galaxies, 6 stars and 2 “just” visible objects.
In sum, in this paper we perform an exercise that few have been able to do with their X-ray data. By being able to SED-classify all objects in our 3364 arcmin$^2$ field of view down to V $\sim$ 23, we can ask ourselves what are the kinds of optical candidates we find within X-ray error circles to those randomly found in the field of view. question is that while all classified objects: QSOs, starburst galaxies, normal galaxies and stars are overrepresented in the X-ray error circles compared to a random distribution, it is only the QSOs that are highly statistically found in these X-ray circles. Yet, at the same time, there are 6.5$\sigma$ more objects within these X-ray circles than if randomly distributed in this image. So, while we know that one of the objects in our X-ray error circles is likely the X-ray source, in absence of independent knowledge, choosing which object it is is still more arbitrary than scientific.
The BATC Survey is supported by the Chinese Academy of Sciences, the Chinese National Natural Science Foundation and the Chinese State Committee of Sciences and Technology. The present work was partially supported by the Chinese National Key Basic Research Science Foundation (NKBRSFG19990754). This research has made use of the NASA/IPAC Extra-galactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
Brandt, W.N., et al., 2001, AJ, 122, 1
Fan, X.H., et al., 1996, AJ, 112, 628
Fischer, J.-U., et al., 1998, AN, 319, 347
Georgantopoulos, I., Stewart, G. C., Shanks, T., Boyle, B. J., & Griffiths, R. E., 1996, MNRAS, 280, 276
Gunn, J.E., Stryker, L.L., 1983, ApJS, 52, 121
Kinney, A.L., Bohlin, R.C., Calzetti, D.,Panagia, N., & Wyse, R.F.G., 1993, ApJS, 86, 5
Kinney, A.L., Calzetti, D., Bohlin, R.C., McQuade, K., Storchi-Bergmann, T., & Schmitt, H.R., 1996, ApJ,467, 38
Kong, X., et al., 2000, AJ, 119, 2745
Lehmann, I., et al., 2000, A&A, 354, 35
Lehmann, I., et al., 2001, A&A, 371, 833
Ma, J., Zhou, X., Wu, H., Chen, J.S., Jiang, Z.J., Xue, S.J., Zhu, J., 2002, ChJAA, 2, 127
Maccacaro, T., Gioia, I.M., Wolter, A., Zamorani, G., & Stocke, J., 1988, ApJ, 326, 680
Madau, P. 1995 ApJ,441,18
McHardy I.M., et al., 1998, MNRAS, 295, 641
Molthagen, K., Wendker, H.J., & Briel, U.G., 1997, A&AS, 126, 509
M$\o$ller, P & Jakobsen, P., 1990 A&A,228,299
Mushotzky, R.F., Cowie, L.L., Barger, A.J.,& Arnaud, K.A. 2000, Nature, 404, 459
Postman, M, Lubin, L.M., Gunn, J.E., Oke, J.B., Hoessel, J.G., Schneider, D.P., & Christensen, J.A. 1996, AJ, 111, 615.
Rutledge, R.E, Brunner, R.J., Prince, T.A.,& Lonsdale, C., 2000, ApJS, 131, 335
Schmidt, M., et al.. 1998, A&A, 329, 495
Shang, Z.H., et al., 1998, ApJL, 504L, 23
Shanks, T., Georgantopoulos, I., Stewart, G.C., Pounds, K.A., Boyle, B.J., & Griffiths, R.E., 1991, Nature, 353, 315
Schneider, D.P. Schmidt, M.,& Gunn, J.E., 1991, AJ, 101, 2004
Sutherland, W. & Saunders, W., 1992, MNRAS, 259, 413
Voges, W., et al., 1999, A&A, 349, 389
Wilkes,B.J., 1986 MNRAS,218,331
Wolf, C. et. al. 2001 A&A,365,681
Yan, H.J., et al., 2000, PASP, 112, 691
Zamorani, G., et al. 1999, A&A, 346, 731
Zheng, Z.Y., et al., 1999, AJ, 117,2757
Zhou, X., Chen, J.S., Xu, W., Zhang, M., Jiang, Z.J., Zheng, Z.Y., Zhu, J, 1999, PASP, 111, 909
Zhou, X., et al., 2003, A&A,397,361
table1 table2
| ArXiv |
---
abstract: 'We study the growth of massive black holes (BH) in galaxies using smoothed particle hydrodynamic simulations of major galaxy mergers with new implementations of BH accretion and feedback. The effect of BH accretion on gas in its host galaxy is modeled by depositing momentum at a rate $\sim \tau L/c$ into the ambient gas, where $L$ is the luminosity produced by accretion onto the BH and $\tau$ is the wavelength-averaged optical depth of the galactic nucleus to the AGN’s radiation (a free parameter of our model). The accretion rate onto the BH is relatively independent of our subgrid accretion model and is instead determined by the BH’s dynamical impact on its host galaxy: BH accretion is thus self-regulated rather than “supply limited.” We show that the [final]{} BH mass and total stellar mass formed during a merger are more robust predictions of the simulations than the [time dependence]{} of the star formation rate or BH accretion rate. In particular, the latter depend on the assumed interstellar medium physics, which determines when and where the gas fragments to form star clusters; this in turn affects the fuel available for further star formation and BH growth. Simulations over a factor of $\sim 30$ in galaxy mass are consistent with the observed $M_{BH}-\sigma$ relation for a mean optical depth of $\tau \sim 25$. This requires that most BH growth occur when the galactic nucleus is optically thick to far-infrared radiation, consistent with the hypothesized connection between ultra-luminous infrared galaxies and quasars. We find tentative evidence for a shallower $M_{BH}-\sigma$ relation in the lowest mass galaxies, $\sigma \lesssim 100 {{\rm \, km \, s^{-1}}}$. Our results demonstrate that feedback-regulated BH growth and consistency with the observed $M_{BH}-\sigma$ relation do not require that BH feedback terminate star formation in massive galaxies or unbind large quantities of cold gas.'
author:
- |
Jackson DeBuhr,$^{1,2}$ Eliot Quataert,$^{1,2}$ and Chung-Pei Ma$^2$\
$^1$Department of Physics, University of California, Berkeley, CA 94720, USA\
$^2$Department of Astronomy and Theoretical Astrophysics Center, University of California, Berkeley, CA 94720, USA
bibliography:
- 'paper\_bib.bib'
- 'main\_bib.bib'
title: The Growth of Massive Black Holes in Galaxy Merger Simulations with Feedback by Radiation Pressure
---
galaxies: evolution – galaxies: active
Introduction
============
Feedback from an active galactic nucleus (AGN) has been invoked to resolve a number of observational problems in galaxy formation: (1) to explain the tight observed [@ferrarese2000; @gebhardt2000; @haring2004] correlations between central black hole (BH) and galaxy properties such as the $M_{BH}-\sigma$ and $M_{BH}-M_*$ relations and the BH “fundamental plane” [@silk1998; @king2003; @murray2005; @di-matteo2005; @sazonov2005; @hopkins2007], (2) to shut off star formation in elliptical galaxies (e.g., by blowing gas out of the galaxy), thereby explaining how ellipticals become “red and dead” (e.g., @springel2005b [@ciotti10]), (3) to heat the hot intracluster plasma (ICM) in groups and clusters, thereby suppressing cooling and star formation in these environments (e.g., @tabor1993 [@ciotti1997; @croton2006]), and (4) to help explain “cosmic downsizing,” namely the fact that both star formation and AGN activity reside in progressively lower mass halos at lower redshifts (e.g., @scannapieco2005).
It is plausible that AGN perform the roles desired of them, but this is by no means certain. Understanding whether this is indeed the case requires developing more sophisticated theoretical models that can be compared quantitatively to observations. There are several key theoretical problems that must be addressed in order to better understand the role of massive BHs in galaxy formation, and to understand the properties of massive BHs themselves. The first is the problem of AGN fueling, i.e., how is gas transferred from galactic scales ($\sim 0.1-1$ kpc) to the vicinity of the massive BH ($\lesssim
0.1$ pc)? A second key problem is the problem of AGN feedback: how do energy and momentum generated by accretion onto a central BH – in the form of radiation and outflows – couple to the surrounding gas, and how does this affect star formation and the growth of the BH itself?
Much of the recent work addressing the impact of BHs on galaxy formation has used qualitatively similar physics (e.g., @2005MNRAS.361..776S [@2009ApJ...690..802J]). For example, many calculations assume that a BH of mass $M_{BH}$ will accrete mass at a rate proportional to the Bondi rate [@1952MNRAS.112..195B]: $$\dot{M}_{Bondi} = \frac{4 \pi f G^2 M_{BH}^2 \rho}{c_s^3}
\label{EquationBondi}$$ where $\rho$ is the density of the surrounding gas, $c_s$ is the sound speed of that gas, and $f \sim 10-100$ is a factor taking into account the possible multi-phase structure of the gas and that the sphere of influence of the BH is often not resolved [@2009MNRAS.398...53B]. There is, however, little justification for using equation \[EquationBondi\]. The Bondi accretion rate estimate assumes that the gas surrounding the BH is spherically symmetric. When the gas is not spherically distributed, the rate of angular momentum transport determines the BH accretion rate (e.g., @shlosman1990). It is generally believed that the progenitors of todays $\gtrsim L^{*}$ ellipticals are gas-rich disk galaxies, the mergers of which lead to luminous starbursts and the growth of the central massive BHs [@1988ApJ...325...74S; @2005ApJ...630..705H]. Most of the gas in disk galaxies, merging galaxies, luminous starbursts [@1998ApJ...507..615D; @2006ApJ...640..228T], and nearby luminous AGN [@ho2008] appears to reside in a rotationally supported disk. There is thus no reason to expect that the spherically symmetric Bondi rate provides a good estimate of the BH accretion rate in gas rich galaxies. Even in the central $\sim$ parsec of own galaxy, where the ambient gas [*is*]{} hot and pressure supported, the Bondi accretion rate fails by orders of magnitude to predict the accretion rate onto the central BH [@sharma07].
There are a number of ways that an AGN can strongly influence its surroundings (e.g., @ostriker10b). Relativistic jets inject energy into intracluster plasma and may be the primary mechanism suppressing cooling flows in galaxy clusters [@mcnamara2007], even though the details of how the energy in the jet couples to the plasma in a volume filling way are not fully understood [@vernaleo2006]. On galactic scales, a wind from an accretion disk around the BH can drive gas out of the galaxy (e.g., @king2003) as could cosmic-ray protons produced by a radio loud AGN [@socrates10]. In addition, the AGN’s radiation can strongly affect the surrounding gas, both by Compton heating/cooling (e.g., @sazonov2005) and by the momentum imparted as UV radiation is absorbed by dust grains [@chang1987; @1988ApJ...325...74S; @murray2005].
This diversity of feedback mechanisms can be roughly separated into two broad classes: energy and momentum injection. We believe that momentum injection is the dominant mode of feedback for most of the gas in a galaxy, largely because of the very short cooling times of dense gas. For example, if a BH radiates at $\sim 10^{46}$ erg $\rm{s}^{-1}$ with a typical quasar spectrum, only gas with $n
\lesssim 1 \, \rm{cm}^{-3}$ can be heated to the Compton temperature within $\sim 100$ pc. However, the mean gas densities in the central $\sim 0.1-1 \, \rm{kpc}$ of luminous star forming galaxies are $\sim
10^{3-5} \rm{cm}^{-3}$ [@1998ApJ...507..615D; @2006ApJ...640..228T]. At these densities, the cooling time of gas is sufficiently short that it is unable to retain much injected energy – be it from the AGN’s radiation or from shocks powered by AGN outflows. Thus it is largely the momentum imparted by AGN outflows and by the absorption and scattering of the AGN’s radiation that dominates the impact of the AGN on dense gas in galaxies. Since it is the dense gas that fuels star formation and the growth of the BH itself, it is critical to understand the impact of momentum feedback on this gas.[^1]
In this paper, we present simulations of major mergers of spiral galaxies using a model for the growth of BHs that includes (1) a BH accretion rate prescription motivated by the physics of angular momentum transport and (2) AGN feedback via momentum injection (e.g., radiation pressure). Some results of this model appear in a companion Letter [@2009arXiv0909.2872D]. The remainder of this paper is organized as follows. Section \[sectionMethods\] presents a summary of our methods, including a description of the model galaxies (§\[sectionICs\]), the model for star formation and the interstellar medium (§\[sectionSFR\]), our BH accretion and feedback model (§\[sectionBlackHoles\]) and a summary of our parameter choices (§\[sectionParam\]). Section \[sectionResults\] shows the results of applying this model to BH growth and star formation in major mergers of gas-rich galaxies. In section \[sectionGalaxyBH\] we show that our model of BH growth and feedback produces a reasonably tight $M_{BH}-\sigma$ correlation similar to that observed. Finally, in section \[sectionDiscussion\] we discuss our results and compare our approach to previous models in the literature. Appendix \[resolution\] presents resolution tests for our fiducial simulation while Appendix \[sectionAppendix\] presents some of the tests used to verify the BH accretion and feedback models that we have implemented.
Methodology {#sectionMethods}
===========
We use a non-public update of the TreeSPH code GADGET-2 [@2005MNRAS.364.1105S] provided by V. Springel to perform simulations of equal-mass mergers of galaxies. This version of the code includes the effective star formation model of [@2003MNRAS.339..289S] but contains no AGN feedback physics. We modified the code further to implement models for massive BH growth and AGN feedback. The details of the simulations are described in the following subsections. The Appendices present resolution tests and some of the tests we performed to verify our implementation of the BH accretion and feedback model.
Initial Conditions and Galaxy Parameters {#sectionICs}
----------------------------------------
Each model galaxy used in our major merger simulations is similar to those in [@2005MNRAS.361..776S]. They include a spherical halo of collisionless dark matter, a centrifugally supported disk of gas and stars, a stellar bulge, and a central point mass representing a black hole. The code used to generate the initial conditions was provided by V. Springel and is identical to that used in [@2005MNRAS.361..776S] except for one change that will be described below.
Table 1 lists the relevant galaxy and simulation parameters for the key merger simulations we focus on in this paper. The simulations are all major mergers of equal mass galaxies. The fiducial simulation (top entry) assumes a mass of $1.94\times 10^{12} M_\odot$ for each merging galaxy, of which 4.1% is assigned to the gas and stars in the disk, 1.36% is assigned to the stars in the bulge, and the rest is in a dark matter halo. The initial mass fraction of gas in the disk is $f_g=0.1$. This run uses a total of $N_p=1.6\times 10^6$ particles with $6\times 10^5$ dark matter particles, $2\times 10^5$ particles each in the gaseous and stellar disk, and $10^5$ particles for the stellar bulge. This run has a Plummer equivalent gravitational force softening of $\epsilon = 47$ pc.
[lcccccccccccc]{} Run Name & $M_{tot}$ & $f_{g,0}$ & $\frac{M_b}{M_d}$ & $N_p$ & $\epsilon$ & $\frac{R_{acc}}{\epsilon}$ & $\alpha$ & $\tau$ & $M_{*,new}$ & $M_{BH,f}$ & $M_{BH,p}$ & $\sigma_f$\
& \[$M_{fid}$\]${}^{a}$ & & & \[$10^6$\] & \[$\rm{pc}$\] & & & & \[$10^{10} M_{\sun}$\] & \[$10^{8} M_{\sun}$\] & \[$10^{8} M_{\sun}$\] & \[$\rm{km} \rm{s}^{-1}$\]\
fid & 1.0 & 0.1 & 0.33 & 1.6 & 47 & 4 & 0.05 & 10 & 1.34 & 1.49 & 1.33 & 169\
fidNof$^{b}$ & 1.0 & 0.1 & 0.33 & 1.6 & 47 & 4 & 0.15 & 0 & 1.36 & 18.1 & 13.5 & 170\
fid3a & 1.0 & 0.1 & 0.33 & 1.6 & 47 & 4 & 0.15 & 10 & 1.34 & 1.03 & 0.90 & 168\
fid6a & 1.0 & 0.1 & 0.33 & 1.6 & 47 & 4 & 0.3 & 10 & 1.35 & 0.86 & 0.77 & 167\
fidTau & 1.0 & 0.1 & 0.33 & 1.6 & 47 & 4 & 0.05 & 3 & 1.36 & 5.05 & 4.31 & 163\
fidt25 & 1.0 & 0.1 & 0.33 & 1.6 & 47 & 4 & 0.05 & 25 & 1.35 & 0.39 & 0.35 & 169\
fid8eps & 1.0 & 0.1 & 0.33 & 1.6 & 47 & 8 & 0.05 & 10 & 1.35 & 2.70 & 1.76 & 163\
fidafg & 1.0 & 0.1 & 0.33 & 1.6 & 47 & 4 & \*$^{c}$ & 10 & 1.32 & 1.21 & 1.02 & 169\
fidq2$^d$ & 1.0 & 0.1 & 0.33 & 1.6 & 47 & 4 & 0.05 & 10 & 1.30 & 1.40 & 1.16 & 168\
fidq07$^e$ & 1.0 & 0.1 & 0.33 & 1.6 & 47 & 4 & 0.05 & 10 & 1.32 & 1.52 & 1.36 & 164\
big & 3.0 & 0.1 & 0.33 & 1.6 & 68 & 4 & 0.05 & 10 & 3.08 & 6.24 & 5.27 & 232\
big6a & 3.0 & 0.1 & 0.33 & 1.6 & 68 & 4 & 0.3 & 10 & 4.17 & 7.86 & 5.15 & 227\
mid & 0.3 & 0.1 & 0.33 & 1.6 & 32 & 4 & 0.05 & 10 & 0.39 & 0.38 & 0.26 & 115\
small & 0.1 & 0.1 & 0.33 & 1.6 & 22 & 4 & 0.05 & 10 & 0.13 & 0.24 & 0.13 & 82\
small6a & 0.1 & 0.1 & 0.33 & 1.6 & 22 & 4 & 0.3 & 10 & 0.13 & 0.25 & 0.24 & 84\
smallq07$^e$ & 0.1 & 0.1 & 0.33 & 1.6 & 22 & 4 & 0.05 & 10 & 0.12 & 0.06 & 0.05 & 81\
fg & 1.0 & 0.3 & 0.33 & 2.4 & 47 & 4 & 0.05 & 10 & 4.41 & 7.10 & 5.53 & 159\
smallfg & 0.1 & 0.3 & 0.33 & 2.4 & 22 & 4 & 0.05 & 10 & 0.36 & 0.31 & 0.23 & 98\
bulge & 1.0 & 0.1 & 0.20 & 1.6 & 47 & 4 & 0.05 & 10 & 1.38 & 1.44 & 1.25 & 161\
LRfid & 1.0 & 0.1 & 0.33 & 0.16 & 102 & 4 & 0.05 & 10 & 1.34 & 1.65 & 0.93 & 164\
MRfid & 1.0 & 0.1 & 0.33 & 0.48 & 70 & 4 & 0.05 & 10 & 1.35 & 2.92 & 2.40 & 168\
MRfidNof$^{b}$ & 1.0 & 0.1 & 0.33 & 0.48 & 70 & 4 & 0.15 & 0 & 1.34 & 13.5 & 11.4 & 167\
LRfidNof$^{b}$ & 1.0 & 0.1 & 0.33 & 0.16 & 102 & 4 & 0.15 & 0 & 1.31 & 13.1 & 11.4 & 175\
fidvol & 1.0 & 0.1 & 0.33 & 1.6 & 47 & 8.62 & 0.05 & 10 & 1.39 & 3.22 & 2.45 & 164\
MRfidvol & 1.0 & 0.1 & 0.33 & 0.48 & 70 & 5.97 & 0.05 & 10 & 1.36 & 3.30 & 1.92 & 164\
Columns are defined as follows: $M_{tot}$ is the total mass in the simulation, $f_{g,0}$ is the initial gas fraction of the disk, $M_b/M_d$ is the bulge to disk mass ratio, $N_p$ is the total number of particles used in the simulation, $\epsilon$ is the Plummer equivalent gravitational force softening, $R_{acc}$, $\alpha$ and $\tau$ are the parameters of the BH accretion and feedback model (§\[sectionBlackHoles\]), $M_{*,new}$ is the total mass of new stars formed during the simulation, $M_{BH,f}$ and $M_{BH,p}$ are the masses of the BH at the end of the simulation and after the peak of accretion (defined to be when the accretion rate drops to one tenth its maximum value), respectively, and $\sigma_f$ is the stellar velocity dispersion of the merger remnant (§\[sectionGalaxyBH\]).
To test the dependence of the results of our fiducial simulation on the model and simulation parameters, we have run a number of additional simulations, varying the gas fraction ($f_g=0.3$ vs 0.1), bulge-to-disk mass ratio (0.2 vs 0.33), total galaxy mass (from 0.1 to 3 of the fiducial value), simulation particle number (from $N_p=1.6\times 10^5$ to $2.4\times
10^6$), force softening ($\epsilon=22$ to 102 pc), as well as the parameters in the black hole model (described in § 2.4 below).
We use a [@1990ApJ...356..359H] density profile for the structure of the dark matter halo: $$\rho_{halo}(r) = \frac{M_{halo}}{2\pi} \frac{a}{r(r+a)^3}.
\label{hernqprof}$$ The scale length $a$ of the halo is set by requiring that the halo enclose the same mass within the virial radius as an NFW profile, and that the densities match at small radii. These conditions yield a relationship among the halo scale length, $a$, the corresponding NFW scale length, $r_s$, and the concentration of the NFW halo, $c$ [@1996ApJ...462..563N; @2005MNRAS.361..776S]: $a = r_s \{2[\ln{(1 + c)}-c/(1+c)]\}^{1/2}$. The halos used in this work all have a concentration of $c = 9$.
The stellar and gaseous disks both initially have exponential surface density profiles: $$\Sigma(R) = \frac{M_i}{2 \pi R_d^2} \exp \left(-\frac{R}{R_d}\right)
\label{EquationExponentialSurfDens}$$
where $M_i$ is the total mass of the component of interest and $R_d$ is the disk scale length, which is initially the same for the stellar and gaseous disks. The disk scale length for the fiducial simulation is $R_d = 3.5$ kpc, which corresponds to the disk having approximately the same angular momentum per unit mass as a halo with a spin parameter of $0.033$. For simulations with different disk masses, we use $R_d \propto M_d^{1/3}$, which is consistent with the observed relation [@2003MNRAS.343..978S]. The stellar disk’s vertical structure is given by the standard ${\rm sech}^2(z/z_0)$ profile, where the vertical scale height $z_0$ is initially set to $z_0 = R_d/5$ at all radii. Unlike the stellar disk, the gaseous disk’s vertical structure is determined by hydrostatic equilibrium given the assumed sound speed/equation of state of the gas (discussed below). Setting up this initial vertical hydrostatic equilibrium requires an iterative procedure that is described in [@2005MNRAS.361..776S].
The stellar bulges also have Hernquist density profiles. The scale length of the bulge $R_b$ is specified as a fraction of the disk scale length, $R_d$. In the fiducial simulation, $R_b = R_d/5$. For different bulge masses, we use the scaling relation $R_b \propto M_b^{1/2}$, which is motivated by the observed mass-radius relation of elliptical galaxies [@2003MNRAS.343..978S].
In our simulations, two galaxies with identical structure are placed on a prograde orbit. For simulations at our fiducial mass of $1.94\times 10^{12} M_\odot$ (for each galaxy), the initial separation of the two galaxies’ centers is $142.8$ kpc. The orbit has approximately zero total energy, which corresponds to an initial velocity for each galaxy of 160 km s$^{-1}$; the velocity is directed at an angle of $28$ degrees from the line connecting the centers of the two galaxies. In order to break the symmetry of the problem, the individual spin axes of the galaxies have a relative angle of about $41$ degrees, with one galaxy of the pair having an inclination with respect to the orbital plane of $10$ degrees. For the simulations with different overall masses, the orbital parameters are scaled by $M^{1/3}$, so that the time to first passage and the time to final merger are similar to those in the fiducial run.
Interstellar Medium Model {#sectionSFR}
-------------------------
The version of GADGET we use includes [@2003MNRAS.339..289S]’s sub-resolution model for the interstellar medium (ISM). This model treats the gas as a two phase medium of cold star forming clouds and a hot ISM. When cooling and star formation are rapid compared to the timescale for adiabatic heating and/or cooling (which is nearly always the case in our calculations), the sound speed of the gas is not determined by its true temperature, but rather by an effective sound speed that averages over the multi-phase ISM, turbulence, etc. The effective sound speed as a function of density can be interpolated freely between two extremes using a parameter ${q_{\rm eos}}$. At one extreme, the gas has an effective sound speed of $10 \, {\rm km\,s^{-1}}$, motivated by, e.g., the observed turbulent velocity in atomic gas in nearby spirals; this is the “no-feedback” case with ${q_{\rm eos}}=0$. The opposite extreme, ${q_{\rm eos}}=1$, represents the “maximal feedback” sub-resolution model of @2003MNRAS.339..289S, motivated by the multiphase ISM model of @mckee77; in this case, $100\%$ of the energy from supernovae is assumed to stir up the ISM. This equation of state is substantially stiffer, with effective sound speeds as high as $\sim200\,{\rm km\,s^{-1}}$. Varying ${q_{\rm eos}}$ between these two extremes amounts to varying the effective sound speed of the ISM, with the interpolation $$c_{s} = \sqrt{{q_{\rm eos}}\,c_{s}^{2}[q=1] + (1-{q_{\rm eos}})\,c_{s}^{2}[q=0]}\ .
\label{eqn:qeos}$$
In addition to this effective equation of state, GADGET models star formation by stochastically converting gas particles into star particles at a rate determined by the gas density, $$\dot{\rho}_{SF} = \frac{1-\beta}{t_{*}^0 \rho_{th}^{1/2}} \rho^{1/2} \rho_c \propto \rho^{3/2}
\label{equationSFRModel}$$ where $\beta = 0.1$ is the fraction of the mass of a stellar population returned to the ISM by stellar evolution. The parameter $t^0_{*}$ is the characteristic timescale for gas to be converted into stars at the threshold density $\rho_{th} = 0.092$ cm$^{-3}$; $\rho_c
\approx \rho$ is the density of the cold clouds, which is related to the density of the SPH particle by equations (17) and (18) of [@2003MNRAS.339..289S]. For a given gas equation of state, the parameters in equation \[equationSFRModel\] can be adjusted to produce a global star formation law similar to the observed Kennicutt-Schmidt relations [@2005MNRAS.361..776S].
For parameters in the equation of state model that have been used in previous work [@2005MNRAS.361..776S] – $T_{SN} = 4 \times 10^8$ K, $A_0 = 4000$, $t^0_{*} = 8.4$ Gyr and $q_{EOS} = 0.5$ – we find that the model overpredicts the sound speed relative to the observed “turbulent” velocities of galaxies, i.e., the non-thermal line widths (see Fig. 1 of @hq10 for a compilation of relevant data). For instance, the above model parameters imply $c_s \sim 30$ km s$^{-1}$ at $n \sim 1$ cm$^{-3}$ and $c_s \sim 110$ km s$^{-1}$ at $n
\sim 10^3$ cm$^{-3}$. These values are too large by a factor of $\sim 2-3$ compared to the random velocities inferred from atomic and molecular line observations [@1998ApJ...507..615D]. To account for this, we set ${q_{\rm eos}}= 0.5$ and then modified GADGET by reducing the pressure everywhere by a factor of $10$. This reduces the effective sound speed by a factor of $\sim 3$ and is thus more consistent with observations. This reduction in ISM pressure is also used in the initial conditions when setting up vertical hydrostatic equilibrium for the gas. Changing the pressure requires changing the equation of state parameters to $T_{SN} = 6.6 \times 10^8$ K, $A_0 = 6600$, and $t_{*}^0 = 13.86$ Gyr to maintain an average star formation rate of $1
\, M_{\sun}$ yr$^{-1}$ for an isolated galaxy with our fiducial Milky Way like mass. In §\[sec:ISM\] we compare our fiducial calculations with this reduction in pressure to models with smaller values of ${q_{\rm eos}}$, $0.07$ and $0.2$; these also have smaller “sound speeds” more comparable to the observed random velocities of galaxies.
The reduction in the sound speed decreases the Jeans length and mass, making it numerically more prohibitive to resolve these critical scales. For the simulations presented here, we are careful to use sufficient numbers of particles so that the Jeans length and mass are always adequately resolved. The higher gas fraction simulations require a higher particle number as a result (see Table \[TableRunParam\]). The reduction in sound speed also makes it more likely that the gas will fragment by gravitational instability into clumps (ala molecular clouds), as we shall discuss in detail later. This fragmentation is real, not numerical; artificially increasing the sound speed to eliminate it is not necessarily physical and could give incorrect results. On the other hand, we do not include sufficient physics in our ISM model to describe the formation and disruption of molecular clouds so our treatment of the resulting clumping is also not correct. In §\[sec:ISM\] we discuss which of our results are the most sensitive to uncertainties related to local gravitational instability in the ISM.
Black Hole Accretion and Feedback {#sectionBlackHoles}
---------------------------------
### Black Hole Accretion Model
We include a BH as an additional collisionless particle at the center of each galaxy. We model the accretion of the surrounding gas onto the BH, via the transport of angular momentum, using $$\dot{M}_{visc} = 3 \pi \alpha \Sigma \frac{c_s^2}{\Omega}
\label{mdotvisceqn}$$ where $\Sigma$ is the mean gas surface density, $\Omega$ is the rotational angular frequency, and $\alpha$ is the dimensionless viscosity (a free parameter of our model). We compute $\Sigma$ and $c_s$ by taking an average of the properties of the SPH particles in a sphere of radius $R_{acc}$ centred on the BH. The radius $R_{acc}$ is typically set equal to four times the gravitational force softening length, i.e., $R_{acc} = 4 \epsilon$, although we explore alternate choices as well. We find that estimating the rotation rate using $\Omega^2 \simeq GM(<R_{acc})/R_{acc}^3$ is more numerically robust than actually calculating the rotation and angular momentum of the gas particles within $R_{acc}$.
Although equation (\[mdotvisceqn\]) is reminiscent of the alpha prescription of , in our formulation $\alpha$ characterizes not only the efficiency of angular momentum transport, but also the uncertainty due to the fraction of the inflowing gas that is turning into stars vs. being accreted onto the central BH. The physical mechanisms driving gas from $\sim$ kpc to $\sim 0.1$ pc are not fully understood, but non-axisymmetric gravitational torques are likely responsible [@1989Natur.338...45S; @hq10]. Using numerical simulations that focus on the nuclei of galaxies (from $\sim 0.1-100$ pc) @hq10 simulate the conditions under which there is significant gas inflow to $\lesssim 0.1$ pc. They argue that the net accretion rate is not a strong function of the gas sound speed (unlike [*both*]{} eqns \[EquationBondi\] and \[mdotvisceqn\]) because non-axisymmetric gravitational perturbations produce orbit crossing and strong shocks in the gas. The resulting inflow rate depends primarily on the non-axisymmetry in the potential, rather than the thermodynamics of the gas. Nonetheless, equation (\[mdotvisceqn\]) evaluated at $\sim
100$ pc and with $\alpha \sim 0.1$ approximates the accretion rate at small radii in their simulations, albeit with substantial scatter (factor of $\sim 10$). Given that one of our key results discussed in §\[sectionResults\] is that the accretion rate is not sensitive to the exact value of $\alpha$, we believe that equation (\[mdotvisceqn\]) is sufficient for the exploratory calculations in this paper.
### Mass of the Black Hole Particle
In our galaxy merger simulations, the two BHs are initially far apart but approach each other in the late stages of the merger. When the BH particles have a separation of less than $R_{acc}$ we consider them to have merged. When this occurs, we sum the individual masses of the two BH particles and set one of the particles to have this mass. This particle is then moved to the center of mass of the two BH system and given the velocity of the center of mass frame. The other BH particle is removed from the region.
The BH particles are subject to stochastic motion due to interaction with the stellar and gaseous particles, which leads to inaccuracy in the position of the BH and noise in the estimate of the accretion rate. To reduce this numerical “Brownian” motion, the BH particles are given a large “tracer” mass of $2 \times 10^8 M_{\sun}$ for the fiducial simulation, and scaled with the overall mass for other simulations. As a result, the BH particle is a factor $\sim 100$ more massive than the halo particles, and a factor $\sim 10^4$ more massive than the stellar and gaseous particles. We artificially increase the BH particle mass solely to reduce numerical relaxation effects. This does not result in spurious dynamical effects on the central stars, gas, and dark matter since the BH’s sphere of gravitational influence extends to $\lesssim 10$ pc for the fiducial simulation, which is significantly smaller than our typical force softening of $\sim 50$ pc.
For the results presented below, the “real” mass of the BH ($\equiv
M_{BH}$) is computed by integrating the accretion rate of equation (\[mdotvisceqn\]) in time. The gas particles are not removed as the BH mass increases. Instead, the gas particles have an additional label that tracks whether or not they have been “consumed.” We track how much mass the BH should have consumed via accretion at a given time, and the mass of gas that has been consumed. When there is a mis-match, we tag a number of gas particles within $R_{acc}$ (chosen at random) as “consumed” until the total mass accreted by the BH is correct. Particles that have been consumed no longer contribute to the accretion rate estimate, even if they are inside $R_{acc}$. This implementation prevents any gas particle from providing more than its mass to the integrated mass of the BH.
### Feedback from the Black Hole
In our simulations, the AGN is assumed to couple to the surrounding gas by depositing momentum into the gas, directed radially away from the BH. This crudely approximates the effects of (1) strong outflows and/or cosmic-ray pressure produced by the AGN [@king2003; @socrates10] and (2) radiation pressure produced by the absorption and scattering of the AGN’s radiation by dust in the ISM [@murray2005]. We focus on the latter when motivating the parameters used in our models.
To accurately account for the impact of the AGN’s radiation on gas in its host galaxy would require a radiative transport calculation, which is beyond the scope of the current work. Instead, we model this radiation pressure by depositing a total momentum per unit time of $$\dot{p} = \tau \frac{L}{c} \quad \mbox{ where } L = {min}\left(\eta \dot{M}_{visc} c^2, L_{Edd}\right)
\label{momdepeqn}$$ radially away from the BH into the SPH particles within a distance of $R_{acc}$ of the BH particle. This momentum is equally distributed among the particles so that each particle experiences the same acceleration. We use a radiative efficiency of $\eta = 0.1$ in all simulations. The physical picture behind our feedback model in equation (\[momdepeqn\]) is that the feedback is produced by the absorption of the ultraviolet light from the AGN by dust in the surrounding gas, and the subsequent reemission of infrared radiation that must diffuse its way out of the nuclear region. As described shortly, the parameter $\tau$ is the total infrared optical depth of the nuclear region.
To motivate equation (\[momdepeqn\]) in more detail, we note that AGN radiate most of their radiation in the ultraviolet. The opacity of dusty gas to UV radiation is $\kappa_{UV} \sim 10^3$ cm$^2$ g$^{-1}$, so that only a surface density of $\sim 10^{-3}$ g cm$^{-2}$ is required to absorb the UV radiation. This is far less than the typical radial column density of gas in the central $\sim 0.1-1$ kpc of luminous star forming galaxies, galaxy mergers, or our simulations (see Fig. \[FigureSigmaFiducial\] below). As a result, the UV radiation is efficiently absorbed, except perhaps along polar lines of sight. The absorption and scattering of the UV radiation deposits a momentum per unit time of $L/c$ into the ambient gas, assuming for simplicity that all of the UV radiation is absorbed. If the infrared optical depth is $\gtrsim 1$, the infrared radiation re-emitted by the dusty gas must diffuse out through the nuclear region; doing so deposits an additional momentum per unit time of $\tau L/c$, where $\tau \sim \kappa_{IR} \Sigma$ is the infrared optical depth and $\kappa_{IR} \sim$ few-10 cm$^2$ g$^{-1}$ is the infrared opacity for the radiation temperatures of interest $\sim 100-1000$ K. The net force due to the UV and infrared radiation is thus $ \dot p \sim (1 + \tau ) L/c \simeq \tau L/c$, i.e. equation (\[momdepeqn\]), for $\tau \gtrsim 1$, which is valid in our calculations near the peak of activity when the BH gains most of its mass.
In our calculations we use a constant value of $\tau$ rather than a time variable $\tau$ given by $\tau = \kappa_{IR} \Sigma$. Given the simplicity of our feedback model relative to a true radiative transfer calculation, this is not an unreasonable approximation. It is also easier to isolate the effects of varying $\tau$ when it is constant in time.
As noted above, we apply the force in equation (\[momdepeqn\]) to all particles within a distance $R_{acc}$ of the BH. A more accurate treatment would be to apply the force out to the point where the column is $\sim \kappa_{IR}^{-1}$, i.e., to where the optical depth to infinity is $\sim 1$. At many times, however, this radius is unresolved. Moreover, it is possible that the photons diffuse primarily along the rotation axis of the gas, rather than in the orbital plane. As a result, the radiation pressure force will be applied primarily at small radii. This is why we apply the force only within $R_{acc}$. One consequence of this is that the number of SPH particles experiencing the feedback, $N$, will change as gas moves in and out of $R_{acc}$. Thus, the strength of feedback felt by an individual particle will change with time. However, because the SPH particles are collisional, they readily share this momentum with neighboring gas particles. In test problems described in Appendix B the effects of our feedback model are essentially independent of $N$ and $R_{acc}$. The results are not quite so clean in our full simulations (see §\[sec:BHmodel\] and Appendix \[resolution\]), but nonetheless none of our major results depend sensitively on the region over which the feedback force is applied.
One might worry that if the number of particles within $R_{acc}$ were too small, the momentum supplied to a single particle would become large enough to artificially accelerate the particle to the escape velocity. The minimum $N$ required to avoid this is actually quite modest for the range of luminosities in our calculations, and for the simulations presented here this concern is never an issue (although it is for some of the test problems in Appendix B).
Parameter Choices for the Black Hole Model {#sectionParam}
------------------------------------------
Our model for BH growth and feedback contains three free parameters: (1) $\alpha$ determines the magnitude of the accretion rate onto the BH; (2) $\tau$ determines the total radiation pressure force produced by accretion onto the BH; it is roughly the optical depth to the far IR in the nuclear region; and (3) $R_{acc}$ is the radius of the spherical region within which the accretion rate is determined and the feedback is applied. Our fiducial values for these parameters are $\alpha = 0.05$, $\tau = 10$, and $R_{acc} = 4 \, \epsilon$ (where $\epsilon$ is the gravitational force softening). We now motivate these particular choices.
The fiducial value of the viscosity used in this work is $\alpha =
0.05$, motivated by the rough consistency between the resulting $\dot
M$ and @hq10’s numerical simulations of gas inflow from $\sim
100$ pc to $\sim 0.1$ pc (although there is factor of $\sim 10$ scatter in the latter that is not captured here). @hq10’s calculations in fact require a more complicated subgrid accretion model that depends on additional parameters such as the bulge to disk ratio of the galaxy (because this influences the strength of non-axisymmetric torques); this will be explored in more detail in future work. In addition to $\alpha = 0.05$, we also carried out simulations with $\alpha = 0.15$ and $\alpha = 0.3$, and found no significant differences, for reasons explained below.
We use a constant value (with time) of $\tau = 10$ in most of our simulations. This is motivated by far infrared opacities of $\kappa_{IR} \sim 3-10$ cm$^{2}$ g$^{-1}$ and surface densities of $\Sigma \sim 1-10$ g cm$^{-2}$ within $R_{acc}$ during the peak of activity in our simulations. These surface densities are also consistent with those directly measured in the nuclei of ultra-luminous infrared galaxies [@1998ApJ...507..615D]. Given the uncertainties associated with the radiative transfer of far infrared photons in galactic nuclei, it is not possible to more accurately estimate the effective value of $\tau$ without detailed radiative transfer calculations. As we shall demonstrate explicitly, however, the exact value of $\tau$ is also not that critical for the qualitative effects of AGN feedback; the value of $\tau$ does, however, strongly affect the final value of the BH mass.
In choosing a value for $R_{acc}$, we must satisfy $R_{acc} > \epsilon$ in order to avoid numerical artifacts. In addition, we find that the BH particle remains within $4 \epsilon$ of the centre of mass of the system at nearly all times, but it can wander around within this region. As a result, $4 \epsilon$ is the smallest we can make $R_{acc}$ without having noise induced by the BHs motion. This choice corresponds to several hundred pc in our typical simulation. Larger values of $R_{acc}$ are unphysical because (1) the accretion rate should only depend on the gas close to the BH; i.e., the transport of gas from, for example, $\sim 8
\epsilon$ to $\sim 2 \epsilon$ is presumably adequately described by our simulations so we should not try to also account for this in our subgrid model, and (2) the radiation pressure force produced by the AGN (and the re-radiated infrared photons) is likely concentrated at relatively small radii, for the reasons described in §\[sectionBlackHoles\].
Galaxy Merger Simulations {#sectionResults}
=========================
Table \[TableRunParam\] summarizes the simulations we focus on in this paper, including the resolution, the parameters that specify the initial conditions for the merging galaxies, the parameters that specify the BH accretion and feedback models, and the final properties of the merger remnants (stellar and BH mass and velocity dispersion). We begin by describing the results from our fiducial simulation (top row in Table \[TableRunParam\]) and then discuss simulations that vary a single parameter of the feedback model relative to the fiducial run. We have also performed simulations at different overall galactic mass scales, initial gas fractions, and numerical resolution. The latter resolution tests are presented in Appendix A.
The Fiducial Simulation {#sec:fid}
-----------------------
![ *Top:* The separation of the black hole particles as a function of time in the fiducial simulation. The blue circles label the times of the images shown in Figure \[figureSFRClump\]. *Middle:* The star formation rate as a function of time for the fiducial simulation (black) and for the run with no feedback (red; run fidNof). *Bottom:* The viscous accretion rate, $\dot{M}_{visc}$ (black), and Eddington rate (grey), as functions of time for the fiducial simulation. The critical $\dot{M}_c$ at which radiation pressure balances gravity (eq. \[EquationMCrit\]) is shown within a radius of $R_{acc}$ (red; solid). The increase in star formation and BH accretion after first passage ($t \sim 0.75$ Gyr) is due to the fragmentation and inspiral of large gaseous/stellar clumps (Fig. \[figureSFRClump\]), while the much larger increase at final coalescence is due to inflow of diffuse gas caused by non-axisymmetric torques. The latter physics dominates the total stellar and BH mass formed during the merger.[]{data-label="FigureFiducialMegaPlot"}](Figure1.eps){width="84mm"}
The top panel of Fig. \[FigureFiducialMegaPlot\] shows the separation of the BH particles for the fiducial simulation, while the middle panel shows the total star formation rate (in both galaxies) for simulations with (black) and without (red) BH feedback. The first close passage of the two galaxies is around $t = 0.33$ Gyr and the system then undergoes a few short oscillations as the BHs finally settle into a merged state around $t = 1.65$ Gyr. The star formation rate increases following the first passage, with a much larger increase in the star formation rate during the final merger of the galaxies. The bottom panel of Fig. \[FigureFiducialMegaPlot\] shows the BH accretion rate determined from equation \[mdotvisceqn\] (black) and the Eddington accretion rate (grey; $\dot M_{edd} \equiv
L_{edd}/0.1c^2$); the initial BH mass is $1.4 \times 10^5 M_\odot$ but as long as it is not too large $\gtrsim 10^8 M_\odot$, the precise initial BH mass is unimportant for our conclusions. In this and similar plots throughout the paper, the value of $\dot{M}$ plotted before the BHs merge is for the BH in the galaxy with the smaller initial inclination relative to the orbital plane; the BH accretion rate for the other galaxy is comparable to that shown here. The evolution of the accretion rate is similar in many of the simulations we have carried out, with an initial period of activity after the first passage of the merging galaxies, and another period of even higher $\dot M$ after the final coalescence of the galaxies and BHs. The latter active episode is when the merged BH gains most of its mass. In particular, the BH reaches the Eddington limit, allowing the mass of the BH to grow exponentially for a few hundred Myr.
[@2009arXiv0909.2872D] showed that the BH accretion and feedback model presented in this work leads to self-regulated BH growth, due to a competition between the (inward) gravitational force produced by the galaxy as a whole and the (outward) radiation pressure force produced by the central AGN (eq. \[momdepeqn\]) [@murray2005]. For a spherically symmetric system, equating these two forces leads to $\tau
L / c = 4 f_g \sigma^4 / G$, where $\sigma^2 = G M_t / 2 R_{acc}$, $M_t$ is the total mass inside $R_{acc}$, and we have evaluated these expressions within $R_{acc}$, where our accretion rate is determined and feedback is implemented. Equivalently, there is a critical accretion rate $\dot{M}_{c}$, analogous to the Eddington rate, at which the two forces balance: $$\dot{M}_c = \frac{4 f_g}{\eta \tau G c} \sigma^4.
\label{EquationMCrit}$$
The bottom panel of Fig. \[FigureFiducialMegaPlot\] shows $\dot{M}_c$ for our fiducial simulation, evaluated within $R_{acc}$ of the BH (solid red). Comparing $\dot M_c$ to the BH accretion rate $\dot M_{visc}$ demonstrates that during the peak episodes of accretion $\dot{M}_{visc} \sim \dot{M}_c$, so that radiation pressure becomes dynamically important. Although it is certainly possible to have accretion rates smaller than $\dot{M}_c$ when there is insufficient gas to fuel the AGN, the accretion rate is limited to a maximum value of $\sim \dot{M}_c$.
Fig. \[FigureSigmaFiducial\] shows the surface density of gas within $R_{acc} = 4 \, \epsilon = 0.19$ kpc for the fiducial simulation and for a higher gas fraction simulation with $f_g = 0.3$. As implied by Fig. \[FigureFiducialMegaPlot\], there are two main epochs during which significant gas is driven into the nuclei of the galaxies: after first passage and at final coalescence. The physical origin of these high nuclear gas densities are, however, somewhat different.
![The mean gas surface density $\Sigma$ interior to the accretion radius $R_{acc} = 4\epsilon = 0.19$ kpc for the fiducial simulation with initial gas fraction $f_g = 0.1$ (solid) and for the simulation with $f_g = 0.3$ (dashed; run fg).[]{data-label="FigureSigmaFiducial"}](Figure2.eps){width="84mm"}
{width="170mm"}
@1996ApJ...464..641M showed that the presence of a bulge like that in our simulation suppresses a nuclear starburst after first passage during galaxy mergers, because the bulge inhibits the non-axisymmetric modes that drive inflow. In our fiducial simulation, the majority of the increase in star formation after first passage is due to gravitational instability and fragmentation of the gas, which produces dense regions of rapid star formation. Fig. \[figureSFRClump\] (left panel) shows the gas density in the vicinity of one of the incoming black holes at $t = 0.74$ Gyr, midway through the first peak in star formation; the companion galaxy is well outside of this image. Two knots of dense gas are clearly seen, both of which will soon enter $R_{acc}$, the BH accretion and feedback region. These two clumps are not the only ones that form after first passage, but they are the only clumps that survive to enter the central region surrounding the BH.[^2] Fig. \[figureSFRClump\] (right panel) also shows an image of the gas density in the nuclear region at $t = 1.71$ Gyr, near the peak of star formation and BH accretion and after the galaxies and BHs have coalesced. At this time, the gas density in the nuclear region is significantly higher than at first passage (see also Fig. \[FigureSigmaFiducial\]) and most of the gas resides in a $\sim 1$ kpc diameter disk. This nuclear gas concentration is the diffuse ISM driven in from larger radii by non-axisymmetric stellar torques during the merger (e.g., @1996ApJ...464..641M).
The galaxies in our fiducial simulation are stable when evolved in isolation. The merger itself drives the gas to fragment by locally exceeding the Jeans/Toomre mass. In reality, the gas in such clumps might disperse after $\sim$ a Myr because of stellar feedback not included in our calculations [@murray10]. This would probably not significantly change our estimate of the star formation rate since we are already normalized to the observed Kennicutt relation; however, such dispersal would lead to little inflow of gas associated with the inspiral of stellar clusters and thus would suppress the first peak in BH accretion (see @hq10 for a more detailed discussion). In §\[sec:ISM\] we will return to these issues and show that the total stellar mass and BH mass formed during the merger are relatively insensitive to the details of our assumed ISM model.
Fig. \[FigureSigmaofRFiducial\] shows the surface density of gas in the fiducial simulation (top panel) and for the run without feedback (bottom panel) as a function of distance from the BH at four times: the initial condition ($t = 0$), shortly after the first close passage of the two galaxies ($t = 0.85$ Gyr), near the peak of accretion ($t = 1.71$ Gyr) and at the end of the simulation ($t =
2.85$ Gyr). Once $\dot M \sim \dot M_c$ at first passage $\sim 0.85$ Gyr, gas is driven out of the nuclear region by the AGN’s radiation pressure. Since at the same time gravitational torques continue to drive gas inwards, the gas begins to pile up at $\sim R_{acc}$. The particular radius at which the pile up occurs of course depends on our choice of $R_{acc}$, and so the particular size of the evacuated region should not be taken too seriously. Qualitatively, however, the behavior in Fig. \[FigureSigmaofRFiducial\] is reasonable: the AGN pushes on the gas in its neighborhood until it deprives itself of fuel.
Near the peak of activity at $t = 1.71$ Gyr, the gas surface density in the central $R_{acc} \simeq 0.19$ kpc is a factor of $\sim 10-30$ larger in the simulations without feedback (bottom panel of Fig. \[FigureSigmaofRFiducial\]). However, the gas density at large radii $\sim 0.5$ kpc is not that different. The radiation pressure force from the BH thus largely affects gas in its immediate environment, rather than the entire gas reservoir of the galaxy. Another indication of this is that the star formation rate is very similar in the simulations with and without feedback (middle panel of Fig. \[FigureFiducialMegaPlot\]).
![Comparison of gas surface density ($\equiv M_g[<r]/\pi r^2$) versus distance from the BH in the fiducial simulation with feedback (top) and without feedback (bottom). Four times are shown: $t = 0$, 0.85 Gyr (first passage), 1.71 Gyr (peak accretion), and 2.85 Gyr (end of simulation). Note that the gas tends to pile up at $R_{acc}
= 0.190$ kpc (shown by the vertical line) in the top panel.[]{data-label="FigureSigmaofRFiducial"}](Figure4.eps){width="84mm"}
Dependence on Parameters of the BH Model {#sec:BHmodel}
----------------------------------------
{width="180mm"}
The models for BH accretion and feedback used here contain uncertain parameters. We have defined the three relevant parameters $\alpha$, $\tau$, and $R_{acc}$ in §\[sectionParam\] and motivated our fiducial values, but it is important to explore how our results change with variations about our fiducial parameters.
The value of $\alpha$ parameterizes the efficiency with which gas accretes from $\sim R_{acc} \sim 190$ pc to smaller radii, encapsulating both the efficiency of angular momentum transport and the effects of star formation on unresolved scales. Naively, a higher value of $\alpha$ would lead to a more massive BH. This is, however, not the case, because during the epochs when the BH gains most of its mass, the accretion rate is set by the efficiency of feedback (eq. \[EquationMCrit\]) not by the available mass supply (see Figs \[FigureFiducialMegaPlot\] & \[FigureSigmaofRFiducial\]). To demonstrate this more explicitly, the top left panel of Fig. \[FigureFourPanel\] compares the BH accretion rates for three simulations with feedback, but differing values of $\alpha$ (0.05, 0.15, and 0.3), to the simulation with no feedback, which has $\alpha = 0.15$. The accretion histories for the three values of $\alpha$ are nearly identical. By contrast, the accretion rate is in general much larger in simulations that neglect feedback (and is $\propto \alpha$). In addition to the constant $\alpha$ runs, we tested a model in which $\alpha$ was time variable, set by the local gas fraction near the BH (fidafg2 in Table \[TableRunParam\]): $\alpha = 3 f_g^2$, with $f_g$ determined within $R_{acc}$ (in practice $\alpha$ varied from $\sim 2
\times 10^{-4}-0.3$). Although this precise functional form is somewhat arbitrary, such a variation is motivated by analytic arguments and numerical simulations which show that instabilities due to self-gravity dominate the transport of gas from $\sim 100$ pc inward [@shlosman1990; @hq10]. For our $\alpha = 3 f_g^2$ simulation, we find that the peak accretion rates and final BH mass are very similar to the constant $\alpha$ simulations. This is consistent with our conclusion that in the limit of large fuel supply, feedback, rather than the efficiency of angular momentum transport, sets the rate at which the BH grows.
The parameter $\tau$ describes the efficacy of the feedback for a given AGN luminosity. The bottom left panel of Fig. \[FigureFourPanel\] compares the BH accretion rate for the fiducial run with $\tau = 10$ (black) and a simulation with a smaller value of $\tau = 3$ (orange). To the extent that the accretion rate is feedback limited and set by $\dot M_c$ in equation \[EquationMCrit\], $\dot M$ should decrease with increasing $\tau$. Physically, this is because larger $\tau$ leads to a larger feedback force, which then requires a smaller accretion rate to provide the luminosity necessary to drive away the surrounding gas. This expectation is borne out by the simulations. To compare the numerical results with the scaling in equation \[EquationMCrit\], the bottom left panel of Fig. \[FigureFourPanel\] also shows $\dot
M$ for the fiducial simulation scaled by a factor of $10/3$ (dashed line). This scaled $\dot M$ of the fiducial simulation is in reasonably good agreement with the $\tau = 3$ simulation, particularly at the first and second peaks in $\dot M$, when most of the BHs mass is accumulated. This demonstrates that the value of $\tau$ does not significantly affect any of the qualitative behavior of how the BH grows, although it does determine the overall value of the BH mass.
In the majority of the simulations presented here, the size of the region over which we apply the feedback and average the gas properties to calculate $\dot M$, $R_{acc}$, is set to $4 \, \epsilon$. The rationale for this choice was given in §\[sectionParam\], but it is important to consider the effects of changing this value. The top right panel of Fig. \[FigureFourPanel\] shows the mass accretion rate for the fiducial simulation and a simulation with $R_{acc} = 8
\epsilon = 380$ pc. The peak values of $\dot M$ and the time of the first and second peaks are reasonably similar in the two cases. The principle difference is that in the simulations with the larger value of $R_{acc}$, the feedback is less effective at clearing gas out of the nuclear region (because the force is distributed over a larger number of particles); this allows a higher level of $\dot M$ to be maintained after the first passage and final coalescence. We suspect that the fiducial simulation better approximates what a higher resolution calculation with radiative transfer would find, but this remains to be demonstrated.
![The star formation rate for the run with no feedback (red) and for runs with various values of the BH accretion and feedback parameters: $\alpha = 0.05, 0.15, 0.3$ (black, green, blue), $\alpha =
3 f_g^2$ (grey), $\tau = 3$ (orange), and $R_{acc} = 8 \epsilon$ (magenta). All of these models have very similar star formation histories.[]{data-label="FigureSFRAlphas"}](Figure6.eps){width="84mm"}
The bottom right panel of Fig. \[FigureFourPanel\] shows the integrated BH mass as a function of time for the fiducial simulation and for the variations in the feedback/accretion model considered in this subsection that have the same value of $\tau$ (but different values of $\alpha$ and/or $R_{acc}$). The key result is that in the presence of feedback (all but the top curve), there is only a factor of $\simeq 3$ change in the BH mass due to differences in how we treat BH accretion and feedback. A factor of $6$ change in $\alpha$ leads to only a $42 \%$ change in the final BH mass. This is because most of the BH mass is gained during the final coalescence of the two galaxies, at which point the BH accretion self-regulates and reaches the Eddington-like value in equation (\[EquationMCrit\]). The run without feedback (top curve), by contrast, has a factor of $\sim 10$ larger BH mass and the BH mass would scale linearly with the assumed value of $\alpha$.
The star formation rates for the simulations with different BH feedback parameters are all shown in Fig. \[FigureSFRAlphas\] (this includes the fiducial simulation with and without feedback and the runs with $\alpha =
0.15, 0.3, 3 f_g^2$, $\tau = 3$, and $R_{acc} = 8\epsilon$). This figure demonstrates that the precise parameters of the BH feedback model have little effect on the galaxy-wide properties such as the star formation rate: the total mass of stars formed in simulations with different BH feedback parameters differ by less than $5\%$.
In previous simulations of BH growth and feedback, AGN feedback acting on dense gas in galaxies has been invoked to quench star formation [@springel2005b]. Our results demonstrate, however, that this is by no means guaranteed (we refer here to ‘quasar’ feedback on cold dense gas, not the effect of AGN on hot dilute gas in galaxy groups and clusters). In our calculations BH growth is self-regulated and closely connected to the properties of the surrounding galaxy (e.g., eq. \[EquationMCrit\]). However, the BHs dynamical influence is centered in the galactic nucleus ($\lesssim 300$ pc); as a result, the BH does not significantly alter the star formation history during a merger. In this scenario, the merger remnant can nonetheless be relatively quiescent (“red and dead”) because the burst of star formation uses up much of the available gas.
Effects of the ISM Model {#sec:ISM}
------------------------
Motivated by observations (e.g., @1998ApJ...507..615D), we have reduced the effective sound speed in GADGET’s subgrid ISM model (see §\[sectionSFR\]). There is nonetheless considerable uncertainty in the accuracy of this (or any other) subgrid model. To study in more detail the effects of the ISM model on our results, we performed two additional simulations at our fiducial galaxy mass with the subgrid interpolation parameter ${q_{\rm eos}}= 0.2$ and $0.07$ (see eq. \[eqn:qeos\]), and without the factor of 10 reduction in pressure used in our fiducial simulation (an additional simulation with ${q_{\rm eos}}= 0.07$ at a lower galaxy mass will be discussed in §\[sectionGalaxyBH\]).[^3] The three different ISM models have $c_s$ and $Q$ within a factor of $\sim 2$ of one another at all radii, with the ${q_{\rm eos}}= 0.2$ model having the largest values of $c_s$ and Q, and our fiducial model having the smallest values. The parameter $Q$ is initially $\sim 3$ for our fiducial simulation at the disk scale length $R_d$, which is why the merger can induce significant fragmentation of the gas (Fig. \[figureSFRClump\]). Given the limited physics included in the subgrid model, we do not believe that it is feasible to unambiguously conclude which of these ISM models is more realistic. These models thus provide an indication of the systematic uncertainty introduced by our treatment of the ISM.
![Comparison of three simulations that differ only in the ISM models: fiducial (black), $q_{EOS} =0.2$ (red), and $q_{EOS} = 0.07$ (blue). The panels show the viscous accretion rate (top), star formation rate (middle), and the integrated black hole mass and mass of new stars formed (bottom). The three different ISM models have $c_s$ and Toomre $Q$ within a factor of $\sim 2$ of one another at all radii; the ${q_{\rm eos}}= 0.2$ model has the largest values of $c_s$ and Q and our fiducial model has the smallest values. []{data-label="figureISMSeries"}](Figure7.eps){width="84mm"}
Fig. \[figureISMSeries\] compares the BH accretion history (top panel), the star formation rate (middle), and the integrated BH mass and mass of new stars formed during the merger (bottom) for the three runs with differing ISM models. For both the fiducial run and the run with $q_{EOS} = 0.07$ there is significant fragmentation after first passage, which generates the first peak in star formation and BH accretion. By contrast, the run with $q_{EOS} = 0.2$ shows no evidence for gas fragmentation or a pronounced peak in activity at first passage. Despite these differing initial histories, the final result of the merger is very similar in all three cases: the large star formation rates and BH accretion rates coincident with the final coalescence of the two galaxies are not due to fragmentation, but are instead largely due to the inflow of diffuse gas to smaller radii. Moreover, the final BH mass and the total amount of new stars formed during the merger are similar in all three cases. Thus, despite uncertainties in the model of the ISM, we find relatively robust integrated quantities (as did the earlier calculations of @1995ApJ...448...41H). The precise time dependence of the star formation and BH accretion (i.e., the lightcurves) are, however, significantly more uncertain and sensitive to the details of the model.
Galaxy Parameters {#sec:gal}
-----------------
Having shown that the final BH mass and new stellar mass do not depend strongly on the uncertain parameters in our accretion, feedback and ISM models, we now examine how our results vary with galaxy properties such as the total mass, gas fraction, and bulge-to-disk ratio.
![Comparison of four simulations that differ only in the total galaxy mass: $M_{fid}=3.88\times 10^{12}M_\odot$ (fiducial; black), $3
M_{fid}$ (red), $0.3 M_{fid}$ (blue), and $0.1 M_{fid}$ (magenta). The three panels show the viscous accretion rate (top), star formation rate (middle), and the integrated black hole mass (bottom). The same parameters are used in the BH accretion and feedback models. []{data-label="FigureMassScale"}](Figure8.eps){width="84mm"}
Fig. \[FigureMassScale\] shows the BH accretion histories (top panel), star formation rate (middle), and integrated BH mass (bottom) for four runs with different total galaxy mass. The models cover a factor of 30 in galaxy mass, from 0.1-3 times our fiducial mass. The BH and star formation parameters are identical in the four simulations, while the gravitational force softening and structural parameters (e.g., disk scale length, bulge radius) change with the total mass (see §\[sectionICs\]).
Fig. \[FigureMassScale\] shows that the BH accretion rates and integrated BH masses increase with galaxy mass as expected from equation \[EquationMCrit\]. However, there is a clear difference between the lower and higher mass simulations: the two higher mass simulations show evidence for the first peak in star formation and BH growth that we have shown is due to fragmentation near first passage, while the lower mass runs do not. This is largely a consequence of the fact that observed disks have $R_D \propto M^{1/3}$ [@2003MNRAS.343..978S], so that more massive galaxies have higher surface densities and are thus more susceptible to gravitational instability (our ISM model counteracts this slightly, but not enough to stabilize the higher mass disks). It is important to reiterate, however, that modest changes to the subgrid sound speed can change whether or not the gas fragments near first passage (§\[sec:ISM\]) so it is not clear if the difference as a function of mass in Fig. \[FigureMassScale\] is robust.
In addition to the systematic change in the importance of fragmentation near first passage, Fig. \[FigureMassScale\] also shows differences in the late-time BH accretion between the low and high mass simulations. In particular the two smaller mass runs each show a period of increased accretion after the main peak during the merger. In these cases the new stars formed around final coalescence develop a bar in the inner $\sim R_{acc}$ of the galaxy. This helps drive some of the remaining gas into the accretion region leading to the increased accretion at late times. There is a milder version of this late-time accretion in the fiducial mass ${q_{\rm eos}}= 0.2$ model without fragmentation in Fig. \[figureISMSeries\]. Interestingly, there is no analogous late-time inflow of gas to within $R_{acc}$ in our low mass galaxy simulations without BH feedback. The late-time activity is also particularly sensitive to the accretion model at a time when the non-axisymmetry produced by the merger has died away (so that $\alpha$ may in reality decrease significantly). For these reasons, we regard the late time growth in Fig. \[FigureMassScale\] as an interesting deviation from self-similarity in the dynamics, but not necessarily a particularly robust one. One important point that this highlights, however, is that because our implementation of BH growth and feedback does not unbind a significant amount of cold gas at late times (unlike calculations by @springel2005b), the predictions of our model are more sensitive to the post galaxy coalescence physics.
In addition to the fiducial gas fraction ($f_g = 0.1$) simulations that we have largely focused on, we performed simulations with an initial gas fraction of $f_g = 0.3$ for our fiducial galaxy mass and at one tenth this mass. The qualitative difference in behavior with galaxy mass in Fig. \[FigureMassScale\] persists in the higher gas fraction runs. In particular, in the low mass $f_g = 0.3$ simulation, the gas does not fragment, while it does in the higher mass $f_g =
0.3$ simulation. Fig. \[FigureSigmaFiducial\] – discussed in §\[sec:fid\] – explicitly shows the increase in the gas surface density within $R_{acc}$ produced by this at early times.
A final property of the galaxy model that we varied was the bulge to disk mass ratio. The majority of our runs include a bulge with one third the mass of the disk; we also ran one simulation with an initial bulge of one fifth the disk mass, at the fiducial galaxy mass. The final BH mass and total mass of stars formed differ by less than $3\%$ each compared to the fiducial simulation.
The $M_{BH}-\sigma$ Correlation {#sectionGalaxyBH}
===============================
Previous numerical studies using models of BH growth and feedback different from those considered here have reproduced a number of the observed correlations between massive BHs and their host galaxies (e.g., @di-matteo2005 [@sazonov2005; @younger08]). @younger08 argue that the galaxy-BH correlations in simulations (in particular, the BH fundamental plane) are relatively independent of the trigger of BH growth, be it minor mergers, major mergers, or global instabilities of galactic disks. Based on the calculations to date, however, it is unclear to what extent the simulated BH-galaxy correlations depend on the details of the BH feedback or accretion models. In this section we assess this question by quantifying the $M_{BH}$ - $\sigma$ relation produced in our models.
We define $\sigma$ of our model galaxies using a method analogous to that of observers: we first project the mass density of the stellar particles into cylindrical bins, and compute the half-mass(light) radius $R_e$. We then compute the velocity dispersion weighted by the surface brightness via $$\sigma^2 = \frac{\int_{R_{min}}^{R_e} \sigma_{los}^2(R) I(R) R dR}{\int_{R_{min}}^{R_e} I(R) R dR}
\label{EquationSigmaLOS}$$ where $I(R)$ is the [projected 2-d stellar]{} mass profile, $\sigma_{los}$ is the line of sight velocity dispersion, and $R_{min} = 2
\epsilon$ to ensure that there are that no artificial effects introduced by the force softening. We repeat this calculation along $1000$ lines of sight with random viewing angles through the center of mass of the merger remnant. The $\sigma$ quoted in this paper and listed in Table 1 is the median value over the 1000 lines of sight.
Fig. \[FigureSigmaPlot\] shows the correlation between the final BH mass $M_{BH,f}$ and the $\sigma$ of the merged galaxy for most of the simulations in Table \[TableRunParam\]: different total galaxy masses (black), different values of the accretion parameter $\alpha$ (red circle), alternate ISM models (red x), higher gas fraction (blue square), alternate bulge mass (red square), different values of $\tau$ (blue circle), and the resolution studies in Appendix A (grey). The solid line indicates the mean relation from the compilation of observational results in [@2009ApJ...698..198G] while the dotted lines are the $1-\sigma$ error bars. We have linearly rescaled all of our final BH masses to a value of $\tau = 25$, using the fact that both the analytic and numerical results are consistent with $\dot
M_{visc}$ and $M_{BH,f}$ being $\propto \tau^{-1}$. The value of $\tau = 25$ is chosen so that the rescaled fiducial simulation lies approximately on the $M_{BH}-\sigma$ relation of [@2009ApJ...698..198G]. For our fiducial simulation carried out with $\tau = 3$ and $\tau = 10$, a linear scaling of $M_{BH,f}$ with $\tau^{-1}$ is accurate to about $2 \%$ (e.g., Table \[TableRunParam\] and Fig. \[FigureFourPanel\]). We also carried out our fiducial simulation with $\tau = 25$; this is consistent with a linear scaling of $M_{BH,f}$ from $\tau = 3$ to $\sim 50 \%$ (Table \[TableRunParam\]). For nearly all of our simulations, rescaling to $\tau = 25$ amounts to dividing the final BH mass by a factor of 2.5.
![The $M_{BH,f}$-$\sigma$ relation for the simulations in this paper, along with the observed relation (solid) and one sigma scatter (dotted) from @2009ApJ...698..198G. The final BH mass $M_{BH,f}$ in all of the simulations has been linearly scaled to $\tau = 25$ from the value used in the simulation (typically $\tau = 10$). The simulations are generally quite consistent with observations; we do find indications of a slight flattening in $M_{BH,f}-\sigma$ at low BH masses.[]{data-label="FigureSigmaPlot"}](Figure9.eps){width="86mm"}
![The ratio of the final BH mass to the BH mass at the peak of accretion for the simulations in Fig. \[FigureSigmaPlot\], using the same symbol types. This quantifies the extent to which late-time accretion increases the BH mass. The late-time increase in BH mass for many of the lower mass systems produces the slight flattening in $M_{BH}-\sigma$ in Fig. \[FigureSigmaPlot\] at low masses; see the text for a discussion of the robustness of this result.[]{data-label="FigureSigmaPlotPeak"}](Figure10.eps){width="86mm"}
Previous analytic arguments were able to reproduce the $M_{BH}-\sigma$ relation with $\tau \sim 1$, rather than requiring $\tau \sim 25$ as we do here (e.g., @king2003 [@murray2005]). These calculations, however, assumed $f_{g} = 0.1$. While perhaps appropriate on average, this is not appropriate in galactic nuclei where the gas densities are higher. The analytic derivations also assumed that BH growth terminated when the system reached the luminosity (accretion rate) at which radiation pressure balances gravity (eq. \[EquationMCrit\]). In reality, however, the luminosity must exceed this critical value by a factor of several in order for gas to be efficiently pushed around in the galactic nucleus (as shown explicitly in the test problems in the Appendix). Moreover, the BH continues to accrete some mass even after reaching $\dot M_c$. Fig. \[FigureSigmaPlotPeak\] shows this explicitly via the ratio of the final BH mass to the BH mass at the peak of activity for all of the simulations in Fig. \[FigureSigmaPlot\].[^4] The net effect of the differences between our simulations and the simple analytic calculations is that a much larger feedback force per unit BH mass ($\tau \sim 25$, not $\sim 1$) is required for consistency with the observed $M_{BH}-\sigma$ relation. The physical implications of this larger value of $\tau$ for models of AGN feedback will be discussed in § \[sectionDiscussion\].
The scatter in BH mass in Fig. \[FigureSigmaPlot\] at our fiducial mass scale of $\sigma \sim 175 {{\rm \, km \, s^{-1}}}$ is reasonably consistent with the observed scatter. In the simulations we have varied the BH accretion model ($\alpha$), the ISM model, numerical resolution, size of the feedback/accretion region $R_{acc}$, and galaxy properties such as the total mass, gas fraction, and bulge to disk ratio. It is encouraging that all of these simulations produce BH masses within a factor of few of each other. The largest BH mass at $\sigma \sim 175 {{\rm \, km \, s^{-1}}}$ is the simulation with an initial gas fraction of $f_g = 0.3$; since this run has a larger gas density at small radii close to the BH (Fig. \[FigureSigmaFiducial\]), it should probably also have a larger $\tau$, which would reduce the BH mass further, in better agreement with the data. It is difficult to make this comparison to the observed scatter more quantitative given the limitation that our simulations are all equal-mass non-cosmological binary mergers on the same orbit.
The numerical results in Fig. \[FigureSigmaPlot\] suggest a slight flattening of the $M_{BH}-\sigma$ relation at $\sigma \lesssim 100
{{\rm \, km \, s^{-1}}}$. This is in large part a consequence of the additional mass gained by the lower mass BHs after their peak of activity (see Figs. \[FigureMassScale\] & \[FigureSigmaPlotPeak\], in particular the fiducial simulations labeled by black squares in Fig. \[FigureSigmaPlotPeak\]). This change in behavior at lower masses is primarily due to the fact that the lower mass galaxies are less prone to fragmentation than the more massive galaxies (§\[sec:gal\]). Without the fragmentation after first passage, more gas is available to feed the BH at late times leading to the slightly higher BH mass. As discussed in §\[sec:gal\], it is unclear how robust this late time accretion is. In fact, a low mass galaxy simulation with an alternate ISM model (${q_{\rm eos}}= 0.07$) does not show significant late-time accretion, leading to a BH mass in good agreement with the extrapolation from higher $\sigma$ (red x at low mass in Figs. \[FigureSigmaPlot\] & \[FigureSigmaPlotPeak\]). We thus regard the case for flattening of $M_{BH}-\sigma$ at low masses in our models as somewhat tentative; our results may instead indicate enhanced scatter at low masses rather than a change in the mean relation. More comprehensive numerical studies of these lower mass systems will be needed to distinguish these two possibilities.
Discussion and Conclusions {#sectionDiscussion}
==========================
We have presented a new method for simulating the growth of massive BHs in galaxies and the impact of AGN activity on gas in its host galaxy (see also our related Letter; @2009arXiv0909.2872D). In this method, we use a local viscous estimate to determine the accretion rate onto a BH given conditions in the surrounding galaxy (eq. \[mdotvisceqn\]), and we model the effect of BH feedback on ambient gas by depositing momentum radially away from the BH into the surrounding gas (eq. \[momdepeqn\]).
Our accretion model qualitatively takes into account the angular momentum redistribution required for accretion of cold gas in galaxies and is thus more appropriate than the spherical accretion estimate that has been used extensively in the literature. In our feedback model, the applied force is given by $\tau L/c$, where the AGN’s luminosity $L$ is determined by our BH accretion model, and the net efficiency of the feedback is determined by the total optical depth $\tau$ of the galactic gas to the AGN’s radiation, which is a free parameter of our model. Previous calculations have demonstrated that only when the gas fraction in a galaxy decreases to $\lesssim 0.01$ can the AGN’s radiation Compton heat matter to high temperatures [@sazonov2005]. More generally, the cooling times in gas-rich galaxies are so short that the primary [dynamical]{} impact of the AGN on surrounding gas is via the momentum imparted by the AGN’s outflows or radiation. It is thus not physically well-motivated to model AGN feedback by depositing energy, but not momentum, into surrounding gas, as many calculations have done (e.g., @di-matteo2005 [@springel2005b; @kawata2005]); see [@ostriker10b] for related points.
Throughout this paper, we have focused on BH growth during major mergers of spiral galaxies. As demonstrated in @2009arXiv0909.2872D, our model leads to a self-regulated mode of BH accretion in which the BH accretion rate is relatively independent of the details of the BH accretion model (see Fig. \[FigureFourPanel\]). This is because the accretion rate self-adjusts so that the radiation pressure force is comparable to the inward gravitational force produced by the host galaxy (see eq. \[EquationMCrit\]). This self-regulated mode of BH accretion is a robust feature of all of our simulations during periods of time when there is a significant nuclear gas reservoir – it thus applies precisely when the BH gains most of its mass.
One important consequence of this self-regulated accretion is that AGN feedback does not drive significant large-scale outflows of gas (in contrast to the models of @springel2005b). For example, the surface density profiles in Fig. \[FigureSigmaofRFiducial\] show that AGN feedback causes gas to pile up at a few hundred pc rather than being completely unbound from the galaxy – this precise radius should not be taken too literally since it is a direct consequence of the fact that we implement feedback and determine the BH accretion rate only within a radius $R_{acc} \sim$ few hundred pc. Nonetheless, we believe that this general result may well be generic: because the BH accretion rate is determined by the gas content close to the BH, the AGN can shut off its own accretion before depositing sufficient energy to unbind all of the gas in the galaxy. If we artificially hold the luminosity of the AGN constant in time at a value exceeding the critical value in equation (\[EquationMCrit\]), then the AGN [ *does*]{} eventually unbind all of the surrounding gas (see, e.g., Figs. \[FigureShellTestEta\] & \[FigIsothermalEta00\] in Appendix B). However, both our isothermal sphere test problem (Fig. \[FigureShellFull\]) and our full merger calculations show that when the BH accretion rate is self-consistently determined by the gas properties in the central $\sim 100$ pc of the galaxy, the AGN simply never stays ‘on’ long enough to unbind all the gas.
Our results do not, of course, preclude that AGN drive galactic winds. For example, some gas may be unbound by a high speed wind/jet produced by the central accretion disk (which is not in our simulations). In addition, at later stages of a merger or at large radii the gas fraction can be sufficiently low ($\lesssim 0.01$) that gas can be Compton heated to high temperatures and potentially unbound (e.g., @ciotti10). This may in fact be sufficient to quench star formation at late times, but only once most of the gas has already been consumed into stars (so that $f_g \lesssim 0.01$). Our results do suggest that AGN feedback does not quench star formation by unbinding a significant fraction of the cold dense gas in a galaxies interstellar medium (in contrast to, e.g., @springel2005b). In future work it will be important to assess whether variability in the accretion rate on smaller scales than we can resolve (e.g., @hq10 [@levine10]) modifies this conclusion; such variability could produce some epochs during which AGN feedback has a significantly larger effect on the surrounding gas. Another improvement would be to carry out radiative transfer calculations and assess what fraction of the AGN’s radiation is absorbed at large radii in a galaxy ($\sim$ kpc) where the gas has a lower surface density and is thus easier to unbind.
Our simulations cover a factor of $\sim 30$ in galaxy mass. The final BH mass in our calculations is $\propto \tau^{-1}$ since a larger value of $\tau$ corresponds to a larger momentum deposition per unit BH mass. We find reasonable consistency with the normalization of the observed $M_{BH}-\sigma$ relation for $\tau \sim 25$. To compare this result to previous work by @di-matteo2005, we note that a momentum deposition of $\dot P$ corresponds to an energy deposition rate of $\dot E \simeq \dot P \sigma$ when the feedback is able to move the gas at a speed comparable to the velocity dispersion $\sigma$ (which is required for efficient self-regulation of the BH growth). For $\tau \simeq 25$, our results thus correspond to $\dot E \simeq 25 \,
L \, \sigma/c \simeq 0.02 \, L \, (\sigma/200 \, {\rm km \, s^{-1}})$. This is similar to the results of @di-matteo2005, who found that depositing $\sim 5 \%$ of the BH accretion energy in the surrounding gas was required to explain the $M_{BH}-\sigma$ relation. It is encouraging that these two different sets of simulations, with different BH accretion and feedback models, agree at the factor of $\sim 2-3$ level on the energetics required to reproduce the $M_{BH}-\sigma$ relation.
The value of $\tau \sim 25$ required to explain the normalization of the $M_{BH}-\sigma$ relation has strong implications for the dominant physics regulating BH growth. The simplest models of super-Eddington winds from an accretion disk close to the BH are ruled out because they typically have $\tau \sim 1$, i.e., a momentum flux comparable to that of the initial radiation field [@king2003]. Similarly, the radiation pressure force produced solely by the scattering and absorption of the AGN’s UV radiation by dust corresponds to $\tau \sim
1$ [@murray2005] and is thus not sufficient to account for the level of feedback required here. Rather, our results suggest that most BH growth happens when the nuclear regions are optically thick to the re-radiated dust emission in the near and far-infrared, so that $\tau \gg 1$. This is consistent with observational evidence in favor of a connection between BH growth, quasars and luminous dust-enshrouded starbursts such as ULIRGs and sub-mm galaxies (e.g., @1988ApJ...325...74S [@dasyra2006; @alexander2008]). Quantitatively, the observed [*stellar*]{} densities at radii $\sim
1-100$ pc in elliptical galaxies reach $\sim 20$ g cm$^{-2}$ [@hopkins10b], implying $\tau \sim 100$ if a significant fraction of the stars were formed in a single gas-rich epoch. It is encouraging that this is within an order of magnitude of (and larger than!) the value of $\tau$ we find is required to explain the observed $M_{\rm BH}-\sigma$ relation.
A fixed value of $\tau \sim 25$ independent of galaxy mass produces an $M_{BH}-\sigma$ relation with a slope and scatter in reasonable agreement with observations (see Fig. \[FigureSigmaPlot\]). Assessing the scatter more quantitatively will require a wider survey of merger orbits. We do find some tentative evidence for a shallower slope in the $M_{BH}-\sigma$ relation at the lowest galaxy masses, corresponding to $\sigma \lesssim 100 {{\rm \, km \, s^{-1}}}$. This range of masses is precisely where the observational situation is particularly unclear, with, e.g., possible differences between the BH-galaxy correlations in classical bulges and pseudo-bulges [@greene08]. It is also unclear whether major mergers are the dominant mechanism for BH growth in these lower mass galaxies (e.g., @younger08).
Our simulations show that fragmentation of a galactic disk into clumps can be efficiently [*induced*]{} by a merger (e.g., Fig. \[figureSFRClump\]), even when an isolated galaxy with same properties is Toomre stable (see, e.g., @2007MNRAS.375..805W for related ideas in the context of dwarf galaxy formation in tidal tails). As Figure \[figureISMSeries\] demonstrates, this fragmentation can produce a significant increase in star formation during the first close passage of galaxies even when there is little inflow of the diffuse ISM (because such inflow is suppressed by a bulge until later in the merger; @1996ApJ...464..641M). In our simulations we often see a corresponding increase in the BH accretion rate due to the inspiral of dense gas-rich clumps (Fig. \[figureSFRClump\]). The inflow of [gas]{} by this process may, however, be overestimated: stellar feedback not included in our simulations can unbind the gas in star clusters on a timescale of $\sim$ a Myr, returning most of the gas to the diffuse ISM (e.g., @murray10 [@hq10]).
Our calculations use subgrid sound speeds motivated by the observed turbulent velocities in galaxies (§\[sectionSFR\]). We thus believe that our ISM model is physically well-motivated, even though the use of a subgrid sound speed necessarily introduces some uncertainty. Overall, the presence/absence of large-scale clumping of the ISM does not significantly change the final BH mass or the mass of new stars formed in our simulations. It can, however, change the star formation rate and BH accretion rate as a function of time, particularly near the first close passage during a merger.
The tentative change in the $M_{BH}-\sigma$ relation we find for lower mass galaxies is largely due to our treatment of the ISM, rather than our BH feedback or accretion model. For a given gas fraction, lower mass galaxies have a lower gas surface density and thus the ISM is less prone to fragmentation (§\[sec:gal\] and Fig. \[FigureMassScale\]). Without the fragmentation after first passage, more gas is available to feed the BH at late times leading to somewhat higher BH mass (Fig. \[FigureSigmaPlotPeak\]).
The BH accretion and feedback models used in this paper can be significantly improved in future work, allowing a more detailed comparison to observations. For example, @hq10 carried out a large number of simulations of gas inflow in galactic nuclei from $\sim 100$ pc to $\lesssim 0.1$ pc (see, e.g., @levine10 for related work). These can be used to provide a more accurate estimate of the BH accretion rate given conditions at larger radii in a galaxy (Hopkins & Quataert, in prep). Another important improvement would be to use a radiative transfer calculation to self-consistently determine the infrared radiation field produced by a central AGN (and distributed star formation). This could then be used to calculate the radiation pressure force on surrounding gas, eliminating the need for our parameterization of the force in terms of the optical depth $\tau$.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Phil Hopkins and Yuval Birnboim for useful conversations. JD and EQ were supported in part by NASA grant NNG06GI68G and the David and Lucile Packard Foundation. Support for EQ was also provided in part by the Miller Institute for Basic Research in Science, University of California Berkeley. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. This work was partially supported by the National Center for Supercomputing Applications under AST080048 and utilized the Intel 64 cluster Abe. The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing HPC resources that have contributed to the research results reported within this paper.
Resolution Studies {#resolution}
==================
{width="180mm"}
In this section, we describe some of our resolution tests both with and without BH feedback. In the absence of feedback, the well-posed questions for resolution studies include both how the gas properties as a function of radius and time depend on the resolution and how integrated properties of the galaxy (e.g., the star formation rate) depend on resolution. However, the feedback, when present, has a nontrivial dependence on the resolution and it is by no means clear that the nonlinear system will in fact converge in a simple way with increasing resolution. Physically, e.g., the AGN’s radiation pressure has the strongest effect on the gas that contributes the most to the optical depth, which is largely determined by the column density (the dust opacity being only a relatively weak function of temperature for the conditions of interest). Higher resolution simulations can resolve higher volume and column densities, largely at smaller radii close to the BH, and thus may change some of the details of the BH feedback. Indeed, Fig. \[FigureSigmaofRFiducial\] shows that the column density increases towards smaller radii in our simulations.
We first consider the question of how the nuclear gas properties depend on numerical resolution *in the absence of feedback*. To this end, the top left panel of Fig. \[FigureResoFourPanel\] shows the BH accretion rate $\dot M_{visc}$ calculated for three different particle numbers $N_p = 1.6 \times 10^5, 4.8 \times 10^5,$ and $1.6
\times 10^6$, with the gravitational force softening $\epsilon \propto
N_p^{1/3}$.[^5] To make a fair comparison, the accretion rate is evaluated within a fixed volume ($R = 406$ pc) and for $\alpha = 0.05$ for all of the simulations. This choice corresponds to $R = 4
\epsilon$ for the lowest resolution run, but is $R \simeq 8.6
\epsilon$ for our fiducial resolution simulation. Fig. \[FigureResoFourPanel\] shows that the lowest resolution simulation (red) does not adequately resolve the fragmentation of the gas, and the resulting peak in the accretion rate, near first passage. The medium and higher (= our fiducial) resolution simulations, however, agree reasonably well, except for a slight difference in the slope of $\dot M_{visc}(t)$ at late times. Computed over a larger volume ($\sim$ kpc), the agreement between these runs improves.
To assess the convergence in the presence of feedback, the top right panel of Fig. \[FigureResoFourPanel\] shows the BH accretion rate $\dot M_{visc}$ evaluated just as in the top left panel, i.e., using a fixed $R_{acc} = 406$ pc, in simulations with the same three particle numbers and force softening. Again the lowest resolution (red) simulation is clearly not adequate, but the medium (blue) and high (black) resolution simulations agree well; the integrated BH mass differs only by 2% in the latter two simulations.
As a final resolution test, the bottom left panel of Fig. \[FigureResoFourPanel\] shows the BH accretion rate as a function of time in simulations with the same three resolutions and force softening, but in which $R_{acc} = 4 \epsilon$. Thus in this case the accretion rate is determined, and the feedback applied, on increasingly small spatial scales in the higher resolution simulations. This is probably the most physically realistic (see §\[sectionParam\]). This panel shows that the large peak of accretion at final coalescence ($t \sim 1.8$ Gyr) is quite similar in all three cases. This is set by the physics of feedback by momentum deposition and is a robust property of all of our simulations. A corollary of this is that the final BH mass, as shown in the bottom right panel of Fig. \[FigureResoFourPanel\], is the same to within a factor of $\sim 2$ for the three different resolutions. However, the results in the lower left panel of Fig. \[FigureResoFourPanel\] also clearly demonstrate that the detailed evolution of the accretion rate is sensitive to the resolution. This is not particularly surprising: at fixed resolution, Fig. \[FigureFourPanel\] has already demonstrated that the details of $\dot M_{visc}(t)$ depend on the value of $R_{acc}$ – although, again, neither the integrated BH mass or star formation rate do. One implication of these results is that it is difficult for current simulations of BH growth to make quantitative predictions about the light curves of AGN activity triggered by mergers.
Code Verification {#sectionAppendix}
=================
We have tested our modifications to GADGET on a number of simplified problems that have answers that can be easily obtained through other methods. §B1 describes tests of the additional momentum feedback force applied to a thin spherical shell of gas. §B2 describes tests in which the force is applied to the gas particles in the central regions of an isothermal sphere. Two ways of implementing the force are tested: to a fixed number of particles around the BH, and to all particles within a fixed region $R_{acc}$ around the BH.
As we are concerned with the performance of our BH accretion and feedback model, in all of the tests presented in this appendix, the multiphase equation of state and star formation model of [@2003MNRAS.339..289S] are *not* used; instead we use an adiabatic equation of state with $\gamma = 5/3$.
Gas shells
----------
To test that the code is applying the radiation pressure force in equation (\[momdepeqn\]) correctly, we have run the code for a simple system containing a black hole particle with a large mass and a thin spherical shell of gas with negligible temperature, pressure and mass. As this gas resides in a thin shell, this problem is more well-posed if we apply the radiation force to a fixed number, $N$, of gas particles.
This system has a critical luminosity defined by the point at which the radiation force balances the inward pull of gravity. As the gas shell is of low temperature and pressure, we can ignore pressure forces. For a black hole of mass $M_{BH}$ and a gas shell of mass $m$ at a radius $r_0$ the critical luminosity $L_C$ satisfies (we take $\tau = 1$ for simplicity)
$$L_C = G \frac{M_{BH} m}{r_0^2} c.
\label{ShellCritLumo}$$
When the luminosity is set to this value, the gas shell should experience no net force. For other values of the luminosity, the expected behaviour can easily be calculated by noting that the gas shell, in the absence of any pressure forces, should have a radius, $r(t)$, that satisfies
$$m\frac{d^2 r(t)}{dt^2} = -\frac{G M_{BH} m}{r(t)^2} + \frac{L}{c}.
\label{ShellDiffEq}$$
This is easily integrated to give the expected behavior of the gas shell.
![Time evolution of the radius of the test shell for three values of radiation force: $\lambda = 0.5, 1.0, 2.0$ (dashed curves). The results match closely with the solutions from integrating eq. (\[ShellDiffEq\]) (superposed grey curves). Here the force is applied to the 25000 innermost gas particles of the $5\times10^4$ that make up the shell. Time is in units of $t_0 =
\sqrt{r_0^3/G M_{BH}}$ and the radius is in units of $r_0$, where $r_0$ is the initial radius of the gas shell.[]{data-label="FigureShellTestEta"}](FigureB1.eps){width="84mm"}
A number of tests of this system were performed with varying luminosities, parameterized by the ratio of the luminosity applied to the critical luminosity,
$$\lambda = \frac{L}{L_C}.$$
Fig. \[FigureShellTestEta\] shows the exact result in grey, with the numerical solution from the modified version of GADGET in black, for runs with $\lambda = 0.5, 1.0 \mbox{ and } 2.0$. For these tests the number of particles in the shell is $N_{shell} = 50000$, and the force was applied to $N = 25000$ of them. In all cases, the numerical solution appears indistinguishable from the exact solution of eq. (\[ShellDiffEq\]).
![Time evolution of the radius of the test shell for three values of $N / N_{shell}$: 0.5 (solid), 0.25 (dashed), and 0.1 (dot-dashed). The numerical solutions are normalized by the exact solution from eq. (\[ShellDiffEq\]). The radiation force is fixed to be $\lambda = 2.0$. The radius $r(t)$ changes by only about 1% as $N$ is changed, indicating that our results are insensitive to the exact number of particles to which the radiation force is applied.[]{data-label="FigureShellTestXis"}](FigureB2.eps){width="84mm"}
We have also tested the dependence of the results on the value of $N/N_{shell}$, the fraction of particles that receives the radiation force. Fig. \[FigureShellTestXis\] shows the ratio of the numerical solution from our code to the exact solution for $N /
N_{shell}= 0.5, 0.25$, and 0.1 for the $\lambda = 2.0$ model. This demonstrates that even though the magnitude of the force on an individual particle increases as $N$ decreases, the overall dynamics of the shell is the same, with the radii differing by only $\sim 1$% in the three cases. This is primarily due to the fact that the SPH particles are collisional and can thus transfer their motion to their neighbors via pressure forces. The extra momentum imparted to the subset of particles is transferred in part to the outer region of the shell, leading to the overall motion that agrees well with the exact solution. By extension, if $N$ were to vary over the duration of the simulation, the results would also not depend strongly on the particular value.
Isothermal Sphere
-----------------
We have performed a second set of tests of the feedback model using an isothermal background given by a King model. The mass of the system is split into two parts. The bulk of the mass makes up the collisionless background that is drawn from the full phase space distribution of the King model. A small fraction of the mass, $f_g=0.05$, is assigned as collisional SPH particles. These gas particles follow the same spatial profile as the collisionless background but are given zero initial velocities and a very low temperature. Both components are realized with $10^5$ particles. Finally, a black hole particle with a small mass is placed at rest at the center of the distribution.
{width="180mm"}
In the absence of feedback, the SPH particles are not in equilibrium by construction and should flow toward the center of the potential provided by the collisionless background. When the feedback is switched on in the isothermal King potential near the center, the feedback will again have a critical value set by force balance: $$\frac{L_c}{c} = 4 \frac{f_g \sigma^4}{G}.
\label{IsoThermCritLEqn}$$ When the luminosity is below this value, we expect the extra momentum to be insufficient to clear the gas out of the center. When the luminosity exceeds this value, the feedback should be strong enough to clear the central regions of the distribution. To test this, we apply feedback with a constant luminosity. Again, we parameterize the strength of the feedback as $\lambda = L / L_c$.
We have tested two ways of assigning the radiation force. In the first case, the force is shared (equally) by a fixed number of gas particles nearest to the black hole. In the second case, the force is shared by all gas particles within a fixed radius of the black hole. We discuss the results separately below.
### Fixed $N$
For the tests in this subsection, the radiation force is applied to a fixed number of gas particles: $N=500$. The King model has $\sigma =
100 \rm{km} \rm{s}^{-1}$, $\Psi/\sigma^2 = 12$ and a total mass of $10^{12} M_{\sun}$.
Fig. \[FigIsothermalEta00\] compare the density and pressure profiles of three runs with $\lambda=0$ (i.e. no feedback; left panels), 1 (middle), and 2 (right). Four timesteps are shown: $t=0$ (black), 0.16 (red), 0.32 (green), and 0.48 Gyr (blue). As expected, the gas flows to the center in the absence of feedback, increasing the density and pressure as the gas begins to equilibrate in the background potential. The middle and right panels show that the feedback clearly has an effect on the gas at the center, providing some support for the incoming gas, allowing the gas to have a lower pressure. For the case with $\lambda=2$, the feedback is strong enough to effectively clear out the central region.
The nature of the feedback allows a calculation of how the size of the evacuated region should grow with time. Ignoring the thickness of the shell swept up as matter begins to be driven out by the feedback, momentum conservation gives
$$\frac{d}{dt}\left[M_{shell}(r) dr/dt \right] = \frac{L}{c} - \frac{G M_{bg}(r) M_{shell}(r)}{r^2}
\label{SnowPlowEqn}$$
where $M_{shell}(r)$ is the initial mass distribution of gas and $M_{bg}(r)$ is the mass distribution of the background. Near the center of the initial distribution, both the gas and background have an isothermal distribution, for which the mass increases linearly with the distance from the centre. This makes the right hand side of Eq. (\[SnowPlowEqn\]) a constant. In this case, the size of the evacuated region, $r(t)$, depends linearly on time: $$r(t) = \sqrt{2 (\lambda - 1) (1 - f_g)} \sigma t + C
\label{SnowPlowSoln}$$
where $C$ is a constant of integration to account for the finite time required to form the shell of swept up gas.
Fig. \[FigIsothermalRwithT\] shows the size of the evacuated region as a function of simulation time for a run with $\lambda = 4$, and the exact solution for a shell moving in an isothermal background (Eq. \[SnowPlowSoln\]) with $C$ chosen to match the position of the shell at $t = 0.1$ Gyr. The size of the evacuated region is defined by the position of the gas particle closest to the black hole. The agreement is very good with only slight deviation at the latest times. For the model employed, the potential is only isothermal near the origin, so when the shell expands sufficiently, the potential shallows and the shell should move faster than the prediction. This is indeed seen at late time in Fig. \[FigIsothermalRwithT\].
![Time evolution of the size of the evacuated region for the isothermal sphere test. The $\lambda = 4$ simulation results shown (solid) match well with the analytic solution (Eq. \[SnowPlowSoln\]; dashed).[]{data-label="FigIsothermalRwithT"}](FigureB4.eps){width="84mm"}
### Fixed $R_{acc}$
For the galaxy merger simulations, we apply the force inside a fixed $R_{acc}$ throughout the simulation. In this section, we run a similar set of tests as in the previous subsection but we hold $R_{acc}$ fixed. When the number of particles inside $R_{acc}$ becomes small, however, the feedback force exerted on individual particles becomes spuriously large. We therefore impose an additional condition of minimum $N$ on the feedback. For the tests in this subsection, the feedback is applied to those particles inside $R_{acc}$, or to the innermost $100$ gas particles if there are fewer than this inside $R_{acc}$. For the simulations in the main paper, however, there were always enough particles inside the accretion and feedback region to avoid the need for such a lower bound on $N$.
Our first test uses a constant $L = 4 L_c$, and holds $R_{acc}$ fixed. We use a King model as in the previous section, but with slightly different parameters to connect more closely to the our fiducial simulation: $\sigma = 160 \rm{km} \rm{s}^{-1}$, $\Psi/\sigma^2 = 12$ and a total mass of $10^{12} M_{\sun}$. We tested this model for three different sizes of the accretion and feedback region: $R_{acc} = 0.7, 1.4$ and $2.8$ kpc. The smallest region has initially $N \sim 500$. Note that the values of $R_{acc}$ used here are larger than those used in our galaxy merger simulations in the main text. These values of $R_{acc}$ were necessary to ensure that $R_{acc}$ contains a reasonable number of particles. In the galaxy merger calculations, the overall larger number of particles in the simulation and the high gas density in the central regions imply that smaller values of $R_{acc}$ can be reliably used. They are also more physical, as we argued in §\[sectionParam\].
Fig. \[FigureShellFixed\] shows the position of the shell of swept up material for the three runs with $R_{acc} = 0.7, 1.4$ and $2.8$ kpc in black, red and blue respectively. Initially, all the gas inside $R_{acc}$ experiences the extra force. As the region becomes more evacuated and the number of particles inside $R_{acc}$ drops, we transition to applying the force to the $N = 100$ particles closest to the BH. The evolution of the shells in this case is quite similar to the evolution in the last section. The model used in this section is smaller in size and so the shell expands past the isothermal core of the King model earlier. As a result, it begins to accelerate outward sooner. However, the tests with different $R_{acc}$ have essentially identical evolution.
![Radius of the swept up shell for the isothermal sphere test with $\lambda = 4$ and fixed $R_{acc}$: 0.7 (black), 1.4 (red), and 2.8 kpc (blue). To avoid numerical problems, the feedback was always applied to at least $N \sim 100$ particles. The numerical results agree well with the dashed curve, which shows a numerical integration of the analytic equation for the shell radius (eq. \[SnowPlowEqn\]).[]{data-label="FigureShellFixed"}](FigureB5.eps){width="84mm"}
![The black hole accretion rate for isothermal sphere simulations in which the full black hole accretion and feedback model are used ($\alpha = 0.1$ and $\tau = 1$). Three different values of $R_{acc}$ are shown: $R_{acc} = 0.7$ (black), 1.4 (red), and 2.8 kpc (blue). All three agree well with each other.[]{data-label="FigureShellFull"}](FigureB6.eps){width="84mm"}
Finally, we run a test in which we determine the luminosity from the accretion rate as in Eq. (\[momdepeqn\]), and increase the BH mass in time accordingly. This test thus employs the full feedback and accretion model of our galaxy merger simulations. We use the same $\sigma = 160$ km s$^{-1}$ King model, and took $\alpha = 0.1$ and $\tau = 1$ for the feedback parameters. The initial mass of the black hole was $M_{BH,i} = 10^5 M_{\sun}$.
Fig. \[FigureShellFull\] shows the accretion history of the BH for the runs with $R_{acc} = 0.7, 1.4$, and 2.8 kpc. In each test, the feedback is initially Eddington limited and it is not until about $t =
0.3$ Gyr that the luminosity approaches that required to evacuate the gas out of $R_{acc}$. At this point, the gas begins to move out of $R_{acc}$ and form a shell of material at $R \sim R_{acc}$. This shell then remains fairly steady as the accretion rate self-regulates around the critical luminosity. As the three values of $R_{acc}$ are all inside the isothermal core of the King model, the critical luminosities (eq. \[IsoThermCritLEqn\]) are the same, and we would thus expect the accretion rate to self-adjust to the same value at late times. This is indeed borne out in the simulations shown in Fig. \[FigureShellFull\]. Of these three runs, only the calculation with $R_{acc} = 0.7$ kpc spends a significant amount of time with fewer than 100 particles inside $R_{acc}$. Despite the large change in the size of the feedback region, Fig. \[FigureShellFull\] shows that the evolution of the gas is quite similar. The black hole masses for these three runs differ by only a factor of $\sim 2$ at the end of the simulation.
[^1]: These conclusions do not apply to dilute plasma in the intracluster or intragroup medium. The densities there are sufficiently low that the plasma can be efficiently heated by an AGN.
[^2]: In the simulation with a higher initial gas density ($f_g = 0.3$), so many fragments form at large radii and spiral into $R_{acc}$ that the surface density in the central region remains elevated from first passage until the merger completes at $t \sim 1.8$ Gyr (see Fig. \[FigureSigmaFiducial\]).
[^3]: We used $T_{SN} = 4 \times
10^8$ K, $A_0 = 4000$ and $t_*^0 = 8.4$ Gyr for these calculations; these values are different from those in our fiducial simulation, and are chosen to fix the total star formation rate for our isolated fiducial galaxy at $1 M_{\sun}$ yr$^{-1}$.
[^4]: To account for the fluctuating nature of the BH accretion rate in some of the simulations, we define the BH mass at “peak” to be the mass when $\dot M$ drops by a factor of 10 from its peak value.
[^5]: In Fig. \[FigureResoFourPanel\], $\dot
M_{visc}$ for the simulations without feedback (upper left) is calculated from the simulation snapshots and the accretion rate is not Eddington limited. The data outputs were relatively infrequent and attempting to integrate the BH mass over such large timesteps was inaccurate.
| ArXiv |
---
abstract: 'We analyze dropout in deep networks with rectified linear units and the quadratic loss. Our results expose surprising differences between the behavior of dropout and more traditional regularizers like weight decay. For example, on some simple data sets dropout training produces negative weights even though the output is the sum of the inputs. This provides a counterpoint to the suggestion that dropout discourages co-adaptation of weights. We also show that the dropout penalty can grow exponentially in the depth of the network while the weight-decay penalty remains essentially linear, and that dropout is insensitive to various re-scalings of the input features, outputs, and network weights. This last insensitivity implies that there are no isolated local minima of the dropout training criterion. Our work uncovers new properties of dropout, extends our understanding of why dropout succeeds, and lays the foundation for further progress.'
author:
- |
David P. Helmbold\
Department of Computer Science\
University of California, Santa Cruz\
Santa Cruz, CA 95064, USA\
`[email protected]`\
- |
Philip M. Long\
Google\
`[email protected]`\
bibliography:
- 'general.bib'
title: |
**Surprising properties of dropout\
in deep networks\
**
---
Properties of the dropout penalty {#s:dropout.penalty}
=================================
Acknowledgments {#acknowledgments .unnumbered}
===============
We are very grateful to Peter Bartlett, Seshadhri Comandur, and anonymous reviewers for valuable communications.
| ArXiv |
---
author:
- 'E. T. Seppälä and M. J. Alava'
date: 'Received: December 12, 2000 / Revised version: April 11, 2001'
title: 'Energy landscapes, lowest gaps, and susceptibility of elastic manifolds at zero temperature'
---
Introduction {#intro}
============
In this paper we study zero-temperature or ground state elastic manifolds that are roughened by bulk disorder, in the presence of an external field. Such objects are relevant in many contexts of condensed matter and statistical physics [@Fis86; @Emig99]. The essential point here is a competition between the elasticity, which prefers flat manifolds, and the disorder, which induces wandering in order to take advantage of the low energy regions in the system. This leads to a glassy, complicated energy landscape and the fact that the quenched randomness dominates thermal effects at low temperatures. In two embedding dimensions (2D) such manifolds are under the name directed polymers (DP) [@KPZ; @HaH95; @Las98] particularly interesting through their connection to the celebrated Kardar-Parisi-Zhang (KPZ) equation of nonlinear surface growth. Directed polymers have an experimental realization as vortex-lines in granular superconductors [@blatter]. In higher dimensions elastic manifolds may be best considered as domain walls (DW) in ferromagnets with quenched impurities. Elastic manifolds have also other connections to charged density waves, the sine-Gordon model with disorder, random substrate problems, and vortex lattices, to name but a few [@blatter; @Cardy82; @Toner90].
Let us start by introducing the classical spin-half random Ising Hamiltonian $${\mathcal H} = - \sum_{\langle ij \rangle} J_{ij} S_i S_j
- \sum_i H_i S_i,
\label{RHamilton}$$ where $J_{ij}$ is the coupling constant between the nearest-neighbor spins $S_i$ and $S_j$, and $H_i$ is a field assigned to each spin. To this system we apply antiperiodic or domain wall -enforcing boundary conditions in one direction. The spins in the opposite boundaries, let us define in $z$-direction, $z=0$ and $z=L_z$ are forced to be up and down, respectively. In the case of ferromagnetic random bond (RB) Ising systems one has $J_{ij} \geq0$ and $H_i=0$. In the minimum energy state the spins prefer to align on each side of the induced domain wall. When $J_{ij} \lessgtr 0$ the spins become frustrated and the task to find the ground state (GS) structure is most often related to spin glass physics [@review; @droplet; @replica]. On the other hand, when for simplicity $J_{ij} = {\rm
const.} = J > 0$, and $H_i \lessgtr 0$ one arrives at random field (RF) Ising systems. The random field Ising model (RFIM) has an experimental realization as a diluted antiferromagnet in a field. In the RFIM the ferromagnetic couplings compete with the random field contribution, which prefers in the ground state to have the spins to be oriented towards the field assigned to them.
Here we concentrate mainly on the random bond Ising Hamiltonian, i.e., $J_{ij}>0$ and $H=0$, and in some special cases extend the discussion to RF domain walls as well. In the simplest case the spins are located in a square lattice, in $d=2$, or a cube in $d=3$, so that the lattice orientation is in the {10} and {100} directions, respectively. The elastic manifold is the interface, with the dimension $D=d-1$, which divides the system in two parts of up and down spins. At $T=0$ the problem of finding the ground state domain wall, which minimizes the path consisting of unsatisfied bonds between the spins on opposite sides of the domain wall becomes “global optimization”. In our case the displacement field is one-dimensional, $n=1$, but one can certainly think about generalizations so that the total dimension of a system $d=(D+n)$, where $n \geq 1$ is the dimension in which the manifold is able to fluctuate.
The continuum Hamiltonian of elastic manifolds with an external field and $n=1$ may be written as $${\mathcal H} = \int \left[ \frac{\Gamma}{2} \{ \nabla
z({\bf x}) \}^2 + V_r \{ {\bf x},z({\bf x})\} + h\{z({\bf x})\}
\right] \, {\rm d}^D{\bf x}. \label{H}$$ The elastic energy is proportional to the area of the interface given by the first term, and $\Gamma$ is the surface stiffness of the interface. The second term of the integrand comes from the random potential, and the last term accounts for the potential caused by the external field. The use of random bond disorder means that the random potential is delta-point correlated, i.e., $\langle V_r ({\bf x},z)V_r
({\bf x'},z') \rangle =2 {\mathcal D} \delta ({\bf x}- {\bf x'})\delta
(z-z')$. In the random field case $\langle V_r ({\bf x},z)V_r ({\bf
x'},z') \rangle \sim \delta ({\bf x}- {\bf x'}) (z-z')$. The Hamiltonian (\[H\]) is also applicable to wetting in a three phase system, where two of the phases are separated by an interface in a random bulk [@Lip86; @Wuttke91; @Dole91]. In that case $h(z)$ is equivalent to the chemical potential, which tries to bind the interface to the wall, and competes with the random potential, in the presence of which the interface tends to wander in the low energy regions of the system.
Below the upper critical dimension the geometric behavior of elastic manifolds is characterized by the spatial fluctuations. For the mean-square fluctuations one has $$w^2 = \left \langle \left[z({\bf x}) - \overline{z({\bf x})} \right ]^2
\right \rangle \sim L^{2 \zeta},
\label{w}$$ where $z({\bf x})$ is the height of the interface with the mean $\overline{z}$, ${\bf x}$ is the $D=d-n$ dimensional internal coordinate of the manifold, $L$ is the linear size of the system, and $\zeta$ is the corresponding roughness exponent. At low temperatures in $(D+n)=(1+1)$ dimensions with RB disorder, i.e., when one actually considers a directed polymer in random media [@KPZ; @HaH95], the roughness exponent is calculated exactly via the KPZ formalism to be $\zeta=2/3$. In higher manifold dimensions $D$ with $n=1$ the functional renormalization group (FRG) calculations give the approximatively values $\zeta \simeq 0.208(4-D)$ for RB disorder and $\zeta = (4-D)/3$ for RF disorder [@Fis86; @new]. The expression for $\zeta$ tells also that the upper critical dimension for the elastic manifold is $D_u=4$. For manifolds with varying $n$ and $D$ Balents and D. Fisher have derived using FRG $\zeta \simeq
[(4-D)/(n+4)]\{1+(1/4e)2^{-[(n+2)/2]}[(n+2)^2/(n+4)]
[1-\ldots]\}$ [@Balents]. At zero temperature the total average energy $\overline{E}$ of an elastic manifold equals its free energy and grows linearly with the system size $L^D$ and its fluctuations scale for all $n$ as $\Delta E = \left \langle ( E- \overline{E} )^2
\right \rangle^{1/2} \sim L^\theta$, where [@HuHe] $$\theta = 2 \zeta +D -2
\label{HuHe}$$ is the first non-analytic correction to the energy. The same hyperscaling law holds for RF manifolds, too, in $D>1$ [@Seppala98]. Having a positive $\theta$ implies that the temperature is an irrelevant variable in the renormalization group (RG) sense and the $T=0$ fixed point dominates. For $D=1$, $n > 2$ there exists a $T_c$, and $T=0$ fixed point dominates only below $T_c$, likewise always for $n\leq 2$ [@Fish91]. At the randomness dominated pinned phase the temperature is “dangerously” irrelevant, which means that in RG calculations the interesting correlation functions cannot be obtained by setting $T$ to zero. Above $T_c$ the fluctuations become random walk -like with $\zeta=1/2$ and $\theta
=0$. For $D=1$ with $n>1$ there is no exact result existing for the roughness exponent below $T_c$, and hence whether $\zeta \to 1/2$ and $\theta \to 0$ for a finite $n_c$ is still an open question – that is, what is the upper critical dimension of the KPZ equation.
Elastic manifolds self-average in the sense that the intensive fluctuations of the roughness and the energy decrease with system size. However, in this paper we will show that, due to the glassiness of the energy landscape, for example the behavior of the mean position of a typical example of a manifold $\overline{z({\bf x})}$ does not coincide with the disorder average, here denoted by $\langle \,
\rangle$, over many realizations with different random configurations. Introducing the external field to a random system induces often drastic changes and one experimental possibility is to study the ground state behavior e.g. by measuring the susceptibility. In disordered superconductors the external field is due to the current density $j$, which drives the vortex lines.
Here we study the elastic manifolds in an external field while being especially interested in the information it gives about the energy landscape of the random system [@Hou1], which is closely related to the susceptibility. The external field is applied to the coupling constants \[see Eq. (\[H\])\] so that $J_\perp(z) = J_{random} +
h(z)$, where $J_\perp$ are ones in the $z$-direction and $h$ is the amplitude of $h(z)$, the strength of the external field. Since we have the field potential linear in height, $h(z)= hz$, in ferromagnetic random systems with a domain wall the external field $h(z)$ may be transformed to a constant external field term $-H\sum_i S_i$ in the Hamiltonian (\[RHamilton\]). Our results are a systematic extension of early numerical work by Mezárd and relate to more recent discussion of the energy landscapes of directed polymers by Hwa and D. Fisher [@Mez90; @Hwa94]. Due to the glassiness of the energy landscape the position changes in “first order type” large jumps, at sample-dependent values of the external field. The second scenario in which the perturbations take place locally via “droplet”-like excitations is ruled out by a scaling argument and numerical results.
We start by studying a specific case where the number of valleys is fixed to a constant. We first consider the case without the external field, and find the scaling behavior of the energy difference of the global energy minimum and the next lowest minimum of a manifold numerically and derive it also from an extreme statistics argument. The scaling of the gap does not only depend on the energy fluctuation scale, but has in addition a logarithmic factor dependent on the number of the low energy minima. The gap scaling is extended to the finite external field case, to derive the susceptibility of the manifolds under the assumption that the zero-field energy landscape is still relevant. The corresponding numerics allows us also to deduce the effective gap probability distribution without any a priori assumptions. We then proceed by constructing a mean field argument for the finite size scaling of the first jump field in a more general case. This agrees with “grand-canonical” numerics in which the manifold is allowed to have an arbitrary ground state position in the system: the number of valleys in the energy landscape fluctuates and the important physics is caught by the simple scaling considerations. Finally, we discuss wetting in random systems in the light of our results, and compare with the necklace theory of M. Fisher and Lipowsky that applies in the limit of large fields. We also consider finite temperatures, and relate the physics at finite external fields to the Kardar-Parisi-Zhang growth problem through the arrival time mapping. The implications of our results in that case concern the [*two first*]{} arrival times and their difference of an interface to a prefixed height. Note that we have published short accounts of some of the work contained here earlier[@Seppala00; @unpublished].
The paper is organized so that it starts with a short review of the thermodynamic behavior of the interfaces in random media at zero temperature in the presence of an applied field, and of the folklore concerning energy landscapes, in Section \[thermo\]. Section \[glassy\] introduces the mean-field level behavior of the interface when the external field is applied. The analytical probability distribution of the first [*jump*]{} field is derived and its relation to the susceptibility is discussed in Section \[argumentti\]. In the section also the lowest energy gaps between the local energy minima and the global minimum are derived from extreme statistics arguments. The numerical method of calculating the first jump field $h_1$ of an interface in a fixed height is introduced in Section \[numerics\]. Section \[results\] contains the numerical results of the first jump field in $(1+1)$ and $(2+1)$ dimensions with the corresponding jump geometry statistics as well as the energy gaps of the first lowest energy minima. The results are compared with the analytical arguments presented in the previous section. In Section \[first\_jump\] a mean field argument for the first jump field of an interface which lies originally in an arbitrary height is derived and compared with the numerical data and the evolution of the jump size distributions is studied. As an application for the finite field case the wetting phenomenon is considered and the corresponding wetting exponents are numerically studied in Section \[wetting\]. The arguments presented in the paper are discussed from the viewpoint of interface behavior at low but finite temperatures in Section \[creep\] and in the context of first arrival times in KPZ growth in Section \[secKPZ\]. The paper is finished with conclusions in Section \[concl\].
Thermodynamic considerations {#thermo}
============================
The interesting behavior arises since in the low temperature phase e.g. directed polymers are found to have [*anomalous*]{} fluctuations resulting from the regions of the random potential with almost degenerate energy minima, which are separated by large energy barriers [@Fish91; @Mez90; @Hwa94]. These spatially large-scale low-energy excitations are rare, but are expected to dominate the thermodynamic properties and cause large variations in the structural properties at low temperatures. We will use the arguments in the next section in order to study the movement of the manifolds in equilibrium when an external field is applied. Hwa and D. Fisher [@Hwa94] derived that for a polymer fixed at one end a small energy excitation with a transverse scale $\Delta$ ($\simeq l^\zeta$, where $l$ is the linear length of the excitation), scales as $\Delta^{\theta/\zeta}$ and gives rise to large sample-to-sample variations in the two-point correlation function and dominates the disorder averages. The probability distribution for the sizes of the excitations was found to be a power-law from the normalization of “density of states” of small energy excitations $W \sim \Delta^{-n} l^{-\theta} \sim
1/\Delta^{n+2-1/\zeta}$, since $\theta =2\zeta -1$, Eq. (\[HuHe\]). This is in contrast to the high temperature phase one, where $W$ is Gaussian. For the finite field case Hwa and D. Fisher argued that the power-law distribution for the excitations is due to the statistical tilt symmetry, which means that the random part of the new potential of the polymer is statistically the same as the old one, when the polymer is excited by an applied field. In Ref. [@Mez90] Mézard showed numerically that the energy difference of two copies of the polymer in the same realization of the random potential scales as $L^\theta$, where $\theta=1/3$ in $d=2$. He also showed that the ground state configuration of a polymer, which is fixed at one end and applied with an external field $h$ to the end point of the polymer, i.e., $h\{z({\bf x})\} = hz\delta(x-L)$ in Hamiltonian (\[H\]), changes abruptly with a distance $L^\zeta, \zeta=2/3$, when the field is increased by $L^{\theta}$. Calculating the variance of the end point of the polymer $var(z) \sim \langle z^2 \rangle - \langle z
\rangle^2$ Mézard found that typically it is zero, i.e., the ground state configuration does not change. With the probability $L^{-\theta}$, there is a sample for which $h=0$ is the critical value of $h$, and the configuration changes with a large difference compared to the original state $L^{2\zeta} \sim L^{1+\theta}$, Eq. (\[HuHe\]). Hence the total average variance becomes $var(z)_{ave} \sim L$. In this paper we generalize such studies as discussed later.
The physics of DP’s have been shown by Parisi to obey weakly broken replica symmetry [@Parisi90], a “baby-spin glass” phenomenon. This means that the replica symmetric solution of a DP is degenerate with the solution with the broken replica symmetry. This is also evidence for an energy landscape with several, far away from each other, nearly degenerate local minima. Mézard and Bouchaud studied later the connection between extreme statistics and one step replica symmetry breaking in the random energy model (REM) [@derrida], in a finite, one-dimensional form [@boumez]. They computed using extreme statistics the probability distribution of the minimum of all the energies $\frac{\gamma}{2}x^2 + E(x)$, where $E(x)$ are random and follow a suitable probability distribution so that its tails decay faster than any power-law, e.g. Gaussian. This can be seen as a toy model for an interface in a random media, in which case the quadratic part plays the role of the elasticity. The minimum of the total energy is easily seen to be distributed according to the Gumbel distribution [@Gumbel] of extreme statistics, and likewise for the position of the interface the shape of the probability distribution to be approximately Gaussian. In Section \[argumentti\] we will use extreme statistics to study the scaling of the lowest minima and energy gaps in elastic manifolds in a situation analogous to that of Bouchaud and Mézard.
The response of the manifolds pinned by quenched impurities to perturbations was discussed in terms of thermodynamic functions by Shapir [@Shapir91], including the susceptibility of the manifolds. The formal definition of the susceptibility for a $D$ dimensional manifold in a $d$ dimensional embedding space reads: $$\chi = \lim_{h \to 0+} \left \langle \frac{\partial m}{\partial h}
\right \rangle,
\label{Eqsuskis}$$ where the change in the magnetization of the whole $d$ dimensional system is calculated in the limit of the vanishing external field from the positive side and the brackets imply disorder average. In Section \[results\] we derive the susceptibility based on energy landscape arguments. By assuming smooth, analytic behavior in the manifolds’ thermodynamic functions, Shapir found that the susceptibility (for a surface of dimension $D$) is proportional to the displacement of the manifold in the limit of small field, which is applied uniformly to the whole manifold, and $\chi_{D} \sim d^2 E /
dh^2 \sim L^{\theta +2 \tilde{\alpha}}$. $\tilde{\alpha}$ results from an argument concerning the energy gap and the external field: $\Delta
E \sim h L^\zeta L^D$, where the external field couples to a droplet of size $L^\zeta$ and the area of the manifold is $L^D$. This should be equal to the energy difference: $\Delta E \sim
L^{\theta+\tilde{\alpha}}$, and thus $\tilde{\alpha} = 2 - \zeta$, since $\theta =2\zeta+D-2$. Hence the susceptibility was derived to be $\chi_D \sim L^{D+2}$ and the susceptibility per unit hyper-surface vary as $L^2$ for a manifold of scale $L$. This surprisingly is [*independent*]{} of the type of the pinning randomness.
The landscapes of DP’s have also been studied by Jögi and Sornette [@Joe98] by moving both end points step by step (with periodic boundary conditions) from $(0,z)$ and $(L,z)$ to $(0,z+1)$ and $(L,z+1)$, and finding the ground state at each step. They studied the sizes of the [*avalanches*]{} as in self-organized systems, i.e., the areas the polymer covers when moving to the next position. They found a power-law for the avalanche sizes $P(S)\, dS
\sim S^{-(1+m)} \, dS$, where $m =2/5$, up to the maximum size $S_{max} \sim L^{5/3}$. $S_{max}$ is actually the size of the largest fluctuation of the polymer in such a controlled movement, i.e., $L
\times L^{\zeta}$. The authors did not however study the energy variations of the manifold during the “dragging process”.
In the limit of a large $h$ the Hamiltonian of Eq. (\[H\]) is applicable to (complete) wetting interface in random systems with three phases. In this case the manifold is the interface between i) non-wet and ii) wet phases near iii) a hard wall at $z=0$. This problem was studied by mean-field arguments by first Lipowsky and M.Fisher and later elaborated numerically [@Lip86; @Wuttke91]. The choice of wetting potential that corresponds to the linear field used here is called in the wetting literature the weak-fluctuation regime (WFL). In the WFL regime one can derive the wetting exponent $\psi$ by a Flory argument, since the mean distance of the interface from the inert wall $\overline{z}$ is of the order of the interface transverse fluctuations $\xi_\perp$, see Fig. \[fig1\], i.e., the field is strong enough to bind the interface to the wall. The Hamiltonian can now be minimized by estimating the total potential to be $V_{tot} =
V_{fl}(\overline{z})+ V_W(\overline{z})$. $V_W \sim h z$ is the wetting potential induced by the external field $h$. The fluctuation-induced potential $V_{fl}$ is the potential of the free interface $V_{el} + V_r = \frac{\Gamma}{2} \{ \nabla z({\bf x}) \}^2 +
V_r ({\bf x},z)$, which gives, when minimized, $\xi_\perp \simeq
\xi_\|^\zeta$. The elastic energy term $V_{el}$ gives the scaling of $V_{fl}$ with respect to the correlation lengths, and expanding the square-term of the gradient of the height gives $\rm{const}_1 +
\rm{const}_2 \left| \frac{\xi_\perp}{\xi_\|} \right |^2 +$ higher order terms. So, $V_{fl} \sim \xi_\perp^{-\tau}$, where the decay exponent $\tau = (2-2\zeta)/\zeta$ defines the scaling between the pinning and elastic energies. Minimizing $V_{tot}$ in WFL, where $\overline{z} \simeq \xi_\perp$ and is much larger than a lattice constant, gives $$\overline{z} \sim h^{-1/(1 +\tau)} \sim h^{-\psi} .
\label{cwexp}$$ Thus the wetting-exponent $\psi = 1/(1+\tau)$ becomes $$\psi = \frac{\zeta}{2-\zeta}.
\label{psi}$$ Huang [*et al.*]{} [@Hua89; @Wuttke91] did numerical calculations for directed polymers at zero temperature using transfer-matrix techniques for slab geometries, $L \gg L_z$, where $L$ is the length of the polymer, and $L_z$ is the height of the system, confirming roughly the expected exponent $\psi \simeq 0.5$. Our results, presented in Section \[wetting\] are in line with the expectations in (1+1) and we also present the first studies in (2+1) dimensions.
Glassy energy landscapes: length scale of perturbations {#glassy}
=======================================================
We next employ a version of the scaling arguments by Hwa and Fisher, and Mézard for the next optimal position of the directed polymer. The goal is to investigate the preferred length scale of the change that takes place, i.e., the size of the “droplet” created with the field. The numerical calculations presented in this paper are done such a way, that first the ground state is searched and the configuration stored. After that the external field is applied and the energy is minimized again. If there is a change in the configuration, the difference in the mean height of the manifold, the so called “jump size” is analyzed. Most of the studies are done in isotropic systems, i.e., the height $L_z$ of the system is of the order of the linear length of the manifold, $L_z \propto L$. One finds that the behavior of the polymer has a [*first order character*]{}: the optimal length scale is such that the whole configuration changes, and this holds for higher dimensional manifolds, too.
Hwa and Fisher defined the next optimal position of the DP fixed at one end and with displacement $\Delta$ to scale as $\Delta \sim
L^{\zeta}$, where $L$ is the length of the polymer. Now one assumes that the energy [*gaps*]{} between two copies of the polymers with overlap, and energies $E_0$ and $E_1$, grows as $E_1 -E_0 \sim
L^\theta$, i.e., is just energy fluctuation exponent of a polymer (as demonstrated by Mézard numerically). Then as a generalization with $n=1$ it follows that $E_1 -E_0 \sim \Delta^{\theta/\zeta}$. The external field has a contribution for the energy differences of interfaces with the dimension $D$ $E_1 -E_0 \sim h L^D \Delta \sim h
\Delta^{1+D/\zeta}$. Assuming that this difference balances the gap, and using the hyperscaling law for the roughness and energy fluctuation exponents $\theta = 2\zeta+D-2$, we get $$h \sim \Delta^{\overline{\alpha}} = \Delta^{(\zeta-2)/\zeta}.
\label{excitation}$$ The exponent $\overline{\alpha}$ is negative assuming that the roughness exponent is below two, which is of course satisfied in the case of both RF and RB domain walls. Hence, the smaller the field the larger the excitations and thus large excitations are the preferred ones, at least below the upper critical dimension. Besides RB systems Eq. (\[excitation\]) works only for RF systems with a ferromagnetic bulk (among others, the relation $E_1 -E_0 \sim L^\theta$ has not been demonstrated to hold for RF interfaces). If the bulk of a RF magnet is paramagnetic, the stiffness of DW’s is expected vanish on large enough length scales, translational invariance is broken, and $\Delta E =
E_1-E_0$ does not scale. The scaling relation (\[excitation\]) should hold also for general $n$. One should note in the case $h$ is applied in one direction but the droplet may extend in $n$ dimensions, in order to Eq. (\[excitation\]) to hold, also the [*projection*]{} of the droplet in the applied field direction has to have the scaling as $\Delta^{\theta/\zeta}$.
In Fig. \[fig2\] it is shown what happens for two different random realizations of DPs, when a perturbing external field is applied. Fig. \[fig2\](a) shows the mean height of the polymer normalized by its original height $\overline{z}/\overline{z}_0$, when the external field is increased. The first realization $1^\circ$ shows a large [*jump*]{} of a size half of the height of its original position at [*first jump field*]{} $h_1 = 8 \times 10^{-5}$. In Fig. \[fig2\](b) the two positions of $1^\circ$, before and after the first jump, $z_0(x)$ and $z_1(x)$, are shown. The other scenario would be that the directed polymer would continuously undergo small geometric adjustments and get meanwhile bound to the wall, $z=0$. An example of such a droplet, is shown in Fig. \[fig2\](b) as the second realization $2^\circ$, see also Fig. \[fig1\]. However, a further field increase, demonstrated in Fig. \[fig2\](a) for $2^\circ$, leads to a global jump too. This leads to a picture in which assuming a starting position far enough from the system boundary, a finite number of large jumps exists from the original position $z_0({\bf x})$ to the positions $z_1({\bf x}), z_2({\bf x}),
\ldots, z_n({\bf x}), \ldots $, closer and closer to the wall, i.e., $z_0({\bf x}) > z_1({\bf x}) > z_2({\bf x}) > \ldots > z_n({\bf x}) >
0$.
The mean jump length is defined to be $\Delta z_n = \overline{z}_{n-1}
- \overline{z}_{n}$, and the jumps take place at the fields $h_1, h_2,
\ldots, h_n, \ldots$. The field value $h_1$ corresponds the jump from $z_0({\bf x})$ to $z_1({\bf x})$. Finally, after a finite number of jumps, with the interface being in the proximity of the wall, around $h
\simeq 2 \times 10^{-2}$ in Fig. \[fig2\](a), see also Fig. \[fig1\], the interface evolves inside the last valley next to the wall continuously, and the wetting behavior, Eqs. (\[cwexp\]) and (\[psi\]), applies. We discuss this picture of consequent jumps before the wetting regime in Section \[first\_jump\] together with the numerical results.
The global changes (large jumps) induce finite changes in the magnetization ($m$), and are reminiscent of first order phase transitions. Note that the field $h_1$ of the first change or jump obeys for any particular ensemble a probability distribution, and thus a co-existence follows between systems that have undergone a jump or change in $m$ and those that are still in the original state. This “transition” is a result of level-crossing between the valleys in the energy landscape for the interface, sketched in Fig. \[fig3\]. To change between the geometrically separated minima, an external field is applied, which at jumps plays the role of a latent heat. Originally the interface lies at height $\overline{z}_0$ and has an energy $E_0$. When the field is applied interface’s energy increases by $h L^D \overline{z}_0$ and at $h_1$ the interface jumps to $\overline{z}_1 < \overline{z}_0$ having energy $E(h_1)=E_1 + h_1
L^D \overline{z}_1 = E_0 + h_1 L^D \overline{z}_0$, where $E_1$ is the energy of the interface at $z_1$ without the field. Similar behavior takes place at $h_n$, $n=2,3,\dots $, when the interface moves from $\overline{z}_{n-1}$ to $\overline{z}_n < \overline{z}_{n-1}$, until the cross-over to the wetting regime.
In the thermodynamic (TD) limit, i.e., for very large systems, when $L, L_z \to \infty$, the interface may originally be located anywhere in the system. Assuming that the roughness exponent $\zeta$ does not define only the width of the manifold, but also the width of the minimum energy valley, i.e., $L^\zeta$ is the only relevant length scale in the transverse direction, gives that the manifold should have $N_z \sim L_z/L^\zeta$ minima from which to choose the global minimum position. For $\zeta <1$ the number of the minima grows with system size, if the geometry is kept isomorphic, $L_z \propto L$. The requirement $\zeta <1$ holds for all RB interfaces; and for RF ones with ferromagnetic bulk, when $D>1$. In the special case that the local minimum closest to the wall coincides with the global minimum, the wetting regime is entered at once without any jumps. The prediction of Eq. (\[excitation\]) that large scale excitations should dominate is an asymptotic one, and for finite system sizes one may find also small excitations, which are less costly in energy (Fig. \[fig2\]). However, the fraction of these cases decreases with system size, see Fig. \[fig4\]. It depicts in $d=2$ up to $L^2
= 1000^2$, with at least $N=3000$ realizations per data point, the probability of finding a non-zero overlap $q$ between the interfaces before and after the first jump. $q$ is the disorder average of the fraction of the samples, which have overlap between the cases before and after the jump, i.e., for at least one $x$ $z_0(x) = z_1(x)$. The probability goes as $q \simeq 0.56L^{-0.23 \pm 0.01}$, and confirms the expected behavior that for isotropic systems in the $L \to \infty$ limit the first jump is always without an overlap, i.e., a macroscopic one. A first guess would give that $q \sim 1/N_z \sim L^{\zeta-1}$. A similar trend seems to exist in $(2+1)$ dimensions, too, but our statistics is not good enough to determine the exponent of $q$ reliably. This is since $q$ is averaged over a binary distribution (overlapping and non-overlapping jumps), and thus very large ensembles would be needed.
Extreme statistics arguments {#argumentti}
============================
Next we compute the field $h_1$ and the susceptibility, Eq. (\[Eqsuskis\]), by taking into accont the “co-existence” of the original ground state and the state after the jump. Assuming that the relevant process in the response of a domain wall is the large-scale jump, the susceptibility per spin of a system with a domain wall, Eq. (\[Eqsuskis\]), may be written in the form $$\chi = \lim_{h \to 0+} \left \langle \frac{\Delta m(h)}{\Delta h}
\right \rangle \simeq \left \langle \frac{\Delta z_1}{L_z}
\right \rangle \lim_{h \to 0+} P(h_1),
\label{Eqsuskis2}$$ where $P(h_1)$ is the probability distribution of the first jump fields with the corresponding first jump size $\Delta z_1$, and the magnetization $m(h) \sim z(h)/L_z$. The limit $h \to 0+$ is taken over the probability of having a jump, and thus the susceptibility will reflect the co-existence phenomenon discussed above. One uses also a plausible assumption, found to be true in the numerics, that the change in the interface does not depend on the threshold field $h_1$. A further simplification is obtained by assuming (this will be shown numerically later) that the jumps have an invariant size distribution, independent of $L$ and $L_z$, after a normalization with $L_z$. This gives $\langle \Delta z_1 \rangle \sim L_z$. Hence the finite size scaling of the susceptibility per spin depends only on the probability distribution of the first jump field in the limit of the vanishing external field, and in particular on the “rare events” measured by such a distribution. The dependence of $h_1$ on the anomalous scaling is found to be true not only for directed polymers but also for the higher dimensional case. Such behavior is contrary to Shapir’s work that assumed smooth, analytic thermodynamic functions and it is thus no surprise that the scaling of the susceptibility, derived in detail in Section \[results\], differs from that.
Next we first derive the distribution for the first jump field and its relation to the lowest energy gap probability distributions. Then the probability distribution and the finite size scaling of the lowest gaps and also the lowest energy level are derived using extreme statistics arguments (see also [@Galambos; @DBL] and our shorter account of parts of this work [@unpublished]). These gaps have a logarithmic dependence on the number of local minima.
In order to derive the probability distribution of the first jump field $P(h_1)$, let us first calculate the probability $P_0$, that with a certain test field $h'<h_1$, the interface has not changed yet, i.e. $$P_0(h_1 >h')= 1-P_0(h_1 \leq h') = 1-\int_0^{h'} P(h) \, dh.
\label{P0}$$ By differentiating $P_0(h_1 >h')$, one gets the probability distribution of the first jump field, since $$P(h_1) = \frac{\partial}{\partial h} \left. P_0(h_1 \leq h')
\right|_{h'=h_1}.
\label{P0derv}$$ We assume all the minima to be non-correlated and well separated from each other, see Fig. \[fig5\](a). There we have a global energy minimum $E_0$ at $z_0$ (note, that here the energy values as well as the field contributions are normalized by $L^D$, constant with a fixed system size), and energy gaps $\Delta E_1 = E_1 - E_0$ and $\Delta
E_{1^*} = E_{1^*} - E_0> \Delta E_1$ with energies $E_1$ at $z_1$ and $E_{1^*}$ at $z_{1^*}$, respectively. Then we apply the field $h$, i.e., we tilt the energy landscape, see Fig. \[fig5\](b). Due to the statistical tilt symmetry the external field $h$ is assumed not to change the shape of the random landscape. At the lowest tilt to move the interface, i.e., at field $h_1$, it moves to $z_{1^*}$, since with the field $h_1$ $E_0(h_1) = E_0 +h_1z_0 = E_{1^*}(h_1)
=E_{1^*}+h_1z_{1^*} < E_1(h_1) = E_1+h_1z_1$. However, sometimes $E_1$ and $E_{1^*}$ coincide, so that the interface jumps to the true second lowest minimum. Now we index the positions of the $N_z$ minima as $z_i$ (need not to be in order), so that $h(z_0-z_i) = h \Delta z_i$, and we have $$\begin{aligned}
P_0(h_1>h') = \prod_{i=1}^{N_z} {\rm prob}_i(\Delta E_i > h\Delta z_i)
\nonumber \\
= \prod_{i=1}^{N_z} \left\{ 1- \int_0^{\Delta z_ih} \hat{P}_i(\Delta E_i) \,
d(\Delta E_i) \right\}.
\label{P0prod}\end{aligned}$$ By assuming all the gap energies from the same probability distribution $\hat{P} (\Delta E)$, setting $\Delta z_k \hat{=}
\frac{k}{N_z}$ and taking the continuum-limit we get for the probability distribution of the first jump field using Eqs. (\[P0\]), (\[P0derv\]) and (\[P0prod\]), $$\begin{aligned}
P(h_1) = \exp\left[-\int_1^{N_z} \int_0^{kh_1/N_z}
\hat{P}(\Delta E) \, d(\Delta E) \, dk \right] \times \nonumber \\
\int_1^{N_z} \frac{ \hat{P}(kh_1/N_z)}{1 -\int_0^{kh_1/N_z}
\hat{P}(\Delta E) \, d(\Delta E)} \, dk.
\label{Ph1}\end{aligned}$$ There are two approaches to go on from the distribution of Eq. (\[Ph1\]): one may either compare its prediction with numerics with a trial distribution for $\hat{P}$, or alternatively proceed by a complete Ansatz for the valley energies, from which the gap naturally follows. We now attempt the latter, and comment on the first one together with the numerics.
In order to calculate the probability distribution for the gap energies $\hat{P}(\Delta E)$, we use a probability distribution for the energy minima suitable for directed polymers, which has the advantage of decaying faster than any power-law, ${\mathcal P}(E) \sim
\exp(-BE^\eta)$, $\eta>0$. For directed polymers with the “single valley” boundary condition case, i.e., one end of the manifold fixed and thus no room for other minima, it is known mostly numerically that the bulk of the distribution of the energy of a directed polymer ${\mathcal P}(E)$ is Gaussian (the exponent $\eta =2$ in the exponential), except for the tails, ($E <E_{min}$ and $E > E_{max}$) in which different cut-offs, ($\eta_- =1.6$ and $\eta_+=2.4$) take over [@KBM; @HaH95]. For the other manifolds that one might be interested in the actual distributions have not generally been studied. In the following we usually approximate the distribution with Gaussian distribution (this is valid for $1/N_z \geq E_{min}$ and $< E_{max}$). The validity of the approximation is discussed in Section \[results\], when the analytical arguments are compared with the numerics.
The distribution ($\eta =2$ for Gaussian) for ${\mathcal P} (E)$ is written in the form: $${\mathcal P}(E) = k \exp\left \{-\left( {|E - \langle E
\rangle |\over \Delta E} \right)^\eta \right \},
\label{distr}$$ where $\langle E \rangle \sim L^D$ is the average energy of the manifold and $k \sim (\Delta E)^{-1} \sim L^{-\theta}$ normalizes the distribution. The extreme statistics argument goes so that in a system with $N_z \sim L_z/L^\zeta$ minima the probability for the lowest energy to be $E$ is $$L_{N_z}(E) = N_z {\mathcal P}(E) \left \{1-C_1(E) \right \}^{N_z-1},
\label{LNdistr}$$ where $C_1(E) = \int_{-\infty}^E {\mathcal P}(e) \, de$ is called the error-function when $\eta =2$. For Gaussian ${\mathcal P}(E)$, likewise for general $\eta>0$, $L_{N_z}(E)$ is known to be Gumbel distributed, $\exp(u-\exp u)$ [@boumez; @Gumbel; @Rowlands]. The gap $\Delta E_1$ follows similarly as $L_{N_z}(E)$. Its distribution, $G_{N_z}(\Delta E_1,E)$ is given by $$\begin{aligned}
G_{N_z}(\Delta E_1,E) = {N_z(N_z-1)\over 2} {\mathcal P}(E)
{\mathcal P}(E+\Delta E_1) \nonumber\\
\{1-C_1(E+\Delta E_1)\}^{N_z-2}.
\label{DeE}\end{aligned}$$ $G_{N_z}(\Delta E_1,E)$ is the probability that if the lowest energy manifold has an energy $E$, then the gap to the next lowest energy level is $\Delta E_1$. $G_{N_z}(\Delta E_1,E)$ is actually a generalization of Gumbel distribution. Integrating Eq. (\[DeE\]) over all energies gives the probability distribution for the $\Delta
E_1$ $$\begin{aligned}
\hat{P}(\Delta E_1) = \int_{-\infty}^{\infty} G_{N_z}(\Delta E_1,E) \, dE.
\label{DE1}\end{aligned}$$ This becomes for Gaussian ($\eta=2$) and $\Delta E_1 \ll \langle E \rangle$ $$\begin{aligned}
\hat{P}(\Delta E_1) = \int_{-\infty}^{\infty} {N_z(N_z-1)\over 2} k^2
\exp \left \{ {- (E- \langle E \rangle)^2 -2\Delta E_1(E- \langle E
\rangle) \over \Delta E^2 } \right \}
\left \{ k\, {\rm erfc} \left( E +\Delta E_1- \langle E \rangle \over
\Delta E \right) \right\}^{N_z-2} \, dE,
\label{DE1_2}\end{aligned}$$ where erf denotes the error-function and erfc $=1-$ erf. Neglecting all ${\mathcal O} \left[ \left\{ k\, {\rm erfc} \left( E
+\Delta E_1 - \langle E \rangle \over \Delta E \right) \right\}^2
\right]$ and higher order terms and using a Taylor expansion around $\Delta
E_1=0$, i.e., Maclaurin-series, gives $$\begin{aligned}
\hat{P}(\Delta E_1) = \int_{-\infty}^{\infty} {N_z(N_z-1)\over 2} k^2
\exp \left \{ {- (E- \langle E \rangle)^2 -2\Delta E_1(E- \langle E
\rangle) \over \Delta E^2 } \right \} \nonumber \\
\left[ 1 -(N_z-2) k\, {\rm erf} \left( E - \langle E \rangle \over \Delta E
\right) +(N_z-2) \Delta E_1 k \exp \left \{ {- (E- \langle E \rangle)^2 \over
\Delta E^2 } \right \} \right ] \, dE.
\label{DE1_3}\end{aligned}$$ The first two terms within the second parenthesis dominate for $\Delta E_1$ small, and thus we see that to first order of $\hat{P}(\Delta E_1) \sim$ const (compare with the numerics presented below). In particular, one should notice that the probability distribution does not vanish for $\Delta E_1 \simeq 0$.
This approach is very similar to the extreme statistics calculation of the one-dimensional version of the random energy model by Bouchaud and Mézard [@boumez] (see Section \[thermo\]). They derived the probability distribution for the location that gives minimum energy for the system, with the difference to our case that Hamiltonian reads $\frac{\gamma}{2}x^2 +
E(x)$. The result becomes such that the distribution for the position is approximately Gaussian, too. The one-dimensional system is close to our example except for the functional form of the external potential which is quadratic instead of a linear one. There is however one essential difference in that in their analysis the “field value” is fixed, whereas in our case we are interested in what happens in any particular sample as the field is varied. Nevertheless such a calculation could be compared to the distribution of the interface locations at a fixed field $h$.
More important than the actual distributions is that the finite size scaling of the gap energies may be computed. Let us start by calculating the typical lowest energy level. The average of it is given by $$\langle E_0 \rangle = \int_{-\infty}^{\infty} E L_{N_z}(E) \,dE,
\label{E0true}$$ and the typical value of the lowest energy is estimated from $$N_z \frac{1}{k} {\mathcal P}(\langle E_0 \rangle) \approx 1.
\label{E0estimate}$$ Note, that in approximating the integral, Eq. (\[E0true\]), with the aid of the distribution, Eq. (\[E0estimate\]) (in the limit $C_1(E)$ in Eq. (\[LNdistr\]) is small) the normalization $1/k$ should be taken into account. Eq. (\[E0estimate\]) gives $$\langle E_0 \rangle \sim \langle E \rangle - \Delta E \left
[ \ln (N_z) \right ]^{1/\eta},
\label{typicalene}$$ where $\Delta E \sim L^\theta$ and for Gaussian $\eta =2$.
To estimate the typical value of the gap, we make a similar approximation as in Eq. (\[E0estimate\]) for $\langle E_0 \rangle$, $${N_z(N_z-1)\over 2 k^2} {\mathcal P}(\langle E_0 \rangle )
{\mathcal P}(\langle E_0\rangle + \langle \Delta E_1 \rangle ) \approx 1,$$ which with (\[typicalene\]) and the fact that $| \langle \Delta E_1
\rangle | \ll | \langle E_0 \rangle| $ yields, $$\langle \Delta E_1 \rangle \approx { \Delta E^\eta \over \eta
(\langle E \rangle - \langle E_0\rangle)^{\eta-1}}
\approx { \Delta E \over \eta \left [ \ln(N_z) \right ]^{(\eta-1)/\eta}}.
\label{gap}$$ We thus find that the gap scales as $\Delta E_1 \sim \Delta E [\ln
(N_z)]^{-1/2} \sim L^{\theta} [\ln (L_z L^{-\zeta})]^{-1/2}$, when $\eta =2$ and assuming as in the previous section, that $N_z \sim
L_z/L^\zeta$. If the interfaces are flat, i.e., $\zeta =0$, which is true for $\{100\}$ RB interfaces below the cross-over roughening scale $L_c$ [@Bouchaud92; @Alava96; @Raisanen98; @unpublished2] if randomness is weak, the same arguments should hold. However, then the energy distribution is pure Gaussian, $\eta
=2$, and $\theta =D/2$ due to Poissonian statistics.
Susceptibility of manifolds {#suskis}
===========================
Numerical method {#numerics}
----------------
For the numerical calculations the RB Hamiltonian (\[RHamilton\]) with $J_{ij} > 0$, $H=0$, has been transformed to a random flow graph. The graph is formed by the Ising lattice and two extra sites: the source and the sink; and the coupling constants $2J_{ij} \equiv
c_{ij}$ between the spins correspond to flow capacities $c_{ij}
\equiv c_{ji}$ from a site $S_i$ to its neighboring one $S_j$ [@Alavaetal]. The graph-theoretical optimization algorithm, a maximum-flow minimum-cut algorithm, enables us to find the bottleneck, which restricts the amount of the flow that can get from the source to the sink given the capacities in such a random graph. This bottleneck, a path $P$ which divides the system into two parts (sites connected to the sink and sites connected to the source) is the minimum cut of the graph and the sum of the capacities belonging to the cut $\sum_P c_{ij}$ equals the maximum flow, the smallest of all cut-paths in the system. The source is connected to the sites in the Ising lattice which are forced to be up and the sink is connected to the sites which are forced to be down. The value of the maximum flow is the total minimum energy of the domain wall, equivalent to the minimum cut. The maximum flow algorithms can be proven to give the exact minimum cut for all the random graphs, in which the capacities are positive semidefinite and with a single source and sink [@network]. The best known maximum flow method is by Ford and Fulkerson and called the augmenting path method [@FordF]. We have used a more sophisticated method called push-and-relabel by Goldberg and Tarjan [@Goltar88], which we have optimized for our purposes. It scales almost linearly, ${\mathcal O}(n^{1.2})$, with the number of spins and gives the ground state DW in about minute for a million of spins in a workstation. Notice that one could use for DP’s the usual transfer matrix method as well, but the max-flow implementation is convenient in the case of systematic perturbations to each sample.
In this study we have done simulations for $(1+1)$ and $(2+1)$ dimensional manifolds. The system sizes extend to $L \times L_z
=1000^2$ and $L^2 \times L_z = 400^2 \times 50$. The number of realizations $N$ ranges from $200$ to $2 \times 10^4$. The random bonds are such that in 2D $J_{ij,z} \in [0,1]$ uniform distribution and $J_{ij,x} = 0.5$ unless otherwise mentioned. In three dimensions we have either $J_{ij} \in [0,1]$ uniform distribution in all $x,y,$ and $z$ directions or dilution type disorder, $P(J_{ij}) = p \,
\delta(J_{ij}-1) + (1-p) \, \delta(J_{ij})$, i.e., a bond has a value of unity or zero with the probability $p$. We have used $p=0.5$ and $p=0.95$. The linear field contribution is applied in the $z$-directional bonds as $J_\perp(z) = J_{random} + hz$. When the jump field values with the corresponding jump distances are searched, the precision is as small as $\Delta h =10^{-5}$ in order to be sure that no smaller changes would occur between the jumps, which could be the case if the precision was much larger. Periodic boundary conditions are used in $x$ and $y$ directions.
When studying the susceptibility, we have controlled the number of the minima in the systems to avoid fluctuations in $N_z$ (and the grand-canonical ensemble). To fix the number of the minima $\langle
N_z \rangle \sim L_z/L^\zeta$ in a certain system size, we have set the initial position of the interface $\overline{z}_0$ in a fixed size window at height $\overline{z}_0/L_z \simeq \rm{const}$. If the ground state interface is originally outside the window, large enough only for a single valley, it is discarded so that the statistics quoted are based on the successful attempts. If the window is well-separated in space from the $z$-directional boundaries it is obvious that this sampling has no effect on the statistical properties as the average energy. After an original ground state is found, with an energy $E_0$, the external field $h$ is applied by increasing the couplings by constant steps until the first jump is observed [@resgraph]. We have also calculated the energy gap between the global minimum and the next lowest minimum. In that case the initial position of the interface $\overline{z}_0$ is also set to be in a fixed size window at height $\overline{z}_0/L_z \simeq \rm{const}$ by discarding all the samples with the global minimum outside the window. After that the lattice is reduced so that bonds in and above the window are neglected and the new ground state, its energy $E_1$, and the corresponding gap energy $\Delta E_1$ are found. Although the discarding procedure is slow we have at least $N=500$ realizations up to system sizes $L=300$, $L_z=500$.
Results
-------
In order to compare the analytical arguments presented in Section \[argumentti\] and in the end to compute the total susceptibility of the manifolds we will first study numerically in $(1+1)$ dimensions the finite size scaling of the average lowest energy level. Then the shape of the energy gap distribution as well as the finite size scaling of the average gap energies are considered. After that the numerical results of the first jump field distribution are presented together with the finite size scaling of the average first jump field. The shape of the first jump field distribution and the finite size scaling of its average are shortly reported for $(2+1)$ dimensional interfaces, too. The susceptibility is derived from the distribution and the finite size scaling of the first jump field. In the end of the section the jump distance distributions are considered.
In order to see the logarithmic correction of the lowest energy level $\langle E_0 \rangle$, Eq. (\[typicalene\]), when the height of a system, and thus the number of the minima, are increased, we have plotted in Fig. \[fig6\] for three different lengths of the directed polymers $L=100,200,$ and $300$ $$\frac{\langle E \rangle - \langle E_0 \rangle}{L^\theta} \sim \left
[ \ln \left( N_z \right) \right ]^{1/2},
\label{typicalene2}$$ where $N_z \sim L_z/L^{\zeta}$, and $\langle E \rangle \simeq
0.365L+1$. The prefactor, 0.365, of the average energy of a polymer with only a single valley in a system, $\langle E \rangle$, we have estimated by calculating systems of sizes $L_z \times L$, where $L_z =
6.5\times w$, $w$ is the average roughness of a polymer, up to system sizes $50 \times 1000$ with $2000$ realizations. One should note, that the prefactor is highly sensitive to the disorder type and boundary conditions, c.f. Ref. [@Hansenetal], and the constant factor (unity) also to the estimate of the size of the single valley. We have used it as a free parameter. Although for the following results the relevant part of the tail of $\langle E \rangle$ should not be a pure Gaussian (at least if we have enough many minima $N_z$, so that the bulk of the distribution is avoided), we have used for $1/\eta = 1/2$ instead of $1/\eta = 1/1.6=0.625$ in the fit of the logarithmic correction. In practice one can not tell these two choices apart in the range of the system sizes used.
The probability distributions for the energy gaps of directed polymers in system sizes $L^2 =100^2$ and $200^2$ are shown in Fig. \[fig7\](a). The distribution has a finite value at $\Delta E_1
=0$ and the tail has approximately a stretched exponential behavior, $\exp(-ax^b)$, with an exponent $b \simeq 1.3$. In the figure there is plotted as a comparison an exponential $\exp(-x)$ line, from which the deviation of the probability distribution is more clearly seen in the inset, where the distribution is in a natural-log scale. The finite value at $\Delta E_1 =0$ is consistent with the weak replica symmetry breaking picture.
To derive the scaling function for the $\Delta E_1$ it is expected that in systems with height $L_z$ small enough to restrict the number of minima $\Delta E_1$ mainly depends on the height of the system $L_z$. On the other hand, when $L_z$ is large enough, there are enough valleys of which to choose the two minima, and one has $\Delta
E_1 \sim \Delta E \sim L^\theta$, hence $$\langle \Delta E_1(L,L_z) \rangle \sim \left\{ \begin{array}{lll}
\tilde {f}(L_z),&\mbox{\hspace{5mm}}&L_z \ll L, \\
L^{\theta},& &L_z \gg L,
\end{array} \right.
\label{limits}$$ when the smaller parameter being varied. Since it is assumed, that $L_z \sim L^\zeta$, a natural scaling form based on these limiting behaviors is, $$\langle \Delta E_1(L,L_z) \rangle\sim L^{\theta} f\left(\frac{L_z}
{L^{\zeta}}\right).
\label{scalingDE}$$ The argument $y = L_z/L^{\zeta}$ for the scaling function $f(y)$ is just a function of the number of the minima, i.e., $L_z/L^\zeta
\sim N_z$, and the scaling function has the form from Eq. (\[gap\]), when $\eta=2$, $$f(y) \sim [\ln y]^{-1/2}.
\label{scalingf}
\label{Nscaling}$$ In Fig. \[fig7\](b) we have plotted the scaling function (\[scalingf\]) by collapsing $\langle \Delta E_1(L,L_z)
\rangle/L^\theta$ versus $L_z/L^{\zeta}$ for various $L$ and $L_z$, and find a nice agreement confirming the scaling behavior Eqs. (\[scalingDE\]), (\[scalingf\]) as well as the analytic form Eq. (\[gap\]) again assuming a Gaussian distribution ($\eta =2$).
Next we explore the first jump fields. Consider the relation of the gap distribution and that of the jump fields given by Eq. (\[Ph1\]). If we approximate $\hat{P}(\Delta E_1)$, Eq. (\[DE1\_3\]), with a uniform distribution, we get for the probability distribution of the first jump field $$P(h_1) = \exp \left[-\frac{N_z h_1}{2}\right] \frac{N_z}{h_1}
\ln \left[ \frac{1- \frac{h_1}{N_z}} {1-h_1}\right] \sim \exp (-h_1).
\label{Ph}$$ The form of $P(h_1)$ implies that it has a finite value at $h_1=0$, which is again consistent with the weakly broken replica symmetry picture [@Parisi90] and also with the discussion in Section \[argumentti\]. We have also tried exponential and power-law type of probability distributions for $\hat{P}(\Delta E_1)$ in Eq. (\[Ph1\]), but all trials with negative exponents vanishes too fast with $h_1$ compared to the numerical data, and all positive exponents have behaviors with $P(h_1 \to 0) \to 0$ and diverge for larger $h_1$.
In Fig. \[fig8\](a) we have plotted the probability distribution of the first jump field for the system sizes $L^2 =100^2$ and $200^2$. The probability distribution of the first jump field is similar to the probability distribution of the gap energies. The analytic formula, Eq. (\[Ph\]), is drawn as a line in the figure. One clearly sees that the line is a pure exponential, $\exp(-x)$, and the deviation of the numerical data from the exponential is similar to Fig. \[fig7\](a) of the energy gap distributions. Hence the shape of the numerical first jump field distribution is approximately a stretched exponential, $\exp(-ax^b)$, with an exponent $b \simeq
1.3$. Based on Figs. \[fig7\](a) and \[fig8\](a) one sees, that the probability distributions of the energy gaps and first jump fields are the same. Thus with the correct $\hat{P}(\Delta E)$ one gets from Eq. (\[Ph1\]) the same $P(h_1)=\hat{P}(\Delta E)$. This distribution may be for small $h_1$ flat, but obviously starts to vanish for larger $h_1$ since $P$ decays faster than exponentially. Another reason can be the fact that by discarding samples with the GS outside of a predefined window we just constrain the number of valleys in the sample so that the expectation value is the same in each one. $N_z$ can however fluctuate from sample to sample. The most important consequence is in any case that $\hat{P}(\Delta E_1=0)=P(h_1=0)
\not=0$. The finite size scaling and the normalization of the probability distribution of $\Delta E_1$ give $\hat{P}(\Delta
E_1=0) \sim L^{-\theta} [\ln(L_z L^{-\zeta})]^{1/\eta}$.
In order to find the scaling relation for the first jump field $h_1$, we make the Ansatz $\langle \Delta E_1 \rangle = \langle h_1 \rangle L
L_z$, since the field contributes to the manifold energy proportional to $L^D$ ($D=1$) and $L_z \sim \langle \Delta z_1 \rangle$ is the difference in the field contributions $hz$ to the energy at finite $h$ at different average valley heights $z_0$, $z_1$. Hence $$\langle h_1(L,L_z) \rangle L L_z \sim L^{\theta}
f\left(\frac{L_z}{L^{\zeta}}\right),
\label{scalingh1}$$ where the scaling function $f(y)$ for the number of the minima $N_z
\sim L_z/L^\zeta \sim y$ has the same scaling function Eq. (\[Nscaling\]). Fig. \[fig8\](b) shows the scaling function (\[scalingf\]) with a collapse of $\langle h_1(L,L_z)
\rangle L^{1-\theta}L_z$ versus $L_z/L^{\zeta}$ for various $L$ and $L_z$ with a good agreement, again.
Next we move over to the $(2+1)$ dimensional manifolds. The inset of Fig. \[fig9\] shows the tail of the distribution for the system size $L^3 =50^3$ with dilution type of disorder and bond occupation probability $p=0.5$. The first non-overlapping jumps are included in the distribution, since due to the anomaly of the dilution disorder (lots of small scale degeneracy), there are typically two adjustments in the interface before the large jump. As a comparison, we plot again the $\exp(-x)$ line, too. One sees that the deviation of the tail of the distribution from the exponential behavior is similar to the $(1+1)$ dimensional case. The finite scaling of the first jump field in Fig. \[fig9\] is shown for interfaces of size $L_x
\times L_y = 50^2$ and systems of height $L_z = 30\--90$. We have fitted for a constant $L$ the formula (\[scalingh1\]), i.e., $\langle
h_1(L_z) \rangle \sim L_z^{-1} [\ln(L_z)]^{-1/2}$ and it works within error bars.
Generalizing the numerical results of $(1+1)$ and $(2+1)$ dimensional calculations and the analytic arguments from the previous section to arbitrary dimensions gives the behavior of $\langle h_1(L,L_z) \rangle
\sim L^{\theta-D} L_z^{-1} [\ln(L_z/ L^{\zeta})]^{-1/2}$. Since the probability distribution has a finite value at $P(h_1 = 0)$ and $\langle h_1(L,L_z) \rangle$ vanishes with increasing system size, one obtains from the normalization factor at $P(h_1 = 0)$ for the scaling of the susceptibility, Eq. (\[Eqsuskis2\]) $$\chi \sim L^{D-\theta} L_z [\ln(L_z/L^{\zeta})]^{1/2},
\label{Eqsuskis3}$$ and in the isotropic limit, $L \propto L_z$, the total susceptibility $\chi_{tot} = L^d \chi$ becomes $$\chi_{tot} \sim L^{2D+1-\theta} [(1-\zeta)\ln(L)]^{1/2}.
\label{Eqsuskistot}$$ Thus the extreme statistics of energy landscapes shows up in the susceptibility of random manifolds or domain walls in a form that Eq. (\[Eqsuskistot\]) has a logarithmic multiplier and the scaling behavior of the first jump field is due to the scaling of the energy minima differences. This is in contrast to Shapir’s result [@Shapir91] and the logarithmic contribution is also missing from our earlier paper [@Seppala00] where the valley energy scale was taken to follow the standard $L^\theta$ assumption. Therefore one can conclude that the effect of extreme statistics is important in this problem: since we look at the finer details of the landscape “typical” differences are not sufficient. Notice that for two dimensional random field Ising magnets $\zeta \simeq 1$ at large scales [@Seppala98] and the susceptibility does not diverge [@Aizenman]: the premise that $N_z > 1$ is broken in that case. If the condition $N_z > 1$ is violated the extreme statistics correction disappears. For flat interfaces so that the effective roughness exponent is zero ($\zeta_{eff}=0$), e.g. {100} oriented RB interfaces with weak disorder and small system sizes $L<L_c$, when $\zeta=0$ [@Bouchaud92; @Alava96; @Raisanen98; @unpublished2], $\theta
=D/2$ and Eq. (\[Eqsuskistot\]) becomes $\chi_{tot} \sim
L^{2D+1-D/2} [\ln(L)]^{1/2} \sim L^4 [\ln(L)]^{1/2}$, when $D=2$.
Finally we report the jump distance distributions of the first jumps. Fig. \[fig10\](a) shows $P(\Delta z_1/\overline{z}_0)$ of the first jump with the field in $(1+1)$ dimensions. The distribution is a superposition of two behaviors: the interface jumps to the lowest minimum as such, and the external field favors the minima close to the wall (remember the differences of $z_1$ and $z_{1^*}$ in Fig. \[fig5\]). Since the non-overlapping cases are excluded and the wall has a repulsive effect, there are cut-offs in the both ends of the distributions. The shape of the distribution does not change with the system size, which is consistent with the assumptions in Eqs. (\[Eqsuskis2\]) and (\[scalingh1\]), $\langle \Delta z_1
\rangle \sim L_z$. Fig. \[fig10\](b) shows the same distribution of $P(\Delta z_1/\overline{z}_0)$ with a field for $(2+1)$ dimensional manifolds and it is clear that the shape of the distribution does not depend on the dimension. In Fig. \[fig10\](c) we have plotted $P(\Delta z_1/\overline{z}_0)$ without a field, i.e., the distance of the true lowest energy minima in $(1+1)$ dimensions. Now there is no field, and thus the shape of the distribution is just a uniform one, again consistent with the predictions made in Section \[argumentti\] in Eq. (\[P0prod\]).
Scaling of the first jumps: field and distance {#first_jump}
==============================================
Following similar arguments as Eq. (\[excitation\]) in Section \[glassy\] for an excitation, a mean field result for the finite size scaling of the first jump field is next derived. Let us have an interface at an arbitrary height $z_0$ with an energy $E_0$ and an isotropic system $L \propto L_z$. The energy gap between the two lowest energy minima, Eq. (\[gap\]), scales as $\Delta E_1 \sim
L^{\theta} [ (1-\zeta) \ln (L)]^{-1/2}$, when the $z_0 \simeq L_z$. However, when the interface is at an arbitrary height the number of the available minima depends on the original position of the interface, and we use only the dominating algebraic behavior so that $\Delta E_1 \sim L^{\theta}$.
On the other hand the energy difference of elastic manifolds at different heights due to the field contribution is $\Delta E \simeq h
\Delta z L^D$. Assuming that $\langle \Delta z \rangle \sim L$ as in the earlier arguments, the field contribution becomes $\Delta E \sim h
L^d$. We expect that the first jump happens, when the gap equals the field energy, and thus the first jump field has a scaling $$\langle h_1 \rangle \sim L^{\alpha} = L^{\theta-d}.
\label{MFh1}$$
In Figs. \[fig11\](a) and \[fig11\](b) we have plotted the average first jump field $\langle h_1 \rangle$ in isotropic systems for $(1+1)$ and $(2+1)$ dimensional manifolds, respectively. The data is taken from non-overlapping jumps, to minimize finite size effects since they are smaller than with the overlapping jumps included. However, the fraction of overlapping jumps is small as seen in Fig. \[fig4\] for $(1+1)$ dimensional case. The data in Figs. \[fig11\] confirms within error bars the expected behavior of $\langle h_1 \rangle \sim L^{-5/3}$ and $\langle h_1
\rangle \sim L^{-2.18}$ for $(1+1)$ and $(2+1)$ dimensional manifolds, respectively. The logarithmic term $[\ln (N_z)]^{-1/2}$ might contribute a downward trend since it decreases with the number of the available minima $N_z$, here disorder averaged due to the arbitrary GS location. This is however not noticeable in the data: the finite size effects in the calculated data are towards smaller absolute value of the exponent. In the insets of Figs. \[fig11\](a) and (b) the linear scaling of jump sizes $\langle \Delta z_1 \rangle \sim L$ is confirmed, which was assumed in Eqs. (\[Eqsuskis2\]) and (\[scalingh1\]).
The evolution of the jump size distribution for succeeding jumps is demonstrated in Fig. \[fig12\]. As a comparison for Fig. \[fig10\] where only the non-overlapping jumps were considered, here we have shown the overlapping ones, too. It is seen as a peak near $\Delta
z_{n}/ \overline{z}_{n-1} = 0$. The first jumps have clearly the most weight in the large jump end of the distribution, but for the following jumps the whole distribution shifts towards $\Delta
z_{n}/\overline{z}_{n-1} = 0$. After a small number of jumps the interface is already in the minimum closest to the wall and then the random-bulk wetting behavior takes over. In order to estimate the number of the jumps the manifold does before binding to the wall, it is assumed that after a jump the system looks statistically the same as before it. From Fig. \[fig10\](a) we infer that the probability distribution for the jump size has a form $P(\Delta z) \sim \Delta
z^2$ (note that we did not attempt to compute this from the analytical valley arguments) and that is then taken to be the same for all (large) jumps $\Delta z_n$. To calculate the average jump size of jumps which [*do not*]{} jump to the closest valley to the wall we have $$\begin{aligned}
\langle \Delta z_n \rangle = \int_0^{z_{n-1}-A_1w} B_{n-1} \Delta z
(\Delta z)^2 \, d(\Delta z) \nonumber \\
= \frac{3(z_{n-1}-A_1w)^4}{4z_{n-1}^3},
\label{Eqjumpsize}\end{aligned}$$ where the upper bound of the integral is to neglect the cases that jump to the closest valley to the wall, $A_1$ is the prefactor to multiply the roughness value to get the valley width, and $B_{n-1} =
3/z_{n-1}^3$ normalizes the probability distribution. Using $\langle z_n \rangle = \langle z_{n-1} \rangle - \langle \Delta z_n
\rangle$ we get the next position of the interface. In order to calculate the probability of an interface to jump with the first jump to the closest valley to the wall, $p_1$, we use the same probability distribution as in Eq. (\[Eqjumpsize\]), $$p_1 = \int_{z_0-A_1w}^{z_0} B_0 (\Delta z)^2 \, d(\Delta z) =
1-\left[1-\frac{A_1w}{z_0}\right]^3 = {\tilde p}_1,
\label{jumpp1}$$ and similarly for the probability of an interface to jump with the second jump to the closest valley to the wall $$\begin{aligned}
p_2 = (1-{\tilde p}_1) {\tilde p}_2 = (1-{\tilde p}_1)
\int_{z_1-A_1w}^{z_1} B_1 (\Delta z)^2 \, d(\Delta z) \nonumber \\
= \left[1-\frac{A_1w}{z_0}\right]^3 \left\{1-\left[1-\frac{A_1w}{z_1
}\right]^3 \right\}
\label{jumpp2}\end{aligned}$$ Due to the hierarchy we finally get for the $n^{th}$ jump $$p_n = {\tilde p}_n \prod_{k=1}^{n-1} (1-{\tilde p}_k) =
\left[1-\left(1-q_{n}\right)^3 \right] \prod_{k=1}^{n-1} (1-q_k)^3,
\label{jumppn}$$ where $$q_n = \frac{A_1w}{z_{n-1}}.
\label{qfrac}$$ To get an estimate for the number of jumps, we write $$\langle {\mathcal N} \rangle = \sum_{n=1}^{\infty} np_n.
\label{n_estim}$$ This can be evaluated, but only approximately since among others the estimate of Eq. (\[jumppn\]) breaks down beyond $n$ small. Using $A_1w \simeq AL^\zeta= 50$, $L = 1000$, which is the case with $A_1
\simeq 6.5$, (see the numerics in Section \[results\]), and taking $\overline{z}_0 =1000$, we get $\langle {\mathcal N} \rangle \simeq
3$. Eq. (\[Eqjumpsize\]) gives for the first jump size, with the above numerical values, $\langle \Delta z_1 \rangle \simeq 0.6
\overline{z}_0$, which is not far from the behavior in the inset of Fig. \[fig11\](a), where $\langle \Delta z_1 \rangle \sim 0.4
L_z \sim 0.8 \overline{z}_0$, since $\langle \overline{z}_0 \rangle
\sim L_z/2$.
Application of a non-zero field: random-bulk wetting {#wetting}
====================================================
In order to see the wetting behavior one just studies the effect of a large external field on the average interface to wall distance. Notice that the low-field physics discussed extensively above can be considered as the eventual outcome in any sample with $L_z > L^\zeta$ so that more than one valley is available. This means that when the field is decreased from a large field value, the interface finally jumps into the bulk. Fig. \[fig13\](a) we show the average mean height $\langle \overline{z}(h) \rangle$ versus the field $h$ for $(1+1)$ and $(2+1)$ dimensional manifolds. The system sizes are $L_z \times L = 100 \times 3000$ and $L_z \times L^2 =50\times
300^2$. To maximize the prefactor, $A_2$, in the scaling of roughness, $w \sim A_2L^\zeta$, and hence the width of the minimum energy valley, the disorder has been chosen to be strong, i.e., dilution type of disorder with small $p$, but above the bond-percolation threshold. However, there are still some deviations in the form of greater exponents than the expected from Eq. (\[psi\]), which gives the values $\psi =1/2$ and $\psi \simeq
0.26$ in $(1+1)$ and $(2+1)$ dimensions from $\zeta_{(1+1)}=2/3$ and $\zeta_{(2+1)}=0.41\pm 0.01$, respectively. We found that the trend is nevertheless clear, with greater $L$ and fixed $L_z$ the exponents become closer to the expected one. The effective exponent $\psi_{eff}(L)$ can be used to extract the asymptotic, $L$-independent exponent in particular in $(2+1)$ dimensions since this case is hampered most by finite-size effects. $\psi_{eff}(L)$ as a function of $1/L$ indicates that the asymptotic value is indeed $0.26 \pm 0.02$ and that the system sizes at which $\psi_{eff}(L)$ approaches that are of the order of $L = 10^4$. That is, only with $10^{10}$ sites it becomes possible to reach the asymptotic regime. When one has $L
\propto L_z$ calculating the average mean height $\langle
\overline{z}(h) \rangle$ with a fixed field $h$ is nothing but averaging over the jumped and not jumped interfaces together with their location, see Fig. \[fig2\](a).
In Fig. \[fig13\](b) we have plotted as a comparison the average mean height $\langle \overline{z}(h) \rangle$ versus the field $h$ for $(2+1)$ dimensional manifolds with a weak disorder. In this case the [*weak*]{} means for system sizes used, that the roughness of the manifolds are not in the asymptotic roughness limit yet, $L<L_c$ [@Bouchaud92; @Alava96; @Raisanen98; @unpublished2]. The system sizes used are $L_z \times L^2 =50\times 80^2\--400^2$ and the behavior is simple: either the interface stays in its original position or jumps directly to the wall. Taking into account the jumped and original interfaces as $\langle z(h) \rangle = \langle z
[1-P(\overline{z}_0,h)] \overline{z}_0 + P(\overline{z}_0,h) \times 0
\rangle = \langle [1-P(\overline{z}_0,h)] \overline{z}_0 \rangle =
\int_0^{L_z} [1-P\{\overline{z}_0(h)\}] \overline{z}_0 \,
d\overline{z}_0$, gives the behavior $\langle z(h) \rangle \sim
h^{-1}$, i.e., the exponent $\psi \simeq 1$, if $P\{\overline{z}_0(h)\}
\sim h^{-1}$. The larger manifolds jump faster to the wall, i.e., they feel the perturbation earlier, since Eq. (\[scalingh1\]) for flat interfaces ($\zeta =0$) becomes $\langle h_1(L,L_z) \rangle \sim
L^{\theta-D} L_z^{-1} [\ln(L_z)]^{-1/2}$. With fixed $L_z$ and $\theta
=D/2$ from Poissonian statistics, we get in $D=2$ $\langle h_1(L)
\rangle \sim L^{-1}$ consistent with the numerical data. This leads to the behavior of the wetting scaling, $\langle \bar{z}(h) \rangle
\sim c(L) h^{-\psi}$ where $c(L) \simeq L^{-1}$ and $\psi=1$. The finite size scaling of the prefactor indicates that at large $L$-limit with fixed $L_z$ the flat interfaces are immediately at the wall, and thus the systems are non-wet. This implies that there is an interesting cross-over around $L_c$ between such a “dry” regime and the bulk wetting that takes over for still larger $L$. In $D>2$ this kind of behavior is relevant even in the thermodynamic limit, if the disorder is weak due to the presence of a bulk roughening transition for bond disorder.
Discussion {#disc}
==========
Finite temperature behavior {#creep}
---------------------------
The movement of the elastic manifolds in random media at low temperatures, when an applied force is much below the depinning threshold $F_c$, is characterized as creep. The dynamics is controlled by thermally activated jumps over pinning energy barriers, which separate the metastable states. D. Fisher and Huse [@Fish91] showed that for a DP at finite temperature $T>0$ the fluctuations of the entropy $(\Delta S)^2$ and the internal energy $(\Delta
E_{int})^2$ scale linearly with the length of the polymer and cancel each other. Hence there are only the fluctuations of the free energy $(\Delta F)^2$, which scale with the zero temperature energy fluctuation exponent $2\theta=2/3$. Since the free energy is the one which should be minimized at finite temperature, it is the one which defines statistically the shape of the energy landscape, although the energy valleys and minima need not to have exactly the same real space structure as at $T=0$. Thus the $\theta=1/3$ exponent should define the energy gaps also when $T>0$, expect in the cases there is a critical temperature $T_c$. Hence, our derivation of the susceptibility and also the first order character in the reorganization of valleys should be relevant also at $T>0$.
First arrival times in nonlinear surface growth {#secKPZ}
-----------------------------------------------
The $(1+n)$ dimensional directed polymers map, in the continuum limit, to the KPZ [@KPZ; @Barab] equation by associating the minimum energy of a DP-configuration with the minimum [*arrival time*]{} $t_1 \equiv
E_0$ of a KPZ-surface to height $H$. The connection is illustrated in Fig. \[fig14\]. The minimal path of the DP with the end point ${\bf
x}_1(t_1)$ equals the path by which the interface reaches $H$, at location ${\bf x}_1$ and at time $t_1$. Thus $t_1$ attains a logarithmic correction, from Eq. (\[typicalene\]), of size $-H^{\beta} \{\ln(L/H^{1/z}) \}^{1/\eta}$, where $L$ is the linear size of the system, $\beta = \theta > 0$ and $z=1/\zeta$ are now the roughening exponent and dynamical exponent of the KPZ universality class and the values of $\theta = 2\zeta-1$ and $\zeta >1/2$ depend on $n$. Notice that if there is a upper critical dimension $n_c$ in the KPZ growth, then the logarithmic correction is not there anymore, and $\theta=0$, $\zeta=1/2$, i.e., a random walk ensues. Consider now the second smallest arrival time $t_2$. In the language of directed polymers, if the path ${\bf x}_2 (t')$ corresponding to $t_2$ is independent of ${\bf x}_1 (t')$ that corresponds to $t_1$ the time and the path are found inside a separate, independent valley. The [*difference*]{} $\Delta t = t_2 - t_1$ then equals $\Delta E_1$, and likewise obeys extreme statistics, so that $\Delta t \sim H^\beta
[\ln (L/H^{1/z})]^{-(\eta-1)/\eta}$. For growing surfaces this limit is the [*early stages*]{} of growth, in which the correlation length $\xi \ll L$, and therefore the arrival times, or directed polymer energies, are independent. On the other hand if we disturb the growing process such a way that it depends on the ${\bf x}$, e.g. linearly with a factor $h$ [@spohn], the polymer ending at ${\bf x}_2$ becomes faster if $h>h_1$, see right hand side of Fig. \[fig14\]. Similarly now the factor $h_1$ has a scaling behavior from Eq. (\[scalingh1\]), $h_1
\sim H^{\beta-1} L^{-1} [\ln (L/H^{1/z})]^{-(\eta-1)/\eta}$.
Conclusions {#concl}
===========
To conclude we have studied the $(1+1)$ and $(2+1)$ dimensional elastic manifolds at zero temperature, when an external field is applied. We have demonstrated that the response of manifolds shows a first order character (“jump”) in the sense that the manifolds change their configuration in the large system size limit completely. This persists in a finite system over a small number of such jumps. The distance that the center-of-mass moves is extensive. The whole picture is based on a level-crossing between two low-energy valleys in the energy landscape. Averaging over the total magnetization of such random magnets with a domain wall or over the positions of manifolds when an external field is applied becomes dependent on whether the DW or manifold has jumped or not. This leads to the problem of self-averaging in random systems. Here a disorder average smooths over the “coexistence” between systems that are not affected by a finite field $h$ and those that have responded.
In order to study the susceptibility of the DW in random media, one has to take into account the probability distribution of the sample-dependent field associated with the global change of the configuration, and take the limit of vanishing fields. This probability distribution has a finite density at $h=0$. The finite size scaling of the density is dependent on the finite size scaling of the number of the low lying nearly degenerate energy minima in the system. The scaling of the number of the energy minima leads to a logarithmic factor in the susceptibility, and can be accounted for by using extreme statistics. Such effects are difficult to study by usual field theoretical means, since one has to have access to the whole probability distribution and not only the few first moments thereof. Notice that the crucial difference to much previous work is the simple fact that we allow for multiple minima in the energy landscape, which is often most excluded by the boundary conditions applied to the problem. Although the derivations and the numerical calculations done here have concentrated on random bond type of randomness similar behavior should be seen in random field cases, too.
The discrete character in the movement of elastic manifolds with an external field results also in that the continuum theory for wetting in random systems works only in slab geometries, where there is room only for a single valley, or in the large external field limit, when the interface is close to the wall. On the other hand the flat interfaces are shown to jump directly to the wall, i.e., to be non-wet. It would be interesting to see if the dynamics of the manifolds at finite temperature reflects the first order character seen here at $T=0$. Through the connection of (1+1) dimensional DW’s to the directed polymers of the KPZ surface growth, we have shown that the logarithmic factor is also present in the statistics of growth times in nonlinear surface growth.
The authors would like to thank the Academy of Finland’s Centre of Excellence Programme for financial support and the Center for Scientific Computing, Espoo, Finland for computing resources. Phil Duxbury and Simone Artz are acknowledged for many valuable discussions.
D. Fisher, Phys. Rev. Lett. **56** (1986), 1964.
T. Emig and T. Nattermann, Eur. Phys. J. B, **8** (1999), 525.
M. Kardar, G. Parisi, and Y.-C. Zhang, Phys. Rev. Lett. **56** (1986), 889.
T. Halpin-Healy and Y.-C. Zhang, Phys. Rep. **254** (1995), 215.
M. Lässig, J. Phys. Cond. Mat. **10** (1998), 9905.
G. Blatter, M. V. Feigel’man, V. B. Geshkenbein, A. I. Larkin, and V. M. Vinokur, Rev. Mod. Phys. **66** (1994), 1125.
J. L. Cardy and S. Ostlund, Phys. Rev. B **25** (1982), 6899.
J. Toner and D. P. DiVincenzo, Phys. Rev. B **41** (1990), 632.
J.-Ph. Bouchaud, L. F. Cugliandolo, J. Kurchan, and M. Mézard in *Spin Glasses and Random Fields*, ed. A. P. Young, (World Scientific, Singapore 1997); T. Giarmarchi and P. Le Doussal, *ibid.*
D. S. Fisher and D. A. Huse, Phys. Rev. B **38** (1988), 386; for recent work see M. Palassini and A. P. Young, Phys. Rev. Lett. **83** (1999), 5126.
M. Mézard, G. Parisi, and M. A. Virasoro, *Spin Glass Theory and Beyond* (World Scientific, Singapore 1987); K. Binder and A. P. Young, Rev. Mod. Phys. **58** (1986), 801.
R. Lipowsky and M. E. Fisher, Phys. Rev. Lett. **56** (1986), 472.
J. Wuttke and R. Lipowsky, Phys. Rev. B **44** (1991), 13 042.
S. Dietrich in *Phase Transitions and Critical Phenomena*, edited by C. Domb and J. L. Lebowitz (Academic Press, San Diego 1988), vol. 12; G. Forgacs, R. Lipowsky and Th. M. Nieuwenhuizen in *Phase Transitions and Critical Phenomena*, edited by C. Domb and J. L. Lebowitz (Academic Press, San Diego 1991), vol. 14.
See for recent studies, e.g., S. Scheidl and Y. Dinçer, cond-mat/0006048; P. Chauve, P. Le Doussal, and K. Wiese, Phys. Rev. Lett. **86** (2001), 1785.
L. Balents and D. S. Fisher, Phys. Rev. B **48** (1993), 5949.
D. A. Huse and C. L. Henley Phys. Rev. Lett. **54** (1985), 2708.
E. T. Seppälä, V. Petäjä, and M. J. Alava, Phys. Rev. E **58** (1998), R5217.
D. S. Fisher and D. A. Huse, Phys. Rev. B **43** (1991), 10 728.
For the spin glass case see J. Houdayer and O. C. Martin, Phys. Rev. Lett. **81** (1998), 2554.
M. Mézard, J. Phys. France **51** (1990), 1831.
T. Hwa and D. S. Fisher, Phys. Rev. B **49** (1994), 3136.
E. T. Seppälä and M. J. Alava, Phys. Rev. Lett. **84** (2000), 3982.
E. T. Seppälä, M. J. Alava, and P. M. Duxbury, Phys. Rev. E, in press, e-print: cond-mat/0102318.
G. Parisi, J. Phys. France **51** (1990), 1595.
B. Derrida, Phys. Rev. B **24** (1981), 2613.
J.-P. Bouchaud and M. Mézard, J. Phys. A **30** (1997), 7997.
E. J. Gumbel, *Statistics of Extremes*, (Columbia University Press, New York 1958).
Y. Shapir, Phys. Rev. Lett. **66** (1991), 1473.
P. Jögi and D. Sornette, Phys. Rev. E **57** (1998), 6936.
M. Huang, M. E. Fisher, and R. Lipowsky, Phys. Rev. B **39** (1989), 2632.
See also J. Galambos, *The Asymptotic Theory of Extreme Order Statistics*, (John Wiley & Sons, New York 1978), section II.
P. M. Duxbury, P. L. Leath, and P. D. Beale, Phys. Rev. B **36** (1987), 367; P. M. Duxbury, P. L. Leath, and P. D. Beale, Phys. Rev. Lett. **57** (1986), 1052.
J. A. Kim, M. A. Moore, and A. J. Bray, Phys. Rev. A **44** (1991), 2345.
S. C. Chapman, G. Rowlands, and N. W. Watkins, cond-mat/0007275.
J.-Ph. Bouchaud and A. Georges, Phys. Rev. Lett. **68** (1992), 3908.
M. J. Alava and P. M. Duxbury, Phys. Rev. B **54** (1996), 14990.
V. I. Räisänen, E. T. Seppälä, M. J. Alava, and P. M. Duxbury, Phys. Rev. Lett. **80** (1998), 329.
E. T. Seppälä, M. J. Alava, and P. M. Duxbury, Phys. Rev. E **63** (2001), 036126.
M. Alava, P. Duxbury, C. Moukarzel, and H. Rieger, in *Phase Transitions and Critical Phenomena*, edited by C. Domb and J. L. Lebowitz (Academic Press, San Diego 2001), vol. 18.
J. C. Picard and H. D. Ratliff, Networks **5** (1975), 357.
L. R. Ford and D. R. Fulkerson, *Flows in Networks*, (Princeton University Press, Princeton 1962) see also any standard book on graph theory and flow problems.
A. V. Goldberg and R. E. Tarjan, J. Assoc. Comput. Mach. **35** (1988), 921.
Note that one can also use the level crossing effect to compute the crossing points exactly, as noticed by Anglès d’Auriac and Sourlas [@Auriac]. This allows to use the residual graph of the original groundstate for the further computations, and makes the solution faster [@Alavaetal].
S. Roux, A. Hansen, and E. Guyon, J. Phys. (Paris) **48** (1987), 2125.
M. Aizenman and J. Wehr, Phys. Rev. Lett. **62** (1989), 2503.
A.-L. Barab[á]{}si and H. E. Stanley, *Fractal concepts in surface growth*, (Cambridge University Press, Cambridge, U.K. 1995).
See an example of effects of boundary conditions: M. Prähofer and H. Spohn, Phys. Rev. Lett. **84** (2000), 4882.
J. C. Anglès d’Auriac and N. Sourlas, Europhys. Lett. **39** (1997), 473.
![A schematic figure of two interfaces in a system of parallel length $L$ and perpendicular one $L_z > L^\zeta$ at heights $\bar{z}_1$ and $\bar{z}_2 \simeq w_2$, with the corresponding roughness values $w_1$ and $w_2$. $\xi_\perp$ denotes the transverse correlation length of a part of the interface of length $\xi_\|$. See the text. A droplet is seen in the interface of height $\bar{z}_1$ as a dotted line.[]{data-label="fig1"}](./fig1.eps){width="9cm"}
![The level-crossing phenomenon for interfaces in random systems in the presence of an external field. Originally the interface lies at height $z_0$ and has an energy $E_0$. When the field is applied its energy increases linearly and at $h_1$ the interfaces jumps to $z_1 < z_0$, with the energy $E_1(h_1)=E_1
+ h_1 L^D z_1$, where $E_1$ is the energy of the interface at $z_1$ without the field. Similar behavior takes place at $h_n$, $n=2,3,\dots $, when the interface moves from $z_{n-1}$ to $z_n < z_{n-1}$. The thick line represents the minimum energy of the interface $E(h)$ as a function of the field $h$, with discontinuities of the derivative at $h_n$.[]{data-label="fig3"}](./fig3.eps){width="7cm"}
![(a) A simplified view of the minima in the energy landscape of a random system. $\Delta E_1 = E_1 -E_0$ is the energy gap between the ground state at $z_0$, denoted with a black circle, and the second lowest minimum at $z_1$ denoted with a gray circle. $\Delta E_{1^*} = E_{1^*} -E_0$ is the energy gap (normalized with $L^D$) between the ground state and the minimum at $z_{1^*}$. (b) The view of the minima in the random system when the field is applied. At $h_1$ the interface moves from $z_0$ to $z_{1^*}$, indicated with a gray circle, since the energy difference $\Delta E_{1^*} = E_{1^*} -E_0 = h_1 z_0 - h_1 z_{1^*} = 0$ while all the other $\Delta E$’s are greater. However, often $E_{1^*}$ and $E_1$ coincide.[]{data-label="fig5"}](./fig5a.eps){width="8cm"}
![(a) A simplified view of the minima in the energy landscape of a random system. $\Delta E_1 = E_1 -E_0$ is the energy gap between the ground state at $z_0$, denoted with a black circle, and the second lowest minimum at $z_1$ denoted with a gray circle. $\Delta E_{1^*} = E_{1^*} -E_0$ is the energy gap (normalized with $L^D$) between the ground state and the minimum at $z_{1^*}$. (b) The view of the minima in the random system when the field is applied. At $h_1$ the interface moves from $z_0$ to $z_{1^*}$, indicated with a gray circle, since the energy difference $\Delta E_{1^*} = E_{1^*} -E_0 = h_1 z_0 - h_1 z_{1^*} = 0$ while all the other $\Delta E$’s are greater. However, often $E_{1^*}$ and $E_1$ coincide.[]{data-label="fig5"}](./fig5b.eps){width="8cm"}
![The relation between directed polymers and growing interfaces. Two directed polymers in independent valleys equal the fastest arrival time $t_1$ at $\vec{x}_1$, solid line, and the second fastest at $\vec{x}_2$ with time $t_2$, dotted line, of a KPZ interface to a fixed height $H$, when the external field $h=0$. In the right hand side figure an external field is added, which increases the growth time depending on the position in the direction of the arrow and the polymer at $\vec{x}_2$ becomes the one corresponding to the fastest time to reach $H$.[]{data-label="fig14"}](./fig14.eps){width="9cm"}
| ArXiv |
---
author:
- Shauna Revay
- Matthew Teschke
bibliography:
- 'sample.bib'
title: Multiclass Language Identification using Deep Learning on Spectral Images of Audio Signals
---
Introduction
============
Recently, voice assistants have become a staple in the flagship products of many big technology companies such as Google, Apple, Amazon, and Microsoft. One challenge for voice assistant products is that the language that a speaker is using needs to be preset. To improve user experience on this and similar tasks such as automated speech detection or speech to text transcription, automatic language detection is a necessary first step.
The technique described in this paper, language identification for audio spectrograms (LIFAS), uses spectrograms of raw audio signals as input to a convolutional neural network (CNN) to be used for language identification. One benefit of this process is that it requires minimal pre-processing. In fact, only the raw audio signals are input into the neural network, with the spectrograms generated as each batch is input to the network during training. Another benefit is that the technique can utilize short audio segments (approximately 4 seconds) for effective classification, necessary for voice assistants that need to identify language as soon as a speaker begins to talk.
LIFAS binary language classification had an accuracy of 97%, and multi-class classification with six languages had an accuracy of 89%.
Background
==========
Finding a dataset of audio clips in various languages sufficiently large for training a network was an initial challenge for this task. Many datasets of this type are not open sourced [@mozilla]. VoxForge [@voxforge], an open-source corpus that consists of user-submitted audio clips in various languages, is the source of data used in this paper.
Previous work in this area used deep networks as feature extractors, but did not use the networks themselves to classify the languages [@conference; @unified]. LIFAS removes any feature extraction performed outside of the network. The network is fed a raw audio signal, and the spectrogram of the data is passed to the neural network during training. The last layer of the network outputs a vector of probabilities with one prediction per language. Thus, the whole process from raw audio signal to prediction of language is performed automatically by the neural network.
In [@lstmpaper], a CNN was combined with a long short-term memory (LSTM) network to classify language using spectrograms generated from audio. The network presented in [@lstmpaper] classified 4 languages using 10-second audio clips for training [@blog], while LIFAS achieves similar performance for 6 languages using 4-second audio clips. This demonstrates the robustness of the architecture and its improvement upon earlier techniques.
Residual and Convolutional Neural Networks
------------------------------------------
CNNs have been shown to give state of the art results for image classification and a variety of other tasks. As neural networks using back propagation were constructed to be deeper, with more layers, they ran into the problem of vanishing gradient [@gradient]. A network updates its weights based on the partial derivatives of the error function from the previous layers. Many times, the derivatives can become very small and the weight updates become insignificant. This can lead to a degradation in performance.
One way to mitigate this problem is the use of Residual Neural Networks (ResNets [@resnet]). ResNets utilize skip connections in layers which connects two non-adjacent layers. ResNets have shown state-of-the-art performance on image recognition tasks, which makes them a natural choice for a network architecture for this task [@imageresidual].
Spectrogram Generation
----------------------
A spectrogram is an image representation of the frequencies present in a signal over time. The frequency spectrum of a signal can be generated from a time series signal using a Fourier Transform.
In practice, the Fast Fourier Transform (FFT) can be applied to a section of the time series data to calculate the magnitude of the frequency spectrum for a fixed moment in time. This will correspond to a line in the spectrogram. The time series data is then windowed, usually in overlapping chunks, and the FFT data is strung together to form the spectrogram image which allows us to see how the frequencies change over time.
Since we were generating spectrograms on audio data, the data was converted to the mel scale, generating “melspectrograms”. These images will be referred to as simply “spectrograms” for the duration of this paper. The conversion from $f$ hertz to $m$ mels that we use is given by,
$$m = 2595 \log_{10} \left( 1 + \frac{f}{700} \right).$$
An example of a spectrogram generated by an English data transmission is shown in figure \[spec\].
![Spectrogram generated from an English audio file.[]{data-label="spec"}](spec.png){width="\textwidth"}
Data Preparation
================
Audio data was collected from VoxForge [@voxforge]. Each audio signal was sampled at a rate of 16kHz and cut down to be 60,000 samples long. In this context, a sample refers to the number of data points in the audio clip. This equates to 3.75 seconds of audio. The audio files were saved as WAV files and loaded into Python using the librosa library and a sample rate of 16kHz.
Each audio file of 60,000 samples was saved separately and is referred to as a clip. The training set consisted of 5,000 clips per language, and the validation set consisted of 2,000 clips per language.
Audio clips were gathered in English, Spanish, French, German, Russian, and Italian. Speakers had various accents and were of different genders. The same speakers may be speaking in more than one clip, but there was no cross contamination in the training and validation sets.
Spectrograms were generated using parameters similar to the process discussed in [@audioblog] which used a frequency spectrum of 20Hz to 8,000Hz and 40 frequency bins. Each FFT was computed on a window of 1024 samples. No other pre-processing was done on the audio files. Spectrograms were generated on-the-fly on a per-batch basis while the network was running (i.e. spectrograms were not saved to disk).
Network
=======
We utilized the fast.ai [@fastai] deep learning library built on PyTorch [@pytorch]. The network used was a pretrained Resnet50. The spectrograms were generated on a per-batch basis, with a batch size of 64 images. Each image was $432 \times 288$ pixels in size.
During training, the 1-cycle-policy described in [@leslie] was used. In this process, the learning rate is gradually increased and then decreased in a linear fashion during one cycle [@onecycleblog]. The learning rate finder within the fast.ai library was first used to determine the maximum learning rate to be used in the 1-cycle training of the network. The maximum learning rate was then set to be $1 \times 10^{-2}$. The learning rate increases until it hits the maximum learning rate, and then it gradually decreases again. The length of the cycle was set to be 8 epochs, meaning that throughout the cycle 8 epochs are evaluated.
Experiments
===========
Binary Classification with Varying Number of Samples
----------------------------------------------------
Binary classification was performed on two languages using clips of 60,000 samples. English and Russian were chosen to use for training and validation. To test the impact of the number of samples on classification while keeping the sample rate constant, binary classification was also performed on clips of 100,000 samples.
Multiple Language Classification
--------------------------------
For each language (English, Spanish, German, French, Russian, and Italian), 5,000 clips were placed in the training set. Each clip was 60,000 samples in length. 2,000 clips per language were placed in the validation set, and no speakers or clips appeared in both the training and validation sets.
Results
=======
Accuracy was calculated for both binary classification and multiclass classification as: $$Accuracy = \frac{Number \; of \; Correct \; Predictions}{Total \;Number \;of \; Predictions}.$$ LIFAS binary classification accuracy for Russian and English clips of length 60,000 samples was 94%. In comparison, LIFAS binary classification accuracy on the clips of 100,000 samples was 97 %. The accuracy totals given in the experiments above are calculated on the total number of clips in the validation set. The accuracy can also be broken up into accuracy for English clips, or accuracy for Russian clips, where there was essentially no difference in the accuracy for English clips and the accuracy for Russian clips.
To confirm that the network performance was not dependent on English and Russian language data, binary classification was tested on other languages with little to no impact on validation accuracy.
LIFAS accuracy for the multi-class network with six languages was 89 %. These results were based on clips of 60,000 samples since a sufficient number of longer clips were unavailable. Results from the 100,000 sample clips in the binary classification model suggest performance could be improved in the multi-class setting with longer clips.
The confusion matrix for the multi-class classification is shown in figure \[confusion\].
![The confusion matrix for the multiclass language identification problem.[]{data-label="confusion"}](confusion.png){width="80.00000%"}
Discussion and Limitations
==========================
Notably, the highest rate of false negative classifications came when Spanish clips were classified as Russian, and when Russian clips were classified as Spanish. Additionally, almost no other language is misclassified as Russian or Spanish. One hypothesis for this observation is the fact that Russian is the only Slavic language in the training set. Therefore, the network may be performing some thresholding at one layer that separates Russian from other languages, and by chance Spanish clips are near the threshold.
One limitation in our findings is that all of the data came from the same dataset. Since audio formats can have a wide variety of parameters such as bit rate, sampling rate, and bits per sample, we would expect clips from other datasets collected in different formats to potentially confuse the network. There is potential for this drawback to be overcome assuming appropriate pre-processing steps were taken for the audio signals so that the spectrograms did not contain artifacts from the dataset itself. This is a problem that should be explored as more data becomes available.
Conclusion
==========
This work shows the viability of using deep network architectures commonly used for image classification in identifying languages from images generated from audio data. Robust performance can be accomplished using relatively short files with minimal pre-processing. We believe that this model can be extended to classify more languages so long as sufficient, representative training and validation data is available. A next step in testing the robustness of this model would be to include test data from additional (e.g. non-VoxForge) datasets.
Additionally, we would want the network to be performant on environments with varying levels of noise. VoxForge data is all user submitted audio clips, so the noise profiles of the clips vary, but more regimented tests should be done to see how robust the network is to different measured levels of noise. Simulated additive white Gaussian noise could be added to the training data to simulate low quality audio, but still might not fully mimic the effect of background noise such as car horns, clanging pots, or multiple speakers in a real life environment.
Another way to potentially increase the robustness of the model would be to implement SpecAugment [@specaugment] which is a method that distorts spectrogram images in order to help overfitting and increase performance of networks by feeding in deliberately corrupted images. This may help to add scalability and robustness to the network, assuming the spectral distortions generated in SpecAugment accurately represent distortions in audio signals observed in the real world.
| ArXiv |
---
abstract: 'It has been proved that almost all $n$-bit Boolean functions have [*exact classical query complexity*]{} $n$. However, the situation seemed to be very different when we deal with [*exact quantum query complexity*]{}. In this paper, we prove that almost all $n$-bit Boolean functions can be computed by an exact quantum algorithm with less than $n$ queries. More exactly, we prove that $\mbox{AND}_n$ is the only $n$-bit Boolean function, up to isomorphism, that requires $n$ queries.'
address: |
$^{1}$Faculty of Informatics, Masaryk University, Brno 60200, Czech Republic\
$^2$ Faculty of Computing, University of Latvia,Rīga, LV-1586, Latvia\
$^3$ School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA\
author:
- 'Andris Ambainis $^{2,3}$'
- 'Jozef Gruska$^{1}$'
- 'Shenggen Zheng$^{1,}$'
title: Exact quantum algorithms have advantage for almost all Boolean functions
---
Quantum computing,Quantum query complexity ,Boolean function ,Symmetric Boolean function ,Monotone Boolean function ,Read-once Boolean function
Introduction
============
[*Quantum query complexity*]{} is the quantum generalization of classical [*decision tree complexity*]{}. In this complexity model, an algorithm is charged for “queries" to the input bits, while any intermediate computation is considered as free (see [@BdW02]). For many functions one can obtain large quantum speed-ups in this model in the case algorithms are allowed a constant small probability of error (bounded error). As the most famous example, Grover’s algorithm [@Gro96] computes the $n$-bit $\mbox{OR}$ function with $O(\sqrt {n})$ queries in the bounded error mode, while any classical (also exact quantum) algorithm needs $\Omega(n)$ queries. More such cases of polynomial speed-ups are known, see[@Amb07; @Bel12; @DHHM06]. For [*partial functions*]{}, even an exponential speed-up is possible, in case quantum resources are used, see [@Shor97; @Sim97]. In the bounded-error setting, quantum complexity is now relatively well understood. The model of [*exact quantum query complexity*]{}, where the algorithms must output the correct answer with certainty for every input, seems to be more intriguing. It is much more difficult to come up with exact quantum algorithms that outperform, concerning number of queries, classical exact algorithms.
Though for partial functions exact quantum algorithms with exponential speed-up are known (for instance in [@AmYa11; @BH97; @DJ92; @GQZ14; @ZQ14; @GQZ14b; @Zhg13]), the results for total functions have been much less spectacular: the best known quantum speed-up was just by a factor of 2 for many years [@CEMM98; @FGGS98]. Recently, in a breakthrough result, Ambainis [@Amb13] has presented the first example of a Boolean function $f:\{0,1\}^n\to \{0,1\}$ for which exact quantum algorithms have superlinear advantage over exact classical algorithms.
In exact classical query complexity ([*decision tree complexity*]{}, [*deterministic query complexity*]{}) model, almost all $n$-bit Boolean functions require $n$ queries [@BdW02]. However, the situation seemed very different for the case of exact quantum complexity. Montanaro et al. [@MJM11] proved that $\mbox{AND}_3$ is the only $3$-bit Boolean function, up to isomorphism, that requires 3 queries and using the semidefinite programming approach, they numerically[^1] demonstrated that all $4$-bit Boolean functions, with the exception of functions isomorphic to the $\mbox{AND}_4$ function, have exact quantum query algorithms using at most 3 queries. They also listed their numerical results for all symmetric Boolean functions on 5 and 6 bits, up to isomorphism.
In 1998, Beals at al. [@BBC+98] proved, for any $n$, that $\mbox{AND}_n$ has exact quantum complexity $n$. Since that time it was an interesting problem whether $\mbox{AND}_n$ is the only $n$-bit Boolean function, up to isomorphism, that has exact quantum complexity $n$. In this paper we approve that this is indeed the case. As a corollary we get that almost all $n$-bit Boolean functions have exact quantum complexity less than $n$.
We prove our main results in four stages. In the first one we give the proof for symmetric Boolean functions, in the second one for monotone Boolean functions and in the third one for the case of read-once Boolean functions. On this basis we prove in the fourth stage the general case. In all four cases proofs used quite different approaches. They are expected to be of a broader interest since all these special classes of Boolean functions are of broad interest.
The paper is organized as follows. In Section 2 we introduce some notation concerning Boolean function and query complexity. In Section 3 we investigate symmetric Boolean functions. In section 4 we investigate monotone Boolean functions. In section 5 we investigate read-once Boolean functions. In Section 6 we prove our main result. Finally, Section 7 contains a conclusion.
Preliminaries
=============
We introduce some basic needed notation in this section. See also [@Gru99; @NC00] for details on quantum computing and see [@BdW02; @BBC+98; @NS94] for more on query complexity models and [*multilinear polynomials*]{}.
Boolean functions
-----------------
An $n$-bit Boolean function is a function $f:\{0,1\}^n\to \{0,1\}$. We say $f$ is total if $f$ is defined on all inputs. For an input $x\in\{0,1\}^n$, we use $x_i$ to denote its $i$-th bit, so $x=x_1x_2\cdots x_n$. Denote $[n]=\{1,2,\ldots,n\}$. For $i\in[n]$, we write $$f_{x_i=b}(x)=f(x_1,\ldots,x_{i-1},b,x_{i+1},\ldots,x_n),$$ which is an $(n-1)$ bit Boolean function. For any $i\in[n]$, we have $$\label{Eq-df(x)}
f(x)=(1-x_i)f_{x_i=0}(x)+x_if_{x_i=1}(x).$$
We say that two Boolean functions $f$ and $g$ are [*query-isomorphic*]{} (by convenience, isomorphic will mean query-isomorphic in this paper) if they are equal up to negations and permutations of the input variables, and negation of the output variable. This relationship is sometimes known as NPN-equivalence [@MJM11].
We will use the sign $(\neg)$ for a possible negation. For example, $\mbox{AND}((\neg)x_1,x_2)$ can denote $x_1\wedge x_2$ or $\neg x_1\wedge x_2$. We use $|x|$ to denote the Hamming weight of $x$ (its number of 1’s).
[**Definition 1:**]{} We call a Boolean function $f:\{0,1\}^n\to \{0,1\}$ symmetric if $f(x)$ depends only on $|x|$.
An $n$-bit symmetric Boolean function $f$ can be fully described by a vector $(b_0,b_1,\ldots,b_n)\linebreak[0]\in\{0,1\}^{n+1}$, where $f(x)=b_{|x|}$, i.e. $b_k$ is the value of $f(x)$ for $|x|=k$ [@ZGR97].
For $x,y\in\{0,1\}^n$, we will write $x\preceq y$ if $x_i\leq y_i$ for all $i\in[n]$. We will write $x\prec y$ if $x\preceq y$ and $x\neq y$.
[**Definition 2:**]{} We call a Boolean function $f:\{0,1\}^n\to \{0,1\}$ monotone if $f(x)\leq f(y)$ holds whenever $x\preceq y$.
Monotonic Boolean functions are precisely those that can be defined by an expression combining the input bits (each of them may appear more than once) using only the operators $\wedge$ and $\vee$ (in particular $\neg$ is forbidden). Monotone Boolean functions have many nice properties. For example they have a unique prime conjunctive normal form (CNF) and a unique prime disjunctive normal form (DNF) in which no negation occurs [@EMG08].
Let $f:\{0,1\}^n\to \{0,1\}$ be a monotone Boolean function, $f$ has a prime CNF $$f(x)=\bigwedge_{I\in C}\bigvee_{i\in I} x_i,$$ where $C$ is the set of some $I\subseteq[n]$. Similarly, $f$ has a prime DNF $$f(x)=\bigvee_{J\in D}\bigwedge_{j\in J} x_j,$$ where $D$ is the set of some $J\subseteq[n]$.
[**Definition 3:**]{} A read-once Boolean function is a Boolean function that can be represented by a Boolean formula in which each variable appears exactly once.
For example $f(x_1,x_2,x_3)=(x_1\vee x_2)\wedge (\neg x_3)$ is a $3$-bit read-once Boolean function and $f'(x_1,x_2,x_3)=(x_1\vee x_2)\wedge (\neg x_1\vee \neg x_3)$ is not read-once.
A Boolean formula over the standard basis $\{\wedge,\vee,\neg \}$ can be represented by a binary tree where each internal node is labeled with $\wedge$ or $\vee$, and each leaf is labeled with a literal, that is, a Boolean variable or its negation. The size of a formula is the number of leaves.
[**Definition 4:**]{} The formula size of a Boolean function $f$, denoted $L(f)$, is the size of the smallest formula which computes $f$.
A read-once Boolean function is a function $f$ such that $L(f)=n$ and $f$ depends on all of its $n$ variables.
Exact query complexity models
-----------------------------
An exact classical (deterministic) query algorithm for computing a Boolean function $f:\{0,1\}^n\to \{0,1\}$ can be described by a decision tree. A decision tree $T$ is a rooted binary tree where each internal vertex has exactly two children, each internal vertex is labeled with a variable $x_i$ and each leaf is labeled with a value 0 or 1. $T$ computes a Boolean function $f$ as follows: Start at the root. If this is a leaf then stop and the output of the tree is the value of the leaf. Otherwise, query the variable $x_i$ that labels the root. If $x_i=0$, then recursively evaluate the left subtree, if $x_i=1$ then recursively evaluate the right subtree. The output of the tree is the value of the leaf that is reached at the end of this process. The depth of $T$ is the maximal length of a path from the root to a leaf (i.e. the worst-case number of queries used on any input). The [*exact classical query complexity*]{} (deterministic query complexity, decision tree complexity) $D(f)$ is the minimal depth over all decision trees computing $f$.
Let $f:\{0,1\}^n\to \{0,1\}$ be a Boolean function and $x = x_1x_2\cdots x_n$ be an input bit string. Each exact quantum query algorithm for $f$ works in a Hilbert space with some fixed basis, called standard. It starts in a fixed starting state, then performs on it a sequence of transformations $U_1$, $Q$, $U_2$, $Q$, …, $U_t$, $Q$, $U_{t+1}$. Unitary transformations $U_i$ do not depend on the input bits, while $Q$, called the [*query transformation*]{}, does, in the following way. Each of the basis states corresponds to either one or none of the input bits. If the basis state $|\psi\rangle$ corresponds to the $i$-th input bit, then $Q|\psi\rangle=(-1)^{x_i}|\psi\rangle$. If it does not correspond to any input bit, then $Q$ leaves it unchanged: $Q|\psi\rangle=|\psi\rangle$. Finally, the algorithm performs a measurement in the standard basis. Depending on the result of the measurement, the algorithm outputs either 0 or 1 which must be equal to $f(x)$. The [*exact quantum query complexity*]{} $Q_E(f)$ is the minimum number of queries used by any quantum algorithm which computes $f(x)$ exactly for all $x$.
Note that if Boolean functions $f$ and $g$ are isomorphic, then $D(f)=D(g)$ and $Q_E(f)=Q_E(g)$.
According to Eq. (\[Eq-df(x)\]), if we query $x_i$ first, suppose that $x_i=b$, then we can compute $f_{x_i=b}(x)$ further. Therefore, for any $i\in[n]$, we have $$\label{Eq-n-1ton}
Q_E(f)\leq \max\{Q_E(f_{x_i=0}),Q_E(f_{x_i=1})\}+1.$$
Some special functions and their exact quantum query complexity
---------------------------------------------------------------
Symmetric, monotone and read-once Boolean functions were well studied in query complexity [@BdW02]. The well known Grover’s algorithm [@Gro96] computes $\mbox{OR}_n$, which is symmetric, monotone and read-once. Read-once functions are also well investigated [@BS04; @SW86; @San95].
Some symmetric functions and their exact quantum query complexity that we will refer to in this paper are as follows:
1. $\mbox{OR}_n(x)=1$ iff $|x|\geq 1$. $Q_E(\mbox{OR}_n)=n$ [@BBC+98].
2. $\mbox{AND}_n(x)=1$ iff $|x|=n$. $Q_E(\mbox{AND}_n)=n$ [@BBC+98].
3. $\mbox{PARITY}_n(x)=1$ iff $|x|$ is odd. $Q_E(\mbox{PARITY}_n)=\lceil\frac{n}{2}\rceil$ [@CEMM98; @FGGS98].
4. $\mbox{EXACT}_n^{k}(x)=1$ iff $|x|=k$. $Q_E(\mbox{EXACT}_n^{k})=\max\{k,n-k\}$ [@AISJ13].
5. $\mbox{Th}_{n}^{k}(x)=1$ iff $|x|\geq k$. $Q_E(\mbox{Th}_n^{k})=\max\{k,n-k+1\}$ [@AISJ13].
$\mbox{OR}_n$ is isomorphic to $\mbox{AND}_n$ since $$\neg\mbox{OR}_n(\neg x_1,\neg x_2,\ldots,\neg x_n)=\mbox{AND}_n(x_1,x_2,\ldots,x_n).$$
Some other functions and their exact quantum query complexity that we will refer to in this paper are as follows:
1. $\mbox{NAE}_{n}(x)=1$ iff there exist $i,j$ such that $x_i\neq x_j$. $Q_E(\mbox{NAE}_{n})\leq n-1$.
2. $f(x_1,x_2,x_3)=x_1\wedge(x_2\vee x_3)$. Its exact quantum query complexity is 2 [@MJM11].
It is easy to prove that $Q_E(\mbox{NAE}_{n})\leq n-1$ since $$\mbox{NAE}_{n}(x_1,\ldots,x_n)=(x_1\oplus x_2)\vee(x_2\oplus x_3)\cdots\vee(x_{n-1}\oplus x_n).$$
Multilinear polynomials
-----------------------
Every Boolean function $f:\{0,1\}^n\to \{0,1\}$ has a unique representation as an $n$-variate multilinear polynomial over the reals, i.e., there exist real coefficients $a_S$ such that $$f(x_1,\ldots,x_n)=\sum_{S\subseteq [n]} a_S \prod_{i\in S} x_i.$$ The degree of $f$ is the degree of its largest monomial: $deg(f)=\max\{|S|:a_S\neq 0\}$.
For example, $\mbox{AND}_2(x_1,x_2)=x_1\cdot x_2$ and $\mbox{OR}_2(x_1,x_2)=x_1+x_2-x_1\cdot x_2$.
$\textrm{deg}(f)$ gives a lower bound on $D(f)$. Indeed, it holds
[*[@BdW02]*]{}\[D(f)-geq-Deg(f)\] $D(f)\geq \textrm{deg}(f)$.
Symmetric Boolean functions
===========================
Let $f:\{0,1\}^n\to \{0,1\}$ be a symmetric Boolean function. $Q_E(f)=n$ iff $f$ is isomorphic to $\mbox{AND}_n$.
If $f$ is isomorphic to $\mbox{AND}_n$, then $Q_E(f)=n$ [@BBC+98].
An $n$-bit symmetric Boolean function can be fully described by a vector $(b_0,b_1,\ldots,b_n)\in\{0,1\}^{n+1}$, where $f(x)=b_{|x|}$, i.e. $b_k$ is the value of $f(x)$ for $|x|=k$.
$(b_0,b_1,b_2,b_3)$ Type of function Query complexity
--------------------- ------------------------------------ ------------------
0 0 0 0 Constant function 0
0 0 0 1 $\mbox{AND}_3$ 3
0 0 1 0 $\mbox{EXACT}_3^{2}$ 2
0 0 1 1 $\mbox{Th}_3^{2}$ 2
0 1 0 0 $\mbox{EXACT}_3^{1}$ 2
0 1 0 1 $\mbox{PARITY}_3$ 2
0 1 1 0 $\mbox{NAE}_3$ 2
0 1 1 1 Isomorphic to $\mbox{AND}_3$ 3
1 0 0 0 Isomorphic to $\mbox{AND}_3$ 3
1 0 0 1 Isomorphic to $\mbox{NAE}_3$ 2
1 0 1 0 Isomorphic to $\mbox{PARITY}_3$ 2
1 0 1 1 Isomorphic to $\mbox{EXACT}_3^{1}$ 2
1 1 0 0 Isomorphic to $\mbox{Th}_3^{2}$ 2
1 1 0 1 Isomorphic to $\mbox{EXACT}_3^{2}$ 2
1 1 1 0 Isomorphic to $\mbox{AND}_3$ 3
1 1 1 1 Constant function 0
: Exact quantum query complexity for $3$-bit symmetric functions.[]{data-label="T1"}
Table \[T1\] contains all 3-bit Boolean functions and their exact quantum query complexity. Four 3-bit Boolean functions that achieve 3 queries are those that can be described by one of the following vectors: $(0,0,0,1),(0,1,1,1),(1,0,0,0), (1,1,1,0)$. They are isomorphic to $\mbox{AND}_3$.
We claim that only $n$-bit Boolean functions that can be described by one of the following vectors $(0,\ldots,0,1),\linebreak[0](0,1,\ldots,1),\linebreak[0](1,0,\ldots,0),\linebreak[0](1,\ldots,1,0)$, which are isomorphisms of $\mbox{AND}_n$, that can achieve $n$ queries. We prove this claim by an induction on $n$ as follows:
[**BASIS**]{}: The result holds clearly for $n=3$.
[**INDUCTION**]{}: Suppose the result holds for $n=k$ ($\geq 3$). We will prove that the result holds also for $n=k+1$. We use vector $(b_0,b_1,\ldots,b_k,b_{k+1})$ to describe the function $f(x_1,\cdots,x_k,x_{k+1})$. Since $$Q_E(f)\leq \max\{Q_E(f_{x_1=0}),Q_E(f_{x_1=1})\}+1,$$ we just need to consider the case that at least one of the functions $f_{x_1=0}$ and $f_{x_1=1}$ is isomorphic to $\mbox{AND}_k$. For other cases we have $Q_E(f)<k+1$.
$b_0b_1\ldots,b_k,b_{k+1}$ Type of function Query complexity
---------------------------- ---------------------------------------- ------------------
$(0,|0,\ldots,0,1)$ $\mbox{AND}_{k+1}$ $k+1$
$(0,|0,1,\ldots,1)$ $\mbox{Th}_{k+1}^{2}$ $k$
$(0,|1,0,\ldots,0)$ $\mbox{EXACT}_{k+1}^1$ $k$
$(0,|1,\ldots,1,0)$ $\mbox{NAE}_{k+1}$ $<k+1$
$(1,|0,\ldots,0,1)$ Isomorphic to $\mbox{NAE}_{k+1}$ $<k+1$
$(1,|0,1,\ldots,1)$ Isomorphic to $\mbox{EXACT}_{k+1}^1$ $k$
$(1,|1,0,\ldots,0)$ Isomorphic to $\mbox{Th}_{k+1}^{2}$ $k$
$(1,|1,\ldots,1,0)$ Isomorphic to $\mbox{AND}_{k+1}$ $k+1$
$(0,\ldots,0,1,|0) $ $\mbox{EXACT}_{k+1}^{k}$ $k$
$(0,1,\ldots,1,|0) $ $\mbox{NAE}_{k+1}$ $<k+1$
$(1,0,\ldots,0,|0)$ Isomorphic to $\mbox{AND}_{k+1}$ $k+1$
$(1,\ldots,1,0,|0) $ Isomorphic to $\mbox{Th}_{k+1}^{k}$ $k$
$(0,\ldots,0,1,|1)$ $\mbox{Th}_{k+1}^{k}$ $k$
$(0,1,\ldots,1,|1) $ Isomorphic to $\mbox{AND}_{k+1}$ $k+1$
$(1,0,\ldots,0,|1) $ Isomorphic to $\mbox{NAE}_{k+1}$ $<k+1$
$(1,\ldots,1,0,|1)$ Isomorphic to $\mbox{EXACT}_{k+1}^{k}$ $k$
: Exact quantum query complexity for $(k+1)$-bit symmetric Boolean functions. []{data-label="T2"}
There are three cases we have to consider according to the value of $b$.
[**Case 1**]{} $b=(0,\ldots,0,1)$. In this case $f=\mbox{AND}_{k+1}$.
[**Case 2**]{} $b=(1,0,\ldots,0)$. In this case $f$ is isomorphic to $\mbox{AND}_{k+1}$.
[**Case 3**]{} Otherwise, $f_{x_1=0}$ can be described by the vector $(b_0,b_1,\ldots,b_{k})$ and $f_{x_1=1}$ can be described by the vector $(b_1,\ldots,b_{k},b_{k+1})$. Thus we just need to consider Boolean functions that can be described by vector $b=(b_0,b_1,\ldots,b_k,b_{k+1})$ such that one of the following vectors $$(\overbrace{0,\ldots,0}^{k},1),\linebreak[0](0,\overbrace{1,\ldots,1}^{k}),\linebreak[0](1,\overbrace{0,\ldots,0}^k),(\overbrace{1,\ldots,1}^k,0)$$ is its prefix or suffix[^2]. There are 16 such Boolean functions and their query complexity are listed in Table \[T2\].
According to Table \[T2\], only $(k+1)$-bit Boolean functions which are isomorphic to $\mbox{AND}_{k+1}$ require $k+1$ queries. Thus, the theorem has been proved.
It is mentioned in [@MJM11; @Aar03] that all non-constant $n$-bit symmetric Boolean functions have exact classical complexity $n$. We give now a rigorous proof of that.
If $f:\{0,1\}^n\to \{0,1\}$ is a non-constant symmetric function, then $D(f)=n$.
Suppose $f$ can be described by the vector $(b_0,b_1,\ldots,b_n)\in\{0,1\}^{n+1}$. Since $f$ is non-constant, there exists a $k\in[n]$ such that $b_{k-1}\neq b_{k}$. If the first $k-1$ queries return $x_i=1$ and the next $n-k$ queries return $x_i=0$, then we will need to query the last variable as well.
Monotone Boolean functions
==========================
Let $f:\{0,1\}^n\to \{0,1\}$ be a monotone Boolean function. $Q_E(f)=n$ iff $f$ is isomorphic to $\mbox{AND}_n$.
Obviously, $\mbox{AND}_n(x)$ and $\mbox{OR}_n(x)$ are the only two $n$-bit monotone Boolean functions that are isomorphic to $\mbox{AND}_n(x)$. If $f$ is isomorphic to $\mbox{AND}_n(x)$, then $Q_E(f)=n$ [@BBC+98]. We prove the other direction by an induction on $n$.
[**BASIS**]{}: Case $n=2$, $\mbox{AND}_2(x_1,x_2)$ is the only $2$-bit function, up to isomorphism, that requires 2 queries. Therefore the result holds for $n=2$.
[**INDUCTION**]{}: Suppose the result holds for all $n\leq k$, we prove that the result holds also for $n=k+1$ in the following way.
For any $i\in[k+1]$, if $Q_E(f_{x_i=0})<k$ and $Q_E(f_{x_i=1})<k$, then $Q_E(f)\leq \max\{Q_E(f_{x_i=0}),\linebreak[0]Q_E(f_{x_i=1})\}+1<k+1$. Therefore, we need to consider only the case that at least one of functions $f_{x_i=0}$ and $f_{x_i=1}$ requires $k$ quires. There are two such cases:
[**Case 1:**]{} $Q_E(f_{x_1=1})=k$. According to the assumption, $f_{x_1=1}$ is isomorphic to $\mbox{AND}_k$. There are now two subcases to consider:
[**Case 1a:**]{} $f_{x_1=1}(x)=\mbox{OR}_k(x_2,\cdots,x_{k+1})=\mbox{OR}_k(x_{-1})$ (For convenience, we write $x_{-i}=x_1,\ldots, x_{i-1},x_{i+1},\linebreak[0]\ldots x_{k+1}$). Let us consider the CNF of $f$: $$f(x)=\bigwedge_{I\in C}\bigvee_{i\in I} x_i=\left(\bigwedge_{I\in C,1\in I}\bigvee_{i\in I} x_i\right)\wedge\left(\bigwedge_{I\in C,1\not\in I}\bigvee_{i\in I} x_i\right)
.$$ Therefore, $$f(x)=(x_1\vee g_1(x_{-1}))\wedge \mbox{OR}_k(x_{-1}),$$ where $x_1\vee g_1(x_{-1})=\left(\bigwedge_{I\in C,1\in I}\bigvee_{i\in I} x_i\right)$ and $g_1$ is also a monotone function. So we have $f(x)=1$ for any $x$ such that $10\cdots 0\prec x$ and $f(x)=0$ for any $x$ such that $x\preceq 10\cdots 0 $.
Let us consider now two subcases. Namely $f_{x_2=1}$ and $f_{x_2=0}$. Since $10\cdots0\preceq 10\cdots 0$, we have $f(10\cdots0)\linebreak[0]=0$ and $f_{x_2=0}(x)\neq \mbox{OR}_k(x_{-2})$. Since $10\cdots 0\prec 1010\cdots 0$, we have $f(1010\cdots 0)\linebreak[0]=1$ and $f_{x_2=0}(x)\neq \mbox{AND}_k(x_{-2})$. Now we have $Q_E(f_{x_2=0})<k$ and therefore $Q_E(f_{x_2=1})\linebreak[0]=k$. Since $10\cdots 0\prec 110\cdots 0$, we have $f(110\cdots 0)=1$ and $f_{x_2=1}(x)\neq \mbox{AND}_k(x_{-2})$. Therefore, $f_{x_2=1}(x)=\mbox{OR}_k(x_{-2})$. Using a similar argument, we can prove that for any $i\geq 2$, $f_{x_i=1}(x)=OR_k(x_{-i})$. Hence, for any $i\in[k+1]$, we have $$f(x)=(x_i\vee g_i(x_{-i}))\wedge \mbox{OR}_k(x_{-i}).$$ So $f(x)=1$ for any $x$ such that $y\prec x$ and $f(x)=0$ for any $x$ such that $x\preceq y$, where $y_i=1$ and $y_j=0$ for any $j\neq i$. It is not hard to see that in this case $f(x)=\mbox{Th}_{k+1}^2(x)$ and therefore $Q_E(f)=k$.
[**Case 1b:**]{} $f_{x_1=1}(x)=\mbox{AND}_k(x_{-1})$. Let us consider the CNF of $f$. We have, $$f(x)=(x_1\vee g'(x_{-1}))\wedge \mbox{AND}_k(x_{-1}),$$ where $g'(x_{-1})$ is also a monotone Boolean function.
If $g'$ is a constant function and $g'(x_{-1})=0$, we have $f(x)=\mbox{AND}_{k+1}(x_1x_2,\cdots,\linebreak[0] x_{k+1})$ and $Q_E(f)=k+1$. Otherwise, $\mbox{AND}_k(x_{-1})\leq g'(x_{-1})$, then $f(x)=\mbox{AND}_k(x_{-1})$ and therefore $Q_E(f)=k$.
[**Case 2:**]{} $Q_E(f_{x_1=0})=k$. There are again two subcases:
[**Case 2a:**]{} $f_{x_1=0}(x)=\mbox{OR}_k(x_{-1})$. Let us consider the DNF of $f$: $$f(x)=\bigvee_{I\in D}\bigwedge_{i\in I} x_i=\left(\bigvee_{I\in D, 1\in I} \bigwedge_{i\in I} x_i \right)\vee\left(\bigvee_{I\in D, 1\not\in I}\bigwedge_{i\in I}x_i\right).$$ We have $$f(x)=(x_1\wedge h'(x_{-1}))\vee \mbox{OR}_{n-1}(x_{-1}),$$ where $h'$ is a monotone Boolean function. If $h'$ is a constant function and $h'(x_{-1})=1$, then $f(x)=\mbox{OR}_{k+1}(x_1x_2,\cdots,x_{k+1})$ and $Q_E(f)=k+1$. Otherwise $h'(x_{-1})\leq OR_k(x_{-1})$ and therefore $f(x)=\mbox{OR}_k(x_{-1})$ and $Q_E(f)=k$.
[**Case 2b:**]{} $f_{x_1=0}(x)=\mbox{AND}_k(x_{-1})$. Let us consider the DNF of $f$. It has the form $$f(x)=(x_1\wedge h_1(x_{-1}))\vee \mbox{AND}_k(x_{-1}),$$ where $h_1(x_{-1})$ is also a monotone Boolean function. Therefore $f(x)=1$ for any $x$ such that $01\cdots 1\preceq x$ and $f(x)=0$ for any $x$ such that $x\prec 01\cdots 1 $.
Let us consider now two subcases: $f_{x_2=1}$ and $f_{x_2=0}$. Since $0110\cdots 0\prec 01\cdots 1$, we have $f(0110\linebreak[0]\cdots 0)\linebreak[0]=0$ and $f_{x_2=1}(x)\neq \mbox{OR}_k(x_{-2})$. Since $01\cdots 1\preceq 01\cdots 1$, we have $f(01\cdots 1)\linebreak[0]=1$ and $f_{x_2=1}(x)\neq \mbox{AND}_k(x_{-2})$. Therefore we have $Q_E(f_{x_2=1})<k$ and $Q_E(f_{x_2=0})\linebreak[0]=k$. Since $0010\cdots 0\prec 01\cdots 1$, we have $f(0010\cdots 0)\linebreak[0]=0$ and $f_{x_2=0}(x)\neq \mbox{OR}_k(x_{-2})$. Therefore, $f_{x_2=0}(x)=\mbox{AND}_k(x_{-2})$. Using a similar argument, we can prove that for any $i\geq 2$, $f_{x_i=0}(x)=\mbox{AND}_k(x_{-i})$. Hence, for any $i\in[k+1]$, we have $$f(x)=(x_i\wedge h_i(x_{-i}))\vee \mbox{AND}_k(x_{-i}).$$
Therefore $f(x)=1$ for any $x$ such that $y\preceq x$ and $f(x)=0$ for any $x$ such that $x\prec y$, where $y_i=0$ and $y_j=1$ for any $j\neq i$. It is now not hard to show that $f(x)=\mbox{Th}_{k+1}^{k}$ and $Q_E(f)=k$.
Therefore, the theorem has been proved.
Read-once Boolean functions
===========================
\[fsize\] If $f:\{0,1\}^n\to \{0,1\}$ is a read-once Boolean function, then $Q_E(f)=n$ iff $f$ is isomorphic to $\mbox{AND}_n$.
If $f$ is isomorphic to $\mbox{AND}_n$, then $Q_E(f)=n$ [@BBC+98]. We prove the other direction as follows.
Since $f$ is a read-once Boolean function, $f$ depends on all $n$ variables and $L(f)=n$, i.e each $(\neg) x_i$ labels once and only once a leaf variable, where $(\neg)$ denotes a possible negation. We prove the result by an induction.
[**BASIS**]{}: $\mbox{AND}_3(x_1,x_2,x_3)$ is the only $3$-bit Boolean function, up to isomorphism, that requires 3 quantum queries [@MJM11]. Therefore the result holds for $n=3$.
[**INDUCTION**]{}: We will suppose the result holds for all $n\leq k$ ($k\geq 3$) and we will prove that the result holds also for all $n\leq k+1$.
Suppose the root of a formula $F$ is labeled with $\wedge$. Without loss of generality, we assume that there exist Boolean functions $g:\{0,1\}^p\to \{0,1\}$ and $h:\{0,1\}^q\to \{0,1\}$ such that $f(x)=g(y)\wedge h(z)$ and $p+q=k+1$, where $x=yz$. Since $f$ depends on all $k+1$ variables and $L(f)=k+1$, we have $L(g)=p$ and $L(h)=q$, where $g$ depends on all $p$ variables and $h$ depends on all $q$ variables. If $Q_E(g)<p$ or $Q_E(h)<q$, then $Q_E(f)\leq Q_E(g)+Q_E(h)<k+1$. Now suppose $Q_E(g)=p$ and $Q_E(h)=q$. According to the assumption, $g$ is isomorphic to $\mbox{AND}_p$ and $h$ is isomorphic to $\mbox{AND}_q$. There are therefore the following four cases to consider.
[**Case 1:**]{} $g(y)=\mbox{AND}_p\left((\neg)x_1,\ldots,(\neg)x_p\right)$ and $h(z)=\mbox{AND}_q\left((\neg)x_{p+1},\linebreak[0]\ldots,\linebreak[0](\neg)x_{k+1}\right)$. Then $f$ is isomorphic to $\mbox{AND}_{k+1}$ and therefore $Q_E(f)=k+1$.
[**Case 2:**]{} $g(y)=\mbox{OR}_p\left((\neg)x_1,\ldots,(\neg)x_p\right)$ and $h(z)=\mbox{OR}_q\left((\neg)x_{p+1},\ldots,\linebreak[0](\neg)x_{k+1}\right)$. Therefore $$f(x)=\mbox{OR}_p\left((\neg)x_1,\ldots,(\neg)x_p\right)\wedge \mbox{OR}_q\left((\neg)x_{p+1},\ldots,(\neg)x_{k+1}\right).$$ Without loss of generality, we suppose that $f(x)=\mbox{OR}_p\left(x_1,\ldots,x_p\right)\wedge \mbox{OR}_q\left(x_{p+1},\ldots,x_{k+1}\right)$. Since $p+k-p+1=k+1>3$, we have $p\geq 2$ or $k-p+1\geq 2$. Without loss of generality, we assume that $k-p+1\geq 2$. Let us query $x_2$ to $x_{k-1}$ first.
1. If $x_i=1$ for some $2\leq i\leq p$ and $x_j=1$ for some $p+1\leq j\leq k-1$, then $f_{x_2\cdots x_{k-1}}(x)=1$.
2. If $x_i=1$ for some $2\leq i\leq p$ and $x_{p+1}=\cdots=x_{k-1}=0$, then $f_{x_2\cdots x_{k-1}}(x)=\mbox{OR}_2\left(x_{k},x_{k+1}\right)$.
3. If $x_2=\cdots=x_p=0$ and $x_i=1$ for some $p+1\leq i\leq k-1$, then $f_{x_2\cdots x_{k-1}}(x)=x_1$.
4. Otherwise, $x_2=\cdots=x_{k-1}=0$ and therefore $f_{x_2\cdots x_{k-1}}(x)=x_1\wedge(x_k\vee x_{k+1})$ and $Q_E(f_{x_2\cdots x_{k-1}})=2$.
Therefore $Q_E(f)\leq k-2+2<k+1$.
[**Case 3:**]{} $g(y)=\mbox{AND}_p\left((\neg)x_1,\ldots,(\neg)x_p\right)$ and $h(z)=\mbox{OR}_q\left((\neg)x_{p+1},\ldots,(\neg)x_{k+1}\right)$. Therefore $f(x)=\mbox{AND}_p\left((\neg)x_1,\ldots,(\neg)x_p\right)\wedge \mbox{OR}_q\left((\neg)x_{p+1},\ldots,(\neg)x_{k+1}\right)$. Without loss of generality, we can now suppose that $$f(x)=\mbox{AND}_p\left(x_1,\ldots,x_p\right)\wedge \mbox{OR}_q\left(x_{p+1},\ldots,x_{k+1}\right).$$ If $p=k$, then $f=\mbox{AND}_{k+1}$ and $Q_E(f)=k+1$. Now we consider the case $p<k$. Let us query $x_2$ to $x_{k-1}$ first.
1. If $x_2\cdots x_{p}\neq 1\cdots1$, then $f(x)=0$.
2. If $x_2\cdots x_{p}= 1\cdots1$ and $x_{p+1}\cdots x_{k-1}\neq 0\cdots0$, then $f_{x_2\cdots x_{k-1}}(x)=x_1$.
3. If $x_2\cdots x_{p}= 1\cdots 1$ and $x_{p+1}\cdots x_{k-1}= 0\cdots0$, then $f_{x_2\cdots x_{k-1}}(x)=x_1\wedge(x_k\vee x_{k+1})$ and $Q_E(f_{x_2\cdots x_{k-1}})=2$.
Therefore $Q_E(f)\leq k-2+2<k+1$.
[**Case 4:**]{} $g(y)=\mbox{OR}_p\left((\neg)x_1,\ldots,(\neg)x_p\right)$ and $h(z)=\mbox{AND}_q\left((\neg)x_{p+1},\ldots,(\neg)x_{k+1}\right)$. This case is analogous to the [**Case 3**]{}.
Symmetrically, we can consider the case that the root of the formula $F$ is labeled with $\vee$. In this case, we will need to deal with functions with the same structure of $f(x_1,x_2,x_3)=x_1\vee (x_2\wedge x_3)$, which is isomorphic to $x_1\wedge(x_2\vee x_3)$. We omit the details here.
It is mentioned in [@San95] that all $n$-bit read-once Boolean functions have exact classical quantum complexity $n$. We give now a rigorous proof of that:
\[Th-readonce\] If $f:\{0,1\}^n\to \{0,1\}$ is a read-once Boolean function, then $D(f)=n$.
Let us consider the multilinear polynomial representation of $f$. It is easy to prove by induction that $\textrm{deg}(f)=n$ and there is just one monomial of $f$ of the degree $n$.
[**BASIS**]{}: If $n=1$, then $f(x)=(\neg) x_1$. Therefore, $\textrm{deg}(f)=1$.
[**INDUCTION**]{}: Suppose the result holds for all $n\leq k$, we will prove the result holds for all $n\leq k+1$.
Without loss of generality, let us assume that three exists an $i\in [n]$ such that $$f(x_1,\ldots,x_{k+1})=g(x_1,\ldots,x_i)\wedge h(x_{i+1},\ldots,x_{k+1})$$ or $$f(x_1,\ldots,x_{k+1})=g(x_1,\ldots,x_i)\vee h(x_{i+1},\ldots,x_{k+1}),$$ where $L(g)=i$, $L(h)=k+1-i$, $g$ and $h$ depend on all their variables. According to assumption of the theorem, we have $\mbox{deg}(g)=i$ and $g(x_1,\ldots,x_i)=(\pm)\prod_{j=1}^i(\neg)x_j+p(x_1,\ldots,x_i)$ where $\mbox{deg}(p)<i$, and $\mbox{deg}(h)=k+1-i$ and $h(x_{i+1},\ldots,x_{k+1})=(\pm)\prod_{j={i+1}}^{k+1}(\neg)x_j+q(x_{i+1},\ldots,x_{k+1})$ where $\mbox{deg}(q)<k+1-i$.
Since $$f(x_1,\ldots,x_{k+1})=g(x_1,\ldots,x_i)\wedge h(x_{i+1},\ldots,x_{k+1})=g\cdot h$$ and $$f(x_1,\ldots,x_{k+1})=g(x_1,\ldots,x_i)\vee h(x_{i+1},\ldots,x_{k+1})=g+h-g\cdot h.$$ Therefore $\textrm{deg}(f)=k+1$ and there is just one monomial of $f$ of the degree $k+1$.
According to Lemma \[D(f)-geq-Deg(f)\], $D(f)\geq \textrm{deg}(f)=n$. Thus, $D(f)=n$.
General $n$-bit Boolean functions
=================================
In this section we prove our main result. Without explicitly pointed out, $n>3$ in this section.
If $f$ is an $n$-bit Boolean function that is isomorphic to $\mbox{AND}_{n}$, then there must exist $b=b_1\ldots b_n\in\{0,1\}^n$ such that every $f_{x_i=b_i}$ is equivalent to $\mbox{AND}_{n-1}$ ($\mbox{OR}_{n-1}$) up to some negations of variables. Moreover $b$ has to be unique. For example, if $f(x)=\mbox{OR}_n(x_1,x_2,\ldots,x_n)$, then we have $f_{x_i=0}(x)=\mbox{OR}_{n-1}(x_1,\ldots,x_{i-1},x_{i+1},\linebreak[0]\ldots,x_n)$ for $i\in[n]$ and $b=0\ldots 0$.
For an $n$-bit Boolean function $f$ that has exact quantum query complexity $n$, we prove the following lemma.
\[Lm-c7-1\] Suppose that $\mbox{AND}_{n-1}$ is the only (n-1)-bit Boolean function, up to isomorphism, has exact quantum query complexity $n-1$. Let $f:\{0,1\}^n\to \{0,1\}$ be an $n$-bit Boolean function that has exact quantum query complexity $n$. There exists one and only one $b=b_1\ldots b_n\in\{0,1\}^n$ for every $i\in[n]$ such that $f_{x_i=b_i}$ is equivalent to $\mbox{AND}_{n-1}$ ($\mbox{OR}_{n-1}$) up to some negations of the variables.
[**Proof:**]{} In order to prove this lemma, we study some properties of exact quantum query complexity of Boolean functions. According to Eq. (\[Eq-n-1ton\]), we have the following lemma:
\[C6-l1\] Let $f:\{0,1\}^n\to \{0,1\}$ be a Boolean function. If there exists an $i\in [n]$ such that both $Q_E(f_{x_i=0})<n-1$ and $Q_E(f_{x_i=1})<n-1$, then $Q_E(f)<n$.
We know from [@MJM11] that $\mbox{AND}_3$ is the only $3$-bit Boolean function, up to isomorphism, that has exact quantum query complexity 3. For any $4$-bit function $f$, if there exists $i\in[4]$ such that neither $f_{x_i=0}$ nor $f_{x_i=1}$ is isomorphic to $\mbox{AND}_{n-1}$, then $Q_E(f)<4$.
\[Lm-both\] Let $f:\{0,1\}^n\to \{0,1\}$ be a Boolean function. If there exists an $i\in [n]$ such that both $f_{x_i=0}$ and $f_{x_i=1}$ are isomorphic to $\mbox{AND}_{n-1}$, then $Q_E(f)<n$.
[**Proof:**]{} Without loss of generality, we can assume that $i=1$. According to Eq. (\[Eq-df(x)\]), we have $$f(x)=\left(\neg x_1\wedge f_{x_1=0}(x_2,\ldots,x_n)\right)\vee\left(x_1\wedge f_{x_1=1}(x_2,\ldots,x_n)\right).$$ Suppose that at least one of the functions $f_{x_1=0}$ and $f_{x_1=1}$ is equivalent to $\mbox{AND}_{n-1}$ up to some negations of the variables. Without loss of generality, we will now assume that $f_{x_1=1}(x)=\mbox{AND}_{n-1}(x_2,\ldots,x_n)$. To prove the theorem, we consider two cases.
[**Case 1:**]{} $f_{x_1=0}(x)=\mbox{AND}_{n-1}((\neg)x_2,\ldots,(\neg)x_n)$. In this case we have two subcases.
[**Case 1a:**]{} $f_{x_1=0}(x)=\mbox{AND}_{n-1}(\neg x_2,\ldots,\neg x_n)$. We have $$f(x)=\mbox{AND}_{n}(\neg x_1,\neg x_2,\ldots,\neg x_n)\vee \mbox{AND}_{n}\linebreak[0](x_1,\linebreak[0]x_2,\ldots,x_n)=\neg \mbox{NAE}(\linebreak[0]x_1\linebreak[0]x_2,\linebreak[0]\ldots,x_n).$$ Therefore, $Q_E(f)<n$.
[**Case 1b:**]{} $f_{x_1=0}(x)\neq \mbox{AND}_{n-1}(\neg x_2,\ldots,\neg x_n)$. Without loss of generality, we can suppose that there exists a $k\in\{2,\ldots,n-1\}$ such that $f_{x_1=0}\linebreak[0](x)=\mbox{AND}_{n-1}(\neg x_2,\linebreak[0]\ldots,\neg x_k,x_{k+1},\ldots,x_n )$. Then $$f(x)=\mbox{AND}_{n}(\neg x_1,\ldots,\neg x_k,x_{k+1}, \ldots,x_n)\vee \mbox{AND}_{n}\linebreak[0](x_1,\linebreak[0]x_2,\ldots,x_n)$$ $$=\left(\mbox{AND}_{k}(\neg x_1,\ldots,\neg x_k)\vee \mbox{AND}_{k}\linebreak[0](x_1,\linebreak[0]\ldots,x_k)\right)\wedge \mbox{AND}_{n-k}(x_{k+1},\ldots,x_n)$$ $$=\neg \mbox{NAE}_k(\neg x_1,\ldots,\neg x_k)\wedge \mbox{AND}_{n-k}(x_{k+1},\ldots,x_n).$$ Therefore, $Q_E(f)< k+n-k=n$.
[**Case 2:**]{} $f_{x_1=0}(x)=\mbox{OR}_{n-1}((\neg)x_2,\ldots,(\neg)x_n)$. This means that we have two subcases.
[**Case 2a:**]{} $f_{x_1=0}(x)=\mbox{OR}_{n-1}(\neg x_2,\ldots,\neg x_n)$. If $g(y)= \mbox{AND}_{n-1}(x_{2},\ldots,x_n)$, then $$f(x)=\left(\neg x_1\wedge \neg g(y)\right)\vee \left( x_1\wedge g(y)\right)=x_1\oplus g(y).$$ Therefore, $Q_E(f)<n$.
[**Case 2b:**]{} $f_{x_1=0}(x)\neq \mbox{OR}_{n-1}(\neg x_2,\ldots,\neg x_n)$. Without loss of generality, we can suppose that $f_{x_1=0}(x)=\mbox{OR}_{n-1}( x_2,(\neg)x_3 \linebreak[0]\ldots,(\neg) x_n )$, then let us query $x_2$ first. If $x_2=0$, then $f_{x_2=0}(x)=\neg x_1\wedge \mbox{OR}_{n-2}( (\neg)x_3 \linebreak[0]\ldots,(\neg) x_n )$. According to Theorem \[fsize\], $Q_E(f_{x_2=0})<n-1$. If $x_2=1$, then $f_{x_2=1}(x)=\neg x_1\vee \mbox{AND}_{n-1}\linebreak[0](x_1,\linebreak[0]x_3,\ldots,x_n)=\neg x_1\vee \mbox{AND}_{n-2}\linebreak[0](x_3,\ldots,x_n)$. According to Theorem \[fsize\], $Q_E(f_{x_2=1})<n-1$. According to Eq. (\[Eq-n-1ton\]), $Q_E(f)< n-1+1=n$.
Now we need to consider the case that both $f_{x_1=0}$ and $f_{x_1=1}$ are $\mbox{OR}_{n-1}$ functions. Without loss of generality, we assume that $f_{x_1=1}(x)=\mbox{OR}_{n-1}(x_2,\ldots,x_n)$. This means that we have again two subcases.
[**Case 3a:**]{} $f_{x_1=0}(x)=\mbox{OR}_{n-1}(x_2,\ldots,x_n)$. In this case, we have $f(x)=\mbox{OR}_{n-1}(x_2,\ldots,x_n)$ and $Q_E(f)=n-1<n.$
[**Case 3b:**]{} $f_{x_1=0}(x)\neq \mbox{OR}_{n-1}(x_2,\ldots,x_n)$. Without loss of generality generality, let us suppose that there exists a $k\in\{2,\ldots,n\}$ such that $f_{x_1=0}(x)=\mbox{OR}_{n-1}(\neg x_2,\linebreak[0]\ldots,\neg x_k,\linebreak[0]x_{k+1},\ldots,x_n )$. In such a case $$f(x)=\left(\neg x_1\wedge\mbox{OR}_{n-1}(\neg x_2,\ldots,\neg x_k,x_{k+1}, \ldots,x_n)\right)\vee \left( x_1\wedge\mbox{OR}_{n-1}\linebreak[0](x_2,\ldots,x_n)\right)$$ Let us query $x_{k+1}$ to $x_n$ first. If $x_{k+1}=\cdots=x_n=0$, let $g(y)=f(x_1,\ldots,x_k,0,\ldots,0)$, then $$g(y)=\left(\neg x_1\wedge\mbox{OR}_{n-1}(\neg x_2,\ldots,\neg x_k,)\right)\vee \left( x_1\wedge\mbox{OR}_{n-1}\linebreak[0](x_2,\ldots,x_k)\right)$$ $$= \mbox{NAE}_{n}(\neg x_1,x_2,\ldots,x_k).$$ Therefore, $Q_E(g)<k.$ Otherwise, there exists a $j\geq k+1$ such that $x_j=1$. It is now easy to show that $f(x)=\neg x_1\vee x_1=1$. Therefore, $Q_E(f)<n-k+k=n.$
\[Lm-and-or\] Let $f:\{0,1\}^n\to \{0,1\}$ be a Boolean function. If there exist an $i\in[n]$ such that $f_{x_i=b}$ is equivalent to $\mbox{AND}_{n-1}$ ($\mbox{OR}_{n-1}$) up to some negations of the variables, then $f_{x_j=c}$ is not equivalent to $\mbox{OR}_{n-1}$ ($\mbox{AND}_{n-1}$) up to some negations of the variables for $j\neq i$, where $b,c\in\{0,1\}$.
[**Proof:**]{} Without loss of generality, we assume that $i=1$, $j=2$ and $f_{x_1=b}(x)=\mbox{AND}_{n-1}(x_2,\linebreak[0]\cdots,x_n)$. In such a case we have $f(bc00*\cdots*)=f(bc01*\cdots*)=0$[^3]. If we fix $c$, then there are more than one inputs such that $f_{x_2=c}(x)=0$. Therefore, $f_{x_2=c}$ is not equivalent to $\mbox{OR}_{n-1}$ up to some negations of the variables.
[**Proof of Lemma \[Lm-c7-1\]:**]{} According to Lemma \[C6-l1\], for every $i\in[n]$, there must exist a $b_i\in\{0,1\}$ such that $f_{x_i=b_i}$ is isomorphic to $\mbox{AND}_{n-1}$, otherwise $Q_E(f)<n$. Without loss of generality, we assume that $f_{x_1=b_1}$ is equivalent to $\mbox{AND}_{n-1}$ ($\mbox{OR}_{n-1}$) up to some negations of the variables. According to Lemma \[Lm-and-or\], no $f_{x_i=b_i}$ is equivalent to $\mbox{OR}_{n-1}$ ($\mbox{AND}_{n-1}$) up to some negations of the variables. Therefore, for every $i>1$, $f_{x_i=b_i}$ is equivalent to $\mbox{AND}_{n-1}$ ($\mbox{OR}_{n-1}$) up to some negations of the variables.
Now, suppose there exists $c=c_1\ldots c_n\neq b$ for every $i\in[n]$ such that $f_{x_i=c_i}$ is equivalent to $\mbox{AND}_{n-1}$ ($\mbox{OR}_{n-1}$) up to some negations of the variables. Since $c\neq b$, there exist $i\in[n]$ such that $b_i\neq c_i$. We have therefore that both $f_{x_i=b_i}$ and $f_{x_i=c_i}$ are isomorphic to $\mbox{AND}_{n-1}$. According to Lemma \[Lm-both\], we have $Q_E(f)<n$, which is a contradiction.
In order to make our main result easier to understand, we consider $4$-bit Boolean functions first.
\[Th6\] If $f$ is a 4-bit Boolean function, then $Q_E(f)=4$ iff $f$ is isomorphic to $\mbox{AND}_4$.
[**Proof:**]{} If $f$ is isomorphic to $\mbox{AND}_4$, then $Q_E(f)=4$ [@BBC+98].
Assume that a $4$-bit Boolean function $f$ such that $Q_E(f)=4$, we prove that $f$ is isomorphic to $\mbox{AND}_4$ as follows. According to Lemma \[Lm-c7-1\], there exists one and only one $b=b_1b_2b_3b_4$ for every $i\in[4]$ such that $f_{x_i=b_i}$ is equivalent to $\mbox{AND}_{3}$ ($\mbox{OR}_{3}$) up to some negations of the variables. Since for any $4$-bit function $f$ with $b=b_1b_2b_3b_4$, there exists a function $f'$ with $b'=0000$ isomorphic to $f$. We can get $f'$ by some negations of the variables $x_i$ whenever $b_i=1$. Therefore, without loss of generality, we assume that $b=0000$ and for every $i\in[4]$ such that $f_{x_i=0}$ is equivalent to $\mbox{OR}_{3}$ up to some negations of the variables.
There are three cases that we need now to consider:
$x_1$ $x_2$ $x_3$ $x_4$ $f(x)$: Case 1 Case 2 Case 3
------- ------- ------- ------- ---------------- -------- --------
0 0 0 0 0 1 1
0 0 0 1 1 1 1
0 0 1 0 1 1 \*
0 0 1 1 1 1 \*
0 1 0 0 1 1 1
0 1 0 1 1 1 1
0 1 1 0 1 1 1
0 1 1 1 1 0 1
1 0 0 0 1 1 1
1 0 0 1 1 1 1
1 0 1 0 1 1 1
1 0 1 1 1 0 1
1 1 0 0 1 1 \*
1 1 0 1 1 0 \*
1 1 1 0 1 0 \*
1 1 1 1 \* \* \*
: Values of $4$-bit Boolean functions. []{data-label="T3"}
[**Case 1:**]{} For every $i\in[4]$, there is no negation variable occurrence in $f_{x_i=0}$, that is $f_{x_1=0}(x)=\mbox{OR}(x_2,x_3,x_4)$, $f_{x_2=0}(x)=\mbox{OR}(x_1,x_3,x_4)$, $f_{x_3=0}(x)=\mbox{OR}(x_1,x_2,x_4)$ and $f_{x_4=0}(x)=\mbox{OR}(x_1,x_2,x_3)$. See Case 1 in Table \[T3\] for values of $f(x)$. We still do not the value of $f(1111)$. If $f(1111)=1$, then $f(x)=\mbox{OR}(x_1,x_2,x_3,x_4)$, which is isomorphic to $\mbox{AND}_4$. If $f(1111)=0$, then $f(x)=\mbox{NAE}(x_1,x_2,x_3,x_4)$ and $Q_E(f)<4$.
[**Case 2:**]{} There are negations of all variables in every $f_{x_i=0}$, that is $f_{x_1=0}(x)=\mbox{OR}(\neg x_2, \neg x_3,\linebreak[0] \neg x_4)$, $f_{x_2=0}(x)=\mbox{OR}(\neg x_1, \neg x_3, \neg x_4)$, $f_{x_3=0}(x)=\mbox{OR}(\neg x_1, \neg x_2,\neg x_4)$ and $f_{x_4=0}(x)=\mbox{OR}(\neg x_1, \linebreak[0]\neg x_2, \neg x_3)$. See Case 2 in Table \[T3\] for values of $f(x)$. If $f(1111)=1$, then $f(x)= \neg \mbox{Th}_4^3$ and $Q_E(f)=3<4$. If $f(1111)=0$, then $f(x)= \neg \mbox{EXACT}_4^3$ and $Q_E(f)=3<4$.
[**Case 3:**]{} There is an $i\in[4]$ such that there is at least one negation variable occurrence and one no negation variable occurrence in $f_{x_i=0}$. Without loss of generality, we can now assume that $f_{x_1=0}(x)=\mbox{OR}( x_2, \neg x_3, (\neg) x_4)$. In order to analyse this case, we prove the following two lemmas first.
\[lm-7\] Let $f$ be an $n$-bit Boolean function and $f_{x_i=0}$ be equivalent to $\mbox{OR}_{n-1}$ up to some negations of the variables for every $i\in[n]$. If $f_{x_1=0}(x)=\mbox{OR}_{n-1}(x_2,\neg x_3, (\neg)x_4,\ldots)$, then $f_{x_2=0}(x)=\mbox{OR}_{n-1}(x_1,\neg x_3, (\neg)x_4,\ldots)$ and $f_{x_3=0}(x)=\mbox{OR}_{n-1}(\neg x_1,\neg x_2, (\neg)x_4,\ldots)$.
[**Proof:**]{} Since $f_{x_1=0}(x)=\mbox{OR}_{n-1}(x_2,\neg x_3, (\neg)x_4,\ldots)$, there exists a $y\in\{0,1\}^{n-3}$ such that $f(001y)=0$. Suppose that $f_{x_2=0}(x)=\mbox{OR}_{n-1}(\neg x_1,(\neg) x_3, (\neg)x_4,\ldots)$ or $f_{x_2=0}(x)=\mbox{OR}_{n-1}((\neg) x_1, x_3, (\neg)x_4,\ldots)$. We have $f(001y)=1$, which is a contradiction. Therefore, $f_{x_2=0}=\mbox{OR}_{n-1}(x_1,\neg x_3, (\neg)x_4,\ldots)$.
Now suppose that $f_{x_3=0}(x)=\mbox{OR}_{n-1}(x_1, (\neg) x_2, (\neg)x_4,\ldots)$. There have to exist $c\in\{0,1\}$ and $z\in\{0,1\}^{n-3}$ such that $f(0c0z)=0$. Since $f_{x_1=0}(x)=\mbox{OR}_{n-1}(x_2,\neg x_3, (\neg)x_4,\ldots)$, we have $f(0c0z)=1$, which is a contradiction. Suppose that $f_{x_3=0}(x)=\mbox{OR}_{n-1}((\neg) x_1, x_2, (\neg)x_4,\linebreak[0]\ldots)$. There exist $c\in\{0,1\}$ and $z\in\{0,1\}^{n-3}$ such that $f(c00z)=0$. Since $f_{x_2=0}(x)=\mbox{OR}_{n-1}(x_1,\neg x_3, \linebreak[0](\neg)x_4,\ldots)$, we have $f(c00z)=1$, which is a contradiction. Therefore, $f_{x_3=0}(x)\linebreak[0]=\linebreak[0]\mbox{OR}_{n-1}(\neg x_1,\neg x_2,\linebreak[0] (\neg)x_4,\ldots)$.
\[lm-8\] Let $f$ be an $n$-bit Boolean function. If there exist 4 distinct inputs $x,y,u,v\in\{0,1\}^n$ such that $f(x)=f(y)=1$ and $f(u)=f(v)=0$, then $f$ is not isomorphic to $\mbox{AND}_n$.
[**Proof:**]{} If $f$ is equivalent to $\mbox{AND}_n$ up to some negations of the variables, then there exists just one $x \in\{0,1\}^n$ such that $f(x)=1$. If $f$ is equivalent to $\mbox{OR}_n$ up to some negations of the variables, then there exists just one $u \in\{0,1\}^n$ such that $f(u)=0$.
According to Lemma \[lm-7\], we have $f_{x_2=0}(x)=\mbox{OR}(x_1, \neg x_3, (\neg) x_4)$, and $f_{x_3=0}(x)=\mbox{OR}(\neg x_1,\linebreak[0] \neg x_2, (\neg) x_4)$. See Case 3 in Table \[T3\] for values of $f(x)$. It is easy to see that if $x_1\oplus x_2=1$, then $f(x)=1$. If $x_1\oplus x_2=0$, then $x_1=x_2$ and $f$ can be represented as a $3$-bit Boolean function $g(x_2,x_3,x_4)$, see Table \[T4\] for its values. Since $f_{x_1=0}(x)=\mbox{OR}( x_2, \neg x_3, (\neg) x_4)$, we have either $g(010)=f(0010)=0$ or $g(011)=f(0011)=0$. Since $f_{x_3=0}(x)=\mbox{OR}(\neg x_1, \neg x_2, (\neg) x_4)$, we have either $g(100)=f(1100)=0$ or $g(101)=f(1101)=0$. We also have $g(000)=f(0000)$ and $g(001)=f(0001)=1$. According to Lemma \[lm-8\], $g(x_2,x_3,x_4)$ is not isomorphic to $\mbox{AND}_3$ and $Q_E(g)<3$.
$x_2$ $x_3$ $x_4$ $g(x_2,x_3,x_4)$
------- ------- ------- ------------------ -- -- --
0 0 0 1
0 0 1 1
0 1 0 \*
0 1 1 \*
1 0 0 \*
1 0 1 \*
1 1 0 \*
1 1 1 \*
: Values of $g(x_2,x_3,x_4)$.[]{data-label="T4"}
Now we give an exact quantum algorithm for $f$ as follows:
1. Evaluate $x_1\oplus x_2$ with one query.
2. If $x_1\oplus x_2=1$, then $f(x)=1$.
3. If $x_1\oplus x_2=0$, then $f(x)=g(x_2,x_3,x_4)$. Evaluate $g$ with exact quantum algorithm.
Therefore, we have $Q_E(f)<1+ Q_E(g)<1+3=4.$ The theorem has been proved.
Finally, we prove the most general case. The main idea of the proof is similar to the proof of the previous theorem.
If $f$ is an $n$-bit Boolean function, then $Q_E(f)=n$ iff $f$ is isomorphic to $\mbox{AND}_n$.
[**Proof:**]{} If $f$ is isomorphic to $\mbox{AND}_n$, then $Q_E(f)=n$ [@BBC+98]. We prove the other direction by an induction on $n$.
[**BASIS**]{}: The result holds for $n=3$.
[**INDUCTION**]{}: Suppose the result holds for $n-1$, we will prove that the result holds for $n$. According to Lemma \[Lm-c7-1\], there exists one and only one $b=b_1\ldots b_n$ for every $i\in[n]$ such that $f_{x_i=b_i}$ is equivalent to $\mbox{AND}_{n-1}$ ($\mbox{OR}_{n-1}$) up to some negations of the variables. Without loss of generality, we assume that $b=0\ldots0$ and for every $i\in[n]$ such that $f_{x_i=0}$ is equivalent to $\mbox{OR}_{n-1}$ up to some negations of the variables.
There are three cases that we need to consider:
[**Case 1:**]{} For every $i\in[n]$, there is no negation variable occurrence in $f_{x_i=0}$, that is $f_{x_i=0}(x)=\mbox{OR}_{n-1}(x_1,\ldots,x_{i-1},x_{i+1},\ldots, x_n)$ for $i\in[n]$. It is easy to see that in such a case $f(0\ldots0)=0$, $f(1\ldots1)=*$ and $f(x)=1$ for $x\not\in\{0\ldots0,1\ldots 1\}$. If $f(1\ldots1)=1$, then $f(x)=\mbox{OR}_n(x_1,\ldots,x_n)$, which is isomorphic to $\mbox{AND}_n$. If $f(1\ldots1)=0$, then $f(x)=\mbox{NAE}(x_1,\ldots,x_n)$ and $Q_E(f)<n$.
[**Case 2:**]{} There are all negation variable occurrences in every $f_{x_i=0}$, that is $f_{x_i=0}(x)=\mbox{OR}_{n-1}(\neg x_1,\ldots,\linebreak[0]\neg x_{i-1},\neg x_{i+1},\ldots, \neg x_n)$ for $i\in[n]$. It is easy to see that $f(x)=1$ for $|x|<n-1$, $f(x)=0$ for $|x|=n-1$ and $f(x)=*$ for $|x|=n$. If $f(1\ldots1)=1$, then $f(x)= \neg \mbox{Th}_n^{n-1}$ and $Q_E(f)=n-1<n$. If $f(1\ldots1)=0$, then $f(x)= \neg \mbox{EXACT}_n^{n-1}$ and $Q_E(f)=n-1<n$.
[**Case 3:**]{} There is an $i\in[n]$ such that there is at least one negation variable occurrence and one no negation variable occurrence $f_{x_i=0}$. Without loss of generality, we assume that $f_{x_1=0}(x)=\mbox{OR}( x_2, \neg x_3, \linebreak[0](\neg) x_4,\ldots)$. According to Lemma \[lm-7\], we have $f_{x_2=0}(x)=\mbox{OR}(x_1, \neg x_3,\linebreak[0] (\neg) x_4, \ldots)$ and $f_{x_3=0}(x)=\mbox{OR}(\neg x_1, \neg x_2, \linebreak[0] (\neg) x_4, \ldots)$. For any $y\in\{0,1\}^{n-2}$, $f(01y)=f(10y)=1$, that is $f(x)=1$ if $x_1\oplus x_2=1$. If $x_1\oplus x_2=0$, then $x_1=x_2$ and $f$ can be represented as an $(n-1)$-bit Boolean function $g(x_2,\ldots,x_n)$. Since $f_{x_1=0}(x)=\mbox{OR}( x_2, \neg x_3, (\neg) x_4,\ldots)$, there must exist a $u\in\{0,1\}^{n-3}$ such that $f(001u)=g(01u)=0$. Since $f_{x_3=0}(x)=\mbox{OR}(\neg x_1, \neg x_2, (\neg) x_4, \ldots)$, there must exist a $v\in\{0,1\}^{n-3}$ such that $f(110v)=g(10v)=0$. We also have $g(00\ldots00)=f(000\ldots00)=1$ and $g(00\ldots01)=f(000\ldots01)=1$. According to Lemma \[lm-8\], we have that $g(x_2,\ldots,x_n)$ is not isomorphic to $\mbox{AND}_{n-1}$ and $Q_E(g)<n-1$.
Now we give an exact quantum algorithm for $f$ as follows:
1. Evaluate $x_1\oplus x_2$ with one query.
2. If $x_1\oplus x_2=1$, then $f(x)=1$.
3. If $x_1\oplus x_2=0$, then $f(x)=g(x_2,\ldots,x_n)$. Evaluate $g$ with exact quantum algorithm.
Therefore, we have $Q_E(f)<1+ Q_E(g)<1+n-1=n.$ The theorem has been proved.
Almost all $n$-bit Boolean functions can be computed by an exact quantum algorithm with less than $n$ queries.
[**Proof:**]{} It is easy to see that there are $2\times 2^n$ $n$-bit Boolean functions which are isomorphic to $\mbox{AND}_n$. Since there are $2^{2^n}$ Boolean functions on $n$ variables, we see that the fraction of functions which have exact quantum query complexity $n$ is $o(1)$. Thus almost all $n$-bit Boolean functions can be computed by an exact quantum algorithm with less than $n$ queries.
Conclusion
==========
We have first shown that $\mbox{AND}_n$ is the only $n$-bit Boolean function in three special classes of Boolean functions, (including symmetric, monotone, read-once functions), up to isomorphism, that has exact quantum query complexity $n$. Finally, we have proved that in general $\mbox{AND}_n$ is the only $n$-bit Boolean function, up to isomorphism, that has exact quantum query complexity $n$. This shows that the advantages for exact quantum query algorithms are more common than previously thought.
In the proof for special classes of Boolean functions, we have used their special properties of different types of Boolean functions. Each approach is different from each other. These approaches that we used in each type of Boolean functions may be helpful in analysis of exact quantum complexity for other interesting functions. In the approach for general case, we have used the properties of the true value table of the Boolean functions.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors are thankful to the anonymous referees for their comments and suggestions on the early version of this paper. The third author would like to thank Alexander Rivosh for his help while visiting University of Latvia. Work of the first author was supported by FP7 FET projects QCS and QALGO and ERC Advanced Grant MQC (at the University of Latvia) and by National Science Foundation under agreement No. DMS-1128155 (at IAS, Princeton). Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Work of the second and third authors was supported by the Employment of Newly Graduated Doctors of Science for Scientific Excellence project/grant (CZ.1.07./2.3.00/30.0009) of Czech Republic.
H. Buhrman and R. de Wolf (2002), [*Complexity measures and decision tree complexity: a survey*]{}, Theoretical Computer Science, 288, pp. 1–43, 2002.
L. K. Grover (1996), [*A fast quantum mechanical algorithm for database search*]{}, in Proceedings of 28th STOC, pp. 212–219. Also arXiv:9605043
A. Ambainis (2007), [*Quantum walk algorithm for element distinctness*]{}, SIAM Journal on Computing, 37, pp. 210–239. Also FOCS’04 and quant-ph/0311001.
A. Belovs (2012), [*Span programs for functions with constant-sized 1-certificates*]{}, in Proceedings of 43rd STOC, pp. 77–84. Also arXiv:1105.4024.
C. D[ü]{}rr, M. Heiligman, P. H[ø]{}yer, and M. Mhalla (2006), [*Quantum query complexity of some graph problems*]{}, SIAM Journal on Computing, 35, pp. 1310–1328 . Earlier version in ICALP’04. Also arXiv:quant-ph/0401091.
P. W. Shor (1997), [*Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer*]{}, SIAM Journal on Computing, 26, pp. 1484–1509. Earlier version in FOCS’94. Also arXiv:9508027.
D. Simon (1997), [*On the power of quantum computation*]{}, SIAM Journal on Computing, 26, pp.1474–1483. Earlier version in FOCS’94.
A. Ambainis and A. Yakary[i]{}lmaz (2012), [*Superiority of exact quantum automata for promise problems*]{}, Information Processing Letters, 112, pp. 289–291. Also arXiv:1101.3837.
G. Brassard and P. H[ø]{}yer (1997), [*An exact quantum polynomial-time algorithm for Simon’s problem*]{}, in Proceedings of the Israeli Symposium on Theory of Computing and Systems, pp. 12–23. Also arXiv:9704027.
D. Deutsch and R. Jozsa (1992), [*Rapid solution of problems by quantum computation*]{}, in Proceedings of the Royal Society of London, volume A439, pp. 553–558.
J. Gruska, D.W. Qiu, and S.G. Zheng (2014), [*Generalizations of the distributed Deutsch-Jozsa promise problem*]{}, arXiv:1402.7254.
S.G. Zheng and D.W. Qiu (2014), [*From quantum query complexity to state complexity*]{}, arXiv:1407.7342.
J. Gruska, D.W. Qiu, and S.G. Zheng (2014), [*Potential of quantum finite automata with exact acceptance*]{}, arXiv:1404.1689.
S.G. Zheng, J. Gruska, and D.W. Qiu (2014), [*On the state complexity of semi-quantum finite automata*]{}, Theoretical Informatics and Applications 48, pp. 187–207. Earlier versions at LATA’14. Also arXiv:1307.2499.
R. Cleve, A. Eckert, C. Macchiavello, and M. Mosca (1998), [*Quantum algorithms revisited*]{}, in Proceedings of the Royal Society of London, volume A454, pp. 339–354. Also arXiv:9708016.
E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser (1998), [*A limit on the speed of quantum computation in determining parity*]{}, Physical Review Letters, 81, pp. 5442–5444. Also arXiv:9802045.
A. Ambainis (2013), [*Superlinear advantage for exact quantum algorithms*]{}, in Proceedings of 45th STOC, pp. 891–900. Also arXiv:1211.0721.
A. Montanaro, R. Jozsa, and G. Mitchison (2013), [*On exact quantum query complexity*]{}, Algorithmica, DOI 10.1007/s00453-013-9826-8. Also arXiv:1111.0475.
R. Beals, H. Buhrman, R. Cleve, M. Mosca, and R. [de]{} Wolf (2001), [*Quantum lower bounds by polynomials*]{}, Journal of the ACM, 48, pp. 778–797. Earlier version in FOCS’98. Also arXiv:9802049.
J. Gruska (1999), [*Quantum Computing*]{}, McGraw-Hill (London).
M. Nielsen and I. Chuang (2000), [*Quantum Computation and Quantum Information*]{}, Cambridge University Press.
N. Nisan and M. Szegedy (1994), [*On the degree of Boolean functions as real polynomials*]{}, Computational Complexity, 4, pp. 301–313. Earlier version in STOC’92.
J. von zur Gathen and J. R. Roche (1997), [*Polynomials with two values*]{}, Combinatorica, 17, pp. 345–362.
T. Eiter, K. Makino, and G. Gottlob (2008), [*Computational aspects of monotone dualization: A brief survey*]{}, Discrete Applied Mathematics, 156, pp. 2035–2049.
H. Barnum and M. Saks (2004), [*A lower bound on the quantum query complexity of read-once functions*]{}, Journal of Computer and System Sciences, 69(2), pp. 244–258. Also arXiv:quant-ph/0201007.
M. Saks and A. Wigderson (1986), [*Probabilistic Boolean decision trees and the complexity of evaluating game trees*]{}, in Proceedings of FOCS’27, pp. 29–38.
M. Santha (1995), [*On the Monte Carlo boolean decision tree complexity of read-once formulae*]{}, Random Structures & Algorithms, 6(1), pp. 75–87.
A. Ambainis, A. Iraids, and J. Smotrovs (2013), [*Exact quantum query complexity of EXACT and THRESHOLD*]{}, in Proceedings of 8th TQC, pp. 263–269. Also arXiv:1302.1235.
S. Aaronson (2003), [*Algorithms for Boolean function query properties*]{}, SIAM Journal on Computing, 32, pp. 1140–1157. Also arXiv:cs/0107010.
[^1]: In their numerical experiments, computation providing correct result with a probability greater than 0.999 is treated as exact.
[^2]: Let $b=(b_0,b_1,\ldots, b_{k+1})$. We say that $(b_0,\ldots, b_{k})$ is a prefix of $b$ and $(b_1,\ldots, b_{k+1})$ a suffix of $b$.
[^3]: \* will denote one bit that can be 0 or 1.
| ArXiv |
---
abstract: 'We modified the modal expansion, which is the traditional method used to calculate thermal noise. This advanced modal expansion provides physical insight about the discrepancy between the actual thermal noise caused by inhomogeneously distributed loss and the traditional modal expansion. This discrepancy comes from correlations between the thermal fluctuations of the resonant modes. The thermal noise spectra estimated by the advanced modal expansion are consistent with the results of measurements of thermal fluctuations caused by inhomogeneous losses.'
author:
- Kazuhiro Yamamoto
- Masaki Ando
- Keita Kawabe
- Kimio Tsubono
title: |
A theoretical approach to thermal noise caused by an inhomogeneously distributed loss\
— Physical insight by the advanced modal expansion
---
Introduction
============
Thermal fluctuation is one of the fundamental noise sources in precise measurements. For example, the sensitivity of interferometric gravitational wave detectors [@LIGO; @VIRGO; @GEO; @TAMA] is limited by the thermal noise of the mechanical components. The calculated thermal fluctuations of rigid cavities have coincided with the highest laser frequency stabilization results ever obtained [@Numata5; @Notcutt]. It is important to evaluate the thermal motion for studying the noise property. The (traditional) modal expansion [@Saulson] has been commonly used to calculate the thermal noise of elastic systems. However, recent experiments [@Yamamoto1; @Harry; @Conti; @Numata3; @Yamamoto3; @Black] have revealed that modal expansion is not correct when the mechanical dissipation is distributed inhomogeneously. In some theoretical studies [@Levin; @Nakagawa1; @Tsubono; @Yamamoto-D], calculation methods that are completely different from modal expansion have been developed. These methods are supported by the experimental results of inhomogeneous loss [@Harry; @Numata3; @Yamamoto3; @Black]. However, even when these method were used, the physics of the discrepancy between the actual thermal noise and the traditional modal expansion was not fully understood.
In this paper, another method to calculate the thermal noise is introduced [@Yamamoto-D]. This method, advanced modal expansion, is a modification of the traditional modal expansion (this improvement is a general extension of a discussion in Ref. [@Majorana]). The thermal noise spectra estimated by this method are consistent with the results of experiments concerning inhomogeneous loss [@Yamamoto1; @Yamamoto3]. It provides information about the disagreement between the thermal noise and the traditional modal expansion. We present the details of these topics in the following sections.
Outline of advanced modal expansion
===================================
Review of the traditional modal expansion
-----------------------------------------
The thermal fluctuation of the observed coordinate, $X$, of a linear mechanical system is derived from the fluctuation-dissipation theorem [@Callen; @Greene; @Landau2], $$\begin{aligned}
G_{X}(f)&=&-\frac{4 k_{\rm B} T}{\omega} {\rm Im}[H_{X}(\omega)],
\label{FDT}\\
H_{X}(\omega)&=&\frac{\tilde{X}(\omega)}{\tilde{F}(\omega)},
\label{transfer function}\\
\tilde{X}(\omega)&=&\frac1{2\pi}
\int^{\infty}_{-\infty}X(t)\exp(-{\rm i}\omega t)dt,
\label{Fourier transform}\end{aligned}$$ where $f(=\omega/2\pi)$, $t$, $k_{\rm B}$ and $T$, are the frequency, time, Boltzmann constant and temperature, respectively. The functions ($G_{X}$, $H_{X}$, and $F$) are the (single-sided) power spectrum density of the thermal fluctuation of $X$, the transfer function, and the generalized force, which corresponds to $X$. In the traditional modal expansion [@Saulson], in order to evaluate this transfer function, the equation of motion of the mechanical system without any loss is decomposed into those of the resonant modes. The details are as follows:
![\[defX\]Example of the definition of the observed coordinate, $X$, in Eq. (\[observed coordinate\]). The mirror motion is observed using a Michelson interferometer. The coordinate $X$ is the output of the interferometer. The vector $\boldsymbol{u}$ represents the displacement of the mirror surface. The field $\boldsymbol{P}$ is parallel to the beam axis. Its norm is the beam-intensity profile [@Levin].](modefig1){width="8.6cm"}
The definition of the observed coordinate, $X$, is described as $$X(t) = \int \boldsymbol{u}(\boldsymbol{r},t)
\cdot \boldsymbol{P}(\boldsymbol{r}) dS,
\label{observed coordinate}$$ where $\boldsymbol{u}$ is the displacement of the system and $\boldsymbol{P}$ is a weighting function that describes where the displacement is measured. For example, when mirror motion is observed using a Michelson interferometer, as in Fig. \[defX\], $X$ and $\boldsymbol{u}$ represent the interferometer output and the displacement of the mirror surface, respectively. The vector $\boldsymbol{P}$ is parallel to the beam axis. Its norm is the beam-intensity profile [@Levin]. The equation of motion of the mechanical system without dissipation is expressed as $$\rho\frac{\partial^2 \boldsymbol{u}}{\partial t^2}
-{\cal L}[\boldsymbol{u}]=F(t)\boldsymbol{P}(\boldsymbol{r}),
\label{eq_mo_continuous}$$ where $\rho$ is the density and ${\cal L}$ is a linear operator. The first and second terms on the left-hand side of Eq. (\[eq\_mo\_continuous\]) represent the inertia and the restoring force of the small elements in the mechanical oscillator, respectively. The solution of Eq. (\[eq\_mo\_continuous\]) is the superposition of the basis functions, $$\boldsymbol{u}(\boldsymbol{r},t)
=\sum_{n}\boldsymbol{w}_n(\boldsymbol{r})q_n(t).
\label{mode decomposition}$$ The functions, $\boldsymbol{w}_n$ and $q_n$, represent the displacement and time development of the $n$-th resonant mode, respectively. The basis functions, $\boldsymbol{w}_n$, are solutions of the eigenvalue problem, written as $${\cal L}[\boldsymbol{w}_n(\boldsymbol{r})]
=-\rho {\omega_n}^2 \boldsymbol{w}_n(\boldsymbol{r}),
\label{eigenvalue problem}$$ where $\omega_n$ is the angular resonant frequency of the $n$-th mode. The displacement, $\boldsymbol{w}_n$, is the component of an orthogonal complete system, and is normalized to satisfy the condition $$\int \boldsymbol{w}_n(\boldsymbol{r})
\cdot \boldsymbol{P}(\boldsymbol{r}) dS = 1.
\label{normalized condition}$$ The formula of the orthonormality is described as $$\int \rho \boldsymbol{w}_n(\boldsymbol{r}) \cdot
\boldsymbol{w}_k(\boldsymbol{r}) dV
= m_n \delta_{nk}.
\label{effective mass}$$ The parameter $m_n$ is called the effective mass of the mode [@Yamamoto1; @Gillespie; @Bondu; @Yamamoto2]. The tensor $\delta_{nk}$ is the Kronecker’s $\delta$-symbol.
Putting Eq. (\[mode decomposition\]) into Eq. (\[observed coordinate\]), we obtain a relationship between $X$ and $q_n$ using Eq. (\[normalized condition\]), $$X(t) = \sum_{n} q_n(t).
\label{observed coordinate decomposition}$$ In short, coordinate $X$ is a superposition of those of the modes, $q_n$. In order to decompose the equation of motion, Eq. (\[eq\_mo\_continuous\]), Eq. (\[mode decomposition\]) is substituted for $\boldsymbol{u}$ in Eq. (\[eq\_mo\_continuous\]). Equation (\[eq\_mo\_continuous\]) is multiplied by $\boldsymbol{w}_n$ and then integrated over all of the volume using Eqs. (\[eigenvalue problem\]), (\[normalized condition\]) and (\[effective mass\]). The result is that the equation of motion of the $n$-th mode, $q_n$, is the same as that of a harmonic oscillator on which force $F(t)$ is applied. After modal decomposition, the dissipation term is added to the equation of each mode. The equation of the $n$-th mode is written as $$-m_n \omega^2 \tilde{q}_n + m_n {\omega_n}^2 [1+{\rm i}\phi_n(\omega)]
\tilde{q}_n=\tilde{F},
\label{traditional1}$$ in the frequency domain. The function $\phi_n$ is the loss angle, which represents the dissipation of the $n$-th mode [@Saulson]. The transfer function, $H_X$, derived from Eqs. (\[transfer function\]), (\[observed coordinate decomposition\]) and (\[traditional1\]) is the summation of those of the modes, $H_n$, $$\begin{aligned}
H_{X}(\omega) &=& \frac{\tilde{X}}{\tilde{F}}
= \sum_n \frac{\tilde{q}_n}{\tilde{F}} \left(= \sum_n H_n \right)\nonumber\\
&=& \sum_{n} \frac1{-m_n \omega^2
+ m_n {\omega_n}^2 [1+{\rm i}\phi_n(\omega)]}.
\label{traditional3}\end{aligned}$$ According to Eqs. (\[FDT\]) and (\[traditional3\]), the power spectrum density, $G_{X}$, is the summation of the power spectrum, $G_{q_n}$, of $q_n$, $$\begin{aligned}
&&G_{X}(f) = \sum_{n} G_{q_n}\nonumber\\
&&= \sum_{n} \frac{4k_{\rm B}T}{m_n \omega}
\frac{{\omega_n}^2\phi_n(\omega)}
{(\omega^2-{\omega_n}^2)^2+{\omega_n}^4{\phi_n}^2(\omega)}.
\label{traditional2} \end{aligned}$$
Equation of motion in an advanced modal expansion
-------------------------------------------------
In the traditional modal expansion, the dissipation term is introduced after decomposition of the equation of motion without any loss. On the contrary, in an advanced modal expansion, the equation with the loss is decomposed [@Yamamoto-D; @optics]. If the loss is sufficiently small, the expansion process is similar to that in the perturbation theory of quantum mechanics [@Sakurai]. The equation of $q_n$ is expressed as $$\begin{aligned}
-m_n \omega^2 \tilde{q}_n + m_n {\omega_n}^2 [1&+&{\rm i}\phi_n(\omega)]
\tilde{q}_n\nonumber\\
&+& \sum_{k \neq n} {\rm i} \alpha_{nk}(\omega) \tilde{q}_k
= \tilde{F},\label{advanced1}\\
\phi_n(\omega) &=& \frac{\alpha_{nn}}{m_n {\omega_n}^2}\label{phi}.
\label{phi_n}\end{aligned}$$ The third term in Eq. (\[advanced1\]) is the difference between the advanced, Eq. (\[advanced1\]), and traditional, Eq. (\[traditional1\]), modal expansions. Since this term is a linear combination of the motions of the other modes, it represents the couplings between the modes. The magnitude of the coupling, $\alpha_{nk}$, depends on the property and the distribution of the loss (described below).
Details of coupling
-------------------
Let us consider the formulae of the couplings caused by the typical inhomogeneous losses, the origins of which exist outside and inside the material (viscous damping and structure damping, respectively) [@Yamamoto-D]. Regarding most of the external losses, for example, the eddy-current damping and residual gas damping are of the viscous type [@Saulson]. The friction force of this damping is proportional to the velocity. Inhomogeneous viscous damping introduces a friction force, ${\rm i} \omega \rho \Gamma(\boldsymbol{r})
\tilde{\boldsymbol{u}}(\boldsymbol{r})$, into the left-hand side of the equation of motion, Eq. (\[eq\_mo\_continuous\]), in the frequency domain. The function $\Gamma (\geq 0)$ represents the strength of the damping. The equation of motion with the dissipation term, ${\rm i} \omega \rho \Gamma(\boldsymbol{r})
\tilde{\boldsymbol{u}}(\boldsymbol{r})$, is decomposed. Since the loss is small, the basis functions of the equation without loss are available [@Sakurai]. Equation (\[mode decomposition\]) is put into the equation of motion along with the inhomogeneous viscous damping. This equation multiplied by $\boldsymbol{w}_n$ is integrated. The coupling of this dissipation is written in the form $$\alpha_{nk} = \omega \int \rho \Gamma(\boldsymbol{r})
\boldsymbol{w}_n(\boldsymbol{r}) \cdot
\boldsymbol{w}_k(\boldsymbol{r}) dV = \alpha_{kn}.
\label{coupling_viscous}$$
In most cases, the internal loss in the material is expressed using the phase lag, $\phi (\geq 0)$, between the strain and the stress [@Saulson]. The magnitude of the dissipation is proportional to this lag. The phase lag is almost constant against the frequency [@Saulson] in many kinds of materials (structure damping). In the frequency domain, the relationship between the strain and the stress (the generalized Hooke’s law) in an isotropic elastic body is written as [@Saulson; @Levin; @Yamamoto-D; @Landau] $$\begin{aligned}
\tilde{\sigma}_{ij}
&=& \frac{E_0[1+{\rm i}\phi(\boldsymbol{r})]}{1+\sigma}
\left(\tilde{u}_{ij} + \frac{\sigma}{1-2\sigma}\sum_{l}\tilde{u}_{ll}
\delta_{ij}\right)\nonumber\\
&=& [1+{\rm i}\phi(\boldsymbol{r})]\tilde{\sigma}'_{ij},
\label{structure_stress}\\
u_{ij} &=& \frac1{2} \left(\frac{\partial u_i}{\partial x_j}
+ \frac{\partial u_j}{\partial x_i}\right),
\label{strain}\end{aligned}$$ where $E_0$ is Young’s modulus and $\sigma$ is the Poisson ratio; $\sigma_{ij}$ and $u_{ij}$ are the stress and strain tensors, respectively. The tensor, $\tilde{\sigma}'_{ij}$, is the real part of the stress, $\tilde{\sigma}_{ij}$. It represents the stress when the structure damping vanishes. The value, $u_i$, is the $i$-th component of $\boldsymbol{u}$. The equation of motion of an elastic body [@Landau] in the frequency domain is expressed as $$-\rho \omega^2 \tilde{u}_i
-\sum_j \frac{\partial \tilde{\sigma}_{ij}}{\partial x_{j}}
= \tilde{F}P_i(\boldsymbol{r}),
\label{eq_mo_elastic_withloss}$$ where $P_i$ is the $i$-th component of $\boldsymbol{P}$. From Eqs. (\[structure\_stress\]) and (\[eq\_mo\_elastic\_withloss\]), an inhomogeneous structure damping term is obtained, $-{\rm i} \sum_j \partial \phi(\boldsymbol{r})
\tilde{\sigma}'_{ij}/\partial x_{j}$. The equation of motion with the inhomogeneous structure damping is decomposed in the same manner as that of the inhomogeneous viscous damping. The coupling is calculated using integration by parts and Gauss’ theorem [@Landau],
$$\begin{aligned}
\alpha_{nk} &=& - \int \sum_{i,j} w_{n,i}
\frac{\partial \phi(\boldsymbol{r}) \sigma_{k,ij}}{\partial x_j} dV
\nonumber\\
&=& - \int \sum_{i,j}
\frac{\partial w_{n,i} \phi(\boldsymbol{r}) \sigma_{k,ij}}
{\partial x_j} dV
+ \int \sum_{i,j} \frac{\partial w_{n,i}}{\partial x_j}
\phi(\boldsymbol{r}) \sigma_{k,ij} dV \nonumber \\
&=& - \int \sum_{i,j} w_{n,i} \phi(\boldsymbol{r}) \sigma_{k,ij} n_j dS
+ \int \sum_{i,j} \frac{\partial w_{n,i}}{\partial x_j}
\phi(\boldsymbol{r}) \sigma_{k,ij} dV \nonumber \\
&=& \int \frac{E_0 \phi(\boldsymbol{r})}{1+\sigma}
\left[\sum_{i,j} \frac{\partial w_{n,i}}{\partial x_j}
\left(w_{k,ij} + \frac{\sigma}{1-2\sigma}
\sum_l w_{k,ll} \delta_{ij}\right)\right] dV\nonumber\\
&=& \int \frac{E_0 \phi(\boldsymbol{r})}{1+\sigma}
\left(\sum_{i,j}w_{n,ij}w_{k,ij}
+ \frac{\sigma}{1-2\sigma}\sum_{l}w_{n,ll}\sum_{l}w_{k,ll} \right) dV
=\alpha_{kn},
\label{coupling_structure}\end{aligned}$$
where $w_{n,i}$ and $n_{i}$ are the $i$-th components of $\boldsymbol{w}_n$ and the normal unit vector on the surface. The tensors, $w_{n,ij}$ and $\sigma_{n,ij}$, are the strain and stress tensors of the $n$-th mode, respectively. In order to calculate these tensors, $w_{n,i}$ is substituted for $u_i$ in Eqs. (\[structure\_stress\]) and (\[strain\]) with $\phi=0$. Equation (\[coupling\_structure\]) is valid when the integral of the function, $\sum_{i,j} w_{n,i} \phi \sigma_{k,ij} n_{j}$, on the surface of the elastic body vanishes. For example, the surface is fixed ($w_{n,i}=0$) or free ($\sum_{j}\sigma_{k,ij}n_{j}=0$) [@Landau].
The equation of motion in the advanced modal expansion coincides with that in the traditional modal expansion when all of the couplings vanish. A comparison between Eqs. (\[effective mass\]) and (\[coupling\_viscous\]) shows that in viscous damping all $\alpha_{nk} (n \neq k)$ are zero when the dissipation strength, $\Gamma(\boldsymbol{r})$, does not depend on the position, $\boldsymbol{r}$. In the case of structure damping, from Eqs. (\[eq\_mo\_continuous\]), (\[mode decomposition\]), (\[eigenvalue problem\]) and (\[eq\_mo\_elastic\_withloss\]), the stress, $\sigma'_{ij}$, without dissipation satisfies $$\sum_j \frac{\partial \tilde{\sigma}'_{ij}}{\partial x_{j}}
= - \sum_n \rho {\omega_n}^2 w_{n,i} \tilde{q}_n.
\label{stress decomposition}$$ According to Eq. (\[effective mass\]), Eq. (\[stress decomposition\]) is decomposed without any couplings. From Eq. (\[stress decomposition\]) and the structure damping term, $-{\rm i} \sum_j \partial \phi(\boldsymbol{r})
\tilde{\sigma}'_{ij}/\partial x_{j}$, the conclusion is derived; all of the couplings in the structure damping vanish when the loss amplitude, $\phi$, is homogeneous. In summary, the inhomogeneous viscous and structure dampings produce mode couplings and destroy the traditional modal expansion.
The reason why the inhomogeneity of the loss causes the couplings is as follows. Let us consider the decay motion after only one resonant mode is excited. If the loss is uniform, the shape of the displacement of the system does not change while the resonant motion decays. On the other hand, if the dissipation is inhomogeneous, the motion near the concentrated loss decays more rapidly than the other parts. The shape of the displacement becomes different from that of the original resonant mode. This implies that the other modes are excited, i.e. the energy of the original mode is leaked to the other modes. This energy leakage represents the couplings in the equation of motion.
It must be noticed that some kinds of “homogeneous” loss cause the couplings. For example, in thermoelastic damping [@Zener; @Braginsky-thermo; @Liu; @Cerdonio], which is a kind of internal loss, the energy components of the shear strains, $w_{n,ij}(i \neq j)$, are not dissipated. The couplings, $\alpha_{nk}$, do not have any terms that consist of the shear strain tensors. The coupling formula of the homogeneous thermoelastic damping is different from Eq. (\[coupling\_structure\]) with the constant $\phi$. The couplings are not generally zero, even if the thermoelastic damping is uniform. The advanced, not traditional, modal expansion provides a correct evaluation of the “homogeneous” thermoelastic damping. In this paper, however, only coupling caused by inhomogeneous loss is discussed.
Thermal-noise formula of advanced modal expansion {#thermal noise of advanced}
-------------------------------------------------
In the advanced modal expansion, the transfer function, $H_{X}$, is derived from Eqs. (\[transfer function\]), (\[observed coordinate decomposition\]), and (\[advanced1\]) (since the dissipation is small, only the first-order of $\alpha_{nk}$ is considered [@alpha2]),
$$H_{X}(\omega) = \sum_n \frac1{-m_n {\omega}^2
+ m_n {\omega_n}^2 (1 + {\rm i}\phi_n)}
- \sum_{k \neq n}
\frac{{\rm i} \alpha_{nk}}
{[-m_n \omega^2+m_n {\omega_n}^2 (1+{\rm i}\phi_n)]
[-m_k \omega^2 +m_k {\omega_k}^2 (1+{\rm i}\phi_k)]}.
\label{advanced3}$$
Putting Eq. (\[advanced3\]) into Eq. (\[FDT\]), the formula for the thermal noise is obtained. In the off-resonance region, where $|-\omega^2+{\omega_n}^2| \gg {\omega_n}^2 \phi_n(\omega)$ for all $n$, this formula approximates the expression $$\begin{aligned}
G_{X}(f)&=&\sum_{n} \frac{4 k_{\rm B} T}{m_n\omega}
\frac{{\omega_n}^2\phi_n(\omega)}
{(\omega^2-{\omega_n}^2)^2}\nonumber\\
&+&\sum_{k \neq n}\frac{4k_{\rm B}T}{m_n m_k \omega}
\frac{\alpha_{nk}}{(\omega^2-{\omega_n}^2)(\omega^2-{\omega_k}^2)}.
\label{advanced2}\end{aligned}$$ The first term is the same as the formula of the traditional modal expansion, Eq. (\[traditional2\]). The interpretation of Eq. (\[advanced2\]) is as follows. The power spectrum density of the thermal fluctuation force of the $n$-th mode, $G_{F_n}$, and the cross-spectrum density between $F_n$ and $F_k$, $G_{F_n F_k}$, are evaluated from Eq. (\[advanced1\]) and the fluctuation-dissipation theorem [@Greene; @Landau2], $$\begin{aligned}
G_{F_n}(f) &=& 4 k_{\rm B} T
\frac{m_n {\omega_n}^2 \phi_n(\omega)}{\omega},
\label{G_F_n}\\
G_{F_n F_k}(f) &=& 4 k_{\rm B} T \frac{\alpha_{nk}(\omega)}{\omega}.
\label{G_F_n_F_k}\end{aligned}$$ The power spectrum density, $G_{F_n}$, is independent of $\alpha_{nk}$. On the other hand, $G_{F_n F_k}$ depends on $\alpha_{nk}$. Having the correlations between the fluctuation forces of the modes, correlations between the motion of the modes must also exist. The power spectrum density of the fluctuation of $q_n$, $G_{q_n}$, and the cross-spectrum density between the fluctuations of $q_n$ and $q_k$, $G_{q_nq_k}$, are described as [@Greene; @Landau2] $$\begin{aligned}
G_{q_n}(f) &=& \frac{4 k_{\rm B} T}{m_n\omega}\frac{{\omega_n}^2\phi_n(\omega)}
{(\omega^2-{\omega_n}^2)^2},
\label{G_q_n}\\
G_{q_n q_k}(f) &=& \frac{4k_{\rm B}T}{m_nm_k\omega}
\frac{\alpha_{nk}}{(\omega^2-{\omega_n}^2)(\omega^2-{\omega_k}^2)},
\label{G_q_n_q_k}\end{aligned}$$ under the same approximation of Eq. (\[advanced2\]). The first and second terms in Eq. (\[advanced2\]) are summations of the fluctuation motion of each mode, Eq. (\[G\_q\_n\]), and the correlations, Eq. (\[G\_q\_n\_q\_k\]), respectively. In conclusion, inhomogeneous mechanical dissipation causes mode couplings and correlations of the thermal motion between the modes. In order to check wheather the formula of the thermal motion in the advanced modal expansion is consistent with the the equipartition principle, the mean square of the thermal fluctuation, $\overline{X^2}$, which is an integral of the power spectrum density over the whole frequency region, is evaluated. This mean square is derived from Eq. (\[FDT\]) using the Kramers-Kronig relation [@Landau2; @KKcomment], $${\rm Re}[H_X(\omega)] = -\frac1{\pi} \int_{-\infty}^{\infty}
\frac{{\rm Im}[H_X(\xi)]}{\xi-\omega}d\xi.
\label{Kramers-Kronig}$$ The calculation used to evaluate the mean square is written as [@Landau2] $$\begin{aligned}
\overline{X^2} &=& \int_0^{\infty} G_{X}(f) df \nonumber\\
&=& \frac1{4\pi} \int_{-\infty}^{\infty} G_{X}(\omega) d\omega \nonumber\\
&=& -\frac{k_{\rm B}T}{\pi} \int_{-\infty}^{\infty}
\frac{{\rm Im}[H_{X}(\omega)]}{\omega} d\omega \nonumber\\
&=& k_{\rm B}T {\rm Re}[H_{X}(0)].
\label{X2}\end{aligned}$$ Since the transfer function, $H_{X}$, is the ratio of the Fourier components of the real functions, the value $H_X(0)$ is a real number. The functions $\phi_n$ and $\alpha_{nk}$, which cause the imaginary part of $H_{X}$, must vanish when $\omega$ is zero [@Landau2]. The correlations do not affect the mean square of the thermal fluctuation. Equation (\[X2\]) is rewritten using Eq. (\[advanced3\]) as $$\overline{X^2}=\sum_n \frac{k_{\rm B}T}{m_n {\omega_n}^2}.
\label{X2_2}$$ Equation (\[X2\_2\]) is equivalent to the prediction of the equipartition principle.
The calculation of the formula of the advanced modal expansion, Eq. (\[advanced2\]), is more troublesome than that of the other methods [@Levin; @Nakagawa1; @Tsubono; @Yamamoto-D], which are completely different from the modal expansion, when many modes contribute to the thermal motion. However, the advanced modal expansion gives clear physical insight about the discrepancy between the thermal motion and the traditional modal expansion, as shown in Sec. \[new insight\]. It is difficult to find this insight using other methods.
Experimental check
==================
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In order to test the advanced modal expansion experimentally, our previous experimental results concerning oscillators with inhomogeneous losses [@Yamamoto1; @Yamamoto3] are compared with an evaluation of the advanced modal expansion [@Yamamoto-D]. In an experiment involving a drum (a hollow cylinder made from aluminum alloy as the prototype of the mirror in the interferometer) with inhomogeneous eddy-current damping by magnets [@Yamamoto3], the measured values agreed with the formula of the direct approach [@Levin], Eq. (6) in Ref. [@Yamamoto3]. This expression is the same as that of the advanced modal expansion [@Yamamoto-D].
Figure \[experiment\] presents the measured spectra of an aluminum alloy leaf spring with inhomogeneous eddy-current damping [@Yamamoto1]. The position of the magnets for the eddy-current damping and the observation point are indicated above each graph. In the figures above each graph, the left side of the leaf spring is fixed. The right side is free. The open circles in the graphs represent the power spectra of the thermal motion derived from the measured transfer functions using the fluctuation-dissipation theorem. These values coincide with the directly measured thermal-motion spectra [@Yamamoto1]. The solid lines are estimations using the advanced modal expansion (the correlations derived from Eqs. (\[coupling\_viscous\]) and (\[G\_q\_n\_q\_k\]) are almost perfect [@Yamamoto-D]). As a reference, an evaluation of the traditional modal expansion is also given (dashed lines). The results of a leaf-spring experiment are consistent with the advanced modal expansion. Therefore, our two experiments support the advanced modal expansion.
Physical insight given by the advanced modal expansion {#new insight}
======================================================
The advanced modal expansion provides physical insight about the disagreement between the real thermal motion and the traditional modal expansion. Here, let us discuss the three factors that affect this discrepancy: the number of the modes, the absolute value and the sign of the correlation.
Number of modes
---------------
Since the difference between the advanced and traditional modal expansions is the correlations between the multiple modes, the number of the modes affects the magnitude of the discrepancy. If the thermal fluctuation is dominated by the contribution of only one mode, this difference is negligible, even when there are strong correlations. On the other hand, if the thermal motion consists of many modes, the difference is larger when the correlations are stronger.
Examples of the one-mode oscillator are given in Fig. \[experiment\]. The measured thermal motion spectra of the leaf spring with inhomogeneous losses below 100 Hz were the same as the estimated values of the “traditional” modal expansion. This is because these fluctuations were dominated by only the first mode (about 60 Hz). As another example, let us consider a single-stage suspension for a mirror in an interferometric gravitational-wave detector. The sensitivity of the interferometer is limited by the thermal noise of the suspensions between 10 Hz and 100 Hz. Since, in this frequency region, this thermal noise is dominated by only the pendulum mode [@Saulson], the thermal noise generated by the inhomogeneous loss agrees with the traditional modal expansion. It must be noticed that the above discussion is valid only when the other suspension modes are negligible. For example, when the laser beam spot on the mirror surface is shifted, the two modes (pendulum mode and mirror rotation mode) must be taken into account. In such cases, the inhomogeneous loss causes a disagreement between the real thermal noise of the single-stage suspension and the traditional modal expansion [@Braginsky].
The discrepancy between the actual thermal motion and the traditional modal expansion in the elastic modes of the mirror [@Yamamoto2] is larger than that of the drum, the prototype of the real mirror in our previous experiment [@Yamamoto3]. One of the reasons is that the thermal motion of the mirror (rigid cylinder) consists of many modes [@Gillespie; @Bondu]. The drum (hollow cylinder) had only two modes [@Yamamoto3]. Since the number of modes that contribute to the thermal noise of the mirror in the interferometer increases when the laser beam radius becomes smaller [@Gillespie; @Bondu], the discrepancy is larger with a narrower beam. This consideration is consistent with our previous calculation [@Yamamoto2].
Absolute value of the correlation
---------------------------------
![\[abs ex\]Example for considering the absolute value of the coupling. There are the $n$-th and $k$-th modes, $\boldsymbol{w}_n$ and $\boldsymbol{w}_k$, of a bar with both free ends. The vertical axis is the displacement. The dashed horizontal lines show the bar that does not vibrate. When only the grey part (A), which is narrower than the wavelengths on the left-hand side, has viscous damping, the absolute value of the coupling, Eq. (\[coupling\_viscous\]), is large. Because the signs of $\boldsymbol{w}_n$ and $\boldsymbol{w}_k$ do not change in this region. If viscous damping exits only in the hatching part (B), which is wider than the wavelengths on the right-hand side, the coupling is about zero, because, in this wide region, the sign of the integrated function in Eq. (\[coupling\_viscous\]), which is proportional to the product of $\boldsymbol{w}_n$ and $\boldsymbol{w}_k$, changes.](modefig3){width="8.6cm"}
In Eq. (\[G\_q\_n\_q\_k\]), the absolute value of the cross-spectrum density, $G_{q_n q_k}$, is proportional to that of the coupling, $\alpha_{nk}$. Equations (\[coupling\_viscous\]) and (\[coupling\_structure\]) show that the coupling depends on the scale of the dissipation distribution. A simple example of viscous damping is shown in Fig. \[abs ex\]. Let us consider the absolute value of $\alpha_{nk}$ when the viscous damping is concentrated (at around $\boldsymbol{r}_{\rm vis}$) in a smaller volume ($\Delta V$) than the wavelengths of the $n$-th and $k$-th modes. An example of this case is (A) in Fig. \[abs ex\]. It is assumed that the vector $\boldsymbol{w}_n(\boldsymbol{r}_{\rm vis})$ is nearly parallel to $\boldsymbol{w}_k(\boldsymbol{r}_{\rm vis})$. The absolute value of the coupling is derived from Eqs. (\[phi\_n\]) and (\[coupling\_viscous\]) as $$\begin{aligned}
|\alpha_{nk}| &\sim& |\omega \rho \Gamma(\boldsymbol{r}_{\rm vis})
\boldsymbol{w}_n(\boldsymbol{r}_{\rm vis}) \cdot
\boldsymbol{w}_k(\boldsymbol{r}_{\rm vis}) \Delta V| \nonumber\\
&\sim& \sqrt{\omega \rho \Gamma(\boldsymbol{r}_{\rm vis})
|\boldsymbol{w}_n(\boldsymbol{r}_{\rm vis})|^2 \Delta V} \nonumber\\
&&\hspace{0.5cm}\times \sqrt{\omega \rho \Gamma(\boldsymbol{r}_{\rm vis})
|\boldsymbol{w}_k(\boldsymbol{r}_{\rm vis})|^2 \Delta V}\nonumber\\
&\sim& \sqrt{\alpha_{nn}\alpha_{kk}}
= \sqrt{m_n {\omega_n}^2 \phi_n m_k {\omega_k}^2 \phi_k}.
\label{coupling_narrow}\end{aligned}$$ The absolute value of the cross-spectrum is derived from Eqs. (\[G\_q\_n\]), (\[G\_q\_n\_q\_k\]), and (\[coupling\_narrow\]) as $$|G_{q_n q_k}| \sim \sqrt{G_{q_n}G_{q_k}}.
\label{maxcorrelation}$$ In short, the correlation is almost perfect [@maxcoupling]. On the other hand, if the loss is distributed more broadly than the wavelengths, the coupling, i.e. the correlation, is about zero, $$|G_{q_n q_k}| \sim 0.$$ The dissipation in the case where the size is larger than the wavelengths is equivalent to the homogeneous loss. An example of this case is (B) in Fig. \[abs ex\]. Although the above discussion is for the case of viscous damping, the conclusion is also valid for other kinds of dissipation. When the loss is localized in a small region, the correlations among many modes are strong. The loss in a narrower volume causes a larger discrepancy between the actual thermal motion and the traditional modal expansion. This conclusion coincides with our previous calculation of a mirror with inhomogeneous loss [@Yamamoto2].
Sign of correlation
-------------------
The sign of the correlation depends on the frequency, the loss distribution, and the position of the observation area. The position dependence provides a solution to the inverse problem: an evaluation of the distribution and frequency dependence of the loss from measurements of the thermal motion.
### Frequency dependence
According to Eq. (\[G\_q\_n\_q\_k\]), the sign of the correlation reverses at the resonant frequencies. For example, in calculating the double pendulum [@Majorana], experiments involving the drum [@Yamamoto3] and a resonant gravitational wave detector with optomechanical readout [@Conti], this change of the sign was found. In some cases, the thermal-fluctuation spectrum changes drastically around the resonant frequencies. A careful evaluation is necessary when the observation band includes the resonant frequencies. Examples are when using wide-band resonant gravitational-wave detectors [@wide1; @wide2; @wide3; @wide4], and thermal-noise interferometers [@Numata3; @Black]. The reason for the reverse at the resonance is that the sign of the transfer function of the mode with a small loss from the force ($F_n$) to the motion ($q_n$), $H_n$ in Eq. (\[traditional3\]) \[$\propto (-\omega^2+{\omega_n}^2)^{-1}$\], below the resonance is opposite to that above it.
Since the sign of the correlation changes at the resonant frequencies, the cross-spectrum densities, the second term of Eq. (\[advanced2\]), make no contribution to the integral of the power spectrum density over the whole frequency region, i.e. the mean square of the thermal fluctuation, $\overline{X^2}$, as shown in Sec. \[thermal noise of advanced\]. Therefore, the consideration in Sec. \[thermal noise of advanced\] indicates that a reverse of the sign of the correlation conserves the equipartition principle, a fundamental principle in statistical mechanics.
### Loss and observation area position dependence
![\[signex\]Example for considering the sign of the coupling. There are the lowest three modes, $\boldsymbol{w}_n$, of a bar with both free ends. The vertical axis is the displacement. The dashed horizontal lines show the bar that does not vibrate. The observation point is at the right-hand side end. The normalization condition is Eq. (\[normalized condition\]) [@normalized; @condition]. The sign and shape of the displacement of all the modes around the observation point are positive and similar, respectively. On the contrary, at the left-hand side end, the sign and shape of the $n$-th mode are different from each other in many cases. From Eqs. (\[normalized condition\]), (\[coupling\_viscous\]), (\[coupling\_structure\]), when the loss is concentrated near to the observation area, most of the couplings are positive. On the other hand, when the loss is localized far from the observation area, the number of the negative couplings is about the same as the positive one. In such a case, most of the couplings between the $n$-th and $(n \pm 1)$-th modes are negative.](modefig4){width="8.6cm"}
According to Eqs. (\[coupling\_viscous\]) and (\[coupling\_structure\]), and the normalization condition, Eq. (\[normalized condition\]) [@normalized; @condition], the sign of the coupling, $\alpha_{nk}$, depends on the loss distribution and the position of the observation area. A simple example is shown in Fig. \[signex\]. Owing to this normalization condition, near the observation area, the basis functions, $\boldsymbol{w}_n$, are similar in most cases. On the contrary, in a volume far from the observation area, $\boldsymbol{w}_n$ is different from each other in many cases. From Eqs. (\[normalized condition\]), (\[coupling\_viscous\]), (\[coupling\_structure\]) and (\[G\_F\_n\_F\_k\]), when the loss is concentrated near to the observation area, most of the couplings (and the correlations between the fluctuation forces of the modes, $G_{F_nF_k}$) are positive. On the other hand, when the loss is localized far from the observation area, the numbers of the negative couplings and $G_{F_nF_k}$ are about the same as the positive ones. In such a case, most of the couplings between the $n$-th and $(n \pm 1)$-th modes (and $G_{F_nF_{n \pm 1}}$) are negative. These are because the localized loss tends to apply to the fluctuation force on all of the modes to the same direction around itself. Equation (\[G\_q\_n\_q\_k\]) indicates that the sign of the correlation, $G_{q_nq_k}$, is the same as that of the coupling, $\alpha_{nk}$, below the first resonance. In this frequency band, the thermal motion is larger and smaller than the evaluation of the traditional modal expansion if the dissipation is near and far from the observation area, respectively. This conclusion is consistent with the qualitative discussion of Levin [@Levin], our previous calculation of the mirror [@Yamamoto2], and the drum experiment [@Yamamoto3].
### Inverse problem
The above consideration about the sign of the coupling gives a clue to solving the inverse problem: estimations of the distribution and frequency dependences of the loss from the measurement of the thermal motion. Since the sign of the coupling depends on the position of the observation area and the loss distribution, a measurement of the thermal vibrations at multiple points provides information about the couplings, i.e. the loss distribution. Moreover, multiple-point measurements reveal the loss frequency dependence. Even if the loss is uniform, the difference between the actual thermal motion and the traditional modal expansion exits when the expected frequency dependence of the loss angles, $\phi_n(\omega)$, is not correct [@Majorana]. The measurement at the multiple points shows whether the observed difference is due to an inhomogeneous loss or an invalid loss angle. This is because the sign of the difference is independent of the position of the observation area if the expected loss angles are not valid.
As an example, our leaf-spring experiment [@Yamamoto1] is discussed. The two graphs on the right (or left) side of Fig. \[experiment\] show thermal fluctuations at different positions in the same mechanical system. The spectrum is smaller than the traditional modal expansion. The other one is larger. Thus, the disagreement in the leaf-spring experiment was due to inhomogeneous loss, not invalid loss angles. When the power spectrum had a dip between the first (60 Hz) and second (360 Hz) modes, the sign of the correlation, $G_{q_1q_2}$, was negative. According to Eq. (\[G\_q\_n\_q\_k\]), the sign of the coupling, $\alpha_{12}$, was positive. The loss was concentrated near to the observation point when a spectrum dip was found. The above conclusion agrees with the actual loss shown in Fig. \[experiment\].
Conclusion
==========
The traditional modal expansion has frequently been used to evaluate the thermal noise of mechanical systems [@Saulson]. However, recent experimental research [@Yamamoto1; @Harry; @Conti; @Numata3; @Yamamoto3; @Black] has proved that this method is invalid when the mechanical dissipation is distributed inhomogeneously. In this paper, we introduced a modification of the modal expansion [@Yamamoto-D; @Majorana]. According to this method (the advanced modal expansion), inhomogeneous loss causes correlations between the thermal fluctuations of the modes. The fault of the traditional modal expansion is that these correlations are not taken into account. Our previous experiments [@Yamamoto1; @Yamamoto3] concerning the thermal noise of the inhomogeneous loss support the advanced modal expansion.
The advanced modal expansion gives interesting physical insight about the difference between the actual thermal noise and the traditional modal expansion. When the thermal noise consists of the contributions of many modes, the loss is localized in a narrower area, which makes a larger difference. When the thermal noise is dominated by only one mode, this difference is small, even if the loss is extremely inhomogeneous. The sign of this difference depends on the frequency, the distribution of the loss, and the position of the observation area. It is possible to derive the distribution and frequency dependence of the loss from measurements of the thermal vibrations at multiple points.
There were many problems concerning the thermal noise caused by inhomogeneous loss. Our previous work [@Yamamoto1; @Yamamoto3; @Yamamoto2] and this research solved almost all of these problems: a modification of the traditional estimation method (in this paper), experimental checks of the new and traditional estimation methods and a confirmation of the new methods ([@Yamamoto1; @Yamamoto3] and this paper), an evaluation of the thermal noise of the gravitational wave detector using the new method [@Yamamoto2], and a consideration of the physical properties of the discrepancy between the actual thermal noise and the traditional estimation method (in this paper).
This research was supported in part by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists, and by a Grant-in-Aid for Creative Basic Research of the Ministry of Education.
[99]{}
A. Abramovici [*et al.*]{}, Science [**256**]{}, 325 (1992).
C. Bradaschia [*et al.*]{}, Nucl. Instrum. Methods Phys. Res., Sect. A [**289**]{}, 518 (1990).
B. Willke [*et al.*]{}, Class. Quant. Grav. [**19**]{}, 1377 (2002).
M. Ando [*et al.*]{}, Phys. Rev. Lett. [**86**]{}, 3950 (2001).
K. Numata, A. Kemery, and J. Camp, Phys. Rev. Lett. [**93**]{}, 250602 (2004).
M. Notcutt [*et al.*]{}, Phys. Rev. A [**73**]{}, 031804(R) (2006).
P.R. Saulson, Phys. Rev. D [**42**]{}, 2437 (1990).
K. Yamamoto, S. Otsuka, M. Ando, K. Kawabe, and K. Tsubono, Phys. Lett. A [**280**]{}, 289 (2001).
G.M. Harry [*et al.*]{}, Class. Quant. Grav. [**19**]{}, 897 (2002).
L. Conti, M. De Rosa, F. Marin, L. Taffarello, and M. Cerdonio, J. Appl. Phys. [**93**]{}, 3589 (2003).
K. Numata, M. Ando, K. Yamamoto, S. Otsuka, and K. Tsubono, Phys. Rev. Lett. [**91**]{}, 260602 (2003).
K. Yamamoto, S. Otsuka, Y. Nanjo, M. Ando, and K. Tsubono, Phys. Lett. A [**321**]{}, 79 (2004).
E.D. Black [*et al.*]{}, Phys. Lett. A [**328**]{}, 1 (2004).
Yu. Levin, Phys. Rev. D [**57**]{}, 659 (1998).
N. Nakagawa, B.A. Auld, E. Gustafson, and M.M. Fejer, Rev. Sci. Instrum. [**68**]{}, 3553 (1997).
K. Tsubono, in preparation.
K. Yamamoto, Ph D. thesis, the University of Tokyo (2001) (http://t-munu.phys.s.u-tokyo.ac.jp/theses/yamamoto[\_]{}d.pdf).
E. Majorana and Y. Ogawa, Phys. Lett. A [**233**]{}, 162 (1997).
H.B. Callen and R.F. Greene, Phys. Rev. [**86**]{}, 702 (1952).
R.F. Greene and H.B. Callen, Phys. Rev. [**88**]{}, 1387 (1952).
Sections 122, 123, 124, and 125 in Chapter 12 of L.D. Landau and E.M. Lifshitz, [*Statistical Physics*]{} Part 1 (Pergamon, New York, 1980).
A. Gillespie and F. Raab, Phys. Rev. D [**52**]{}, 577 (1995).
F. Bondu and J-Y. Vinet, Phys. Lett. A [**198**]{}, 74 (1995).
K. Yamamoto, M. Ando, K. Kawabe, and K. Tsubono, Phys. Lett. A [**305**]{}, 18 (2002).
Within the optics community, another revision of the modal expansion, the quasinormal modal expansion, is being discussed: A.M. van den Brink, K. Young, and M.H. Yung, J. Phys. A: Math. Gen. [**39**]{}, 3725 (2006) and its references.
Chapter 5 of J.J. Sakurai, [*Modern Quantum Mechanics*]{} (Benjamin/Cummings, California, 1985).
Chapters 1 and 3 of L.D. Landau and E.M. Lifshitz, [*Theory of Elasticity*]{} (Pergamon, New York, 1986).
C. Zener, Phys. Rev. [**52**]{}, 230 (1937); [**53**]{}, 90 (1938).
V.B. Braginsky, M.L. Gorodetsky, and S.P. Vyatchanin, Phys. Lett. A [**264**]{}, 1 (1999).
Y.T. Liu and K.S. Thorne, Phys. Rev. D [**62**]{}, 122002 (2000).
M. Cerdonio, L. Conti, A. Heidmann, and M. Pinard, Phys. Rev. D [**63**]{}, 082003 (2001).
In the off-resonance region, where $|-\omega^2+{\omega_n}^2| \gg {\omega_n}^2 \phi_n(\omega)$ for all $n$, this approximation is always appropriate because the maximum of ${\alpha_{nk}}^2$ is $m_n {\omega_n}^2 \phi_n m_k {\omega_k}^2 \phi_k$ [@maxcoupling]. In the calculation to derive Eq. (\[advanced3\]), Cramer’s rule is useful. The sign on the right-hand side of the Kramers-Kronig relation in Ref. [@Landau2] is positive. The definition of the Fourier transformation in Ref. [@Landau2] is conjugate to that of this paper, Eq. (\[Fourier transform\]).
V.B. Braginsky, Yu. Levin, and S.P. Vyatchanin, Meas. Sci. Technol. [**10**]{}, 598 (1999).
This discussion suggests that $|\alpha_{nk}|$ is never larger than $\sqrt{m_n {\omega_n}^2 \phi_n m_k {\omega_k}^2 \phi_k}$. The validity of this conjecture is supported by the Cauchy-Schwarz inequality [@Yamamoto-D].
M. Cerdonio [*et al.*]{}, Phys. Rev. Lett. [**87**]{}, 031101 (2001).
T. Briant [*et al.*]{}, Phys. Rev. D [**67**]{}, 102005 (2003).
M. Bonaldi [*et al.*]{}, Phys. Rev. D [**68**]{}, 102004 (2003).
M. Bonaldi [*et al.*]{}, Phys. Rev. D [**74**]{}, 022003 (2006).
The results of the discussion presented in this section are valid under arbitrary normalization conditions. If a normalization condition other than Eq. (\[normalized condition\]) is adopted, the signs of some couplings change. The signs of the right-hand sides of Eqs. (\[observed coordinate decomposition\]) and (\[advanced1\]) also change. Equations (\[observed coordinate\]), (\[eq\_mo\_continuous\]), and (\[mode decomposition\]) say that the right-hand sides of Eqs. (\[observed coordinate decomposition\]) and (\[advanced1\]) in the general style are $\sum_{n} q_n(t) \int \boldsymbol{w}_n(\boldsymbol{r})
\cdot \boldsymbol{P}(\boldsymbol{r}) dS$ and $\tilde{F} \int \boldsymbol{w}_n(\boldsymbol{r})
\cdot \boldsymbol{P}(\boldsymbol{r}) dS$, respectively. The change in the sign cancels each other in the process of the calculation for the transfer function, $H_X$, and the power spectrum of the thermal motion, $G_{X}$.
| ArXiv |
---
abstract: 'Stochastic resetting is prevalent in natural and man-made systems giving rise to a long series of non-equilibrium phenomena. Diffusion with stochastic resetting serves as a paradigmatic model to study these phenomena, but the lack of a well-controlled platform by which this process can be studied experimentally has been a major impediment to research in the field. Here, we report the experimental realization of colloidal particle diffusion and resetting via holographic optical tweezers. This setup serves as a proof-of-concept which opens the door to experimental study of resetting phenomena. It also vividly illustrates why existing theoretical models must be improved and revised to better capture the real-world physics of stochastic resetting.'
author:
- 'Ofir Tal-Friedman$^{1}$'
- 'Arnab Pal$^{2,3}$'
- 'Amandeep Sekhon$^{2}$'
- 'Shlomi Reuveni$^{2,3}$'
- 'Yael Roichman$^{1,2}$'
title: Experimental realization of diffusion with stochastic resetting
---
=1
Stochastic resetting is ubiquitous in nature, and has recently been the subject of vigorous studies in physics [@Evans2011_1; @Evans2011_2; @Evans2011_3], chemistry [@Restart-Biophysics1; @Restart-Biophysics2; @Restart-Biophysics6], biological physics [@Restart-Biophysics3; @Restart-Biophysics8], computer science [@restart-CS1; @restart-CS2], queuing theory [@queue1; @queue2] and other cross-disciplinary fields (see [@review] for extensive account of recent developments). A stylized model to study resetting phenomena was proposed by Evans and Majumdar in 2011 [@Evans2011_1]. The model, which considers a diffusing particle subject to stochastic resetting, exhibits many rich properties e.g., the emergence of a non-equilibrium steady state and interesting relaxation dynamics [@Evans2011_1; @Evans2011_2; @Evans2011_3; @Evans2014_3; @Pal2016_1; @relaxation1; @relaxation2; @local] which were also observed in other systems subject to stochastic resetting [@restart_conc3; @SEP; @return1; @return2; @return3; @return4; @return5; @Bod1; @Bod2]. The model is also pertinent to the study of search and first-passage time (FPT) questions [@RednerBook; @Schehr-review]. In particular, it was used to show that resetting can significantly reduce the mean FTP of a diffusing particle to a target by mitigating the deleterious effect of large FPT fluctuations that are intrinsic to diffusion in the absence of resetting [@Evans2011_1; @Evans2011_2; @Evans2011_3; @review; @Pal2016_1; @Ray; @interval]. Interestingly, this beneficial effect of resetting also extends beyond free diffusion and applies to many other stochastic processes [@review; @return5; @Bod1; @Bod2; @return3; @return4; @ReuveniPRL; @PalReuveniPRL; @branching_II; @Restart-Search1; @Restart-Search2; @Chechkin; @Landau; @HRS]; and further studies moreover revealed a genre of universality relations associated with optimally restarted processes as well as the existence of a globally optimal resetting strategy [@Restart-Biophysics1; @Restart-Biophysics2; @ReuveniPRL; @PalReuveniPRL; @Chechkin; @branching_II; @Landau; @HRS].
![Experimental realization of diffusion with stochastic resetting. a) A sample trajectory of a silica particle diffusing (blue) near the bottom of a sample cell. The particle sets off from the origin and is stochastically reset at a rate $r=0.05 s^{-1}$. Following a resetting epoch, the particle is driven back to the origin at a constant radial velocity $v=0.8 \mu m/s$ using HOTs (red). After the particle arrives at the origin it remains trapped there for a short period of time to improve localization (green). Inset shows a schematic illustration of the experiment. b) Projection of the particle’s trajectory onto the $x$-axis.[]{data-label="Fig:expt"}](Fig1_slow.pdf)
Despite a long catalogue of theoretical studies dedicated to stochastic resetting, no attempt to experimentally study resetting in a controlled environment has been made to date. This is needed as resetting in the real world is never ‘clean’ as in theoretical models which glance over physical complications for the sake of analytical tractability and elegance. In this letter, we report the experimental realization of diffusion with stochastic resetting ([Fig. \[Fig:expt\]]{}). Our setup comprises of a colloidal particle suspended in fluid (in quasi-two dimensions) and resetting is implemented via holographic optical tweezers (HOTs) [@Dufresne2001; @Polin2005; @Grier06; @Crocker1996]. We study two, physically amenable, resetting protocols in which the particle is returned to the origin: (i) at a constant velocity, and (ii) within a constant time. In both cases, resetting is stochastic and time intervals between resetting events come from an exponential distribution with mean $1/r$.
In what follows, we utilize the setup in [Fig. \[Fig:expt\]]{} to study two different statistical measures of diffusion with stochastic resetting. First, we study the long time position distribution of a tagged particle and how it depends on the resetting protocol. Then, we study the mean FPT of a tagged particle to a region in space. Finally, we also consider the work and energy required to implement resetting in our system. In all cases, we discover that existing theoretical models must be extended and revised to better capture the physics of stochastic resetting in the real world. We conclude with discussion and outlook on the future of experimental studies of stochastic resetting.
Our experimental setup is based on a home built holographic optical tweezers (HOTs) system. It uses a spatial light modulator (Hamamtsu, X10468-04) to imprint a computer generated phase pattern on an expanded laser beam (Coherent, Verdi $\lambda=532$nm). The beam is then projected on the back aperture of a 100x objective (oil immersion, NA = 1.4) mounted on an Olympus microscope (IX71). Samples consist of a dilute colloidal suspension of spherical silica particles with a diameter of $d=1.5\pm 0.02\mu m$ and a refractive index of $n_p=1.46$ (Kisker Biotech, lot\# GK0611140 02) in double distilled water sealed between a glass slide and a coverslip with a sample thickness of approximately $20\mu$m. Motion of the particle (confined to a quasi two dimensional geometry) is recorded by a CMOS camera (Grasshopper 3, Point Gray) at a rate of 20 fps. Particle position is extracted using conventional video microscopy algorithms [@Crocker1996] with an accuracy of approximately 30nm. A laser power of $1$W was used to ensure sufficient trapping of the particle. We utilize in-house developed programs for hardware control and data analysis.
We start our experiments by realizing diffusion with stochastic resetting in the following manner. Every experiment starts by drawing a series of random resetting times $\{t_1,t_2,t_3,...\}$ taken from an exponential distribution with mean $1/r$. At time zero, the particle is trapped at the origin and the experiment, which consists of a series of statistically identical steps, begins. At the $i$-th step of the experimental protocol, the particle is allowed to diffuse for a time $t_i$ eventually arriving at a position $(x_i,y_i)$. At this time, an optical trap is projected onto the particle and the particle is dragged by the trap to its initial position. A typical trajectory of a colloidal particle performing diffusion under stochastic resetting with $r=0.05s^{-1}$ is shown in [Fig. \[Fig:expt\]]{}a (see also Supplementary movie 1). Note that the trajectory is composed of three phases of motion: diffusion, return, and a short waiting time to allow for optimal localization at the origin ([Fig. \[Fig:expt\]]{}b). To collect sufficient statistics, we perform approximately 450 resetting events for each constant velocity experiment, and 305 events for the constant return time experiment.
*Stochastic resetting with instantaneous returns.—* We first utilize our setup to study a canonical (yet non-physical) case in which upon resetting the particle is teleported back to the origin in zero time. This case was the first to be analysed theoretically [@Evans2011_1], thus providing a benchmark for experimental results. To obtain trajectories of diffusion with stochastic resetting and instantaneous returns we digitally remove the return (red) and wait (green) phases of motion from the experimentally measured trajectories ([Fig. \[Fig:expt\]]{}b). A sample trajectory obtained via this procedure is shown in Fig. S1.
![Steady-state distribution of diffusion with stochastic resetting and instantaneous returns. a) Distribution of the position along the $x$-axis. Markers come from experiments and the dashed line is the theoretical prediction of Eq. (1). b) The radial position distribution. Markers come from experiments and the dashed line is the theoretical prediction $\rho(R)=\alpha_0^2RK_0(\alpha_0R)$ [@SM] with $K_n(z)$ standing for the modified Bessel function of the second kind [@Stegun]. In both panels no fitting procedure was applied: $D=0.18\pm 0.02 \mu m^2/s$ was measured independently and $r=0.05s^{-1}$ was set by the operator.[]{data-label="Fig:ssDist"}](1DinstantPDFv2.pdf "fig:"){width="4.25cm" height="3.25cm"} ![Steady-state distribution of diffusion with stochastic resetting and instantaneous returns. a) Distribution of the position along the $x$-axis. Markers come from experiments and the dashed line is the theoretical prediction of Eq. (1). b) The radial position distribution. Markers come from experiments and the dashed line is the theoretical prediction $\rho(R)=\alpha_0^2RK_0(\alpha_0R)$ [@SM] with $K_n(z)$ standing for the modified Bessel function of the second kind [@Stegun]. In both panels no fitting procedure was applied: $D=0.18\pm 0.02 \mu m^2/s$ was measured independently and $r=0.05s^{-1}$ was set by the operator.[]{data-label="Fig:ssDist"}](1DinstantPDFradialv2.pdf "fig:"){width="4.25cm" height="3.25cm"}
A particle undergoing free Brownian motion is not bound in space. It has a Gaussian position distribution with a variance that grows linearly with time. Repeated resetting of the particle to its initial position will, however, result in effective confinement and in a non-Gaussian steady state distribution [@Evans2011_1; @Evans2011_2]. Estimating the steady state distribution of the particle’s position along the $x$-axis from recorded trajectories (see SI for details), we find (Fig. 2a) that the experimentally measured results conform with the theoretical result derived by Evans and Majumdar [@Evans2011_1; @Evans2011_2] $$\rho(x)=\frac{\alpha_0}{2}e^{-\alpha_0|x|}~,
\label{Eq:SS}$$ where $\alpha_0=\sqrt{r/D}$ is an inverse length scale corresponding to the typical distance diffused by the particle in the time between two resetting events, and $D$ is the diffusion constant. The steady state radial density of the particle can also be extracted from the experimental trajectories by looking at the steady-state distribution of the distance $R=\sqrt{x^2+y^2}$ from the origin. Here too, we find excellent agreement with the theoretical result (Fig. 2b).
*Stochastic resetting with non-instantaneous returns.—* We now turn our attention to more realistic pictures of diffusion with stochastic resetting. These have just recently been considered theoretically in attempt to account for the non-instantaneous returns and waiting times that are seen in all physical systems that include resetting [@Restart-Biophysics1; @Restart-Biophysics2; @Restart-Biophysics6; @HRS; @return1; @return2; @return3; @return4; @return5]. First, we consider a case where upon resetting HOTs are used to return the particle to the origin at a constant radial velocity $v=\sqrt{v_x^2+v_y^2}$ ([Fig. \[Fig:expt\]]{}). This case naturally arises for resetting by constant force in the over-damped limit. We find that the radial steady state density is then given by [@SM] (R)=p\_D\^[c.v.]{}\_(R)+(1-p\_D\^[c.v.]{})\_(R), \[radial-constant-velocity\] where $p_D^{c.v.}=\left(1+\frac{\pi r}{2 \alpha_0 v} \right)^{-1}$ is the steady-state probability to find the particle in the diffusive phase. $\rho_{\text{diff}}(R)=\alpha_0^2 R K_0(\alpha_0R)$ and $\rho_{\text{ret}}(R)=\frac{2\alpha_0^2}{\pi}RK_1(\alpha_0R)$ stand for the conditional probability densities of the particle’s position when in the diffusive and return phases respectively. Here $K_{n}(z)$ is once again the modified Bessel function of the second kind [@Stegun]. The result in [Eq. (\[radial-constant-velocity\])]{} is in very good agreement with experimental data as shown in [Fig. \[non-inst\]]{}a and Fig. S3.
![Steady-state distributions of diffusion with stochastic resetting and non-instantaneous returns. a) The radial position distribution, $\rho(R)$, as a function of the distance $R$ and the radial return velocity $v$ as given by [Eq. (\[radial-constant-velocity\])]{}. Experimental results of a realization with $v=0.8\mu m/s$ are superimposed on the theoretical prediction (black spheres). b) The radial position distribution as a function of $R$ and the return time $\tau_0$ as given by [Eq. (\[radial-constant-time\])]{}. Experimental results of a realization with $\tau_0=3.79 s$ are superimposed on the theoretical prediction (black spheres).[]{data-label="non-inst"}](3Dradialjoint1.pdf){width="8.5cm"}
Next, we consider a case where upon resetting HOTs are used to return the particle to the origin at a constant time $\tau_0$ — irrespective of the particle’s position at the resetting epoch. This case is appealing due to its simplicity and ease of experimental implementation. Here too, we find that the radial steady-state position distribution can be put in a closed form which reads [@SM] (R)=p\_D\^[c.t.]{}\_(R)+(1-p\_D\^[c.t.]{})\_(R), \[radial-constant-time\] where $ p_D^{c.t.}=(1+r \tau_0)^{-1}$ is the steady-state probability to find the particle in the diffusive phase, and with $\rho_{\text{diff}}(R)=\alpha_0^2 R K_0(\alpha_0R)$ and $\rho_{\text{ret}}(R)=\frac{\pi \alpha_0^2}{2}\left[ \frac{1}{\alpha _0}-R \left[K_0\left( \alpha _0 R \right) \pmb{L}_{-1}\left( \alpha _0 R \right)
+K_1\left( \alpha _0 R \right) \pmb{L}_0\left( \alpha _0 R \right)\right] \right]$, standing for the conditional probability densities of the particle’s radial position when in the diffusive and return phases respectively. Here, $\pmb{L}_n$ is the modified Struve function of order $n$ [@Stegun]. The result in [Eq. (\[radial-constant-velocity\])]{} is in very good agreement with experimental data as shown in [Fig. \[non-inst\]]{}b and Fig. S5.
Comparing the steady-state distributions for the constant time and constant velocity cases, we find that they are almost identical for short return times and high return speeds. Indeed, in these limits the two protocols are virtually indistinguishable as returns are effectively instantaneous. On the other extreme, i.e., for long return times and slow return speeds, marked differences are found between the distributions (Fig. S4 and S6).
*First Passage under stochastic resetting.—* Having realized diffusion with stochastic resetting and analyzed its stationary properties, we now turn to study how resetting affects the first-passage statistics of a Brownian particle. First-passage processes have numerous applications in natural sciences as they are used to describe anything from chemical reactions to single‐cell growth and division, and everything from transport dynamics to search and animal foraging [@Evans2011_3; @Restart-Biophysics1; @Restart-Biophysics2; @Restart-Biophysics6; @Restart-Biophysics3; @Restart-Biophysics8; @RednerBook; @Schehr-review; @ReuveniPRL; @PalReuveniPRL; @branching_II; @Restart-Search1; @Restart-Search2; @Chechkin; @Landau; @HRS; @Frinkes2010; @Branton2010; @Tu2013; @Bezrukov2000; @Grunwald2010; @Ghale2014; @Ma2013; @Iyer2014; @Ingraham1983; @Amir2014; @Osella2014; @cooper1991; @MetzlerBook]. To this end, it is known that while the mean first-passage time (MFPT) of a Brownian particle to a stationary target diverges [@RednerBook; @Schehr-review], resetting will render it finite [@Evans2011_1] even if the returns are non-instantaneous [@Restart-Biophysics1; @Restart-Biophysics2; @Restart-Biophysics6; @HRS; @return3; @return5].
To experimentally study first-passage under stochastic resetting, we consider the setup illustrated in [Fig. \[Fig:MFPT\]]{}a. In this set of experiments, similarly to the previous set, the experiment starts at time zero with the particle positioned at the origin. Resetting is conducted stochastically with rate $r$, and HOTs are used to return the particle to the origin at a constant return time $\tau_0$. However, we now also define a target, set to be a virtual infinite absorbing wall located at $x=L$, i.e., parallel to the $y$-axis. The particle is allowed to diffuse with stochastic resetting until it hits the target, and the hitting times (first-passage times) are recorded. Experiments were performed at 5 different resetting rates: $r=0.05s^{-1}$, $0.0667s^{-1}$, $0.125s^{-1}$, $0.5s^{-1}$, and $1s^{-1}$ with a constant return time of $\tau_0= 3.79s$. A typical trajectory extracted from such an experiment with $L = 1 \mu$m and $r=0.05s^{-1}$ is shown in Fig. \[Fig:MFPT\]b, Fig. S7, and Supplementary movie 2. We extract the FPTs from this and other trajectories from the duration of paths that start at the origin and end at the first crossing of the virtual wall (Fig. \[Fig:MFPT\]b). Several hundreds of resetting events were performed to gather enough FPT statistics (see SI for details).
To check agreement between data coming from FPT experiments and theory, we derived a formula for the mean FPT of diffusion with stochastic resetting and constant time returns to the origin. This is given by [@SM] T\_r =( +\_0 ) . \[Eq:MFPT\_return\] Equation \[Eq:MFPT\_return\] is in excellent agreement with the experimental data as shown in [Fig. \[Fig:MFPT\]]{}c, including accurate prediction of the optimal resetting rate which minimizes the mean FPT of the particle to the target.
*Work and energy.—*A central, and previously unexplored, aspect of stochastic resetting in physical systems concerns the energetic cost associated with the resetting process itself. As discussed above, stochastic resetting prevents a diffusing particle from spreading over the entire available space as it normally would. Instead, a localized, non-equilibrium, steady-state is formed; but the latter can only be maintained by working on the system continuously.
![Energetic cost of resetting. a) The radial distance from the origin vs. time for a particle diffusing with stochastic resetting at rate $r=0.05s^{-1}$ and constant radial return velocity $v=0.8\mu m/s$. b) The cumulative energy expenditure for the trajectory in panel a) (neglecting the cost of the wait period). c) The distribution of energy spent per resetting event. Red disks come from experiments and the theoretical prediction of [Eq. (\[Eq:Energy\])]{} is plotted as a solid blue line. d) Normalized energy spent per resetting event at constant power vs. the normalized radial return velocity as given by [Eq. (\[minmax\])]{}. The minimal energy is attained at a maximal velocity for which the trap is just barely strong enough to overcome the fluid drag force and prevent the particle from escaping the trap.[]{data-label="Fig:work"}](work_const_v.pdf)
In our experiments, work is done by the laser to capture the particle in an optical trap and drag it back to the origin. The total energy spent per resetting event is then simply given by $E=\mathcal{P}\tau(R)$, where $\mathcal{P}$ is the laser power fixed at 1W and $\tau(R)$ is the time required for the laser to trap the particle at a distance $R$ and bring it back to the origin. As the particle’s distance at the resetting epoch fluctuates randomly from one resetting event to another ([Fig. \[Fig:work\]]{}a), the energy spent per resetting event is also random ([Fig. \[Fig:work\]]{}b). To compute its distribution, we note that $E$ is proportional to the return time whose probability density function is in turn given by [@SM] (t)=\_[-]{}\^ d\_0\^ dR R \_0\^dt’ G\_0(R,t’)f(t’) , where $f(t)=re^{-rt}$ is the resetting time density and $G_0(R,t)=\frac{1}{4\pi Dt}e^{-R^2/4Dt}$ is the diffusion propagator in polar coordinates. For the case of constant radial return velocity, $v$, we have $\tau(R)=R/v$. A simple derivation then yields the probability density of the energy spent per resetting event [@SM] (E)= K\_0(E/E\_0) ,\[Eq:Energy\] with $E_0=\alpha_0^{-1}v^{-1}\mathcal{P}$; and note that this is a special case of the K-distribution [@Redding; @Long]. The mean energy spent per resetting event can be computed directly from [Eq. (\[Eq:Energy\])]{} and is given by $\langle E \rangle=\pi E_0/2$. Equation (\[Eq:Energy\]) demonstrates good agreement with experimental data ([Fig. \[Fig:work\]]{}c).
As $\langle E \rangle \propto v^{-1}$, the average energy spent per resetting event in our experiment can be made smaller by working at higher return velocities. Note, however, that the stiffness of the optical trap should be strong enough to oppose the drag force acting on the particle so as to keep it in the trap. Assuming that the maximum allowed displacement of a particle in the trap is $\approx0.5\mu m$ [@Roichman07], we find that working conditions must obey $k\ge2\gamma v$. As the stiffness is proportional to the laser power, $k=\mathcal{C} \mathcal{P}$ (where $\mathcal{C}$ is the conversion factor), the maximal working velocity is given by $v_{\text{max}}\approx \frac{1}{2} \mathcal{C} \mathcal{P}/\gamma$ which—independent of laser power— minimizes energy expenditure to $E_{\text{min}} \approx \pi \gamma \mathcal{C}^{-1} \alpha_0^{-1}$. Going to dimensionless variables we find E /E\_=v\_/v \[minmax\] for $v<v_{\text{max}}$ ([Fig. \[Fig:work\]]{}d).
*Discussion and future outlook.—*In this study, we have demonstrated a unique and versatile method to realize experimentally a resetting process in which many parameters can be easily controlled. We have used this technique to verify an array of theoretical predictions, and further motivate the derivation of new results which come to address novel consideration that arise from experimental realization of diffusion with stochastic resetting. Of prime importance in this regard is the energetic cost of resetting [@thermo1; @thermo2; @thermo3].
The optical trapping method used herein is far from being the most efficient way to apply force to a colloidal particle. In fact, in our experiments we used $1W$ of power at the laser output to create a trap of $k=30pN/\mu m$ for a silica bead of radius $a=0.75\mu m$. For experiments with a constant return velocity $v=0.8\mu m/s$ and resetting rate $r=0.05s^{-1}$, the average return time was $ \langle \tau(R) \rangle=3.68s$. This translates to an average energy expenditure of $\langle E\rangle=\mathcal{P}\langle \tau(R) \rangle=3.68\pm0.05J$ per resetting event. In contrast, the work done against friction to drag the particle at a constant velocity $v$ for a distance $R$ is given by $W_{\text{drag}}=\gamma v R$ where $\gamma=6\pi\eta a$ is the the Stokes drag coefficient. Taking averages, we find that the work required per resetting event is given by $\langle W_{\text{drag}} \rangle= \gamma v \langle R \rangle = \pi \alpha_0^{-1} \gamma v/2$, which translates into $3.4\cdot10^{-20}J$ or $8.3 k_BT$ per resetting event. We thus see that $\langle W_{\text{drag}} \rangle \ll \langle E \rangle$, i.e., that the work required to reset the particle’s position is orders of magnitude smaller than the actual amount of energy spent when resetting is done using HOTs.
Concluding, we note that experimental research of stochastic resetting is still in its infancy with many open questions left to be answered by consecutive studies. To this end, the setup described above along with its future extensions provide a promising platform.
**Acknowledgments** The authors acknowledge Gilad Pollack for his help in coding the resetting protocol of the HOTs. Arnab Pal acknowledges support from the Raymond and Beverly Sackler Post-Doctoral Scholarship at Tel-Aviv University; and Somrita Ray for many fruitful discussions. Amandeep Sekhon acknowledges support from the Ratner center for single molecule studies. Shlomi Reuveni acknowledges support from the Azrieli Foundation, from the Raymond and Beverly Sackler Center for Computational Molecular and Materials Science at Tel Aviv University, and from the Israel Science Foundation (grant No. 394/19). Yael Roichman acknowledges support from the Israel Science Foundation (grant No. 988/17).
[99]{}
Evans, M.R. and Majumdar, S.N., 2011. Diffusion with stochastic resetting. Physical review letters, 106(16), p.160601.
Evans, M.R. and Majumdar, S.N., 2011. Diffusion with optimal resetting. Journal of Physics A: Mathematical and Theoretical, 44(43), p.435001.
Evans, M.R., Majumdar, S.N. and Mallick, K., 2013. Optimal diffusive search: nonequilibrium resetting versus equilibrium dynamics. Journal of Physics A: Mathematical and Theoretical, 46(18), p.185001.
Reuveni, S., Urbakh, M. and Klafter, J., 2014. Role of substrate unbinding in Michaelis-Menten enzymatic reactions. Proceedings of the National Academy of Sciences, 111(12), pp.4391-4396.
Rotbart, T., Reuveni, S. and Urbakh, M., 2015. Michaelis-Menten reaction scheme as a unified approach towards the optimal restart problem. Physical Review E, 92(6), p.060101.
Robin, T., Reuveni, S. and Urbakh, M., 2018. Single-molecule theory of enzymatic inhibition. Nature communications, 9(1), p.779.
Roldan, E., Lisica, A., Sanchez-Taltavull, D. and Grill, S.W., 2016. Stochastic resetting in backtrack recovery by RNA polymerases. Physical Review E, 93(6), p.062411.
Budnar, S., Husain, K.B., Gomez, G.A., Naghibosadat, M., Varma, A., Verma, S., Hamilton, N.A., Morris, R.G. and Yap, A.S., 2019. Anillin promotes cell contractility by cyclic resetting of RhoA residence kinetics. Developmental cell, 49(6), pp.894-906.
Luby, M., Sinclair, A. and Zuckerman, D., 1993. Optimal speedup of Las Vegas algorithms. Information Processing Letters, 47(4), pp.173-180.
Gomes, C.P., Selman, B. and Kautz, H., 1998. Boosting combinatorial search through randomization. AAAI/IAAI, 98, pp.431-437.
Di Crescenzo, A., Giorno, V., Nobile, A.G. and Ricciardi, L.M., 2003. On the M/M/1 queue with catastrophes and its continuous approximation. Queueing Systems, 43(4), pp.329-347.
Kumar, B.K. and Arivudainambi, D., 2000. Transient solution of an M/M/1 queue with catastrophes. Computers & Mathematics with applications, 40(10-11), pp.1233-1240.
Evans, M.R., Majumdar, S.N. and Schehr, G., 2020. Stochastic resetting and applications. Journal of Physics A: Mathematical and Theoretical.
Evans, M.R. and Majumdar, S.N., 2014. Diffusion with resetting in arbitrary spatial dimension. Journal of Physics A: Mathematical and Theoretical, 47(28), p.285001.
Majumdar, S.N., Sabhapandit, S. and Schehr, G., 2015. Dynamical transition in the temporal relaxation of stochastic processes under resetting. Physical Review E, 91(5), p.052131.
Pal, A., 2015. Diffusion in a potential landscape with stochastic resetting. Physical Review E, 91(1), p.012113.
Pal, A., Kundu, A. and Evans, M.R., 2016. Diffusion under time-dependent resetting. Journal of Physics A: Mathematical and Theoretical, 49(22), p.225001.
Pal, A., Chatterjee, R., Reuveni, S. and Kundu, A., 2019. Local time of diffusion with stochastic resetting. Journal of Physics A: Mathematical and Theoretical, 52(26), p.264002.
Eule, S. and Metzger, J.J., 2016. Non-equilibrium steady states of stochastic processes with intermittent resetting. New Journal of Physics, 18(3), p.033006.
Evans, M.R. and Majumdar, S.N., 2018. Effects of refractory period on stochastic resetting. Journal of Physics A: Mathematical and Theoretical. 52 01LT01.
Bodrova, A.S., Chechkin, A.V. and Sokolov, I.M., 2019. Scaled Brownian motion with renewal resetting. Physical Review E, 100(1), p.012120.
Bodrova, A.S., Chechkin, A.V. and Sokolov, I.M., 2019. Nonrenewal resetting of scaled Brownian motion. Physical Review E, 100(1), p.012119.
Basu, U., Kundu, A. and Pal, A., 2019. Symmetric exclusion process under stochastic resetting. Physical Review E, 100(3), p.032136.
Pal, A., Kusmierz, L. and Reuveni, S., 2019. Time-dependent density of diffusion with stochastic resetting is invariant to return speed. Physical Review E, 100(4), p.040101.
Pal, A., Kusmierz, L. and Reuveni, S., 2019. Invariants of motion with stochastic resetting and space-time coupled returns. New Journal of Physics, 21(11), p.113024.
Bodrova, A.S. and Sokolov, I.M., 2019. Resetting processes with non-instanteneous return. arXiv preprint arXiv:1907.12326.
Maso-Puigdellosas, A., Campos, D. and Mendez, V., 2019. Transport properties of random walks under stochastic noninstantaneous resetting. Physical Review E, 100(4), p.042104.
Redner, S., 2007. A Guide to First-Passage Processes. A Guide to First-Passage Processes, by Sidney Redner, Cambridge, UK: Cambridge University Press, 2007.
Bray, A.J., Majumdar, S.N. and Schehr, G., 2013. Persistence and first-passage properties in nonequilibrium systems. Advances in Physics, 62(3), pp.225-361.
Ray, S., Mondal, D. and Reuveni, S., 2019. Peclet number governs transition to acceleratory restart in drift-diffusion. Journal of Physics A: Mathematical and Theoretical, 52(25), p.255002.
Pal, A. and Prasad, V.V., 2019. First passage under stochastic resetting in an interval. Physical Review E, 99(3), p.032123.
Kusmierz, L., Majumdar, S.N., Sabhapandit, S. and Schehr, G., 2014. First order transition for the optimal search time of Lévy flights with resetting. Physical review letters, 113(22), p.220602.
Kusmierz, L., & Gudowska-Nowak, E. (2015). Optimal first-arrival times in Lévy flights with resetting. Physical Review E, 92(5), 052127.
Reuveni, S., 2016. Optimal stochastic restart renders fluctuations in first passage times universal. Physical review letters, 116(17), p.170601.
Pal, A. and Reuveni, S., 2017. First Passage under Restart. Physical review letters, 118(3), p.030603.
Chechkin, A. and Sokolov, I.M., 2018. Random search with resetting: a unified renewal approach. Physical review letters, 121(5), p.050601.
Pal, A., Eliazar, I. and Reuveni, S., 2019. First passage under restart with branching. Physical review letters, 122(2), p.020602.
Pal, A. and Prasad, V.V., 2019. Landau-like expansion for phase transitions in stochastic resetting. Physical Review Research, 1(3), p.032001.
Pal, A., Kusmierz, L. and Reuveni, S., 2019. Home-range search provides advantage under high uncertainty. arXiv preprint arXiv:1906.06987.
Dufresne, E.R., Spalding, G.C., Dearing, M.T., Sheets, S.A. and Grier, D.G., 2001. Computer-generated holographic optical tweezer arrays. Review of Scientific Instruments, 72(3), pp.1810-1816.
Polin, M., Ladavac, K., Lee, S.H., Roichman, Y. and Grier, D.G., 2005. Optimized holographic optical traps. Optics Express, 13(15), pp.5831-5845.
Grier, D.G. and Roichman, Y., 2006. Holographic optical trapping. Applied optics, 45(5), pp.880-887.
Crocker, J.C. and Grier, D.G., 1996. Methods of digital video microscopy for colloidal studies. Journal of colloid and interface science, 179(1), pp.298-310.
See supplemental material for the detailed derivation.
Abramowitz, M. and Stegun, I.A. eds., 1948. Handbook of mathematical functions with formulas, graphs, and mathematical tables (Vol. 55). US Government printing office.
Firnkes, M., Pedone, D., Knezevic, J., Doblinger, M. and Rant, U., 2010. Electrically facilitated translocations of proteins through silicon nitride nanopores: conjoint and competitive action of diffusion, electrophoresis, and electroosmosis. Nano letters, 10(6), pp.2162-2167.
Branton, D., Deamer, D.W., Marziali, A., Bayley, H., Benner, S.A., Butler, T., Di Ventra, M., Garaj, S., Hibbs, A., Huang, X. and Jovanovich, S.B., 2010. The potential and challenges of nanopore sequencing. In Nanoscience and technology: A collection of reviews from Nature Journals (pp. 261-268).
Tu, L.C., Fu, G., Zilman, A. and Musser, S.M., 2013. Large cargo transport by nuclear pores: implications for the spatial organization of FG‐nucleoporins. The EMBO journal, 32(24), pp.3220-3230.
Bezrukov, S.M., Kullman, L. and Winterhalter, M., 2000. Probing sugar translocation through maltoporin at the single channel level. FEBS letters, 476(3), pp.224-228.
Ghale, G., Lanctot, A.G., Kreissl, H.T., Jacob, M.H., Weingart, H., Winterhalter, M. and Nau, W.M., 2014. Chemosensing ensembles for monitoring biomembrane transport in real time. Angewandte Chemie International Edition, 53(10), pp.2762-2765.
Grünwald, D. and Singer, R.H., 2010. In vivo imaging of labelled endogenous $\beta$-actin mRNA during nucleocytoplasmic transport. Nature, 467(7315), p.604.
Ma, J., Liu, Z., Michelotti, N., Pitchiaya, S., Veerapaneni, R., Androsavich, J.R., Walter, N.G. and Yang, W., 2013. High-resolution three-dimensional mapping of mRNA export through the nuclear pore. Nature communications, 4, p.2414.
Iyer-Biswas, S., Wright, C.S., Henry, J.T., Lo, K., Burov, S., Lin, Y., Crooks, G.E., Crosson, S., Dinner, A.R. and Scherer, N.F., 2014. Scaling laws governing stochastic growth and division of single bacterial cells. Proceedings of the National Academy of Sciences, 111(45), pp.15912-15917.
Ingraham, J.L., Maaloe, O. and Neidhardt, F.C., 1983. Growth of the bacterial cell. Sinauer Associates.
Amir, A., 2014. Cell size regulation in bacteria. Physical Review Letters, 112(20), p.208102.
Osella, M., Nugent, E. and Lagomarsino, M.C., 2014. Concerted control of Escherichia coli cell division. Proceedings of the National Academy of Sciences, 111(9), pp.3431-3435.
Cooper, S., 1991. Bacterial growth and division: biochemistry and regulation of the division cycle of prokaryotes and eukaryotes.
Metzler, R., Redner, S. and Oshanin, G., 2014. First-Passage Phenomena and Their Applications (Vol. 35). Singapore: World Scientific.
Redding, N.J., 1999. Estimating the parameters of the K distribution in the intensity domain (No. DSTO-TR-0839). ELECTRONICS RESEARCH LAB SALISBURY (AUSTRALIA).
Long, M.W., 1975. Radar reflectivity of land and sea. Lexington, Mass., DC Heath and Co., 1975. 390 p. Roichman, Y., Wong, V. and Grier, D., 2007. Colloidal transport through optical tweezer arrays. Phys. Rev. E., 75(1), p011407. Fuchs, J., Goldt, S. and Seifert, U., 2016. Stochastic thermodynamics of resetting. EPL (Europhysics Letters), 113(6), p.60009.
Pal, A. and Rahav, S., 2017. Integral fluctuation theorems for stochastic resetting systems. Physical Review E, 96(6), p.062135.
Gupta, D., Plata, C.A. and Pal, A., 2020. Work fluctuations and Jarzynski equality in stochastic resetting. Physical Review Letters, 124(11), p.110608.
| ArXiv |
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abstract: 'Synthetic spectra generated with the parameterized supernova synthetic–spectrum code SYNOW are compared to photospheric–phase spectra of Type Ib supernovae (SNe Ib). Although the synthetic spectra are based on many simplifying approximations, including spherical symmetry, they account well for the observed spectra. Our sample of SNe Ib obeys a tight relation between the velocity at the photosphere, as determined from the Fe II features, and the time relative to that of maximum light. From this we infer that the masses and the kinetic energies of the events in this sample were similar. After maximum light the minimum velocity at which the He I features form usually is higher than the velocity at the photosphere, but the minimum velocity of the ejected helium is at least as low as 7000 . Previously unpublished spectra of SN 2000H reveal the presence of hydrogen absorption features, and we conclude that hydrogen lines also were present in SNe 1999di and 1954A. Hydrogen appears to be present in SNe Ib in general, although in most events it becomes too weak to identify soon after maximum light. The hydrogen–line optical depths that we use to fit the spectra of SNe 2000H, 1999di, and 1954A are not high, so only a mild reduction in the hydrogen optical depths would be required to make these events look like typical SNe Ib. Similarly, the He I line optical depths are not very high, so a moderate reduction would make SNe Ib look like SNe Ic.'
author:
- '[David Branch]{}, [S. Benetti]{}, [Dan Kasen]{}, [E. Baron]{}, [David J. Jeffery]{}, [Kazuhito Hatano]{}, [R. A. Stathakis]{}, [Alexei V. Filippenko]{}, [Thomas Matheson]{}, [A. Pastorello]{}, [G. Altavilla]{}, [E. Cappellaro]{}, [L. Rizzi]{}, [M. Turatto]{}, [Weidong Li]{}, [Douglas C. Leonard]{}, and [Joseph C. Shields]{}'
title: Direct Analysis of Spectra of Type Ib Supernovae
---
INTRODUCTION
============
Supernovae of Type II are those that have obvious hydrogen lines in their optical spectra. Type IIb supernovae have obvious hydrogen lines around the time when they reach their maximum brightness (hereafter just “maximum light”) but later the hydrogen lines become weak or even disappear. Type Ib supernovae do not have obvious hydrogen lines but they do develop conspicuous He I lines after maximum light. Neither hydrogen nor He I lines are conspicuous in the spectra of Type Ic supernovae. Most or all events of these four types — II, IIb, Ib, and Ic — are thought to result from core collapse in massive stars. (Type Ia supernovae, whose spectra lack hydrogen and have a strong absorption feature produced by Si II $\lambda\lambda6347,6371$, are thought to have a fundamentally different origin, as thermonuclear disruptions of accreting or merging white dwarfs.) For a recent review of supernova spectral classification, including its historical development and with illustrations of spectra of each type, see Filippenko (1997).
In this paper we are concerned with optical photospheric–phase spectra of Type Ib supernovae (SNe Ib). Until recently, good photospheric–phase spectra had been published for only two SNe Ib: SN 1983N (Richtler & Sadler 1983; Harkness et al. 1987) and SN 1984L (Harkness et al. 1987). The situation has improved substantially now that Matheson et al. (2001) have published spectra of the SNe Ib that were observed at the Lick Observatory during the 1990s. These newly available spectra, together with some additional previously unpublished spectra that are presented in this paper, motivated us to carry out a comparative study of SN Ib spectra. (Matheson et al. also present spectra of SNe IIb and Ic, the study of which we defer to separate papers.)
Our method is to compare the observed SN Ib spectra with synthetic spectra that we generate with the fast, parameterized, supernova spectrum–synthesis code, SYNOW. We refer to this approach, in which the goal is to extract some constraints on the ejected matter from the observations in an empirical spirit, as “direct” analysis — to distinguish it from the process of making very detailed non–local–thermodynamic–equilibrium (non–LTE) calculations of synthetic spectra based on supernova hydrodynamical models (e.g., Baron et al. 1999). Issues that we can explore by means of our direct analysis include (1) line identifications; (2) the extent to which synthetic spectra calculated on the basis of simple assumptions can or cannot account for observed SN Ib spectra; (3) the degree to which the rather homogeneous appearance of SN Ib spectra, pointed out by Matheson et al. (2001), reflects a genuine physical homogeneity; (4) the velocities at which the He I lines form, compared to the velocity at the photosphere as determined by the Fe II lines; (5) whether hydrogen lines are present and, when they are, the velocities at which they form.
Previous work on the interpretation of photospheric–phase spectra of SNe Ib is briefly summarized in §2. The observed spectra that were selected for this project are discussed in §3 and those that have not been published previously are displayed. The synthetic spectrum calculations are described in §4, and comparisons with observed spectra are presented in §5. The results are summarized and discussed in §6.
PREVIOUS WORK
=============
The classic early paper on the interpretation of the photospheric–phase spectra of SNe Ib was that of Harkness et al. (1987), who calculated local thermondynamic equilibrium (LTE) synthetic spectra for parameterized model supernovae having power–law density structures and homogeneous chemical compositions, and compared them to observed spectra of SNe 1984L and 1983N. Some observed features that appear in spectra of all supernova types were readily attributed to Ca II and Fe II lines. Most of the remaining conspicuous features were convincingly attributed to lines of He I, even though large ad hoc overpopulations of the highly excited lower levels of the He I lines had to be invoked in order to account for their presence. The explanation for the overpopulations was later shown to be nonthermal excitation and ionization caused by the decay products of radioactive $^{56}$Ni and $^{56}$Co (Lucy 1991; Swartz et al. 1993).
A feature of special interest in the spectra of SNe 1984L and 1983N was an absorption near 6300 Å that could not be attributed to Fe II, Ca II, or He I lines. \[Harkness et al. (1987) referred to this as “the 6300 Å absorption” and so will we, although its wavelength can be as short as 6200 Å at early times.\] Harkness et al. tentatively attributed this absorption to C II $\lambda6580$, forming in outer high–velocity ($\ge$14,000 ) layers of the ejected matter. Later, on the basis of LTE synthetic spectra calculated for radially stratified chemical compositions, Wheeler et al. (1994) suggested the absorption to be H$\alpha$, forming at $\ge$13,000 . The issue of whether hydrogen is present in SNe Ib is very important because of its implications for the nature and appearance of the progenitor stars, but it has been a difficult issue to resolve because C II 6580, being less than 800 to the red of H$\alpha$, is usually a plausible alternative identification.
Woosley & Eastman (1997) presented a comparison of a non–LTE synthetic spectrum based on a particular explosion model with a photospheric–phase spectrum of SN 1984L. Overall, the synthetic spectrum accounted rather well for the major features in the observed spectrum. The explosion model did not contain any hydrogen, and in the synthetic spectrum the absorption nearest to the 6300 Å feature was produced by Si II, but as we will see below this cannot be the actual identification of the 6300 Å absorption. This is the only comparison of an non–LTE synthetic spectrum with an observed SN Ib photospheric–phase spectrum to be published so far.
No other SN Ib was well observed until SN 1999dn. Deng et al. (2000) used the same SYNOW code that we use in this paper to make a detailed study of line identifications in three spectra that were obtained at the Beijing Astronomical Observatory at times of about 10 days before, at, and 14 days after maximum light. In addition to Fe II, Ca II, and He I lines, Deng et al. explored the possible role of lines of other ions (C I, O I, C II, \[O II\], Na I, Mg II, Si II, Ca I, and Ni II) in shaping the spectra of SN 1999dn. They attributed the 6300 Å absorption in the latest of their three spectra of SN 1999dn to C II $\lambda6580$, forming at $\ge$10,000 , but they suggested that in the earlier two spectra the observed feature was more likely to be H$\alpha$, forming at higher velocity.
DATA
====
The 11 SNe Ib that were selected for this study are listed in Table 1. An asterisk preceding the recession velocity, $cz$, indicates that it is the value given by Matheson et al. (2001) for an H II region near the site of the supernova; otherwise the listed value is that of the parent galaxy, from the Asiago Supernova Catalog (Barbon et al. 1999; updates are available at [*http://merlino.pd.astro.it/supern/*]{}). All observed spectra displayed in this paper are corrected for redshift using the values of $cz$ listed in Table 1. An asterisk preceding the date of maximum light in the $V$ band, $t_{max}$, indicates that only the date of discovery is listed, because the date of maximum light is unknown.
Six of these events — SNe 1991ar, 1997dc, 1998T, 1998dt, 1999di, and 1999dn — were selected from Matheson et al. (2001) because they don’t have obvious hydrogen lines while they do have conspicuous He I lines. \[SN 1991D, which also may be a Type Ib but with fairly weak He I lines, is discussed in a separate paper (S. Benetti et al., in preparation).\] SNe 1998dt, 1999di, and 1999dn are especially useful for our study because on the basis of photometry obtained at the Lick Observatory, Matheson et al. were able to estimate the dates of maximum light in the $R$ band, which we will assume to peak at the same time as the $V$ band \[as was the case for the Type IIb SN 1996cb (Qiu et al. 1999)\]. The three spectra of SN 1999dn that appeared in Deng et al. (2000) also are included in this study.
The spectra of SN 1983N are from Richter & Sadler (1983) and Harkness et al. (1987), and the adopted date of maximum light, 1983 July 17, (in the [*IUE*]{} FES band, which is roughly like the $V$ band) is from an unpublished manuscript that was circulated by N. Panagia et al. in 1984. The spectra of SN 1984L are from Harkness et al. (1987). Tsvetkov (1987) estimated that SN 1984L reached maximum light in the $B$ band on 1984 August $20
\pm 4$ days. We assume that the $V$ band peaked two days later (as was the case for SN 1996cb) and adopt August 22 as the date of maximum light in the $V$ band.
SN 1954A is a special case because only photographic spectra, obtained by N. U. Mayall at the Lick Observatory and by R. Minkowski at the Mount Wilson and Palomar Observatories, are available. Microphotometer tracings of the spectra of SN 1954A and many other supernovae observed at the Lick Observatory and the Mount Wilson and Palomar Observatories between 1937 and 1971 have been digitized and displayed by Casebeer et al. (2000) and Blaylock et al. (2000). In the Asiago Catalog the date of maximum light of SN 1954A in the $B$ band is estimated as 1954 April 19, so we will adopt April 21 for the $V$ band.
One previously unpublished spectrum of SN 1996N is included in this study. The spectrum, obtained at the Anglo–Australian Telescope on 1996 March 23 (Germany et al. 2000), 11 days after discovery, is shown in Figure 1. The date of maximum light of SN 1996N is unknown. The spectrum appears to be that of a typical SN Ib not long after maximum light.
Six previously unpublished spectra[^1] of SN 2000H (Pastorello et al. 2000; Benetti et al. 2000) also are included. These are shown in Figure 2. The spectra of SN 2000H resemble those of a typical SN Ib except for an unusually deep 6300 Å absorption in the first four spectra, as well as a weak, narrow absorption near 4650 Å in at least the second and third spectra. Benetti et al. attributed these absorptions to H$\alpha$ and H$\beta$. (The H$\beta$ feature will be seen more clearly in subsequent figures.) This identification of hydrogen lines might raise the question of whether SN 2000H should be regarded as a Type IIb, but we do not favor such a classification because even at the earliest observed times the presence of hydrogen lines was not obvious, as evidenced by initial classifications of SN 2000H as a peculiar Type Ia (Garnavich et al. 2000) and a Type Ic (Pastorello et al. 2000). From unpublished ESO photometry of SN 2000H we estimate that the date of maximum light in the $B$ band was 2000 February $9\pm2$ days, so we adopt February 11 as the date of maximum light in the $V$ band.
CALCULATIONS
============
Calculations have been carried out with the fast, parameterized supernova spectrum–synthesis code, SYNOW. Recent applications to Type Ic supernovae and brief descriptions of SYNOW can be found in Millard et al. (1999) and Branch (2001), and technical details of the code are in Fisher (2000). An extensive discussion and illustration of the elements of supernova line formation appears in Jeffery & Branch (1990). The basic assumptions of SYNOW are spherical symmetry; velocity proportional to radius; a sharp photosphere; and line formation by resonant scattering, treated in the Sobolev approximation.
Various fitting parameters are available. The parameter $T_{bb}$ is the temperature of the blackbody continuum from the photosphere. The values used in this paper range from 8500 to 3600 K and typically are $\sim$6500 K around the time of maximum light and $\sim$5000 K beginning roughly two weeks after maximum. We do not attach much physical significance to these values because (for one thing) the observed spectra have not been corrected for interstellar extinction.
For each ion whose lines are introduced, the optical depth at the photosphere of a “reference line” is a fitting parameter, and the optical depths of the other lines of the ion are calculated assuming Boltzmann excitation at excitation temperature $T_{exc}$. In this paper, to reduce the number of free parameters, we simply fix $T_{exc}$ at a nominal SN Ib value of 7000 K. The relevant lines of a given ion don’t have widely differing excitation potentials so their relative optical depths don’t depend strongly on $T_{exc}$, within the range of temperatures that are relevant here.
The line optical depths are taken to vary with ejection velocity as $v^{-n}$. Again for simplicity, in this paper we always use $n = 8$, except for one illustration of the effects of using $n = 5$ instead. In the analysis of the SN 1999dn spectra by Deng et al. (2000), the line optical depths were taken to vary as $e^{-v/v_e}$, with $v_e = 1000$ . Since the exponential distribution has an effective power–law index of $n = v/v_e$, the distribution used by Deng et al. falls off more steeply than $n = 8$ for $v > 8000$ and less steeply for $v < 8000$ . This leads to some differences in the values of $v_{phot}$ and the reference–line optical depths used by Deng et al. and by us to match the same observed spectra.
The maximum velocity of the line–forming region is set high enough so that effectively there is no outer boundary. The default minimum velocity of the line–forming region is the velocity at the photosphere; when an ion is assigned a higher minimum velocity, that ion is said to be detached from the photosphere.
Reasons that SYNOW spectra cannot be expected to provide exact fits to observed spectra are numerous and obvious: the calculations are based on many simplifiying assumptions, including spherical symmetry, and the oscillator strengths (Kurucz 1993) are good but not perfect. In this paper we are not concerned with proposing a line identification for every weak observed feature. We are more interested in establishing the identities of the major features and then concentrating on a comparative analysis — to investigate the degree to which the SNe Ib of our sample are physically similar, and to look for differences.
COMPARISONS
===========
The Fiducial SN Ib Spectrum: SN1999dn, 17 Days After Maximum
------------------------------------------------------------
We begin the comparisons of observed and synthetic spectra by concentrating on a “fiducial” SN Ib spectrum — a spectrum of a typical SN Ib that has good signal–to–noise ratio and broad wavelength coverage, and in which most of the major spectral features are well developed. The best available spectrum for this purpose is the Matheson et al. (2001) spectrum of SN 1999dn obtained on 1999 September 17, 17 days after maximum light. In Figure 3, this spectrum is compared with a synthetic spectrum that has =6000 and =4800 K, and contains lines of Fe II, He I, O I, Ca II, Ti II, and Sc II. Almost all of the features in the observed spectrum can be attributed to these ions. The discrepancies will be discussed as we look at the contribution of each ion to the synthetic spectrum. As always when fitting observed spectra with SYNOW spectra, we are more concerned with discrepancies in the wavelengths of absorption features than with discrepancies in flux; the latter are inevitable given the simplicity of our spectrum calculations.
Figure 4 is like Figure 3 but with nothing but the Fe II lines in the synthetic spectrum. The optical depth of the reference line, 5018, is 7. The Fe II lines are mainly responsible for the spectral features from about 4300 to 5300 Å, and they have additional effects at shorter wavelengths. At this post–maximum time they may also be responsible for the observed absorptions near 6100 and 6300 Å. (These two features are not strong enough in this particular synthetic spectrum, but a higher value of $T_{exc}$ would increase their strengths relative to the reference line.) For this reason we don’t use H$\alpha$ or C II 6580 to account for the weak 6300 Åabsorption in this observed spectrum. Around maximum light, however, Fe II lines are not strong enough to account for the 6300 Åabsorption that is observed at that time.
In this paper we always determine the value of $v_{phot}$ on the basis of the Fe II features in the region from about 4300 to 5300 Å. Figure 5 shows a comparison of two Fe II synthetic spectra that have = 5000 and 10,000 . This figure shows that the spectral signature of Fe II in this wavelength range is quite sensitive to . Our fitting uncertainty in $v_{phot}$ is about 1000 .
The top panel of Figure 6 is like Figure 3 but with only the He I lines in the synthetic spectrum. The He I lines are detached at 8000 (recall that =6000 ), where the optical depth of the reference line, 5876, is 10. It is likely that two optical He I lines, 6678 and 7065, are almost entirely responsible for their corresponding observed features. Two other lines, 5876 and 4472, are mainly responsible for their corresponding observed features but they may be blended with the Na I D lines (5890, 5896) and Mg II 4481, respectively. In this spectrum He I 7281 accounts very nicely for an observed feature, but in some other spectra the fit is not so good. The remaining optical He I lines are weaker and in the synthetic spectrum of Figure 3 they are overwhelmed by lines of other ions. In Figures 3 and 6 the blue edge of the synthetic absorption produced by He I 10830 is not blue enough to account for the sharp drop in the observed spectrum near 9000 Å, but we show how to remedy this below.
In Figure 3 and the top panel of Figure 6 the He I lines are detached at 8000 in order to fit the wavelengths of the corresponding observed absorption features. The detachment causes the flat tops of the synthetic He I emission components (which are superimposed on a sloping continuum). The rounded emission peak that is observed near 5900 Å could easily be achieved in the synthetic spectrum by including undetached Na I D lines. To illustrate the necessity of detaching the He I lines, the bottom panel of Figure 6 shows how they appear when they are undetached, i.e., when they are allowed to form down to the photospheric velocity of 6000 . These synthetic absorptions obviously are insufficently blueshifted.
The top panel of Figure 7 is like Figure 3 but with only the O I lines. The optical depth of the reference line, 7773, is 1. The 7773 line accounts for at least most of an observed feature; in some of the other observed spectra this feature may be partly produced by Mg II 7890. The O I 9264 line may be responsible for a weak observed feature, while the 8446 feature usually is overwhelmed by the Ca II infrared triplet in SNe Ib. Whenever we use O I lines in the synthetic spectra of this paper, the optical depth of the reference line is near 1.
The bottom panel of Figure 7 is like Figure 3 but with only the Ca II lines. The optical depth of the reference line, 3933, is 300. Only the H&K lines (3933, 3968) and the infrared triplet (8542, 8662, 8498) produce observable features, both of which are very strong. The notch in the synthetic spectrum near 8400 Ånicely matches an observed feature. In this synthetic spectrum the Ca II lines are detached to 7000 to match the infrared triplet, but in most of our synthetic spectra the Ca II lines are undetached.
The top panel of Figure 8 is like Figure 3 but with only the Ti II lines. The optical depth of the reference line, 4550, is 1. The Ti II lines are used to help match the broad observed absorption trough between 4100 and 4500 Å. We consider the presence of Ti II lines in the observed spectrum to be probable but not definite. \[We do consider them to be definite in peculiar subluminous SNe Ia such as SN 1991bg (Filippenko et al. 1992) and SN 1999by (Garnavich et al. 2001), and in the Type Ic SN 1994I (Millard et al. 1999)\]. Whenever we use Ti II lines in this paper, the optical depth is near 1. None of our other conclusions would be affected by omitting the Ti II lines.
The bottom panel of Figure 8 is like Figure 3 but with only the Sc II lines. The optical depth of the reference line, 4247, is 0.5. The Sc II lines are used mainly to get a peak in the synthetic spectrum near 5500 Å. The price to be paid is an overly strong synthetic absorption produced by 4247. Sc II lines are plausibly present in SNe Ib because they are expected (in LTE) to appear at low temperatures (Hatano et al. 1999) and they appear in SNe II. Harkness et al. (1987) suggested that in SN 1984L the observed emission peak near 5470 Å was caused by the early emergence of blueshifted \[O I\] 5577 nebular–phase emission, but Swartz et al. (1993) found this to be unlikely for the Type Ic SN 1987M. For a discussion of the possibility of an early emergence of blueshifted \[O I\] 5577 emission in the Type IIb SN 1996cb, see Qiu et al. (1999) and Deng, Qiu, & Hu (2001). In our view the 5500 Å emission in the fiducial spectrum of SN 1999dn probably, but not definitely, is produced by Sc II lines. Whenever we use them, the optical depth of the reference line is near 1. None of our other conclusions would be affected by omitting the Sc II lines.
Figure 9 is like Figure 3 except that the power–law index $n$ has been reduced from 8 to 5 (and the optical depths of the reference lines have been correspondingly reduced to keep the synthetic features from becoming too strong). Now the blue wings of the synthetic absorptions produced by the Ca II infrared triplet, Ca II H&K, and He I 10830 fit better than in Figure 3, but the synthetic absorptions produced by 5876 and 6678 extend too far to the blue. This reflects the fact that in real supernovae, contrary to our assumption, the line optical depths do not all follow the same power law (or any power law). A slower radial decline of the optical depth of 10830 line, compared to the optical He I lines, is expected on the basis of the nonthermal–excitation calculations of Lucy (1991; his Figure 3) and Swartz et al. (1993; their Figure 11). Note that if the blue edge of the 10830 absorption really extends to 9000 Å, as it does in the synthetic spectrum of Figure 9, then the line is forming all the way out to 50,000 .
Matheson et al. (2001) demonstrated that the absorption produced by 6678 (the lower level of which is 1s2p $^1$P$^0$) becomes weaker with time relative to the absorptions produced by 5876 and 7065 (both 1s2p $^3$P$^0$); this also is to be expected on the basis of the results of Lucy and of Swartz et al., because the singlet resonance transitions to the ground state become less opaque as the ejecta density decreases through expansion.
As mentioned above, Deng et al. (2000) identified C II lines in their September 14 spectrum of SN 1999dn, obtained only three days before our fiducial spectrum of September 17. The identification of C II 6580 for the 6300 Å absorption was supported by attributing a weak absorption near 4580 Å to C II 4738, 4745. The reasons that we don’t introduce C II lines for the fiducial spectrum (apart from the fact that the 4580 Å absorption doesn’t appear distinctly in the fiducial spectrum) are that (1) as mentioned above, Fe II lines could be responsible for the 6300 Å absorption, and (2) the absorption near 4580 Å in the Beijing spectrum of September 14 might be produced by He I 4731 (see the top panel of Figure 6) and/or lines of Sc II (see the bottom panel of Figure 8).
SNe 2000H, 1999di, and 1954A: Hydrogen in SNe Ib
------------------------------------------------
Now we turn to SN 2000H, an event that has conspicuous He I lines but that according to Benetti et al. (2000) also has hydrogen lines. Figure 10 compares the $+5$ day spectrum of SN 2000H with a synthetic spectrum that has =8000 and =6500 K, and contains hydrogen lines in addition to the ions used above for the fiducial spectrum of SN 1999dn. The He I lines are detached at 9000 , where the optical depth of the reference line is 2. The hydrogen lines are detached at 13,000 where the optical depth of the reference line, H$\alpha$, is 2.5. With this detachment velocity, H$\alpha$ accounts for at least most of the 6300 Å absorption and H$\beta$ accounts for the unusual notch in the emission peak near 4650 Å. A closer view of the H$\beta$ region is provided in Figure 11. The presence of an absorption that is attributable to H$\beta$ provides strong support for the presence of hydrogen in SN 2000H.
Matheson et al. (2001) noted the presence of an unusually deep 6300 Å absorption in SN 1999di, and mentioned that it could be Si II 6355, C II 6580, or H$\alpha$. Si II can now be rejected because its absorption would be blueshifted by only 2600 , which is too much lower than than our value of =6000 . Figure 12 compares our earliest spectrum of SN 1999di, obtained 21 days after maximum, with the $+19$ day spectrum of SN 2000H. The similarity of these two spectra is remarkable (apart from differences at wavelengths longer than 9000 Å where both spectra are noisy). The narrow H$\beta$ absorption of SN 2000H also can be seen in SN 1999di. Figure 13 compares the $+21$ day spectrum of SN 1999di with a synthetic spectrum that has =7000 and =4500 K and contains the same ions as Figure 10 for SN 2000H. Hydrogen is detached at 12,000 . Figure 14 shows a closer view of the H$\beta$ region.
Could the 6300 Å absorption in SNe 2000H and 1999di be produced by C II 6580 rather than H$\alpha$? We think not. It would be surprising to see a deep C II 6580 absorption in SNe Ib, when even in SNe Ic this line never forms a deep absorption and, if present at all, is hard to identify. It also would be surprising that the C II feature would be so detached in SNe Ib. The top panel of Figure 15 is like Figure 13 except that only C II lines, detached at 13,000 , are used. Although 6580 can account for the 6300 Å absorption as well as H$\alpha$ does, and 7236, 7231 are not a problem because, being more detached than He I, they fall near or within the strong feature produced by He I 7065, the absorption produced by C II 4738, 4745 is much too strong (at least in LTE at 7000 K).
Could the 6300 Å absorption in SNe 2000H and 1999di be produced by Ne I $\lambda$6402 rather than H$\alpha$? No, it cannot. The bottom panel of Figure 15 shows that undetached Ne I fails in two ways: (1) the absorption produced by $\lambda$6402 is too far to the blue, and (2) even though the synthetic absorption produced by $\lambda$6402 has not been made strong enough, other unwanted features already are present.
In view of these difficulties with C II and Ne I, and the apparent presence of H$\beta$ in the observed spectra, we consider the identification of hydrogen lines in SNe 2000H and 1999di to be definite.
SN 1954A appears to have been spectroscopically akin to SNe 2000H and 1999di. McLaughlin (1963) and Branch (1972) identified He I lines in photographic spectra of SN 1954A obtained at the Lick Observatory and the Mount Wilson and Palomar Observatories, respectively. Consequently SN 1954A usually has been regarded to have been a Type Ib supernova, but on occasion doubts has been expressed about the classification because of the low quality of the photographic spectra compared to modern observations. Figure 16 compares microphotometer tracings of the two earliest spectra of SN 1954A (Blaylock et al. 2000), obtained on blue– and red–sensitive emulsions 46 days after maximum light, with the $+19$ day spectrum of SN 2000H. Only this one spectrum of SN 1954 was obtained on a red–sensitive emulsion. The SN 1954A spectra are not actually relative flux but merely a measure of the transmission through the photographic plate. Parts of the spectra were overexposed so the features are distorted; in particular, emission peaks tend to be suppressed. The shapes of these spectra also are strongly influenced by the wavelength dependences of the emulsion sensitivities; e.g., neither emulsion is sensitive around 5200 Åand the sensitivity falls off steeply between 6500 and 7000 Å. Nevertheless, some of the absorptions in SN 1954A can be located and they correspond well with the absorptions in SN 2000H, including the two attributed to H$\alpha$ and H$\beta$. Branch (1972) considered the possibility of H$\alpha$ and H$\beta$ absorptions in SN 1954A, blueshifted by 10,800 in the observer’s frame; the value of $cz$ for the parent galaxy is now known to be 291 , so the hydrogen lines are blueshifted by about 11,000 in the supernova frame. Branch also considered the possibility of Ne I lines in SN 1954A. This is perhaps still not ruled out, but given the apparent resemblance of SN 1954A to SNe 2000H, we prefer the hydrogen identification.
In terms of apparent magnitude, SN 1954A was the fourth brightest supernova of the twentieth century, surpassed only by the Type II SN 1987A and the Type Ia SNe 1972E and 1937C (Barbon et al. 1999). SN 1954A occurred in the star–bursting dwarf galaxy NGC 4214, at a distance of only about 4 Mpc (Leitherer et al. 1996), more than 10 times closer than SNe 2000H and 1999di, so it is more amenable to studies of the environment in which it exploded. We note that Van Dyk, Hamuy, & Filippenko (1996) found SN 1954A to be unusual among SNe Ib in that it was not near any visible H II region, and that a deep VLA search for radio emission at the site of SN 1954A carried out by J. Cowan and D. Branch in May, 1986, resulted in a three–sigma upper limit to the flux density at 20 cm of 0.068 mJy (Eck 1998), which corresponds to a monochromatic luminosity of less than one twentieth of Cas A.
Other Selected Comparisons
--------------------------
Figure 17 compares a spectrum of SN 1984L obtained 9 days after maximum with a synthetic spectrum that has =8000 and = 6500 K, and includes hydrogen lines detached at 15,000 . The Fe II lines fit very well, and the fit to the other features is satisfactory. Given the presence of H$\alpha$ in SNe 2000H, 1999di, and 1954A, we assume that the 6300 Å absorption is produced by H$\alpha$. However, because the H$\alpha$ optical depth at the detachment velocity is only 0.6, there is no support for this identification from H$\beta$ because it is too weak to see. (The oscillator strength of H$\beta$ is about one fifth that of H$\alpha$.) Figure 18 compares the $+9$ day spectrum of SN 1984L with the $+5$ day spectrum of SN 2000H. The spectra are similar except that in SN 1984L the 6300 Å absorption is weaker and the 4560 Å absorption is not visible. This shows that while our assumption that the 6300 Å absorption in SN 1984L is produced by H$\alpha$ is reasonable, it is not proven. It is conceivable that C II 6580 or (more plausibly because it wouldn’t need to be detached) Ne I $\lambda$6402 could be responsible for the 6300 Åabsorption in SN 1984L and other typical SNe Ib, and be overwhelmed by H$\alpha$ only in events such as SNe 2000H, 1999di, and 1954A. We proceed on the assumption that at early times the 6300 Å absorption is produced by H$\alpha$ in all the SNe Ib of our sample.
As an example of a comparison at a earlier time when is higher, Figure 19 compares the earliest spectrum for which good wavelength coverage is available, the Beijing spectrum of SN 1999dn 10 days before maximum, with a synthetic spectrum that has =14,000 and = 6500 K. Now hydrogen lines are detached at 18,000 , where the H$\alpha$ optical depth is 1.3. The fit is good, except near 6600 Å.
Figure 20 compares a spectrum of SN 1998dt obtained 32 days after maximum with a synthetic spectrum that has = 9000 and = 5000 K. This comparison is shown because, as will be seen below, although the spectrum has a typical SN Ib appearance the inferred value of is unusually high for a SN Ib this long after maximum. The fit to the Fe II features is unusually poor, but this value of does give the best fit in the 4300 to 5300 Å region, and a significantly lower value would give a noticeably worse fit. (The narrow H$\alpha$ emission from an H II region in the parent galaxy is very close to 6563 Å, which shows that the high required value of is not due to the observed spectrum having been inadvertently overcorrected for parent galaxy redshift.)
Now we briefly consider the events of our sample for which the times of maximum light are unknown. The spectrum of SN 1996N indicates that it was discovered not long after maximum. Because we use hydrogen lines in the synthetic spectrum for SN 1996N, the comparison with the observed spectrum is shown in Figure 21. Helium lines are undetached and the optical depth of the reference line is 5; hydrogen lines are detached at 17,000 where the optical depth of H$\alpha$ is 0.5. Overall, the fit is good.
SNe 1991ar and 1997dc do not appear to be unusual provided that they were discovered well after maximum light. As discussed by Matheson et al. (2001), the spectra of SN 1998T are seriously contaminated by light from the parent galaxy; taking this into consideration, SN 1998T does not appear to be unusual provided that it was discovered about a week after maximum. As far as we can tell, SNe 1991ar, 1997dc, and 1998T were typical SNe Ib, but they were not observed early enough to check on the presence of H$\alpha$.
RESULTS AND DISCUSSION
======================
The most important of the fitting parameters that have been used for the synthetic spectra are collected in Table 2. The spectra are listed in order of time with respect to maximum light so only the supernovae for which we have an estimate of the time of maximum light appear in the table. (The date 0703, for example, refers to July 3.)
For the six SNe Ib for which we have estimates of the time of maximum light, is plotted against time in Figure 22. The tightness of the relationship is striking. SN 1998dt at 32 days after maximum seems to stand out; otherwise, the scatter about the mean curve is about what should be expected from our nominal errors of 1000 in and a few days in the dates of maximum light. For the simple case of constant opacity and a $v^{-n}$ density distribution, the velocity at the photosphere would decrease with time as $v_{phot}
\propto t^{-2/(n-1)}$. The line in Figure 22, the best power–law fit to the data (excluding SN 1998dt at 32 days), corresponds to $n=3.6$. Considering that the opacity is not really constant, that the actual density distribution does not really follow a single power law over a wide velocity range, and that the best power–law index for fitting the spectra is not well constrained, not much significance should be attched to the difference between $n=3.6$ and $n=8$.
Our adopted values of can be used to make rough estimates of the mass and kinetic energy above the photosphere. For spherical symmetry and a $v^{-n}$ density distribution, the mass (in $M_\odot$) and the kinetic energy (in 10$^{51}$ ergs) above the electron–scattering optical depth $\tau_{es}$ are (Millard et al. 1999)
$$M=1.2 \times 10^{-4}\ v_4^2\ t_d^2\ \mu_e\ {{n-1}\over{n-3}}\ \tau_{es},
\eqno (1)$$
$$E=1.2 \times 10^{-4}\ v_4^4\ t_d^2\ \mu_e\ {{n-1}\over{n-5}}\ \tau_{es},
\eqno (2)$$
where $v_4$ is in units of 10,000 , $t_d$ is the time since explosion in days, $\mu_e$ is the mean molecular weight per free electron, and the integration is carried out to arbitrarily high velocity. If we assume that maximum light occurs 20 days after explosion, that $n=8$, that $\mu_e = 8$ (e.g., half–ionized helium or singly ionized oxygen), and that is at $\tau_{es}=1$, then at maximum light =10,000 (Figure 22) gives $M=0.5~M_\odot$ and $E=0.9 \times
10^{51}$ ergs. At 20 days after maximum, using =7000 and keeping the other parameters the same gives $M=1.1~M_\odot$ and $E=0.9
\times 10^{51}$ ergs. In reality, of course, the kinetic energy above the 7000 photosphere must be greater than that above the 10,000 photosphere. If we use $n=4$ instead of $n=8$ between 7000 and 10,000 then we obtain a total of $M=1.5~M_\odot$ and $E=1.4 \times 10^{51}$ ergs above the 7000 photosphere.
Figure 22 provides some constraints on models of SNe Ib. The hydrodynamics must account for the velocity at the photosphere as a function of time, and the ensemble of SN Ib progenitors must be consistent with the tightness of the relationship. The small scatter suggests that the masses and the kinetic energies of the SNe Ib of our sample are similar, and it does not leave much room for the influence on of departures from spherical symmetry.
Figure 23 shows the minimum velocities of the He I lines (squares when undetached and diamonds when detached). There appears to be a standard pattern (again with the possible exception of SN 1998dt at 32 days). Before and near the time of maximum the He I lines tend to be undetached. After maximum the lines tend to be detached, but the detachment velocities tend to decrease with time, from about 10,000 to 7000 . This means that the fraction of helium in this velocity range that is in the lower levels of the optical He I lines is increasing with time faster than $t^2$. (The matter density is decreasing as $t^{-3}$ by expansion but the Sobolev optical depth also is proportional to $t$ because it is inversely proportional to the velocity gradient.) The increasing fraction of helium in the excited levels may be understandable in terms of the decreasing column depth between the nickel core and the helium layers, and the decreasing detachment velocity may mean that the fractional helium abundance is lower at lower velocities. In any case, some helium is present at least down to 7000 . Our estimate above for the total mass above the 7000 photosphere was 1.1 to 1.5 M$_\odot$, which is a rough upper limit on the mass of helium above 7000 . There could be more helium below 7000 .
Figure 23 provides more constraints on models of SNe Ib. The radial profile of the helium abundance, together with that of the $^{56}$Ni that is responsible for exciting it, should account for the He I velocities and optical depths (Table 2).
Figure 23 also shows the minimum velocities of the hydrogen lines (circles), which always are detached. In the three events for which we are convinced of the hydrogen identifications — SNe 2000H, 1999dn, and 1954A — the minimum hydrogen velocity is between 11,000 and 13,000 . In the events in which we assume H$\alpha$ to be present at early times — SNe 1983N, 1984L, 1999dn, and 1996N (the latter is not shown in Figure 23 because the date of maximum light is unknown) — the detachment velocities tend to decrease with time but they are consistent with similar minimum velocities of the hydrogen. Thus the available evidence is consistent with the proposition that SNe Ib in general have hydrogen down to $11,000 - 13,000$ . \[For comparison, in the Type IIb SN 1993J the characteristic velocity of the ejected hydrogen was about 9000 (Patat et al. 1995; Utrobin 1996; Houck & Fransson 1996), and in the Type IIb SN 1996cb it was about 10,000 (Deng et al. 2001).\] A challenge for those who study the complicated evolution of massive stars in binary systems (e.g., Podsiadlowski, Joss, & Hsu 1992; Nomoto, Iwamoto, & Suzuki 1995; Wellstein, Langer, & Braun 2001) is to understand why many or perhaps even all stellar explosions that develop strong He I lines should eject at least a small amount of hydrogen.
The hydrogen mass that is required to give an H$\alpha$ optical depth of unity depends on the fraction of hydrogen that is in the Balmer level. In LTE, with an electron density of $10^9$ cm$^{-3}$, the Balmer fraction peaks around $10^{-9}$ near 6000 K. In this case we estimate that a hydrogen mass on the order of $10^{-2}~M_\odot$ would be required. Non–LTE calculations that take nonthermal excitation into account are needed for a more reliable estimate of the hydrogen mass.
The optical depths of the helium lines, and especially the hydrogen lines, are not very high, even though the corresponding absorption features are distinct and fairly deep. This reflects a simple geometric aspect of supernova line formation: as explained in Jeffery & Branch (1990), absorption features formed by detached lines are deeper than those formed by undetached lines. This point is illustrated in Figure 24, which shows that when hydrogen is detached from the photosphere by a factor of two and H$\alpha$ has $\tau=2$, its absorption feature is deeper than that of an undetached H$\alpha$ that has $\tau=10$. The H$\alpha$ optical depths in our synthetic spectra for SNe 2000H, 1999di, and 1954A are not high, so if they were only mildly lower these events would look like typical SNe Ib. Similarly, moderately lower He I line optical depths would transform a SN Ib into a SN Ic.
Figure 24 also illustrates how undetached hydrogen lines are more “obvious” than detached lines. First, undetached lines have conspicuous narrow, rounded emission peaks while detached lines have inconspicuous broad, flat peaks. Note also that although the H$\alpha$ absorption is deeper in the detached spectrum, the H$\beta$ absorption is deeper in the undetached spectrum. This is because in the undetached spectrum the optical depth at the photosphere of H$\beta$ is about 2 while in the detached spectrum the optical depth at the detachment velocity is only about 0.4. An optical depth as low as 0.4 can produce only a shallow absorption, even when the line is detached. For these reasons, supernovae that have undetached hydrogen lines have obvious hydrogen lines and are classified as Type II. Supernovae that have detached hydrogen lines are classified as Type Ib because the presence of hydrogen is not immediately obvious, even when the H$\alpha$ absorption is as deep as it is in SNe 2000H, 1999di, and 1954A. SNe IIb are those that have undetached hydrogen lines when they are first observed. In some cases, whether an event is classified as Ib or IIb may depend on how early the first spectrum is obtained.
The implication of the previous paragraphs is that the spectroscopic differences between SNe IIb, the SNe Ib that have deep H$\alpha$ absorptions, and typical SNe Ib may be caused mainly by mild differences in the hydrogen mass. For a given kinetic energy, the lower the hydrogen mass the higher the minimum velocity of the ejected hydrogen. Similarly, the spectroscopic differences between typical SNe Ic and SNe Ib could be caused mainly by moderate differences in the helium mass. For example, Matheson et al. (2001) found higher blueshifts of the O I $\lambda$7773 line in SNe Ic than in SNe Ib. For a given kinetic energy, the lower the helium mass the higher the minimum velocity of the ejected helium, and therefore the higher the velocity of the ejected oxygen. These suggestions are not original to this paper, but they are strengthened by our finding that the H I and He I optical depths in SNe Ib are not very high. These suggestions also are not inconsistent with arguments, based on light curves, for the existence of different physical classes of hydrogen–poor events that cut across the conventional spectroscopic types (e.g., Clocchiatti & Wheeler 1997).
The number of SNe Ib for which good spectral coverage is available is still relatively small. More events should be observed to explore the degree of the spectral homogeneity and to find out whether there is a continuum of hydrogen line strengths. Also needed are detailed non–LTE spectrum calculations for supernova models having radially stratified compositions, — to determine the the hydrogen and helium masses and the distribution of the $^{56}$Ni that is required to excite the helium. The possibility that nonthermally excited Ne I can produce spectral features strong enough to be seen needs to be investigated. Detailed non–LTE calculations for parameterized SN Ib models, using the PHOENIX code (e.g., Baron et al. 1999), are underway.
This material is based upon work supported by the National Science Foundation under Grants No. AST–9986965 and AST–9731450 at Oklahoma and AST–9987438 at Berkeley. A.V.F. is grateful to the Guggenheim Foundation for a Fellowship.
Barbon, R., Buondi, V., Cappellaro, E., & Turatto, M. 1999, A&AS, 139, 531
Baron, E., Branch, D., Hauschildt, P. H., Filippenko, A. V., & Kirshner, R. P. 1999, ApJ, 527, 739
Benetti, S., Cappellaro, E., Turatto, M., & Pastorello, A. 2000, IAU Circ. 7375
Blaylock, M., Branch, D., Casebeer, D., Millard, J., Baron, E., Richardson, D., & Ancheta, C. 2000, PASP, 112, 1439
Branch, D. 1972, A&A, 16, 247
Branch, D. 2001, in Supernovae and Gamma–Ray Bursts: The Largest Explosions in the Universe, ed. M. Livio (Cambridge: Cambridge University Press), in press
Casebeer, D., Branch, D., Blaylock, M., Millard, J., Baron, E., Richardson, D., & Ancheta, C. 2000, PASP, 112, 1433
Clocchiatti, A. & Wheeler, J. C. 1997, in Thermonuclear Supernovae, ed. P. Ruiz–Lapuente, R. Canal, & J. Isern (Dordrecht: Kluwer), p. 863
Deng, J. S., Qiu, Y. L., Hu, J. Y., Hatano K., & Branch D. 2000, ApJ, 540, 452
Deng, J., Qiu, Y., & Hu, J. 2001, preprint
Eck, C. R. 1998, PhD Thesis, University of Oklahoma
Filippenko, A. V. 1997, ARAA, 35, 309
Filippenko, A. V., et al., 1992, AJ, 104, 1543
Fisher, A. 2000, PhD Thesis, University of Oklahoma
Garnavich, P. M., et al., 2001, ApJ, submitted
Garnavich, P., Challis, P., Jha, S., & Kirshner, R. 2000, IAU Circ. 7366
Germany, L., Schmidt, B., Stathakis, R., & Johnston, H. 2000, IAU Circ. 6351
Harkness, R. P., et al., 1987, ApJ, 317, 355
Hatano, K., Branch, D., Fisher, A., Millard, J., & Baron, E. 1999, ApJS, 121, 233
Houck, J. C., & Fransson, C. 1996, ApJ, 456, 811
Jeffery, D. J., & Branch, D. 1990, in Supernovae, ed. J. C. Wheeler, T. Piran, & S. Weinberg (Singapore: World Scientific), p. 149
Kurucz, R. L. 1993, CDROM No. 1: Atomic Data for Opacity Calculations, Cambridge, Smithsonian Astrophysical Observatory
Leitherer, C., Vacca, W. D., Conti, P. S., Filippenko, A. V., Robert, C., & Sargent, W. L. W. 1996, ApJ, 465, 717
Lucy, L. B. 1991, ApJ, 383, 308
Matheson, T., Filippenko, A. V., Li, W., Leonard, D. C., & Shields, J. C. 2001, AJ, 121, 1648
McLaughlin, D. B. 1963, PASP, 75, 133
Millard, J., et al., 1999, ApJ, 527, 746
Nomoto, K., Iwamoto, K., & Suzuki, T. 1995, Phys. Rep., 256. 173
Pastorello, A., Altavilla, G., Cappellaro, E., & Turatto, M. 2000, IAU Circ. 7367
Patat, F., Chugai, N., & Mazzali, P. A. 1995, A&A, 299, 715
Podsiadlowski, Ph., Joss, P. C., & Hsu, J. J. L. 1992, ApJ, 391, 246
Qiu, Y., Li, W., Qiao, Q., & Hu, J. 1999, AJ, 117, 736
Richter, T., & Sadler, E. M. 1983, A&A, 128, L3
Swartz, D. A., Filippenko, A. V., Nomoto, K., & Wheeler, J. C. 1993, ApJ, 411, 313
Tsvetkov, D. Yu. 1987, Sov. Astron. Lett., 13, 376
Utrobin, V. P. 1996, A&A, 306, 219
Van Dyk, S., Hamuy, M., & Filippenko, A. V. 1996, AJ, 111, 2017
Wellstein, S., Langer, N., & Braun, H. 2001, A&A, 369, 939
Wheeler, J. C., Harkness, R. P., Clocchiati, A., Benetti, S., Brotherton, M. S., DePoy, D. L., & Elias, J. 1994, ApJ, 436, L135
Woosley, S. E., & Eastman, R. G. 1997, in Thermonuclear Supernovae, ed. P. Ruiz–Lapuente, R. Canal, & J. Isern (Dordrecht: Kluwer), p. 821
[lrrcrr]{}
1954A & NGC 4214 & 291 & April 21\
1983N & NGC 5236 & 513 & July 17\
1984L & NGC 991 & 1534 & August 22\
1991ar & IC 49 & \*4520 & \*September 2\
1996N & NGC 1398 & 1491 & \*March 12\
1997dc & NGC 7678 & 3480 & \*August 5\
1998T & NGC 3690 & \*3080 & \*March 3\
1998dt & NGC 945 & \*4580 & September 12\
1999di & NGC 776 & \*4920 & July 27\
1999dn & NGC 7714 & 2700 & August 31\
2000H & IC 454 & 3894 & February 11\
[lrrcrrrr]{}
1983N & 0703 & -14 & 17000 &&&&\
1983N &0706 &-11 & 13000 &&&&\
1999dn& 0821& -10 & 14000 & 2.0 & 14000 & 1.5& 18000\
1983N & 0713 & -4 & 11000 &&&&\
1983N & 0717 & 0 & 11000 & 2.5& 11000 & 0.8 & 15000\
2000H & 0211 & 0 & 11000 & 1 & 11000 & 6.0 & 13000\
1999dn &0831 & 0 & 10000 & 5.0 &11000 & 1.0 & 14000\
1983N & 0719 & 2 & 10000 & 4.0 & 10000& 0.7 & 14000\
2000H & 0216 & 5 & 8000 & 2.0 & 9000 & 2.5 & 13000\
1984L& 0830 & 8 & 9000 & 2.0 & 10000 & 0.6 & 15000\
1998dt& 0920 & 8 & 9000 & 4.0 &11000 &&\
1984L & 0831& 9 & 8000 & 1.5 & 10000 & 0.6& 15000\
1983N & 0727 & 10 & 7000 & 7.0 & 8000 & 0.6 & 12000\
1999dn & 0910 & 10 & 7000 & 3.0 & 8000 &&\
1984L & 0903 & 12 & 7000 & 1.0& 9000 & 0.5& 14000\
1999dn& 0914 &14 & 6000 & 10 & 7000 &&\
1999dn &0917 &17 &6000 & 10 & 8000 &&\
2000H &0302 &19 &6000 & 5.0& 8000& 2.0& 13000\
1999di &0817 &21 &7000 & 10 &8500 & 4.0 & 12000\
1984L &0919 &28 &5000 &&&&\
2000H &0313 &30 &5000 & 3.0& 7000& 1.5 &13000\
1984L &0923 &32 &5000 & 5.0& 7000 &&\
1998dt &1015 & 33 &9000 & 10 & 9000 &&\
1984L &0928 &37 &5000 & 10 & 6000 &&\
1999dn &1008 &38 & 6000 & 10 & 7000&&\
1999di &0910 & 45& 6000 & 10 & 7000 & 2.0 & 12000\
2000H &0330 &47 &5000 & 1.0& 7000 & 0.5& 12000\
1999di &0917 &52 &6000 & 10 & 7000& 1.0& 12000\
2000H &0408 &56 &4000 &&&&\
1984L &1018 &57 &4000 &&&&\
[^1]: These spectra are partially based on observations collected at the European Southern Observatory, Chile, ESO N$^0$65.H–0292, and at the Asiago Observatory
| ArXiv |
---
author:
- |
Alain Chenciner & Hugo Jiménez-Pérez\
Observatoire de Paris, IMCCE (UMR 8028), ASD\
77, avenue Denfert-Rochereau, 75014 Paris, France\
`[email protected], [email protected]`
title: 'Angular momentum and Horn’s problem'
---
We prove a conjecture made in [@C1]: given an $n$-body central configuration $X_0$ in the euclidean space $E$ of dimension $2p$, let $Im{\cal F}$ be the set of ordered real $p$-tuples $\{\nu_1,\nu_2,\cdots,\nu_p\}$ such that $\{\pm i\nu_1,\pm i\nu_2,\cdots,\pm i\nu_p\}$ is the spectrum of the angular momentum of some (periodic) relative equilibrium motion of $X_0$ in $E$. Then $Im {\cal F}$ is a convex polytope. The proof consists in showing that there exist two, generically $(p-1)$-dimensional, convex polytopes ${\cal P}_1$ and ${\cal P}_2$ in ${\ensuremath{\mathbb{R}}}^{p}$ such that ${\cal P}_1\subset Im{\cal F}\subset {\cal P}_2$ and that these two polytopes coincide.
${\cal P}_1$, introduced in [@C1], is the set of spectra corresponding to the hermitian structures $J$ on $E$ which are “adapted" to the symmetries of the inertia matrix $S_0$; it is associated with Horn’s problem for the sum of $p\times p$ real symmetric matrices with spectra $\sigma_-$ and $\sigma_+$ whose union is the spectrum of $S_0$.
${\cal P}_2$ is the orthogonal projection onto the set of “hermitian spectra" of the polytope ${\cal P}$ associated with Horn’s problem for the sum of $2p\times 2p$ real symmetric matrices having each the same spectrum as $S_0$.
The equality ${\cal P}_1={\cal P}_2$ follows directly from a deep combinatorial lemma, proved in [@FFLP], which characterizes those of the sums $C=A+B$ of two $2p\times 2p$ real symmetric matrices $A$ and $B$ with the same spectrum, which are hermitian for some hermitian structure.
Origin of the problem: $N$-body relative equilibria and their angular momenta
=============================================================================
We recall here the results of [@AC; @C1; @C2] which are needed in order to understand the mechanical origin of the purely algebraic conjecture solved in the present paper: given a configuration $x_0=(\vec r_1,\cdots,\vec r_N)\in E^N$ of $N$ punctual positive masses in the euclidean space $E$, a [*rigid motion*]{} of the configuration under Newton’s attraction is a motion in which the mutual distances $||\vec r_i-\vec r_j||$ between the bodies stay constant. It is proved in [@AC] (see also [@C2]) that such a motion is necessarily a [*relative equilibrium*]{}. This implies that the motion takes place in a space of even dimension $2p$, which can be supposed to coincide with $E$, and that, in a galilean frame fixing the center of mass at the origin, it is of the form $x(t)=(e^{\Omega t}\vec r_1,e^{\Omega t}\vec r_2,\cdots,e^{\Omega t}\vec r_N)$, where $\Omega$ is a $2p\times 2p$-antisymmetric endomorphism of the euclidean space $E$ which is non degenerate. Choosing an orthonormal basis where $\Omega$ is normalized, this amounts to saying that there exists a hermitian structure on the space $E$ of motion and an orthogonal decomposition $E\equiv{\ensuremath{\mathbb{C}}}^p={\ensuremath{\mathbb{C}}}^{k_1}\times\cdots\times{\ensuremath{\mathbb{C}}}^{k_r}$ such that $$x(t)=(x_1(t),\cdots,x_r(t))=(e^{i\omega_1t}x_1,\cdots,e^{i\omega_rt}x_r),$$ where $x_m$ is the orthogonal projection on ${\ensuremath{\mathbb{C}}}^{k_m}$ of the $N$-body configuration $x$ and the action of $e^{i\omega_mt}$ on $x_m$ is the diagonal action on each body of the projected configuration. Such quasi-periodic motions exist only for very special configurations, called [*balanced configurations*]{} (see [@AC; @C2] for their characterization). The most degenerate balanced configurations are the [*central configurations*]{} for which all the frequencies $\omega_i$ are the same; this means that $\Omega=\omega J$, with $J$ a hermitian structure on $E$, and the motion is $$x(t)=(\vec r_1(t),\cdots,\vec r_N(t))=e^{i\omega t}x_0=(e^{i\omega t}\vec r_1,\cdots,e^{i\omega t}\vec r_N)$$ in the hermitian space $E\equiv{\ensuremath{\mathbb{C}}}^{p}$; in particular, it is periodic. In a space of dimension at most 3, $E$ is necessarily of dimension 2 and the configuration of any relative equilibrium is central.
Given a configuration $x=(\vec r_1,\cdots,\vec r_N)$ and a configuration of velocities $y=\dot x=(\vec v_1,\cdots, \vec v_N)$, both with center of mass at the origin: $\sum_{k=1}^Nm_k\vec r_k=\sum_{k=1}^Nm_k\vec v_k=0$, the [*angular momentum*]{} of $(x,y)$ is the bivector ${\mathcal C}=\sum_{k=1}^Nm_k\vec r_k\wedge\vec v_k$. If we represent $x$ and $y$ by the $2p\times N$ matrices $X$ and $Y$ whose $i$th column are respectively made of the components $(r_{1i},\cdots,r_{2pi})$ and $(v_{1i},\cdots,v_{2pi})$ of $\vec r_i$ and $\vec v_i$ in an orthonormal basis of $E$ and if $M=\hbox{diag}(m_1,\cdots,m_N)$, this bivector is represented by the antisymmetric matrix [*(we use the french convention $^{t\!}X$ for the transposed of $X$)*]{} $$C=-XM^{t\!}Y+YM^{t\!\!}X\;\;\hbox{with coefficients}\;\; c_{ij}
=\sum_{k=1}^Nm_k(-r_{ik}v_{jk}+r_{jk}v_{ik}).$$ The dynamics of a solid body is determined by its [*inertia tensor*]{} (with respect to its center of mass), represented in the case of a point masses configuration $X$ by the symmetric matrix $$S=XM^{t\!\!}X\;\;\hbox{with coefficients}\;\;
s_{ij}=\sum_{k=1}^Nm_kr_{ik}r_{jk},$$ whose trace is the [*moment of inertia of the configuration $x$ with respect to its center of mass*]{}. In particular, the angular momentum of a relative equilibrium solution $X(t)=e^{t\Omega}X_0$ is represented by the antisymmetric matrix $C=S_0\Omega+\Omega S_0$, where $S_0=X_0M^{t\!\!}X_0$. Restricting to the case of central configurations, that is $\Omega=\omega J$, and making $\omega=1$, we consider in what follows the spectrum of $J$-skew-hermitian matrices of the form $S_0J+JS_0$ or, what amounts to the same, the spectrum of $J$-hermitian matrices[^1] of the form $J^{-1}S_0J+S_0$.
[*In the following, we identify $E$ with ${\ensuremath{\mathbb{R}}}^{2p}$ by the choice of some orthonormal basis. ${\ensuremath{\mathbb{R}}}^{2p}$ is endowed with its canonical basis $e_i=(0,\cdots,1,\cdots,0)$ and its canonical euclidean scalar product $x\cdot y=\sum_{i=1}^{2p}x_iy_i$; this allows identifying linear endomorphisms of $E={\ensuremath{\mathbb{R}}}^{2p}$ and $2p\times 2p$ matrices with real coefficients. When we say that $J$ is a hermitian structure, we mean that the standard euclidean structure is given and that $J$ is a complex structure which is orthogonal.*]{}
The frequency map
=================
We recall the definition, given in [@C1], of the [*frequency map*]{} ${\cal F}$ from the set of hermitian structures on ${\ensuremath{\mathbb{R}}}^{2p}$ to the positive Weyl chamber $W_p^+\subset {\ensuremath{\mathbb{R}}}^p$: given some $2p\times 2p$ real symmetric matrix $S_0$, we consider the mapping $J\mapsto J^{-1}S_0J+S_0$ from the space of hermitian structures on $E$ to the set of $2p\times 2p$ real symmetric matrices. We are only interested in the spectra of these matrices, hence choosing an orientation for $J$ is harmless and we shall consider only those of the form $J=R^{-1}J_0R$, where $J_0$ is the “standard" structure defined by $J_0=\begin{pmatrix}0&-Id\\ Id&0\end{pmatrix}$ and $R\in SO(2p)$. Two elements $R'$ and $R''$ of $SO(2p)$ defining the same $J$ if and only if there exists an element $U\in U(p)$ such that $R''=UR'$, it follows that the space of (oriented) hermitian structures is identified to the homogeneous space $U(p)\backslash SO(2p)$. The symmetric matrix $J^{-1}S_0J+S_0$ is $J$-hermitian, that is, it commutes with $J$. This implies that its spectrum is real, of the form $\{\nu_1,\nu_2,\cdots,\nu_p\}$ if considered as a $p\times p$ complex matrix (for the identification of ${\ensuremath{\mathbb{R}}}^{2p}$ to ${\ensuremath{\mathbb{C}}}^p$ defined by $J$) and of the form $\{(\nu_1, \nu_2, \cdots, \nu_p), (\nu_1, \nu_2, \cdots, \nu_p)\}$ if considered as a $2p\times 2p$ real matrix (see the next section for the trivial proof).
The [*frequency mapping*]{} $${\mathcal F}:U(p)\backslash SO(2p)\to W_p^+=\{(\nu_1,\cdots\nu_p)\in{\ensuremath{\mathbb{R}}}^p, \nu_1\ge\cdots\ge\nu_p\}$$ associates to each hermitian structure $J$ the ordered spectrum $(\nu_1,\cdots,\nu_p)$ of the $J$-hermitian matrix $J^{-1}S_0J+S_0$.
Hermitian spectra
=================
Let $C:{\ensuremath{\mathbb{R}}}^{2p}\to{\ensuremath{\mathbb{R}}}^{2p}$ be a symmetric endomorphism. The following assertions are equivalent:
1\) There exists a hermitian structure $J=R^{-1}J_0R$ such that $C$ be $J$-hermitian (i.e. $JC=CJ$);
2\) The spectrum $\sigma(C)$ of $C$ is of the form $$\sigma(C)=\{(\nu_1, \nu_2, \cdots, \nu_p), (\nu_1, \nu_2, \cdots, \nu_p)\}.$$
Let $J=R^{-1}J_0R$; the mapping $C$ is $J$-hermitian if and only if $RCR^{-1}$ is $J_0$-hermitian. This is equivalent to the existence of an isomorphism $U\in U(p)\subset SO(2p)$ such that $$U^{-1}RCR^{-1}U=\hbox{diag}\bigl((\nu_1,\cdots,\nu_p),(\nu_1,\cdots,\nu_p)\bigr).$$ Conversely, the identity $$R^{-1}CR=\hbox{diag}\bigl((\nu_1,\cdots,\nu_p),(\nu_1,\cdots,\nu_p)\bigr)$$ implies the commutation of $R^{-1}CR$ with $J_0$ and hence the commutation of $C$ with $J=R^{-1}J_0R$.
[**Notations.**]{} We shall call [*hermitian*]{} the spectra of this form and [*the diagonal*]{} the linear subspace $\Delta$ of $W_{2p}^+$ defined by $$\Delta=\{(\mu_1\ge\mu_2\ge\cdots\ge\mu_{2p}),\; \mu_1=\mu_2,\, \mu_3=\mu_4,\, \, \cdots,\mu_{2p-1}=\mu_{2p}\}.$$ Hence the ordered hermitian spectra are the ones belonging to $\Delta$.
Two convex polytopes
====================
Let $S_0:{\ensuremath{\mathbb{R}}}^{2p}\to{\ensuremath{\mathbb{R}}}^{2p}$ be a symmetric endomorphism with spectrum $$\sigma(S_0)=\{\sigma_1\ge\sigma_2\ge\cdots\ge\sigma_{2p}\}.$$ To $S_0$ we associate the subsets ${\cal P}_1$ and ${\cal P}_2$ of ${\ensuremath{\mathbb{R}}}^p$ (in fact of the positive Weyl chamber $W_p^+$ of ordered real $p$-tuples), defined as follows:
1\) ${\cal P}_1$ is the set of ordered spectra $$\sigma(c)=\{\nu_1\ge\nu_2\ge\cdots\ge\nu_p\}$$ of symmetric endomorphisms $c$ of ${\ensuremath{\mathbb{R}}}^p$ of the form $c=a+b$, where $a:{\ensuremath{\mathbb{R}}}^p\to{\ensuremath{\mathbb{R}}}^p$ and $b:{\ensuremath{\mathbb{R}}}^p\to{\ensuremath{\mathbb{R}}}^p$ are arbitrary symmetric endomorphisms with respective spectra $$\sigma(a)=\sigma_-:= \{\sigma_1,\sigma_3,\cdots,\sigma_{2p-1}\},\quad \sigma(b)=\sigma_+:=\{\sigma_2,\sigma_4,\cdots,\sigma_{2p}\} ;$$
2\) ${\cal P}_2$ is the set of $p$-tuples $\{\nu_1\ge\nu_2\ge\cdots\ge\nu_p\}$ such that $$\{(\nu_1,\nu_2\cdots,\nu_{p}),(\nu_1,\nu_2\cdots,\nu_{p})\}$$ is the spectrum of some symmetric endomorphism $C$ of ${\ensuremath{\mathbb{R}}}^{2p}$ of the form $C=A+B$, where $A:{\ensuremath{\mathbb{R}}}^{2p}\to{\ensuremath{\mathbb{R}}}^{2p}$ and $B:{\ensuremath{\mathbb{R}}}^{2p}\to{\ensuremath{\mathbb{R}}}^{2p}$ are arbitrary symmetric endomorphisms with the same spectrum $$\sigma(A)=\sigma(B)=\sigma(S_0).$$ In other words, identifying canonically the diagonal $\Delta$ with ${\ensuremath{\mathbb{R}}}^p$, one can write $${\cal P}_2={\cal P}\cap\Delta,$$ where ${\cal P}$ is the $(2p-1)$-dimensional Horn polytope which describes the ordered spectra of sums $C=A+B$ of $2p\times 2p$ real symmetric matrices $A,B$ with the same spectrum as $S_0$.
${\cal P}_1$ and ${\cal P}_2$ are both contained in the hyperplane of ${\ensuremath{\mathbb{R}}}^p$ with equation $$\sum_{i=1}^p\nu_i=\sum_{j=1}^{2p}\sigma_j.$$ Moreover, ${\cal P}_1$ and ${\cal P}_2$ are both $(p-1)$-dimensional convex polytopes and $${\cal P}_1\subset Im {\cal F} \subset{\cal P}_2.$$
The first identity comes from the additivity of the trace function. The fact that both ${\cal P}_1$, ${\cal P}$ and hence ${\cal P}_2={\cal P}\cap\Delta$, are convex polytopes is a general fact coming from the interpretation of the Horn problem as a moment map problem. Finally, the second inclusion comes from the very definition of ${\cal F}$ and the first comes from Lemma 1 and the following identity, where $\sigma_-$ and $\sigma_+$ are considered as $p\times p$ diagonal matrices and $\rho\in SO(p)$: $$\begin{cases}
&\begin{pmatrix}
\sigma_-&0\\
0&\sigma_+
\end{pmatrix}+
\begin{pmatrix}
0&-\rho^{-1}\\
\rho&0
\end{pmatrix}^{-1}
\begin{pmatrix}
\sigma_-&0\\
0&\sigma_+
\end{pmatrix}
\begin{pmatrix}
0&-\rho^{-1}\\
\rho&0
\end{pmatrix}\\
=&
\begin{pmatrix}
\sigma_-+\rho^{-1}\sigma_+\rho&0\\
0&\rho\sigma_-\rho^{-1}+\sigma_+
\end{pmatrix}\cdot
\end{cases}$$
[**Remark.**]{} The choice of the partition $\sigma=\sigma_-\cup\sigma_+$ of $\sigma$ is dictated by the following theorem, which proves that any other partition of $\sigma$ into two subsets with $p$ elements will lead to a smaller polytope ${\cal P}_1$:
Let $A$ and $B$ be $p\times p$ Hermitian matrices. Let $\sigma_1\ge\sigma_2\ge\cdots\ge\sigma_{2p}$ be the eigenvalues of $A$ and $B$ arranged in descending order. Then there exist Hermitian matrices $\tilde A$ and $\tilde B$ with eigenvalues $\sigma_1\ge\sigma_3\ge\cdots\ge\sigma_{2p-1}$ and $\sigma_2\ge\sigma_4\ge\cdots\ge\sigma_{2p}$ respectively, such that $\tilde A+\tilde B=A+B$.
This was used in [@C1] to prove that ${\cal P}_1= Im {\cal A}$ is the image under the frequency map ${\cal F}$ of the [*adapted*]{} hermitian structures, i.e. those $J$ which send some $p$-dimensional subspace of ${\ensuremath{\mathbb{R}}}^{2p}$ generated by eigenvectors of $S_0$ onto its orthogonal.
The projection property
=======================
In this section, we prove the
The two polytopes ${\cal P}_1$ and ${\cal P}_2$ coincide.
${Im {\cal F}}={\cal P}_1= Im {\cal A}$. In other words, the image by the frequency map ${\cal F}$ of the adapted structures is already the full image $Im {\cal F}$.
We need recall the inductive definition of the Horn inequalities which define the Horn polytopes (see [@F]). For a sum $a+b=c$ of symmetric $p\times p$ matrices with respective (ordered in decreasing order) spectra $$\alpha=(\alpha_1,\cdots,\alpha_p),\; \beta=(\beta_1,\cdots,\beta_p),\; \gamma=(\gamma_1,\cdots,\gamma_p),$$ they read $$(^*IJK)\quad\quad\quad \forall r<p,\; \forall (I,J,K)\in T^p_r,\quad \sum_{k\in K}\gamma_k\le \sum_{i\in I}\alpha_i+\sum_{j\in J}\beta_j,$$ where $T^p_r$ (notation of [@F], noted $LR^p_r$ by reference to Littlewood-Richardson coefficients in [@FFLP]) is defined as follows: let $U^p_r$ be the set of triples $(I,J,K)$ of subsets of cardinal $r$ of $\{1,2,\cdots, p\}$ such that $$\sum_{i\in I}i+\sum_{j\in J}j=\sum_{k\in K}k+\frac{r(r+1)}{2}.$$ Then set $T^p_1=U^p_1$ and define recursively $T^p_r$ by $$T^p_r=
\begin{bmatrix}
(I,J,K)\in U^p_r, \forall s<r,\, \forall (F,G,H)\in T^r_s, \\
\sum_{f\in F}i_f+\sum_{g\in G}j_g\le \sum_{h\in H}k_h+\frac{s(s+1)}{2}
\end{bmatrix}$$
An immediate computation gives the following
Let $$\begin{cases}
I_2&=(2i_1-1,2i_2-1,\cdots,2i_r-1,2j_1,2j_2,\cdots,2j_r),\\
J_2&=(2i_1-1,2i_2-1,\cdots,2i_r-1,2j_1,2j_2,\cdots,2j_r),\\
K_2&=(2k_1-1,2k_1,2k_2-1,2k_2,\cdots,2k_r-1,2k_r),
\end{cases}$$
Then $(I_2,J_2,K_2)\in U^{2p}_{2r}$.
It suffices to check that $$2\left[\sum_{i\in I}(2i-1)+\sum_{j\in J}(2j)\right]=\sum_{k\in K}(2k-1)+\sum_{k\in K}2k+\frac{2r(2r+1)}{2}\cdot$$
Now, comes the key fact:
For any triple $(I,J,K)$ in $T^p_r$, the triple $(I_2,J_2,K_2)$ is in $T^{2p}_{2r}$
The proof of this theorem, which concerns the so-called “domino-decomposable Young diagrams", is based on a version of the Littlewood-Richardson rule due to Carré and Leclerc [@CL].
It implies that, for any a sum $A+B=C$ of real symmetric $2p\times 2p$ matrices with respective (ordered in decreasing order) spectra $$\hat\alpha=(\hat\alpha_1,\cdots,\hat\alpha_{2p}),\; \hat\beta=(\hat\beta_1,\cdots,\hat\beta_{2p}),\; \hat\gamma=(\hat\gamma_1,\cdots,\hat\gamma_{2p}),$$ $(^*I_2,J_2,K_2)$ holds, that is $$\sum_{k\in K}(\hat\gamma_{2k-1}+\hat\gamma_{2k})\le \sum_{i\in I}(\hat\alpha_{2i-1}+\hat\beta_{2i-1})+\sum_{j\in J}(\hat\alpha_{2j}+\hat\beta_{2j}).$$ In particular, if $$\hat\alpha=\hat\beta=\sigma=(\sigma_1,\sigma_2,\cdots,\sigma_{2p}),$$ we get that $$\sum_{k\in K}\frac{\hat\gamma_{2k-1}+\hat\gamma_{2k}}{2}\le \sum_{i\in I}\sigma_{2i-1}+\sum_{j\in J}\sigma_{2j}.$$ Note that the mapping $$(\hat\gamma_1,\hat\gamma_2,\cdots,\hat\gamma_{2p-1},\hat\gamma_{2p})\mapsto (\frac{\hat\gamma_1+\hat\gamma_2}{2},\frac{\hat\gamma_1+\hat\gamma_2}{2},\cdots, \frac{\hat\gamma_{2p-1}+\hat\gamma_{2p}}{2},\frac{\hat\gamma_{2p-1}+\hat\gamma_{2p}}{2})$$ is the orthogonal projection of the ordered set $(\hat\gamma_1,\hat\gamma_2,\cdots,\hat\gamma_{2p-1},\hat\gamma_{2p})$ on the [*diagonal*]{} $\Delta$ of ${\ensuremath{\mathbb{R}}}^{2p}$ defined by the equations $\hat\gamma_1=\hat\gamma_2,\cdots,\hat\gamma_{2p-1}=\hat\gamma_{2p}$, that is on the subset of “hermitian" spectra. Hence a paraphrase of the above theorem is
Let $C=A+B$ be the sum of two $2p\times 2p$ real symmetric matrices with the same spectrum $\{\sigma_1\ge\sigma_2\ge\cdots\ge\sigma_{2p}\}$.
If $\{\nu_1\ge\cdots\ge\nu_p\}$ is the orthogonal projection on the diagonal $\Delta\equiv{\ensuremath{\mathbb{R}}}^p$ of the spectrum $\{\hat\gamma_1\ge\hat\gamma_2\ge\cdots\ge\hat\gamma_{2p}\}$ of $C$, that is if $\nu_k=\frac{\hat\gamma_{2k-1}+\hat\gamma_{2k}}{2}$, the triple of ordered spectra $$\alpha=(\sigma_1,\sigma_3,\cdots,\sigma_{2p-1}),\; \beta=(\sigma_2,\sigma_4,\cdots,\sigma_{2p}),\; \gamma=(\nu_1,\nu_2,\cdots,\nu_p)$$ satisfies the Horn inequality $(^*I,J,K)$.
This implies the following extremal property of the subset of “hermitian" spectra:
The orthogonal projection on the diagonal $\Delta$ of the $(2p-1)$-dimensional Horn polytope ${\cal P}\subset {\ensuremath{\mathbb{R}}}^{2p}$ associated with the spectra $$\sigma(A)=\sigma(B)=\{\sigma_1\ge\sigma_2\ge\cdots\ge\sigma_{2p}\}$$ coincides with the $(p-1)$-dimensional Horn polytope ${\cal P}_1\in{\ensuremath{\mathbb{R}}}^p$ associated with the spectra $$\sigma(a)=(\sigma_1,\sigma_3,\cdots,\sigma_{2p-1}),\; \sigma(b)=(\sigma_2,\sigma_4,\cdots,\sigma_{2p}).$$
In particular, the intersection ${\cal P}_2={\cal P}\cap\Delta$ corresponding to the hermitian spectra, that is those such that $\hat\gamma_1=\hat\gamma_2,\cdots,\hat\gamma_{2p-1}=\hat\gamma_{2p}$, coincides with ${\cal P}_1$. Indeed, ${\cal P}_2$ coincides with the projection of ${\cal P}$, which itself coincides with ${\cal P}_1$.
[**Remark.**]{} The equality $Im {\cal F}={\cal P}_2$ implies the following
Let $C:{\ensuremath{\mathbb{R}}}^{2p}\to{\ensuremath{\mathbb{R}}}^{2p}$ be the sum $C=A+B$ of two symmetric endomorphisms $A$ and $B$ with the same spectrum $\sigma(A)=\sigma(B)=\sigma(S_0)$. Then $C$ is $J$-hermitian for some hermitian structure $J$ on ${\ensuremath{\mathbb{R}}}^{2p}$ if and only if it is conjuguate by an element of $SO(2p)$ to a matrix of the form $S_0+\tilde J^{-1}S_0\tilde J$, where $\tilde J$ is a hermitian structure on $R^{2p}$.
[**Note.**]{} The proof of the above results has been written by the first author after he was convinced by the numerical experiments made by the second author that the equality ${\cal P}_1={\cal P}_2$ was plausible when $p=3$ and more precisely that not only the intersection ${\cal P}_2={\cal P}\cap\Delta$ but also the orthogonal projection of the Horn polytope ${\cal P}$ on $\Delta$ was contained in ${\cal P}_1$ after the canonical identification of $\Delta$ with ${\ensuremath{\mathbb{R}}}^p$. This led first to a proof when $p=2$ or $3$, obtained by coping directly with Horn’s inequalities and then to the discovery that the general case followed from a lemma which turned out to be exactly the lemma 1.18 of [@FFLP]. The numerical experiments are described in the next section.
Numerical experiments
=====================
The numerical checking of the conjecture that $Im {\cal F}={\cal P}_1$, was made on the matrix $S_0= \frac{1}{32}{\rm diag}\left\{ 13,8,5,3,2,1 \right\}$, whose spectrum satisfies the inequalities in [@C1] (section 8). We wrote a program in TRIP [@GL11] producing different rotation matrices $R\in SO(2p)$ in a random way. Starting from the canonical basis $\xi=\{\xi_1,\dots,\xi_m\},$ $m=p(2p-1)$, of $\mathfrak{so}(2p)$, we created a list containing the $m$ one-parameter subgroups $G_i(t)= e^{t\xi_i}\subset SO(2p)$. We created a second list of $m$ random values $[t_i]_{i=1}^m$ and a random permutation $[1,2,\dots,m]\to [i_1,i_2,\dots ,i_m]$. The random rotation matrix was defined as $$\begin{aligned}
R = \prod_{j=1}^m G_{i_j}(t_j),\quad 0\leq t_i \leq 2\pi.
\label{eqn:R}\end{aligned}$$ The program which plots $\mathcal P_1$ is very simple (the fact that we replaced the conjugation of $J_0$ by the conjugation of $S_0$ is immaterial and comes from the formulation of the conjecture in [@C1]):\
--------------------------------------------
`create` $S_0$ `and` $J_0$
`for` $i=1$ `to` $N_{max}$ `do`
$\quad$ `create` $R$ `and` $R^{-1}$;
$\quad$ $S = RS_0R^{-1}$;
$\quad$ $C = S - J_0 S J_0$;
$\quad$ `lst = eigenvalues(`$C$`)`;
$\quad$ `plot ( lst[5], lst[3], lst[1] )`;
`end for`.
--------------------------------------------
We have assigned the value $N_{max}=25000$ obtaining the results shown in Figure \[fig:AA\]. The figure shows also the simplex $\gamma_1+\gamma_2+\gamma_3=1$ and its intersection with $W_3^+$.
![${\cal P}_1=Im {\cal A}$: 25000 random adapted hermitian structures[]{data-label="fig:AA"}](fig1)
The modified algorithm to estimate the shape of $\mathcal P_2=\mathcal
P\cap\Delta$ in $W^+_3$ is similar. For a random $R$ in $SO(6)$, the ordered spectrum $spec(C)=\left( \gamma_1,\dots,\gamma_6 \right)$ of $C=S_0+R^{-1}S_0R$ is projected orthogonally onto the diagonal $\Delta$ by the map $\pi_\Delta:\mathbb R^6 \to \Delta$: $$\begin{aligned}
(\gamma_1,\gamma_2,\dots,\gamma_6)&\mapsto& \left( \frac{\gamma_1+\gamma_2}{2},
\frac{\gamma_3+\gamma_4}{2},
\frac{\gamma_5+\gamma_6}{2}
\right).\end{aligned}$$ At first, when $spec(C)$ was $\varepsilon$-close to $\Delta$ *i.e.*, if $\sum_{k=1}^3 |\gamma_{2k-1}-\gamma_{2k}|^2 < 2\varepsilon^2$ for $\varepsilon$ small, the projected point was plotted in green; otherwise it was plotted in red. No particular pattern was found for the green points meanwhile the red ones were all contained in $\mathcal P_1$. The algorithm to plot $\pi_\Delta(\mathcal P)$ is
------------------------------------------
`create` $S_0$
`for` $i=1$ `to` $N_{max}$ `do`
$\quad$ `create` $R$ `and` $R^{-1}$;
$\quad$ $C = S_0 + R^{-1} S_0 R$;
$\quad$ `lst = eigenvalues(`$C$`)`;
$\quad$ `sort( lst )`;
$\quad$ `plot` $\left(
\frac{lst[6]+lst[5]}{2},
\frac{lst[4]+lst[3]}{2},
\frac{lst[2]+lst[1]}{2}\right)$`;`
`end for`.
------------------------------------------
The results of the projection $\pi_\Delta(spec(C))$ for $50000$ random rotation matrices are shown in Figure \[fig:BB\].
![Projection of ${\cal P}$: 50000 random rotations[]{data-label="fig:BB"}](fig2)
The matrix $S_0$ and hence the polytope $\mathcal P_1$ are the same as in Figure \[fig:AA\] (the interior lines correspond to the polytopes associated to different partitions of the spectrum of $S_0$, as depicted in the corresponding figure in [@C1]). Recall that the polytope ${\cal P}$ has dimension 5; this explains that in order to get a better filling one should have taken many more points. This was not done because the evidence was sufficiently convincing to ask for a proof.
Acknowledgements
================
Warm thanks to Sun Shanzhong and Zhao Lei for their interest in this work and numerous discussions.
[99]{}
A. Albouy, A. Chenciner [*Le Problème des $N$ corps et les distances mutuelles*]{}, Inventiones mathematicae 131 (1998), 151-184.
A. Chenciner [*The angular momentum of a relative equilibrium*]{}, arXiv:1102.0025v1, final version to appear in D.C.D.S.
A. Chenciner [*The Lagrange reduction of the N-body problem: a survey*]{}, to appear in Acta Mathematica Vietnamica
C. Carré, B. Leclerc [*Splitting the Square of a Schur Function into its Symmetric and Antisymmetric Parts*]{}, Journal of Algebraic Combinatorics 4 (1995), 201-231
W. Fulton [*Eigenvalues, invariant factors, highest weights, and Schubert calculus*]{}, Bull. Amer. Math. Soc. (N.S.) [**37**]{} no. 3, 209-249 (2000)
S. Fomin, W. Fulton, C.K. Li, Y.T. Poon, [*Eigenvalues, singular values, and Littlewood-Richardson coefficients*]{}, Amer. J. Math. [**127**]{}, no. 1, 101–127 (2005)
M. Gastineau and J. Laskar, 2011. [TRIP]{} 1.1.18, *TRIP Reference manual*. IMCCE, Paris Observatory. [http://www.imcce.fr/trip/]{}.
[^1]: Notice that this is the same as the spectrum of the $J_0$-hermitian matrix $\Sigma=J_0^{-1}SJ_0+S$, where $S=RS_0R^{-1}$, which was considered in [@C1].
| ArXiv |
---
abstract: 'The control of the spatial distribution of micrometer-sized dust particles in capacitively coupled radio frequency discharges is relevant for research and applications. Typically, dust particles in plasmas form a layer located at the sheath edge adjacent to the bottom electrode. Here, a method of manipulating this distribution by the application of a specific excitation waveform, i.e. two consecutive harmonics, is discussed. Tuning the phase angle $\theta$ between the two harmonics allows to adjust the discharge symmetry via the Electrical Asymmetry Effect (EAE). An adiabatic (continuous) phase shift leaves the dust particles at an equilibrium position close to the lower sheath edge. Their levitation can be correlated with the electric field profile. By applying an abrupt phase shift the dust particles are transported between both sheaths through the plasma bulk and partially reside at an equilibium position close to the upper sheath edge. Hence, the potential profile in the bulk region is probed by the dust particles providing indirect information on plasma properties. The respective motion is understood by an analytical model, showing both the limitations and possible ways of optimizing this sheath-to-sheath transport. A classification of the transport depending on the change in the dc self bias is provided, and the pressure dependence is discussed.'
address: |
$^1$ Institute for Plasma and Atomic Physics, Ruhr University Bochum, 44780 Bochum, Germany\
$^2$ Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, Hungarian Academy of Sciences, H-1525 Budapest POB 49, Hungary\
$^3$ Department of Electronics, Kyushu University, 819-0395 Fukuoka, Japan
author:
- 'Shinya Iwashita$^1$, Edmund Schüngel$^1$, Julian Schulze$^1$, Peter Hartmann$^2$, Zoltán Donkó$^2$, Giichiro Uchida$^3$, Kazunori Koga$^3$, Masaharu Shiratani$^3$, Uwe Czarnetzki$^1$'
title: 'Transport control of dust particles via the Electrical Asymmetry Effect: experiment, simulation, and modeling'
---
Introduction {#Introduction}
============
Dusty plasmas exhibit interesting physical phenomena [@DustyPlasmasBasic; @FortovPysRep2005] such as the interaction of the plasma sheath [@ParticleSheath1; @ParticleSheath2; @ParticleSheath3; @Melzer] and bulk [@ParticleBulk] with the dust particles, the occurrence of waves [@ParticleWaves1] and instabilities [@ParticleInstab1; @ParticleInstab2; @ParticleInstab3], phase transitions [@Phase1; @Phase2; @Phase3; @Phase4; @Phase5], and the formation of Coulomb crystals [@Thomas; @Chu; @Hayashi; @Arp]. They have drawn a great attention for industrial application because dust particles in plasmas play various roles: on one hand the accumulation of dust particles is a major problem for device operation in fusion plasma reactors as well as for semiconductor manufacturing [@Bonitz; @Shukla; @Bouchoule; @Krasheninnikov; @Selwyn], i.e. they are impurities to be removed. On the other hand, they are of general importance for deposition purposes [@DustDepo1; @DustDepo2] and it is well known that an enhanced control of such dust particles in plasmas has the potential to realize the bottom up approach of fabricating novel materials, e.g., microelectronic circuits, medical components, and catalysts [@ShirataniJPD11; @Koga; @Wang; @Yan; @Fumagalli; @Kim]. In all cases the manipulation of dust particles, which is realized by controlling forces exerted on them such as electrostatic, thermophoretic, ion drag, and gravitational forces, or externally applied ones, e.g., created by a laser beam [@Nosenkoa; @MorfillPoP10; @Laser1; @Laser2], is crucially important. Furthermore, the use of dust particles as probes of these forces revealing plasma properties is a current topic of research [@MorfillPRL04; @DustProbes2].\
We have developed a novel method to control the transport of dust particles in a capacitively coupled radio frequency (CCRF) discharge by controlling the electrical symmetry of the discharge [@Iwashita]. Alternative dust manipulation methods using electrical pulses applied to wires have also been reported [@SamsonovPRL2002; @PustylnikPRE2006; @KnapekPRL2007; @PustylnikPoP2009]. Our dust manipulation method is based on the Electrical Asymmetry Effect (EAE) [@Heil]. The EAE allows to generate and control a dc self bias, $\eta$, electrically even in geometrically symmetric discharges. It is based on driving one electrode with a particular voltage waveform, $\phi_{\sim}(t)$, which is the sum of two consecutive harmonics with an adjustable phase shift, $\theta$: $$\label{EQappvol}
\phi_{\sim}(t)=\frac{1}{2}\phi_0[\cos(2\pi f t+\theta)+\cos(4 \pi f t) ].$$ Here, $\phi_0$ is the identical amplitude of both harmonics. In such discharges, $\eta$ is an almost linear function of $\theta$. In this way, separate control of the ion mean energy and flux at both electrodes is realized in an almost ideal way. At low pressures of a few Pa, the EAE additionally allows one to control the maximum sheath voltage and width at each electrode by adjusting $\theta$ [@Heil], resulting in the control of forces exerted on dust particles, such as electrostatic and ion drag forces. In contrast to the pulsing methods mentioned above, the change in the phase angle does not require a change in the applied power or RF amplitude. Furthermore, it is a radio frequency technique, i.e. no DC voltage is applied externally and the EAE is, therefore, applicable to capacitive discharge applications with dielectric electrode surfaces, without the need for additional electrodes or power supplies for the pulsing. The EAE can be optimized with respect to the control range of the dc self-bias by choosing non-equal voltage amplitudes for the individual harmonics [@EAE7] or by adding more consecutive harmonics to the applied voltage waveform [@EAE11; @BoothJPD12]. In this study we intend to describe the basic mechanisms of the manipulation of the dust particle distribution in electrically asymmetric CCRF discharges. Thus, we restrict ourselves to the simplest case described by Eq. (\[EQappvol\]). It is important for the analysis carried out in this work that the dust density is sufficiently low so that the plasma parameters are not disturbed by the dust particles. A large concentration of dust particles disturbs the electron density and can cause a significant change of the dc self bias when distributed asymmetrically between the sheaths [@Boufendi2011; @Watanabe1994; @EddiJPD2013]. The critical parameter for the disturbance is Havnes’ value: $P = 695 T_e r_d n_d / n_i$, where $T_e$, $r_d$, $n_d$ and $n_i$ are electron temperature, radius of dust particles, their number density and ion density, respectively [@Thomas; @Havnes1990]. $P$ is basically the ratio of the charge density of dust particles to that of ions. The concentration of dust particles disturbs the electron density for $P > 1$, while it does not for $P << 1$. In the critical region ${P_c} = 0.1-1$ the charge of the dust particles becomes significant in the total charge balance [@Havnes1990]. We calculate $P \approx 10^{-3}$ for our experiment, which is well below the $P_c$. For this estimation, direct images of dust particles were analyzed and a mean distance between particles of about 1 mm is determined. Thus, the concentration of dust particles is quite low in this study and they do not disturb the plasma.\
This paper is structured in the following way: this introduction is followed by a description of the methods used in this work. There, information on the experimental setup as well as the numerical simulation method is provided, and the analytical approaches on the RF sheath driven by non-sinusoidal voltage waveforms and the motion of dust particles in the plasma bulk region are explained. The results, which are presented and discussed in the third section, include the control of the dc self bias in dusty plasmas via the EAE, the change of the dust levitation position when changing the phase angle adiabatically (continuously), the motion of dust particles through the plasma bulk when tuning the phase angle abruptly, and a classification of the dust particle transport depending on the change in the dc self bias and the discharge conditions. Finally, concluding remarks are given in section four.
Methods
=======
Experiment
----------
![Sketch of the experimental setup.[]{data-label="FIGsetup"}](fig1.eps)
Figure \[FIGsetup\] shows the experimental setup. The experiments are carried out using a CCRF discharge operated in argon gas at $p$ = 2 - 13 Pa, excited by applying $\phi_{\sim}(t)$ according to Eq. (\[EQappvol\]) with $f$ = 13.56 MHz and $\phi_0$ = 200 - 240 V. The applied voltage and the dc self bias are measured using a high voltage probe. Details of the electrical circuit have been provided in previous papers [@Julian; @Iwashita]. The lower (powered) and upper (grounded) electrodes of 100 mm diameter are placed at a distance of $d=22$ mm. The plasma is confined radially between the electrodes by a glass cylinder to improve the discharge symmetry. Both the grounded chamber and the powered electrode are water cooled to eliminate the influence of the thermophoretic force. The upper electrode has a 20 mm diameter hole sealed with a fine sieve in the center for injecting SiO$_2$ dust particles of 1.5 $\mu$m in size, from a dispenser situated above the upper electrode. The gap between the upper electrode and the dispenser, which is located at the center of the upper electrode, is sealed with a teflon ring to prevent any disturbances due to gas flowing through the gap. The supply of argon gas inside the glass cylinder is realized through slits of a teflon ring, which is placed between the glass cylinder and the grounded electrode. An aluminum ring (100 mm outer diameter, 60 mm inner diameter, 2 mm height) is set on the lower electrode to confine the dust particles radially. The injected dust particles initially tend to reside relatively near the edge inside the aluminum ring, therefore the observation area is taken to be in the region of 2 mm $\leq z \leq$ 22 mm and 18 mm $\leq r \leq $ 25 mm using a two dimensional laser light scattering (2DLLS) method [@Bouchoule; @ShirataniJPD11; @Koga; @XuLLS] as shown in Fig. \[FIGsetup\]. A vertical laser sheet passes between the two electrodes, with height and width of 20 mm and 1 mm, respectively. The laser power is 150 mW at 532 nm. The light scattered by the dust particles is detected through a side window using a CCD camera equipped with an interference filter and running at a frame rate of 30 pictures per second.
PIC/MCC simulation
------------------
The rf discharge is described by a simulation code based on the Particle-In-Cell approach combined with Monte Carlo treatment of collision processes, PIC/MCC [@DonkoJPD09; @DonkoAPL09; @DonkoPSST11]. The code is one-dimensional in space and three-dimensional in velocity space. The simulations are performed in pure argon, although PIC/MCC simulations of dusty plasmas have already been reported [@Choi; @Schweigert; @Matyash]. Our approximation is based on the assumption that the dust particles represent only a minor perturbation to the plasma, which is justified for low concentration of dust particles as it is the case in this study. It has been proven that the simulations can be used to explain the motion of dust particles qualitatively as described in [@Iwashita], and the forthcoming analysis also shows the applicability. The PIC/MCC simulations are performed at pressures between 4 and 12 Pa. Although our simulations are not capable of accounting for any two dimensional effects, the simulation data are helpful to understand the experimental findings, which are analyzed in the direction perpendicular to the electrode surfaces only. In the simulations the discharge is driven by a voltage specified by Eq. (\[EQappvol\]). Electrons are reflected from the electrode surfaces with a probability of 0.2 and the secondary electron emission coefficient is set to $\gamma$ = 0.1. Based on the simulation results, the time averaged forces acting on dust particles, i.e. the ion drag force, $F_i$, electrostatic force, $F_e$ and gravity, $F_g$, are calculated as a function of the position between the electrodes [@Iwashita]. Here, the model of $F_i$ provided by Barnes et al [@Barnes] is applied. $F_e$ and $F_g$ are simply expressed as $F_e = Q_dE$ and $F_g = m_dg$, where $E$ and $m_d$ are the time averaged electric field and mass of dust particles, respectively. The charge of dust particles is calculated based on the standard formula: $Q_d = 1400 r_d T_e$ for isolated dust particles, e.g., by Bonitz [@Bonitz] or Piel [@Piel], to be $Q_d \approx -3300e$ in the plasma bulk (see Fig. \[Dustcharge\]), which is close to the typical value reported elsewhere [@ParticleBulk]. Here $e$ is the elementary charge. The typical error in the plasma bulk due to the spatial inhomogeneity is estimated to be about 10 %. Finally, the spatial profiles of the potential energy are derived from the net forces exerted on dust particles.
![Estimated spatial profile of the dust charge based on the standard formula [@Bonitz; @Piel] (Ar, 8 Pa, $\phi_0$ = 200 V, $\theta$ = $0^{\circ}$). The dashed line shows the spatial average in the plasma bulk, that is used in the manuscript. The location of dust particles in equilibrium near the lower electrode, which is obtained experimentally, is also shown.[]{data-label="Dustcharge"}](fig2.eps)
Analytical model of the RF sheath driven by an arbitrary voltage waveform {#SectionSheathModel}
-------------------------------------------------------------------------
In this section a model of CCRF discharges is combined with the Child-Langmuir approximation to obtain the main properties of the RF sheath, i.e. the time dependent sheath width and the spatio-temporal distribution of the potential and electric field inside the sheath, in an electrically asymmetric capacitive discharge. The goal is to calculate the time averaged sheath electric field and correlate this field with the levitation of the dust particles above the powered electrode in case of an adiabatic phase shift, discussed in section \[adiabatic\]. The dynamics of the sheath in a “classical” dual frequency discharge driven by two substantially different frequencies has been modeled using similar approaches [@DFsheath1; @DFsheath2; @DFsheath3]. According to the model, which has been introduced in [@Heil; @VQ; @Czarnetzki11], we find the following expression for the sheath voltage at the powered electrode normalized by $\phi_0$: $$\bar{\phi}_{sp}(t) = - \left[\frac{-{\varepsilon}q_t + \sqrt{{\varepsilon}{q_t}^2 - (1-\varepsilon)[\bar\eta + \bar\phi_{\sim}(t)]}}{1 - \varepsilon}\right]^2.
\label{EQphibar}$$ Here $\varepsilon$, $q_t$, $\bar\eta$ and $\bar\phi_{\sim}(t)$ are the symmetry parameter as defined and discussed in [@Heil], normalized total charge, the dc self bias as well as the applied voltage normalized by $\phi_0$, respectively. Eq. (\[EQphibar\]) provides the sheath voltage as a function of time. In order to obtain a spatio-temporal model of the sheath electric field, the collisionless Child-Langmuir sheath theory [@Lieberman] can be applied at low pressures of a few Pa. To simplify the analysis, we restrict ourselves to a one-dimensional scenario. In this approximation, the maximum width of the sheath adjacent to the powered electrode is expressed as $s_{max,p} = \frac{\sqrt{2}}{3} \lambda_{De} \left(2 \left| \hat{\phi}_{sp} \right| e / T_e\right)^\frac{3}{4}$, where $\hat{\phi}_{sp}$, $\lambda_{De}$ and $T_e$ are the maximum of the sheath voltage at the powered electrode, the Debye length and the electron temperature at the sheath edge (in eV), respectively. The time dependent sheath width is given by the scaling with the sheath voltage: $s_p(t) = s_{max,p} \left( \phi_{sp}(t) / \hat{\phi}_{sp} \right)^{\frac{3}{4}}$. The minimum voltage drop across the powered sheath, $\hat{\phi}_{sp} <0$, is found from the voltage balance: $\phi_{\sim}(t) + \eta = \phi_{sp} + \phi_{sg} + \phi_b$ at the time of minimum applied voltage. Here $\phi_{sg}$ and $\phi_b$ are the sheath voltage at the grounded electrode and the bulk voltage, respectively. Neglecting the floating potential at the grounded sheath and $\phi_b$ yields $\hat{\phi}_{sp} \approx \tilde\phi_{min}+\eta,$ so that the minimum sheath voltage can easily be deduced from experimentally measured values, for instance. Here $\tilde\phi_{min}$ is the minimum of the applied voltage. Assuming that both the electric field and the potential are zero at the sheath edge the spatio-temporal profile of the electric potential in the sheath region at the powered electrode ($0 \leq z \leq s_p(t)$) is expressed by [@Oksuz] $$\label{EQsheathvoltage}
\phi_{sp}(z,t) = - \frac{T_e}{2e} \left( \frac{3}{\sqrt{2}}\frac{s_p(t)-z}{\lambda_{De}} \right)^{\frac{4}{3}}.$$ Here $z$ = 0 is the position of the powered electrode. Finally, the spatio-temporal profile of the electric field in the sheath region is found by differentiation: $$\label{Efield}
E_{sp}(z,t)=-\frac{\partial \phi_{sp}(z)}{\partial z}=-\frac{\sqrt{2} T_e}{e \lambda_{De}}\left( \frac{3}{\sqrt{2}}\frac{s_p(t)-z}{\lambda_{De}} \right)^{\frac{1}{3}}$$ Eq. (\[Efield\]) is used to understand the dust motion as a consequence of the adiabatic (continuous) phase change and to determine the electron density in section \[adiabatic\].
Model to describe the motion of dust particles {#SectionTransportModel}
----------------------------------------------
![Spatial profile of electrostatic force, $F_e$, ion drag force, $F_i$, and gravity, $F_g$, exerted on dust particles. The spatial profile is obtained from PIC/MCC simulation (Ar, 8 Pa, $\phi_0$ = 200 V, $\theta$ = $0^{\circ}$).[]{data-label="Forces"}](fig1a.eps)
The motion of dust particles in plasmas is determinded by the forces exerted on them [@Bouchoule; @Barnes; @Garrity; @Morfill; @Goree; @Piel]. Here, we propose a simple analytical model to describe the one-dimensional transport of dust particles between both sheaths through the plasma bulk. Models of the dust motion based on the force balance have already been reported [@Chu; @Couedel; @NefedovNJP2003; @LandNJP2007; @Zhakhovskii; @Graves]. We would like to emphasize again that the concentration of dust particles is quite low in this study and they do not disturb the plasma, which is different from the condition under which these models have been provided. Our approach focuses on analyzing the particular dust transport which has been obtained experimentally when changing the phase angle abruptly, and in fact the model proposed here can explain the experimental results.
Further studies are required to investigate non-Hamiltonian effects [@Tuckerman; @Kompaneets] and clarify their role for the physics presented in this work. f In reactors with horizontal plane parallel electrodes separated by a discharge gap, $d$, and in the absence of thermophoretic forces, negatively charged dust particles tend to be confined at the sheath edges, where the forces exerted on them balance. Right after introducing the dust particles into the discharge volume, they are typically located around the lower sheath edge due to gravity. Let us focus on the motion of dust particles between the sheath edge of the bottom (powered) electrode $(z = s_p)$ and the upper (grounded) one $(z = d-s_g)$, e.g., after applying an upward force at the lower equilibium position. Later on, we will approximate the electrostatic force around the sheath edges as hard walls, i.e. the particles are instantaneously reflected without any change in their kinetic energy. This assumption is justified due to the fact that the electrostatic force caused by the bulk electric field (see Fig. \[Forces\]) or the interaction between dust particles is negligible under our condition. One reason for this quite small bulk electrostatic force is the relatively high ion density in the bulk, which is also realized in the void formation in dusty plasmas [@Bouchoule; @Bonitz]. In contrast to our situation, the electrostatic force is of vital importance in complex plasmas, where the major contribution of negative charges to the total charge balance in the bulk is given by the dust particles and not by the electrons (see e.g., [@Chu; @Couedel; @Takahashi]). The inter-particle force, i.e. Coulomb force can be comparable to the sheath electrostatic force under certain conditions [@Hwang]. This becomes crucial particularly when the lateral motion of dust particles is discussed. This study is, however, focused only on their vertical motion. Additionally, dust particles are initially located only at the lower sheath edge due to the balance between the sheath electrostatic force and the ion drag force, suggesting that these two forces are dominantly exerted on the dust particles in this study. Thus, the vertical component of the Coulomb force is much smaller than the respective component of the sheath electrostatic force and the ion drag force. In our model, small errors occur only at the bulk side of the sheath edge (equilibrium position of dust particles) where the electrostatic force is neither close to zero nor represents a hard wall. The dust particles are assumed not to perturb the plasma. Within the plasma bulk region, the dust particle motion is associated with the following force balance: $$\label{EQmombal}
m_d\ddot{z} = -m_dg - m_d\nu\dot{z} + F_i(z).$$ Here, $m_d$, $g$, $\nu$, and $F_i$ are the mass of a dust particle, the acceleration of gravity, the frequency of momentum loss due to collisions between dust particles and gas atoms [@Piel; @Epstein], and the ion drag force, respectively. Note that the gas friction force $m_d\nu\dot{z}$ is derived from the assumption that the velocity of dust particles is much smaller than the thermal velocity of gas molecules. Therefore, the dependence of $\nu$ on the particle velocity can be neglected. Any interaction between the dust particles, e.g., a repulsive Coulomb force [@Thomas; @Chu; @Hayashi; @Arp; @Takahashi; @LinIJPD94], is not taken into account. Although the force profiles shown in Fig. \[Forces\] suggest that gravity can be neglected, we keep the corresponding term in the force balance to ensure the applicability of the resulting formulae for all types of particles, e.g., different sizes and/or mass densities (materials).\
There are several models of the ion drag force [@Barnes; @KhrapakPRL2003; @Fortov2] and the analytical description of this force remains an interesting research topic in itself. There are discussions in the literature on the validity of the different models. Although more sophisticated models are available, the Barnes model [@Barnes] is applied here in order to calculate the ion drag force in a simple way. The formula is generally considered to be accurate at low dust densities as pointed out e.g., in [@Bouchoule; @Piel], which is the case in this study. We assume that $n_i$ as well as the ion velocity, $v_i$, are expressed by trigonometric functions, as it results from the basic diffusion estimation in a steady state CCRF discharge [@Lieberman]: $$\begin{aligned}
\label{iondensityprofile}
n_i (z) & = & n_{i0} \cos\left[\left(z-\frac{d}{2}\right)\frac{\pi}{\Lambda_{i}}\right], \\
v_i (z) & = & v_{i0} \tan\left[\left(z-\frac{d}{2}\right)\frac{\pi}{\Lambda_{i}}\right].
\label{ionvelocityprofile}\end{aligned}$$ Here, the maximum ion density, $n_{i0}$, and ion velocity, $v_{i0}$, are constants. $\Lambda_{i}$ is the ion diffusion length; the value is actually close to the distance between the discharge center and the sheath edges. These input parameters are determined by fitting to the PIC/MCC simulation data as shown in Fig. \[Fitting\].
![Spatial profile of ion density and velocity obtained from the PIC/MCC simulation and fit functions of the analytical model (Ar, 8 Pa, $\phi_0$ = 200 V, $\theta$ = $0^{\circ}$).[]{data-label="Fitting"}](fig3.eps)
The estimated model quantities from this fitting are $n_{i0}=6.6 \cdot 10^{15}$ m$^{-3}$, $v_{i0}=344$ m s$^{-1}$, $\Lambda_i=15.5$ mm, and $d=23.0$ mm, respectively. The ion drag force consists of the collection force due to ions hitting the particle surface and the orbit force due to Coulomb collisions with the drifting ions. In low pressure CCRF discharges the orbit force [@Barnes], $$F_{i,orb}= 4 \pi n_i v_s m_i b_{\pi/2}^2 \Gamma,$$ typically dominates. Here, $v_s$, $m_i$, $b_{\pi/2}$ and $\Gamma$ are the mean ion velocity, the ion mass, the impact parameter and the Coulomb logarithm [@Barnes], respectively: $$\begin{aligned}
b_{\pi/2} & = & \frac{eQ_d}{4\pi\epsilon_0m_iv_s^2}, \\
\Gamma & =& \frac{1}{2}\ln\left(\frac{\lambda_{De}^2+b_{\pi/2}^2}{r_d^2(1-\frac{2e\phi_f}{m_i v_s^2})+b_{\pi/2}^2}\right).\end{aligned}$$ Note that these quantities depend on the radius ($r_d$), floating potential ($\phi_f$), and charge ($Q_d$) of the dust particles. In this paper, we use the simplifying assumption of the dust particle charge to be negative and constant: $Q_d \approx -3300e$ as shown in Fig. \[Dustcharge\].\
In our approach, we neglect the thermal motion of the ions, i.e. the mean ion velocity $v_s$ is given by the drift component, $v_i$: $$v_s=\left(\frac{8k_BT_i}{\pi m_i}+v_i^2\right)^{\frac{1}{2}} \approx v_i.$$ Applying the approximation $F_i \approx F_{i,orb} \propto n_iv_i$, the ion drag force becomes $$F_i (z) = \bar{F}_{i0} \sin\left[\left(z-\frac{d}{2}\right)\frac{\pi}{\Lambda_i}\right].$$ Here, the maximum ion drag force ($\bar{F}_{i0}$) is a constant. In order to solve Eq. (\[EQmombal\]) analytically only the linear variation of the sine function is considered here: $$\label{EQiondrag}
F_i (z) \approx F_{i0} \left(z-\frac{d}{2}\right) \frac{\pi}{\Lambda_{i}},$$ with $F_{i0}= 4 \pi m_i n_{i0} v_{i0} b_{\pi/2}^2 \Gamma$. The input parameters obtained from Fig. \[Fitting\] provide $F_{i0}=3.8 \cdot 10^{-13}$ N. Equation \[EQiondrag\] corresponds to a strong simplification of $F_i (z)$ and deviations from the exact solution appear, particularly in the regions close to the sheath edges. However, our aim is to explain the transport of dust particles through the plasma bulk with this model. In the bulk region, the model is a reasonable approach, since it includes the most relevant forces in this region. Furthermore, the forthcoming analysis shows that the basic features of particle motion and the experimental observation of the dust transport can be explained reasonably well by this approach.\
After inserting Eq. (\[EQiondrag\]) into Eq. (\[EQmombal\]) a second order linear ordinary differential equation $$\label{EQmombalsimple}
m_d\ddot{z} + m_d\nu\dot{z} - F_{i0} \left[\left(z-\frac{d}{2}\right)\frac{\pi}{\Lambda_{i}}\right] + m_dg = 0$$ needs to be solved. Note that Eq. (\[EQmombalsimple\]) represents a harmonic oscillator in the space coordinate $(z-d/2)\pi/\Lambda_{i}$ with frequency $\sqrt{F_{i0}/m_d}$, which is externally driven by gravity and damped by collisions. Finally, using the boundary conditions $z(0)=z_0$ and $\dot{z}(0)=u_{0}$, which corresponds to the initial velocity of dust particles, the trajectory of dust particles is given by $$\label{EQtrajectory}
z(t) = \left[\beta_1 \cosh \left(\alpha t \right) + \beta_2 \sinh \left(\alpha t\right)\right] e^{-\frac{\nu}{2}t} + \delta.$$ Here, $\alpha$, $\beta_1$, $\beta_2$, and $\delta$ are: $$\begin{aligned}
\alpha & = & \sqrt{\left( \frac{\nu}{2}\right)^2 + \frac{\pi F_{i0}}{m_d \Lambda_{i}}}, \\
\beta_1 & = & x_0 - \frac{d}{2} - \frac{m_d \Lambda_{i} g}{\pi F_{i0}}, \\
\beta_2 & = & \left( u_0 + \beta_1 \frac{\nu}{2} \right) \alpha^{-1}, \\
\delta & = & \frac{m_d \Lambda_{i} g}{\pi F_{i0}} + \frac{d}{2}. \end{aligned}$$ From this trajectory of the dust particles, the kinetic energy is obtained: $$\label{EQenergy}
W(t) = \frac{1}{2} m_d\dot{z}^2(t) = \frac{m_d}{8\alpha^2} \left( -A e^{\alpha t} + B e^{-\alpha t} \right)^2 e^{-\nu t},$$ where A and B are defined as $$\begin{aligned}
A & = & g + \frac{F_{i0} \pi d}{2 m_d \Lambda_{i}} - x_0 \frac{F_{i0} \pi}{m_d \Lambda_{i}} + u_0 \left( \frac{\nu}{2} - \alpha \right), \\
B & = & g + \frac{F_{i0} \pi d}{2 m_d \Lambda_{i}} - x_0 \frac{F_{i0} \pi}{m_d \Lambda_{i}} + u_0 \left( \frac{\nu}{2} + \alpha \right).\end{aligned}$$ Eq. (\[EQenergy\]) is used to describe the dust energy as a consequence of the abrupt phase change in section \[abrupt\]. This rather complex result will be compared to the simple assumption that the kinetic energy of the dust particles is not affected by the particular shape of the potential profile and that the loss of the energy of the dust particles is only due to gas friction. Then, the velocity and kinetic energy of the dust particles can be estimated as $$\begin{aligned}
u_{d}(t) = u_{0}e^{-\frac{\nu}{2}t}, \\
\label{EQenergysimple0}
W(t) = \frac{1}{2}m_{d}u_{d}^{2}(t) = W_0e^{-{\nu}t} .
\label{EQenergysimple}\end{aligned}$$ Here $W_0$ is the initial kinetic energy of dust particles. Eq. (\[EQenergysimple\]) is used to determine the potential profile experimentally using the spatial profile of the laser light scattering (LLS) intensity from dust particles in section \[abrupt\]. It should be noted that practically the dust charge fluctuates and the reflection of the dust particles at the sheath edge is “soft”. Again, our model aims to describe the dust transport observed in this study in a simple way, and thus the simple assumption, e.g., a constant dust charge and a rough approximation of the electrostatic force as a hard wall, is applied here.
Results and Discussion
======================
dc self bias control via the EAE in a plasma containing a small amount of dust
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![Experimentally obtained dc self bias as a function of the phase angle $\theta$ with and without dust particles for different neutral gas pressures. The applied voltage amplitude is kept constant at $\phi_0$ = 200 V. Solid symbols relate to discharges without and open symbols to ones with dust particles. Square: 2 Pa, triangle: 4 Pa, inverted triangle: 8 Pa.[]{data-label="FIGbias"}](fig4.eps)
Fig. \[FIGbias\] shows the dc self bias, $\eta$, obtained from the experiment, as a function of the phase angle, $\theta$. $\eta$ is generated as a monotonic function of $\theta$. As described in details before [@Heil; @EAE7; @EAE11; @Julian; @Czarnetzki11; @Eddi; @DonkoJPD09; @DonkoAPL09], the EAE allows to control the discharge symmetry electrically. The control range for gas pressures between 2 and 8 Pa and an applied voltage amplitude of $\phi_0$ = 200 V is found to be close to about 45 % of the applied voltage amplitude. Therefore, a strong change in both the time averaged sheath voltages ($\eta=\left\langle \phi_{sp}(t)\right\rangle+\left\langle \phi_{sg}(t)\right\rangle$) and the maximum sheath voltages as a function of $\theta$ can be expected. $\eta$ is shifted towards negative values because the discharge setup becomes effectively geometrically asymmetric due to the parasitic effect of capacitive coupling between the glass cylinder and the grounded chamber walls [@Coburn; @Savas; @Julian; @Booth10; @Booth12; @Booth12_2]. This effect tends to be stronger at higher pressures. It is important to note that in this study no significant difference of $\eta$ in cases with and without dust particles is observed, indicating that the presence of a low dust concentration does not influence the plasma significantly. Therefore, the models described in the previous section are indeed applicable as pointed out already in section \[Introduction\] by estimating Havnes’ value $P$.
Adiabatic phase change {#adiabatic}
----------------------
![Spatial profile of the measured LLS intensity from the dust particles around the lower electrode as a function of the phase angle $\theta$ combined with the electric field calculated from the analytical model (Ar, 2 Pa, $\phi_0$ = 200 V). The observation of the LLS intensity within the lower region (0 mm $\leq z \leq$ 2 mm) is blocked by the aluminum ring.[]{data-label="FIGllsfield"}](fig5.eps)
The dust particles injected into the discharge are initially located at the sheath edge adjacent to the lower electrode. Any adiabatic (continuous) change of $\theta$ leaves the dust particles at an equilibrium position close to this lower sheath edge as shown in Fig. \[FIGllsfield\]. By increasing the phase angle from $0^{\circ}$ to $90^{\circ}$ adiabatically, the time averaged sheath width becomes smaller and both the mean and the maximum sheath voltages at the lower electrode decrease. Therefore, the equilibrium position of the dust particles is shifted closer towards the electrode. This change of the equilibrium position can be understood by the electric field profile obtained from the analytical model described in section \[SectionSheathModel\] using input parameters of $T_e$ = 3 eV and $\lambda_{De}$ = 644 $\mu$m calculated under the assumption of $n_e$ = $4 \times 10^{14}$ m$^{-3}$ (see lines in Fig. \[FIGllsfield\]). Electron density and temperature are taken from the PIC/MCC simulations because we applied a glass cylinder to confine the plasma. Thus, performing Langmuir probe measurements is not possible. We find very good agreement between the measured LLS and the part of the electric field distribution at a strength of about -4 kV/m, i.e. where forces exerted on dust particles balance.
![Distribution of the electric field at (a) $\theta=0^\circ$ and (b) $\theta=90^\circ$ as a function of spatial position within the phase angle dependent maximum sheath width and time resulting from the model shown in Eq. (\[Efield\]) (Ar, 2 Pa, $\phi_0$ = 200 V). $T_{rf}$ = 74 ns. The sheath reaches the region above dashed line only once per rf period.[]{data-label="FIGfield00field90"}](fig6.eps)
![Strength of the time averaged electric field as a function of position corresponding to Fig. \[FIGfield00field90\]. The dashed line is drawn according to that in Fig. \[FIGfield00field90\] (b). The gradient of the time averaged electric field changes at the boundary indicated by the dashed line.[]{data-label="FIGfield00field90_2"}](fig7.eps)
When $\theta$ is changed from $0^{\circ}$ to $90^{\circ}$, the maximum of the time averaged electric field in the powered electrode sheath, i.e. $\left\langle E\right\rangle_{max}$ found at the electrode, becomes smaller due to the decrease in the mean sheath voltage. In addition, the change in the shape of the applied voltage as a function of $\theta$ leads to a change in the sheath voltage, $\phi_{sp}(t)$, which causes a change in the spatial distribution of the time averaged electric field. As it becomes clear from Fig. \[FIGfield00field90\] and \[FIGfield00field90\_2\], the slope of $\left\langle E\right\rangle(z)$ becomes flatter in the upper part of the sheath with increasing $\theta$, i.e. the time averaged voltage drop over this region becomes smaller. In particular, the field is relatively small during the second half of the rf period (see dashed line in Fig. \[FIGfield00field90\] (b)). Thus, the broadening of the equilibrium position (region of bright LLS) is well understood by the analytical model. This correlation analysis of the dust equilibrium position combined with the spatial electric field profile is applicable as a diagnostic tool to estimate plasma parameters, i.e. the dust particles can serve as electrostatic probes [@MorfillPRL04; @Kersten09; @DustProbes2; @DustProbes3; @DustProbes4]. The correlation analysis yields the maximum sheath extension as the only free fitting parameter, which depends on electron temperature and density ($s_{max,p} \propto \lambda_{De} / T_e^{3/4} \propto n_e^{-1/2} T_e^{-1/4}$). Hence, $s_{max,p}$ is more sensitive to changes in the electron density and, if the electron temperature is known, $n_e$ can be obtained assuming that these plasma parameters are constant, independently of $\theta$. In our discharge configuration, it is not possible to measure $T_e$. However, estimating $T_e \approx 3$ eV, for instance, results in an electron density of about $n_e \approx 4 \cdot 10^{14}$ $m^{-3}$ at the sheath edge under the condition of Fig. \[FIGllsfield\] (Ar, 2 Pa and $\phi_0$ = 200 V). Note that the charge of dust particles becomes smaller than that in the plasma bulk when they are closer to the sheath edge as shown in Fig. \[Dustcharge\], i.e. the charge of the dust particles observed in Fig. \[FIGfield00field90\_2\] might be smaller than -3300e which is assumed as the dust charge in this paper. Further study is required to discuss this topic in detail.
Abrupt phase change {#abrupt}
-------------------
When the phase angle is changed abruptly from $90^{\circ}$ to $0^{\circ}$, i.e. much faster than the reaction time scale of the particles, all dust particles are transported upwards into the plasma bulk and undergo rapid oscillations between the sheaths. Thereafter, a fraction of the particles reaches the upper sheath region and settles there (see Fig. \[FIGtransport\](a)). In this way, sheath-to-sheath transport is realized [@Iwashita]. Before discussing the conditions, under which sheath-to-sheath transport is possible, in more detail, this particle motion should be understood. As in the case of the adiabatic phase change, dust particles injected into the discharge are initially located at the sheath edge adjacent to the lower electrode. If the phase is changed abruptly from $90^{\circ}$ to $0^{\circ}$, the dust particles are suddenly located in a region of high potential due to their inertia. Consequently, they bounce back and forth between both sheaths, while being decelerated by gas friction (see Fig. \[Transportmodel\]) [@Iwashita]. As described in section \[SectionTransportModel\], the motion of dust particles is determined by gravity, the ion drag force pushing the particles out of the bulk towards the sheaths, deceleration due to friction by collisions with the neutral gas, as well as electrostatic forces due to the sheath electric field, which basically can be regarded as boundaries, thus spatially confining the particle motion. Afterwards, they reside inside the potential well at either the upper or the lower sheath edge [@Iwashita]. The shape of the potential profile consists of a peak close to the discharge center, two minima located around the sheath edges and steep rises inside the sheaths. The difference in the height of the two minima is mainly caused by gravity in the absence of thermophoretic forces. The term “potential” is valid only, if the result does not depend on the particle velocity, i.e. if the time scale of the dust particle motion is the slowest of all time scales of interest here. This condition is fulfilled: for instance, the thermal motion of both the neutral and the ionized gas atoms is about two orders of magnitude faster compared to the dust particle motion (the maximum dust velocity estimated from the experimental results (Fig. \[FIGtransport\]) is a few m/s at most). Therefore, the potential profile is provided independently from the dust velocity.\
![Spatiotemporal profiles of the measured LLS intensity by the dust particles within the discharge gap (Ar, (a) 8 Pa and (b) 12 Pa, $\phi_0$ = 200 V). The abrupt phase change takes place at [*t $\approx$*]{} 0 ms. Observation of the lower region (0 mm $\leq z \leq$ 2 mm) is blocked by the aluminum ring. The upper (diamond and triangle) and lower (circle and square) points are taken to obtain the upper and lower potential wells in Fig. \[FIGpotexp\], respectively. The arrow illustrates the estimation of an initail velocity of $u_0 \approx 1$ m/s.[]{data-label="FIGtransport"}](fig8.eps)
![Model of sheath-to-sheath transport of dust particles[@Iwashita]. The potential profile is calculated from PIC/MCC simulation data (Ar, 4 Pa, $\phi_0$ = 200 V). $L_1$ and $L_2$ are the widths of the upper and lower potential wells, respectively, at $\theta = 0^\circ$.[]{data-label="Transportmodel"}](fig9.eps)
It is possible to determine this potential distribution qualitatively from the experimental results. Hence, information on basic plasma properties might be achievable from this analysis. The shapes of the potential wells at the upper and lower sheath edges are obtained from the LLS profile (see four kinds of points in Fig. \[FIGtransport\](a)). The points are taken at the contour line, which is both existent in the entire plasma bulk region and shows a reasonably high intensity. Note that the resulting data points are close to the region of maximum gradient of the LLS intensity, as well. The upper (diamond and triangle) and lower (circle and square) points correspond to the confinement regions of dust particles in the potential wells at the upper and lower sheath edges, respectively. In order to deduce the potential distribution from them, the temporal evolution of the energy of the dust particles needs to be known. The simplest model of the dust motion is applied here, i.e. dust particles lose their kinetic energy only due to gas friction. Using this approximation allows an analytical treatment of $W(t)$ by using Eq. (\[EQenergysimple\]). Using the data points shown in Fig. \[FIGtransport\] (a) and replacing the time scale by the corresponding energy, the potential profile shown in Fig. \[FIGpotexp\] is obtained. Here, the potential energy scale is normalized by the initial energy of the dust particles. An estimation yields $W_0 \approx m_d u_0^2 /2 \approx 1.8 \times 10^{-15}$ J (11 keV) for an initial velocity of $u_0 \approx 1$ m/s, which was obtained from the spatiotemporal profile of the LLS intensity by the dust particles (see arrow in Fig. \[FIGtransport\]). Taking into account the uncertainty in $W_0$, we restrict ourselves to a qualitative discussion of the potential profile in this study. Comparing this profile to the one calculated from the simulation data shown in Fig. \[FIGpotpic\], we see that the position of the lower potential minimum agrees well between the experiment and the PIC simulation ($z \approx 5.7$ mm). In the experiment the upper minimum is located at 18.6 mm, whereas the position in the simulation is 16.9 mm. This difference is probably caused by the effective geometrical asymmetry of the discharge in the experiment, which is also indicated by the self bias voltage, $\eta$ (see 8 Pa case in Fig. \[FIGbias\]). In the PIC simulation the discharge is geometrically symmetric, thus yielding a symmetric dc self bias curve ($\eta(\theta=0^\circ)=-\eta(\theta=90^\circ) \approx -52$ V) and a wider sheath compared to the experiment at the grounded side for all $\theta$. The lowest part of the potential curve resulting from the experimental data cannot be obtained by this approach (see the curve at around z = 5 mm in Fig. \[FIGpotexp\]), since the residual spatial distribution is caused by the residual energy, $W_r$, of dust particles in equilibrium position due to thermal motion and Coulomb interaction, respectively, as well as the spatial resolution of the optical measurements (see the LLS intensity from dust particles after 100 ms in Fig. \[FIGtransport\]), which are neglected in our simple model. Except for this region, the dust particles can be used as probes to determine the potential, which depends on plasma properties via $F_i (z)$ and $F_{e}(z)$, in a major part of the discharge region. The probability for the trapping of dust particles at the upper sheath, $P_{trans}$ might be roughly estimated by the width of the upper potential well divided by the sum of the widths of the lower and upper potential wells , which is expressed as $L_1 / (L_1 + L_2)$, in the simple approximation made above (see Fig. \[Transportmodel\]) [@Iwashita]. Here $L_1$ and $L_2$ are the widths of the upper and lower potential wells, respectively. The probability calculated this way is about 0.5 for the experiment for 8 Pa and $\phi_0$=200 V, which agrees well with that calculated for the simulation potential profile.\
Furthermore, the potential profile can be used to obtain input parameters for the analytical model of dust transport described in section \[SectionTransportModel\]. For this model the potential profile in the plasma bulk is obtained by integrating Eq. (\[EQmombalsimple\]). Due to the small-angle approximation for the ion drag force (Eq. (\[EQiondrag\])) the potential profile is expressed by a simple parabola: $U(z)=U_0 - \left[F_{i,0} \left( z - d \right) \frac{z \pi}{2 \Lambda_i} - m_d g z \right]$, where $U_0$ is an integration constant. The model curve resulting from fits of equations \[iondensityprofile\] and \[ionvelocityprofile\] to PIC simulation data is shown in Fig. \[FIGpotpic\]. One can find a difference of the central maxima of the potential profile obtained from PIC/MCC simulation for $\theta = 90^{\circ}$ and $0^{\circ}$. This is derived from the spatial profiles of the ion drag force (mainly orbit force), i.e. the direction of the ion drag force changes at the center of the plasma bulk [@Iwashita] and the gradient of the force profile for $\theta = 90^{\circ}$ in this region is steeper than that for $\theta = 0^{\circ}$, resulting in the difference of the central maxima for $\theta = 90^{\circ}$ and $0^{\circ}$. The model shows reasonable agreement with the potential profile using the exact values from the PIC/MCC simulation within the plasma bulk. As discussed above, deviations can be observed close to the sheath edges, e.g., due to the simplified treatment of the electrostatic force as a hard wall.
![Potential profile at $\theta = 0^\circ$ obtained from the measured 2DLLS intensity shown in Fig. \[FIGtransport\] (a) using a simple model. The potential energy scale is normalized by a rough estimation of the initial energy of the dust particles.[]{data-label="FIGpotexp"}](fig10.eps)
![Potential profile calculated from PIC/MCC simulations data (Ar, 8 Pa, $\phi_0$=200 V). The model curve resulting from fits of equations \[iondensityprofile\] and \[ionvelocityprofile\] to PIC simulation data is shown, as well.[]{data-label="FIGpotpic"}](fig11.eps)
Figure \[FIGtrajectory\] shows the trajectories of dust particles calculated from Eq. (\[EQtrajectory\]) and using the input parameters given above, for different values of the initial velocity. Right after the time of the abrupt phase shift all dust particles gain a certain initial velocity. If the initial velocity is below $u_0 \approx 1.0$ m/s, they cannot overcome the central maximum of the potential and bounce only inside the lower potential well. Dust particles with the initial velocity above $u_0 \approx 1.25$ m/s travel through the whole plasma bulk just after the phase shift. Dust particles with an initial velocity of $u_0 \approx 1.5$ m/s oscillate back and forth in the bulk region. However, their final equilibrium position is again located around the lower sheath. Therefore, from the model the initial velocity to realize the sheath-to-sheath transport is found at certain intervals, e.g., dust particles having $u_0$ = 2.0 m/s end up in the upper potential minimum while those having $u_0$ = 1.75 m/s do not. The conclusion obtained from Fig. \[FIGtrajectory\] can be summarized by introducing the number of passages of dust particles through the plasma bulk, $N_{trans}$.
$u_0$ (m/s) 1.00 1.25 1.5 1.75 2.0
------------- ------ ------ ----- ------ -----
$N_{trans}$ 0 1 2 2 3
: \[table1\] Summary of the effective transport of dust particles through the plasma bulk obtained in this study, depending on the initial velocity (Ar, 8 Pa, $\phi_0=200$ V). Odd number of $N_{trans}$ realizes sheath-to-sheath transport, while even number of $N_{trans}$ does not.
Any odd number of $N_{trans}$ means that sheath-to-sheath transport is realized, whereas even numbers of $N_{trans}$ correspond to a final position close to the initial position at the lower sheath edge (table \[table1\]). We also note that the trajectory of $u_0 \approx$ 1.25 m/s obtained from the model agrees well with the experimental result (Fig. \[FIGtransport\](a)).
![Trajectory of dust particles calculated from the model for different initial velocities (Ar, 8 Pa, $\phi_0$=200 V). The input parameter fitted on the data calculated from PIC/MCC simulations (see Fig. \[FIGpotpic\]) are used.[]{data-label="FIGtrajectory"}](fig12.eps)
![Time evolution of the kinetic energy of dust particles after the abrupt phase shift according to the $u_0 = 1.25$ m/s case in Fig. \[FIGtrajectory\] (Ar, 8 Pa, $\phi_0$ = 200 V).[]{data-label="FIGenergy"}](fig13.eps)
Using Eq. (\[EQenergy\]) the time evolution of the kinetic energy of the dust particles after the abrupt phase shift is obtained as shown in Fig. \[FIGenergy\]. An anharmonic oscillation is superimposed on the simple assumption of an exponential decay of the dust velocity (Eq. (\[EQenergysimple\])) as a function of time. The sharp edges in this oscillations are due to the treatment of the electrostatic forces as hard walls. When the dust particles bounce between the sheath edges, they do not just lose their kinetic energy on long timescales, but they also gain kinetic energy temporarily due to the ion drag force while moving from the discharge center towards the sheaths. However, the kinetic energy stays below $W_0 e^{-\nu t}$ between $t=$ 0 and the time of trapping in one of the two potential wells. This is because the potential profile leads to a deceleration of the dust particles just after the abrupt phase change. Therefore, the dust particles spend even more time on their way to the upper sheath and undergo more collisions with the neutral gas, resulting in enhanced friction losses. The information on the trajectory and energy provided by the analytical model of dust transport is useful for the optimization of their transport: it can be understood that a monoenergetic initial distribution within one of the velocity intervals allowing sheath-to-sheath transport, e.g., $u_0 \approx 1.25$ m/s in the case discussed here, is favorable to transport as many particles as possible to the upper sheath. Moreover, the outcome of the model suggests that the rough estimation of the probability of successful particle transport, $P_{trans}$, given above might overestimate the fraction of particles residing at the upper sheath edge, because the energy loss on the way from the upper sheath to the potential peak is much smaller than the energy loss occuring on the way from the lower sheath to the peak. In general, this model only requires the peak ion density in the discharge center and the electron temperature as input parameters, which could be measured by other diagnostic methods. However, there is no simple access to apply such methods in our experimental setup. The upgrading of the experimental setup to obtain these key parameters is required for our further study.
Classification of transport conditions
--------------------------------------
![Experimentally obtained classification of the dust particle transport as a function of $\Delta \bar{\eta}$ and pressure. The voltage amplitude is kept at $\phi_0$ = 200 V for $p<$10 Pa and $\phi_0$ = 200-240 V for $p\geq$10 Pa, respectively.[]{data-label="FIGtransclass"}](fig14.eps)
![Normalized measured LLS intensity from dust particles around the upper sheath edge ($I_{upper} / I_{all}$) as a parameter of $\Delta \bar{\eta}$ for the abrupt phase shift (Ar, 4 Pa, $\phi_0$ = 200 V). $I_{upper} / I_{all}$ is obtained by dividing the sum of the LLS intensity from dust particles around the upper sheath edge by that from dust particles around both sheath edges. []{data-label="Velocity"}](fig15.eps)
We now turn to the discussion of conditions, under which sheath-to-sheath transport is possible. The key parameter for this transport is the rapid change of the dc self bias, $\Delta \eta$, which can be easily controlled between $\Delta {\eta}_{min}=0$ and $\Delta {\eta}_{max}= {\eta}(90^\circ) - {\eta}(0^\circ)$ by choosing certain intervals of the change in the phase angle (see Fig. \[FIGbias\]). As shown in Fig. \[FIGtransclass\], a threshold value of $\Delta \bar{\eta}$ is apparently required to achieve the transport of a fraction of the particles to the upper equilibrium position. Here the difference of normalized dc self bias $\Delta \bar{\eta}$ is given by $\Delta \bar{\eta} = [\eta(\theta_2) - \eta(\theta_1)] / \phi_0 $ in case of the phase shift from $\theta_1$ to $\theta_2$. The threshold increases with pressure, due to the increasing collisionality and, even more important, a stronger ion drag force, i.e. the central peak in the potential distribution becomes higher with increasing pressure. Therefore, it becomes more difficult for the particles to overcome this potential barrier. If $\Delta \bar{\eta}$ is smaller than the threshold, sheath-to-sheath transport is not realized: the dust particles reach a certain position below this potential peak and are forced towards the equilibrium position around the lower electrode sheath again (see Fig. \[FIGtransport\](b)). In this case, similar to the adiabatic phase change, information on the local plasma properties might be gained from this disturbance of the particle distribution. In particular, we observe that the maximum displacement of the dust particles strongly depends on global parameters, such as pressure and voltage, in the experiment. However, a very good spatio-temporal resolution of the LLS measurements is required, which is not provided in our experiment. At low pressures, the sheath-to-sheath transport is possible within a wide range of $\Delta \bar{\eta}$ (see Fig. \[FIGtransclass\]). However, as it has been motivated by the model results shown in Fig. \[FIGtrajectory\], the fraction of dust particles might vary as a function of $\Delta \bar{\eta}$. Figure \[Velocity\] shows the normalized LLS intensity from dust particles around the upper sheath edge ($I_{upper} / I_{all}$) as a function of $\Delta \bar{\eta}$, for the abrupt phase shift. A low pressure of 4 Pa has been applied here. $I_{upper} / I_{all}$ is obtained by dividing the sum of the LLS intensity from dust particles around the upper sheath edge by that from dust particles around both sheath edges. The maximum of $I_{upper} / I_{all}$ is seen at $\Delta \bar{\eta}$ = 23%, and sheath-to-sheath transport is not achieved for $\Delta \bar{\eta}$ $<$ 16%. These results indicate that the optimum initial velocity for sheath-to-sheath transport is slightly above the minimum value where sheath-to-sheath transport is realized. It also becomes clear that the change in the dc self bias, $\Delta \bar{\eta}$, for the [*efficient*]{} sheath-to-sheath transport is found at a certain interval, e.g., dust particles are transported efficiently for $\bar{\eta}$ = 48% and $\bar{\eta}$ = 23%, while they are not for $\bar{\eta}$ = 41% (see Fig. \[Velocity\]). The initial velocity of dust particles, $u_0$, is controlled by changing $\Delta \bar{\eta}$, since the temporally averaged sheath voltage depends almost linearly on the dc self bias [@EAEpower] and it can be approximated that the initial energy of the dust particles is proportional to the change of the mean sheath voltage. Hence, $u_0 \propto \sqrt{\Delta \bar{\eta}}$ and these results support the model of the dust motion described above (Fig. \[FIGtrajectory\]).\
Conclusion
==========
The opportunities of controlling the transport of dust particles via the EAE have been discussed using the results of experiment, simulations, and analytical models. For these models, it has been confirmed that the dust particles do not significantly perturb the electrical properties of the discharge. In the case of an adiabatic tuning of the phase angle between the applied harmonics the dust particles are kept at an equilibrium position close to the lower sheath edge and their levitation is correlated with the time averaged electric field profile. This might provide the opportunity to estimate the electron density by using the dust particles as electrostatic probes. In the case of an abrupt phase shift ($90^{\circ}$ $\rightarrow$ $0^{\circ}$) the dust particles are transported upwards, i.e. they move between both sheaths through the plasma bulk. The trajectory as well as the temporal evolution of the dust particle energy are well understood using an analytical model. It is found that an initial velocity of the dust particles of about 1.25 m/s is required to push them over the potential hill located around the center of the plasma bulk. Thus, changing the applied voltage waveform via the EAE allows transporting a fraction of the dust particles from the equilibrium position around the lower sheath edge to the one at the upper electrode sheath, i.e. sheath-to-sheath transport is realized. The model also predicts that the initial velocity to realize sheath-to-sheath transport is found at certain intervals, which is in agreement with the dependence of the probability of sheath-to-sheath transport (fraction of LLS intensity at the upper sheath edge) on the change in the dc self bias found in the experiment. Furthermore, a certain threshold value of the rapid change of the dc self bias is required to achieve sheath-to-sheath transport. If the change in the dc self bias lies below the threshold value, the dust particles move within the lower potential well. Due to an increase in the collisionality and in the height of the potential peak, the threshold increases and the displacement decreases as a function of neutral gas pressure.
This research was supported by the German Federal Ministry for the Environment (0325210B), the Alexander von Humboldt Foundation, the RUB Research Department Plasma, and the Hungarian Scientific Research Fund (OTKA-K-77653+IN-85261, K-105476, NN-103150).
References {#references .unnumbered}
==========
[27]{} Fortov V E, Khrapak A G, Khrapak S A, Molotkov V I and Petrov O V 2004 [*Phys. Usp.*]{} [**47**]{} 447 Fortov V E, Ivlev A V, Khrapak S A, Khrapak A G, Morfill G E 2005 [*Phys. Rep.*]{} [**421**]{} 1 Nitter T 1996 [*Plasma Sources Sci. Technol.*]{} [**5**]{} 93 Melzer A, Trottenberg T and Piel A 1994 [*Phys. Lett. A*]{} [**191**]{} 301 Ivlev A V, Sütterlin R, Steinberg V, Zuzic M and Morfill G 2000 [*Phys. Rev. Lett.*]{} [**85**]{} 4060 H[ü]{}bner S and Melzer A 2009 [*Phys. Rev. Lett.*]{} [**102**]{} 215001 Khrapak S A, Ratynskaia S V, Zobnin A V, Usachev A D, Yaroshenko V V, Thoma M H, Kretschmer M, Hofner M, Morfill G E, Petrov O F and Fortov V E 2005 [*Phys. Rev. E*]{} [**72**]{} 016406 Kalman G, Rosenberg M and DeWitt H E 2000 [*Phys. Rev. Lett.*]{} [**84**]{} 6030 Nunomura S, Goree J, Hu S, Wang X and Bhattacharjee A 2002 [*Phys. Rev. E*]{} [**65**]{} 066402 Couedel L, Mikikian M, Samarian A A and Boufendi L 2010 [*Phys. Plasmas*]{} [**17**]{} 083705 Cavarroc M, Jouanny M C, Radouane K, Mikikian M and Boufendi L 2006 [*J. Appl. Phys.*]{} [**99**]{} 064301 Hamaguchi S, Farouki R T and Dubin D H E 1997 [*Phys. Rev. E*]{} [**56**]{} 4671 Melzer A, Homann A and Piel A 1996 [*Phys. Rev. E*]{} [**53**]{} 2757 Meijer E J and Frenkel D 1991 [*J. Chem. Phys.*]{} [**94**]{} 2269 Schweigert V A, Schweigert I V, Melzer A, Homann A and Piel A 1998 [*Phys. Rev. Lett.*]{} [**80**]{} 5345 Aschinger A and Winter J 2012 [*New J. Phys.*]{} [**14**]{} 093036 Thomas H, Morfill G E and Demmel V 1994 [*Phys. Rev. Lett.*]{} [**73**]{} 652 Chu J H and I Lin 1994 [*Phys. Rev. Lett.*]{} [**72**]{} 4009 Hayashi Y and Tachibana K 1994 [*Jpn. J. Appl. Phys.*]{} [**33**]{} L804 Arp O, Block D and Piel A 2004 [*Phys. Rev. Lett.*]{} [**93**]{} 165004 Bonitz M, Henning C and Block D 2010 [*Rep. Prog. Phys.*]{} [**73**]{} 066501 Shukla P K and Eliasson B 2009 [*Rev. Mod. Phys.*]{} [**81**]{} 25 Bouchoule A 1999 [*Dusty Plasmas*]{} (Chichester: John Wiley & Sons) Krasheninnikov S I and Soboleva T K 2005 [*Plasma Phys. Contr. Fusion*]{} [**47**]{} A339 Selwyn G S, Singh J and Bennett R S 1989 [*J. Vac. Sci. Technol.*]{} A [**7**]{} 2758 Cavarroc M, Mikikian M, Tessier Y and Boufendi L 2008 [*IEEE Trans. Plasma Sci.*]{} [**36**]{} 1016 Roca i Cabarrocas P, Nguyen-Tran Th, Djeridane Y, Abramov A, Johnson E and Patriarche G 2007 [*J. Phys. D: Appl. Phys.*]{} [**40**]{} 2258 Shiratani M, Koga K, Iwashita S, Uchida G, Itagaki N and Kamataki K 2011 [*J. Phys. D: Appl. Phys.*]{} [**44**]{} 174038 Koga K, Iwashita S and Shiratani M 2007 [*J. Phys. D: Appl. Phys.*]{} [**40**]{} 2267 Wang X, Ocola L E, Divan R S and Sumant A V 2012 [*Nanotechnology*]{} [**23**]{} 075301 Yan H, Choe H S, Nam S W, Hu Y, Das S, Klemic J F, Ellenbogen J C and Lieber C M 2011 [*Nature*]{} [**470**]{} 240 Fumagalli F, Kylián O, Amato L, Hanǔs J and Rossi F 2012 [*J. Phys. D: Appl. Phys.*]{} [**45**]{} 135203 Kim H H, Ogata A, Schiorlin M, Marotta E and Paradisi C 2011 [*Catal. Lett.*]{} [**141**]{} 277 Nosenko V, Goree J and Piel A 2006 [*Phys. Plasmas*]{} [**13**]{} 032106 Nosenko V, Ivlev A V, and Morfill G E 2010 [*Phys. Plasmas*]{} [**17**]{} 123705 Piel A and Melzer A 2002 [*Adv. Space Res.*]{} [**29**]{} 1255 Klindworth M, Melzer A, Piel A and Schweigert V A 2000 [*Phys. Rev. B*]{} [**61**]{} 8404 Annaratone B M, Antonova T, Thomas H M, and Morfill G E 2004 [*Phys. Rev. Lett.*]{} [**93**]{} 185001 Beckers J, Ockenga T, Wolter M, Stoffels W W, van Dijk J, Kersten H and Kroesen G M W 2011 [*Phys. Rev. Lett.*]{} [**106**]{} 115002 Iwashita S, Uchida G, Schulze J, Sch[ü]{}ngel E, Hartmann P, Shiratani M, Donkó Z and Czarnetzki U 2012 [*Plasma Sources Sci. Technol.*]{} [**21**]{} 032001 Samsonov D, Ivlev A V, Quinn R A, Morfill G and Zhdanov S 2002 [*Phys. Rev. Lett.*]{} [**88**]{} 095004 Pustylnik M Y, Ohno N, Takamura S, and Smirnov R 2006 [*Phys. Rev. E*]{} [**74**]{} 046402 Knapek C A, Samsonov D, Zhdanov S, Konopka U, and Morfill G E 2007 [*Phys. Rev. Lett.*]{} [**98**]{} 015004 Pustylnik M Y, Ivlev A V, Thomas H M, Morfill G E, Vasilyak L M, Vetchinin S P, Polyakov D N, and Fortov V E 2009 [*Phys. Plasmas*]{} [**16**]{} 113705 Heil B G, Czarnetzki U, Brinkmann R P and Mussenbrock T 2008 [*J. Phys. D: Appl. Phys.*]{} [**41**]{} 165202 Schulze J, Schüngel E, Donkó Z and Czarnetzki U 2009 [*J. Appl. Phys.*]{} **106** 063307 Schulze J, Schüngel E, Donkó Z and Czarnetzki U 2011 [*Plasma Sources Sci. Technol.*]{} [**20**]{} 015017 Lafleur T and Booth J P 2012 [*J. Phys. D: Appl. Phys.*]{} [**45**]{} 395203 Boufendi L, Jouanny M Ch, Kovacevic E, Berndt J and Mikikian M 2011 [*J. Phys. D: Appl. Phys.*]{} [**44**]{} 174035 Watanabe Y, Shiratani M, Fukuzawa T and Kawasaki H 1994 [*Plasma Sources Sci. Technol.*]{} [**3**]{} 355 Schüngel E, Mohr S, Iwashita S, Schulze J and Czarnetzki U 2013 [*J. Phys. D: Appl. Phys.*]{} [**46**]{} 175205 Havnes O, Aanesen T K, and Melandso F 1990 [*J. Geophys. Res.*]{} [**95**]{} 6581 Schulze J, Sch[ü]{}ngel E and Czarnetzki U 2009 [*J. Phys. D: Appl. Phys.*]{} [**42**]{} 092005 Xu R, 2002 [*Particle Characterization: Light Schattering Methods*]{} (Dordrecht: Kluwer Academic Publishers) Donkó Z, Schulze J, Heil B G, Czarnetzki U 2009 [*J. Phys. D: Appl. Phys.*]{} [**42**]{} 025205 Donkó Z, Schulze J, Czarnetzki U and Luggenh[ö]{}lscher D 2009 [*Appl. Phys. Lett.*]{} [**94**]{} 131501 Donkó Z 2011 [*Plasma Sources Sci. Technol.*]{} [**20**]{} 024001 Choi J S, Ventzew P L G, Hoekstra R J and Kushner M J 1994 [*Plasma Sourc. Sci. Technol.*]{} [**3**]{} 419 Schweigert I V, Alexandrov A L, Ariskin D A, Peeters F M, Stefanovi[ć]{} V, Kova[c]{}evi[ć]{} E, Berndt J and Winter J 2008 [*Phys. Rev. E*]{} [**78**]{} 026410 Matyash K, Schneider R, Taccogna F, Hatayama A, Longo S, Capitelli M, Tskhakaya D and Bronold F X 2007 [*Contrib. Plasma Phys.*]{} [**47**]{} 595 Barnes M S, Keller J H, Forster J C, O’Neill J A and Coultas D K 1992 [*Phys. Rev. Lett.*]{} [**68**]{} 313 Piel A 2010 [*Plasma Physics*]{} (Berlin: Springer) Robiche J, Boyle P C, Turner M M and Ellingboe A R 2003 [*J. Phys. D: Appl. Phys.*]{} **36** 1810 Franklin R N 2003 [*J. Phys. D: Appl. Phys.*]{} **36** 2660 Jiang W, Mao M and Wang Y N 2006 [*Phys. Plasmas*]{} [**13**]{} 113502 Schulze J, Heil B G, Luggenhölscher D, Brinkmann R P and Czarnetzki U 2008 [*J. Phys. D: Appl. Phys.*]{} **41** 195212 Czarnetzki U, Schulze J, Sch[ü]{}ngel E and Donkó Z 2011 [*Plasma Sources Sci. Technol.*]{} [**20**]{} 024010 Lieberman M A and Lichtenberg A J 2005 [*Principles of Plasma Discharges and Materials Processing, 2nd Ed.*]{} (Hoboken: John Wiley & Sons) Oksuz L and Hershkowitz N 2005 [*Plasma Sources. Sci. Technol.*]{} [**14**]{} 201 Garrity M P, Peterson T W, Garrett L M, and O’Hanlon J F, 1995 [*J. Vac. Sci. Technol.*]{} A [**13**]{} 2939 Rothermel H, Hagl T, Morfill G E, Thoma M H and Thomas H M 2002 [*Phys. Rev. Lett.*]{} [**89**]{} 175001 Liu B, Goree J, Fortov V E, Lipaev A, Molotkov V I, Petrov O F, Morfill G E, Thomas H M and Ivlev A V 2010 [*Phys. Plasmas*]{} [**17**]{} 053701 Cou[ë]{}del L, Nosenko V, Zhdanov S K, Ivlev I V, Thomas H M, and Morfill G E 2009 [*Phys. Rev. Lett.*]{} [**103**]{} 215001 Nefedov A P, Morfill G E, Fortov V E, Thomas H M, Rothermel H, Hagl T, Ivlev A V, Zuzic M, Klumov B A, Lipaev A M, Molotkov V I, Petrov O F, Gidzenko Y P, Krikalev S K, Shepherd W, Ivanov A I, Roth M, Binnenbruck H, Goree J A and Semenov Y P 2003 [*New J. Phys.*]{} [**5**]{} 33 Land V and Goedheer W J 2007 [*New J. Phys.*]{} [**9**]{} 246 Zhakhovski V V, Molotkov V I, Nefedov A P, Torchinski V M, Khrapak A G, and Fortov V E 1997 [*JETP Lett.*]{} [**66**]{} 419 Graves D B, Daugherty J E, Kilgore M D and Porteous R K 1994 [*Plasma Sources Sci. Technol.*]{} [**3**]{} 433 Tuckerman M E, Mundy C J, and Martyna G J 1999 [*Europhys. Lett.*]{} [**45**]{} 149 Kompaneets R, Vladimirov S V, Ivlev A V, Tsytovich V, Morfill G 2006 [*Phys. Plasmas*]{} [**13**]{} 072104 Takahashi K, Oishi T, Shimomai K, Hayashi Y, and Nishino S 1998 [*Phys. Rev. E*]{} [**58**]{} 7805
Hwang H H and Kushner M 1997 [*J. Appl. Phys.*]{} [**82**]{} 2106
Epstein P S 1924 [*Phys. Rev.*]{} [**23**]{} 710 Chu J H, Du Ji-Bin and Lin I, 1994 [*J Phys. D: Appl. Phys.*]{} [**27**]{} 296 Khrapak S A, Ivlev A V, Morfill G E, Zhdanov S K 2003 [*Phys. Rev. Lett.*]{} [**90**]{} 225002 Fortov V E and Morfill G E 2009 [*Complex and Dusty Plasmas*]{} (Boca Raton: CRC Press) Sch[ü]{}ngel E, Zhang Q Z, Iwashita S, Schulze J, Hou L J, Wang Y N and Czarnetzki U 2011 [*J. Phys. D: Appl. Phys.*]{} [**44**]{} 285205 Coburn J W and Kay E 1972 [*J. Appl. Phys.*]{} [**43**]{} 4965 Lieberman M A and Savas S E 1990 [*J. Vac. Sci. Technol. A*]{} [**8**]{} 1632 Johnson E V, Verbeke T, Vanel J C and Booth J P 2010 [*J Phys. D: Appl. Phys.*]{} [**43**]{} 412001 Johnson E V, Delattre P A, and Booth J P 2012 [*Appl. Phys. Lett.*]{} [**100**]{} 133504 Lafleur T, Boswell R W, and Booth J P 2012 [*Appl. Phys. Lett.*]{} [**100**]{} 194101 Basner R, Sigeneger F, Loffhagen D, Schubert G, Fehske H and Kersten H 2009 [*New J. Phys.*]{} [**11**]{} 013041 Schubert G, Basner R, Kersten H and Fehske H 2011 [*Eur. Phys. J. D*]{} [**63**]{} 431 Maurer H R, Schneider V, Wolter M, Basner R, Trottenberg T and Kersten H 2011 [*Contrib. Plasma Phys.*]{} [**51**]{} 218 Schüngel E, Schulze J, Donkó Z and Czarnetzki U 2011 [*Phys. Plasmas*]{} **18** 013503
| ArXiv |
---
abstract: 'Let $I$ and $J$ be homogeneous ideals in a standard graded polynomial ring. We study upper bounds of the Hilbert function of the intersection of $I$ and $g(J)$, where $g$ is a general change of coordinates. Our main result gives a generalization of Green’s hyperplane section theorem.'
address:
- ' Department of Mathematics, Purdue University, West Lafayette, IN 47901, USA. '
- ' Satoshi Murai, Department of Mathematical Science, Faculty of Science, Yamaguchi University, 1677-1 Yoshida, Yamaguchi 753-8512, Japan. '
author:
- Giulio Caviglia
- Satoshi Murai
title: |
On Hilbert functions of\
general intersections of ideals
---
[^1]
Introduction
============
Hilbert functions of graded $K$-algebras are important invariants studied in several areas of mathematics. In the theory of Hilbert functions, one of the most useful tools is Green’s hyperplane section theorem, which gives a sharp upper bound for the Hilbert function of $R/hR$, where $R$ is a standard graded $K$-algebra and $h$ is a general linear form, in terms of the Hilbert function of $R$. This result of Green has been extended to the case of general homogeneous polynomials by Herzog and Popescu [@HP] and Gasharov [@Ga]. In this paper, we study a further generalization of these theorems.
Let $K$ be an infinite field and $S=K[x_1,\dots,x_n]$ a standard graded polynomial ring. Recall that the *Hilbert function* $H(M,-) : \mathbb{Z} \to \mathbb{Z}$ of a finitely generated graded $S$-module $M$ is the numerical function defined by $$H(M,d)=\dim_K M_d,$$ where $M_d$ is the graded component of $M$ of degree $d$. A set $W$ of monomials of $S$ is said to be *lex* if, for all monomials $u,v \in S$ of the same degree, $u \in W$ and $v>_{{\mathrm{lex}}}u$ imply $v \in W$, where $>_{{\mathrm{lex}}}$ is the lexicographic order induced by the ordering $x_1> \cdots > x_n$. We say that a monomial ideal $I \subset S$ is a *lex ideal* if the set of monomials in $I$ is lex. The classical Macaulay’s theorem [@Ma] guarantees that, for any homogeneous ideal $I \subset S$, there exists a unique lex ideal, denoted by $I^{{{\mathrm{lex}}}}$, with the same Hilbert function as $I$. Green’s hyperplane section theorem [@Gr] states
\[green\] Let $I \subset S$ be a homogeneous ideal. For a general linear form $h \in S_1$, $$H(I \cap (h),d) \leq H(I^{{\mathrm{lex}}}\cap (x_n),d) \ \ \mbox{for all } d \geq 0.$$
Green’s hyperplane section theorem is known to be useful to prove several important results on Hilbert functions such as Macaulay’s theorem [@Ma] and Gotzmann’s persistence theorem [@Go], see [@Gr]. Herzog and Popescu [@HP] (in characteristic $0$) and Gasharov [@Ga] (in positive characteristic) generalized Green’s hyperplane section theorem in the following form.
\[hpg\] Let $I \subset S$ be a homogeneous ideal. For a general homogeneous polynomial $h \in S$ of degree $a$, $$H(I \cap (h),d) \leq H(I^{{\mathrm{lex}}}\cap(x_n^a),d) \ \ \mbox{for all } d \geq 0.$$
We study a generalization of Theorems \[green\] and \[hpg\]. Let $>_{{\mathrm{{oplex}}}}$ be the lexicographic order on $S$ induced by the ordering $x_n> \cdots > x_1$. A set $W$ of monomials of $S$ is said to be *opposite lex* if, for all monomials $u,v \in S$ of the same degree, $u \in W$ and $v>_{{\mathrm{{oplex}}}}u$ imply $v \in W$. Also, we say that a monomial ideal $I \subset S$ is an *opposite lex ideal* if the set of monomials in $I$ is opposite lex. For a homogeneous ideal $I \subset S$, let $I^{{\mathrm{{oplex}}}}$ be the opposite lex ideal with the same Hilbert function as $I$ and let ${\ensuremath{\mathrm{Gin}}}_\sigma(I)$ be the generic initial ideal ([@Ei §15.9]) of $I$ with respect to a term order $>_\sigma$.
In Section 3 we will prove the following
\[intersection\] Suppose $\mathrm{char}(K)=0$. Let $I\subset S$ and $J \subset S$ be homogeneous ideals such that ${\ensuremath{\mathrm{Gin}}}_{{\mathrm{lex}}}(J)$ is lex. For a general change of coordinates $g$ of $S$, $$H(I \cap g(J),d) \leq H(I^{{\mathrm{lex}}}\cap J^{{\mathrm{{oplex}}}},d)
\ \ \mbox{for all } d\geq 0.$$
Theorems \[green\] and \[hpg\], assuming that the characteristic is zero, are special cases of the above theorem when $J$ is principal. Note that Theorem \[intersection\] is sharp since the equality holds if $I$ is lex and $J$ is oplex (Remark \[rem1\]). Note also that if ${\ensuremath{\mathrm{Gin}}}_\sigma(I)$ is lex for some term order $>_\sigma$ then ${\ensuremath{\mathrm{Gin}}}_{{\mathrm{lex}}}(J)$ must be lex as well ([@Co1 Corollary 1.6]).
Unfortunately, the assumption on $J$, as well as the assumption on the characteristic of $K$, in Theorem \[intersection\] are essential (see Remark \[example\]). However, we prove the following result for the product of ideals.
\[product\] Suppose $\mathrm{char}(K)=0$. Let $I\subset S$ and $J \subset S$ be homogeneous ideals. For a general change of coordinates $g$ of $S$, $$H(I g(J),d) \geq H(I^{{\mathrm{lex}}}J^{{\mathrm{{oplex}}}},d)
\ \ \mbox{for all } d\geq 0.$$
Inspired by Theorems \[intersection\] and \[product\], we suggest the following conjecture.
\[conj\] Suppose $\mathrm{char}(K)=0.$ Let $I\subset S$ and $J \subset S$ be homogeneous ideals such that ${\ensuremath{\mathrm{Gin}}}_{{\mathrm{lex}}}(J)$ is lex. For a general change of coordinates $g$ of $S$, $$\dim_K {\ensuremath{\mathrm{Tor}}}_i(S/I,S/g(J))_d \leq \dim_K {\ensuremath{\mathrm{Tor}}}_i(S/I^{{\mathrm{lex}}},S/J^{{\mathrm{{oplex}}}})_d
\ \ \mbox{for all } d\geq 0.$$
Theorems \[intersection\] and \[product\] show that the conjecture is true if $i=0$ or $i=1.$ The conjecture is also known to be true when $J$ is generated by linear forms by the result of Conca [@Co Theorem 4.2]. Theorem \[2.5\], which we prove later, also provides some evidence supporting the above inequality.
Dimension of ${\ensuremath{\mathrm{Tor}}}$ and general change of coordinates
============================================================================
Let ${{GL}}_n(K)$ be the general linear group of invertible $n \times n$ matrices over $K$. Throughout the paper, we identify each element $h=(a_{ij}) \in {{GL}}_n(K)$ with the change of coordinates defined by $h(x_i)=\sum_{j=1}^n a_{ji}x_j$ for all $i$.
We say that a property (P) holds for a general $g \in {{GL}}_n(K)$ if there is a non-empty Zariski open subset $U \subset {{GL}}_n(K)$ such that (P) holds for all $g \in U$.
We first prove that, for two homogeneous ideals $I \subset S$ and $J \subset S$, the Hilbert function of $I \cap g(J)$ and that of $I g(J)$ are well defined for a general $g \in {{GL}}_n (K)$, i.e. there exists a non-empty Zariski open subset of ${{GL}}_n(K)$ on which the Hilbert function of $I \cap g(J)$ and that of $I g(J)$ are constant.
\[2-0\] Let $I \subset S$ and $J \subset S$ be homogeneous ideals. For a general change of coordinates $g \in {{GL}}_n(K)$, the function $H(I \cap g(J),-)$ and $H(I g(J),-)$ are well defined.
We prove the statement for $I \cap g(J)$ (the proof for $Ig(J)$ is similar). It is enough to prove the same statement for $I+g(J)$. We prove that ${\ensuremath{\mathrm{in}}}_{{\mathrm{lex}}}(I+g(J))$ is constant for a general $g \in {{GL}}_n(K)$.
Let $t_{kl}$, where $1 \leq k,l \leq n$, be indeterminates, $\tilde K=K(t_{kl}: 1 \leq k,l \leq n)$ the field of fractions of $K[t_{kl}: 1 \leq k,l \leq n]$ and $A=\tilde K [x_1,\dots,x_n]$. Let $\rho: S \to A$ be the ring map induced by $\rho(x_k)= \sum_{l=1}^n t_{lk} x_l$ for $k=1,2,\dots,n$, and $\tilde L= I A + \rho(J)A \subset A$. Let $L \subset S$ be the monomial ideal with the same monomial generators as ${\ensuremath{\mathrm{in}}}_{{\mathrm{lex}}}(\tilde L)$. We prove ${\ensuremath{\mathrm{in}}}_{{\mathrm{lex}}}(I+g(J))=L$ for a general $g \in {{GL}}_n(K)$.
Let $f_1,\dots,f_s$ be generators of $I$ and $g_1,\dots,g_t$ those of $J$. Then the polynomials $f_1,\dots,f_s,\rho(g_1),\dots,\rho(g_t)$ are generators of $\tilde L$. By the Buchberger algorithm, one can compute a Gröbner basis of $\tilde L$ from $f_1,\dots,f_s,\rho(g_1),\dots,\rho(g_t)$ by finite steps. Consider all elements $h_1,\dots,h_m \in K(t_{kl}:1 \leq k,l \leq n)$ which are the coefficient of polynomials (including numerators and denominators of rational functions) that appear in the process of computing a Gröbner basis of $\tilde L$ by the Buchberger algorithm. Consider a non-empty Zariski open subset $U \subset {{GL}}_n(K)$ such that $h_i(g) \in K \setminus \{0\}$ for any $g \in U$, where $h_i(g)$ is an element obtained from $h_i$ by substituting $t_{kl}$ with entries of $g$. By construction ${\ensuremath{\mathrm{in}}}_{{\mathrm{lex}}}(I+g(J))=L$ for every $g \in U$.
\[ConstantHF\] The method used to prove the above lemma can be easily generalized to a number of situations. For instance for a general $g \in {{GL}}_n(K)$ and a finitely generated graded $S$-module $M,$ the Hilbert function of ${\ensuremath{\mathrm{Tor}}}_i(M,S/g(J))$ is well defined for every $i$. Let $\mathbb F: 0 \stackrel{\varphi_{p+1}}{\longrightarrow}
\mathbb F_p \stackrel{\varphi_p}{\longrightarrow}
\cdots
\longrightarrow
\mathbb F_1 \stackrel{\varphi_1}{\longrightarrow}
\mathbb F_0
\stackrel{\varphi_0}{\longrightarrow}0$ be a graded free resolution of $M.$ Given a change of coordinates $g$, one first notes that for every $i=0,\dots,p$, the Hilbert function $H({\ensuremath{\mathrm{Tor}}}_i(M,S/g(J)),-)$ is equal to the difference between the Hilbert function of $\rm{Ker}(\pi_{i-1} \circ \varphi_i)$ and the one of $\varphi_{i+1}(F_{i+1}) + F_i \otimes_S g(J)$ where $\pi_{i-1}: F_{i-1} \rightarrow F_{i-1} \otimes_S S/g(J)$ is the canonical projection. Hence we have $$\begin{aligned}
\label{H-TOR}
\nonumber H({\ensuremath{\mathrm{Tor}}}_i & (M,S/g(J)),-)= \\
&H(F_i, -) -H(\varphi_i(F_i)+ g(J) F_{i-1},-)
+ H(g(J) F_{i-1},-)\\
\nonumber &- H(\varphi_{i+1}(F_{i+1}) + g(J) F_i,-).\end{aligned}$$ Clearly $H(F_i,-)$ and $H(g(J) F_{i-1},-)$ do not depend on $g.$ Thus it is enough to show that, for a general $g$, the Hilbert functions of $\varphi_i(F_i)+g(J) F_{i-1}$ are well defined for all $i=0,\dots,p+1.$ This can be seen as in Lemma \[2-0\].
Next, we present two lemmas which will allow us to reduce the proofs of the theorems in the third section to combinatorial considerations regarding Borel-fixed ideals.
The first Lemma is probably clearly true to some experts, but we include its proof for the sake of the exposition. The ideas used in Lemma \[lemma2\] are similar to that of [@Ca1 Lemma 2.1] and they rely on the construction of a flat family and on the use of the structure theorem for finitely generated modules over principal ideal domains.
\[lemma1\] Let $M$ be a finitely generated graded $S$-module and $J \subset S$ a homogeneous ideal. For a general change of coordinates $g \in {{GL}}_n(K)$ we have that $\dim_K {\ensuremath{\mathrm{Tor}}}_i(M,S/g(J))_j \leq \dim_K {\ensuremath{\mathrm{Tor}}}_i(M,S/J)_j$ for all $i$ and for all $j.$
Let $\mathbb F$ be a resolution of $M,$ as in Remark \[ConstantHF\]. Let $i$, $0\leq i \leq p+1$ and notice that, by equation , it is sufficient to show: $H(\varphi_i(F_i)+g(J) F_{i-1},-)\geq
H(\varphi_i(F_i)+JF_{i-1},-).$ We fix a degree $d$ and consider the monomial basis of $ (F_{i-1})_d.$ Given a change of coordinates $h=(a_{kl}) \in {{GL}}_n(K)$ we present the vector space $V_d=(\varphi_i(F_i)+h(J)F_{i-1})_d$ with respect to this basis. The dimension of $V_d$ equals the rank of a matrix whose entries are polynomials in the $a_{kl}$’s with coefficients in $K.$ Such a rank is maximal when the change of coordinates $h$ is general.
For a vector ${\mathbf w}=(w_1,\ldots,w_n) \in { \ensuremath{\mathbb{Z}}}_{\geq 0}^n$, let ${\ensuremath{\mathrm{in}}}_{\mathbf w}(I)$ be the initial ideal of a homogeneous ideal $I$ with respect to the weight order $>_{\mathbf w}$ (see [@Ei p. 345]). Let $T$ be a new indeterminate and $R=S[T]$. For $\mathbf a=(a_1,\dots,a_n) \in { \ensuremath{\mathbb{Z}}}_{\geq 0}^n$, let $x^{\mathbf a}=x_1^{a_1} x_2^{a_2} \cdots x_n^{a_n}$ and $(\mathbf a, {\mathbf w})= a_1w_1 + \cdots + a_n w_n$. For a polynomial $f= \sum_{\mathbf a \in { \ensuremath{\mathbb{Z}}}_{\geq 0}^n} c_{\mathbf a} x^{\mathbf a}$, where $c_{\mathbf a} \in K$, let $b= \max \{ (\mathbf a,{\mathbf w}) : c_{\mathbf a} \ne 0\}$ and $$\tilde f = T^b \left(\sum_{\mathbf a \in { \ensuremath{\mathbb{Z}}}_{\geq 0}^n} T^{-(\mathbf a,{\mathbf w})}c_{\mathbf a} x^{\mathbf a}\right) \in R.$$ Note that $\tilde f$ can be written as $\tilde f={\ensuremath{\mathrm{in}}}_{\mathbf w}(f) + T g$ where $g \in R$. For an ideal $I \subset S$, let $\tilde I =(\tilde f :f \in I) \subset R$. For $\lambda \in K \setminus\{0\}$, let $D_{\lambda,{\mathbf w}}$ be the diagonal change of coordinates defined by $D_{\lambda,{\mathbf w}}(x_i)=\lambda^{-w_i} x_i$. From the definition, we have $$R/\big(\tilde I +(T)\big) \cong S/ {\ensuremath{\mathrm{in}}}_{\mathbf w}(I)$$ and $$R/\big(\tilde I +(T-\lambda)\big) \cong S/D_{\lambda,{\mathbf w}}(I)$$ where $\lambda \in K \setminus \{0\}$. Moreover $(T-\lambda)$ is a non-zero divisor of $R/\tilde I$ for any $\lambda \in K$. See [@Ei §15.8].
\[lemma2\] Fix an integer $j$. Let ${\mathbf w}\in { \ensuremath{\mathbb{Z}}}_{\geq 0}^n$, $M$ a finitely generated graded $S$-module and $J \subset S$ a homogeneous ideal. For a general $\lambda \in K$, one has $$\dim_K {\ensuremath{\mathrm{Tor}}}_i \big(M,S/{\ensuremath{\mathrm{in}}}_{\mathbf w}(J)\big)_j\geq \dim_K {\ensuremath{\mathrm{Tor}}}_i\big(M, S/D_{\lambda,{\mathbf w}}(J)\big)_j
\ \mbox{ for all $i$.}$$
Consider the ideal $\tilde {J} \subset R$ defined as above. Let $\tilde M = M \otimes_S R$ and $T_i={\ensuremath{\mathrm{Tor}}}_i^{R}(\tilde M,R/\tilde{J})$. By the structure theorem for modules over a PID (see [@La p. 149]), we have $$(T_i)_j\cong K[T]^{a_{ij}}
\bigoplus A_{ij}$$ as a finitely generated $K[T]$-module, where $a_{ij} \in { \ensuremath{\mathbb{Z}}}_{\geq 0}$ and where $A_{ij}$ is the torsion submodule. Moreover $A_{ij}$ is a module of the form $$A_{ij}\cong \bigoplus_{h=1}^{b_{ij}} K [T]/(P^{i,j}_{h}),$$ where $P^{i,j}_h$ is a non-zero polynomial in $K[T]$. Set $l_{\lambda}=T-\lambda$. Consider the exact sequence $$\begin{aligned}
\label{aa}
\begin{CD} 0 @>>> R/\tilde{J} @>\cdot l_{\lambda}>>
R/\tilde{J} @>>> R/\big((l_{\lambda})+\tilde{J} \big) @>>> 0.
\end{CD}\end{aligned}$$ By considering the long exact sequence induced by ${\ensuremath{\mathrm{Tor}}}^R_i(\tilde M,-),$ we have the following exact sequence $$\label{bo} 0\longrightarrow T_i/l_{\lambda} T_i \longrightarrow
{\ensuremath{\mathrm{Tor}}}_i^{R}\big(\tilde M,R/\big((l_{\lambda})+\tilde{J}\big)\big) \longrightarrow
K_{i-1} \longrightarrow 0,
$$ where $K_{i-1}$ is the kernel of the map $T_{i-1} \xrightarrow{\cdot l_{\lambda}} T_{i-1}$. Since $l_{\lambda}$ is a regular element for $R$ and $\tilde M$, the middle term in (\[bo\]) is isomorphic to $$\begin{aligned}
{\ensuremath{\mathrm{Tor}}}_i^{R/(l_\lambda)} \big(\tilde M /l_\lambda \tilde M, R/\big((l_{\lambda})+\tilde J \big)\big)
=\left\{
\begin{array}{lll}
{\ensuremath{\mathrm{Tor}}}_i^S \big(M,S/{\ensuremath{\mathrm{in}}}_{\mathbf w}(J)\big), & \mbox{ if } \lambda=0,\\
{\ensuremath{\mathrm{Tor}}}_i^S \big(M,S/D_{\lambda,{\mathbf w}}(J)\big), & \mbox{ if } \lambda\ne0
\end{array}
\right.\end{aligned}$$ (see [@Mat p. 140]). By taking the graded component of degree $j$ in (\[bo\]), we obtain $$\begin{aligned}
\label{banngou}
\begin{array}{lll}
\dim_K
{\ensuremath{\mathrm{Tor}}}_i^{S}\big(M,S/{\ensuremath{\mathrm{in}}}_{\mathbf w}(J) \big)_j &=& a_{ij} +
\# \{P^{ij}_h : P^{i,j}_h(0)=0\}\\
&& + \# \{P^{i-1,j}_h : P^{i-1,j}_h(0)=0\},
\end{array}\end{aligned}$$ where $\# X$ denotes the cardinality of a finite set $X$, and $$\begin{aligned}
\label{yon}
\dim_K
{\ensuremath{\mathrm{Tor}}}_i^{S}\big(M,S/D_{\lambda,{\mathbf w}}(J) \big)_j &=& a_{ij}\end{aligned}$$ for a general $\lambda \in K$. This proves the desired inequality.
\[add\] With the same notation as in Lemma \[lemma2\], for a general $\lambda \in K$, $$\dim_K {\ensuremath{\mathrm{Tor}}}_i \big(M,{\ensuremath{\mathrm{in}}}_{\mathbf w}(J)\big)_j \geq \dim_K {\ensuremath{\mathrm{Tor}}}_i \big(M, D_{\lambda,{\mathbf w}}(J) \big)_j
\mbox{ for all }i.$$
For any homogeneous ideal $I \subset S$, by considering the long exact sequence induced by ${\ensuremath{\mathrm{Tor}}}_i(M,-)$ from the short exact sequence $0 \longrightarrow I \longrightarrow S \longrightarrow S/I \longrightarrow 0$ we have $${\ensuremath{\mathrm{Tor}}}_i(M,I) \cong {\ensuremath{\mathrm{Tor}}}_{i+1}(M,S/I)
\mbox{ for }i \geq 1$$ and $$\dim_K {\ensuremath{\mathrm{Tor}}}_0(M,I)_j = \dim_K {\ensuremath{\mathrm{Tor}}}_1(M,S/I)_j + \dim_K M_j - \dim_K {\ensuremath{\mathrm{Tor}}}_0(M,S/I)_j.$$ Thus by Lemma \[lemma2\] it is enough to prove that $$\begin{aligned}
&&\dim_K {\ensuremath{\mathrm{Tor}}}_1\big(M,S/{\ensuremath{\mathrm{in}}}_{\mathbf w}(J)\big)_j -\dim_K {\ensuremath{\mathrm{Tor}}}_1\big(M,S/D_{\lambda,{\mathbf w}}(J)\big)_j\\
&&\geq \dim_K {\ensuremath{\mathrm{Tor}}}_0\big(M,S/{\ensuremath{\mathrm{in}}}_{\mathbf w}(J)\big)_j -\dim_K {\ensuremath{\mathrm{Tor}}}_0\big(M,S/D_{\lambda,{\mathbf w}}(J)\big)_j.\end{aligned}$$ This inequality follows from (\[banngou\]) and (\[yon\]).
\[2.3\] Fix an integer $j$. Let $I \subset S$ and $J \subset S$ be homogeneous ideals. Let ${\mathbf w},{\mathbf w}' \in { \ensuremath{\mathbb{Z}}}_{\geq 0}^n$. For a general change of coordinates $g \in {{GL}}_n(K)$,
- $\dim_K {\ensuremath{\mathrm{Tor}}}_i(S/I,S/g(J))_j
\leq \dim_K {\ensuremath{\mathrm{Tor}}}_i (S/{\ensuremath{\mathrm{in}}}_{{\mathbf w}}(I), S/{{\ensuremath{\mathrm{in}}}_{{\mathbf w}'}}(J))_j
\ \mbox{ for all }i.$
- $\dim_K {\ensuremath{\mathrm{Tor}}}_i(I,S/g(J))_j
\leq \dim_K {\ensuremath{\mathrm{Tor}}}_i ({\ensuremath{\mathrm{in}}}_{{\mathbf w}}(I), S/{{\ensuremath{\mathrm{in}}}_{{\mathbf w}'}}(J))_j
\ \mbox{ for all }i.$
We prove (ii) (the proof for (i) is similar). By Lemmas \[lemma1\] and \[lemma2\] and Corollary \[add\], we have $$\begin{aligned}
\dim_K {\ensuremath{\mathrm{Tor}}}_i \big({\ensuremath{\mathrm{in}}}_{{\mathbf w}}(I), S/{\ensuremath{\mathrm{in}}}_{{\mathbf w}'}(J)\big)_j
&\geq& \dim_K {\ensuremath{\mathrm{Tor}}}_i \big(D_{\lambda_1,{\mathbf w}}(I), S/D_{\lambda_2,{\mathbf w}'}(J)\big)_j \\
&=& \dim_K {\ensuremath{\mathrm{Tor}}}_i \big(I, S/D^{-1}_{\lambda_1,{\mathbf w}} \big(D_{\lambda_2,{\mathbf w}'}(J)\big)\big)_j\\
&\geq& \dim_K {\ensuremath{\mathrm{Tor}}}_i\big(I,S/g(J)\big)_j,\end{aligned}$$ as desired, where $\lambda_1,\lambda_2$ are general elements in $K$.
Let ${\mathbf w}'=(1,1,\dots,1)$ and note that the composite of two general changes of coordinates is still general. By replacing $J$ by $h(J)$ for a general change of coordinates $h,$ from Proposition \[2.3\](i) it follows that $$\dim_K {\ensuremath{\mathrm{Tor}}}_i(S/I,S/h(J))_j \leq \dim_K {\ensuremath{\mathrm{Tor}}}_i\big(S/{\ensuremath{\mathrm{in}}}_{>_{\sigma}}(I),S/h(J))_j$$ for any term order $>_\sigma$.
The above fact gives, as a special case, an affirmative answer to [@Co Question 6.1]. This was originally proved in the thesis of the first author [@Ca2]. We mention it here because there seem to be no published article which includes the proof of this fact.
\[2.5\] Fix an integer $j$. Let $I \subset S$ and $J \subset S$ be homogeneous ideals. For a general change of coordinates $g \in {{GL}}_n(K)$,
- $\dim_K {\ensuremath{\mathrm{Tor}}}_i(S/I,S/g(J))_j \leq \dim_K {\ensuremath{\mathrm{Tor}}}_i(S/{\ensuremath{\mathrm{Gin}}}_{{\mathrm{lex}}}(I),S/{\ensuremath{\mathrm{Gin}}}_{{\mathrm{{oplex}}}}(J))_j
\ \ \mbox{for all }i.$
- $\dim_K {\ensuremath{\mathrm{Tor}}}_i(I,S/g(J))_j \leq \dim_K {\ensuremath{\mathrm{Tor}}}_i({\ensuremath{\mathrm{Gin}}}_{{\mathrm{lex}}}(I),S/{\ensuremath{\mathrm{Gin}}}_{{\mathrm{{oplex}}}}(J))_j
\ \ \mbox{for all }i.$
Without loss of generality, we may assume ${\ensuremath{\mathrm{in}}}_{{\mathrm{lex}}}(I)={\ensuremath{\mathrm{Gin}}}_{{\mathrm{lex}}}(I)$ and that ${\ensuremath{\mathrm{in}}}_{{\mathrm{{oplex}}}}(J)={\ensuremath{\mathrm{Gin}}}_{{\mathrm{{oplex}}}}(J)$. It follows from [@Ei Propositin 15.16] that there are vectors ${\mathbf w}, {\mathbf w}' \in { \ensuremath{\mathbb{Z}}}_{\geq 0}^n$ such that ${\ensuremath{\mathrm{in}}}_{\mathbf w}(I)={\ensuremath{\mathrm{in}}}_{{\mathrm{lex}}}(I)$ and ${\ensuremath{\mathrm{in}}}_{{\mathbf w}'}(g(J))={\ensuremath{\mathrm{Gin}}}_{{\mathrm{{oplex}}}}(J)$. Then the desired inequality follows from Proposition \[2.3\].
Since ${\ensuremath{\mathrm{Tor}}}_0(S/I,S/J)\cong S/(I+J)$ and ${\ensuremath{\mathrm{Tor}}}_0(I,S/J)\cong I/IJ$, we have the next corollary.
\[2.6\] Let $I \subset S$ and $J \subset S$ be homogeneous ideals. For a general change of coordinates $g \in {{GL}}_n(K)$,
- $H(I \cap g(J) ,d) \leq H({\ensuremath{\mathrm{Gin}}}_{{\mathrm{lex}}}(I)\cap {\ensuremath{\mathrm{Gin}}}_{{\mathrm{{oplex}}}}(J),d)$ for all $d \geq 0$.
- $H(Ig(J),d) \geq H({\ensuremath{\mathrm{Gin}}}_{{\mathrm{lex}}}(I){\ensuremath{\mathrm{Gin}}}_{{\mathrm{{oplex}}}}(J),d)$ for all $d \geq 0$.
We conclude this section with a result regarding the Krull dimension of certain Tor modules. We show how Theorem \[2.5\] can be used to give a quick proof of Proposition \[MiSp\], which is a special case (for the variety $X=\mathbb{P}^{n-1}$ and the algebraic group ${{SL}}_n$) of the main Theorem of [@MS].
Recall that generic initial ideals are *Borel-fixed*, that is they are fixed under the action of the Borel subgroup of ${{GL}}_n(K)$ consisting of all the upper triangular invertible matrices. In particular for an ideal $I$ of $S$ and an upper triangular matrix $b\in {{GL}}_n(K)$ one has $b({\ensuremath{\mathrm{Gin}}}_{{\mathrm{lex}}}(I))= {\ensuremath{\mathrm{Gin}}}_{{\mathrm{lex}}}(I).$ Similarly, if we denote by $op$ the change of coordinates of $S$ which sends $x_i$ to $x_{n-i}$ for all $i=1,\dots,n,$ we have that $b( op ({\ensuremath{\mathrm{Gin}}}_{{\mathrm{{oplex}}}}(I)))= op ({\ensuremath{\mathrm{Gin}}}_{{\mathrm{{oplex}}}}(I)).$
We call *opposite Borel-fixed* an ideal $J$ of $S$ such that $op(J)$ is Borel-fixed (see [@Ei §15.9] for more details on the combinatorial properties of Borel-fixed ideals).
It is easy to see that if $J$ is Borel-fixed, then so is $(x_1,\dots,x_i)+J$ for every $i=1,\dots,n.$ Furthermore if $j$ is an integer equal to $\min \{i : x_i\not \in J \}$ then $J:x_j$ is also Borel-fixed; in this case $I$ has a minimal generator divisible by $x_j$ or $I=(x_1,\dots,x_{j-1}).$ Analogous statements hold for opposite Borel-fixed ideals.
Let $I$ and $J$ be ideals generated by linear forms. If we assume that $I$ is Borel fixed and that $J$ is opposite Borel fixed, then there exist $1\leq i,j \leq n $ such that $I=(x_1,\dots,x_i)$ and $J=(x_j,\dots,x_n).$ An easy computation shows that the Krull dimension of ${\ensuremath{\mathrm{Tor}}}_i(S/I,S/J)$ is always zero when $i>0.$
More generally one has
\[MiSp\] Let $I$ and $J$ be two homogeneous ideals of $S.$ For a general change of coordinates $g$, the Krull dimension of ${\ensuremath{\mathrm{Tor}}}_i(S/I,S/g(J))$ is zero for all $i>0.$
When $I$ or $J$ are equal to $(0)$ or to $S$ the result is obvious. Recall that a finitely generated graded module $M$ has Krull dimension zero if and only if $M_d=0$ for all $d$ sufficiently large. In virtue of Theorem \[2.5\] it is enough to show that ${\ensuremath{\mathrm{Tor}}}_i(S/I,S/J)$ has Krull dimension zero whenever $I$ is Borel-fixed, $J$ opposite Borel-fixed and $i>0.$ By contradiction, let the pair $I,J$ be a maximal counterexample (with respect to point-wise inclusion). By the above discussion, and by applying $op$ if necessary, we can assume that $I$ has a minimal generator of degree greater than 1. Let $j=\min \{h : x_h\not \in I \}$ and notice that both $(I:x_j)$ and $(I+(x_j))$ strictly contain $I.$ For every $i>0$ the short exact sequence $ 0 \rightarrow S/(I:x_j) \rightarrow S/I \rightarrow S/(I+(x_j)) \rightarrow 0$ induces the exact sequence $${\ensuremath{\mathrm{Tor}}}_i(S/(I:x_j),S/J)
\rightarrow
{\ensuremath{\mathrm{Tor}}}_i(S/I,S/J)
\rightarrow
{\ensuremath{\mathrm{Tor}}}_i(S/(I+(x_j)),S/J).$$ By the maximality of $I,J$, the first and the last term have Krull dimension zero. Hence the middle term must have dimension zero as well, contradicting our assumption.
General intersections and general products
==========================================
In this section, we prove Theorems \[intersection\] and \[product\]. We will assume throughout the rest of the paper $\mathrm{char}(K)=0.$
A monomial ideal $I \subset S$ is said to be *$0$-Borel* (or *strongly stable*) if, for every monomial $u x_j \in I$ and for every $1 \leq i <j$ one has $ux_i \in I$. Note that $0$-Borel ideals are precisely all the possible Borel-fixed ideals in characteristic $0$. In general, the Borel-fixed property depends on the characteristic of the field and we refer the readers to [@Ei §15.9] for the details. A set $W \subset S$ of monomials in $S$ is said to be *$0$-Borel* if the ideal they generate is $0$-Borel, or equivalently if for every monomial $u x_j \in W$ and for every $1 \leq i <j$ one has $ux_i \in W$. Similarly we say that a monomial ideal $J \subset S$ is *opposite $0$-Borel* if for every monomial $ux_j \in J$ and for every $j < i \leq n$ one has $ux_i \in J$.
Let $>_{{\mathrm{{rev}}}}$ be the reverse lexicographic order induced by the ordering $x_1 > \cdots >x_n$. We recall the following result [@Mu Lemma 3.2].
\[3-1\] Let $V=\{v_1,\dots,v_s\} \subset S_d$ be a $0$-Borel set of monomials and $W =\{w_1,\dots,w_s\} \subset S_d$ the lex set of monomials, where $v_1 \geq_{{{\mathrm{{rev}}}}} \cdots \geq_{{{\mathrm{{rev}}}}} v_s$ and $w_1 \geq_{{{\mathrm{{rev}}}}} \cdots \geq _{{{\mathrm{{rev}}}}} w_s$. Then $v_i \geq_{{{\mathrm{{rev}}}}} w_i$ for all $i=1,2,\dots,s$.
Since generic initial ideals with respect to $>_{{\mathrm{lex}}}$ are $0$-Borel, the next lemma and Corollary \[2.6\](i) prove Theorem \[intersection\].
\[3-2\] Let $I \subset S$ be a $0$-Borel ideal and $P \subset S$ an opposite lex ideal. Then $\dim_K(I\cap P)_d \leq \dim_K (I^{{\mathrm{lex}}}\cap P)_d$ for all $d\geq 0$.
Fix a degree $d$. Let $V,W$ and $Q$ be the sets of monomials of degree $d$ in $I$, $I^{{\mathrm{lex}}}$ and $P$ respectively. It is enough to prove that $\# V \cap Q \leq \# W \cap Q$.
Observe that $Q$ is the set of the smallest $\#Q$ monomials in $S_d$ with respect to $>_{{\mathrm{{rev}}}}$. Let $m=\max_{>_{{\mathrm{{rev}}}}} Q$. Then by Lemma \[3-1\] $$\# V \cap Q = \# \{ v \in V: v \leq_{{{\mathrm{{rev}}}}} m\}
\leq \# \{ w \in W: w \leq_{{{\mathrm{{rev}}}}} m\} = \# W \cap Q,$$ as desired.
Next, we consider products of ideals. For a monomial $u \in S$, let $\max u$ (respectively, $\min u$) be the maximal (respectively, minimal) integer $i$ such that $x_i$ divides $u$, where we set $\max 1 = 1$ and $\min 1 = n$. For a monomial ideal $I \subset S$, let $I_{(\leq k)}$ be the K-vector space spanned by all monomials $u \in I$ with $\max u \leq k$.
\[3-4\] Let $I \subset S$ be a $0$-Borel ideal and $P \subset S$ an opposite $0$-Borel ideal. Let $G(P)=\{u_1,\dots,u_s\}$ be the set of the minimal monomial generators of $P$. As a $K$-vector space, $IP$ is the direct sum $$IP=\bigoplus_{i=1}^s (I_{(\leq \min u_i)})u_i.$$
It is enough to prove that, for any monomial $w \in IP$, there is the unique expression $w=f(w)g(w)$ with $f(w) \in I$ and $g(w) \in P$ satisfying
- $\max f(w) \leq \min g(w)$.
- $g(w) \in G(P)$.
Given any expression $w=fg$ such that $f \in I$ and $g \in P$, since $I$ is $0$-Borel and $P$ is opposite $0$-Borel, if $\max f > \min g$ then we may replace $f$ by $f \frac{x_{\min g}} {x_{\max f}} \in I$ and replace $g$ by $g \frac{x_{\max f}} {x_{\min g}} \in P$. This fact shows that there is an expression satisfying (a) and (b).
Suppose that the expressions $w=f(w)g(w)$ and $w=f'(w)g'(w)$ satisfy conditions (a) and (b). Then, by (a), $g(w)$ divides $g'(w)$ or $g'(w)$ divides $g(w)$. Since $g(w)$ and $g'(w)$ are generators of $P$, $g(w)=g'(w)$. Hence the expression is unique.
\[3-5\] Let $I \subset S$ be a $0$-Borel ideal and $P \subset S$ an opposite $0$-Borel ideal. Then $\dim_K(IP)_d \geq \dim_K (I^{{{\mathrm{lex}}}}P)_d$ for all $d\geq 0$.
Lemma \[3-1\] shows that $\dim_K {I_{(\leq k)}}_d \geq \dim_K {I^{{\mathrm{lex}}}_{(\leq k)}}_d$ for all $k$ and $d \geq 0$. Then the statement follows from Lemma \[3-4\].
Finally we prove Theorem \[product\].
Let $I'={\ensuremath{\mathrm{Gin}}}_{{\mathrm{lex}}}(I)$ and $J'={\ensuremath{\mathrm{Gin}}}_{{\mathrm{{oplex}}}}(J)$. Since $I'$ is $0$-Borel and $J'$ is opposite $0$-Borel, by Corollary \[2.6\](ii) and Lemmas \[3-5\] $$H(Ig(J),d) \geq H(I'J',d) \geq H(I^{{{\mathrm{lex}}}} J',d) \geq H(I^{{\mathrm{lex}}}J^{{\mathrm{{oplex}}}},d)$$ for all $d \geq 0$.
\[rem1\] Theorems \[intersection\] and \[product\] are sharp. Let $I \subset S$ be a Borel-fixed ideal and $J \subset S$ an ideal satisfying that $h(J)=J$ for any lower triangular matrix $h \in {{GL}}_n(K)$. For a general $g \in {{GL}}_n(K)$, we have the LU decomposition $g=bh$ where $h \in {{GL}}_n(K)$ is a lower triangular matrix and $b \in {{GL}}_n(K)$ is an upper triangular matrix. Then as $K$-vector spaces $$I \cap g(J) \cong b^{-1}(I) \cap h(J)= I\cap J
\mbox{ and }
I g(J) \cong b^{-1}(I) h(J)= I J.$$ Thus if $I$ is lex and $J$ is opposite lex then $H(I\cap g(J),d)=H(I\cap J,d)$ and $H(Ig(J),d)=H(I J,d)$ for all $d\geq 0$.
\[example\] The assumption on ${\ensuremath{\mathrm{Gin}}}_{{\mathrm{lex}}}(J)$ in Theorem \[intersection\] is necessary. Let $I=(x_1^3,x_1^2x_2,x_1x_2^2,x_2^3) \subset K[x_1,x_2,x_3]$ and $J=(x_3^2,x_3^2x_2,x_3x_2^2,x_2^3)\subset K[x_1,x_2,x_3]$. Then the set of monomials of degree $3$ in $I^{{\mathrm{lex}}}$ is $\{x_1^3,x_1^2x_2,x_1^2x_3,x_1x_2^2\}$ and that of $J^{{\mathrm{{oplex}}}}$ is $\{x_3^3,x_3^2x_2,x_3^2x_1,x_3x_2^2\}$. Hence $H(I^{{\mathrm{lex}}}\cap J^{{\mathrm{{oplex}}}},3)=0$. On the other hand, as we see in Remark \[rem1\], $H(I\cap g(J),3)=H(I\cap J,3)=1$. Similarly, the assumption on the characteristic of $K$ is needed as one can easily see by considering $\mathrm{char}(K)=p>0$, $I=(x_1^p,x_2^p)\subset K[x_1,x_2]$ and $J=x_2^p.$ In this case we have $H(I^{{\mathrm{lex}}}\cap J^{{\mathrm{{oplex}}}},p)=0$, while $H(I\cap g(J),p)=H(g^{-1}(I)\cap J,p)=1$ since $I$ is fixed under any change of coordinates.
Since ${\ensuremath{\mathrm{Tor}}}_0(S/I,S/J) \cong S/(I+J)$ and ${\ensuremath{\mathrm{Tor}}}_1(S/I,S/J) \cong (I\cap J)/ IJ$ for all homogeneous ideals $I \subset S$ and $J \subset S$, Theorems \[intersection\] and \[product\] show the next statement.
\[cor\] Conjecture \[conj\] is true if $i=0$ or $i=1.$
[1]{}
A. Aramova and J. Herzog, Koszul cycles and Eliahou-Kervaire type resolutions, *J. Algebra* **181** (1996), 347–370.
A. Bigatti, Upper bounds for the Betti numbers of a given Hilbert function, *Comm. Algebra* **21** (1993), 2317–2334.
G. Caviglia, The pinched Veronese is Koszul, *J. Algebraic Combin.* **30** (2009), 539–548.
G. Caviglia, Kozul algebras, Castelnuovo–Mumford regularity and generic initial ideals, Ph.d Thesis, 2004.
A. Conca, Reduction numbers and initial ideals, *Proc. Amer. Math. Soc.* **131** (2003), 1015–1020.
A. Conca, Koszul homology and extremal properties of Gin and Lex, *Trans. Amer. Math. Soc.* **356** (2004), 2945–2961.
D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Grad. Texts in Math., vol. 150, Springer-Verlag, New York, 1995.
V. Gasharov, Hilbert functions and homogeneous generic forms, II, *Compositio Math.* **116** (1999), 167–172.
G. Gotzmann, Einige einfach-zusammenhängende Hilbertschemata, *Math. Z.* [**180**]{} (1982), 291–305.
M. Green, Restrictions of linear series to hyperplanes, and some results of Macaulay and Gotzmann, in: Algebraic Curves and Projective Geometry (1988), in: Trento Lecture Notes in Math., vol. 1389, Springer, Berlin, 1989, pp. 76–86.
J. Herzog and D. Popescu, Hilbert functions and generic forms, *Compositio Math.* **113** (1998), 1–22.
H. Hulett, Maximum Betti numbers of homogeneous ideals with a given Hilbert function, *Comm. Algebra* **21** (1993), 2335–2350.
S. Lang, Algebra, Revised third edition, Grad. Texts in Math., vol. 211, Springer-Verlag, New York, 2002.
F.S. Macaulay, Some properties of enumeration in the theory of modular systems, *Proc. London Math. Soc.* **26** (1927), 531–555.
H. Matsumura, Commutative Ring Theory, Cambridge University Press, 1986.
E. Miller, D. Speyer, A Kleiman-Bertini theorem for sheaf tensor products. *J. Algebraic Geom.* **17** (2008), no. 2, 335–340.
S. Murai, Borel-plus-powers monomial ideals, *J. Pure Appl. Algebra* **212** (2008), 1321–1336.
S. Murai, Free resolutions of lex-ideals over a Koszul toric ring, *Trans. Amer. Math. Soc.* **363** (2011), 857–885.
K. Pardue, Deformation classes of graded modules and maximal Betti numbers, *Illinois J. Math.* **40** (1996), 564–585.
[^1]: The work of the first author was supported by a grant from the Simons Foundation (209661 to G. C.). The work of the second author was supported by KAKENHI 22740018.
| ArXiv |
---
abstract: 'We study electronic structure of hole- and electron-doped Mott insulators in the two-dimensional Hubbard model to reach a unified picture for the normal state of cuprate high-$T_{\text{c}}$ superconductors. By using a cluster extension of the dynamical mean-field theory, we demonstrate that structure of coexisting zeros and poles of the single-particle Green’s function holds the key to understand Mott physics in the underdoped region. We show evidence for the emergence of non-Fermi-liquid phase caused by the topological quantum phase transition of Fermi surface by analyzing low-energy charge dynamics. The spectra calculated in a wide range of energy and momentum reproduce various anomalous properties observed in experiments for the high-$T_{\text{c}}$ cuprates. Our results reveal that the pseudogap in hole-doped cuprates has a $d$-wave-like structure only below the Fermi level, while it retains non-$d$-wave structure with a fully opened gap above the Fermi energy even in the nodal direction due to a zero surface extending over the entire Brillouin zone. In addition to the non-$d$-wave pseudogap, the present comprehensive identifications of the spectral asymmetry as to the Fermi energy, the Fermi arc, and the back-bending behavior of the dispersion, waterfall, and low-energy kink, in agreement with the experimental anomalies of the cuprates, do not support that these originate from (the precursors of) symmetry breakings such as the preformed pairing and the $d$-density wave fluctuations, but support that they are direct consequences of the proximity to the Mott insulator. Several possible experiments are further proposed to prove or disprove our zero mechanism.'
author:
- 'Shiro Sakai,$^{1,2}$ Yukitoshi Motome,$^2$ and Masatoshi Imada$^2$'
title: |
Doped high-$T_{\text{c}}$ cuprate superconductors\
elucidated in the light of zeros and poles of electronic Green’s function
---
INTRODUCTION
============
Anomalous behaviors of high-$T_{\text{c}}$ cuprates observed in the normal metallic state above $T_{\text{c}}$ hold the key not only to understanding the mechanism of the superconductivity but also to a possible manifestation of an unexplored metallic phase distinguished from the Fermi liquid.[@ts99] Extraordinary electronic structure is induced by a small density of carrier doping into the Mott insulator. Angle-resolved photoemission spectroscopies (ARPES) have in fact revealed detailed anomalies of the normal-state spectra, such as momentum-dependent excitation gap (pseudogap), a truncated Fermi surface (Fermi arc), and kinks in the dispersion.[@dh03]
Toward the understanding of the anomalous metals, especially the pseudogap formation, many theoretical proposals have been made so far.[@yj03] The proposals include a Cooper paring without phase coherence,[@ek95] and hidden orders or its fluctuations competing with the superconductivity, such as antiferromagnetism,[@ks90; @p97; @vt97] charge or stripe orderings,[@kb03] and $d$-density wave.[@v97; @cl01] Mechanisms attributing the origin of the pseudogap to a direct consequence of the proximity to the Mott insulator have also been proposed.[@rice05; @sp03; @sk06; @p09; @sm09]
Among the theoretical efforts, recent development of the dynamical mean-field theory (DMFT) [@gk96] and its cluster extensions [@ks01; @mj05] has enabled studies on dynamics of microscopic models without any [*ad hoc*]{} approximation. In particular, studies on the two-dimensional (2D) Hubbard model using the cluster DMFT (CDMFT) have offered many useful insights into the electronic structure of cuprates, by identifying the pseudogap, Fermi arc,[@mp02; @st04; @sm09; @cc05; @kk06; @sk06; @mj06; @aa06; @hk07; @lt09; @fc09] and high-energy kink [@mj07] in the calculated spectra. The CDMFT is specifically suited and powerful for this problem because of its nonperturbative framework, namely, it is based on neither weak nor strong coupling expansions. Furthermore, it takes account of short-range spatial correlations within a cluster explicitly. These are big advantages in exploring momentum-resolved dynamics in the intermediate coupling region, which is relevant to physics of the cuprates. In particular, recent CDMFT studies on doped Mott insulators have revealed emergence of non-Fermi-liquid phases, characterized by unexpected coexistence of two singularities at the Fermi level, one characteristic to the weak-coupling and the other to the strong-coupling regions.[@sk06; @sm09; @sm09-2; @lt09; @wg09]
The singularity characterizing weakly interacting metals, namely, Fermi liquids, is a pole of the single-particle Green’s function $G$. Energy dependence of its locus in the momentum space determines band dispersions, particularly, the Fermi surface at the Fermi level. On the other hand, a crucial singularity in the strong coupling region is a zero of $G$. The situations are illustrated in Fig. \[fig:zero\], which schematically shows how the real part of $G$ changes the sign in the energy-momentum space. Because Re$G$ must be positive at high energy while negative at low energy, it has to change its sign at least once between these two regions. The sign change also has to take place in the Brillouin zone at the Fermi level, say, between the zone center and the zone boundary, if the dispersion far from the Fermi level ensures positive Re$G$ in some part of momenta and negative Re$G$ in the other part. In the normal metals, the sign change occurs at the poles of $G$ (i.e., the band dispersions), where Re$G$ goes to $-\infty$ on the lower side of the band and comes back from $+\infty$ on the upper side \[Fig. \[fig:zero\](a)\]. On the other hand, what happens when correlation effects induce a gap in the band at the Fermi level? Because of the absence of poles inside the gap by definition, there is no way for Re$G$ to change its sign at the Fermi level except for getting through zeros \[Fig. \[fig:zero\](b)\].[@d03; @sp07] Re$G=0$ means the divergence of the self-energy, so that a zero of $G$ is a singular point of the self-energy.
The non-Fermi liquids discovered in CDMFT show a coexistence of these two characteristics — poles and zeros of $G$ at the Fermi level.[@sk06; @sm09] In a lightly hole-doped region, the poles form a hole-pocket Fermi surface around the nodal direction \[from $\Vec{k} = (0,0)$ to $(\pi,\pi)$\], while the zeros form a surface enclosing $(\pi,\pi)$. It has been suggested that the Fermi arc emerges because of such peculiar electronic structure: The pocket loses the spectral intensity on the side closer to the zero surface, leaving the other side as an arc.[@sk06; @sm09; @sm09-2] Similar mechanisms were proposed, based on an assumed functional form of Green’s functions [@kr06; @ks06] or a weakly-coupled chain model.[@et02; @bg06] The pseudogap is also characterized by the same zero surface crossing the Fermi level. In the previous paper [@sm09] we clarified the structure of poles and zeros in the whole energy-momentum space for hole-doped and undoped Mott insulators. We thus proposed a unified picture for understanding various puzzling features of Mott physics in the light of reconstructions of pole-zero structure under progressive doping from the Mott insulator to the Fermi liquid in the overdoped region.
The aim of the present paper is to show how and to what extent we can understand experimental findings within the ‘pole-zero mechanism’. We implement CDMFT calculations at zero and finite temperatures, clarifying the relationship between zero-temperature pole-zero structure and the spectra observed in experiments at finite temperatures. We find that a number of non-Fermi-liquid aspects, i.e., spectral asymmetry as to the Fermi level, back-bending or incoherent feature of the dispersion, high- and low-energy kinks, and Fermi arc or pockets in the hole- and electron-doped cuprates, are comprehensively understood within the pole-zero mechanism. In addition to a coherent picture for various experiments, our numerical results predict a distinctive feature of the pseudogap: The numerical data support a fully opened gap above the Fermi level even in the nodal direction, which is incompatible with scenarios on the basis of the $d$-wave gap in the zero-temperature limit. Our result offers a mechanism distinct from the scenarios based on the preformed pairing or $d$-density-wave order.
The paper is organized as follows. In Sec. \[sec:method\] we introduce the 2D Hubbard model and present the central ideas of the CDMFT. We also describe the essence of numerical solvers for the effective cluster problem that we used in the present study. The numerical results are presented and discussed in comparison with various experiments in Sec. \[sec:result\]. From the low-energy pole-zero structure, we derive a simple interpretation of several unusual features in hole-doped cuprates (Sec. \[ssec:nond\]-\[ssec:back\]). We also find several anomalies in the band dispersion, which are consistent with ARPES data (Sec. \[ssec:edc\]-\[ssec:kink\]). In Sec. \[ssec:arc\] we discuss the results of the Fermi surface in comparison with the recent ARPES observation of the hole pocket. The comparison is further made in Sec. \[ssec:tp\] by changing the next-nearest-neighbor hopping to reach more quantitative understanding of the experimental results. In Sec. \[ssec:tp\], \[ssec:ele2\], and \[ssec:ele\], we extend our theory to electron-doped cuprates and find good agreements with ARPES data. We summarize our results and make concluding remarks in Sec. \[sec:summary\].
MODEL AND METHOD {#sec:method}
================
As a simplest model for high-$T_{\text{c}}$ cuprates we take the Hubbard Hamiltonian, $$\begin{aligned}
H= \sum_{\Vec{k}\s}\e(\Vec{k})c_{\Vec{k}\s}^\dagger c_{\Vec{k}\s}
-\mu \sum_{i\s}n_{i\s}+ U\sum_{i}n_{i\ua}n_{i\da},
\label{eq:hubbard}\end{aligned}$$ on a square lattice. Here $c_{\Vec{k}\s}$ $(c_{\Vec{k}\s}^\dagger)$ annihilates (creates) an electron of spin $\s$ with momentum $\Vec{k} = (k_x, k_y)$, $c_{i\s}$ $(c_{i\s}^\dagger)$ is its Fourier component at site $i$, and $n_{i\s}\equiv c_{i\s}^\dagger c_{i\s}$. $U$ represents the onsite Coulomb repulsion, $\mu$ the chemical potential, and $$\begin{aligned}
\e(\Vec{k})\equiv -2t(\cos k_x + \cos k_y) -4t' \cos k_x\cos k_y,
\label{eq:disp}\end{aligned}$$ where $t$ $(t')$ is the (next-)nearest-neighbor transfer integral.
Based on a first principles calculation, the value of $t$ was estimated to be $\sim 0.4$eV for La$_2$CuO$_4$.[@hs90] $-t'/t$ is considered to be $\sim 0.2$ and $\sim 0.4$ for La$_{2-x}$Sr$_x$CuO$_4$ and Bi$_2$Sr$_2$CaCu$_2$O$_{8+\d}$, respectively. We adopt $U=8t$ or $12t$, which are realistic values for the cuprates and indeed reproduce the Mott insulating state for undoped case.
In the CDMFT [@ks01] we map the system (\[eq:hubbard\]) onto a model consisting of an $N_{\text{c}}$-site cluster C and bath degrees of freedom B. The bath is determined in a self-consistent way to provide an $N_{\text{c}}\times N_{\text{c}}$ dynamical mean-field matrix $\hat{g}_0(i\w_n)$ at temperature $T(\equiv 1/\b)$, where $\w_n\equiv (2n+1)\pi T$.
After the self-consistency loop converges, we calculate a quantity $Q^{\text{L}}$ defined on the original lattice from those on the cluster, $Q^{\text{C}}$.[@ks01] This periodization procedure is based on the Fourier transformation truncated by the cluster size $N_{\text{c}}$, $$\begin{aligned}
Q^{\text{L}}(\Vec{k})=\frac{1}{N_{\text{c}}}\sum_{ij\in \text{C}}
[Q^{\text{C}}]_{ij}e^{i\Vec{k}\cdot\Vec{r}_{ij}},
\label{eq:periodize}\end{aligned}$$ where $\Vec{k}$ is defined on the entire Brillouin zone of the original lattice and $\Vec{r}_{ij}$ is the real-space vector connecting two cluster sites $i$ and $j$.
The truncation by a small $N_{\text{c}}$ gives a good approximation to the thermodynamic limit of $N_{\text{c}} \to \infty$ if $Q^{\text{C}}$ is well localized within the cluster. In reality it is difficult to find a quantity which is short ranged in the entire parameter range of the interaction strength and the doping concentration. Therefore we need to choose an appropriate quantity to periodize, according to situations. For example, $Q=\S$ ($\S$: self-energy) is a good choice for weakly interacting normal metals,[@ks01] but it becomes highly nonlocal and long ranged in the strong coupling region (e.g., in the Mott insulator). This is due to an appearance of zeros of $G$, i.e., poles of $\S$ in the momentum space (see APPENDIX A for further detail). On the other hand, in the Mott insulator $Q=G$ is more appropriate because it is nearly local in the strong coupling regime.[@kk06]
Another choice for the periodization is the cumulant $$M=[i\w_n+\mu-\S]^{-1}.$$ It was pointed out that the cumulant periodization works well in a wide range of $U$ including the strong coupling, because it is similar in the functional form to the atomic Green function.[@sk06] Moreover, the periodization by the cumulant has an important feature, i.e., it can describe both poles and zeros of $G$ at the same time, while the periodization by using $\S$ ($G$) describes only poles (zeros). This opens up the intriguing possibility of exploring the coexistence of poles and zeros at the Fermi level, as substantiation of anomalous metals.
In the present study, we adopt the cumulant periodization scheme, $Q=M$, to investigate [*metals in the vicinity of the Mott insulator*]{}. In fact, $M$ is highly local even in the doped metallic states. In APPENDIX B we demonstrate this local nature by CDMFT calculations for $N_{\text{c}}=4\times4$ cluster; we find that $M^{\text{C}}$ is nearly localized already within the inner $2\times 2$ cluster for a parameter region relevant to the present study. After obtaining the lattice cumulant $M^{\text{L}}$ through Eq. (\[eq:periodize\]), we calculate the self-energy $\S^{\text{L}}$, Green’s function $G^{\text{L}}$, and spectral function $A^{\text{L}}$ on the original lattice with $$\begin{aligned}
\S^{\text{L}}(\Vec{k},\w)\equiv
\left[\w+\mu-{M^{\text{L}}}^{-1}(\Vec{k},\w)\right]^{-1},\nonumber\\
G^{\text{L}}(\Vec{k},\w)\equiv
\left[\w+\mu-\e(\Vec{k})-\S^{\text{L}}(\Vec{k},\w)\right]^{-1},\end{aligned}$$ and $$\begin{aligned}
A^{\text{L}}(\Vec{k},\w)\equiv
-\frac{1}{\pi}\text{Im}G^{\text{L}}(\Vec{k},\w).\end{aligned}$$ In the following calculations, we employ an $N_{\text{c}}=2\times 2$ cluster, and concentrate on the paramagnetic metallic solution.
We numerically solve the effective cluster problem by means of the exact diagonalization (ED) method at $T=0$ and the continuous-time quantum Monte Carlo (CTQMC) method at $T>0$, as we elaborate below. We hereafter omit the superscript L in $\S^{\text{L}}$, $G^{\text{L}}$ and $A^{\text{L}}$.
Exact diagonalization method {#ssec:ed}
----------------------------
Although the pseudogap state in cuprates is experimentally detectable only above $T_{\text{c}}$, it is still significant to elucidate the nature of the pseudogap state in the zero temperature limit, by assuming the paramagnetic metal for the doped Mott insulator. This is a circumstance similar to the normal metal, where the concept of the Fermi liquid justified only in the zero temperature limit in the strict sense has proven to be fruitful. For this purpose of clarifying the zero temperature limit, we employ the Lanczos ED method [@l50] for the cluster problem.
In this scheme we take a large but finite value of pseudo inverse temperature $\b'$ ($=100/t$ or $200/t$ throughout the paper), which practically represents the ground state with an energy resolution corresponding to $1/\b'$. We then represent the dynamical mean field $\hat{g}_0$ with a finite number $N_{\text{B}}$ of bath degrees of freedom, which constitute together with C sites the effective Hamiltonian to be diagonalized. We take $N_{\text{B}}=8$ throughout the paper.
The optimization of B sites is done by minimizing the distance function defined by $$\begin{aligned}
d\equiv \sum_{ij\in \text{C}}\sum_n
\left|\left[\hat{g}_0(i\w_n)\right]_{ij}
-\left[\hat{g}_{0,N_{\text{B}}}(i\w_n)\right]_{ij}\right|^2
e^{-\w_n/t},
\label{eq:distance}\end{aligned}$$ where $\hat{g}_{0,N_{\text{B}}}$ is the non-interacting Green’s function for the effective Hamiltonian and we have introduced the exponential weight factor $e^{-\w_n/t}$ with $\w_n\equiv (2n+1)\pi/\b'$ to reproduce more precisely the important low-frequency part. We have examined several other types of distance functions and confirmed that qualitative feature of the results obtained in this paper does not depend on the choice.
An advantage of the Lanczos method is that Green’s function is obtained as a function of real frequency $\w$ by the continued-fraction expansion, $$\begin{aligned}
G(\w)&=\frac{\langle 0| c c^\dagger | 0\rangle}
{\w+i\eta-a_0^>-\displaystyle{ \frac{{b_1^>}^2}{\w+i\eta-a_1^>
- \displaystyle{\frac{{b_2^>}^2}{\w+i\eta-a_2^>-\cdots}}}}}\nonumber\\
&+\frac{\langle 0| c^\dagger c | 0\rangle}
{\w+i\eta-a_0^<-\displaystyle{\frac{{b_1^<}^2}{\w+i\eta-a_1^<
-\displaystyle{\frac{{b_2^<}^2}{\w+i\eta-a_2^<-\cdots}}}}},
\label{eq:cf}\end{aligned}$$ where $|0\rangle$ is the ground-state vector and the coefficients $a_i^{>,<}$ and $b_i^{>,<}$ are the elements of the tridiagonal matrix appearing in the Lanczos algorithm.[@gb87] We take account of up to 2000th order in the expansion. A small positive $\eta$ is introduced to satisfy the causality. In principle, $\eta$ is taken to be infinitesimal, but in practice, it is useful to consider $\eta$ as a parameter, which serves as a resolution in energy or mimics an infinite-size effect not incorporated into the ED calculation.
In the following study, we pursue a further benefit of the parameter $\eta$: We use $\eta$ as a mimic of the source of incoherence in the electronic structure, such as thermal or impurity scattering. This is based on the observation that $\eta$ does not substantially change the location of poles and zeros of $G$ but changes only the sharpness of these singularities.[@sm09; @sm09-2] Indeed, as long as $\eta$ is sufficiently smaller than the typical energy scale of the system, the location of the poles and zeros is virtually determined only by $a_i^{>,<}$ and $b_i^{>,<}$ in Eq. (\[eq:cf\]), which are calculated in the self-consistency loop performed on the Matsubara-frequency axis and independent from $\eta$. Thus, $\eta$ provides an opportunity to get insight into how the electronic structure at finite temperatures evolves from the pole-zero structure at zero temperature. The effect of $\eta$ is confirmed by a direct comparison with the finite-temperature results obtained by QMC introduced below. Note that this smearing technique by $\eta$ is very useful because it is difficult for QMC to obtain the precise electronic structure at real frequencies because it requires an analytic continuation of the numerical data. We use the smearing technique in Sec. \[sec:result\] to compare the CDMFT+ED results with ARPES ones.
Quantum Monte Carlo method {#ssec:ctqmc}
--------------------------
In order to discuss thermal effects directly, we implement QMC calculations for the effective cluster problem. Since the scheme takes into account infinite bath degrees of freedom, the results complement the limitation in the bath size in the CDMFT+ED results as well. We adopt the algorithm based on a weak-coupling series expansion and auxiliary-field transformation. The idea was first proposed by Rombouts [*et al.*]{},[@rh99] applied to DMFT by Sakai [*et al.*]{},[@sa06] and recently formulated in a sophisticated fashion by Gull [*et al.*]{}[@gw08], which enabled a study at sufficiently low temperatures.
In the QMC, we first expand the many-body partition function in the interaction part of the Hamiltonian and apply the Hubbard-Stratonovich decoupling [@rh98; @h83] to the interaction part. This decomposes the many-body partition function into a sum of single-particle systems, which we collect.
While the algorithm is based on the series expansion up to infinite order, it is feasible to obtain a numerically exact result because, after numerical convergence, all the orders in the expansion are virtually taken into account.[@rh99]
We implement the CTQMC sampling for the auxiliary fields, employing the updating algorithm proposed in Ref. . We typically take $2\times10^5$ steps for each QMC simulation, and after convergence in the self-consistency loop, we average over 30 data starting from $g_0$’s which are different within a statistical error bar.
RESULT AND DISCUSSION {#sec:result}
=====================
Non-[*d*]{}-wave pseudogap {#ssec:nond}
--------------------------
First of all, we make a remark on the structure of pseudogap obtained in Fig. \[fig:fig2\](a) in Ref. \[reproduced in Fig. \[fig:fig2\](a)\], which depicts the low-energy pole and zero surfaces calculated by CDMFT+ED for 9% hole doping to the Mott insulator at $U=8t$ and $t'=0$. The pseudogap is formed by the zero surface (red) connecting two separated pole surfaces (green). Here we define the $\Vec{k}$-dependent amplitude $\D(\Vec{k})$ of the pseudogap as the energy difference between the upper and the lower poles at each $\Vec{k}$. Then we see in Figs. \[fig:fig2\](a) and (c) that the gap in the direct transition opens in the entire Brillouin zone, though it is somewhat larger in the antinodal region \[$\D((\pi,0))=0.35t$\] than in the nodal region \[$\D(\Vec{k}_{\text{F1}}\equiv(0.55\pi,0.55\pi))=0.31t$\]. This is one of the central results of this paper, and clearly different from previous scenarios of pseudogap which assume a $d$-wave gap above $T_{\text{c}}$, such as preformed pair,[@ek95] [*d*]{}-density-wave,[@v97; @cl01] resonating valence-bond,[@rice05; @kr06] and nodal-liquid theory,[@bf98; @ft01] since the direct transition gap closes in the nodal direction in these scenarios.
Nevertheless the electronic structure in Fig. \[fig:fig2\](a), which was calculated without any assumption on the gap structure for the microscopic model, is consistent with the ARPES data, as described below. To start with, it should be noted that ARPES observes only the spectra below the Fermi energy $E_{\text{F}}(\equiv 0)$ if $T$ is low, so that the gap amplitude is often estimated by symmetrizing the spectra below and above $E_{\text{F}}$.[@nd98] This is a misleading, artificial procedure because it assumes a symmetric structure of the gap as to the Fermi energy and neglects the structure above $E_{\text{F}}$ if any. Suppose we have the result in Fig. \[fig:fig2\](a) only below $E_{\text{F}}$ and symmetrize it as in the ARPES procedure to estimate the pseudogap, we end up with a $d$-wave like gap since the gap in the part below $E_{\text{F}}$ is larger in the antinodal region while it is smaller or even zero in the nodal region \[this is more clearly seen in Fig. \[fig:fig2\](c) where we plot Re$G^{-1}(\Vec{k},\w)$ against energy and momentum along symmetric lines\]. The situation is schematically shown in Fig. \[fig:fig2\](b). Our results suggest that the “$d$-wave structure" is an artifact of the symmetrizing analysis, and in reality, the pseudogap has a non-$d$-wave (namely, full-gap) structure. In APPENDIX C we confirm that the gap in the nodal direction persists for a larger cluster, by implementing an $N_\text{c}=8$ CDMFT+CTQMC calculation at a low temperature.
It is interesting to examine this interpretation experimentally. Our theory predicts that a gap exists above $E_{\text{F}}$ even in the nodal direction if the paramagnetic metal persists down to zero temperature. This means that the symmetrization [@nd98] used in ARPES breaks down in the pseudogap state due to the spectral asymmetry as to the Fermi energy. This may be examined by spectroscopic or scattering probes to observe unoccupied electronic states above the Fermi level, such as the inverse photoemission spectroscopy (IPES). It is highly desired to reveal the pseudogap structure by combining both PES and IPES without any symmetrization procedure. Other possible experimental probes suited for this purpose may be electron energy loss spectroscopy, resonant inelastic X-ray scattering, and time-resolved photoemission. In addition, another possible study is ARPES on electron-doped cuprates. When we interpret the result with the electron-hole transformation, it provides information on the spectra of unoccupied states in a hole-doped system. We will discuss this in Sec. \[ssec:ele2\] and \[ssec:ele\], comparing the results with existing ARPES data on electron-doped cuprates.
Spectral symmetry and asymmetry around the Fermi energy {#ssec:eh}
-------------------------------------------------------
Although a thorough comparison of the calculated pole-zero structure with experiments is not possible at present because of the lack of the experimental spectra above $E_{\text{F}}$, it is still significant to make comparison with available experimental data which partly elucidated them. The scanning tunnelling microscopy (STM) [@hl04] and the ARPES [@yr08] reported such data, where they found an electron-hole asymmetry of the spectra around $E_{\text{F}}$. The asymmetry provides a clue to the mechanism of the pseudogap, especially to the relation between the preformed pairing and the pseudogap, because the pairing will lead to a symmetric spectrum as established in the Bardeen-Cooper-Schrieffer (BCS) theory.[@bc57]
Figure \[fig:asym\](a) shows the low-energy part of the density of states (DOS) for $U=8t$, $t'=0$, and $n=0.94$ and for $U=8t$, $t'=-0.4t$, and $n=0.95$. The data show that the weight of the low-energy occupied states is significantly larger than that of unoccupied ones, in accord with other CDMFT studies.[@sk06; @kk06] The asymmetry is consistent with the spatially averaged electron-tunnelling spectra \[Fig. \[fig:asym\](b), reproduced from Fig. 1(c) in Ref. \] measured by the STM for lightly hole-doped cuprates, Na$_{0.12}$Ca$_{1.88}$CuO$_2$Cl$_2$ and Bi$_2$Sr$_2$CaCu$_2$O$_{8+\d}$, where the observed probability of electron extraction is considerably greater than that of injection. Moreover, the numerical data show a peak around $\w=0.3t(0.4t)$ for $t'=0(-0.4t)$, which roughly agrees with the peak injection energies (100-300meV) seen in Fig. \[fig:asym\](b). We note that the numerical data have the minimum above the Fermi level while the STM spectra have a V-shape gap with the minimum at zero bias. The V-shape gap might be attributed to the soft Coulomb gap [@es75] or soft Hubbard gap [@si09] caused by an interplay of electron correlations and randomness, as indicated by the strong charge inhomogeneity observed by the STM at surfaces.[@hl04]
ARPES allows a more detailed comparison of the spectra. Yang [*et al.*]{} [@yr08] succeeded in deriving low-energy ($\w\lesssim 0.03$eV) spectra above the Fermi level by carefully analyzing the ARPES data for Bi$_2$Sr$_2$CaCu$_2$O$_{8+\d}$. This was done by using the fact that ARPES data at finite temperatures contains information of unoccupied states because of the smeared tail of the Fermi distribution function for $\w > 0$. In the superconducting state they observed electron-hole symmetric spectra both in the nodal and antinodal region, in accordance with the BCS spectral function.[@bc57] Meanwhile in the pseudogap state they found that (i) around the antinode the spectrum is nearly symmetric with intense peaks below and above $E_{\text{F}}$ separated by a gap of $\sim 0.06$eV, and that (ii) as approaching the node the peak below $E_{\text{F}}$ goes up and eventually crosses $E_{\text{F}}$ while the peak above $E_{\text{F}}$ disappears from the measured energy range. While (ii) clearly shows the asymmetry as to $E_{\text{F}}$, Yang [*et al.*]{} [@yr08] interpreted (i) as an evidence of preformed pairing.
Liebsch and Tong [@lt09] obtained an asymmetry similar to (ii) with the CDMFT. Nevertheless it is still worthwhile to see if the symmetry (i) can be reproduced since it is relevant to the mechanism of the pseudogap. Figure \[fig:asym\](c) shows the normal-state spectral function $A(\Vec{k},\w)$, calculated with the CDMFT+ED using $\eta=0.03t$. We find that around the antinode, the two intense peaks reside nearly symmetric with opening of a gap. As approaching to nodal region, the gap below the Fermi level monotonically decreases and vanishes when the lower peak reaches the Fermi level \[right panel of Fig. \[fig:asym\](c); see also Fig. \[fig:fig2\](a)\]. These behaviors are consistent with the experimental observations (i) and (ii). We note that in the $t'=0$ case \[Fig. \[fig:fig2\](c)\] the gap at $(\pi,0)$ above $E_{\text{F}}$ is larger than that below $E_{\text{F}}$, whereas the spectrum is more symmetric for $t'=-0.4t$ \[Fig. \[fig:asym\](c)\], which is more appropriate for Bi$_2$Sr$_2$CaCu$_2$O$_{8+\d}$.
In the nodal direction, the gap lies above $\w\simeq 0.1t$. This is again consistent with the ARPES [@yr08] which saw the region below $0.03\text{eV} \sim 0.08t$ and observed no gap in this direction. We note that this is distinct from the picture in Ref. , where the gap was assumed to close in the nodal direction.
Thus we have shown that the ARPES data [@yr08] does not necessarily indicate a preformed pairing, but is rather naturally interpreted as a consequence of the dispersive zero surface.
Back-bending behavior of dispersion {#ssec:back}
-----------------------------------
Another important aspect captured in Fig. \[fig:fig2\](a) is the back-bending behavior of the band cutting the Fermi level. Namely, the pole surface, which monotonically goes up to $(\pi,\pi)$ in the bare dispersion, is bent back below $E_{\text{F}}$ around $(\pi,\pi)$. This is more clearly seen in Fig. \[fig:back\](a), which is an enlarged view of Fig. \[fig:fig2\](a) in a lower energy range. The back bending can be seen below $E_{\text{F}}$ around the antinodes while it is above $E_{\text{F}}$ around the nodal direction.
Actually the ARPES [@kc08] observed a similar behavior. Figure \[fig:back\](b) is a reproduction of Fig. 2(d) in Ref. , where the back-bending behavior is seen only around the antinode. In the light of Fig. \[fig:back\](a) the absence of the back bending around the node will be simply because the top of the band is located at a higher energy than the maximum energy of the measurement. Note that $t'<0$ lifts (lowers) the band around the nodes (antinodes). Enhancing $-t'$ shifts the Fermi surface to $(0,0)$ as well, as we will discuss in Sec. \[ssec:tp\]. Then a better agreement with the ARPES data may be reached.
In Ref. the back-bending behavior around the antinodes was interpreted as an evidence of preformed Cooper pairs in the pseudogap state because it resembles a band dispersion in the BCS superconductors.[@bc57] However, our result implies another simple interpretation: The band is pushed down by the neighboring zero surface, which cuts the Fermi level around $(\pi,\pi)$, due to the large self-energy around it. This picture is confirmed in Fig. \[fig:fig2\](d), where we plot the energy dependence of the self-energy at the Fermi momentum $\Vec{k}_{\text{F2}}$ closest to the zero surface. We see that the real part of the self-energy is negatively large below the zero at $\w\simeq0.06t$, which should push down the band around the momentum. The result indicates that the pair formation is not necessary and the zeros of $G$ resulting directly from strong correlation effects give an alternative interpretation of the back-bending dispersions.
Recently, an evidence against the scenario that the back bending is a consequence of preformed pair was further reported.[@hh10] In this ARPES observation, the back-bending momentum, i.e., the location of the pole at the lowest binding energy along momentum cuts, is clearly deviated from the Fermi momentum at above the pseudogap opening temperature. In contrast, we note that the two momenta should agree in the BCS theory and in the preformed pair scenario as well. Furthermore, the ARPES reported that the back-bending momentum along $(\pi,0)$-$(\pi,\pi)$ shifts closer to $(\pi,\pi)$ than the Fermi momentum. This indicates a zero surface around $(\pi,\pi)$ at the Fermi level and a center of the gap residing above $E_\text{F}$, in full consistency with our pole-zero structure Fig.2(a).
We note that a back-bending dispersion was already observed in an early QMC study,[@ph97] where antiferromagnetic fluctuations were proposed as the origin of the pseudogap. Although our numerical data do not exclude the antiferromagnetic fluctuations from the possible mechanisms, the less-$\Vec{k}$-dependent pseudogap as well as the asymmetric location of the hole pocket (see Sec. \[ssec:arc\]) seems to oppose the mechanism; instead it rather supports that the pseudogap is a direct consequence of the proximity to the Mott insulator.
Energy-distribution curve {#ssec:edc}
-------------------------
Zeros of $G$ are not directly seen in spectra. Their footprints may, however, be detected through a sudden suppression of spectra due to a large Im$\S$ around the zeros. Figure \[fig:edc\](a) shows energy-distribution curves (EDC) of the spectral function along momentum cuts $(0,0)\text{-}(\pi,0)\text{-}(\pi,\pi)$, calculated by the CDMFT+ED with $\eta=0.1t$ for $t'=-0.2t$, $U=12t$, and $n=0.93$. The results exhibit a coherent peak around $(\pi,0)$ just below the Fermi level, its shift to lower energy from $(\pi,0)$ to $(\frac{\pi}{2},0)$, and the incoherent feature around $(0,0)$ and $(\pi,\pi)$. All these features are consistent with ARPES data \[Fig. 5 in Ref. , reproduced in Fig. \[fig:edc\](b)\]. This agreement supports our zero mechanism and shows that the incoherent feature can be interpreted as the effect of zeros of $G$: Since the zero surface exists just above the band around $(\pi,\pi)$ \[Fig. \[fig:fig2\](a)\] and since another one is located just below the band around $(0,0)$ \[not shown in Fig. \[fig:fig2\](a), but can refer to Fig. \[fig:rkw\_tp\](b) below\], the spectrum associated with the band is smeared around these momenta due to the large Im$\S$ around the zeros. While the suppression around $(\pi,\pi)$ is relevant to the emergence of Fermi arc,[@sk06; @sm09; @sm09-2] that around $(0,0)$ is related to the waterfall behavior discussed in the next subsection.
Waterfall {#ssec:wf}
---------
Recent ARPES [@kb05; @gg07; @xy07; @vk07; @mz07] observed an anomalous spectral structure, called “waterfall", at high binding energies $\sim 0.3\text{-}0.4$eV, where the band cutting the Fermi level suddenly loses the spectral weight and starts falling down to $\sim -0.7$eV with a suppressed intensity. Below the energy a strong intensity emerges again from nearly the same momentum as that the waterfall starts. This high-energy anomaly has been found quite generally in hole-doped cuprates, irrespective of the presence or absence of superconductivity, under, optimally or overdoped, and detailed compositions. Moreover similar features have been reported in other transition-metal compounds such as SVO$_3$ [@yt05] and LaNiO$_3$.[@ec09] These suggest a universality of the phenomenon in strongly-correlated metals.[@bk07]
The dynamical cluster approximation (DCA) + QMC study [@mj07] for the 2D Hubbard model with $t'=0$ reported a similar structure in the spectra. Based on the similarity of the DCA results to those with a perturbative calculation incorporating antiferromagnetic spin fluctuations, the authors proposed that the waterfall results from high-energy spin fluctuations. Here we implement a CDMFT+ED study, and closely examine how the pole-zero structure underlies the waterfall phenomenon and how the energy scale of the waterfall depends on model parameters. Our analysis indicates alternative interpretation for the phenomenon.
Figures \[fig:wf\](a)-(d) plot the spectral intensity calculated along the momentum cut from $(0,\frac{\pi}{8})$ to $(\frac{7\pi}{8},\pi)$ for various parameter sets. The results show nice resemblances with the ARPES result \[Fig. 1(c) in Ref. , reproduced in Fig. \[fig:wf\](e)\], i.e., (i) an abrupt change in the slope of the band around $\w=E_1$, accompanied by the simultaneous reduction of the intensity, (ii) nearly vertical dispersion with suppressed intensity in the waterfall region, and (iii) reemergence of a strong intensity around $\w=E_2$. The energy scale also agrees roughly with the experimental value, if one keeps in mind $t\sim 0.4$ eV.
We investigate how $E_1$ and $E_2$ depend on $t'$, $n$, and $U$. First, comparing Figs. \[fig:wf\](a) and (b), we see that $t'<0$ reduces both $|E_1|$ and $|E_2|$. Second, comparison of Figs. \[fig:wf\](b) and (c) shows that doping also reduces the energies. This doping dependence is qualitatively consistent with the DCA results [@mj07] for 16 sites cluster at a finite temperature. Third, the decrease of $U$ from $12t$ \[Fig. \[fig:wf\](a)\] to $8t$ \[Fig. \[fig:wf\](d)\] increases the energies.
These parameter dependences can be naturally explained as follows. In general, strong correlation effects make the width of coherent band (i.e., the band above $E_1$) narrower by increasing the mass of the low-energy particles. This leads to the decrease of $|E_1|$ as $U$ increases. Indeed the slope of the coherent band decreases from $1.3ta$ ($a$: the lattice constant) to $1.0ta$ as $U$ increases from $8t$ to $12t$. Meanwhile $t'$ lifts the band around the nodal direction, as will be discussed in Sec. \[ssec:tp\]. This elevates $E_1$ and $E_2$. The doping dependence seen in Figs. \[fig:wf\](b) and (c) can be qualitatively understood as a downward shift of the chemical potential with doping. As we showed in Ref. , in hole-doped Mott insulators doping causes a rigid-band-like shift of the chemical potential, though it cannot be described within the single-electron picture. In fact, the low-doping phase is not adiabatically connected with the Fermi liquid phase at higher dopings because of the intervening Lifshitz transition and the zero-surface emergence.
The above interpretation suggests that the energy scale of the waterfall is not necessarily related to the antiferromagnetic spin fluctuations [@mj07] but is considered to be a rather direct consequence of electron correlation, i.e., the mass renormalization of the coherent band. The picture is corroborated by the fact that a similar behavior can be seen within the single-site DMFT.[@bk07]
In view of the pole-zero structure in Fig. 1(b) in Ref. , the waterfall emerges in the energy region $-2t\lesssim\w\lesssim-t$, where many smeared pole and zero surfaces pile. The congestion of poles and zeros within the finite-cluster calculation indicates an incoherent nature of this energy-momentum region, which would result in the waterfall. Notice that the waterfall obtained in this paper has neither a gap nor dispersive features [@mj07], but a vertical structure with a suppressed intensity. This unusual structure comes out thanks to the use of the cumulant periodization (see Sec. \[sec:method\]) which can describe the strong momentum dependence of the self-energy. Below the pile we see a relatively coherent pole surface extending down to $\w\sim -3.5t$. This corresponds to the band reemerging at the high binding energy \[(iii)\]. Note that a comparison at higher binding energies is difficult because the ARPES spectra are overlapped by other Cu-$d$ or O-$p$ bands.
Low-energy kink in dispersion {#ssec:kink}
-----------------------------
In prior to the discovery of the waterfall, another anomaly (kink) in dispersion has been found in ARPES at lower binding energies [@bl00] and its origin has been in dispute. The kink is located at a binding energy around $0.05$eV, where the band cutting the Fermi level sharply changes its slope. In the normal state the kink is clearly seen around the nodal region while, as approaching the antinodal region, the band becomes flatter and then the kink becomes less visible. The band structure continuously changes from the nodal to antinodal region where a pseudogap exists just above the band.[@kr01; @sm03]
The CDMFT+ED results in Fig. \[fig:kink\] show similar behaviors: (i) The band suddenly changes the slope at a kink around $\w=-0.1t\sim -0.04$eV in the nodal region \[Fig. \[fig:kink\](a)\], and (ii) as approaching the antinode, the band becomes flatter and the slope change becomes weaker \[Fig. \[fig:kink\](b)\].
In general, the zero surface which generates the pseudogap pushes down the dispersion near the Fermi level. This makes a quick change of the slope of the dispersion distinct from the part deeply below the Fermi level, which may be the underlying origin of the kink formation. However, the sudden change of the dispersion observed in Fig. \[fig:kink\](a) naturally requires a precursory formation of a tiny zero surface around $\w=-0.1t$. This means a coupling of the quasiparticle with some other excitations with this energy, whichever bosonic or fermionic. The energy resolution of the present cluster size is not obvious and it could be an artifact of the present small cluster calculation. Nevertheless, it is remarkable to note that the observed kink in Fig. \[fig:kink\](a) is rather universal and we see similar kinks in other part of the quasiparticle dispersions, for instance in Fig. \[fig:rkw\_tp\](b) below. This may be alternatively interpreted that, in the strongly correlated region, the quasiparticle is strongly renormalized and may couple to various intrinsic electronic modes of charge and spin origins.
In high-$T_{\text{c}}$ cuprates the mechanism of the kink has been extensively discussed in the literature from the viewpoint of a coupling of electrons with some bosonic mode, such as phonon [@lb01] and magnetic ones.[@jv01; @kr01] Our result, however, implies that the kink may be a rather general phenomenon in the proximity to the Mott insulator,[@bk07] while the specific mechanism of kink formation is left for future studies.
Fermi arc and hole-pocket Fermi surface {#ssec:arc}
---------------------------------------
Next we shift our focus on the zero-energy electronic structure, namely, Fermi surface. The spectra in hole-doped cuprates have been extensively studied by ARPES.[@dh03] One of the most remarkable findings in these studies is the observation of truncated Fermi surfaces, called “Fermi arc".[@nd98]
Several authors have discussed the Fermi arc in terms of coexisting poles and zeros of $G$.[@sm09; @kr06; @et02; @bg06; @ks06; @sk06; @sm09-2; @sp07] In this mechanism hole-pocket Fermi surfaces around the nodal directions coexist with a zero surface around $(\pi,\pi)$. Because Im$\S$ is large around the zero surface, the pockets lose the spectral intensity more on the side closer to the zero surface, leaving arc-like spectra on the opposite side. This is reproduced in Figs. \[fig:pocket\](a) and (b). Figure \[fig:pocket\](a) shows the underlying pole-zero structure calculated by the CDMFT+ED at $T=0$ without any substantial smearing to the singularities, where a clear hole-pocket Fermi surface coexists with a zero surface. When we introduce a smearing to the singularities (see Sec. \[ssec:ed\]), we obtain the spectral map, Fig. \[fig:pocket\](b), in which the arc-like structure is observed.
We confirm the zero mechanism by directly calculating a spectral weight at finite temperatures by employing the CTQMC method as the solver for the CDMFT. To obtain the spectrum without through any analytic continuation procedure which inevitably suffers from a large error bar, we calculate $$\begin{aligned}
\label{eq:ghb}
-\b G(\Vec{k},\t=\beta/2)=\frac{\b}{2}
\int_{-\infty}^{\infty}\frac{A(\Vec{k},\w)}{\cosh(\b\w/2)}d\w\end{aligned}$$ as a function of $\Vec{k}$. This quantity is an integral of the spectral weight over a width $\sim T$ around $\w=0$ and approaches $A(\Vec{k},\w=0)$ for large $\beta$, so that gives an estimate of $A(\Vec{k},\w=0)$ at low temperatures.[@footnote2] In Figs. \[fig:pocket\](c) and (d) we plot the results at $T=0.025t$. For $n=0.95$ \[Fig. \[fig:pocket\](c)\] we see an arc at a location similar to the one in Fig. \[fig:pocket\](b) while for $n=0.9$ \[Fig. \[fig:pocket\](d)\] the spectra extend to the antinodal regions. The doping evolution of the spectra is qualitatively consistent with that obtained with the CDMFT+ED in Fig. 2(c)-(e) in Ref. . The qualitative agreement between the CDMFT+ED result with broadening $\eta$ and the CDMFT+CTQMC result at finite temperatures supports that the phenomenological smearing factor $\eta$ well simulates the thermal effects. Moreover it corroborates the above-mentioned zero mechanism for the emergence of the Fermi arc. We note that the zero surface around $(\pi,\pi)$ is also consistent with the results by other cluster schemes such as DCA, which observed a strong scattering amplitude in the momentum patch around $(\pi,\pi)$ in both $N_\text{c}=4$ (Ref. ) and $N_\text{c}=8$ (Ref. ) calculations.
Interestingly, recent high-resolution ARPES [@ml09] reported the existence of hole-pocket Fermi surfaces around the nodal directions in underdoped Bi$_2$Sr$_{2-x}$La$_x$CuO$_{6+\d}$. The observed hole pockets have much less intensity on the side closer to $(\pi,\pi)$ than the opposite. This is consistent with the zero mechanism in Fig. \[fig:pocket\](b). Important findings in the ARPES are that (i) the hole pockets are not located symmetrically with respect to the antiferromagnetic Brillouin zone boundary, i.e., $(\pi,0)\text{-}(0,\pi)$ line (and its symmetrically-related ones), and that (ii) the spectral intensity on the $(\pi,\pi)$ side is finite even in the nodal directions. (i) excludes several scenarios for the pseudogap, which attribute the pockets to symmetry breakings [@ks90; @v97; @cl01] because in these theories hole pockets should be centered symmetrically as to the $(\pi,0)\text{-}(0,\pi)$ line.[@ml09] Meanwhile (i) is consistent with our zero mechanism that does not assume any symmetry breaking. (ii) is at odds with the theory based on an ansatz that gives a zero intensity in the outer point of the pocket crossing the nodal direction,[@ml09] while it agrees well with our numerical data in Fig. \[fig:pocket\](b). This is more clearly seen in Fig. \[fig:pocket\](e), where we plot $A(\Vec{k},0)$ along the momentum cut in the nodal direction: For $\eta=0.01t$, in addition to the main peak at $\Vec{k}\simeq(0.56\pi,0.56\pi)$, we see the secondary peak at $\simeq (0.7\pi,0.7\pi)$, corresponding to the hole pocket structure. The second peak is, however, not visible for $\eta=0.1t$ and only the broadened main peak can be seen there. Namely, in our theory, the spectrum looks either a pocket like or an arc like depending on how the incoherence due to the zero surface is strong. The strength of the incoherence is controlled by the value of $\eta$ or temperature, or the distance between zeros and poles, as partly demonstrated in Fig. 4 in Ref. . Figure \[fig:pocket\](e) also suggests that the energy resolution required to detect the pocket in the example of the pole-zero structure as Fig. \[fig:pocket\](a) is in between $0.01t(\sim 4\text{meV})$ and $0.1t(\sim 40\text{meV})$, which roughly corresponds to the energy resolution in ARPES. This consideration on the incoherence explains why the pocket had not been detected until the recent high-resolution ARPES.[@ml09]
One obvious difference between Fig. \[fig:pocket\](b) and the ARPES [@ml09] is the location of the pocket: The pocket in Fig. \[fig:pocket\](b) resides closer to $(\pi,\pi)$ than that observed in the ARPES. This may be attributed to the difference between the model parameters we used and realistic ones for Bi$_2$Sr$_{2-x}$La$_x$CuO$_{6+\d}$. In particular, we show in the next subsection that the next-nearest-neighbor transfer $t'$, which is zero in Fig. \[fig:pocket\](b), indeed shifts the pocket in the direction to $(0,0)$.
Effect of the next nearest-neighbor transfer $t'$ {#ssec:tp}
-------------------------------------------------
Here we systematically study the effect of $t'$ on the electronic structure in lightly hole- and electron-doped regions. Figure \[fig:arc\_tp\] shows $t'$ dependence of the integrated spectra, Eq. (\[eq:ghb\]), calculated with CDMFT+CTQMC at $T=0.05t$ for (a)-(c) $U=8t$ and $n=0.93$ and (d)-(f) $U=12t$ and $n=0.90$. We see that the arc shifts to $(0,0)$ as $-t'/t$ increases, and that at $t'=-0.4t$, which is a reasonable value for Bi$_2$Sr$_{2-x}$La$_x$CuO$_{6+\d}$, the arc resides inside of the antiferromagnetic Brillouin zone, in consistency with experiments.[@ml09] We also see that $t'$ increases the curvature of the arc. A comparison with ARPES in further detail will require a more realistic determination of the model parameters, longer-range transfer integrals and Coulomb interactions, and a calculation on a larger cluster. These remain for future researches.
The above shift of the arc with $t'$ can be understood by plotting the underlying pole-zero structures. Figures \[fig:rkw\_tp\](a) and (b) compare the structures for $t'=0$ and $t'=-0.2t$ in a lightly hole-doped region. The figures show that at low energies $t'$ lifts the poles in the nodal direction while lowers them around the antinodes, as already expected from the bare dispersion Eq. (\[eq:disp\]). Then the hole pocket expands and the Fermi surface with the stronger intensity shifts to $(0,0)$ direction. We note that, as we discuss in the next subsection, the third-neighbor transfer integral, which is not taken into account in the present calculation but exists in real materials, further enhances the pocket around $(\frac{\pi}{2},\frac{\pi}{2})$.
The effect of $t'$ is totally different for electron-doped cases. Figures \[fig:rkw\_tp\](c) and (d) show the structures for $t'=0$ and $-0.2t$, respectively, in an electron-doped region. The effect of $t'$ at low energies is again understood by Eq. (\[eq:disp\]): $t'$ lifts the poles in the nodal direction while lowers around the antinodes. However, the consequence on the Fermi surface completely differs from that in hole-doped cases. For the particle-hole symmetric $t'=0$ case, the electron pockets appear around the nodes for electron doping, corresponding to the hole pockets in hole-doped cases, however, for finite $t'$, the electron pockets appear around the antinodes, as shown in Fig. \[fig:rkw\_tp\](d). This is because the low-energy zero surface pushes up the dispersion around $(0,\frac{\pi}{2})$. We compare the Fermi surface structure with ARPES for the electron-doped cuprates in detail in Sec. \[ssec:ele\].
High-energy spectra in electron-doped cuprates {#ssec:ele2}
----------------------------------------------
Recently high-energy spectra of an electron-doped cuprate Nd$_{1.85}$Ce$_{0.15}$CuO$_4$ were studied by ARPES.[@iy09] To discuss the results we first emphasize that here the ARPES observes a totally different energy region from that for hole-doped cuprates. This is illustrated in Figs. \[fig:wf2\](a) and (b): ARPES observes the lower Hubbard band (LHB) in hole-doped cases while in electron-doped cases it does ingap states between the Mott gap and the pseudogap, in addition to the pseudogap structure itself. As mentioned in Sec. \[ssec:eh\], with the electron-hole transformation the ARPES result can be interpreted as that for unoccupied states in a hole-doped system. Therefore the results in Ref. provide a precious opportunity to compare the calculated spectra of ingap states with experiments. The pseudogap and the Fermi surface are discussed in the next section.
The ARPES spectra and its second derivative in $\w$ (Fig. 1 in Ref. ) are reproduced in Figs. \[fig:wf2\](c)(d) and (e)(f), respectively. Around the antinode the spectra \[Fig. \[fig:wf2\](d)\] suddenly lose the intensity at $\w \simeq -0.4$eV. The analysis on the second derivative \[Fig. \[fig:wf2\](f)\] shows that the low-energy band reaches the bottom at this energy. On the other hand, around the node the low-energy band persists down to $\w \simeq -0.7$eV \[Fig. \[fig:wf2\](c)(e)\].
Figures \[fig:wf2\](g) and (h) depict the spectra calculated with the CDMFT+ED for $t'=-0.4t$, $U=12t$, and $n=1.11$ along momentum cuts, $(0,0)\text{-}(\pi,\pi)$ and $(0,\pi)\text{-}(\pi,\pi)$, respectively. For $\w<0$ we see that the band bottom is located at $\w\simeq -0.7t$ around $(0,\pi)$ while it is located at a higher binding energy $\w\simeq -t$ around $(0,0)$. This is consistent with the ARPES data. Note that the numerical data do not show any intensity below the band bottoms, $\w < -0.7t$ ($-t$) around $(0,\pi)$ \[$(0,0)$\], which is within the gap region between LHB and the ingap states, whereas the ARPES spectra show waterfall-like structures in $-1\text{eV}\lesssim \w \lesssim -0.7\text{eV}$ ($\w \lesssim -0.4\text{eV}$) around $(0,\pi)$ \[$(0,0)$\]. The origin of this discrepancy is not clear, but one possibility is a multiband effect, which is not considered in the present single-band model.
Meanwhile for $\w>0$, we see a waterfall in the dispersion at $\w\simeq t\text{-}2t$. This corresponds, with the electron-hole transformation, to the waterfall in hole-doped cases. The result predicts that this anomaly will be observed in electron-doped cuprates when the unoccupied spectra become available up to the high energy.[@footnote5]
Pseudogap and Fermi pockets in electron-doped cuprates {#ssec:ele}
------------------------------------------------------
Lastly we discuss low-energy electronic structures in electron-doped cases. The pseudogap has been experimentally observed also in electron-doped cuprates,[@ot01] although the doping range is much more limited than that in hole-doped cases due to the wider antiferromagnetic region near the undoped Mott insulator. The mechanism of the pseudogap has extensively been discussed within weak-coupling theories [@kr03; @kh04], which has successfully reproduced various experimental results such as the doping evolution of the Fermi surface. In these theories the mechanism has been ascribed to the antiferromagnetic long-range correlations. On the other hand, in the strong-coupling regime the cluster-perturbation theory [@st04] and the CDMFT [@cc05; @kk06] found that the pseudogap results from nonlocal but short-ranged dynamics without any long-range order or correlations. As pointed out in Ref. and , the gap amplitude does not scale as $J\sim t^2/U$, so that the short-ranged dynamics is not simply ascribed to the AF correlation. Because it is unknown which mechanism, weak-coupling or strong-coupling, is relevant to electron-doped cuprates, it is worthwhile to see whether and how experimental data are understood in the strong-coupling picture, especially from the perspective of zeros of $G$.
The available ARPES data for the electron doped cuprates seems to have relatively poor resolutions and a clear dispersion has not been reported. Nevertheless, a recent ARPES [@iy09-2] measurement appears to have revealed the structure of the pseudogap for a family of electron-doped cuprates, $Ln_{2-x}$Ce$_x$CuO$_4$ ($Ln=$ Nd, Sm, and Eu). In Fig. 2 in Ref. the EDC peak position jumps from $\w\sim-0.03$eV to $\w\sim-0.15$eV around antinode, which implies the presence of the pseudogap opening below the electron pocket. This highly asymmetric position of the pseudogap is a feature which cannot be seen by ARPES for the hole-doped cuprates but observable for electron-doped ones: Because in electron-doped cases the zero surface forming the pseudogap mainly extends below $E_\text{F}$ \[see Fig. \[fig:rkw\_tp\](d)\], the ARPES can observe the major part of the pseudogap, in contrast to the hole-doped cases. Around the node, the EDC peak position saturates at $\w\sim-0.05$eV for $Ln=$Eu compound, implying the pseudogap at $\omega \gtrsim -0.05$eV, while the pseudogap is not clear for $Ln=$Nd in EDC. This is consistent with the observation in Fig. \[fig:rkw\_tp\] that the pseudogap position in the nodal region is lowered for smaller $|t'/t|$ and that the Eu compound appears to have a relatively small $|t'/t|$. Namely, the main part of the pseudogap in the nodal region of the Nd compound is located in the positive energy side, while it becomes visible below $E_\text{F}$ for the Eu compound. This suggests a gap around the node, which can be interpreted as an indirect evidence of a gap opening above $E_\text{F}$ in hole-doped cases via the electron-hole transformation. These observations are qualitatively consistent with the pole-zero structure in Fig. \[fig:rkw\_tp\](d), supporting the non-$d$-wave (fully-opened) pseudogap proposed in Sec. \[ssec:nond\]. We propose to perform ARPES measurements of the electron doped compounds with a better resolution than Ref. , because the pseudogap appears to be suggested only by the jump in EDC and the detailed pseudogap structure in the momentum space is not very clear so far.
The ARPES [@ar02; @iy09-2] has also revealed a characteristic evolution of the Fermi surface with doping. Figures \[fig:edope\](a)-(c) are the reproductions of the ARPES results (Fig. 3 in Ref. ) on Nd$_{2-x}$Ce$_x$CuO$_4$. The ARPES study found that (i) at small doping ($x=0.04$) strong intensity emerges around $(\pi,0)$ and its symmetrically-related points \[Fig. \[fig:edope\](a)\], (ii) as doping is increased, additional intensity develops around $(\frac{\pi}{2},\frac{\pi}{2})$ \[Fig. \[fig:edope\](b)\], and (iii) at large doping the spectra merge, forming a single large surface around $(\pi,\pi)$ \[Fig. \[fig:edope\](c)\]. In Ref. it is also seen that the band cutting the Fermi level at around $(\frac{\pi}{2},\frac{\pi}{2})$ evolves with doping from a high to low binding energy while that around $(\pi,0)$ emerges first around the Fermi level and extends to a high binding energy. This suggests that the Fermi surface around $(\frac{\pi}{2},\frac{\pi}{2})$ is a hole pocket while the one around $(\pi,0)$ is an electron pocket.
The above features (i) and (iii) were qualitatively reproduced by the cluster perturbation theory [@st04] and by CDMFT with the $\S$ [@cc05] and $G$ [@kk06] periodizations on the 2D Hubbard model at $T=0$. In Ref. and the suppression of the intensity around the node at low dopings was understood by the presence of the hot spot, where Im$\S$ is large, in this region.
In the following we clarify how the above observations (i)-(iii) can be understood in terms of the underlying zero surface. First, we show the CDMFT+CTQMC results at $T=0.05t$. Figures \[fig:edope\](d)-(f) depict the integrated low-energy spectra, Eq. (\[eq:ghb\]), for $t'=-0.4t$, $U=12t$, at $5$, $10$, and $15$% electron dopings, respectively. At $n=1.05$ a strong intensity emerges only around the antinodal points, in accord with (i). A large Fermi surface around $(\pi,\pi)$ at $n=1.15$ agrees with (iii). As to (ii), our results do not show strong intensity around $(\frac{\pi}{2},\frac{\pi}{2})$; this will be attributed to the absence of the third-neighbor transfer integral $t''$ in the present calculation, as discussed below.
Second, Figs. \[fig:edope\](g)-(i) show the CDMFT+ED results for $t'=-0.4t$ and $U=12t$ at $n=1.14$, $1.17$, and $1.19$, respectively. We see that a zero surface exists at the Fermi level, surrounding $(0,0)$. For low dopings electron-pocket Fermi surfaces reside around the antinodal points. As the doping is increased, the pockets expand. Then they merge around the nodal point and change the topology into two Fermi surfaces, one around $(\pi,\pi)$ and the other around $(0,0)$. Further doping makes the inner Fermi surface annihilate in pair with the zero surface, resulting in the Fermi liquid with the single large Fermi surface around $(\pi,\pi)$.[@footnote3] Thus in electron doping there occur, at least, two phase transitions, the Lifshitz transition [@ko07] and a pole-zero annihilation transition, on the way from the Mott insulator to the Fermi liquid.
The presence of the zero surface accounts for the weak spectral intensity around the nodal direction in Figs. \[fig:edope\](d) and (e). Then how is the intensity around the node in Fig. \[fig:edope\](b) and (c) explained? To answer this we consider how the electronic structure, Fig. \[fig:rkw\_tp\](d), changes with $t''$, which is not included in our calculations based on the $2\times2$ cluster. Although $|t''|$ is usually several times smaller than $|t'|$ and therefore negligible for discussing global electronic structures, it can be crucial to understand the nodal intensity. This is because around $(\frac{\pi}{2},\frac{\pi}{2})$ the pole surface at $\w\simeq -0.15t$ in Fig. \[fig:rkw\_tp\](d) moves up with $-t'/t$, as seen in comparison with Fig. \[fig:rkw\_tp\](c), and is located just below $E_{\text{F}}$ for $-t'/t=0.4$ (not shown). Since to the first approximation $t''(>0)$ elevates dispersions around $(\frac{\pi}{2},\frac{\pi}{2})$ while lowers around $(0,0)$, $(\pi,\pi)$, $(\pi,0)$, and $(0,\pi)$, according to $$\begin{aligned}
\e(\Vec{k}) \rightarrow \e(\Vec{k})-2t''[\cos(2k_x)+\cos(2k_y)],\end{aligned}$$ a hole pocket can emerge around $(\frac{\pi}{2},\frac{\pi}{2})$. This hole pocket will show up just inside the zero surface in Figs. \[fig:edope\](g) and (h). Recent experimental observation [@hk09] of the Shubnikov-de Haas oscillation in Nd$_{2-x}$Ce$_x$CuO$_4$ also indicates a presence of the hole pocket around optimal doping. Therefore, to confirm the above scenario with a larger cluster calculation incorporating $t''$ is an intriguing future problem.
We note some additional indications of the zero surface in experiments. In Fig. 2(c)-(e) in Ref. , which plotted the ARPES EDCs for Nd$_{2-x}$Ce$_x$CuO$_4$ along the underlying Fermi surface, a clear jump of the peak positions can be seen between the nodal and antinodal directions. This indicates the presence of the zero surface around the jump, in accord with the above picture \[see Fig. \[fig:rkw\_tp\](d)\]. In Fig. 1 in Ref. , which plotted the ARPES spectra at the Fermi level, a similar structure to Fig. \[fig:edope\](c) was found while it clearly shows a finite intensity in some regions on the $(\pi,0)\text{-}(0,\pi)$ line. This appears to be inconsistent with the zero surface on the $(\pi,0)\text{-}(0,\pi)$ line assumed in Ref. for hole-doped cases, but compatible with our zero surface around $(0,0)$ as seen in Figs. \[fig:edope\](g) and (h).
SUMMARY AND CONCLUSION {#sec:summary}
======================
In summary, we have shown that various spectral anomalies observed in the pseudogap states of hole- or electron-doped cuprates are naturally understood in terms of underlying pole-zero structure of electronic Green’s function. The pole-zero structure has been calculated for the paramagnetic metallic phase in the 2D Hubbard model with employing the CDMFT+ED at $T=0$ and the CDMFT+CTQMC at $T>0$. We have confirmed that the cumulant periodization scheme [@sk06] is suited for the parameter region of our interest, namely lightly doped Mott insulator in the intermediate to strong coupling region, since the cumulant is well localized within the cluster that we employed, as demonstrated in APPENDIX B. This corroborates the convergence and reliability of our momentum resolution in the studies on Fermi surface topology and the differentiation in the momentum space. Furthermore, the cumulant periodization is useful since it enables to study the coexistence of pole and zero surfaces.
The result calculated by CDMFT+ED shows a pseudogap in the lightly doped region around the Mott insulator. The pseudogap is characterized by a low-energy zero surface, which connects the bifurcated low-energy bands \[Fig. \[fig:fig2\](a)\]. In hole-doped cases the lower band cuts the Fermi level while the upper one resides at the energy higher than and adjacent to the pseudogap above the Fermi level. This directly leads to (i) the non-$d$-wave character of the fully opened pseudogap and (ii) the spectral asymmetry around the Fermi level. Looking into the structure around $(\pi,\pi)$ we have also found that a large Re$\S$ around the zero surface causes (iii) the back-bending dispersion. All of (i), (ii), and (iii) are consistent with experiments semiquantitatively. The agreement imposes a strong constraint on theories for pseudogap: Although (ii) and (iii) have been interpreted as evidences of the preformed pairing in the literatures, (i) is in sharp contrast to $d$-wave-gap scenarios including those by preformed pair. It is compatible neither with other precursory or real $d$-wave-type symmetry breakings such as $d$-density wave nor with commensurate antiferromagnetism. We have confirmed a full gap formation at the nodal point in a larger-cluster calculation in APPENDIX C. Moreover, the non-$d$-wave gap is supported by the agreement of our result on the pseudogap structure with that observed by ARPES for electron-doped cuprates.
The zero mechanism also enables a simple and unified understanding of various spectral anomalies: In hole-doped cases the same zero surface which causes the back-bending behavior around the antinode induces the incoherence around $(\pi,\pi)$, Fermi pockets and Fermi arcs, while a pile of pole and zero surfaces at a higher binding energy results in the high-energy kink (waterfall) and the incoherence around $(0,0)$. Moreover we have found a low-energy kink structure in the nodal dispersion. On the other hand, in electron-doped cases the zero surface which constitutes the pseudogap crosses the Fermi level around $(0,0)$, making electron pockets around the antinodes. All these features are consistent with experimental results. We would like to emphasize that the zero surface does not result from symmetry breakings, but is a direct consequence of the strong correlation effect, i.e., proximity to the Mott insulator.
To elucidate the effect of zeros at finite temperatures, we have also implemented CDMFT+CTQMC calculations. The results qualitatively agree with those of CDMFT+ED with a smearing factor $\eta$, which confirms that the thermal scatterings broaden the effect of zeros and indeed create the characteristic spectra similar to those observed in the cuprates.
To conclude, the origins of various anomalies in the electronic structure of the normal state in the high-$T_{\text{c}}$ cuprates are unified into the presence of the low-energy zero surface which persists against doping. The zero surface is also expected to cause anomalous metallic behaviors in other physical quantities, such as the specific heat, electronic resistivity, and Hall coefficient.[@ts99] Comparisons of these quantities with experiments remain for future studies.
Although the microscopic mechanism creating the zero surface has not been discussed in this paper, it has emerged clearly as the proximity to the Mott insulator. The similarity in the structure of the zero surfaces between the pseudogap and the Mott gap in the undoped system \[see Fig. 1(a) in Ref. \] also implies the Mott origin of the pseudogap. At the same time, however, the pseudogap does not appear as that directly or continuously connected to the zero surface which forms the Mott gap in the undoped state. This is because, for example in the hole-doped case, the zero surface creating the fully-opened pseudogap is bounded above by the existence of low-energy excitations far below the upper Hubbard band (UHB). The reason why the ingap states are separated by the pseudogap from the main quasiparticle band is left for an important future issue. In Ref. we proposed a scenario in which the emergence of such a zero surface in the low-energy scale and the resultant pseudogap are a remnant of the binding of doubly occupied site (doublon) and empty site (holon) drastically weakened by the screening of Coulomb repulsions by mobile carriers. This has something to do neither with any symmetry breaking nor with its precursor. Recently, it has been proposed that a gap arising from hybridization between the quasiparticle and a composite fermion excitation is responsible for the pseudogap and the related zero surface. [@yamaji10] Possible supplementary role of strong antiferromagnetic fluctuations are left for future studies. Further study on the mechanism of pseudogap will be described elsewhere.
ACKNOWLEDGMENT {#acknowledgment .unnumbered}
==============
S. S. thanks Masafumi Udagawa, Giorgio Sangiovanni and Alessandro Toschi, and M. I. thanks Youhei Yamaji for useful discussions. S. S. is also grateful to Karsten Held for the hospitality. This work is supported by a Grant-in-Aid for Science Research on Priority Area “Physics of New Quantum Phases in Superclean Materials" (Grant No.17071003) from MEXT, Japan. S. S. is supported by Research Fellowship of the Japan Society for the Promotion of Science for Young Scientists. The calculations are partly performed at the Supercomputer Center, ISSP, University of Tokyo.
APPENDIX A: Breakdown of $\S$ periodization for the Mott insulator {#appendix-a-breakdown-of-s-periodization-for-the-mott-insulator .unnumbered}
==================================================================
Here we note that a standard periodization technique, which substitutes $Q=\S$ to Eq. (\[eq:periodize\]), cannot reproduce the Mott gap for large $U$. Because Im$G=0$ in the Mott gap, Im$\S$ must be 0 or $\infty$ in the whole Brillouin zone. However, this situation does never occur in Eq. (\[eq:periodize\]) with $Q=\S$ unless $^\forall i,j \in \text{C}, {\rm Im}\S_{ij}^{\text{C}}=0$ or $^\exists (ij), \S_{ij}^{\text{C}}=\infty$ for all $\w$ inside the gap, and these conditions are never satisfied as far as both $t$ and $U$ are nonzero and finite.
As a matter of fact, we see the electronic structure shown in Fig. \[fig:speriodize\](a) when we use the $\S$-periodization procedure for the same Mott insulator shown in Fig. 1(a) in Ref. \[and reproduced in Fig. \[fig:speriodize\](b) for comparison\]. We see that zeros of $G$ exist only at the Fermi level without a dispersion, and that poles of $G$ extend to the Fermi level around $(0,0)$ from the positive $\w$ side and around $(\pi,\pi)$ from the negative $\w$ side, making the density of states half metallic. This failure of the $\S$ periodization in the Mott insulator is ascribed to the fact that $\S$ is not localized within the $2\times 2$ cluster.[@sk06] The nonlocality of $\S$ is a direct consequence of the presence of dispersive zeros of $G$, i.e., momentum-dependent divergence of $\S$.
As we have discussed so far, zeros of $G$ still persist in doped Mott insulators up to a critical doping level beyond which the Fermi liquid emerges. Therefore $\S$ should be highly nonlocal also in the non-Fermi-liquid region and the $\S$ periodization again breaks down there. On the contrary, the cumulant $M$ is well localized within the $2\times 2$ cluster in this region, as we mentioned in Sec. \[sec:method\]. In APPENDIX B we give another numerical evidence for the locality of the cumulant in doped Mott insulators.
APPENDIX B: Locality of cumulant in doped Mott insulators {#appendix-b-locality-of-cumulant-in-doped-mott-insulators .unnumbered}
=========================================================
Here we present a CDMFT+QMC result for an $N_{\text{c}}=4\times 4$ cluster in a parameter region of doped Mott insulators. We use the Hirsch-Fye algorithm [@hf86] and calculate the cluster cumulant $M^{\text{C}}$ for $t'=0$, $U=8t$, $n=0.91$, and $T=0.1t$.
Figures \[fig:cum\](a) and (b) show $\w_n$ dependence of the real and imaginary parts of the cumulant $M(\Vec{r},\w_n)$, respectively, where $\Vec{r} = (i,j)$ is the real-space vector connecting the sites $i$ and $j$ in the cluster \[the inset of Fig. \[fig:cum\](b)\]. We notice that $\Vec{r}=(0,0)$, (1,0), and (1,1) components are much larger than the other components at longer distances. This can be more clearly seen in Fig. \[fig:cum\](c), where the cumulant at the lowest Matsubara frequency, $\w_0$, is plotted against the Euclidean distance, $\sqrt{i^2+j^2}$. We find that the cumulant is well localized within the $2\times 2$ cluster, which justifies the $M$ periodization.
Although the cumulant at $T=0$ might be more extended than that at $T=0.1t$, we do not have a tractable way to examine it for a cluster larger than $2\times 2$. It is worthwhile, however, to note that the finite-temperature results obtained by our CDMFT+CTQMC calculation are consistent with the CDMFT+ED results with finite $\eta$’s, as discussed in Sec. \[ssec:arc\]. This fact implies that the $M$ periodization with $2\times 2$ cluster still remains a good approximation even at $T=0$.
APPENDIX C: Nodal spectra obtained by 8-site CDMFT+CTQMC {#appendix-c-nodal-spectra-obtained-by-8-site-cdmftctqmc .unnumbered}
========================================================
To demonstrate the fully-opened pseudogap in a larger cluster, we implement an 8-site cluster calculation with CDMFT+CTQMC. We calculate the spectral function $A(\Vec{k},\w)$ at $\Vec{k}=\Vec{k}_\text{node}\equiv(\pi/2,\pi/2)$, where the $d$- and $s$-wave (fully-opened) pseudogaps are most distinguishable. The spectral function is obtained from $G(\Vec{k},i\w_n)$ through the maximum-entropy method.
Figure \[fig:8site\] shows $A(\Vec{k}_\text{node},\w)$ for $t'=-0.2t$, $U=8t$, and $n=0.95$. The small but clear gap just above the Fermi level at $T=0.05t$ proves the fully-opened pseudogap.[@footnote4] The gap vanishes at a higher temperature $T=0.1t$. This is why the previous DCA study [@mj06] at a high temperature $T=0.12t$ did not observe the gap. Note that, a very recent study with 8-site DCA [@lg10] at $T=0.05t$ also detected a wispy reduction of the spectral weight at $\Vec{k}_\text{node}$ at low doping though the signal was so weak that it was not explicitly analyzed in their paper. The weakness of the signal would be due to the flat momentum dependence of the self-energy in each momentum patch in the DCA. Although the suppression of the spectrum is more subtle at higher doping in their result, it is reasonable because the pseudogap temperature decreases with doping.
The gap formation at the nodal point cannot be reproduced by the assumption of the $d$-wave pseudogap. We have confirmed this by employing the assumed form of the Green function employed in Ref. .
T. Timusk and B. Statt, Rep. Prog. Phys. [**62**]{}, 61 (1999). A. Damascelli, Z. Hussain, and Z.-X. Shen, Rev. Mod. Phys. [**75**]{}, 473 (2003). Y. Yanase, T. Jujo, T. Nomura, H. Ikeda, T. Hotta, and K. Yamada, Phys. Rep. [**387**]{}, 1 (2003). V. J. Emery and S. A. Kivelson, Nature (London) [**374**]{}, 434 (1995). A. Kampf and J. R. Schrieffer, Phys. Rev. B [**41**]{}, 6399 (1990). D. Pines, Z. Phys. B [**103**]{}, 129 (1997). Y. M. Vilk and A.-M. S. Tremblay, J. Phys. I France [**7**]{}, 1309 (1997). S. A. Kivelson, I. P. Bindloss, E. Fradkin, V. Oganesyan, J. M. Tranquada, A. Kapitulnik, and C. Howald, Rev. Mod. Phys. [**75**]{}, 1201 (2003). C. M. Varma, Phys. Rev. B [**55**]{}, 14554 (1997). S. Chakravarty, R. B. Laughlin, D. K. Morr, and C. Nayak, Phys. Rev. B [**63**]{}, 094503 (2001). T. M. Rice, Prog. Theor. Phys. Suppl. [**160**]{}, 39 (2005). T. D. Stanescu and P. Phillips, Phys. Rev. Lett. [**91**]{}, 017002 (2003). T. D. Stanescu and G. Kotliar, Phys. Rev. B [**74**]{}, 125110 (2006); T. D. Stanescu, M. Civelli, K. Haule, and G. Kotliar, Ann. Phys. (N. Y.) [**321**]{}, 1682 (2006). P. Philips, T.-P. Choy, and R. G. Leigh, Rep. Prog. Phys. [**72**]{}, 036501 (2009). S. Sakai, Y. Motome, and M. Imada, Phys. Rev. Lett. [**102**]{}, 056404 (2009).
A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. [**68**]{}, 13 (1996). G. Kotliar, S. Y. Savrasov, G. Palsson, and G. Biroli, Phys. Rev. Lett. [**87**]{}, 186401 (2001). T. Maier, M. Jarrell, T. Pruschke, and M. H. Hettler, Rev. Mod. Phys. [**77**]{}, 1027 (2005). Th. A. Maier, Th. Pruschke, and M. Jarrell, Phys. Rev. B [**66**]{}, 075102 (2002). D. Seńećhal and A.-M. S. Tremblay, Phys. Rev. Lett. [**92**]{}, 126401 (2004). M. Civelli, M. Capone, S. S. Kancharla, O. Parcollet, and G. Kotliar, Phys. Rev. Lett. [**95**]{}, 106402 (2005). B. Kyung, S. S. Kancharla, D. Seńećhal, A.-M. S. Tremblay,M. Civelli, and G. Kotliar, Phys. Rev. B [**73**]{}, 165114 (2006). A. Macridin, M. Jarrell, T. Maier, P. R. C. Kent, and E. D’Azevedo, Phys. Rev. Lett [**97**]{}, 036401 (2006). M. Aichhorn, E. Arrigoni, M. Potthoff, and W. Hanke, Phys. Rev. B [**74**]{}, 024508 (2006). K. Haule and G. Kotliar, Phys. Rev. B [**76**]{}, 104509 (2007). A. Liebsch and N.-H. Tong, Phys. Rev. B [**80**]{}, 165126 (2009). M. Ferrero, P. S. Cornaglia, L. De Leo, O. Parcollet, G. Kotliar, and A. Georges, Phys. Rev. B [**80**]{}, 064501 (2009).
A. Macridin, M. Jarrell, T. Maier, and D. J. Scalapino, Phys. Rev. Lett. [**99**]{}, 237001 (2007). P. Werner, E. Gull, O. Parcollet, and A. J. Millis, Phys. Rev. B [**80**]{}, 045120 (2009); E. Gull, O. Parcollet, P. Werner, and A. J. Millis, [*ibid*]{} [**80**]{}, 245102 (2009). S. Sakai, Y. Motome, and M. Imada, Physica B [**404**]{}, 3183 (2009). I. Dzyaloshinskii, Phys. Rev. B [**68**]{}, 085113 (2003). T. D. Stanescu, P. Phillips, and T.-P. Choy, Phys. Rev. B [**75**]{}, 104503 (2007). R. M. Konik, T. M. Rice, and A. M. Tsvelik, Phys. Rev. Lett. [**96**]{}, 086407 (2006); K.-Y. Yang, T. M. Rice, and F.-C. Zhang, Phys. Rev. B [**73**]{}, 174501 (2006); K.-Y. Yang, H.-B. Yang, P. D. Johnson, T. M. Rice, and F.-C. Zhang, Europhys. Lett. [**86**]{}, 37002 (2009); K. Le Hur and T. M. Rice, Ann. Phys. [**324**]{}, 1452 (2009). E. Z. Kuchinskii and M. V. Sadovskii, JETP [**103**]{}, 415 (2006). F. H. L. Essler and A. M. Tsvelik, Phys. Rev. B [**65**]{}, 115117 (2002); [**71**]{}, 195116 (2005). C. Berthod, T. Giamarchi, S. Biermann, and A. Georges, Phys. Rev. Lett. [**97**]{}, 136401 (2006). M. S. Hybertsen, E. B. Stechel, M. Schluter, and D. R. Jennison, Phys. Rev. B [**41**]{}, 11068 (1990).
C. Lanczos, J. Research NBS [**45**]{}, 255 (1950). E. R. Gagliano and C. A. Balseiro, Phys. Rev. Lett. [**59**]{}, 2999 (1987). S. M. A. Rombouts, K. Heyde, and N. Jachowicz, Phys. Rev. Lett. [**82**]{}, 4155 (1999). S. Sakai, R. Arita, and H. Aoki, Physica B [**378-380**]{}, 288 (2006); S. Sakai, R. Arita, K. Held, and H. Aoki, Phys. Rev. B [**74**]{}, 155102 (2006). E. Gull, P. Werner, O. Parcollet, and M. Troyer, Eur. Phys. Lett. [**82**]{}, 57003 (2008). S. Rombouts, K. Heyde, and N. Jachowicz, Phys. Lett. A [**242**]{}, 271 (1998). J. E. Hirsch, Phys. Rev. B [**28**]{}, 4059 (1983); [**29**]{}, 4159 (1984). L. Balents, M. P. A. Fisher, and C. Nayak, Int. J. Mod. Phys. B [**12**]{}, 1033 (1998). M. Franz and Z. Tesanovic, Phys. Rev. Lett. [**87**]{}, 257003 (2001). M. R. Norman, H. Ding, M. Randeria, J. C. Campuzano, T. Yokoya, T. Takeuchi, T. Takahashi, T. Mochiku, K. Kadowaki, P. Guptasarma, and D. G. Hinks, Nature [**392**]{}, 157 (1998). T. Hanaguri, C. Lupien. Y. Kohsaka, D.-H. Lee, M. Azuma, M. Takano, H.Takagi, and J. C. Davis, Nature [**430**]{}, 1001 (2004). H.-B. Yang, J. D. Rameau, P. D. Johnson, T. Valla, A. Tsvelik, and G. D. Gu, Nature [**456**]{}, 77 (2008). J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. [**108**]{}, 1175 (1957). A. L. Efros and B. I. Shklovskii, J. Phys. C [**8**]{}, L49 (1975). H. Shinaoka and M. Imada, Phys. Rev. Lett. [**102**]{}, 016404 (2009); J. Phys. Soc. Jpn. [**78**]{}, 094708 (2009). A. Kanigel, U. Chatterjee, M. Randeria, M. R. Norman, G. Koren, K. Kadowaki, and J. C. Campuzano, Phys. Rev. Lett. [**101**]{}, 137002 (2008). M. Hashimoto, R.-H. He, K. Tanaka, J.-P. Testaud, W. Meevasana, R. G. Moore, D. Lu, H. Yao, Y. Yoshida, H. Eisaki, T. P. Devereaux, Z. Hussain, and Z.-X. Shen, Nature Phys. NPHYS1632 (2010). R. Preuss, W. Hanke, C. Grober, and H. G. Evertz, Phys. Rev. Lett. [**79**]{}, 1122 (1997). T. Yoshida, X. J. Zhou, D. H. Lu, S. Komiya, Y. Ando, H. Eisaki, T. Kakeshita, S. Uchida, Z. Hussain, Z.-X. Shen, and A. Fujimori, J. Phys.: Condens. Matter [**19**]{}, 125209 (2007). A. A. Kordyuk, S. V. Borisenko, A. Koitzsch, J. Fink, M. Knupfer, and H. Berger, Phys. Rev. B. [**71**]{}, 214513 (2005). J. Graf, G.-H. Gweon, K. McElroy, S. Y. Zhou, C. Jozwiak, E. Rotenberg, A. Bill, T. Sasagawa, H. Eisaki, S. Uchida, H. Takagi, D.-H. Lee, and A. Lanzara, Phys. Rev. Lett. [**98**]{}, 067004 (2007). B. P. Xie, K. Yang, D. W. Shen, J. F. Zhao, H. W. Ou, J. Weil, S. Y. Gu, M. Arita, S. Qiao, H. Namatame, M. Taniguchi, N. Kaneko, H. Eisaki, K. D. Tsuei, C. M. Cheng, I. Vobornik, J. Fujii, G. Rossi, Z. Q. Yang, and D. L. Feng, Phys. Rev. Lett. [**98**]{}, 147001 (2007). T. Valla, T. E. Kidd, W.-G. Yin, G. D. Gu, P. D. Johnson, Z.-H. Pan, and A. V. Fedorov, Phys. Rev. Lett. [**98**]{}, 167003 (2007). W. Meevasana, X. J. Zhou, S. Sahrakorpi, W. S. Lee, W. L. Yang, K. Tanaka, N. Mannella, T. Yoshida, D. H. Lu, Y. L. Chen, R. H. He, Hsin Lin, S. Komiya, Y. Ando, F. Zhou, W. X. Ti, J. W. Xiong, Z. X. Zhao, T. Sasagawa, T. Kakeshita, K. Fujita, S. Uchida, H. Eisaki, A. Fujimori, Z. Hussain, R. S. Markiewicz, A. Bansil, N. Nagaosa, J. Zaanen, T. P. Devereaux, and Z.-X. Shen, Phys. Rev. B. [**75**]{}, 174506 (2007). T. Yoshida, K. Tanaka, H. Yagi, A. Ino, H. Eisaki, A. Fujimori, and Z.-X. Shen, Phys. Rev. Lett. [**95**]{}, 146404 (2005). R. Eguchi, A. Chainani, M. Taguchi, M. Matsunami, Y. Ishida, K. Horiba, Y. Senba, H. Ohashi, and S. Shin, Phys. Rev. B [**79**]{}, 115122 (2009). K. Byczuk, M. Kollar, K. Held, Y.-F. Yang, I. A. Nekrasov, Th. Pruschke, and D. Vollhardt, Nature Phys. [**3**]{}, 168 (2007). P. V. Bogdanov, A. Lanzara, S. A. Kellar, X. J. Zhou, E. D. Lu, W. J. Zheng, G. Gu, J.-I. Shimoyama, K. Kishio, H. Ikeda, R. Yoshizaki, Z. Hussain, and Z. X. Shen, Phys. Rev. Lett. [**85**]{}, 2581 (2000). A. Kaminski, M. Randeria, J. C. Campuzano, M. R. Norman, H. Fretwell, J. Mesot, T. Sato, T. Takahashi, and K. Kadowaki, Phys. Rev. Lett. [**86**]{}, 1070 (2001). T. Sato, H. Matsui, T. Takahashi, H. Ding, H.-B. Yang, S.-C.Wang, T. Fujii, T. Watanabe, A. Matsuda, T. Terashima, and K. Kadowaki, Phys. Rev. Lett. [**91**]{}, 157003 (2003). A. Lanzara, P. V. Bogdanov, X. J. Zhou, S. A. Kellar, D. L. Feng, E. D. Lu, T. Yoshida, H. Eisaki, A. Fujimori, K. Kishio, J.-I. Shimoyama, T. Nodak, S. Uchidak, Z. Hussain, and Z.-X. Shen, Nature [**412**]{}, 510 (2001). P. D. Johnson, T. Valla, A. V. Fedorov, Z. Yusof, B. O. Wells, Q. Li, A. R. Moodenbaugh, G. D. Gu, N. Koshizuka, C. Kendziora, Sha Jian, and D. G. Hinks, Phys. Rev. Lett. [**87**]{}, 177007 (2001). Note that ARPES spectra are also usually integrated around the Fermi level and that a typical width of the energy window is $\sim 10\text{meV} \sim 0.025t$, which is similar to the $T$ window used in Eq. (\[eq:ghb\]). E. Gull, P. Werner, X. Wang, M. Troyer, and A. J. Millis, Europhys. Lett. [**84**]{}, 37009 (2008). J. Meng, G. Liu, W. Zhang, L. Zhao, H. Liu, X. Jia, D. Mu, S. Liu, X. Dong, J. Zhang, W. Lu, G. Wang, Y. Zhou, Y. Zhu, X. Wang, Z. Xu, C. Chen, and X. J. Zhou, Nature [**462**]{}, 335 (2009). M. Ikeda, T. Yoshida, A. Fujimori, M. Kubota, K. Ono, Y. Kaga, T. Sasagawa, and H. Takagi, Phys. Rev. B. [**80**]{}, 184506 (2009). The spectral feature that the band bottom is at a higher energy around $(\pi,0)$ than around $(0,0)$, as well as a waterfall-like structure at $\w>0$, can also be seen in Fig. 5(b) in Ref. . Y. Onose, Y. Taguchi, K. Ishizaka, and Y. Tokura, Phys. Rev. Lett. [**87**]{}, 217001 (2001). H. Kusunose and T. M. Rice, Phys. Rev. Lett. [**91**]{}, 186407 (2003). B. Kyung, V. Hankevych, A.-M. Dare, and A.-M. S. Tremblay, Phys. Rev. Lett. [**93**]{}, 147004 (2004). M. Ikeda, T. Yoshida, A. Fujimori, M. Kubota, K. Ono, H. Das, T. Saha-Dasgupta, K. Unozawa, Y. Kaga, T. Sasagawa, and H. Takagi, Phys. Rev. B [**80**]{}, 014510 (2009). N. P. Armitage, F. Ronning, D. H. Lu, C. Kim, A. Damascelli, K. M. Shen, D. L. Feng, H. Eisaki, Z.-X. Shen, P. K. Mang, N. Kaneko, M. Greven, Y. Onose, Y. Taguchi, and Y. Tokura, Phys. Rev. Lett. [**88**]{}, 257001 (2002). The doping evolution of the Fermi surface and zero surface in hole-doped cases was discussed in Ref. . M. M. Korshunov and S. G. Ovchinnikov, Eur. Phys. J. B [**57**]{}, 271 (2007); M. M. Korshunov, E. V. Zakharova, I. A. Nekrasov, Z. V. Pchelkina, and S. G. Ovchinnikov, J. Phys.: Condens. Matter [**22**]{}, 015701 (2010). T. Helm, M. V. Kartsovnik, M. Bartkowiak, N. Bittner, M. Lambacher, A. Erb, J. Wosnitza, and R. Gross, Phys. Rev. Lett. [**103**]{}, 157002 (2009). N. Lin, E. Gull, and A. J. Millis, arXiv:1004.2999. Y. Yamaji and M. Imada, unpublished. J. E. Hirsch and R. M. Fye, Phys. Rev. Lett. [**56**]{}, 2521 (1986). The density of states also shows a gap at the corresponding energy.
| ArXiv |
---
abstract: 'Periodic control systems used in spacecrafts and automotives are usually period-driven and can be decomposed into different modes with each mode representing a system state observed from outside. Such systems may also involve intensive computing in their modes. Despite the fact that such control systems are widely used in the above-mentioned safety-critical embedded domains, there is lack of domain-specific formal modelling languages for such systems in the relevant industry. To address this problem, we propose a formal visual modeling framework called as a concise and precise way to specify and analyze such systems. To capture the temporal properties of periodic control systems, we provide, along with , a property specification language based on interval logic for the description of concrete temporal requirements the engineers are concerned with. The statistical model checking technique can then be used to verify the models against desired properties. To demonstrate the viability of our approach, we have applied our modelling framework to some real life case studies from industry and helped detect two design defects for some spacecraft control systems.'
author:
- |
Zheng Wang$^{1, 5}$, Geguang Pu$^{1}$, Shenchao Qin$^{2}$, Jianwen Li$^{1}$,\
Kim G. Larsen$^{3}$, Jan Madsen$^{4}$, Bin Gu$^{5}$, Jifeng He$^{1}$
bibliography:
- 'main.bib'
title: |
: A Mode Diagram Modeling Framework for\
Periodic Control Systems
---
$^{1}$ [<[email protected]>]{}, [<[email protected]>]{},\
Shanghai Key Laboratory of Trustworthy Computing,\
East China Normal University\
$^{2}$ [<[email protected]>]{}, University of Teesside\
$^{3}$ [<[email protected]>]{}, Aalborg University of Denmark\
$^{4}$ [<[email protected]>]{}, Technical University of Denmark\
$^{5}$ [<[email protected]>]{}, Beijing Institute of Control Engineering
intro.tex
lang.tex
property.tex
verification.tex
experiment.tex
related.tex
| ArXiv |
---
author:
- 'O. Cucciati'
- 'B. C. Lemaux'
- 'G. Zamorani'
- 'O. Le Fèvre'
- 'L. A. M. Tasca'
- 'N. P. Hathi'
- 'K-G. Lee'
- 'S. Bardelli'
- 'P. Cassata'
- 'B. Garilli'
- 'V. Le Brun'
- 'D. Maccagni'
- 'L. Pentericci'
- 'R. Thomas'
- 'E. Vanzella'
- 'E. Zucca'
- 'L. M. Lubin'
- 'R. Amorin'
- 'L. P. Cassarà'
- 'A. Cimatti'
- 'M. Talia'
- 'D. Vergani'
- 'A. Koekemoer'
- 'J. Pforr'
- 'M. Salvato'
bibliography:
- 'biblio.bib'
date: 'Received - ; accepted -'
title: 'The progeny of a Cosmic Titan: a massive multi-component proto-supercluster in formation at z=2.45 in VUDS[^1]'
---
Introduction {#intro}
============
Proto-clusters are crucial sites for studying how environment affects galaxy evolution in the early universe, both in observations (see e.g. [@steidel05; @peter07; @miley08; @tanaka10; @strazzullo13]) and simulations (e.g. [@chiang17; @muldrew18]). Moreover, since proto-clusters mark the early stages of structure formation, they have the potential to provide additional constraints on the already well established probes on standard and non-standard cosmology based on galaxy clusters at low and intermediate redshift (see e.g. [@allen11; @heneka18; @schmidt09; @roncarelli15], and references therein).
Although the sample of confirmed or candidate proto-clusters is increasing in both number (see e.g. the systematic searches in [@diener2013_list; @chiang2014_list; @franck16_CCPC; @lee2016_colossus; @toshikawa18_goldrush]) and maximum redshift (e.g. [@higuchi18_silverrush]), our knowledge of high-redshift ($z>2$) structures is still limited, as it is broadly based on heterogeneous data sets. These structures span from relaxed to unrelaxed systems, and are detected by using different, and sometimes apparently contradicting, selection criteria. As a non-exhaustive list of examples, high-redshift clusters and proto-clusters have been identified as excesses of either star-forming galaxies (e.g. [@steidel00; @ouchi05; @lemaux09; @capak11]) or red galaxies (e.g. [@kodama07; @spitler12]), as excesses of infrared(IR)-luminous galaxies [@gobat11], or via SZ signatures [@foley11_SZ] or diffuse X-ray emission [@fassbender11]. Other detection methods include the search for photometric redshift overdensities in deep multi-band surveys [@salimbeni09; @scoville13_env] or around active galactic nuclei (AGNs) and radio galaxies [@pentericci00; @galametz12], the identification of large intergalactic medium reservoirs via Ly$\alpha$ forest absorption [@cai16_method; @lee2016_colossus; @cai17_z23], and the exploration of narrow redshift slices via narrow band imaging [@venemans02; @lee14_NB].
The identification and study of proto-structures can be boosted by two factors: 1) the use of spectroscopic redshifts, and 2) the use of unbiased tracers with respect to the underlying galaxy population. On the one hand, the use of spectroscopic redshifts is crucial for a robust identification of the overdensities themselves, for the study of the velocity field, especially in terms of the galaxy velocity dispersion which can be used as a proxy for the total mass, and finally for the identification of possible sub-structures. On the other hand, if such proto-structures are found and mapped by tracers that are representative of the dominant galaxy population at the epoch of interest, we can recover an unbiased view of such environments.
In this context, we used the VUDS (VIMOS Ultra Deep Survey) spectroscopic survey [@lefevre2015_vuds] to systematically search for proto-structures. VUDS targeted approximately $10000$ objects presumed to be at high redshift for spectroscopic observations, confirming over $5000$ galaxies at $z>2$. These galaxies generally have stellar masses $\gtrsim 10^{9} {\rm M}_{\odot}$, and are broadly representative in stellar mass, absolute magnitude, and rest-frame colour of all star-forming galaxies (and thus, the vast majority of galaxies) at $2 \lesssim z \lesssim 4.5$ for $i\leq25$. We identified a preliminary sample of $\sim50$ candidate proto-structures (Lemaux et al, in prep.) over $2<z<4.6$ in the COSMOS, CFHTLS-D1 and ECDFS fields (1 deg$^2$ in total). With this ‘blind’ search in the COSMOS field we identified the complex and rich proto-structure at $z\sim2.5$ presented in this paper.
This proto-structure, extended over a volume of $\sim60\times60\times150$ comoving Mpc$^3$, has a very complex shape, and includes several density peaks within $2.42<z<2.51$, possibly connected by filaments, that are more dense than the average volume density. Smaller components of this proto-structure have already been identified in the literature from heterogeneous galaxy samples, like for example Ly$\alpha$ emitters (LAEs), three-dimensional (3D) Ly$\alpha$-forest tomography, sub-millimetre starbursting galaxies, and CO-emitting galaxies (see [@diener2015_z245; @chiang2015_z244; @casey2015_z247; @lee2016_colossus; @wang2016_z250]). Despite the sparseness of previous identifications of sub-clumps, a part of this structure was already dubbed “Colossus” for its extension [@lee2016_colossus].
With VUDS, we obtain a more complete and unbiased panoramic view of this large structure, placing the previous sub-structure detections reported in the literature in the broader context of this extended large-scale structure. The characteristics of this proto-structure, its redshift, its richness over a large volume, the clear detection of its sub-components, the extensive imaging and spectroscopy coverage granted by the COSMOS field, provide us the unique possibility to study a rich supercluster in its formation.
From now on we refer to this huge structure as a ‘proto-supercluster’. On the one hand, throughout the paper we show that it is as extended and as massive as known superclusters at lower redshift. Moreover, it presents a very complex shape, which includes several density peaks embedded in the same large-scale structure, similarly to other lower-redshift structures defined superclusters. In particular, one of the peaks has already been identified in the literature [@wang2016_z250] as a possibly virialised structure. On the other hand, we also show that the evolutionary status of some of these peaks is compatible with that of overdensity fluctuations which are collapsing and are foreseen to virialise in a few gigayears. For all these reasons, we consider this structure a proto-supercluster.
In this work, we aim to characterise the 3D shape of the proto-supercluster, and in particular to study the properties of its sub-components, for example their average density, volume, total mass, velocity dispersion, and shape. We also perform a thorough comparison of our findings with the previous density peaks detected in the literature on this volume, so as to put them in the broader context of a large-scale structure.
The paper is organised as follows. In Sect. \[data\] we present our data set and how we reconstruct the overdensity field. The discovery of the proto-supercluster, and its total volume and mass, are discussed in Sect. \[supercluster\]. In Sect. \[3D\_peaks\] we describe the properties of the highest density peaks embedded in the proto-supercluster (their individual mass, velocity dispersion, etc.) and we compare our findings with the literature. In Sect. \[discussion\] we discuss how the peaks would evolve according to the spherical collapse model, and how we can compare the proto-supercluster to similar structures at lower redshifts. Finally, in Sect. \[summary\] we summarise our results.
Except where explicitly stated otherwise, we assume a flat $\Lambda$CDM cosmology with $\Omega_m=0.25$, $\Omega_{\Lambda}=0.75$, $H_0=70\kms {\rm Mpc}^{-1}$ and $h=H_0/100$. Magnitudes are expressed in the AB system [@oke74; @fukugita96]. Comoving and physical Mpc(/kpc) are expressed as cMpc(/ckpc) and pMpc(/pkpc), respectively.
The data sample and the density field {#data}
=====================================
VUDS is a spectroscopic survey performed with VIMOS on the ESO-VLT [@lefevre2003], targeting approximately $10000$ objects in the three fields COSMOS, ECDFS, and VVDS-2h to study galaxy evolution at $2 \lesssim z \lesssim 6$. Full details are given in [@lefevre2015_vuds]; here we give only a brief review.
VUDS spectroscopic targets have been pre-selected using four different criteria. The main criterion is a photometric redshift ($z_p$) cut ($z_p+1\sigma \geq 2.4$, with $z_p$ being either the $1^{st}$ or $2^{nd}$ peak of the $z_p$ probability distribution function) coupled with the flux limit $i\leq 25$. This main criterion provided 87.7% of the primary sample. Photometric redshifts were derived as described in [@ilbert2013] with the code [*Le Phare[^2]*]{} [@arnouts99; @ilbert2006_pz]. The remaining targets include galaxies with colours compatible with Lyman-break galaxies, if not already selected by the $z_p$ criterion, as well as drop-out galaxies for which a strong break compatible with $z>2$ was identified in the $ugrizYJHK$ photometry. In addition to this primary sample, a purely flux-limited sample with $23 \leq i \leq 25$ has been targeted to fill-up the masks of the multi-slit observations.
VUDS spectra have an extended wavelength coverage from 3600 to $9350\AA$, because targets have been observed with both the LRBLUE and LRRED grisms (both with R$\sim230$), with 14h integration each. With this integration time it is possible to reach S/N $\sim 5$ on the continuum at $\lambda\sim8500$Å (for $i = 25$), and for an emission line with flux $F = 1.5\times10^{-18}$erg s$^{-1}$ cm$^{2}$. The redshift accuracy is $\sigma_{zs}= 0.0005 (1 + z)$, corresponding to $\sim150\kms$ (see also [@lefevre2013a]).
We refer the reader to [@lefevre2015_vuds] for a detailed description of data reduction and redshift measurement. Concerning the reliability of the measured redshifts, here it is important to stress that each measured redshift is given a reliability flag equal to X1, X2, X3, X4, or X9[^3], which correspond to a probability of being correct of 50-75%, 75-85%, 95-100%, 100%, and $\sim80$% respectively. In the COSMOS field, the VUDS sample comprises 4303 spectra of unique objects, out of which 2045 have secure spectroscopic redshift (flags X2, X3, X4, or X9) and $z\ge 2.$
Together with the VUDS data, we used the zCOSMOS-Bright [@lilly2007; @lilly2009] and zCOSMOS-Deep (Lilly et al, in prep., [@diener2013_list]) spectroscopic samples. The flag system for the robustness of the redshift measurement is basically the same as in the VUDS sample, with very similar flag probabilities (although they have never been fully assessed for zCOSMOS-Deep). In the zCOSMOS samples, the spectroscopic flags have also been given a decimal digit to represent the level of agreement of the spectroscopic redshift ($z_s$) with the photometric redshift ($z_p$). A given $z_p$ is defined to be in agreement with its corresponding $z_s$ when $|z_s-z_p|<0.08(1+z_s)$, and in these cases the decimal digit of the spectroscopic flag is ‘5’. For the zCOSMOS samples, we define secure $z_s$ those with a quality flag X2.5, X3, X4, or X9, which means that for flag X2 we used only the $z_s$ in agreement with their respective $z_p$, while for higher flags we trust the $z_s$ irrespectively of the agreement with their $z_p$. With these flag limits, we are left with more than 19000 secure $z_s$, of which 1848 are at $z\geq 2$. We merged the VUDS and zCOSMOS samples, removing the duplicates between the two surveys as follows. For each duplicate, that is, objects observed in both VUDS and zCOSMOS, we retained the redshift with the most secure quality flag, which in the vast majority of cases was the one from VUDS. In case of equal flags, we retained the VUDS spectroscopic redshift. Our final VUDS$+$zCOSMOS spectroscopic catalogue consists of 3822 unique secure $z_s$ at $z\geq 2$.
We note that we did not use spectroscopic redshifts from any other survey, although other spectroscopic samples in this area are already publicly available in the literature (see e.g. [@casey2015_z247; @chiang2015_z244; @diener2015_z245; @wang2016_z250]). These samples are often follow-up of small regions around dense regions, and we did not want to be biased in the identification of already known density peaks. Unless specified otherwise, our spectroscopic sample always refers only to the good quality flags in VUDS and zCOSMOS discussed above. We also did not include public $z_s$ from more extensive campaigns, like for example the COSMOS AGN spectroscopic survey [@trump09], the MOSDEF survey [@kriek15], or the DEIMOS 10K spectroscopic survey [@hasinger18].
We matched our spectroscopic catalogue with the photometric COSMOS2015 catalogue [@laigle2016]. The matching was done by selecting the closest source within a matching radius of $0.55^{\prime \prime}$. Objects in the COSMOS2015 have been detected via an ultra-deep $\chi^2$ sum of the $YJHK_s$ and $z^{++}$ images. $YJHK_s$ photometry was obtained by the VIRCAM instrument on the VISTA telescope (UltraVISTA-DR2 survey[^4], [@mcCracken12]), and the $z^{++}$ data, taken using the Subaru Suprime-Cam, are a (deeper) replacement of the previous $z-$band COSMOS data [@taniguchi2007; @taniguchi2015]. With this match with the COSMOS2015 catalogue we obtained a uniform target coverage of the COSMOS field down to a given flux limit (see Sect.\[method\]), using spectroscopic redshifts for the objects in our original spectroscopic sample or photometric redshifts for the remaining sources. The photometric redshifts in COSMOS2015 are derived using $3^{\prime \prime}$ aperture fluxes in the 30 photometric bands of COSMOS2015. According to Table 5 of [@laigle2016], a direct comparison of their photometric redshifts with the spectroscopic redshifts of the entire VUDS survey in the COSMOS field (median redshift $z_{\rm med}=2.70$ and median $i^+-$band $i^+_{\rm med}=24.6$) gives a photometric redshift accuracy of $\Delta z = 0.028(1+z)$. The same comparison with the zCOSMOS-Deep sample (median redshift $z_{\rm med}=2.11$ and median $i^+-$band $i^+_{\rm med}=23.8$) gives $\Delta z = 0.032(1+z)$.
{width="6.cm"} {width="6.cm"} {width="6.cm"}
The method to compute the density field and identify the density peaks is the same as described in [@lemaux2018_z45]; we describe it here briefly. The method is based on the Voronoi Tessellation, which has already been successfully used at different redshifts to characterise the local environment around galaxies and identify the highest density peaks, including the search for groups and clusters (see e.g. [@marinoni02; @coooper05; @cucciati10; @gerke12; @scoville13_env; @darvish15; @smolcic17]). Its main advantage is that the local density is measured both on an adaptive scale and with an adaptive filter shape, allowing us to follow the natural distribution of tracers.
In our case, we worked in two dimensions in overlapping redshift slices. We used as tracers the spectroscopic sample complemented by a photometric sample which provides us with the photometric redshifts of all the galaxies for which we did not have any $z_s$ information.
For each redshift slice, we generated a set of Monte Carlo (MC) realisations. Galaxies (with $z_s$ or $z_p$) to be used in each realisation were selected observing the following steps, in this order:
- irrespectively of their redshift, galaxies with a $z_s$ were retained in a percentage of realisations equal to the probability associated to the reliability flag; namely, in each realisation, before the selection in redshift, for each galaxy we drew a number from a uniform distribution from 0 to 100 and retained that galaxy only if the drawn number was equal to or less than the galaxy redshift reliability;
- galaxies with only $z_p$ were first selected to complement the retained spectroscopic sample (i.e. the photometric sample comprises all the galaxies without a $z_s$ or for which we threw away their $z_s$ for a given iteration), then they were assigned a new photometric redshift $z_{\rm p,new}$ randomly drawn from an asymmetrical Gaussian distribution centred on their nominal $z_p$ value and with negative and positive sigmas equal to the lower and upper uncertainties in the $z_p$ measurement, respectively; with this approach we do not try to correct for catastrophic redshift errors, but only for the shape of the PDF of each $z_p$;
- among the samples selected at steps 1 and 2, we retained all the galaxies with $z_s$ (from step 1) or $z_{\rm p,new}$ (from step 2) falling in the considered redshift slice.
We performed a 2D Voronoi tessellation for each $i^{th}$ MC realisation, and assigned to each Voronoi polygon a surface density $\Sigma_{VMC,i}$ equal to the inverse of the area (expressed in Mpc$^2$) of the given polygon. Finally, we created a regular grid of $75\times75$ pkpc cells, and assigned to each grid point the $\Sigma_{VMC,i}$ of the polygon enclosing the central point of the cell. For each redshift slice, the final density field $\Sigma_{VMC}$ is computed on the same grid, as the median of the density fields among the realisations, cell by cell. As a final step, from the median density map we computed the local over-density at each grid point as $\delta_{\rm gal} = \Sigma_{VMC}/ \tilde{\Sigma}_{VMC} -1 $, where $\tilde{\Sigma}_{VMC}$ is the mean $\Sigma_{VMC}$ for all grid points. In our analysis we are more interested in $\delta_{\rm gal}$ than in $\Sigma_{VMC}$ because we want to identify the regions that are overdense with respect to the mean density at each redshift, a density which can change not only for astrophysical reasons but also due to characteristics of the imaging/spectroscopic survey. Moreover, as we see in the following sections, the computation of $\delta_{\rm gal}$ is useful to estimate the total mass of our proto-cluster candidates and their possible evolution.
Proto-cluster candidates were identified by searching for extended regions of contiguous grid cells with a $\delta_{\rm gal}$ value above a given threshold. The initial systematic search for proto-clusters in the COSMOS field (which will be presented in Lemaux et al., in prep.) was run with the following set of parameters: redshift slices of 7.5 pMpc shifting in steps of 3.75 pMpc (so as to have redshift slices overlapping by half of their depth); 25 Monte Carlo realisations per slice; and spectroscopic and photometric catalogues with $[3.6]\leq 25.3$ (IRAC Channel 1). With this ‘blind’ search we re-identified two proto-clusters at $z\sim3$ serendipitously discovered at the beginning of VUDS observations [@lemaux2014_z33; @cucciati2014_z29], together with other outstanding proto-structures presented separately in companion papers ([@lemaux2018_z45], Lemaux et al. in prep.).
Discovery of a rich extended proto-supercluster {#supercluster}
===============================================
The preliminary overdensity maps showed two extended overdensities at $z\sim2.46$, in a region of $0.4\times0.25$ deg$^2$. Intriguingly, there were several other smaller overdensities very close in right ascension (RA), declination (Dec), and redshift. We therefore explored in more detail the COSMOS field by focusing our attention on the volume around these overdensities. This focused analysis revealed the presence of a rich extended structure, consisting of density peaks linked by slightly less dense regions.
The method {#method}
----------
We re-ran the computation of the density field and the search for overdense regions with a fine-tuned parameter set (see below), in the range $2.35 \lesssim z \lesssim 2.55$, which we studied by considering several overlapping redshift slices. Concerning the angular extension of our search, we computed the density field in the central $\sim1\times1$ deg$^2$ of the COSMOS field, but then used only the slightly smaller 0.91 deg$^2$ region at $149.6\leq RA \leq 150.52$ and $1.74 \leq Dec \leq2.73$ to perform any further analysis (computation of the mean density etc.). This choice was made to avoid the regions close to the field boundaries, where the Voronoi tessellation is affected by border effects. In this smaller area, considering a flux limit at $i=25$, about 24% of the objects with a redshift ($z_s$ or $z_p$) falling in the above-mentioned redshift range have a spectroscopic redshift. If we reduce the area to the region covered by VUDS observations, which is slightly smaller, this percentage increases to about 28%.
We also verified the robustness of our choices for what concerns the following issues:
[**Number of Monte Carlo realisations**]{}. With respect to Lemaux et al. (in prep.), we increased the number of Monte Carlo realisations from the initial 25 to 100 to obtain a more reliable median value (similarly to, e.g. [@lemaux2018_z45]). We verified that our results did not significantly depend on the number of realisations $n_{\rm MC}$ as long as $n_{\rm MC}\geq 100$, and, therefore, all analyses presented in this paper are done on maps which used $n_{\rm
MC}=100$. This high number of realisations allowed us to produce not only the median density field for each redshift slice, but also its associated error maps, as follows. For each grid cell, we considered the distribution of the 100 $\Sigma_{VMC}$ values, and took the $16^{th}$ and $84^{th}$ percentiles of this distribution as lower and upper limits for $\Sigma_{VMC}$. We produced density maps with these lower and upper limits, in the same way as for the median $\Sigma_{VMC}$, and then computed the corresponding overdensities that we call $\delta_{\rm gal,16}$ and $\delta_{\rm gal,84}$.
[**Spectroscopic sample.**]{} As in [@lemaux2018_z45], we assigned a probability to each spectroscopic galaxy to be used in a given realisation equal to the reliability of its $z_s$ measurement, as given by its quality flag. Namely, we used the quality flags X2 (X2.5 for zCOSMOS), X3, X4, and X9 with a reliability of 80%, 97.5%, 100% and 80% respectively (see Sect. \[data\]; here we adopt the mean probability for the flags X2 and X3, for which [@lefevre2015_vuds] give a range of probabilities). These values were computed for the VUDS survey, but we applied them also to the zCOSMOS spectroscopic galaxies in our sample, as discussed in Sect. \[data\]. We verified that our results do not qualitatively change if we choose slightly different reliability percentages or if we used the entire spectroscopic sample (flag=X2/X2.5, X3, X4, X9) in all realisations instead of assigning a probability to each spectroscopic galaxy. The agreement between these results is due to the very high flag reliabilities, and to the dominance of objects with only $z_p$. With the cut in redshift at $2.35 \leq z \leq 2.55$, the above-mentioned quality flag selection, and the magnitude limit at $i\leq 25$ (see below), we are left with 271 spectroscopic redshifts from VUDS and 309 from zCOSMOS, for a total of 580 spectroscopic redshifts used in our analysis. This provides us with a spectroscopic sampling rate of $\sim24\%$, considering the above mentioned redshift range and magnitude cut. We remind the reader that we use only VUDS and zCOSMOS spectroscopic redshift, and do not include in our sample any other $z_s$ found in the literature.
[**Mean density.**]{} To compute the mean density $\tilde{\Sigma}_{VMC}$ we proceeded as follows. Given that $\Sigma_{VMC}$ has a log-normal distribution [@coles91], in each redshift slice we fitted the distribution of ${\rm log}(\Sigma_{VMC})$ of all pixels with a $3\sigma$-clipped Gaussian. The mean $\mu$ and standard deviation $\sigma$ of this Gaussian are related to the average density $\langle \Sigma_{VMC} \rangle$ by the equation $\langle \Sigma_{VMC}\rangle = 10^{\mu}e^{2.652\sigma^{2}}$. We used this $\langle \Sigma_{VMC} \rangle$ as the average density $\tilde{\Sigma}_{VMC}$ to compute the density contrast $\delta_{gal}$. $\tilde{\Sigma}_{VMC}$ was computed in this way in each redshift slice.
[**Overdensity threshold.**]{} In each redshift slice, we fitted the distribution of ${\rm log}(1+\delta_{\rm gal})$ with a Gaussian, obtaining its $\mu$ and $\sigma$. We call these parameters $\mu_{\rm \delta}$ and $\sigma_{\rm \delta}$, for simplicity, although they refer to the Gaussian fit of the ${\rm log}(1+\delta_{\rm gal})$ distribution and not of the $\delta_{\rm gal}$ distribution. We then fitted $\mu_{\rm \delta}$ and $\sigma_{\rm \delta}$ as a function of redshift with a second-order polynomial, obtaining $\mu_{\rm \delta,fit}$ and $\sigma_{\rm \delta,fit}$ at each redshift. Our detection thresholds were then set as a certain number of $\sigma_{\rm \delta,fit}$ above the mean overdensity $\mu_{\rm \delta,fit}$, that is, as $ {\rm log}(1+\delta_{\rm gal}) \geq \mu_{\rm \delta,fit}(z_{\rm
slice}) + n_{\sigma}\sigma_{\rm \delta,fit}(z_{\rm slice})$, where $z_{\rm slice}$ is the central redshift of each slice, and $n_{\sigma}$ is chosen as described in Sects. \[3D\] and \[3D\_peaks\]. From now, when referring to setting a ‘$n_{\sigma}\sigma_{\rm \delta}$ threshold’ we mean that we consider the volume of space with $ {\rm log}(1+\delta_{\rm gal}) \geq \mu_{\rm \delta,fit}(z_{\rm
slice}) + n_{\sigma}\sigma_{\rm \delta,fit}(z_{\rm slice})$.
[**Slice depth and overlap.**]{} We used overlapping redshift slices with a full depth of 7.5 pMpc, which corresponds to $\delta z \sim 0.02$ at $z\sim2.45$, running in steps of $\delta z \sim 0.002$. We also tried with thinner slices (5 pMpc), but we adopted a depth of 7.5 pMpc as a compromise between i) reducing the line of sight (l.o.s.) elongation of the density peaks (see Sect. \[3D\]) and ii) keeping a low noise in the density reconstruction. We define ‘noise’ as the difference between $\delta_{\rm gal}$ and its lower and upper uncertainties $\delta_{\rm gal,16}$ and $\delta_{\rm gal,84}$[^5]. The choice of small steps of $\delta z \sim 0.002$ is due to the fact that we do not want to miss the redshift where each structure is more prominent.
[**Tracers selection**]{} We fine-tuned our search method (including the $\delta_{\rm gal}$ thresholds etc...) for a sample of galaxies limited at $i=25$. We verified the robustness of our findings by using also a sample selected with $K_S\leq 24$ and one selected with ${\rm[3.6]}\leq24$ (IRAC Channel 1). With these two latter cuts, in the redshift range $2.3\leq z \leq 2.6$ we have a number of galaxies with spectroscopic redshift corresponding to $\sim87\%$ and $\sim94\%$ of the number of spectroscopic galaxies with $i\leq25$, respectively, but not necessarily the same galaxies, while roughly 65% and 85% more objects, respectively, with photometric redshifts entered in our maps than did with $i\leq25$. Although the $K_S\leq 24$ and ${\rm[3.6]}\leq24$ samples might be distributed in a different way in the considered volume because of the different clustering properties of different galaxy populations, with these samples we recovered the overdensity peaks in the same locations as with $i\leq25$. Clearly, the $\delta_{\rm gal}$ distribution is slightly different, so the overdensity threshold that we used to define the overdensity peaks (see Sect. \[3D\_peaks\]) encloses regions with slightly different shape with respect to those recovered with a sample flux-limited at $i\leq25$. We defer a more precise analysis of the kind of galaxy populations which inhabit the different density peaks to future work.
Figure \[2D\_maps\] shows three 2D overdensity ($\delta_{\rm gal}$) maps obtained as described above, in the redshift slices $2.422<z<2.444$, $2.438<z<2.460$, and $2.454<z<2.476$. We can distinguish two extended and very dense components at two different redshifts and different RA-Dec positions: one at $z\sim 2.43$, in the left-most panel, that we call the “South-West” (SW) component, and the other at $z\sim 2.46$, at higher RA and Dec, that we call here the “North-East” (NE) component (right-most panel). The NE and SW components seem to be connected by a region of relatively high density, shown in the middle panel of the figure. This sort of filament is particularly evident when we fix a threshold around $2\sigma_{\rm \delta}$, as shown in the figure. For this reason, we retained the $2\sigma_{\rm \delta}$ threshold as the threshold used to identify the volume of space occupied by this huge overdensity. As a reference, a $2\sigma_{\rm \delta}$ threshold corresponds to $\delta\sim0.65$, while 3, 4, and 5$\sigma_{\rm \delta}$ thresholds correspond to $\delta\sim1.1$, $\sim1.7$, and $\sim2.55$, respectively
To better understand the complex shape of the structure, we performed an analysis in three dimensions, as described in the following sub-section.
The 3D matter distribution {#3D}
--------------------------
We built a 3D overdensity cube in the following way. First, we considered each redshift slice to be placed at $z_{\rm slice}$ along the line of sight, where $z_{\rm slice}$ is the central redshift of the slice. All the 2D maps were interpolated at the positions of the nodes in the 2D grid of the lowest redshift ($z=2.35$). This way we have a 3D data cube with RA-Dec pixel size corresponding to $\sim75\times75$ pkpc at $z=2.38$, and a l.o.s. pixel size equal to $\delta z \sim 0.002$ (see Sect. \[method\]). From now on we use ‘pixels’ and ‘grid cells’ with the same meaning, referring to the smallest components of our data cube. We smoothed our data cube in RA and Dec with a Gaussian filter with sigma equal to 5 pixels. Along the l.o.s., we used instead a boxcar filter with a depth of 3 pixels. The shape and dimension of the smoothing in RA-Dec was chosen as a compromise between the two aims of i) smoothing the shapes of the Voronoi polygons and ii) not washing away the highest density peaks. The smoothing along the l.o.s. was done to link each redshift slice with the previous and following slice. Different choices on the smoothing filters do not significantly affect the 2D maps in terms of the shapes of the over-dense regions, and have only a minor effect on the values of $\delta_{\rm gal}$, even if the highest-density peaks risk to be washed away in case of excessive smoothing. We produced data cubes for the lower and upper limits of $\delta_{\rm gal}$ ($\delta_{\rm gal,16}$ and $\delta_{\rm gal,84}$) in the same way. These two latter cubes are used for the treatment of uncertainties in our following analysis.
Figure \[2D\_maps\] shows that around the main components of the proto-supercluster there are less extended density peaks. Since we wanted to focus our attention on the proto-supercluster, we excluded from our analysis all the density peaks not directly connected to the main structure. To do this, we proceeded as follows: we started from the pixels of the 3D grid which are enclosed in the $2\sigma_{\rm \delta}$ contour of the “NE” region in the redshift slice $2.454<z<2.476$ (right panel of Fig. \[2D\_maps\]). Starting from this pixel set, we iteratively searched in the 3D cube for all the pixels, contiguous to the previous pixels set, with a ${\rm log}(1+\delta_{\rm gal})$ higher than $2\sigma_{\rm \delta}$ above the mean, and we added those pixels to our pixel set. We stopped the search when there were no more contiguous pixels satisfying the threshold on ${\rm log}(1+\delta_{\rm gal})$. In this way we define a single volume of space enclosed in a $2\sigma_{\rm \delta}$ surface, and we define our proto-supercluster as the volume of space comprised within this surface. The final 3D overdensity map of the proto-supercluster is shown in Fig. \[3D\_cube\], with the three axes in comoving megaparsecs.
The 3D shape of the proto-supercluster is very irregular. The NE and SW components are clearly at different average redshifts, and have very different 3D shapes. Figure \[3D\_cube\] also shows that both components contain some density peaks (visible as the reddest regions within the $2\sigma_{\rm \delta}$ surface) with a very high average $\delta_{\rm gal}$. We discuss the properties of these peaks in detail in Sect. \[3D\_peaks\].
The volume occupied by the proto-supercluster shown in Fig. \[3D\_cube\] is about $9.5\times 10^4$ cMpc$^3$ (obtained by adding up the volume of all the contiguous pixels bounded by the 2$\sigma_{\rm \delta}$ surface), and the average overdensity is $\langle \delta_{\rm gal} \rangle \sim 1.24$. We can give a rough estimate of the total mass $M_{\rm tot}$ of the proto-supercluster by using the formula (see [@steidel98]):
$$\displaystyle
M_{\rm tot}=\rho_{\rm m} V (1+\delta_{\rm m}),
\label{eq_mass}$$
where $\rho_m$ is the comoving matter density, $V$ the volume[^6] that encloses the proto-cluster and $\delta_{\rm m}$ the matter overdensity in our proto-cluster. We computed $\delta_{\rm m}$ by using the relation $\delta_{\rm m}=\langle \delta_{\rm gal} \rangle/b$, where $b$ is the bias factor. Assuming $b=2.55$, as derived in [@durkalec15b] at $z\sim2.5$ with roughly the same VUDS galaxy sample we use here, we obtain $M_{\rm tot} \sim 4.8\times10^{15}{\rm M}_{\odot}$. There are at least two possible sources of uncertainty in this computation[^7]. The first is the chosen $\sigma_{\rm \delta}$ threshold. If we changed our threshold by $\pm0.2\sigma_{\rm \delta}$ around our adopted value of $2\sigma_{\rm \delta}$, $\langle \delta_{\rm gal} \rangle$ would vary by $\sim \pm 10\%$ and the volume would vary by $\sim \pm 17\%$, for a variation of the estimated mass of $\sim \pm 15\%$ (a higher threshold means a higher $\langle \delta_{\rm gal} \rangle$ and a smaller volume, with a net effect of a smaller mass; the opposite holds when we use a lower threshold). Another source of uncertainty is related to the uncertainty in the measurement of $\delta_{\rm gal}$ in the 2D maps. If we had used the 3D cube based on $\delta_{\rm gal,16}$(/$\delta_{\rm gal,84}$), we would have obtained $\langle \delta_{\rm gal} \rangle \sim 1.23(/1.26)$ and a volume of 1.06(/0.75)$\times 10^5$ cMpc$^3$, for an overall total mass $\sim10$% larger (/ $\sim20$% smaller). If we sum quadratically the two uncertainties, the very liberal global statistical error on the mass measurement is of about $+18\%/-25\%$. Irrespectively of the errors, it is clear that this structure has assembled an immense mass ($> 2\times10^{15}{\rm M}_{\odot}$) at very early times. This structure is referred to hereafter as the “Hyperion proto-supercluster”[^8] or simply “Hyperion” (officially PSC J1001$+$0218) due to its immense size and mass and because one of its subcomponents (peak \[3\], see Sect. \[peak3\]) is broadly coincident with the Colossus proto-cluster discovered by [@lee2016_colossus].
We remark that the volume computed in our data cube is most probably an overestimate, at the very least because it is artificially elongated along the l.o.s. This elongation is mainly due to 1) the photometric redshift error ($\Delta z\sim0.1$ for $\sigma_{zp}=0.03(1+z)$ at $z=2.45$), 2) the depth of the redshift slices ($\Delta z\sim0.02$) used to produce the density field, and 3) the velocity dispersion of the member galaxies, which might create the feature known as the Fingers of God ($\Delta z\sim0.006$ for a velocity dispersion of $500\kms$). Although the velocity dispersion should be important only for virialised sub-structures, these three factors should all work to surreptitiously increase the dimension of the structure along the l.o.s. and at the same time decrease the local overdensity $\delta_{\rm gal}$. In this transformation there is no mass loss (or, equivalently, the total galaxy counts remain the same, with galaxies simply spread on a larger volume). Therefore, the total mass of our structure, computed with Eq. \[eq\_mass\], would not change if we used the real (smaller) volume and the real (higher) density instead of the elongated volume and its associated lower overdensity.
We also ran a simple simulation to verify the effects of the depth of the redshift slices on the elongation. We built a simple mock galaxy catalogue at $z=2.5$ following a method similar to that described in [@tomczak17], a method which is based on injecting a mock galaxy cluster and galaxy groups onto a sample of mock galaxies that are intended to mimic the coeval field. As in [@tomczak17], the three dimensional positions of mock field galaxies are randomly distributed over the simulated transverse spatial and redshift ranges, with the number of mock field galaxies set to the number of photometric objects within an identical volume in COSMOS at $z\sim2.5$ that is devoid of known proto-structures. Galaxy brightnesses were assigned by sampling the $K-$band luminosity function of [@cirasuolo10], with cluster and group galaxies perturbed to slightly brighter luminosities (0.5 and 0.25 mag, respectively). Member galaxies of the mock cluster and groups were assigned spatial locations based on Gaussian sampling with $\sigma$ equal to 0.5 and 0.33 h$_{70}^{-1}$ pMpc, respectively, and were scattered along the l.o.s. by imposing Gaussian velocity dispersions of 1000 and 500 km s$^{-1}$, respectively. We then applied a magnitude cut to the mock catalogue similar to that used in our actual reconstructions, applied a spectroscopic sampling rate of 20%, and, for the remainder of the mock galaxies, assigned photometric redshifts with precision and accuracy identical to those in our photometric catalogue at the redshift of interest. We then ran the exact same density field reconstruction and method to identify peaks as was run on our real data, each time varying the depth of the redshift slices used. Following this exercise, we observed a smaller elongation for decreasing slice depth, with a $\sim40\%$ smaller elongation observed when dropping the slice size from 7.5 to 2.5 pMpc. This result confirmed that we need to correct for the elongation if we want to give a better estimate of the volume and/or the density of the structures in our 3D cube. We will apply a correction for the elongation to the highest density peaks found in the Hyperion proto-supercluster, as discussed in Sect. \[3D\_peaks\].
![3D overdensity map of the Hyperion proto-supercluster, in comoving megaparsecs. Colours scale with $\rm log(\sigma_{\rm \delta})$, exactly as in Fig. \[2D\_maps\], from blue ($2\sigma_{\rm \delta}$) to the darkest red ($\sim8.3\sigma_{\rm \delta}$, the highest measured value in our 3D cube). The $x-$, $y-$ and $z-$axes span the ranges $149.6 \leq RA \leq 150.52$, $1.74 \leq Dec \leq2.73$ and $2.35 \leq z \leq 2.55$. The NE and SW components are indicated. We highlight the fact that this figure shows only the proto-supercluster, and omits other less extended and less dense density peaks which fall in the plotted volume (see discussion in Sect. \[3D\].)[]{data-label="3D_cube"}](./fig2.ps){width="9.0cm"}
![Zoom-in of Fig. \[3D\_cube\]. The angle of view is slightly rotated with respect to Fig. \[3D\_cube\] so as to distinguish all the peaks. The colour scale is the same as in Fig. \[3D\_cube\], but here only the highest density peaks are shown, that is, the 3D volumes where ${\rm log}(1+\delta_{\rm gal})$ is above the $5\sigma_{\rm \delta}$ threshold discussed in Sect. \[3D\_peaks\]. Peaks are numbered as in Fig. \[3D\_sph\_peaks\] and Table \[peaks\_tab\]. []{data-label="3D_cube_peaks"}](./fig3.ps){width="8.0cm"}
The highest density peaks {#3D_peaks}
=========================
We identified the highest density peaks in the 3D cube by considering only the regions of space with ${\rm log}(1+\delta_{\rm gal})$ above $5\sigma_{\rm \delta}$ from the mean density. In our work, this threshold corresponds to $\delta_{\rm gal} \sim 2.6$, which corresponds to $\delta_m\sim 1 $ when using the bias factor $b=2.55$ found by [@durkalec15b]. We also verified, [*a posteriori*]{}, that with this choice we select density peaks which are about to begin or have just begun to collapse, after the initial phase of expansion (see Sect. \[discussion\]). This is very important if we want to consider these peaks as proto-clusters.
With the overdensity threshold defined above, we identified seven separated high-density sub-structures. We show their 3D position and shape in Fig. \[3D\_cube\_peaks\]. We computed the barycenter of each peak by weighting the $(x,y,z)$ position of each pixel belonging to the peak by its $\delta_{\rm gal}$. For each peak, we computed its volume, its $\langle \delta_{\rm gal} \rangle$, and derived its $M_{\rm tot}$ using Eq. \[eq\_mass\] (the bias factor is always $b=2.55$, found by [@durkalec15b] and discussed in Sect. \[3D\]). Table \[peaks\_tab\] lists barycenter, $\langle \delta_{\rm gal} \rangle$, volume, and $M_{\rm tot}$ of the seven peaks, numbered in order of decreasing $M_{\rm tot}$. We applied the same peak-finding procedure on the data cubes with $\delta_{\rm gal,16}$ and $\delta_{\rm gal,84}$, and computed the total masses of their peaks in the same way. We used these values as lower and upper uncertainties for the $M_{\rm tot}$ values quoted in the table.
From Table \[peaks\_tab\] we see that the overall range of masses spans a factor of $\sim30$, from $\sim0.09$ to $\sim2.6$ times $10^{14}$M$_\odot$. The total mass enclosed within the peaks ($\sim5.0 \times 10^{14}$M$_\odot$) is about 10% of the total mass in the Hyperion proto-supercluster, while the volume enclosing all the peaks is a lower fraction of the volume of the entire proto-supercluster ($\sim6.5\%$), as expected given the higher average overdensity within the peaks. The most massive peak (peak \[1\]) is included in the NE structure, together with peak \[4\] which has one fifth the total mass of peak \[1\]. Peak \[2\], which corresponds to the SW structure, has a $M_{\rm tot}$ comparable to peak \[4\], and it is located at lower redshift. Peak \[3\], with a $M_{\rm tot}$ similar to peaks \[2\] and \[4\], is placed in the sort of filament shown in the middle panel of Fig. \[2D\_maps\]. At smaller $M_{\rm tot}$ there is peak \[5\], with the highest redshift ($z=2.507$), and peak \[6\], at slightly lower redshift. They both have $M_{\rm tot} \sim0.2 \times 10^{14}$M$_\odot$. Finally, peak \[7\] is the least massive, and is very close in RA-Dec to peak \[2\], and at approximately the same redshift. In Appendix \[app\_Mtot\_sigma\] we show that the computation of $M_{\rm tot}$ is relatively stable if we slightly change the overdensity threshold used to define the peaks, with the exception of the least massive peak (peak \[7\]).
Figure \[3D\_cube\_peaks\] shows that the peaks have very different shapes, from irregular to more compact. We verified that their shape and position are not possibly driven by spectral sampling issues, by checking that the peaks persist through the $2^{\prime}$ gaps between the VIMOS quadrants from VUDS. This also implies that we are not missing high-density peaks that might fall in the gaps. We remind the reader that the zCOSMOS-Deep spectroscopic sample, which we use together with the VUDS sample, has a more uniform distribution in RA-Dec, and does not present gaps.
Concerning the shape of the peaks, we tried to take into account the artificial elongation along the l.o.s.. As mentioned at the end of Sect. \[3D\], this elongation is probably due to the combined effect of the velocity dispersion of the member galaxies, the depth of the redshift slices, and the photometric redshift error (although we refer the reader to e.g. [@lovell18] for an analysis of the shapes of proto-clusters in simulations). We used a simple approach to give an approximate statistical estimate of this elongation, starting from the assumption that on average our peaks should have roughly the same dimension in the $x$, $y,$ and $z$ dimensions[^9], and any measured systematic deviation from this assumption is artificial. In each of the three dimensions we measured a sort of effective radius $R_e$ defined as $R_{e,x}=\sqrt{ \sum _{i}w_i(x_{i}-x_{peak})^2 / \sum_i(w_i) }$ (and similarly for $R_{e,y}$ and $R_{e,z}$), where the sum is over all the pixels belonging to the given peak, the weight $w_i$ is the value of $\delta_{\rm gal}$, $x_{i}$ the position in cMpc along the $x-$axis and $x_{peak}$ is the barycenter of the peak along the $x-$axis, as listed in Table \[peaks\_tab\]. We defined the elongation $E_{\rm z/xy}$ for each peak as the ratio between $R_{e,z}$ and $R_{e,xy}$, where $R_{e,xy}$ is the mean between $R_{e,x}$ and $R_{e,y}$. The effective radii and the elongations are reported in Table \[peaks\_elongation\_tab\]. If the measured volume $V_{\rm meas}$ of our peaks is affected by this artificial elongation, the real corrected volume is $V_{\rm corr} = V_{\rm meas} / E_{\rm z/xy}$. Moreover, given that the elongation has the opposite and compensating effects of increasing the volume and decreasing $\delta_{\rm gal}$, as discussed at the end of Sect. \[3D\], $M_{\rm tot}$ remains the same. For this reason, inverting Eq. \[eq\_mass\] it is possible to derive the corrected (higher) average overdensity $\langle \delta_{\rm gal, corr} \rangle$ for each peak, by using $V_{\rm corr}$ and the mass in Table \[peaks\_tab\]. $V_{\rm corr}$ and $\langle \delta_{\rm gal, corr} \rangle$ are listed in Table \[peaks\_elongation\_tab\]. We note that by definition $R_e$ is smaller than the total radial extent of an overdensity peak, because it is computed by weighting for the local $\delta_{\rm gal}$, which is higher for regions closer to the centre of the peak. For this reason, the $V_{\rm corr}$ values are much larger than the volumes that one would naively obtain by using $R_{e,xy}$ as intrinsic total radius of our peaks. We use $\langle \delta_{\rm gal, corr} \rangle$ in Sect. \[discussion\] to discuss the evolution of the peaks. We refer the reader to \[app\_elongation\] for a discussion on the robustness of the computation of $E_{\rm z/xy}$ and its empirical dependence on $R_{e,xy}$.
![Same volume of space as Fig. \[3D\_cube\_peaks\], but in RA-Dec-$z$ coordinates. Each sphere represents one of the overdensity peaks, and is placed at its barycenter (see Table \[peaks\_tab\]). The colour of the spheres scales with redshift (blue = low $z$, dark red = high $z$), and the dimension scales with the logarithm of $M_{\rm tot}$ quoted in Table \[peaks\_tab\]. Small blue dots are the spectroscopic galaxies which are members of each overdensity peak, as described in Sect.\[3D\_peaks\].[]{data-label="3D_sph_peaks"}](./fig4.ps){width="8.0cm"}
We also assigned member galaxies to each peak. We defined a spectroscopic galaxy to be a member of a given density peak if the given galaxy falls in one of the $\geq 5 \sigma_{\delta}$ pixels that comprise the peak. The 3D distribution of the spectroscopic members is shown in Fig. \[3D\_sph\_peaks\], where each peak is schematically represented by a sphere placed in a $(x,y,z)$ position corresponding to its barycenter. It is evident that the 3D distribution of the member galaxies mirrors the shape of the peaks (see Fig. \[3D\_cube\_peaks\]). The number of spectroscopic members $n_{\rm zs}$ is quoted in Table \[peaks\_tab\]. The most extended and massive peak, peak \[1\], has 24 spectroscopic members. All the other peaks have a much smaller number of members (from 7 down to even only one member). We remind the reader that these numbers depend on the chosen overdensity threshold used to define the peaks, because the threshold defines the volume occupied by the peaks. Moreover, here we are counting only spectroscopic galaxies with good quality flags (see Sect. \[data\]) from VUDS and zCOSMOS, excluding other spectroscopic galaxies identified in the literature (but see Sect. \[vel\_disp\] for the inclusion of other samples to compute the velocity dispersion).
Velocity dispersion and virial mass {#vel_disp}
-----------------------------------
We computed the l.o.s. velocity dispersion $\sigma_{\rm v}$ of the galaxies belonging to each peak. For this computation we used a more relaxed definition of membership with respect to the one described above, so as to include also the galaxies residing in the tails of the velocity distribution of each peak. Basically, we used all the available good-quality spectroscopic galaxies within $\pm2500\kms$ from $z_{\rm peak}$ comprised in the RA-Dec region corresponding to the largest extension of the given peak on the plane of the sky. Moreover, we did not impose any cut in $i-$band magnitude, because, in principle, all galaxies can serve as reliable tracers of the underlying velocity field. We also included in this computation the spectroscopic galaxies with lower quality flag (flag = X1 for VUDS, all flags with X1.5$\leq$flag$<$2.5 for zCOSMOS), but only if they could be defined members of the given peak, with membership defined as at the end of the previous section. This less restrictive choice allows us to use more galaxies per peak than the pure spectroscopic members, although we still have only $\leq 4$ galaxies for three of the peaks. We quote these larger numbers of members in Table \[peaks\_tab\_veldisp\].
With these galaxies, we computed $\sigma_{\rm v}$ for each peak by applying the biweight method (for peak \[1\]) or the gapper method (for all the other peaks), and report the results of these computations in Table \[peaks\_tab\_veldisp\]. The choice of these methods followed the discussion in [@beers90], where they show that for the computation of the scale of a distribution the gapper method is more robust for a sample of $\lesssim 20$ objects (all our peaks but peak \[1\]), while it is better to use the biweight method for $\gtrsim 20$ objects (our peak \[1\]). We computed the error on $\sigma_{\rm v}$ with the bootstrap method, which was taken as the reference method in [@beers90]. In the case of peak \[7\], with only three spectroscopic galaxies available to compute $\sigma_{\rm v}$, we had to use the jack-knife method to evaluate the uncertainty on $\sigma_{\rm v}$; see also Sect. \[app\_vdisp\_sigma\] for more details on $\sigma_{\rm v}$ of peak \[7\].
We found a range of $\sigma_{\rm v}$ between $320\kms$ and $731\kms$. The most massive peak, peak \[1\], has the largest velocity dispersion, but for the other peaks the ranking in $M_{\rm tot}$ is not the same as in $\sigma_{\rm v}$. The uncertainty on $\sigma_{\rm v}$ is mainly driven by the number of galaxies used to compute $\sigma_{\rm v}$ itself, and it ranges from $\sim12$% for peak \[1\] to $\sim65$% for peak \[7\], for which we used only three galaxies to compute $\sigma_{\rm v}$. As we see below, other identifications in the literature of high-density peaks at the same redshift cover broadly the same $\sigma_{\rm v}$ range.
As we already mentioned, there are some works in the literature that identified/followed up some overdensity peaks in the COSMOS field at $z\sim2.45$, such as for example [@casey2015_z247], [@diener2015_z245], [@chiang2015_z244], and [@wang2016_z250]. Moreover, the COSMOS field has also been surveyed with spectroscopy by other campaigns, such as for example the COSMOS AGN spectroscopic survey [@trump09], the MOSDEF survey [@kriek15], and the DEIMOS 10K spectroscopic survey [@hasinger18]. We collected the spectroscopic redshifts of these other samples (including in this search also much smaller samples, like e.g. the one by [@perna15]), removed the possible duplicates with our sample and between samples, and assigned these new objects to our peaks, by applying the same membership criterion as applied to our VUDS$+$zCOSMOS sample. We re-computed the velocity dispersion using our previous sample plus the new members found in the literature. We note that many objects in the COSMOS field have been observed spectroscopically multiple times, and in most of the cases the new redshifts were concordant with previous observations. This is a further proof of the robustness of the $z_s$ we use here.
In the literature we only find new members for the peaks \[1\], \[3\], \[4\], and \[5\]. For each of these peaks, Table \[peaks\_tab\_veldisp\] reports the number $n_{\rm lit}$ of spectroscopic redshifts added to our original sample, together with the new estimates of $\sigma_{\rm v}$ and $M_{\rm vir}$. The new $\sigma_{\rm v}$ is always in very good agreement (below $1\sigma$) with our previous computation, but it has a smaller uncertainty. We will see that this translates into new $M_{\rm vir}$ values which are in very good agreement with those based on the original $\sigma_{\rm v}$.
As a by-product of the use of the spectroscopic member galaxies, we also computed a second estimate of the redshift of each peak (after the barycenter, see above). [@beers90] show that the biweight method is the most robust to compute the central location of a distribution of objects (in our case, the average redshift) also in the case of relatively few objects ($5-50$). This central redshift, $z_{\rm BI}$, is reported in Table \[peaks\_tab\_veldisp\], and is in excellent agreement with $z_{\rm peak}$, that is, the barycenter along the l.o.s. quoted in Table \[peaks\_tab\].
The use of the gapper and/or biweight methods is to be favoured when estimating the scale of a distribution also because they apply when the distribution is not necessarily a Gaussian, and certainly the shape of the galaxy velocity distribution in a proto-cluster may not follow a Gaussian distribution. In addition, it is questionable to assume that proto-clusters are virialised systems. Nevertheless, a crude way to estimate the mass of the peaks is to assume the validity of the virial theorem. In this way we can estimate the virial mass $M_{\rm vir}$ by using the measured velocity dispersion and some known scaling relations. We follow the same procedure as [@lemaux12], where $M_{\rm vir}$ is defined as:
$$\displaystyle
M_{\rm vir}=\frac{3 \sqrt{3} \sigma_{\rm v}^{3}}{\alpha~ 10~ G~ H(z)}.
\label{mvir}$$
In Eq. \[mvir\], $\sigma_{\rm v}$ is the line of sight velocity dispersion, $G$ is the gravitational constant, and $H(z)$ is the Hubble parameter at a given redshift. Equation \[mvir\] is derived from i) the definition of the virial mass,
$$\displaystyle
M_{\rm vir}=\frac{3}{G}\sigma_{\rm v}^{2}~R_{\rm v} ,
\label{vir_theo}$$
where $R_{\rm v}$ is the virial radius; ii) the relation between $R_{\rm 200}$ and $R_{\rm v}$,
$$\displaystyle
R_{\rm 200}=\alpha ~ R_{\rm v},
\label{rvir_r200}$$
where $R_{\rm 200}$ is the radius within which the density is 200 times the critical density, and iii) the relation between $R_{\rm 200}$ and $\sigma_{\rm v}$,
$$\displaystyle
R_{\rm 200}=\frac{\sqrt{3}~\sigma_{\rm v}}{10~H(z)}.
\label{r200_sigma}$$
Equations \[vir\_theo\] and \[r200\_sigma\] are from [@carlberg97]. Differently from [@lemaux12], we use $\alpha\simeq0.93$, which is derived comparing the radii where a NFW profile with concentration parameter $c=3$ encloses a density 200 times ($R_{\rm 200}$) and 173 times ($R_{\rm v}$) the critical density at $z\simeq2.45$. Here we consider a structure to be virialised when its average overdensity is $\Delta_{\rm v} \simeq 173$, which corresponds, in a $\Lambda$CDM Universe at $z\simeq2.45$, to the more commonly used value $\Delta_{\rm v} \simeq 178$, constant at all redshifts in an Einstein-de Sitter Universe (see the discussion in Sect. \[collapse\]).
The virial masses of our density peaks, computed with Eq. \[mvir\], are listed in Table \[peaks\_tab\_veldisp\], together with the virial masses obtained from the $\sigma_{\rm v}$ computed by using also other spectroscopic galaxies in the literature. Figure \[virial\_mass\] shows how our $M_{\rm vir}$ compared with the total masses $M_{\rm tot}$ obtained with Eq. \[eq\_mass\]. For four of the seven peaks, the two mass estimates basically lie on the 1:1 relation. In the three other cases, the virial mass is higher than the mass estimated with the overdensity value: namely, for peaks \[4\] and \[5\] the agreement is at $<2\sigma$, while for peak \[7\] the agreement is at less than $1\sigma$ given the very large uncertainty on $M_{\rm vir}$.
The overall agreement between the two sets of masses is surprisingly good, considering that $M_{\rm vir}$ is computed under the strong (and probably incorrect) assumption that the peaks are virialised, and that $M_{\rm tot}$ is computed above a reasonable but still arbitrary density threshold. Indeed, although the adopted density threshold corresponds to selecting peaks which are about to begin or have just begun to collapse (see Sect. \[3D\_peaks\]), the evolution of a density fluctuation from the beginning of collapse to virialisation can take a few gigayears (see Sect. \[discussion\]). Moreover, the galaxies used to compute $\sigma_v$ and hence $M_{\rm vir}$ are drawn from slightly larger volumes than the volumes used to compute $M_{\rm
tot}$, because we included galaxies in the tails of the velocity distribution along the l.o.s., outside the peaks’ volumes. We also find that $M_{\rm tot}$ continuously varies by changing the overdensity threshold to define the peaks (see Appendix \[app\_Mtot\_sigma\]), while the computation of the velocity dispersion in our peaks is very stable if we change this same threshold (see Appendix \[app\_vdisp\_sigma\]). As a consequence, we do not expect the estimated $M_{\rm vir}$ to change either. In addition to these caveats, peaks \[1\], \[2\] and \[3\] show an irregular 3D shape (see Appendix \[app\_peaks\]), and they might be multi-component structures. In these cases, the limited physical meaning of $M_{\rm
vir}$ is evident.
We also note that peak \[5\] has already been identified in the literature as a virialised structure (see [@wang2016_z250] and our discussion in Sect. \[peak5\]), meaning that its $M_{\rm vir}$ is possibly the most robust among the peaks, but in our reconstruction it is the most distant from the 1:1 relation between $M_{\rm vir}$ and $M_{\rm tot}$. This might suggest that our $M_{\rm tot}$ is underestimated, at least for this peak.
We also remark that there is not a unique scaling relation between $\sigma_{\rm v}$ and $M_{\rm vir}$. For instance, [@munari13] study the relation between the masses of groups and clusters and their 1D velocity dispersion $\sigma_{\rm 1D}$. Clusters are extracted from $\Lambda$CDM cosmological N-body and hydrodynamic simulations, and the authors recover the velocity dispersion by using three different tracers, that is, dark-matter particles, sub-halos, and member galaxies. They find a relation in the form:
$$\displaystyle
\sigma_{\rm 1D}=A_{1D}\left[ \frac{h(z)~M_{\rm 200}}{10^{15}M_{\odot}} \right]^{\alpha} ,
\label{m200_munari}$$
where $A_{1D} \simeq 1180 \kms$ and $\alpha \simeq 0.38$, as from their Fig. 3 for $z=2$ (the highest redshift they consider) and by using galaxies as tracers for $\sigma_{\rm 1D}$. [@evrard08] find a relation based on the same principle as Eq. \[m200\_munari\], but they use DM particles to trace $\sigma_{\rm 1D}$. On the observational side, [@sereno15] find a relation in perfect agreement with [@munari13] by using observed data, with cluster masses derived via weak lensing. We also used Eq. \[m200\_munari\] to compute $M_{\rm vir}$[^10]. We found that the $M_{\rm vir}$ computed via Eq. \[m200\_munari\] are systematically smaller (by 20-40%) than the previous ones computed with Eq. \[mvir\]. This change would not appreciably affect the high degree of concordance between $M_{\rm vir}$ and $M_{\rm tot}$ for our peaks.
In summary, the comparison between $M_{\rm vir}$ and $M_{\rm tot}$ is meaningful only if we fully understand the evolutionary status of our overdensities and know their intrinsic shapes (and we remind the reader that in this work the shape of the peaks depends at the very least on the chosen threshold, and it is not supposed to be their intrinsic shape). On the other hand, it would be very interesting to understand whether it is possible to use this comparison to infer the level of virialisation of a density peak, provided that its shape is known. This might be studied with simulations, and we defer this analysis to a future work.
![Virial mass $M_{\rm vir}$ of the seven identified peaks, as in Table \[peaks\_tab\_veldisp\], vs. the total mass $M_{\rm tot}$ as in Table \[peaks\_tab\]. We show both the virial mass computed only with our spectroscopic sample (red dots, column 6 of Table \[peaks\_tab\_veldisp\]) and how it would change if we add to our sample other spectroscopic sources found in the literature (black crosses, column 9 of Table \[peaks\_tab\_veldisp\]). Only peaks \[1\], \[3\], \[4\], and \[5\] have this second estimate of $M_{\rm vir}$. The dotted line is the bisector, as a reference.[]{data-label="virial_mass"}](./fig5.ps){width="9.0cm"}
The many components of the proto-supercluster {#peak_list}
---------------------------------------------
The COSMOS field is one of the richest fields in terms of data availability and quality. It was noticed early on that it contains extended structures at several redshifts (see e.g. [@scoville07_env; @guzzo07_z07; @cassata07_z07; @kovac10_density; @delatorre10_clustering; @scoville13_env; @iovino16_wall]). Besides using galaxies as direct tracers, as in the above-mentioned works, the large-scale structure of the COSMOS field has been revealed with other methods like weak lensing analysis (e.g. [@massey07]) and Ly$\alpha$-forest tomography [@lee2016_colossus; @lee2017_clamato]. Systematic searches for galaxy groups and clusters have also been performed up to $z\sim1$ (for instance [@knobel09_groups] and [@knobel12_groups20k]), and in other works we find compilations of candidate proto-groups [@diener2013_list] and candidate proto-clusters [@chiang2014_list; @franck16_CCPC; @lee2016_colossus] at $z\gtrsim1.6$. In some cases, the search for (proto-)clusters was focused around a given class of objects, like radio galaxies (see e.g. [@castignani14]).
In particular, it has been found that the volume of space in the redshift range $2.4 \lesssim z \lesssim 2.5 $ hosts a variety of high-density peaks, which have been identified by means of different techniques/galaxy samples, and in some cases as part of dedicated follow-ups of interesting density peaks found in the previous compilations. Some examples are the studies by [@diener2015_z245], [@chiang2015_z244], [@casey2015_z247], [@lee2016_colossus], and [@wang2016_z250]. In this paper, we generally refer to the findings in the literature as density peaks when referring to the ensemble of the previous works; we use the definition adopted in each single paper (e.g. ‘proto-groups’, ‘proto-cluster candidates’, etc.) when we mention a specific study.
We note that in the vast majority of these previous works there was no attempt to put the analysed density peaks in the broader context of a large-scale structure. The only exceptions are the works by [@lee2016_colossus] and [@lee2017_clamato], based on the Ly$\alpha$-forest tomography. [@lee2016_colossus] explore an area of $\sim14\times16$ h$^{-1}$ cMpc, which is roughly one ninth of the area covered by Hyperion, while [@lee2017_clamato] extended the tomographic map up to an area roughly corresponding to one third of the area spanned by Hyperion. Both these works do mention the complexity and the extension of the overdense region at $z\sim 2.45$, and the fact that it embeds three previously identified overdensity peaks [@diener2015_z245; @casey2015_z247; @wang2016_z250]. Nevertheless, they did not expand on the characteristics of this extended region, and were unable to identify the much larger extension of Hyperion, because of the smaller explored area.
In this section we describe the characteristics of our seven peaks, and compare our findings with the literature. The aim of this comparison is to show that some of the pieces of the Hyperion proto-supercluster have already been sparsely observed in the literature, and with our analysis we are able to add new pieces and put them all together into a comprehensive scenario of a very large structure in formation. We also try to give a detailed description of the characteristics (such as volume, mass, etc.) of the structures already found in the literature, with the aim to show that different selection methods are able to find the same very dense structures, but these methods in some cases are different enough to give disparate estimates of the peaks’ properties. For this comparison, we refer to Fig. \[3D\_map\_lit\] and Table \[literature\_tab\], as detailed below. Moreover, in Appendix \[app\_peaks\] we show more details on our four most massive peaks, which we dub “Theia”, “Eos”, “Helios”, and “Selene”[^11]. Among the previous findings, we discuss only those falling in the volume where our peaks are contained. We remind that we did not make use of the samples used in these previous works. The only exception is that the zCOSMOS-Deep sample, included in our data set, was also used by [@diener2013_list].
### Peak \[1\] - “Theia” {#peak1}
Peak \[1\] is by far the most massive of the peaks we detected. Figure \[3D\_cube\_peaks\] shows that its shape is quite complex. The peak is composed of two substructures that indeed become two separated peaks if we increase the threshold for the peak detection from $5\sigma_{\delta}$ to $6.6\sigma_{\delta}$. In Fig. \[peak1\_fig\] of Appendix \[app\_peaks\] we show two 2D projections of peak \[1\], which indicate the complexity of the 3D structure of this peak.
Figure \[3D\_map\_lit\] is the same as Fig. \[3D\_cube\_peaks\], but we also added the position of the overdensity peaks found in the literature. We verified that our peak \[1\] includes three of the proto-groups in the compilation by [@diener2013_list], called D13a, D13b, and D13d in our figure. Proto-goups D13a and D13b are very close to each other ($\sim 3$ arcmin on the RA-Dec plane) and together they are part of the main component of our peak \[1\]. D13d corresponds to the secondary component of peak \[1\], which detaches from the main component when we increase the overdensity threshold to $6.6\sigma_{\delta}$. Another proto-group (D13e) found by [@diener2013_list] falls just outside the westernmost and northernmost border of peak\[1\]. It is not unexpected that our peaks (see also peaks \[3\] and \[4\]) have a good match with the proto-groups found by [@diener2013_list], given that their density peaks have been detected using the zCOSMOS-Deep sample, which is also included in our total sample[^12]. In our peak \[1\] we find 24 spectroscopic members (see Table \[peaks\_tab\]), 14 of which come from the VUDS survey and 10 from the zCOSMOS-Deep sample.
The shape of peak\[1\] (a sort of ‘L’, or triangle) is mirrored by the shape of the proto-cluster found by [@casey2015_z247], as shown in their Fig. 2. In our Fig. \[3D\_map\_lit\] their proto-cluster is marked as Ca15, and we placed it roughly at the coordinates of the crossing of the two arms of the ‘L’ in their figure, where they found an X-ray detected source. In their figure, the S-N arm extends to the north and has a length of $\sim14$ arcmin, and the E-W arm extends towards east and its length is about 10 arcmin. They also show that their proto-cluster encloses the three proto-groups D13a, D13b, and D13d.
Although we found a correspondence between the position/extension of our peak \[1\] and the position/extension of some overdensities in the literature, it is harder to compare the properties of peak \[1\] and such overdensities. This difficulty is given mainly by the different detection techniques. We attempted this comparison and show the results in Table \[literature\_tab\]. In this table, for each overdensity in the literature we show its redshift, $\delta_{\rm gal}$, velocity dispersion, and total mass, when available in the respective papers. We also computed its total volume, based on the information in its respective paper, and computed its $\delta_{\rm gal}$ and total mass (using Eq. \[eq\_mass\]) in that same volume in our 3D cube. In the case of a 1:1 match with our peak (like in the case of Ca15 and our peak \[1\]), we also reported the properties of our matched peak.
In the case of the proto-groups D13a, D13b, D13d and D13e, we found in the literature only their $\sigma_{\rm v}$, which we cannot compare directly with our peak \[1\] given that there is not a 1:1 match. The $\delta_{\rm gal}$ recovered in our 3D cube in the volumes corresponding to the four proto-groups are broadly consistent with the typical $\delta_{\rm gal}$ of our peaks, with the exception of D13e which in fact falls outside our peak \[1\]. These proto-groups have all relatively small volumes and masses compared to our peaks. At most, the largest one (D13a) is comparable in volume and mass with our smallest peaks (\[5\],\[6\], and \[7\]). The average difference in volume between our peaks and the proto-groups found in [@diener2013_list] might be due to the fact that they identified groups with a Friend-of-Friend algorithm with a linking length of 500 pkpc, i.e. $\sim1.7$ cMpc at $z=2.45$, which is smaller than the effective radius of our largest peaks (although their linking lengths and our effective radii do not have the same physical meaning).
The properties of Ca15 were computed in a volume almost three times as large as our peak \[1\]. Nevertheless, its $\delta_{\rm gal}$ is much higher, probably because of the different tracers (they use dusty star forming galaxies, ‘DSFGs’). Despite our lower density in the Ca15 volume, we find a higher total mass ($M_{\rm tot} = 4.82 \times 10^{14} M_{\odot}$ instead of their total mass of $>0.8 \times 10^{14} M_{\odot}$). This is probably due to the different methods used to compute $M_{\rm tot}$: we use Eq. \[eq\_mass\], while [@casey2015_z247] use abundance matching techniques to assign a halo mass to each galaxy, and then sum the estimated halo masses for each galaxy in the structure. Moreover, they state that their mass estimate is a lower limit.
### Peak \[2\] - “Eos” {#peak2}
As peak \[1\], this peak seems to be composed by two sub-structures, as shown in details in Fig. \[peak2\_fig\]. The two substructures detach from each other when we increase the overdensity threshold to $5.3\sigma_{\delta}$. On the contrary, by decreasing the overdensity threshold to $4.5\sigma{\delta}$ we notice that this peak merges with the current peak \[7\].
We did not find any direct match of peak \[2\] with previous detections of proto-structures in the literature. We note that this part of the COSMOS field is only partially covered by the tomographic search performed by [@lee2016_colossus] and [@lee2017_clamato]. This could be the reason why they do not find any prominent density peak there.
### Peak \[3\] - “Helios” {#peak3}
The detailed shape of peak \[3\] is shown in Fig. \[peak3\_fig\]. From our density field, it is hard to say whether its shape is due to the presence of two sub-structures. Even by increasing the overdensity threshold, the peak does not split into two sub-components.
Peak \[3\] is basically coincident with the group D13f from [@diener2013_list], and its follow-up by [@diener2015_z245], which we call D15 in our Fig. \[3D\_map\_lit\]. The barycenter of our peak \[3\] is closer to the position of D13f than to the position of D15, on both the RA-Dec plane ($<8^{\prime\prime}$ to D13f, $\sim50^{\prime\prime}$ on the Dec axis to D15) and the redshift direction ($\Delta z \sim 0.004$ with D13f, and $\Delta z \sim 0.05$ with D15). This very good match is possibly due also to the fact that our sample includes the zCOSMOS-Deep data (see comment in Sect. \[peak1\]). Indeed, out of the seven spectroscopic members that we identified in peak \[3\], five come from the zCOSMOS-Deep sample and two from VUDS. We note that the list of candidate proto-clusters by [@franck16_CCPC] includes a candidate that corresponds, as stated by the authors, to D13f. Interestingly, [@diener2015_z245] mention that D15 might be linked to the radio galaxy COSMOS-FRI 03 [@chiaberge09], around which [@castignani14] found an overdensity of photometric redshifts. Although the overdensity of photometric redshifts surrounding the radio galaxy is formally at slightly lower redshift than D15 (see also [@chiaberge10]), it is possibly identifiable with D15, given the photometric redshift uncertainty.
Table \[literature\_tab\] shows that the velocity dispersion found by [@diener2015_z245] for D15 is very similar to the one we find for our peak \[3\], although the density that they recover is much larger ($\delta_{\rm gal}= 10$ vs $\delta_{\rm gal} \sim 3.$). We note that D15 is defined over a volume which is almost twice as large as peak \[3\]. The velocity dispersion of F16 is instead almost double the one we recover for peak \[3\]. Their search volume is huge ($\sim10000$ cMpc$^3$) compared to the volume of peak \[3\]. Considering that they also find quite high $\delta_{\rm gal}$, they compute a total mass of $\sim15\times10^{14}$ M$_\odot$, which is approximately three times larger than the one we find in our data in their same volume ($4.89\times10^{14}$ M$_\odot$), but about a factor of 30 larger than the mass of our peak \[3\].
Very close to peak \[3\] there are the three components of the extended proto-cluster dubbed ‘Colossus’ in [@lee2016_colossus][^13]. Here we call the three sub-structures L16a, L16b and L16c, in order of decreasing redshift. This proto-cluster was detected by IGM tomography (see also [@lee2017_clamato]) performed by analysing the spectra of galaxies in the background of the proto-cluster. The three peaks form a sort of chain from $z\sim2.435$ to $z\sim2.45$, which extends over $\sim2^{\prime}$ in RA and $\sim6^{\prime}$ in Dec. We derived the positions of the first and third peaks from Fig. 12 of [@lee2016_colossus], and assumed that the intermediate peak was roughly in between (see their Figs. 4 and 13). Neither L16a, L16b, or L16c coincide precisely with one of our peaks, but they fall roughly 3 arcmin eastwards of the barycenter of our peak \[3\]. The declination and redshift of the intermediate component correspond to those of our peak \[3\]. Given the extension of the three peaks in RA-Dec (they have a radius from $\sim2$ to $\sim4$ arcmin) and the extension of our peak \[3\] ($\sim2$ arcmin radius), the ‘Colossus’ overlaps with, and it might be identified with, our peak \[3\].
[@lee2016_colossus] compute the total mass of their overdensity, and find that it is $1.6\pm0.9\times10^{14}$ M$_\odot$. Computing the overall mass in the volumes of the three components L16a, L16b, and L16c in our data cube, we find a smaller mass ($0.83\times10^{14}$ M$_\odot$), which is still consistent with the value found by [@lee2016_colossus].
We additionally compared our results with those by [@lee2016_colossus] by directly using the smoothed IGM overdensity, $\delta_F^{\rm sm} $, estimated from the latest tomographic map [@lee2017_clamato]. We measured their average $\delta_F^{\rm sm}$ in the volume enclosing our peak \[3\] and found that this volume of space corresponds to an overdense region with respect to the mean intergalactic medium (IGM) density at these redshifts. Specifically, using the definition in [@lee2016_colossus], for which negative values of $\delta_F^{\rm sm} $ signify overdense regions, we found that our peak has $\langle \delta_F^{\rm sm} \rangle \sim -2.4\sigma_{\rm sm}$, with $\sigma_{\rm sm}$ denoting the effective sigma of the $\delta_F^{\rm sm} $ distribution. We repeated the same analysis in the volumes enclosed by our other peaks (with the exception of peak \[2\], which lies almost entirely outside the tomographic map), and we found that their $\langle \delta_F^{\rm sm} \rangle$ fall in the range from $-1.9\sigma_{\rm sm}$ to $-1\sigma_{\rm sm}$ meaning that all of our peaks appear overdense with respect to the mean IGM density at these redshifts. This persistent overdensity measured across the six peaks that we are able to measure in the tomographic map strongly hint at a coherent overdensity also present in the IGM maps. Further, all peaks have measured $\langle \delta_F^{\rm sm} \rangle$ values consistent with the expected IGM absorption signal due to the presence of at least some fraction of simulated massive ($M_{\rm tot, z=0}>10^{14} M_{\odot}$) proto-clusters (see section 4 of [@lee2016_colossus]). We note, however, that none of our peaks have $\langle \delta_F^{\rm sm} \rangle < -3\sigma_{\rm sm}$, which is the threshold suggested by [@lee2016_colossus] to safely identify proto-clusters (see their Fig. 6) with IGM tomography. Additionally, the level of the galaxy overdensity or $M_{\rm tot}$ from our galaxy density reconstruction does not necessarily correlate well with the $\langle \delta_F^{\rm sm} \rangle$ measured for the ensemble of proto-supercluster peaks likely due to a variety of astrophysical reasons as well as reasons drawing from the slight differences in the samples employed and reconstruction method. Regardless, this comparison demonstrates the complementarity of our method and IGM tomography to identify proto-clusters. This comparison will be expanded in future work to investigate differences in the signals in the two types of maps according to physical properties (like gas temperature, etc.) of individual proto-clusters.
[@lee2016_colossus] identify their proto-cluster with one of the candidate proto-clusters found by [@chiang2014_list] (proto-cluster referred to here as Ch14). These latter authors systematically searched for proto-clusters using photometric redshifts and [@chiang2015_z244] presented a follow-up of Ch14, presenting a proto-cluster that we refer to here as Ch15. From [@chiang2015_z244], it is not easy to derive an official RA-Dec position of Ch15, so we assume it is at the same RA-Dec coordinates as Ch14. The redshifts of Ch14 and Ch15 are slightly different ($z=2.45$ and $z=2.445$, respectively). Our peak \[3\] is $\lesssim5$ arcmin away on the plane of the sky from Ch14 and Ch15, and this is in agreement with the distance that [@chiang2015_z244] mention from their proto-cluster to the proto-group D15, which matches with our peak \[3\]. Moreover, [@chiang2015_z244] associate a size of $\sim10\times7$ arcmin$^2$ to Ch15, which makes Ch15 overlap with peak \[3\]. According to [@chiang2015_z244], Ch15 has an overdensity of LAEs of $\sim4$ , computed over a volume of $\sim12000$ cMpc$^3$. Over this volume, the overdensity in our data cube is very low ($\delta_{\rm gal} = 0.53$), because it encompasses also regions well outside the highest peaks and even outside the proto-supercluster. Despite the low density, the volume is so huge that the mass of Ch15 that we compute in our data cube exceeds $5\times10^{14}$ M$_\odot$. [@chiang2015_z244] do not mention any mass estimate for Ch15.
### Peak \[4\] - “Selene” {#peak4}
Peak \[4\] seems to be composed of a main component, which includes most of the mass/volume, and a tail on the RA-Dec plane, which is as long as about twice the length of the main component. This is shown in Fig. \[peak4\_fig\]. We did not find spectroscopic members in the tail.
The barycenter of peak \[4\], centred on its main component, is coincident with the position of the proto-group D13c from [@diener2013_list]. Their distance on the plane of the sky is $\lesssim 30^{\prime\prime}$ arcsec, and they have the same redshift. Also in this case, this perfect agreement might be due to our use of the zCOSMOS-Deep sample (see Sect. \[peak1\]), although only half (2 out of 4) of the spectroscopic members of peak \[4\] come from the zCOSMOS-Deep survey.
[@diener2013_list] compute a velocity dispersion of 239 km s$^{-1}$ for D13c, while we measured $\sigma_{\rm v}=672$ km s$^{-1}$ for peak \[4\]. This discrepancy, which holds even if we consider our uncertainty of $\sim 150$ km s$^{-1}$, might be due to the larger number of galaxies that we use to compute $\sigma_{\rm v}$ (9 vs. their 3 members). Moreover, the volume over which their proto-group is defined is much smaller (one seventh) than the volume covered by peak \[4\].
### Peak \[5\] {#peak5}
Peak \[5\] has a regular roundish shape on the RA-Dec plane, so we do not show any detailed plot in Appendix \[app\_peaks\]; it corresponds to the cluster found by [@wang2016_z250], which we call W16 in this work. We remark that [@wang2016_z250] find an extended X-ray emission associated to this cluster, and indeed they define W16 as a ‘cluster’ and not a ‘proto-cluster’ because they claim that there is evidence that it is already virialised. We refer to their paper for a more detailed discussion. The RA-Dec coordinates of W16 are offset by $\sim30^{\prime\prime}$ on the RA axis and $\sim5^{\prime\prime}$ on the Dec axis from peak \[5\]. The redshift of our peak \[5\] is $\Delta z=0.001$ higher than the redshift of W16.
The velocity dispersion of our peak \[5\] is in remarkably good agreement with the one computed by [@wang2016_z250] (533 and 530 km s$^{-1}$, respectively), and, as a consequence, there is a very good agreement between the two virial masses. We note that peak \[5\] is one of the cases in our work where the total mass computed from $\delta_{\rm gal}$ is much smaller than the virial mass computed from the $\sigma_{\rm v}$. What is interesting in W16 is that it is extremely compact: the extended X-ray detection has a radius of about $24^{\prime \prime}$, and the majority of its member galaxies are also concentrated on the same area. Should we consider this small radius, its volume would be five times smaller than the one of our peak \[5\]. Instead, in Table \[literature\_tab\] we used a larger volume for the comparison (429 cMpc$^3$), derived from the maximum RA-Dec extension of the member galaxies quoted in [@wang2016_z250].
### Peak \[6\] {#peak6}
Peak \[6\] has a regular shape on the plane of the sky. We did not find any other overdensity peak or proto-cluster detected in the literature matching its position.
### Peak \[7\] {#peak7}
Peak \[7\] has also a roughly round shape on the RA-Dec plane. It merges with peak \[2\] if we decrease the overdensity threshold to $4.5\sigma{\delta}$. We could not match it with any previous detection of proto-structures in the literature.
---------------------------- ----------------- ------------------ ---------------- -------------- ------------------------------------- -------------- -------------------------
ID RA$_{\rm peak}$ Dec$_{\rm peak}$ $z_{\rm peak}$ n$_{\rm zs}$ $\langle \delta_{\rm gal} \rangle $ Volume $M_{\rm tot}$
(Fig. \[3D\_cube\_peaks\]) \[deg\] \[deg\] \[cMpc$^3$\] \[$10^{14}$ M$_\odot$\]
(1) (2) (3) (4) (5) (6) (7) (8)
1 150.0937 2.4049 2.468 24 3.79 3134 2.648$_{-1.39}^{+0.56}$
2 149.9765 2.1124 2.426 7 2.89 951 0.690$_{-0.51}^{+0.84}$
3 149.9996 2.2537 2.444 7 3.03 805 0.598$_{-0.37}^{+0.24}$
4 150.2556 2.3423 2.469 4 3.20 720 0.552$_{-0.30}^{+0.40}$
5 150.2293 2.3381 2.507 1 3.11 252 0.190$_{-0.16}^{+0.09}$
6 150.3316 2.2427 2.492 4 3.12 251 0.190$_{-0.13}^{+0.06}$
7 149.9581 2.2187 2.423 1 2.58 134 0.092$_{-0.09}^{+0.11}$
---------------------------- ----------------- ------------------ ---------------- -------------- ------------------------------------- -------------- -------------------------
---------------------------- ---------------- ----------- ----------- ----------- ---------------- ------------------------------------------- ----------------
ID $z_{\rm peak}$ R$_{e,x}$ R$_{e,y}$ R$_{e,z}$ $E_{\rm z/xy}$ $\langle \delta_{\rm gal, corr} \rangle $ V$_{\rm corr}$
(Fig. \[3D\_cube\_peaks\]) cMpc cMpc cMpc \[cMpc$^3$\]
(1) (2) (3) (4) (5) (6) (7) (8)
1 2.468 3.37 4.07 7.76 2.09 10.84 1500
2 2.426 2.31 3.25 5.18 1.87 7.74 509
3 2.444 1.94 1.82 6.15 3.26 15.92 247
4 2.469 2.77 2.12 6.00 2.45 11.73 294
5 2.507 1.05 1.27 4.07 3.52 17.70 72
6 2.492 0.88 1.05 5.83 6.03 32.29 42
7 2.423 1.22 0.90 2.71 2.55 10.73 53
---------------------------- ---------------- ----------- ----------- ----------- ---------------- ------------------------------------------- ----------------
---------------------------- ---------------- ----------------- -------------- --------------------- ------------------------- --------------- --------------------- ------------------------- -------
ID $z_{\rm peak}$ n$_{zs,\sigma}$ $z_{\rm BI}$ $\sigma_{\rm v}$ $M_{\rm vir}$ n$_{\rm lit}$ $\sigma_{\rm v}$ $M_{\rm vir}$ Ref.
(Fig. \[3D\_cube\_peaks\]) \[km s$^{-1}$\] \[$10^{14}$ M$_\odot$\] \[km s$^{-1}$\] \[$10^{14}$ M$_\odot$\]
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
1 2.468 29 2.467 731$_{-92}^{+88}$ 2.16$_{-0.71}^{+0.88}$ 11 737$_{-86}^{+85}$ 2.21$_{-0.69}^{+0.85}$ 1,2,3
2 2.426 8 2.426 474$_{-144}^{+129}$ 0.60$_{-0.40}^{+0.63}$ - - - -
3 2.444 7 2.445 417$_{-121}^{+91}$ 0.41$_{-0.26}^{+0.33}$ 7 500$_{-87}^{+79}$ 0.70$_{-0.30}^{+0.39}$ 4,5,6
4 2.469 9 2.467 672$_{-162}^{+145}$ 1.68$_{-0.94}^{+1.33}$ 1 644$_{-158}^{+142}$ 1.47$_{-0.84}^{+1.21}$ 1
5 2.507 4 2.508 533$_{-163}^{+87}$ 0.82$_{-0.55}^{+0.49}$ 13 472$_{-80}^{+86}$ 0.57$_{-0.24}^{+0.37}$ 7
6 2.492 4 2.490 320$_{-151}^{+56}$ 0.18$_{-0.15}^{+0.11}$ - - - -
7$^*$ 2.423 3 2.428 461$_{-304}^{+304}$ 0.55$_{-0.53}^{+1.97}$ - - - -
---------------------------- ---------------- ----------------- -------------- --------------------- ------------------------- --------------- --------------------- ------------------------- -------
![Same as Fig. \[3D\_cube\_peaks\], but in RA-Dec-z coordinates. Moreover, we overplot the location of the overdensity peaks/proto-clusters/proto-groups detected in other works in the literature (blue and green cubes, and blue and cyan crosses). Different colours and shapes are used for the symbols for clarity purposes only. Labels correspond to the IDs in Table \[literature\_tab\]. The dimensions of the symbols are arbitrary and do not refer to the extension of the overdensity peaks found in the literature. []{data-label="3D_map_lit"}](./fig6.ps){width="9.0cm"}
------------------------- ------ ------- ------------------------ ------------------ ------------------------- ------------- ------------------------------------ ------------------------- ------------ ------------------------------------ ---------- ------------------------- ------------------------ -------------------------
ID Ref. z $\delta_{\rm gal}$ $\sigma_{\rm v}$ M$_{\rm tot}$ Volume $\langle \delta_{\rm gal} \rangle$ M$_{\rm tot}$ Match with $\langle \delta_{\rm gal} \rangle$ Volume M$_{\rm tot}$$^{e}$ $\sigma_{\rm v}$$^{e}$ M$_{\rm vir}$$^{e}$
(Fig. \[3D\_map\_lit\]) \[km s$^{-1}$\] \[$10^{14}$ M$_\odot$\] cMpc$^3$ \[$10^{14}$ M$_\odot$\] this work cMpc$^3$ \[$10^{14}$ M$_\odot$\] \[km s$^{-1}$\] \[$10^{14}$ M$_\odot$\]
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15)
L16a 4 2.450 - - 1.6$\pm$0.9$^{b}$ 1568$^{b}$ 1.50$^{b}$ 0.83$^{b}$ \[3\]\* - - - - -
L16b 4 2.443 - - 1.6$\pm$0.9$^{b}$ 1568$^{b}$ 1.50$^{b}$ 0.83$^{b}$ \[3\]\* - - - - -
L16c 4 2.435 - - 1.6$\pm$0.9$^{b}$ 1568$^{b}$ 1.50$^{b}$ 0.83$^{b}$ \[3\]\* - - - - -
W16 5 2.506 - 530$\pm120$ 0.79$^{+0.46}_{-0.29}$ 429 2.46 0.29 \[5\] 3.11 252 0.190 533 0.82
F16 8 2.442 9.27$\pm$4.93 770 15.5/14.1$^{d}$ $\sim10000$ 1.04 4.89 \[3\] 3.03 805 0.598 417 0.41
D15 1 2.450 10 426 - 1513 1.99 0.92 \[3\] 3.03 805 0.598 417 0.41
Ca15 2 2.472 11$^{c}$ - $>0.8\pm0.3$ 8839 1.55 4.82 \[1\] 3.79 3134 2.648 731 2.16
Ch15 3 2.440 4$^{a}$ - - $\sim12000$ 0.53 $\sim5.6$ \[3\]\* - - - - -
Ch14 7 2.450 1.34$^{+0.49}_{-0.40}$ - - $\sim23000$ 0.37 $\sim9.1$ \[3\]\* - - - - -
D13a 6 2.476 - 264 - 87 3.12 0.07 \[1\]\* - - - - -
D13b 6 2.469 - 488 - 253 3.73 0.21 \[1\]\* - - - - -
D13c 6 2.469 - 239 - 108 4.26 0.10 \[4\] 3.20 720 0.552 672 1.68
D13d 6 2.463 - 30 - 26 4.08 0.02 \[1\]\* - - - - -
D13e 6 2.452 - 476 - 38 0.89 0.02 \[1\]\* - - - - -
D13f 6 2.440 - 526 - 425 2.87 0.31 \[3\] 3.03 805 0.598 417 0.41
------------------------- ------ ------- ------------------------ ------------------ ------------------------- ------------- ------------------------------------ ------------------------- ------------ ------------------------------------ ---------- ------------------------- ------------------------ -------------------------
Discussion
==========
The detection of such a huge, massive structure, caught during its formation, poses challenging questions. On the one hand, one would like to know whether we can predict the evolution of its components. On the other, it would be interesting to understand whether at least some of these components are going to interact with one another, or at the very least, how much they are going to interact with the surrounding large-scale structure as a whole. Moreover, the existence of superclusters at lower redshifts begs the question of whether this proto-structure will evolve to become similar to one of these closer superclusters. We address these issues below in a qualitative way, and defer any further analysis to a future work.
The evolution of the individual density peaks. {#collapse}
----------------------------------------------
Assuming the framework of the spherical collapse model, we computed the evolution of our overdensity peaks as if they were isolated spherical overdensities. This is clearly a significant assumption (see e.g. [@despali13] for the evolution of ellipsoidal halos), but it can help us in roughly understanding the evolutionary status of these peaks, and how peaks with similar overdensities would evolve with time.
According to the spherical collapse model, any spherical overdensity will evolve like a sub-universe, with a matter-energy density higher than the critical overdensity at any given epoch. In our case, we reasonably assume that the average matter overdensity $\langle \delta_{\rm m} \rangle $ in our peaks corresponds to a non-linear regime, because it is already well above 1. We report $\langle \delta_{\rm m} \rangle $ in Table \[peaks\_evol\_tab\] as $\langle \delta_{\rm m,corr} \rangle $, given that we define it as $\langle \delta_{\rm m,corr} \rangle = \langle \delta_{\rm gal,corr}
\rangle / b$, with $\langle \delta_{\rm gal,corr} \rangle $ as reported in Table \[peaks\_elongation\_tab\] and $b$ the bias measured by [@durkalec15b] as in Sect. \[3D\].
Given that it is much easier to compute the evolution of an overdensity in linear regime than in non-linear regime, we transform [@padmanabhan] our $\langle \delta_{\rm NL} \rangle $ into their corresponding values in linear regime, $\langle \delta_{\rm L} \rangle $, and make them evolve according to the spherical linear collapse model.
In particular, the overdense sphere passes through three specific evolutionary steps. The first one is the point of turn-around, when the overdense sphere stops expanding and starts collapsing, becoming a gravitationally bound structure. This happens when the overdensity in linear regime is $\delta_{\rm L,ta}\simeq 1.062$ (in non-linear regime it would be $\delta_{\rm NL,ta}\simeq 4.55$). After the turn-around, when the radius of the sphere becomes half of the radius at turn-around, the overdense sphere reaches the virialisation. In this moment, we have $\delta_{\rm L,vir}\simeq 1.58$ and $\delta_{\rm NL,vir}\simeq 146$. The sphere then continues the collapse process, till the moment of maximum collapse which theoretically happens when its radius becomes zero with an infinite density. In the real universe the collapse stops before the density becomes infinite, and at that time the system, which still satisfies the virial theorem, reaches $\delta_{\rm L,c}\simeq 1.686$ ($\delta_{\rm NL,c}\simeq 178$).
In our work we are interested in the moments of turn-around and collapse. Here we will follow the formalism as in [@pace10], and we will use the symbol $\delta_{\rm c}$ for $\delta_{\rm L,c}\simeq 1.686$ and the symbol $\Delta_{\rm V}$ for $\delta_{\rm NL,c}\simeq 178$. When we refer to the time(/redshift) of turn-around and collapse, we use $t_{\rm ta}$(/$z_{\rm ta}$) and $t_{\rm c}$(/$z_{\rm c}$).
We reiterate that $\delta_{\rm c}$ and $\Delta_{\rm V}$ are constant with redshift in an Einstein - de Sitter (EdS) Universe, while they evolve with time in a $\Lambda$CDM cosmology, and their evolution depends on the relative contribution of $\Omega_{\rm \Lambda}(z)$ and $\Omega_{\rm m}(z)$ to $\Omega_{\rm tot}(z)$. At high redshift (e.g. $z=5$) when $\Omega_{\rm \Lambda}(z)$ is small, $\delta_{\rm c}$ and $\Delta_{\rm V}$ are close to their EdS counterparts. As time goes by, $\Omega_{\rm \Lambda}(z)$ increases and both $\delta_{\rm c}$ and $\Delta_{\rm V}$ decrease. This is shown, for instance, in [@pace10], where they show that $\delta_{\rm c}$ decreases by less than 1% from $z=5$ to $z=0$, while in the same timescale $\Delta_{\rm V}$ decreases from $\sim178$ to $\sim 100$ (see also [@bryan_norman98], where they use the symbol $\Delta_{\rm c}$ instead of $\Delta_{\rm V}$). In our work we allow our overdensities to evolve in the linear regime, so we are interested at the time when they reach $\delta_{\rm c}$. Given its small evolution with redshift, we consider it a constant, set as in the EdS universe.
The evolution of a fluctuation is given by its growing mode $D_{\rm +}(z)$. At a given redshift $z_{\rm 2}$, the overdensity $\delta_{\rm L}(z_{\rm 2})$ can be computed knowing the overdensity at another redshift $z_{\rm 1}$ and the value of the growing mode at the two redshifts, as follows:
$$\displaystyle
\delta_{\rm L}(z_{\rm 2}) = \delta_{\rm L}(z_{\rm 1}) \frac{D_{\rm +}(z_{\rm 2})}{D_{\rm +}(z_{\rm 1})}
\label{delta_evol}
.$$
In a $\Lambda$CDM universe, we define the linear growth factor $g$ as $g \equiv D_{\rm +}(z)/a$, where $a=(1+z)^{-1}$ is the cosmic scale factor. By using an approximate expression for $g$ (see e.g. [@carroll92] and [@hamilton01]), which depends on $\Omega_{\rm \Lambda}(z)$ and $\Omega_{\rm m}(z)$, we can recover $D_{\rm +}(z)$ and with equation \[delta\_evol\] derive the time when our peaks reach $\delta_{\rm L,ta}$ and $\delta_{\rm c}$, starting from the measured values of $\delta_{\rm L}(z_{\rm obs})$, with $z_{\rm obs}$ being the redshifts given in Table \[peaks\_tab\]. Figure \[delta\_evol\_ps\] shows the evolution of the density contrast of our peaks. In Table \[peaks\_evol\_tab\] we list the values of $z_{\rm ta}$ and $z_{\rm c}$, together with the time elapsed from $z_{\rm obs}$ to these two redshifts. As a very rough comparison, if we considered the entire Hyperion proto-supercluster with its $\langle \delta_{\rm gal} \rangle \sim 1.24$ (Sect. \[3D\]), and assumed an elongation equal to the average elongation of the peaks to derive its $\langle \delta_{\rm gal,corr} \rangle$ and then its $\langle \delta_{\rm m,corr} \rangle$, the proto-supercluster would have $\delta_{\rm L} \lesssim 0.8$ at $z=2.46$ (to be compared with the y-axis of Fig. \[delta\_evol\_ps\]).
We note that the evolutionary status of the peaks depends by definition on their average density, that is, the higher the density, the more evolved the overdensity perturbation. The most evolved is peak \[6\], which has $\langle \delta_{\rm m,corr} \rangle = 12.66$, almost twice as large as the second densest peak (peak \[5\]). According to the spherical collapse model, peak \[6\] will be a virialised system by $z\sim1.7$, that is, in $1.3$ Gyr from the epoch of observation. The least evolved is peak \[2\], that will take 0.6 Gyr to reach the turn-around and then another $\sim3.8$ Gyr to virialise.
This simple exercise, which is based on a strong assumption, shows that the peaks are possibly at different stages of their evolution, and will become virialised structures at very different times. In reality, the peaks’ evolution will be more complex, given that they will possibly accrete mass/subcomponents/galaxies during their lifetimes, and these results make it desirable to study how we can combine the density-driven evolution of the individual peaks with the overall evolution of the Hyperion proto-supercluster as a whole. Moreover, by comparing the evolutionary status of each peak with the average properties of its member galaxies, it will be possible to study the co-evolution of galaxies and the environment in which they reside. We defer these analyses to future works.
![Evolution of $\delta_m$ for the seven peaks listed in Table \[peaks\_evol\_tab\], with different line styles as in the legend. The evolution is computed in a linear regime for a $\Lambda$CDM Universe. For each peak, we start tracking the evolution from the redshift of observation (column 2 in Table \[peaks\_evol\_tab\]), and we consider as starting $\delta_{\rm m}$ the one computed from the corrected $\langle \delta_{\rm gal,corr} \rangle$ (column 7 in Table \[peaks\_elongation\_tab\]) and transformed into linear regime. The horizontal lines represent $\delta_{\rm L,ta}\simeq 1.062$, $\delta_{\rm L,vir}\simeq 1.58$ and $\delta_{\rm L,c}\simeq 1.686$. See Sect. \[collapse\] for more details.[]{data-label="delta_evol_ps"}](./fig7.ps){width="9.0cm"}
----- ------- ---------------------------------------- ---------------- ------------- --------------------- --------------------
ID $z$ $\langle \delta_{\rm m,corr} \rangle $ $z_{\rm ta}$ $z_{\rm c}$ $\Delta t_{\rm ta}$ $\Delta t_{\rm c}$
\[Gyr\] \[Gyr\]
(1) (2) (3) (4) (5) (6) (7)
1 2.468 4.25 2.402 1.054 0.08 3.16
2 2.426 2.04 2.001 0.781 0.60 4.37
3 2.444 6.24 $>z_{\rm obs}$ 1.282 - 2.32
4 2.469 4.60 $>z_{\rm obs}$ 1.108 - 2.95
5 2.507 6.94 $>z_{\rm obs}$ 1.388 - 2.07
6 2.492 12.66 $>z_{\rm obs}$ 1.675 - 1.33
7 2.423 4.21 2.347 1.017 0.10 3.26
----- ------- ---------------------------------------- ---------------- ------------- --------------------- --------------------
: Evolution of the density peaks according to the spherical collapse model in linear regime. Columns (1) and (2) are the ID and the redshift of the peak, as in Table \[peaks\_elongation\_tab\]. Column (3) is the average matter overdensity derived from the average galaxy overdensity of column (7) of Table \[peaks\_elongation\_tab\]. Columns (4) and (5) are the redshifts when the overdensity reaches the overdensity of turn-around and collapse, respectively. Columns (6) and (7) are the corresponding time intervals $\Delta t$ since the redshift of observation $z_{\rm obs}$ (column 2) to the redshifts of turn-around and collapse. When $z_{\rm ta} < z_{\rm obs}$ the turn-around has already been reached before the redshift of observation, and in these cases the corresponding $\Delta t $ have not been computed. See Sect. \[collapse\] for more details.[]{data-label="peaks_evol_tab"}
The proto-supercluster as a whole. {#whole_psc}
----------------------------------
In the previous section we pretended that the peaks were isolated density fluctuations and traced their evolution in the absence of interactions with other components of the proto-supercluster. This is an oversimplification, because several kinds of interactions are likely to happen in such a large structure, such as for example accretion of smaller groups along filaments onto the most dense peaks, as expected in a $\Lambda$CDM universe.
For instance, for what concerns merger events between proto-clusters, [@lee2016_colossus] examined the merger trees of some of the density peaks that they identified in realistic mock data sets by applying the same 3D Ly$\alpha$ forest tomographic mapping that they applied to the COSMOS field. They found that in the examined mocks, very few of the proto-structures identified by the tomography at $z\sim2.4$ and with an elongated shape (such as the ‘chain’ of their peaks L16a, L16b, and L16c discussed in Sect. \[peak3\]) are going to collapse to one single cluster at z=0. Similarly, [@topping18] analysed the Small MultiDark Planck Simulation in search for $z\sim3$ massive proto-clusters with a double peak in the galaxy velocity distribution and with the two peaks separated by about $2000\kms$, like the one they identified in previous observations [@topping16]. They found that such double-peaked overdensities are not going to merge into a single cluster at $z=0$.
The structures found by [@lee2016_colossus] and [@topping16] are much smaller and with simpler shapes compared to the Hyperion proto-supercluster, and yet they are unlikely to form a single cluster at z=0, according to simulations. Interestingly, [@topping18] also found that in their simulation the presence of two massive peaks separated by $2000\kms$ is a very rare event (one in $7.4h^3$Gpc$^{-3}$) at $z\sim3$. These findings indicate that the evolution of the Hyperion proto-supercluster cannot be simplified as series of merging events, and that the identification of massive/complex proto-clusters at high redshift could be useful to give constraints on dark matter simulations.
Indeed, it would be interesting to know whether or not Hyperion could be the progenitor of known lower-redshift superclusters. One difficulty is that there is no unique definition of a supercluster (but see e.g. [@chon15] for an attempt), and the taxonomy of known superclusters up to $z\sim1.3$ spans wide ranges of mass (from a few $10^{14}M_{\odot}$ as in [@swinbank07] to $>10^{16}M_{\odot}$ as in [@bagchi17]), dimension (a few cMpc as in [@rosati99_lynx] or $\sim 100$ cMpc as in [@kim16_sc]), morphology (compact as in [@gilbank08], or with multiple overdensities as in [@lubin00; @lemaux12]), and evolutionary status (embedding collapsing cores as in [@einasto16_SGW] or already virialised clusters as in [@rumbaugh18]). This holds also for the well-known superclusters in the local universe (see e.g. [@shapley30; @shapley34_hercules; @delapparent86_greatwall; @haynes86_pp]), not to mention the category of the so-called Great Walls, which are sometimes defined as systems of superclusters (like e.g. the Sloan Great Wall, [@vogeley04; @gott05], and the Boss Great Wall, [@lietzen16_sc]).
Clearly, Hyperion shares many characteristics with the above-mentioned superclusters, making it likely that its eventual fate will be to become a supercluster. A further step would be identifying which known supercluster is most likely to be similar to the potential descendant(s) of Hyperion. This would be surely an important step in understanding how the large-scale structure of the universe evolves and how it affects galaxy evolution. On the other hand, it is also interesting to study the likelihood of such (proto-) superclusters existing in a given cosmological volume, given their volumes and masses (see e.g. [@sheth11]). For instance, [@lim14] show that the relative abundance of rich superclusters at a given epoch could be used as a powerful cosmological probe.
From [@lim14] we can qualitatively assess how many superclusters of the kind that we detect are expected in the volume probed by VUDS. [@lim14] derive the mass function of superclusters, defined as clusters of clusters according to a Friend of Friend algorithm. Since the supercluster mass function at $z\sim2.5$ was not explicitly studied, we adopt here expectations from their study of the $z=1$ supercluster mass function keeping in mind that this expectation will be a severe upper limit given that the halo mass function at the high-mass end decreases by a factor of $\ga100$ from $z=1$ to $z=2.5$ (see, e.g. [@percival05]). With this in mind, we estimate, using those results of [@lim14] that employ a similar cosmology to the one used in this study, the extreme upper limit to the number of superclusters with a total mass $>5\times10^{14}$ M$_{\odot}$ expected within the RA-Dec area studied in this paper and in the redshift range $2<z<4$ to be $\sim$4. We consider this mass limit, $>5\times10^{14}$ M$_{\odot}$, because it is the sum of the masses of our peaks, similarly to how they compute the masses of their superclusters. The extremeness of this upper limit is such that much more precise comparisons need to be made. We defer the detailed analysis of number counts and evolution of proto-superclusters at $z\sim2.5$ in simulated cosmological volumes to a future work.
Summary and conclusions {#summary}
=======================
Thanks to the spectroscopic redshifts of VUDS, together with the zCOSMOS-Deep spectroscopic sample, we unveiled the complex shape of a proto-supercluster at $z\sim2.45$ in the COSMOS field. We computed the 3D overdensity field over a volume of $\sim100\times100\times250$ comoving Mpc$^3$ by applying a Voronoi tessellation technique in overlapping redshift slices. The tracers catalogue comprises our spectroscopic sample complemented by photometric redshifts for the galaxies without spectroscopic redshift. Both spectroscopic and photometric redshifts were treated statistically, according to their quality flag or their measurement error, respectively. The main advantage of the Voronoi Tessellation is that the local density is measured both on an adaptive scale and with an adaptive filter shape, allowing us to follow the natural distribution of tracers. In the explored volume, we identified a proto-supercluster, dubbed “Hyperion" for its immense size and mass, extended over a volume of $\sim60\times60\times150$ comoving Mpc$^3$. We estimated its total mass to be $\sim 4.8\times 10^{15}{\rm M}_{\odot}$. Within this immensely complex structure, we identified seven density peaks in the range $2.40<z<2.5$, connected by filaments that exceed the average density of the volume. We analysed the properties of the peaks, as follows:
- We estimated the total mass of the individual peaks, $M_{\rm tot}$, based on their average galaxy density, and found a range of masses from $\sim 0.1\times 10^{14}{\rm M}_{\odot}$ to $\sim 2.7\times 10^{14}{\rm M}_{\odot}$.
- By assigning spectroscopic members to each peak, we estimated the velocity dispersion of the galaxies in the peaks, and then their virial mass $M_{\rm vir}$ (under the admittedly strong assumption that they are virialised). The agreement between $M_{\rm vir}$ and $M_{\rm tot}$ is surprisingly good, considering that (almost all) the peaks are most probably not yet virialised.
- If we assume that the peaks are going to evolve separately, without accretion/merger events, the spherical collapse model predicts that these peaks have already started or are about to start their collapse phase (‘turn-around’), and they will all be virialised by redshift $z\sim0.8$.
- We finally performed a careful comparison with the literature, given that some smaller components of this proto-supercluster had previously been identified in other works using heterogeneous galaxy samples (LAEs, 3D Ly$\alpha$ forest tomography, sub-mm starbursting galaxies, CO emitting galaxies). In some cases we found a one-to-one match between previous findings and our peaks, in other cases the match is disputable. We note that a direct comparison is often difficult because of the different methods/filters used to identify proto-clusters.
In summary, with VUDS we obtained, for the first time across the central $\sim1$ deg$^2$ of the COSMOS field, a panoramic view of this large structure that encompasses, connects, and considerably expands on all previous detections of the various sub-components. The characteristics of the Hyperion proto-supercluster (its redshift, its richness over a large volume, the clear detection of its sub-components), together with the extensive band coverage granted by the COSMOS field, provide us the unique possibility to study a rich supercluster in formation 11 billion years ago.
This impressive structure deserves a more detailed analysis. On the one hand, it would be interesting to compare its mass and volume with similar findings in simulations, because the relative abundance of superclusters could be used to probe deviations from the predictions of the standard $\Lambda$CDM model. On the other hand, it is crucial to obtain a more complete census of the galaxies residing in the proto-supercluster and its surroundings. With this new data, it would be possible to study the co-evolution of galaxies and the environment in which they reside, at an epoch ($z\sim2-2.5$) when galaxies are peaking in their star-formation activity.
We thank the referee for his/her comments, which allowed us to clarify some parts of the paper. This work was supported by funding from the European Research Council Advanced Grant ERC-2010-AdG-268107-EARLY and by INAF Grants PRIN 2010, PRIN 2012 and PICS 2013. This work was additionally supported by the National Science Foundation under Grant No. 1411943 and NASA Grant Number NNX15AK92G. OC acknowledges support from PRIN-INAF 2014 program and the Cassini Fellowship program at INAF-OAS. This work is based on data products made available at the CESAM data center, Laboratoire d’Astrophysique de Marseille. This work partly uses observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at TERAPIX and the Canadian Astronomy Data Centre as part of the Canada–France–Hawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS. This paper is also based in part on data products from observations made with ESO Telescopes at the La Silla Paranal Observatory under ESO programme ID 179.A-2005 and on data products produced by TERAPIX and the Cambridge Astronomy Survey Unit on behalf of the UltraVISTA consortium. OC thanks M. Roncarelli, L. Moscardini, C. Fedeli, F. Marulli, C. Giocoli, and M. Baldi for useful discussions, and J.R. Franck and S.S. McGaugh for their kind help in unveiling the details of their work.
Stability of the peaks properties {#app_stability}
=================================
We investigated the extent to which the choice of a 5$\sigma_{\delta}$ threshold affects some of the properties of the identified peaks. Namely, we varied the overdensity threshold from 4.5$\sigma_{\delta}$ to 5.5$\sigma_{\delta}$, and verified the variation of $M_{\rm tot}$ (Table \[peaks\_tab\]), velocity dispersion (Table \[peaks\_tab\_veldisp\]) and elongation (Table \[peaks\_elongation\_tab\]) as a function of the used threshold.
Total mass {#app_Mtot_sigma}
----------
Figure \[Mtot\_vs\_sigma\] shows the fractional variation of $M_{\rm tot}$ (Table \[peaks\_tab\]) as a function of the adopted threshold, which is expressed in terms of the corresponding multiple of $\sigma_{\delta}$. Five peaks out of seven show roughly the same variation, while peak \[1\] has a much smaller variation and peak \[7\] a much steeper one. This might imply that the (baryonic) matter distribution within peak \[7\] is less peaked toward the centre with respect to the other peaks, while the matter distribution within peak \[1\] is more peaked.
Given that we are probing very dense peaks (they are about to collapse, see Sect. \[discussion\]), we expect the total mass enclosed above a given overdensity threshold to have large variations if we vary the overdensity threshold by much. If instead we focus on a small $n_{\sigma}$ range around our nominal value of $n_{\sigma}=5$, for instance the interval $5\pm0.2$, we see that the variation of the total mass is much smaller than the uncertainty on the total mass quoted in Table \[peaks\_tab\], which was computed by using the density maps obtained with $\delta_{\rm gal,16}$ and $\delta_{\rm gal,84}$ (see Sect. \[method\]).
This means that, although the total mass of our peaks depends on the chosen overdensity threshold, because of the very nature of the mass distribution in these peaks, at the chosen threshold the uncertainty is dominated by the uncertainty on the reconstruction of the density field and not by our precise definition of ‘overdensity peak’.
![ Fractional variation of the total mass $M_{\rm tot}$ (Table \[peaks\_tab\]) for the seven peaks as a function of the overdensity threshold adopted to identify them, expressed in terms of the corresponding multiples $n_{\sigma}$ of $\sigma_{\delta}$. The reference total mass value is the one at the 5$\sigma_{\delta}$ threshold. The different lines correspond to the different peaks as in the legend. The filled symbols on the right, with their error bars, correspond to the fractional variation of $M_{\rm tot}$ calculate at $5\sigma_{\delta}$ resulting from the uncertainties on the density reconstruction quoted in Table \[peaks\_tab\]. The position of the error bars on the x-axis is arbitrary. In all cases, these errors are much larger than the uncertainty resulting from slightly modulating the overdensity threshold employed.[]{data-label="Mtot_vs_sigma"}](./figA1.ps){width="9.0cm"}
Velocity dispersion {#app_vdisp_sigma}
-------------------
Similarly to the variation of the total mass, we verified how the velocity dispersion $\sigma_{\rm v}$ varies as a function of the adopted overdensity threshold, for the seven identified peaks. For each threshold, the velocity dispersion and its error are computed as described in Sect. \[vel\_disp\], and only when we could use at least three spectroscopic galaxies. For all the peaks, $\sigma_{\rm v}$ is relatively stable in the entire range of the explored overdensity thresholds, and its small variations (due to the increasing or decreasing number of spectroscopic members) are always much smaller than the uncertainties computed on the velocity dispersion itself, at fixed $n_{\sigma}$. For this reason we consider the virial masses quoted in Table \[peaks\_tab\_veldisp\] to be independent from small variations of the overdensity threshold.
We remind the reader that for the computation of the velocity dispersion we used a more relaxed definition of galaxy membership within each peak so as to increase the number of the available galaxies (see Sect. \[vel\_disp\]). Even with this broader definition, for peak \[7\] we had only two galaxies available if we used $n_{\sigma}=5$ to define the peak, while their number increased to four by using $n_{\sigma}=4.9$. For this reason, we decided that the most reliable value of $\sigma_{\rm v}$ for peak \[7\] is the one computed using $n_{\sigma}=4.9$, and we quote this $\sigma_{\rm v}$ in Table \[peaks\_tab\_veldisp\].
Elongation {#app_elongation}
----------
Here we approximately estimate how the elongation depends on the typical dimension of our density peaks. Our estimation is based on the following simplistic assumptions: 1) the intrinsic shape of a proto-cluster is a sphere with radius $r_{\rm int}$, and its measured dimensions on the $x-$ and $y-$axis ($r_x$ and $r_y$) correspond to the intrinsic dimension $r_{int}$, i.e. $r_x=r_y=r_{\rm int}$, and 2) the measured dimension on the $z-$axis ($r_z$) corresponds to $r_{int}$ plus a constant factor $\Delta r$, which is the result of the complex interaction among the several factors that might cause the elongation (the depth of the redshift slices, the photometric redshift error etc), i.e. $r_z=r_{int}+\Delta r$. From these assumptions it follows:
$$\displaystyle
\frac{r_z}{r_{xy}} = 1 + \frac{\Delta r}{r_{int}},
\label{elongation_eq}$$
where $r_{xy}$ is the average between $r_x$ and $r_y$, and in our example we have $r_{xy}=r_x=r_y=r_{\rm int}$. If we substitute $r_x$, $r_y$ and $r_z$ with $R_{e,x}$, $R_{e,y}$ and $R_{e,z}$ as defined in Sect. \[3D\_peaks\], from Eq. \[elongation\_eq\] follows:
$$\displaystyle
E_{\rm z/xy} = 1 + \frac{\Delta r}{R_{e,xy}},
\label{elongation_eq2}$$
with $E_{\rm z/xy}$ and $R_{e,xy}$ as defined in Sect. \[3D\_peaks\]. This means that the measured elongation depends on the circularised 2D effective radius as $y=1+A/x$.
To verify this dependence, we measured $E_{\rm z/xy}$ and $R_{e,xy}$ for our seven peaks for different thresholds, expressed in terms of the multiples $n_{\sigma}$ of $\sigma_{\rm \delta}$. In this case, we made the threshold vary from 4.1 to 7 $\sigma_{\rm \delta}$, because the two peaks \[1\] and \[4\] merge in one huge structure if we use a threshold $<4.1\sigma_{\delta}$. We notice that peak \[5\] disappears for $\sigma_{\rm \delta}>5.8$ above the mean density, and peak \[7\] for $\sigma_{\rm \delta}>5.4$. The peaks \[1\], \[2\] and \[4\] are split into two smaller peaks when $\delta_{\rm gal}$ is above 6.5$\sigma_{\rm \delta}$, 5.2$\sigma_{\rm \delta}$ and 5.1$\sigma_{\rm \delta}$ above the mean density, respectively. Figure \[elongation\_fig\] shows how $E_{\rm z/xy}$ varies as a function of $R_{e,xy}$. The three curves with equation $y=1+A/x$ are shown to guide the eye, with $A$ tuned by eye to match the normalisation of some of the observed trends. It is evident that the foreseen dependence of $E_{\rm z/xy}$ on $R_{e,xy}$ is confirmed. In the Figure, $A$ increases by a factor of $\sim3$ from the lowest curve (corresponding e.g. to peak \[7\]) to the highest one (matching e.g. peak \[6\]). The specific value of $A$ is likely due to a complex combination of peculiar velocities, spectral sampling, reconstruction methods (e.g. slice size relative to the true l.o.s. extent), and photometric redshift errors. It is beyond the scope of this paper to precisely quantify the contribution of each for each individual peak. Nevertheless, although in some cases $E_{\rm z/xy}$ quickly vary for small changes of $R_{e,xy}$ (i.e. small changes in the threshold), this plot confirms that its measured values are reasonably consistent with our expectations.
![Elongation $E_{\rm z/xy}$ as a function of $R_{e,xy}$. The different colours refer to the different peaks as in the legend. $E_{\rm z/xy}$ and $R_{e,xy}$ are measured by fixing different thresholds (number of $\sigma_{\rm \delta}$ above the mean density) to define the peaks themselves, ranging from 4.1 to 7 $\sigma_{\rm \delta}$. $E_{\rm z/xy}$ and $R_{e,xy}$ measured at the $5\sigma_{\rm \delta}$ threshold are highlighted with a filled circle, and are the same quoted in Table \[peaks\_elongation\_tab\]. The peaks \[1\] and \[2\] are split into two smaller peaks when $\delta_{\rm gal}$ is above 5.5$\sigma_{\rm \delta}$ and 5.7$\sigma_{\rm \delta}$ above the mean density, respectively: this is shown in the plot by splitting the curve of the two peaks into two series of circles (filled and empty). The three dotted lines corresponds to the curves $y=1+A/x$, with $A=4.3,2.9,1.5$ from top to bottom. The values of A are chosen to make the curves overlap with some of the data, to guide the eye.[]{data-label="elongation_fig"}](./figA2.ps){width="9.0cm"}
Details on individual peaks {#app_peaks}
===========================
We show here the projections on the RA-Dec and $z$-Dec planes of the four most massive peaks (“Theia”, “Eos”, “Helios”, and “Selene”), to highlight their complex shape. The remaining peaks have very regular shapes on the RA-Dec and $z$-Dec planes, so we do not show them here. The projections that we show include the peak isodensity contours in the 3D cube and the position of the spectroscopic member galaxies. The $z$-Dec projection is associated to the velocity distribution of the spectroscopic members.
![For peak \[1\], “Theia”, the [*top-left*]{} panel show the projection on the RA-Dec plane of the $5\sigma_{\rm \delta}$ contours which identify the peak in the 3D overdensity cube; the different colours indicate the different redshift slices (from blue to red, they go from the lowest to the highest redshift). Filled circles are the spectroscopic galaxies which are members of the peak (flag=X2/X2.5, X3, X4, X9), with the same colour code as the the contours. The black cross is the RA-Dec barycenter of the peak. In the top-right and bottom-left corners we show the scale in pMpc and cMpc, respectively, for both RA and Dec. [*Top-right.*]{} Projections on the $z$-Dec plane of the same contours shown in the top-left panel, with the same colour code. The filled circles and the black cross are as in the top-left panel. On the top and on the bottom of the panel we show the scale in pMpc and cMpc, respectively. [ *Bottom-right*]{}. The black histogram represents the velocity distribution of the spectroscopic galaxies which fall in the same RA-Dec region as the proto-cluster. The red histogram includes only VUDS and zCOSMOS galaxies with reliable quality flag, and flags X1/X1.5 for galaxies within the peak volume (see Sect. \[vel\_disp\] for details). The vertical solid green line indicates the barycenter along the l.o.s (the top x-axis is the same as the one in the top-right panel), and the two dashed vertical lines the maximum extent of the peak. The dotted-dashed blue vertical line is the $z_{\rm BI}$ of Table \[peaks\_tab\_veldisp\], around which we center the Gaussian (blue solid curve) with the same $\sigma_v$ as in Table \[peaks\_tab\_veldisp\]. The two dotted blue curves are the uncertainties on the Gaussian due to the uncertainties on $\sigma_v$. In the [*bottom-left*]{} corner of the figure we summarise some of the peak properties, which are all already mentioned in the Tables or in the text. []{data-label="peak1_fig"}](./figB1.ps){width="9.0cm"}
![As in Fig.\[peak1\_fig\], but for Peak \[2\], “Eos”.[]{data-label="peak2_fig"}](./figB2.ps){width="9.0cm"}
![As in Fig.\[peak1\_fig\], but for Peak \[3\], “Helios”. []{data-label="peak3_fig"}](./figB3.ps){width="9.0cm"}
![As in Fig.\[peak1\_fig\], but for Peak \[4\], “Selene”.[]{data-label="peak4_fig"}](./figB4.ps){width="9.0cm"}
[^1]: Based on data obtained with the European Southern Observatory Very Large Telescope, Paranal, Chile, under Large Program 185.A-0791.
[^2]: http://www.cfht.hawaii.edu/$\sim$arnouts/LEPHARE/lephare.html
[^3]: $X=0$ is for galaxies, $X=1$ for broad line AGNs, and $X=2$ for secondary objects falling serendipitously in the slits and spatially separable from the main target. The case $X=3$ is as $X=2$ but for objects not separable spatially from the main target.
[^4]: https://www.eso.org/sci/observing/phase3/data\_releases/uvista\_dr2.pdf
[^5]: In this work we neglect the correlations in the noise between the cells in the same slice and those in different slices.
[^6]: In [@cucciati2014_z29] we corrected the volume of the proto-cluster under analysis by a factor which took into account the Kaiser effect, which causes the observed volume to be smaller than the real one, due to the coherent motions of galaxies towards density peaks on large scales. Here we show that we are concerned rather by an opposite effect, i.e. our volumes might be artificially elongated along the l.o.s..
[^7]: Excluding the possible uncertainty on the bias factor $b$, which does not depend on our reconstruction of the overdensity field. For instance, if we assume $b=2.59$, as derived in [@bielby13] at $z\sim3$, we obtain a total mass $<1\%$ smaller.
[^8]: Hyperion, one of the Titans according to Greek mythology, is the father of the sun god Helios, to whom the Colossus of Rhodes was dedicated.
[^9]: This assumption is more suited for a virialised object than for a structure in formation. Nevertheless, our approach does not intend to be exhaustive, and we just want to compute a rough correction.
[^10]: First we computed $M_{\rm 200}$ as in Eq. \[m200\_munari\], then converted $M_{\rm 200}$ into $M_{\rm vir}$ based on the same assumptions as for the conversion between $R_{\rm 200}$ and $R_{\rm v}$. This gives $M_{\rm vir} =1.06~M_{\rm 200}$.
[^11]: According to Greek mythology, Theia is a Titaness, sister and spouse of Hyperion. Eos, Helios, and Selene are their offspring.
[^12]: In our case the zCOSMOS-Deep sample, used together with the VUDS sample, is cut at $I=25$. Moreover we do not use the zCOSMOS-Deep quality flag 1.5. [@diener2013_list] used also flag=1.5 and did not apply any magnitude cut.
[^13]: [@lee2016_colossus] mention that from their unsmoothed tomographic map this huge overdensity is composed of several lobes (see e.g. their Figs. 4 and 13), but it is more continuous after applying a smoothing with a $4 h^{-1}$Mpc Gaussian filter.
| ArXiv |
---
abstract: 'We propose to synthesize feasible caging grasps for a target object through computing *Caging Loops*, a closed curve defined in the *shape embedding space* of the object. Different from the traditional methods, our approach *decouples* caging loops from the surface geometry of target objects through working in the embedding space. This enables us to synthesize caging loops encompassing multiple topological holes, instead of always tied with one specific handle which could be too small to be graspable by the robot gripper. Our method extracts caging loops through a topological analysis of the distance field defined for the target surface in the embedding space, based on a rigorous theoretical study on the relation between caging loops and the field topology. Due to the decoupling, our method can tolerate incomplete and noisy surface geometry of an unknown target object captured on-the-fly. We implemented our method with a robotic gripper and demonstrate through extensive experiments that our method can synthesize reliable grasps for objects with complex surface geometry and topology and in various scales.'
author:
- 'Jian Liu$^{1}$, Shiqing Xin$^{1}$, Zengfu Gao$^{1}$, Kai Xu$^{2}$, Changhe Tu$^{1}$ and Baoquan Chen$^{1}$[^1]'
bibliography:
- 'reference.bib'
title: 'Caging Loops in Shape Embedding Space: Theory and Computation'
---
Introduction {#sec:intro}
============
As an important type of robot grasping, caging grasps [@Rodriguez-2011FromCT; @Wan-2013AN; @Diankov-2008ManipulationPW], as compared to force-closure grasps [@Zhu-PlanningFG2004; @Ding-Computing3O2000; @Borst-GraspingTD2003; @Ferrari-PlanningOG1992], are advantageous in handling target objects with unknown or uncertain surface geometry and/or friction properties. This makes caging grasps more practically applicable in a wide spectrum of real scenarios. We are especially interested in a simple yet effective type of caging grasp formed by *caging loops*. A caging loop is a closed curve in three dimensional space computed around some part of the target object and used to guide robot grippers to form a caging grasp.
Existing methods on 3D caging grasp are based either on the geometric (e.g. [@Zarubin-2013CagingCO]) or the topological (e.g. [@Pokorny-2013GraspingOW]) information of the target surface, or even both [@Kwok-2016RopeCA]. A common issue to these methods is that the computed caging curves seriously depend on topological and geometrical features of objects, while being oblivious to the relative size between the target object and the gripper. Taking the genus-4 Indonesian-Lady model in Fig. \[fig:teaser\] for example. The six handles on the model are all seemingly good candidates for grasping. However, when the size of the model is too small compared to the robot gripper, these handles will no longer be graspable since the holes may be too small for the fingers to pass through. In such case, a more feasible grasp would be enclosing the object with a loop encompassing multiple handles (see Fig. \[fig:teaser\](top) and Fig. \[fig:scale\](a)).
Another issue with geometry-based caging curves is that they easily lead to non-convex spatial curves which are not suited for guiding the gripper configuration. The example in Fig. \[fig:scale\](d) demonstrates such case, where the gripper penetrates into the object due to the non-convexity of the caging loop. Estimating a convex hull for the spatial loop still cannot guarantee a penetration-free configuration.
Motivation And Contribution
---------------------------
![Grasping a 3D-printed Indonesian-Lady model (the top and middle row) in two sizes. Our method is able to synthesize caging loops (red circles) encompassing multiple topological handles, when the object is too small to be grasped on one handle (top row). When the object is large, our method naturally grasps one handle (middle row). The two cases are integrated seamlessly in method. The bottom row shows how a caging loop computed in the embedding encloses the two handles of a pliers. The 3D objects are acquired by two RGBD cameras and reconstructed on-the-fly (middle column). As a reference, a human grasp is shown to the left for each object.[]{data-label="fig:teaser"}](teaser.png){width="0.95\columnwidth"}
These examples motivated us in seeking to “fill up” those small topological holes and “smooth out” the geometric details on the target surface, before computing caging curves. Therefore, we advocate computing caging loops in the *embedding space* of the target surface, through a topological analysis of the distance field defined for the target surface in the embedding space.
We conduct a theoretical study on the fundamental relationship between caging loops and Morse singularities (including minimal, maximal and saddle points) of a spatial distance function. Based on that, we develop an algorithm of caging loop extraction through saddle point detection and analysis, within a proper grasping space defined in account of the gripper size. Working with a distance function defined in the embedding space naturally decouples the shape of caging loops from the geometric details of the target surface, while still keeping them aware of the overall shape of the target object. The caging loops are properly placed and scaled based on the relative size of the gripper against the target shape, rather than always tied with a specific handle as in traditional approaches. Another benefit of working in embedding space is the tolerance of incomplete and noisy surface geometry of the target object. This makes our method especially suited for synthesizing grasps for unknown objects which are captured and reconstructed on-the-fly, with a minimal effort of robot observation. In our implementation (see Fig. \[fig:overview\]), two depth cameras are deployed to capture the target object from two (front and back) views. Even with such a sparse capturing and low-quality reconstruction, our method can still synthesize feasible caging loops for robust grasping. We found this simple idea leads to a robust and efficient algorithm, with theoretical guarantees. We implemented our algorithm in a grasping system composed of a Barrett WAM robotic arm with a three-finger gripper and two Xtion Pro RGB-D cameras. Only depth images are used for reconstructing the target surface based on the depth fusion technique [@Newcombe-2011KinectFusionRD]. We have conducted numerous evaluations with both synthetic and real examples to evaluate the performance of our method. We show that our system is able to robustly grasp objects with complex surface geometry and topology and in various scales.
Our work makes the following contributions:
- We propose a novel method for caging grasp synthesis through topological analysis of shape-aware distance field defined in shape embedding space. The method is able to generate relative-scale-aware caging loops for unknown objects captured on-the-fly.
- We provide a rigorous study on the relation between the topology of distance field and caging loops, based on Morse theory, and derive a robust algorithm for caging loop estimation. We also provide a handful of provably effective techniques to reduce the computational cost of our method.
- We implement our method in a grasping system using robot gripper, and conduct thorough evaluations and comparisons with both synthetic and real objects.
![By decoupling caging loops from target surfaces, our method synthesizes feasible caging grasps for objects containing tiny topological handles (a) or presenting concave surface geometry (b); see the red circles and the corresponding grasps to the right. In contrast, the loops (yellow circles in c and d) computed over the target surfaces incur gripper-object collision; see the gripper parts in red color in the bottom row.[]{data-label="fig:topologyAndGeometry"}](topologyAndGeometry1.png "fig:"){width="0.95\linewidth"}\
\
![By decoupling caging loops from target surfaces, our method synthesizes feasible caging grasps for objects containing tiny topological handles (a) or presenting concave surface geometry (b); see the red circles and the corresponding grasps to the right. In contrast, the loops (yellow circles in c and d) computed over the target surfaces incur gripper-object collision; see the gripper parts in red color in the bottom row.[]{data-label="fig:topologyAndGeometry"}](topologyAndGeometry2.png "fig:"){width="0.95\linewidth"}\
\[fig:scale\]
Related Work
------------
Robot grasping is a long-standing yet actively studied research topic in the fields of robotics and vision. Force-closure and caging are two typical approaches that have been developed to synthesize grasps. Force-closure methods [@Bicchi-OnTC1995; @Miller-GraspitAV2004; @Liu-QualitativeTA1999; @Howard-OnTS1996; @Ding-Computing3O2000; @Zhu-PlanningFG2004] concentrate on finding a stable grasping configuration for the grippers where a mechanical equilibrium is achieved. The advantage of such approach is that the synthesized grasps are usually physically feasible. The method, however, requires that the 3D shape of the target model is known *a priori* and cannot tolerate much the surface defect such as missing data. Furthermore, the contact area between the gripper and the target surface is often small, leading to unsteady grasps.
Caging grasps [@Diankov-2008ManipulationPW], as compared to force-closure ones, seeks for a sufficiently large contact area and thus are deemed to have better stability, although they are not designed to directly reach a mechanical equilibrium. The key benefit of caging [@Diankov-2008ManipulationPW] is that it is robust to surface uncertainty and imperfection. This makes it especially applicable to unknown objects being captured and reconstructed on-the-fly. Some works studied the computation of planar cages in 2D space for planar objects [@Rimon-1999CagingPB; @Pipattanasomporn-2006TwofingerCO; @Pipattanasomporn-2011TwoFingerCO; @Vahedi-2008CagingCP]. Most existing approaches to 3D object caging rely on the topological structure of the target surface [@Pokorny-2013GraspingOW; @Dey-2010ApproximatingLI; @Stork-2013IntegratedMA]. Some further take geometry information into account [@Kwok-2016RopeCA]. However, such approaches cannot compute a caging loop encompassing multiple handles or deal with different relative scales between the gripper and the target object.
The Morse theory, as a connection between geometry and topology, has been widely utilized in the graphics and visualization fields [@Bremer-2004ATH; @Ni-2004FairMF]. In our approach, the core algorithmic step is to find a caging loop according to a $p$-based distance field, where $p$ is a point in the grasping space. At this point, the Morse theory is used to build a fundamental relationship between caging loops and Morse saddle points.
Theory {#sec:theory}
======
Grasping Strategy
-----------------
In order to define a caging loop, we have to [consider at least geometric and mechanical aspects.]{} The geometric considerations include:
- A caging loop encompasses the target object - any penetration into the target shape is not allowed.
- A caging loop encloses some part of the target object tightly, i.e., cannot be shortened with respect to a slight perturbance (i.e., [*stable grasp*]{}) or at least goes around the target object like a great circle enclosing a sphere (i.e., [*unstable grasp*]{}).
- A caging loop should roughly match the real robot gripper size.
On the other side, the mechanical considerations include
- The center point of a caging loop should be as close as possible to the center of gravity of the target shape so as to minimize the moment of intertia.
- A caging loop should be roughly horizontal so that the target object can be taken up steadily.
Our strategy is to compute a collection of caging loop candidates in consideration of the above-mentioned geometric principles. For purpose of efficient computation, we also invent a set of filtering techniques to reduce the number of loop candidates as far as possible.
Mathematical Formulation
------------------------
Imagine the scenario of a caging grasp where the fingers of the gripper stretch to two opposite directions, roughly forming a loop; See Fig. \[fig:teaser\] and Fig. \[fig:topologyAndGeometry\]. In the following, we shall formally characterize in which space we extract caging loops and systematically establish properties of caging loops.
The surface $S$ of the target object, typically represented as a watertight mesh, divides the whole $\mathbb{R}^3$ space into interior parts and exterior parts, where only the visible free space (the outmost surface exterior space) is helpful to determine a real grasp configuration. Rather than constrain caging loops lying on the target surface $S$, we relax caging loops from $S$ to the shape embedding space.
The visible free space separated by the target surface $S$ is called the [*grasping space*]{}.
Generally speaking, a stable grasp is desired, i.e., the caging loop encloses some part of the target object tightly and cannot be shortened even with a slight perturbance. In some rare cases, however, an unstable grasp like a great circle enclosing a sphere is also acceptable. Both cases imply that there is a fundamental relationship between caging loops and locally shortest loops in the grasping space.
\[defn:grasping\_loop\]Suppose the target object $S$ defines a grasping space $\mathbf{G}$. A closed curve $L\in \mathbf{G}$ is called a [*caging loop candidate*]{} if and only if $L$ is [*locally shortest everywhere*]{}, i.e., for any point $p\in L$, any sufficiently short segment of $L$ around $p$ cannot be shortened any more. All such loop candidates constitute a [*caging loop space*]{}, denoted by $\mathbf{L}$.
\[property:three\]Each caging loop $L\in{\mathbf{L}}$ touches the target surface at three or more points.
[*Proof.*]{} Without loss of generality, we assume that $L$ touches the target surface $S$ at only one point $p$. Then the open curve ${{L}}\backslash p$ lies in the grasping space but doesn’t touch $S$. Considering that a locally shortest curve in $\mathbb{R}^3$ must be a straight line segment, ${{L}}\backslash p$ cannot include a bending point. Furthermore, $p$ is not only the start point of the straight line segment ${{L}}\backslash p$ but also its endpoint. Therefore, $L$ degenerates into a single point under the above assumption, which contradicts to the given condition that $L$ is a caging loop. Similarly, it can be shown that the case of two touching points is impossible. We can further show that each caging loop consists of an alternative sequence of straight line segments in the grasping space and geodesics on the target surface.
Let ${S}$ be the target surface. Each caging loop $L$ consists of an alternative sequence of geodesic paths lying on ${S}$ and straight line segments in the grasping space $\mathbf{G}$, where a geodesic segment may degenerate into a single point.
In fact, the loop space $\mathbf{L}$ includes all geodesic loops constrained on the surface $S$ and thus cannot be empty, which can be easily verified from the Lusternick-Schnirelmann theorem [@Lusternik1934M].
${\mathbf{L}}$ is non-empty.
However, it is difficult to directly extract a caging loop without any further hint. Therefore, we consider a type of relaxed caging loops.
\[defn:relaxed\_loop\]Suppose the target object $S$ defines a grasping space $\mathbf{G}$. Let $p$ be a point in $\mathbf{G}$. A closed curve $L\in \mathbf{G}$ is called a [*$p$-based caging loop candidate*]{} if and only if $L$ is [ locally shortest everywhere]{} except at $p$. When $p$ is taken over all points in $\mathbf{G}$, all such loop candidates constitute a different [caging loop space]{}, denoted by $\widetilde{\mathbf{L}}$.
We immediately have the following property.
$\widetilde{\mathbf{L}}$ is a superset of ${\mathbf{L}}$.
[**Remark:**]{} Each loop $L\in\widetilde{\mathbf{L}}$ carries a base point $p$. If we eliminate those loops that are not locally shortest at the corresponding base point, then $\widetilde{\mathbf{L}}$ becomes ${\mathbf{L}}$. Therefore, the above property implies that we can select the desirable caging loop from $\widetilde{\mathbf{L}}$.
Methodology {#sec:methodology}
===========
Computing Loop Candidates
-------------------------
![\[fig:morse\_saddle\]An example of Morse-Smale saddle point of the $p$-based distance field restricted in a grasping space $\mathbf{G}$. ](morseSaddle.pdf){width="0.85\columnwidth"}
Let $p$ be a point in the grasping space $\mathbf{G}$. For each point $x$ in $\mathbf{G}$, we use $\mathbf{D}_p(x): G \rightarrow \mathbb{R}$ to denote the length of the shortest path connecting $p$ and $x$. $\mathbf{D}_p$ is called the distance field rooted at $p$. Note that the distance is measured in $\mathbf{G}$ rather than on the target surface.
Suppose that $L_p$ is a $p$-based loop in the caging loop space $\widetilde{\mathbf{L}}$. It is easy to know that there is a point $q\in L_p$ such that $q$ divides $L_p$ into two equal-length parts. Obviously, both the two sub-curves are locally shortest paths in the grasping space $\mathbf{G}$. In the following, we shall reveal the fact that there is a fundamental relationship between Morse theory and caging loops.
\[thm:saddle\]Suppose that the target object defines a grasping space $\mathbf{G}$. Let $\mathbf{D}_p$ be the distance field rooted at $p\in\mathbf{G}$. Each Morse saddle or maximal point of $\mathbf{D}_p$ is able to define a $p$-based caging loop.
[*Proof.*]{} Let $q$ be an Morse saddle (or maximal) point of $\mathbf{D}_p$. Since there exists a pair of shortest paths $\Pi_1, \Pi_2$ that go along opposite directions at $q$. By combining $\Pi_1, \Pi_2$, we get a $p$-based caging loop. Although the above discussion is assumed in the continuous setting, Morse-Smale theory is also well defined in the discrete setting; see more details in [@Ni-2004FairMF]. We can inherit the spirit in [@Ni-2004FairMF] and distinguish Morse saddle points, minimal points and maximal points by considering the relative magnitude at a voxel and its neighboring voxels. As Fig. \[fig:morse\_saddle\] shows, we label $p$’s neighbor with a “+” if the neighbor has a higher value and a “-” otherwise. It is easy to know that there are at most $2^6$ possible configurations. Fig. \[fig:morse\_saddle\] gives a typical situation of Morse saddle point, where two opposite neighbors are labeled with “-” while the other four neighbors are labeled with “+”. Similarly, a voxel is classified as a maximal point if all 6 neighbors are labeled with “-”.
Based on Theorem \[thm:saddle\], it is natural to devise a naïve algorithm (see Algorithm \[alg:naive\]) to build the $p$-based caging loop space $\widetilde{\mathbf{L}}$.
Initialize $\widetilde{\mathbf{L}}$ to be empty.\
Compute a sample set $P$ in the grasping space $\mathbf{G}$.\
Filtering Rules
---------------
However, $\widetilde{\mathbf{L}}$ is very large generally. We need to invent a handful of filtering rules to reduce the computational cost. First of all, the reduction of the grasping space $\mathbf{G}$ is much helpful to filter out redundant caging loops.
\[thm:convexhull\]Let ${H}$ be the convex hull of the target surface $S$. Any caging loop must lie between $S$ and $H$.
[Secondly]{}, Property \[property:three\] asserts that the base point $p$ can be constrained on the target surface $S$, which cannot cause missing any useful caging loop. In fact, the location of $p$ can be more restricted; See the following theorem.
\[thm:principal\]For a point $p$ on the target surface $S$, if both the principal curvatures are negative, $p$ cannot determine a caging loop.
[*Proof.*]{} Suppose $L_p$ is a caging loop. The sufficiently short segment of $L_p$ around $p$ can be viewed as the intersection between a normal plane at $p$ and the target surface $S$. Since both the principal curvatures at $p$ are negative, the loop can be shortened by moving $p$ toward $\mathbf{G}$ a little bit, leading to a contradiction.
[Finally]{}, even if the base point $p$ is given, there is no need to compute the entire distance field $\mathbf{D}_p$ since an overly long loop is no use for grasping. It is sufficient to limit the sweep process in an appropriate range comparable to the gripper size. In practice, it is reasonable to require that the total stretching length of the robot gripper (twice as long as the gripper finger), denoted by $2h$, should be larger than one half of the length of the caging loop. Therefore, during the computation of the $p$-based distance field, we can terminate the sweep process when the sweep radius amounts to $2h$ since at this moment, any $p$-based caging loop longer than $4h$ has been found.
Taking the above speedup techniques simultaneously into consideration, we give an advanced algorithm for computing caging loops; See Algorithm \[alg:advanced\] (the difference from Algorithm \[alg:naive\] is underlined).
Initialize $\widetilde{\mathbf{L}}$ to be empty.\
Compute the grasping space $\mathbf{G}$ .\
Implementation {#sec:implementation}
==============
In a real grasping scenario, the gripper thickness cannot be negligible - a gripper cannot stretch into small topological holes or gaps. A commonly used technique is to filter out those infeasible caging loops by checking interference. Rather than leave it to ex post interference analysis, in this paper, we take each gripper finger as a skeleton curve equipped with a sweep radius $r$. In implementation, we offset the target surface outward in a distance of $r$ and require any caging loop to be lying in the grasping space $\mathbf{G}_r$ separated by the $r$-offset surface $S_r$. Fig. \[fig:overview\] shows an implementation details of our approach.
{width="1.0\linewidth"}\
\
{width="1.0\linewidth"}\
Grasping Space $\mathbf{G}_r$ {#subsec:GraspingSpace}
-----------------------------
Given a scanned point cloud $\{(x_i,\mathbf{n}_i)\}$ of the target shape, we shall adapt the radial basis function (RBF) technique [@Carr-2001ReconstructionAR] to represent the $r$-offset surface $S_r$, which is central to define the grasping space $\mathbf{G}_r$. The general RBF with regard to $\{(x_i,\mathbf{n}_i)\}$ is defined as follows: $$f(x) = \sum_{i}^n w_i\phi(\|x-x_i\|)+P(x),$$ where $\phi(t) = t$ is the basis function used in our experiments, and the weighting coefficients $\{w_i\}$ and the low-degree (typically linear) polynomial $P(x)$ is undetermined. Taking $x_j$ into the RBF, we have $$\sum_{i}^n w_i\phi(\|x_j-x_i\|)+(1,x_j)^\text{T}\mathbf{c}=f(x_j) = 0.$$ Furthermore, considering that the point $x_j+r\mathbf{n}_j$ that lies on the $r$-offset surface $S_r$,, we have $$\sum_{i}^n w_i\phi(\|x_j+r\mathbf{n}_j-x_i\|)+(1,x_j)^\text{T}\mathbf{c}=f(x_j+r\mathbf{n}_j) = r,$$ where $\mathbf{c}=(c_0,c_1,c_2,c_3)$ is unknown. At the same time, in the RBF based approach, the four side conditions are $$\sum_{i}^n w_i = \sum_{i}^n w_i x_i = \sum_{i}^n w_i y_i = \sum_{i}^n w_i z_i = 0.$$ The above formulation can be finally transformed into a linear system from which we can immediately compute $\{w_i\}$ and $P(x)$. To this end, we find an implicit surface $f(x)=r$ to represent the $r$-offset surface $S_r$. If the shape embedding space is discretized into voxels, it is very easy to identify outside voxels that meet $f(x)\geq r$, which can be viewed as a discrete representation of the grasping space $\mathbf{G}_r$.
Gripper Configuration {#subsec:GripperConfiguration}
---------------------
Upon obtaining a desirable caging loop $L$, we need to determine the origin position $o$ of the gripper, as well as an orthogonal frame to set the gripper orientation, which facilitates a real grasp. Suppose that $L$ is represented by a point sequence $\{p_1,p_2,\cdots,p_n\}$. Imagine that there is an inward cone rooted at $p\in\{p_1,p_2,\cdots,p_n\}$ and the center line of the cone coincides with the normal vector at $p$; See Fig. \[fig:gripperConfiguration\]. (If $p$ is not located on $S_r$, the normal vector is discussed later.) We further define $\theta_p$ to be the maximum open angle under the condition that no penetration occurs between the inward cone and the target shape. The origin point $o$ is then selected from $\{p_1,p_2,\cdots,p_n\}$ so as to maximize the opening angle. Let $c=\frac{p_1+p_2+\cdots+p_n}{n}$ be the center point of $L$. We then define the first direction $Dir_1$ as follows: $$Dir_1 = \frac{o-c}{\|o-c\|},$$ which roughly means the forward direction of the gripper. Considering that the loop $L$ is roughly a planar curve, we can fit $L$ using a plane $
\mathbf{n}\cdot x = b,
$ where $\mathbf{n}$ is a unit vector. Finally, we define $Dir_2$ as follows: $$Dir_2 = Dir_1 \times \mathbf{n}.$$ If the triple $(o, Dir_1, Dir_2)$ is able to define a valid grasping configuration (no global interference happens), we use it to guide the orientation of the gripper and launch a real grasp. In our experiments, the gripper will spread its fingers and move to the point $o$ first. It then wraps the target object tightly with the hint of $Dir_1$ and $Dir_2$.
In Fig. \[fig:gripperConfiguration\], we show some examples on how to find a valid grasping configuration assuming that a desirable caging loop has been found. The key step is to check global interference in the simulation environment of OpenRAVE. It can be observed that our approach can report different gasping configurations on the same model with different sizes.
[**Remark:** ]{}If $p$ is not located on $S_r$, $Dir_1$ is given by $\mathbf{t}_p\times \mathbf{n}$, where $\mathbf{t}_p$ is the tangent direction of the loop $L$ at $p$ and $\mathbf{n}$ is the normal to the fitting plane of $L$.
![\[fig:gripperConfiguration\] Inferring the gripper configuration based on a caging loop. In order to define a valid caging configuration, we check global interference in the simulation environment of OpenRAVE. Row 1: Large-size Kitten; Row 2: Small-size Kitten; Row 3: Large-size Yoga; Row 4: Small-size Yoga. ](gripperConfigurationResult.jpg){width="0.95\columnwidth"}
Experimental Results {#sec:experimental}
====================
We conducted both simulation and mechanical experiments to validate our approach. [In this section, we first test the effectiveness of our algorithm on a variety of complex objects. We then show that our algorithm can be applied to 3D shapes with various levels of noise and geometric features. After that, we demonstrate the superior caging ability of our approach on real grasping scenarios (the digital models of the target objects are unknown in advance). Finally, we give the timing statistics of the main computational steps.]{}
Test on High-genus Models in Various Sizes
------------------------------------------
![\[fig:adaptive\] Caging loops generated on the Fertility and Yoga models with various sizes. (a) Caging loops (yellow) produced by the method in [@Zarubin-2013CagingCO]. (b) Caging loops (red) computed by our method. ](adaptive.png "fig:"){width="0.95\columnwidth"}\
{width="0.95\linewidth"}
There are a number of research works on caging a 3D shape, which largely fall into one of the following three categories: topology guided [@Pokorny-2013GraspingOW], geometry guided [@Zarubin-2013CagingCO] and topology & geometry guided [@Kwok-2016RopeCA; @Varava-2016CagingGO]. Existing approaches, whether topology guided or geometry guided, consider only those loops constrained on the target surface. Therefore, it is hard for them to deal with small topological holes or small gaps. As shown by the comparison in Fig. \[fig:adaptive\], our algorithm handles 3D shapes with small topological holes and is aware of the relative size between the gripper and the shape. More caging loop examples can be found in Fig. \[fig:MoreGraspingLoops\].
Test on Models with Various Levels of Noise and Geometric Feature
-----------------------------------------------------------------
![\[fig:venusbodynoise\] [Top row: Caging loops on the Venus model with varying levels of Gaussian noise (relative to the whole model size): 0.01, 0.03, 0.05, 0.07. Bottom row: Caging loops computed by our method on the Pillar models are insensitive to varying levels of geometric details.]{} ](venusbodynoise.png "fig:"){width="\columnwidth"}\
![\[fig:venusbodynoise\] [Top row: Caging loops on the Venus model with varying levels of Gaussian noise (relative to the whole model size): 0.01, 0.03, 0.05, 0.07. Bottom row: Caging loops computed by our method on the Pillar models are insensitive to varying levels of geometric details.]{} ](FeatureScale.png "fig:"){width="\columnwidth"}
In real grasping scenarios, the target object is often scanned into a point cloud with noise. Therefore, a key criteria to evaluate a caging algorithm is whether it is robust to geometric noise or variations. In Fig. \[fig:venusbodynoise\], the top row shows a group of caging loops on the Venus models with various levels of noise, while bottom row shows a group of caging loops on the Pillar models with various levels of geometric details. Both of them exhibit the robustness of our algorithm against geometric noise and details.
Test on Real Objects
--------------------
![\[fig:realDataResult\] Grasping household objects by our system. From left to right: Scanned point clouds, offset surfaces and caging loops, simulation results, progressive demonstration of real grasping. ](RealDataResult.pdf "fig:"){width="1.0\columnwidth"}\
In order to validate our approach on real data, we build a platform with a 7-DoF Barrett WAM arm, a Barrett BH8-282 three-finger gripper and two Xtion Pro depth cameras. For each object shown in Fig. \[fig:realDataResult\], we keep the depth cameras unchanged while rotating the target shape repeatedly for $10$ times. In this way, we recorded of the grasp success rates. A grasp is regarded to be successful if the target shape does not escape from the gripper during a large-scale movement that is about $10$cm off the ground. We report the success rate of grasping in Table \[tab:GraspSuccessfulRate\]. It shows that our approach can synthesize reliable caging grasps, as compared to traditional approaches whose success rate is generally about $85\%$.
\[tab:GraspSuccessfulRate\]
Performance
-----------
In order to accomplish a grasping task, we have to perform a sequence of shape analysis operations and then send a grasp instruction to the robot. Recall that we define a grasping space and search the best caging loop in that space. In our implementation, we discreticize the bounding box enclosing the target shape into $50 \times 50 \times 50$ voxels and label each voxel between the $r$-offset surface $S_r$ and its convex hull $H_r$ with “1”. After that, we extract a set of $500$ uniformly distributed sample points from $S_r$ and eliminate those sample points that do not help determine a caging loop at all (see Theorem \[thm:principal\]). Finally, the loop candidate pool is generated based on Morse theory. Among these steps, the most time-consuming steps include the grasping space computation, a distance field generation for a collection of base points and topological analysis based on Morse theory. The mesh models shown in this paper are discretized into 2K vertices, and the average computation time for each model is about 1.5 seconds. It can be seen that our algorithm runs very fast at this level of resolution. Therefore, if we simultaneously execute the computation task and the move of gripper, it does not introduce a noticeable delay.
Conclusion and Discussion
=========================
In this work, we propose to synthesize feasible caging grasps in the shape embedding space of the target object. Our caging loops are able to encompass multiple small topological handles and concave regions, which are relatively too small to be grasped, through decoupling their computation from surface geometry of the target object. This also facilitates grasp synthesis for unknown objects which are acquired and reconstructed on-the-fly. Extensive experimental results exhibit that our approach can deal with real objects with complex surface geometry and topology, being aware of the relative size between objects and gripper.
Our current solution has several limitations. First, the method used for measuring the physical feasibility is merely a preliminary solution which can definitely be replaced by other alternatives. Our core method for caging loop computation, however, ensures the candidate loops are mostly feasible with respect to the gripper size. Second, the caging loops computed by our method mainly reflect the geometric aspect of graspability and do not account for the high level information of semantics or functionality. For example, an object can be grasped in different ways for different purposes. Synthesizing function related grasps is an interesting venue for future study.
[^1]: $^{1}$Jian Liu, Shiqing Xin, Zengfu Gao, Changhe Tu and Baoquan Chen are with School of Computer, Shandong University, 266237 Qingdao, P. R. China. $^{2}$Kai Xu is with School of Computer, National University of Defense Technology, 410073 Changsha, P. R. China. This work was supported by National 973 Program (2015CB352501), and NSFC (61332015, 61772318, 61572507, 61532003, 61622212, 61772016).
| ArXiv |
---
abstract: |
We study $\gamma\gamma$ scattering in noncommutative QED (NCQED) where the gauge field has Yang-Mills type coupling, giving new contributions to the scattering process and making it possible for it to occur at tree level. The process takes place at one loop level in the Standard Model (SM) and could be an important signal for physics beyond SM. But it is found that the Standard Model contribution far exceeds the tree level contribution of the noncommutative case.\
\
[**Keywords**]{}: Noncommutative, gamma-gamma scattering\
\
[**PACS**]{}: 12.60.-i, 13.40.-f
author:
- |
Namit Mahajan[^1]\
[*Department of Physics and Astrophysics,*]{}\
[*University of Delhi, Delhi-110 007, India.*]{}
title: 'Noncommutative QED and $\gamma\gamma$ scattering'
---
=cmr10
\*[Introduction]{} Noncommutativity of a pair of conjugate variables forms the central theme of quantum mechanics in terms of the Uncertainty Principle. We are quite familiar with the noncommutativity of rotations in ordinary Euclidean space. The idea of noncommutative (NC) space-time can be traced back to the work of Snyder [@snyder]. But more recently, string theory arguments have motivated an extensive study of Quantum Field Theory (QFT) on NC spaces [@douglas]. The noncommutativity of space-time is realised by the coordinate operators, $x_{\mu}$, satisfying $$[x_{\mu},x_{\nu}] = \iota\Theta_{\mu\nu}$$ with $\Theta_{\mu\nu}
= \theta \epsilon_{\mu\nu}$. $\theta$ is the noncommutativity parameter with dimensions $(mass)^{-2}$ and $\epsilon
_{\mu\nu}$ is a dimensionless antisymmetric matrix with elements ${\mathcal O}
(1)$. The field theories formulated on such spaces are non-local and violate Lorentz symmetry. The deviation from the standard theory manifests as violation of Lorentz invariance. We can still expect manifest Lorentz invariance for energies satisfying $E^2\theta << 1$. In the limit $\theta \rightarrow 0$, one expects to recover the standard theory. This is true for the theory at classical level. But at the quantum level, the limit $\theta \rightarrow 0$ does not lead to the commutative theory [@armoni]. The theory of electrons in a strong magnetic field, projected to the lowest Landau level, is a classic example of NC field theory.\
Various attempts, both theoretical and phenomenological, have been made to study QFT on NC spaces. The study of perturbative behaviour and divergence structure [@sheraz], $C$, $P$ and $T$ properties and renormalisability [@shiekh] of such theories has been undertaken. It has been shown that quantum theories with time-like noncommuatativities are not unitary [@gomis]. We shall therefore restrict our discussion to the theories with space-like noncomutativities, although it has been shown that light-like noncommutative theories are also free of pathologies [@gomis1]. To this end, the coordinate commutator simply reads $$[x_i,x_j] = \iota\theta\epsilon_{ij}$$
There have been attempts to write down particle physics models, in particular SM, on such NC spaces [@connes]. From a phenomenological point of view, various scattering processes have been analysed [@pheno; @hewett] along with the attempts to calculate additional contributions to the precisely measured quantities like anomalous magnetic moment [@sh1] and Lamb shift [@sh2] in the noncommutative version of QED.
\*[$\gamma\gamma$ scattering in NCQED]{} Consider NCQED i.e. a $U(1)$ noncommutative theory coupled to fermions. The noncommutative version of a theory can be written by replacing the field products by what is called the [*’star product’*]{}. The star ($\ast$) product for any two functions is given by $$f(x)\ast g(x) = f(x)e^{\frac{\iota}{2}\overleftarrow{\partial_{\alpha}}
\Theta^{\alpha\beta}
\overrightarrow{\partial_{\beta}}}g(x)$$
The NCQED action, using the above line of reasoning, is $$S_{NCQED} = \int d^Dx \Bigg( -\frac{1}{4g^2}F^{\mu\nu}(x)\ast F_{\mu\nu}(x)
+ \iota\bar{\psi}(x)\gamma^{\mu}\ast D_{\mu}\psi(x)
- m\bar{\psi}(x)\ast\psi(x)\Bigg)$$ where $g$ is the coupling and $$F_{\mu\nu} = \partial_{\mu}A_{\nu}(x) - \partial_{\nu}A_{\mu}(x)
+ \iota g[A_{\mu}(x),A_{\nu}(x)]_{\ast}$$ The covariant derivative is given by $$D_{\mu}\psi(x) = \partial_{\mu}\psi(x) + \iota gA_{\mu}(x)\ast\psi(x)$$ The action is invariant under the noncommutative $U(1)$ transformations obtained by replacing all the products in the standard transformations by the corresponding star products.
The noncommutativity is encoded in the star product and from the above expressions it is quite evident that the field strength, even in the case of $U(1)$, theory is nonlinear in gauge field and it is precisely this nonlinearity that gives rise to additional vertices for the gauge field. It is now a straight forward task to derive the Feynman rules from the above action [@sh1], Arfaei et.al [@pheno]. It is found that apart from generating the three and four point vertices for the gauge field self interaction, each interaction vertex picks up a momentum dependent phase factor, whose argument typically has the structure $\frac{\iota}{2} p \wedge k$. The $\wedge$ product, in general, is defined as $$p \wedge k = p_{\mu} \Theta^{\mu\nu} k_{\nu}$$ In the case of theories with only space-like noncommutativities, only the space-space elements contribute and using Eq.(2) it simply reduces to the usual vector cross-product of the two three momenta i.e. $$p \wedge k = \vec{p}\times \vec{k}$$ The process, $\gamma\gamma \longrightarrow \gamma\gamma$ takes place at the one loop level in standard QED as well as SM and thus is quite suppressed. But the presence of Yang-Mills type coupling for the photon field in NCQED enables the process to take place at the tree level. This makes the above process a plausible candidate to look for physics beyond SM at the tree level.\
The diagrams contributing to the scattering process are
(155,120)(-5.0,-20) (0,45)(45,0)[3]{}[6]{} (15,50)\[c\][$k_1$]{} (20,40)(30,30) (0,-45)(45,0)[3]{}[6]{} (15,-50)\[c\][$k_2$]{} (20,-40)(30,-30) (45,0)(90,0)[3]{}[5]{} (90,0)(135,45)[3]{}[6]{} (120,50)\[c\][$p_1$]{} (105,30)(115,40) (90,0)(135,-45)[3]{}[6]{} (120,-50)\[c\][$p_2$]{} (105,-30)(115,-40)
1.5cm
(155,120)(-5.0,-20) (0,45)(45,0)[3]{}[6]{} (15,50)\[c\][$k_1$]{} (20,40)(30,30) (0,-45)(45,0)[3]{}[6]{} (15,-50)\[c\][$p_1$]{} (30,-30)(20,-40) (45,0)(90,0)[3]{}[5]{} (90,0)(135,45)[3]{}[6]{} (120,50)\[c\][$k_2$]{} (115,40)(105,30) (90,0)(135,-45)[3]{}[6]{} (120,-50)\[c\][$p_2$]{} (105,-30)(115,-40)
1.5cm
(155,120)(-5.0,-20) (0,45)(45,0)[3]{}[6]{} (15,50)\[c\][$k_1$]{} (20,40)(30,30) (0,-45)(45,0)[3]{}[6]{} (15,-50)\[c\][$p_2$]{} (30,-30)(20,-40) (45,0)(90,0)[3]{}[5]{} (90,0)(135,45)[3]{}[6]{} (120,50)\[c\][$p_1$]{} (105,30)(115,40) (90,0)(135,-45)[3]{}[6]{} (120,-50)\[c\][$k_2$]{} (115,-40)(105,-30)
1.5cm
(155,120)(-5.0,-20) (0,45)(45,0)[3]{}[6]{} (15,50)\[c\][$k_1$]{} (20,40)(30,30) (0,-45)(45,0)[3]{}[6]{} (15,-50)\[c\][$k_2$]{} (20,-40)(30,-30) (45,0)(90,45)[3]{}[6]{} (75,50)\[c\][$p_1$]{} (60,30)(70,40) (45,0)(90,-45)[3]{}[6]{} (75,-50)\[c\][$p_2$]{} (60,-30)(70,-40)
Due to the noncommutative nature of the coordinates, the theory is not Lorentz invariant and the results are frame dependent. In writing down the amplitudes corresponding to each of the above diagrams, we assume that $\theta \ll 1$ and make the substitution $\sin(a\theta) \longrightarrow a\theta$, where $a$ is used to generically denote the quantity appearing in the argument of the sine function multiplied to $\theta$.\
Choosing to work in the center of mass frame, we find that the s-channel diagram vanishes. The square of the matrix amplitude reads $$|{\mathcal{M}}_{NC}|^2 = \left(\frac{e\theta}{16}\right)^4~[100s^4 + 96t^4 +
204st^3 + 360s^2t^2 + 250s^3t]$$ and the total (unpolarised) cross section is $$\sigma_{NC} = (1.5 \times 10^{-3}) \alpha_{em}^2s^3\theta^4$$ which for $\sqrt{s}\sim$ TeV and $\theta \leq (10^4~GeV)^{-2}$ as in [@sh2] gives $\sigma_{NC}\sim 10^{-10}$ fb, to be compared with the SM contribution, $\sigma_{SM}\sim$ fb at the same center of mass energy [@jikia]. It is found that at low energies the fermion contribution dominates the SM cross-section while at higher $\sqrt{s}$ ($>$ 100 GeV), it is the W contribution that becomes important. The SM contribution gradually decreases as $\sqrt{s}$ crosses the 500 GeV range. Although, in contrast to SM, the NC cross-section increases monotonously with $\sqrt{s}$, it can never catch up with the SM cross-section for the same energy.
\*[Conclusions]{} In this article we have computed the NCQED contribution to the $\gamma\gamma$ scattering and found that even though in this case the process occurs at the tree level as opposed to SM, where it takes place at the one loop level, the SM contribution far exceeds the NC contribution. It is clear that the NCQED contribution will start showing up only when $\theta$ is much larger than the value used here.\
The process has been studied in context of NCQED by Hewett et.al [@hewett] but the authors argue that inspired by recent theories of extra dimensions [@nima], where the effective scale of gravity is ${\mathcal{O}} \sim$ TeV as opposed to the Planck scale, the scale of noncommutativity can, too, be chosen to be around TeV. But a more physical approach would be to use the value of $\theta$, as obtained from studies like Lamb shift [@sh2], to calculate the new contributions. Also the authors have taken into account time-like noncommutativity that may lead to possible non-unitary S-matrix elements. Even for $\theta~\sim~(TeV)^{-2}$ as taken by the authors, the SM contribution is still overwhelmingly large. Thus with the present day and near future experiments, it doesn’t seem possible to get a signal of NCQED from $\gamma\gamma$ scattering.
\*[Acknowledgements]{} The author would like to thank University Grants Commission, India for fellowship.
[99]{} H. S. Snyder, Phys. Rev. [**71**]{}, 38 (1947). For a recent review see M. R. Douglas and N. A. Nekrasov, hep-th/0106048 and references therein. A. Armoni, Nucl. Phys. [**B 593**]{}, 229 (2001). S. Minwalla, M. V. Raamsdonk and N. Sieberg, hep-th/9912072; T. Krajewski and R. Wulkenhaar, Int. J. Mod. Phys. [**A15**]{}, 1011 (2000); M.Hayakawa, Phys. Lett. [**B 478**]{}, 394 (2000); A. Micu and M. M. Shiekh-Jabbari, hep-th/0008057; C. E. Carlson, C. D. Carone and R. F. Lebed, Phys. Lett. [**B 518**]{}, 201 (2001). C. P. Martin and D. Sanchez-Ruiz, Phys. Rev. Lett. [**83**]{}, 476 (1999); M. M. Shiekh-Jabbari, hep-th/9903107; M. M. Shiekh-Jabbari, Phys. Rev. Lett. [**84**]{}, 5265 (2000). J. Gomis and T. Mehen, hep-th/0005129. O. Aharony, J. Gomis and T. Mehen, hep-th/0006236. A. Connes and J. Lott, Nucl. Phys. Proc. Suppl. [**18B**]{}, 29 (1991); M. Chaichian, P. Presnajder, M. M. Shiekh-Jabbari and A. Tureanu, hep-th/0107055. N. Chair and M. M. Shiekh-Jabbari, hep-th/0009037; P. Mathews, Phys. Rev. [**D 63**]{}, 075007 (2001); S. Baek, D. K. Ghosh, X. G. He and W-Y. P. Hwang, Phys. Rev. [**64**]{}, 056001 (2001); H. Arfaei and M. H. Yavartanoo, hep-th/0010244; S. Godfrey and M. A. Doncheski, hep-ph/0108268. J. L. Hewett, F. J. Petriello and T. G. Rizzo, hep-ph/0010354.
I. F. Riad and M. M. Shiekh-Jabbari, hep-th/0008132. N. Chair, M. M. Shiekh-Jabbari and A. Tureanu, Phys. Rev. Lett. [**86**]{}, 2716 (2001). N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. [**B 429**]{}, 263 (1998); I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phy. Lett. [**B 436**]{}, 257 (1998); L. Randall and R. Sundrum, Phys. Rev. Lett [**83**]{}, 3370 (1999); Phys. Rev. Lett. [**83**]{}, 4690 (1999). G. Jikia and A. Tkabladze, Phys. Lett. [**B 323**]{} 453 (1994).
[^1]: E–mail : [email protected], [email protected]
| ArXiv |
---
abstract: 'We study the convexity of mutual information along the evolution of the heat equation. We prove that if the initial distribution is log-concave, then mutual information is always a convex function of time. We also prove that if the initial distribution is either bounded, or has finite fourth moment and Fisher information, then mutual information is eventually convex, i.e., convex for all large time. Finally, we provide counterexamples to show that mutual information can be nonconvex at small time.'
author:
-
bibliography:
- 'mi\_arxiv\_v2.bib'
title: |
Convexity of mutual information\
along the heat flow
---
Introduction
============
The heat equation plays a fundamental role in many fields. In thermodynamics, it describes the diffusion of heat in a body due to temperature differences. In probability theory, it describes the evolution of the Brownian motion. In information theory, it describes the additive white Gaussian noise channel, which is one of the most important communication channels. In general, the heat equation can be used to model the transport of any quantity in a medium via a diffusion process. It also forms the basis for more general stochastic processes, such as the Ornstein-Uhlenbeck process or the Fokker-Planck process. Therefore, the heat equation has found applications in diverse scientific disciplines—from explaining the evolution of zebra stripes [@Tur52] to modeling stock prices via the Black-Scholes formula [@BlaSch73]. We are interested in the heat flow, which is the flow of the heat equation in the space of random variables.
The properties of the heat flow are closely linked to entropy. Indeed, one important interpretation of the heat flow is as the flow that increases entropy as fast as possible. More precisely, heat flow is the gradient flow (i.e., the steepest descent flow) of negative entropy in the space of probability distributions with the Wasserstein metric structure [@JKO98]. In this paper we will not need this result, but only use a certain key identity in our calculation. Nevertheless, this relation suggests an intricate connection between entropy and the heat flow.
The behavior of entropy along the heat flow has been long studied. The gradient flow interpretation above shows that entropy is increasing along the heat flow. In particular, De Bruijn’s identity [@Sta59] states that the time derivative of entropy along the heat flow is given by the Fisher information, which is always positive. Moreover, entropy is a concave function of time along the heat flow. This is because the second time derivative of entropy along the heat flow is the negative of the second-order Fisher information [@McKean66; @Tos99; @Vil00]; the latter identity also implies the concavity of entropy power along the heat flow [@Cos85; @Dembo89; @Dembo91]. It is further conjectured that the higher derivatives of entropy along the heat flow have alternating signs [@McKean66; @Vil02; @Che15]. In one dimension, this has been verified up to the fourth derivative [@Che15]; in multi dimension, this is true for the third derivative when the initial distribution is log-concave [@Tos15].
On the other hand, the behavior of mutual information along the heat flow has been less explored. Clearly mutual information is decreasing along the heat flow by the data processing inequality, since the heat flow is a Markov chain. De Bruijn’s identity implies that the time derivative of mutual information along the heat flow is the negative of the mutual Fisher information; the latter is proportional to the minimum mean square error (mmse) of estimating the initial from the final distribution, thus recovering the I-MMSE relation for the additive Gaussian channel [@GuoEtAl05]. Similarly, the second time derivative of mutual information along the heat flow is the mutual version of the second-order Fisher information; unfortunately, it does not always have a definite sign.
In this paper we study the convexity of mutual information along the heat flow. This amounts to determining when the mutual second-order Fisher information is positive along the heat flow. We show that in general, the mutual second-order Fisher information is positive whenever the final distribution is log-concave. Since the heat flow preserves log-concavity, this implies our first main result: If the initial distribution is log-concave, then mutual information is always convex along the heat flow. In some cases, for example when the initial distribution is bounded, the heat flow implies eventual log-concavity, which means the final distribution eventually becomes log-concave; this implies mutual information is eventually convex along the heat flow for these cases. Furthermore, we prove that in general, regardless of log-concavity, mutual information is eventually convex along the heat flow whenever the initial distribution has finite fourth moment and Fisher information.
Unlike entropy, however, we show that mutual information can be nonconvex along the heat flow. We provide explicit counterexamples, namely mixtures of point masses and mixtures of Gaussians, for which mutual information along the heat flow is nonconvex at small time; furthermore, by scaling we can arrange the region of nonconvexity to engulf any finite time. We elaborate on these results below.
Background and problem setup
============================
The heat flow
-------------
The heat equation in ${\mathbb{R}}^n$ is the partial differential equation: $${\frac{\partial \rho}{\partial t}} = \frac{1}{2} \Delta \rho$$ where $\rho = \rho(x,t)$ for $x \in {\mathbb{R}}^n$, $t \ge 0$, and $\Delta = \sum_{i=1}^n {\frac{\partial ^2}{\partial x_i^2}}$ is the Laplacian operator. This equation conserves mass, so if $\rho_0 = \rho(\cdot,0)$ is a probability distribution, then so is $\rho_t = \rho(\cdot,t)$ for all $t > 0$. The heat equation admits a closed-form solution via convolution: $$\rho_t = \rho_0 \ast \gamma_t$$ where $\gamma_t(x) = (2\pi t)^{-\frac{n}{2}} e^{-\frac{\|x\|^2}{2t}}$ is the heat kernel at time $t$. Probabilistically, if $X_0 \sim \rho_0$ is a random variable in ${\mathbb{R}}^n$, then $X_t \sim \rho_t$ that evolves following the heat equation is given by $$X_t = X_0 + \sqrt{t} Z$$ where $Z \sim {\mathcal{N}}(0,I)$ is the standard Gaussian random variable in ${\mathbb{R}}^n$ independent of $X_0$. We call this the heat flow. (Note that the true solution to the heat equation is the Brownian motion, but at each time $t$ it has the same distribution as $X_t$ above.) Observe that even when $X_0 \sim \rho_0$ has a singular density, $X_t \sim \rho_t$ has a smooth positive density for all $t > 0$.
If $X_0 \sim \delta_{a}$ is a point mass at some $a \in {\mathbb{R}}^n$, then $X_t \sim {\mathcal{N}}(a, tI)$ is Gaussian with mean $a$ and covariance $tI$.
If $X_0 \sim {\mathcal{N}}(\mu,\Sigma)$ is Gaussian, then $X_t \sim {\mathcal{N}}(\mu,\Sigma+tI)$ is also Gaussian with the same mean and increasing covariance.
If $X_0 \sim \sum_{i=1}^k p_i \delta_{a_i}$ is a mixture of point masses, then $X_t \sim \sum_{i=1}^k p_i {\mathcal{N}}(a_i,tI)$ is a mixture of Gaussians with the same covariance $tI$.
If $X_0 \sim \sum_{i=1}^k p_i {\mathcal{N}}(a_i, \Sigma_i)$ is a mixture of Gaussians, then $X_t \sim \sum_{i=1}^k p_i {\mathcal{N}}(a_i,\Sigma_i+tI)$ is also a mixture of Gaussians with the same means and increasing covariance.
Entropy and Fisher information
------------------------------
Let $X$ be a random variable in ${\mathbb{R}}^n$ with a smooth positive density $\rho$.
The (differential) [*entropy*]{} of $X \sim \rho$ is $$H(X) = -\int_{{\mathbb{R}}^n} \rho(x) \log \rho(x) \, dx.$$
The [*Fisher information*]{} of $X \sim \rho$ is $$J(X) = \int_{{\mathbb{R}}^n} \rho(x) \|\nabla \log \rho(x)\|^2 \, dx.$$
The [*second-order Fisher information*]{} of $X \sim \rho$ is $$K(X) = \int_{{\mathbb{R}}^n} \rho(x) \|\nabla^2 \log \rho(x)\|_{{\mathrm{HS}}}^2 \, dx.$$ Here $\|A\|_{{\mathrm{HS}}}^2 = \sum_{i,j=1}^n A_{ij}^2 = \sum_{i=1}^n \lambda_i(A)^2$ is the Hilbert-Schmidt (or Frobenius) norm of a symmetric matrix $A = (A_{ij}) \in {\mathbb{R}}^{n \times n}$ with eigenvalues $\lambda_i(A) \in {\mathbb{R}}$.
In general we have the inequality $$\begin{aligned}
\label{Eq:KJ}
K(X) \ge \frac{J(X)^2}{n}\end{aligned}$$ which is equivalent to the entropy power inequality [@Cos85; @Dembo89; @Dembo91; @Vil00].
If $X \sim {\mathcal{N}}(\mu,\Sigma)$ is Gaussian, then $$\begin{aligned}
H(X) &= \frac{1}{2} \log \det (2 \pi e \Sigma) = \frac{1}{2} \sum_{i=1}^n \log (2 \pi e \lambda_i) \\
J(X) &= \operatorname{Tr}(\Sigma^{-1}) = \sum_{i=1}^n \frac{1}{\lambda_i} \\
K(X) &= \|\Sigma^{-1}\|^2_{{\mathrm{HS}}} = \sum_{i=1}^n \frac{1}{\lambda_i^2}\end{aligned}$$ where $\lambda_1,\dots,\lambda_n > 0$ are the eigenvalues of $\Sigma \succ 0$.
Our interest in the first and second-order Fisher information is because they are the first and second derivatives of entropy along the heat flow.
\[Lem:DerEnt\] Along the heat flow $X_t = X_0 + \sqrt{t} Z$, $$\begin{aligned}
\frac{d}{dt} H(X_t) &= \frac{1}{2} J(X_t) \\
\frac{d^2}{dt^2} H(X_t) &= -\frac{1}{2} K(X_t).\end{aligned}$$
Note that since $J(X_t) \ge 0$, the first derivative of entropy is positive, which means entropy is increasing along the heat flow. Similarly, since $K(X_t) \ge 0$, the second derivative of entropy is negative, which means entropy is a concave function along the heat flow.
Mutual information and mutual Fisher information {#Sec:Mut}
------------------------------------------------
Let $(X,Y)$ be a joint random variable in ${\mathbb{R}}^n \times {\mathbb{R}}^n$ with a joint density $\rho_{XY}$, which we can factorize into a product of marginal and conditional densities: $$\rho_{XY}(x,y) = \rho_X(x) \, \rho_{Y|X}(y\,|\,x) = \rho_Y(y) \, \rho_{X|Y}(x\,|\,y).$$ We assume $\rho_Y$ and $\rho_{Y|X}(\cdot\,|\,x)$ are smooth and positive for all $x \in {\mathbb{R}}^n$.
The [*mutual information*]{} of $(X,Y)$ is $$I(X;Y) = H(Y) - H(Y\,|\,X)$$ where $H(Y\,|\,X) = \int \rho_X(x) H(\rho_{Y|X}(\cdot\,|\,x))\,dx$ is the expected entropy of the conditional densities.
The [*mutual Fisher information*]{} of $(X,Y)$ is $$J(X;Y) = J(Y\,|\,X) - J(Y)$$ where $J(Y\,|\,X) = \int \rho_X(x) J(\rho_{Y|X}(\cdot\,|\,x))\,dx$ is the expected Fisher information of the conditional densities.
The [*mutual second-order Fisher information*]{} of $(X,Y)$ is $$K(X;Y) = K(Y\,|\,X) - K(Y)$$ where $K(Y\,|\,X) = \int \rho_X(x) K(\rho_{Y|X}(\cdot\,|\,x))\,dx$ is the expected second-order Fisher information of the conditional densities.
Mutual information is symmetric: $I(X;Y) = I(Y;X)$. However, mutual first and second-order Fisher information are not symmetric: in general, $J(X;Y) \neq J(Y;X)$ and $K(X;Y) \neq K(Y;X)$.
The mutual Fisher information $J(X;Y)$ can be shown to be equal to the [*backward (statistical) Fisher information*]{} $\Phi(X\,|\,Y)$, which is manifestly positive. The mutual second-order Fisher information $K(X;Y)$, on the other hand, is not always positive, but it can be represented in terms of the [*backward (statistical) second-order Fisher information*]{} $\Psi(X\,|\,Y)$; see Appendix \[App:ProofKJMut\] for detail.
Analogous to the basic (non-mutual) inequality , we have the following result. Recall that a smooth probability distribution $\rho$ in ${\mathbb{R}}^n$ is [*$\alpha$-log-semiconcave*]{} for some $\alpha \in {\mathbb{R}}$ if $$-\nabla^2 \log \rho(x) \succeq \alpha I~~~~\forall \, x \in {\mathbb{R}}^n.$$ When $\alpha \ge 0$, we say $\rho$ is log-concave.
\[Lem:KJMut\] If $Y \sim \rho_Y$ is $\alpha$-log-semiconcave for some $\alpha \in {\mathbb{R}}$, then $$K(X;Y) \ge \frac{J(X;Y)^2}{n} + 2\alpha J(X;Y).$$
In particular, if $\rho_Y$ is log-concave, then $K(X;Y) \ge 0$.
Mutual information along the heat flow {#Sec:MutHeat}
--------------------------------------
Now consider when $Y = X_t$ is the heat flow from $X = X_0$.
By the linearity of the channel, the identities for the derivatives of entropy in Lemma \[Lem:DerEnt\] imply the following identities for the derivatives of mutual information along the heat flow.
\[Lem:DerMut\] Along the heat flow $X_t = X_0 + \sqrt{t} Z$, $$\begin{aligned}
\frac{d}{dt} I(X_0;X_t) &= -\frac{1}{2} J(X_0;X_t) \\
\frac{d^2}{dt^2} I(X_0;X_t) &= \frac{1}{2} K(X_0;X_t).\end{aligned}$$
Since $J(X_0;X_t) = \Phi(X_0\,|\,X_t) \ge 0$, the first identity above shows that mutual information is decreasing along the heat flow. In fact along the heat flow $\Phi(X_0\,|\,X_t) = \frac{1}{t^2} \operatorname{Var}(X_0\,|\,X_t)$ is proportional to the mmse of estimating $X_0$ from $X_t$, thus recovering the I-MMSE relation for Gaussian channel [@GuoEtAl05; @WibisonoJL17]. From the second identity above, we see that the convexity of mutual information along the heat flow is equivalent to the positivity of $K(X_0;X_t)$, for which Lemma \[Lem:KJMut\] will be useful.
Finally, we note that since $X_t \,|\, X_0$ is Gaussian, the various mutual quantities in Lemma \[Lem:DerMut\] are simply comparisons against a baseline Gaussian: $I(X_0;X_t) = H(X_t) - \frac{n}{2} \log (2 \pi t e)$, $$J(X_0;X_t) = \frac{n}{t} - J(X_t), ~~\text{ and }~~
K(X_0;X_t) = \frac{n}{t^2} - K(X_t).$$ In the opposite order, mutual information stays the same: $I(X_t;X_0) = I(X_0;X_t)$. On the other hand, the mutual first and second-order Fisher information can be computed explicitly and do not depend on $X_t$: $$J(X_t;X_0) = \frac{n}{t} ~~~~\text{ and }~~~~ K(X_t;X_0) = \frac{n}{t^2} + \frac{2}{t} J(X_0).$$ See Appendix \[App:DetMutHeat\] for detail.
Convexity of mutual information
===============================
We present our main results on the convexity of mutual information along the heat flow. Throughout, let $X_t = X_0 + \sqrt{t} Z$ denote the heat flow.
Perpetual convexity when initial distribution is log-concave
------------------------------------------------------------
Recall from Lemma \[Lem:KJMut\] and \[Lem:DerMut\] that mutual information is convex whenever the final distribution is log-concave. Since the heat flow preserves log-concavity, this implies mutual information is always convex when the initial distribution is log-concave.
\[Thm:PerpConv\] If $X_0 \sim \rho_0$ has a log-concave distribution, then mutual information $t \mapsto I(X_0;X_t)$ is convex for all $t \ge 0$.
Eventual convexity when initial distribution is bounded
-------------------------------------------------------
Next, we ask when the final distribution is eventually convex under the heat flow, which also implies the eventual convexity of mutual information. We can show that if the initial distribution is bounded, then the final distribution is eventually log-concave; this fact has also been observed in [@Lee03].
We say a probability distribution $\rho$ is [*$D$-bounded*]{} for some $D \ge 0$ if it is supported on a domain of diameter at most $D$.
\[Thm:EventConv\] If $X_0 \sim \rho_0$ has a $D$-bounded distribution, then mutual information $t \mapsto I(X_0;X_t)$ is convex for all $t \ge D^2$.
Since convolution with log-concave distribution preserves log-concavity, we also have the following corollary. Note that when the bounded part is a point mass (with diameter $D = 0$) this recovers Theorem \[Thm:PerpConv\] above.
\[Cor:EventConv\] If $X_0 \sim \rho_0$ is a convolution of a $D$-bounded and a log-concave distribution, then mutual information $t \mapsto I(X_0;X_t)$ is convex for all $t \ge D^2$.
For example, if $X_0 \sim \sum_{i=1}^k p_i {\mathcal{N}}(a_i, \Sigma)$ is a mixture of Gaussians with the same covariance, then the bounded part $\sum_{i=1}^k p_i \delta_{a_i}$ has diameter $D = \max_{i \neq j} \|a_i-a_j\|$.
Eventual convexity when Fisher information is finite
----------------------------------------------------
We now investigate when mutual information is eventually convex in general, regardless of the log-concavity of the distributions. We show that if the initial distribution has finite fourth moment and Fisher information, then mutual information is eventually convex.
For $p \ge 0$, let $M_p(X) = {\mathbb{E}}[\|X-\mu\|^p]$ denote the $p$-th moment of a random variable $X$ with mean ${\mathbb{E}}[X] = \mu \in {\mathbb{R}}^n$.
\[Thm:EventConvFI\] If $X_0 \sim \rho_0$ has finite fourth moment $M_4(X_0) < \infty$ and Fisher information $J(X_0) < \infty$, then mutual information $t \mapsto I(X_0;X_t)$ is convex for all $t \ge \frac{1}{n^2} J(X_0)M_4(X_0)$.
Thus, we see that under a wide variety of conditions, mutual information is eventually convex along the heat flow. However, it turns out mutual information is [*not*]{} always convex along the heat flow, in contrast to the concavity of entropy or entropy power along the heat flow.
Nonconvexity of mutual information {#Sec:NonConv}
==================================
We present some counterexamples for which mutual information along the heat flow is not convex at some small time. Concretely, we study mixtures of point masses and mixtures of Gaussians as initial distribution of the heat flow.
Mixture of two point masses {#Sec:MixtPoint}
---------------------------
Let $X_0 \sim \frac{1}{2} \delta_{-a} + \frac{1}{2} \delta_a$ be a uniform mixture of two point masses centered at $a$ and $-a$, for some $a \in {\mathbb{R}}^n$, $a \neq 0$. Along the heat flow, $X_t \sim \frac{1}{2} {\mathcal{N}}(-a,tI) + \frac{1}{2} {\mathcal{N}}(a,tI)$ is a uniform mixture of two Gaussians with equal covariance $tI$.
For $u > 0$, let $$V_u = {\mathcal{N}}(u,u) \in {\mathbb{R}}$$ denote the one-dimensional Gaussian random variable with mean and variance both equal to $u$. Then by direct calculation: $$\begin{aligned}
I(X_0;X_t) &= \frac{\|a\|^2}{t} - {\mathbb{E}}[\log \cosh(V_{\frac{\|a\|^2}{t}})] \\
J(X_0;X_t) &= \frac{\|a\|^2}{t^2} {\mathbb{E}}[\operatorname{sech}^2(V_{\frac{\|a\|^2}{t}})] \\
K(X_0;X_t) &= \frac{2\|a\|^2}{t^3} {\mathbb{E}}[\operatorname{sech}^2(V_{\frac{\|a\|^2}{t}})] \! - \! \frac{\|a\|^4}{t^4} {\mathbb{E}}[\operatorname{sech}^4(V_{\frac{\|a\|^2}{t}})].\end{aligned}$$ Note the dependence on dimension is only implicit via $\|a\|^2$.
The behavior of these quantities is illustrated in Figure \[Fig:MixtPoint\]. Mutual information is not convex at small time since it starts at some finite value (in fact $\log 2$), and stays flat for a while before decreasing. Its second derivative, the mutual second-order Fisher information, starts at $0$ and becomes negative before eventually becoming positive. Thus, mutual information is concave for all small time. Furthermore, by scaling $\|a\|^2$ we can stretch the region of nonconvexity to cover any finite time interval.
Mixture of two Gaussians {#Sec:MixtGaus}
------------------------
Let $X_0 \sim \frac{1}{2} {\mathcal{N}}(-a,sI) + \frac{1}{2} {\mathcal{N}}(a,sI)$ be a uniform mixture of two Gaussians with the same covariance $sI$ for some $s > 0$, centered at $-a$ and $a$ for some $a \in {\mathbb{R}}^n$, $a \neq 0$. Note, the limit $s \to 0$ recovers the mixture of two point masses above. Along the heat flow, $X_t \sim \frac{1}{2} N(-a,(s+t)I) + \frac{1}{2} {\mathcal{N}}(a,(s+t)I)$ is also a mixture of two Gaussians with increasing covariance.
Then with $V_u = {\mathcal{N}}(u,u)$ as above, we have: $$\begin{aligned}
I(X_0;X_t) &= \frac{n}{2} \log\left(1+\frac{s}{t}\right) + \frac{\|a\|^2}{s+t} - {\mathbb{E}}[\log \cosh(V_{\frac{\|a\|^2}{s+t}})] \\
J(X_0;X_t) &= \frac{ns}{t(s+t)} + \frac{\|a\|^2}{(s+t)^2} {\mathbb{E}}[\operatorname{sech}^2(V_{\frac{\|a\|^2}{s+t}})] \\
K(X_0;X_t) &= \frac{ns(s+2t)}{t^2(s+t)^2} + \frac{2\|a\|^2}{(s+t)^3} {\mathbb{E}}[\operatorname{sech}^2(V_{\frac{\|a\|^2}{s+t}})] \\
&~~~~ - \frac{\|a\|^4}{(s+t)^4} {\mathbb{E}}[\operatorname{sech}^4(V_{\frac{\|a\|^2}{s+t}})].\end{aligned}$$ Note the explicit dependence on the dimension $n$.
The behavior of these quantities is illustrated in Figure \[Fig:MixtGaus\] for $n=1$. Mutual information initially starts at $+\infty$, but it decreases quickly and exhibits a similar pattern of nonconvexity as the mixture of point masses. Its second derivative, the mutual second-order Fisher information, also starts at $+\infty$, but decreases quickly and becomes negative for some time before eventually becoming positive. Thus, mutual information is concave at some small time, and by scaling $\|a\|^2$ we can enlarge the region of nonconvexity.
[0.2311]{} ![Behavior of mutual information and its two derivatives along the heat flow. (a) Left: $X_0 \sim \frac{1}{2} \delta_{-1} + \frac{1}{2} \delta_1$. (b) Right: $X_0 \sim \frac{1}{2} {\mathcal{N}}(-1,s) + \frac{1}{2} {\mathcal{N}}(1,s)$ with $s = 10^{-3}$.[]{data-label="Fig:Mixt"}](IMixtPoint.eps "fig:"){width="\textwidth"}
[0.2311]{} ![Behavior of mutual information and its two derivatives along the heat flow. (a) Left: $X_0 \sim \frac{1}{2} \delta_{-1} + \frac{1}{2} \delta_1$. (b) Right: $X_0 \sim \frac{1}{2} {\mathcal{N}}(-1,s) + \frac{1}{2} {\mathcal{N}}(1,s)$ with $s = 10^{-3}$.[]{data-label="Fig:Mixt"}](IMixtGaus.eps "fig:"){width="\textwidth"}
[0.22]{} ![Behavior of mutual information and its two derivatives along the heat flow. (a) Left: $X_0 \sim \frac{1}{2} \delta_{-1} + \frac{1}{2} \delta_1$. (b) Right: $X_0 \sim \frac{1}{2} {\mathcal{N}}(-1,s) + \frac{1}{2} {\mathcal{N}}(1,s)$ with $s = 10^{-3}$.[]{data-label="Fig:Mixt"}](JMixtPoint.eps "fig:"){width="\textwidth"}
[0.22]{} ![Behavior of mutual information and its two derivatives along the heat flow. (a) Left: $X_0 \sim \frac{1}{2} \delta_{-1} + \frac{1}{2} \delta_1$. (b) Right: $X_0 \sim \frac{1}{2} {\mathcal{N}}(-1,s) + \frac{1}{2} {\mathcal{N}}(1,s)$ with $s = 10^{-3}$.[]{data-label="Fig:Mixt"}](JMixtGaus.eps "fig:"){width="\textwidth"}
[0.22]{} ![Behavior of mutual information and its two derivatives along the heat flow. (a) Left: $X_0 \sim \frac{1}{2} \delta_{-1} + \frac{1}{2} \delta_1$. (b) Right: $X_0 \sim \frac{1}{2} {\mathcal{N}}(-1,s) + \frac{1}{2} {\mathcal{N}}(1,s)$ with $s = 10^{-3}$.[]{data-label="Fig:Mixt"}](KMixtPoint.eps "fig:"){width="\textwidth"}
[0.22]{} ![Behavior of mutual information and its two derivatives along the heat flow. (a) Left: $X_0 \sim \frac{1}{2} \delta_{-1} + \frac{1}{2} \delta_1$. (b) Right: $X_0 \sim \frac{1}{2} {\mathcal{N}}(-1,s) + \frac{1}{2} {\mathcal{N}}(1,s)$ with $s = 10^{-3}$.[]{data-label="Fig:Mixt"}](KMixtGaus.eps "fig:"){width="\textwidth"}
General mixture of point masses
-------------------------------
Let $X_0 \sim \sum_{i=1}^k p_i \delta_{a_i}$ be a mixture of point masses centered at distinct $a_i \in {\mathbb{R}}^n$, with mixture probabilities $p_i > 0$, $\sum_{i=1}^k p_i = 1$. Along the heat flow, $X_t \sim \sum_{i=1}^k p_i {\mathcal{N}}(a_i,tI)$ is a mixture of Gaussians with increasing covariance $tI$ at the same centers.
We show that mutual information starts at a finite value which is equal to the discrete entropy of the mixture probability, and it is exponentially concentrated at small time.
Let $\|p\|_\infty = \max_{i,j} p_i/p_j$ and $m = \min_{i \neq j} \|a_i-a_j\| > 0$. Let $h(p) = -\sum_{i=1}^k p_i \log p_i$ denote the discrete entropy.
\[Thm:GenMixt\] For all $0 < t \le \frac{m^2}{676\|p\|_\infty^2}$, $$0 \,\le\, h(p) - I(X_0;X_t) \,\le\, 3(k-1)\|p\|_\infty e^{-0.085 \frac{m^2}{t}}.$$
The theorem above implies that $$\lim_{t \to 0} I(X_0;X_t) = h(p).$$ In particular, the initial value of mutual information does not depend on the locations of the centers, as long as they are distinct. This is interesting, because by moving the centers and merging them we can obtain discontinuities of the mutual information with respect to the initial random variable at the origin (moving the centers changes the mutual information curve but preserves the starting point, while merging the centers makes the starting point jump).
Furthermore, if a function converges exponentially fast, then all its derivatives must converge to zero exponentially fast. Thus, we have the following corollary.
\[Cor:Last\] For all $\ell \in \mathbb{N}$, $\lim_{t \to 0} \frac{d^\ell}{dt^\ell} I(X_0;X_t) = 0$.
In particular, the first derivative of mutual information, which is negative mutual Fisher information, starts at $0$. Since the initial distribution is bounded, mutual information is eventually convex by Theorem \[Thm:EventConv\], which means mutual Fisher information is eventually decreasing. Since mutual Fisher information is always nonnegative, this means it must initially increase, during which mutual information is concave; this is similar to the behavior observed in $\S\ref{Sec:MixtPoint}$.
Moreover, by the continuity of the second-order Fisher information, this suggests that when the initial distribution is a mixture of Gaussians, mutual information may be also be concave at some small time.
Discussion and future work
==========================
In this paper we have studied the convexity of mutual information along the heat flow. We have shown that under a wide variety of conditions mutual information is eventually convex, and we have shown examples where mutual information may be concave at some small time. Many questions remain.
One question is how much we can extend the results to general stochastic processes. We can show most of our results still hold for the Ornstein-Uhlenbeck process [@WJOU18]. For general Fokker-Planck processes the situation is more complicated, but at least there are explicit formulae for the second derivatives [@Vil08].
Another question is whether there are other conditions that imply eventual log-concavity under the heat flow. Currently we only know it for when the initial distribution is a convolution of a bounded and a log-concave distribution. It is interesting to study what happens for a larger class of initial distributions, for example sub-Gaussian.
Alternatively, for each point in space we can define the notion of a “time to log-concavity,” after which the final distribution is log-concave at that point. In general, this time is finite for each fixed point, and eventual log-concavity occurs if the supremum of this time over space is finite. There is a generic bound for this time to log-concavity in terms of the variance, and we can prove a slightly better bound under sub-Gaussian assumption, but not much is known.
We are seeking a proof of the nonconvexity of mutual information for the examples presented in $\S$\[Sec:NonConv\]. The nonconvexity is clear from Figure \[Fig:Mixt\], and we have explicit formulae for the second derivatives, but it is desirable to have a formal proof.
It is also interesting to study the effect of dimension in this problem, whether it makes convexity of mutual information easier or more difficult to occur. From Theorem \[Thm:EventConvFI\], and taking into account the growth of Fisher information and fourth moment with dimension, we see that the effect of dimension seems to be to delay the eventual convexity.
Finally, for mixtures of point masses, we have shown that the definition of self-information under the heat flow remembers the discrete initial data. We can show this also holds for the Ornstein-Uhlenbeck process [@WJOU18]. It is interesting to study whether the self-information limit is the same under more general flows such as the Fokker-Planck process.
Proofs
======
Proof of Lemma \[Lem:DerEnt\]
-----------------------------
These identities follow by direct calculation and integration by parts (and Bochner’s formula for the second identity). The first derivative of entropy along the heat flow is De Bruijn’s identity [@Sta59]. The second derivative of entropy along the heat flow is by McKean [@McKean66] in one dimension, and by Toscani [@Tos99] in multi dimension; see also Villani [@Vil00] for a clean proof.
Proof of Lemma \[Lem:KJMut\] {#App:ProofKJMut}
----------------------------
We first introduce some definitions. We view the joint distribution $\rho_{XY}(x,y) = \rho_Y(y) \rho_{X|Y}(x\,|\,y)$ as a family of probability distributions $\rho_{X|Y}(\cdot\,|\,y)$ parameterized by $y \in {\mathbb{R}}^n$, which has distribution $\rho_Y$. We also assume the density $\rho_{X|Y}(\cdot\,|\,y)$ is smooth with respect to $y$.
The [*pointwise backward Fisher information matrix*]{} of $X$ given $Y=y$ is $$\begin{gathered}
\widetilde \Phi(X\,|\,Y=y) = \\ \int_{{\mathbb{R}}^n} \rho_{X|Y}(x\,|\,y) (\nabla_y \log \rho_{X|Y}(x\,|\,y))(\nabla_y \log \rho_{X|Y}(x\,|\,y))^\top dx.\end{gathered}$$ By integration by parts (assuming boundary terms vanish), we can also write $$\widetilde \Phi(X\,|\,Y\!=\!y) = -\int_{{\mathbb{R}}^n} \rho_{X|Y}(x\,|\,y) \nabla^2_y \log \rho_{X|Y}(x\,|\,y) dx.$$ The [*pointwise backward Fisher information*]{} of $X$ given $Y=y$ is $$\begin{aligned}
\Phi(X\,|\,Y\!=\!y) &= \operatorname{Tr}(\widetilde \Phi(X\,|\,Y\!=\!y)) \\
&= \int_{{\mathbb{R}}^n} \rho_{X|Y}(x\,|\,y) \|\nabla_y \log \rho_{X|Y}(x\,|\,y)\|^2 dx.\end{aligned}$$ The [*backward Fisher information matrix*]{} of $X$ given $Y$ is $$\widetilde \Phi(X\,|\,Y) = \int_{{\mathbb{R}}^n} \rho_Y(y) \, \widetilde \Phi(X\,|\,Y\!=\!y) \, dy.$$ The [*backward Fisher information*]{} of $X$ given $Y$ is $$\Phi(X\,|\,Y) = \operatorname{Tr}(\widetilde \Phi(X\,|\,Y)).$$ Note $\widetilde \Phi(X\,|\,Y\!=\!y) \succeq 0$ and $\Phi(X\,|\,Y\!=\!y) \ge 0$ for all $y \in {\mathbb{R}}^n$, so $\widetilde \Phi(X\,|\,Y) \succeq 0$ and $\Phi(X\,|\,Y) \ge 0$.
Similarly, the [*pointwise backward second-order Fisher information*]{} of $X$ given $Y=y$ is $$\Psi(X\,|\,Y\!=\!y) =
\int_{{\mathbb{R}}^n} \rho_{X|Y}(x\,|\,y) \|\nabla^2_y \log \rho_{X|Y}(x\,|\,y)\|^2_{{\mathrm{HS}}} \, dx.$$ The [*backward second-order Fisher information*]{} of $X$ given $Y$ is $$\Psi(X\,|\,Y) = \int_{{\mathbb{R}}^n} \rho_Y(y) \, \Psi(X\,|\,Y\!=\!y) \, dy.$$ Note that $\Psi(X\,|\,Y=y) \ge 0$ for all $y \in {\mathbb{R}}^n$, so $\Psi(X\,|\,Y) \ge 0$.
Finally, the [*Fisher information matrix*]{} of $Y$ is $$\widetilde J(Y) = \int_{{\mathbb{R}}^n} \rho_Y(y) (\nabla_y \log \rho_Y(y))(\nabla_y \log \rho_Y(y))^\top dy.$$ By integration by parts (assuming boundary terms vanish), we can also write $$\widetilde J(Y) = -\int_{{\mathbb{R}}^n} \rho_Y(y) \nabla^2_y \log \rho_Y(y) dy.$$ Note that $\widetilde J(Y) \succeq 0$ and Fisher information is its trace: $J(Y) = \operatorname{Tr}(\widetilde J(Y))$.
As stated in $\S\ref{Sec:Mut}$, mutual Fisher information is in fact equal to the backward Fisher information.
\[Lem:JPhi\] For any joint random variable $(X,Y)$, $$J(X;Y) = \Phi(X\,|\,Y).$$
From the factorization $$\rho_X(x) \rho_{Y|X}(y\,|\,x) = \rho_Y(y) \rho_{X|Y}(x\,|\,y)$$ we have $$-\nabla^2_y \log \rho_{Y|X}(y\,|\,x) = -\nabla^2_y \log \rho_Y(y) - \nabla^2_y \log \rho_{X|Y}(x\,|\,y).$$ We integrate both sides with respect to $\rho_{XY}(x,y)$. The left-hand side gives the expected Fisher information matrix $\widetilde J(Y\,|\,X)$. The first term on the right-hand side gives $\widetilde J(Y)$, while the second term gives the $\widetilde \Phi(X\,|\,Y)$. That is, $\widetilde J(Y\,|\,X) = \widetilde J(Y) + \widetilde \Phi(X\,|\,Y)$, or equivalently, $$\widetilde J(X;Y) = \widetilde J(Y\,|\,X) - \widetilde J(Y) = \widetilde \Phi(X\,|\,Y).$$ Taking trace gives $$J(X;Y) = \operatorname{Tr}(\widetilde J(X;Y)) = \operatorname{Tr}(\widetilde \Phi(X\,|\,Y)) = \Phi(X;Y)$$ as desired.
Similarly, mutual second-order Fisher information can be represented in terms of the backward second-order Fisher information, albeit in a more complicated way.
\[Lem:KPsi\] For any joint random variable $(X,Y)$, $$\begin{gathered}
K(X;Y) = \Psi(X\,|\,Y) \, + \\ 2 \int_{{\mathbb{R}}^n} \rho_Y(y) \langle -\nabla^2 \log \rho_Y(y), \, \widetilde \Phi(X\,|\,Y\!=\!y)\rangle_{{\mathrm{HS}}} \, dy.\end{gathered}$$
As before we have the decomposition $$-\nabla^2_y \log \rho_{Y|X}(y\,|\,x) = -\nabla^2_y \log \rho_Y(y) - \nabla^2_y \log \rho_{X|Y}(x\,|\,y).$$ Taking the squared norm on both sides and expanding, we get $$\begin{aligned}
&\|\nabla^2_y \log \rho_{Y|X}(y\,|\,x)\|^2_{{\mathrm{HS}}} \\
&~~~~= \|\nabla^2_y \log \rho_Y(y)\|^2_{{\mathrm{HS}}} + \|\nabla^2_y \log \rho_{X|Y}(x\,|\,y)\|^2_{{\mathrm{HS}}} \\
&~~~~~~~~ + 2 \langle \nabla^2_y \log \rho_Y(y), \nabla^2_y \log \rho_{X|Y}(x\,|\,y) \rangle_{{\mathrm{HS}}}.\end{aligned}$$ We integrate both sides with respect to $\rho_{XY}(x,y)$. On the left-hand side we get $K(Y\,|\,X)$. The first term on the right-hand side gives $K(Y)$; the second term gives $\Psi(X\,|\,Y)$; for the third term, by first integrating over $\rho_{X|Y}(x\,|\,y)$ we obtain an inner product with $\widetilde \Phi(X\,|\,Y\!=\!y)$. That is, $$\begin{gathered}
K(Y\,|\,X) = K(Y) + \Psi(X\,|\,Y) \\
+ 2 \int_{{\mathbb{R}}^n} \rho_Y(y) \langle -\nabla^2_y \log \rho_Y(y), \widetilde \Phi(X\,|\,Y\!=\!y)\rangle_{{\mathrm{HS}}} \, dy.\end{gathered}$$ This implies the desired expression for $K(X;Y) = K(Y\,|\,X)-K(Y)$.
We can prove a lower bound for $K(X;Y)$ under log-semiconcavity assumption on $Y$.
\[Lem:KLC\] If $Y \sim \rho_Y$ is $\alpha$-log-semiconcave for some $\alpha \in {\mathbb{R}}$, then $$K(X;Y) \ge \Psi(X\,|\,Y) + 2\alpha \Phi(X\,|\,Y).$$
Since $-\nabla^2 \log \rho_Y(y) \succeq \alpha I$ and $\widetilde \Phi(X\,|\,Y=y) \succeq 0$ for all $y \in {\mathbb{R}}^n$, we have $$\begin{aligned}
\langle -\nabla^2 \log \rho_Y(y), \, \widetilde \Phi(X\,|\,Y\!=\!y) \rangle_{{\mathrm{HS}}}
&\ge \, \langle \alpha I, \, \widetilde \Phi(X\,|\,Y\!=\!y) \rangle_{{\mathrm{HS}}} \\
&=\, \alpha \operatorname{Tr}(\widetilde \Phi(X\,|\,Y\!=\!y)) \\
&=\, \alpha \, \Phi(X\,|\,Y\!=\!y).\end{aligned}$$ Integrating with respect to $\rho_Y(y)$ gives $$\begin{aligned}
&\int_{{\mathbb{R}}^n} \rho_Y(y) \langle -\nabla^2 \log \rho_Y(y), \, \widetilde \Phi(X\,|\,Y\!=\!y) \rangle_{{\mathrm{HS}}} \, dy \\
&~~~~~~~ \ge\, \alpha \int_{{\mathbb{R}}^n} \rho(y) \Phi(X\,|\,Y\!=\!y) \, dy
\,=\, \alpha \, \Phi(X\,|\,Y).\end{aligned}$$ Adding $\Psi(X\,|\,Y)$ and using Lemma \[Lem:KPsi\] gives the result.
Furthermore, we have the following result which is reminiscent of the inequality between first and second-order Fisher information.
\[Lem:PsiPhi\] For any joint random variable $(X,Y)$ in ${\mathbb{R}}^n \times {\mathbb{R}}^n$, $$\Psi(X\,|\,Y) \ge \frac{\Phi(X\,|\,Y)^2}{n}.$$
Let $A_{x,y} = -\nabla^2_y \log \rho_{X|Y}(x\,|\,y)$. By Cauchy-Schwarz inequality, $$\|A_{x,y}\|^2_{{\mathrm{HS}}} = \operatorname{Tr}(A_{x,y}^2) \ge \frac{(\operatorname{Tr}(A_{x,y}))^2}{n}.$$ Taking expectation over $(X,Y) \sim \rho_{XY}$ and applying Cauchy-Schwarz again, we get the desired result $$\begin{aligned}
\Psi(X\,|\,Y) &= {\mathbb{E}}[\|A_{X,Y}\|^2_{{\mathrm{HS}}}] \\
&\ge \frac{{\mathbb{E}}[(\operatorname{Tr}(A_{X,Y}))^2]}{n} \\
&\ge \frac{({\mathbb{E}}[\operatorname{Tr}(A_{X,Y})])^2}{n} \\
&= \frac{\Phi(X\,|\,Y)^2}{n}.\end{aligned}$$
Finally, we are ready to prove Lemma \[Lem:KJMut\].
By Lemma \[Lem:KLC\] and \[Lem:PsiPhi\], $$K(X;Y) \ge \frac{\Phi(X\,|\,Y)^2}{n} + 2\alpha \Phi(X\,|\,Y).$$ Since $J(X;Y) = \Phi(X\,|\,Y)$ by Lemma \[Lem:JPhi\], the result follows.
Proof of Lemma \[Lem:DerMut\]
-----------------------------
These identities follow from Lemma \[Lem:DerEnt\] and the linearity of the heat flow channel.
Concretely, recall by Lemma \[Lem:DerEnt\] that $\frac{d}{dt} H(X_t) = \frac{1}{2} J(X_t)$. We apply this result to the conditional density $\rho_{X_t|X_0}(\cdot\,|\,x_0)$ to get $\frac{d}{dt} H(X_t\,|\,X_0=x_0) = \frac{1}{2} J(X_t\,|\,X_0=x_0)$ for each $x_0 \in {\mathbb{R}}^n$. Taking expectation over $X_0 \sim \rho_0$ and interchanging the order of expectation and time differentiation yields $\frac{d}{dt} H(X_t\,|\,X_0) = \frac{1}{2} J(X_t\,|\,X_0)$. Combining this with the earlier result above yields $\frac{d}{dt} I(X_0;X_t) = \frac{1}{2} J(X_0;X_t)$, as desired. The proof for $\frac{d^2}{dt^2} I(X_0;X_t) = -\frac{1}{2} K(X_0;X_t)$ proceeds identically using the second identity in Lemma \[Lem:DerEnt\].
Detail for $\S\ref{Sec:MutHeat}$ {#App:DetMutHeat}
--------------------------------
We compute $J(X_t;X_0)$ and $K(X_t;X_0)$ along the heat flow $X_t = X_0 + \sqrt{t} Z$. Let $X_0 \sim \rho_0$, $X_t \sim \rho_t$, $(X_0,X_t) \sim \rho_{0t}$, and we write the conditionals as $$\rho_0(x) \rho_{t|0}(y\,|\,x) = \rho_{0t}(x,y) = \rho_t(y) \rho_{0|t}(x\,|\,y).$$ Then $$-\nabla_x \log \rho_{0|t}(x\,|\,y) = -\nabla_x \log \rho_0(x) - \nabla_x \log \rho_{t|0}(y\,|\,x).$$ Along the heat flow $X_t\,|\,X_0$ is Gaussian with covariance $tI$, so we have explicitly $-\nabla_x \log \rho_{t|0}(y\,|\,x) = \frac{x-y}{t}$. Therefore, $$\begin{aligned}
\label{Eq:DetMutHeatCalc}
-\nabla_x \log \rho_{0|t}(x\,|\,y) = -\nabla_x \log \rho_0(x) +\frac{x-y}{t}.\end{aligned}$$ Take the squared norm on both sides and expand: $$\begin{aligned}
\|\nabla_x \log \rho_{0|t}(x\,|\,y)\|^2 &= \|\nabla_x \log \rho_0(x)\|^2 + \frac{\|x-y\|^2}{t^2} \\
&~~~~ + \frac{2}{t} \langle -\nabla_x \log \rho_0(x), x-y \rangle.\end{aligned}$$ Now we take expectation of both sides over $(X_0,X_t)$. The left-hand side gives $J(X_0\,|\,X_t)$. The first term on the right-hand side gives $J(X_0)$; the second term gives $\frac{1}{t^2}{\mathbb{E}}[\|X_0-X_t\|^2] = \frac{1}{t^2} {\mathbb{E}}[\|\sqrt{t}Z\|^2] = \frac{n}{t}$ where $Z \sim {\mathcal{N}}(0,I)$; while the third term gives $0$ by integrating over $y$ first for each fixed $x$. That is, $$\begin{aligned}
\label{Eq:JXYX}
J(X_0\,|\,X_t) = J(X_0) + \frac{n}{t}.\end{aligned}$$ Therefore, $$J(X_t;X_0) = J(X_0\,|\,X_t) - J(X_0) = \frac{n}{t}.$$
Next, we differentiate again with respect to $x$ to get $$-\nabla^2_x \log \rho_{0|t}(x\,|\,y) = -\nabla^2_x \log \rho_0(x) +\frac{I}{t}.$$ Take the squared norm on both sides and expand: $$\begin{aligned}
&\|\nabla^2_x \log \rho_{0|t}(x\,|\,y)\|_{{\mathrm{HS}}}^2 \\
&= \|\nabla^2_x \log \rho_0(x)\|_{{\mathrm{HS}}}^2 + \frac{\|I\|^2_{{\mathrm{HS}}}}{t^2}
+ \frac{2}{t} \langle -\nabla^2_x \log \rho_0(x), I \rangle_{{\mathrm{HS}}} \\
&= \|\nabla^2_x \log \rho_0(x)\|_{{\mathrm{HS}}}^2 + \frac{n}{t^2}
- \frac{2}{t} \Delta_x \log \rho_0(x).\end{aligned}$$ Now we take expectation of both sides over $(X_0,X_t)$. The left-hand side gives $K(X_0\,|\,X_t)$. The first term on the right-hand side gives $K(X_0)$; the second term is a constant; while the third term gives $\frac{2}{t}J(X_0)$. That is, $$K(X_0\,|\,X_t) = K(X_0) + \frac{n}{t^2} + \frac{2}{t} J(X_0).$$ Therefore, $$K(X_t;X_0) = K(X_0\,|\,X_t) - K(X_0) = \frac{n}{t^2} + \frac{2}{t} J(X_0).$$
Proof of Theorem \[Thm:PerpConv\]
---------------------------------
Recall that the heat flow preserves log-concavity. This is because the Gaussian density (the heat kernel) is log-concave, and convolution with a log-concave distribution preserves log-concavity by the Prékopa-Leindler inequality.
By assumption $X_0 \sim \rho_0$ is log-concave, so $X_t \sim \rho_t$ is also log-concave for all $t \ge 0$. By Lemma \[Lem:KJMut\] and \[Lem:DerMut\], this implies $\frac{d^2}{dt^2} I(X_0;X_t) = K(X_0;X_t) \ge 0$, which means mutual information is always convex.
Proof of Theorem \[Thm:EventConv\] {#App:ProofEventConv}
----------------------------------
Throughout, let $X_t = X_0 + \sqrt{t} Z$ denote the heat flow. Let $X_0 \sim \rho_0$, $X_t \sim \rho_t$, $(X_0,X_t) \sim \rho_{0t}$, and we write the conditionals as $$\rho_0(x) \rho_{t|0}(y\,|\,x) = \rho_{0t}(x,y) = \rho_t(y) \rho_{0|t}(x\,|\,y).$$
We first establish the following result to help us determine when we have eventual log-concavity under the heat flow; see Appendix \[App:ProofHesHeat\] for the proof.
\[Lem:HesHeat\] Along the heat flow, for all $y \in {\mathbb{R}}^n$, $$\begin{aligned}
\label{Eq:HesHeat}
-\nabla^2_y \log \rho_{t}(y) = \frac{1}{t} \left(I - \frac{1}{t} \operatorname{Cov}(\rho_{0|t}(\cdot\,|\,y))\right).\end{aligned}$$
In particular, for bounded initial distribution we have the following eventual log-concavity.
\[Lem:BddLC\] If $X_0 \sim \rho_0$ is $D$-bounded, then along the heat flow, $X_t \sim \rho_t$ is log-concave for all $t \ge D^2$.
Since $X_0 \sim \rho_0$ is $D$-bounded, the conditional distributions $\rho_{0|t}(\cdot\,|\,y)$ are also $D$-bounded for all $y \in {\mathbb{R}}^n$. In particular, $\operatorname{Cov}(\rho_{0|t}(\cdot\,|\,y)) \preceq D^2 I.$ Therefore, by Lemma \[Lem:HesHeat\], $$-\nabla^2_y \log \rho_{t}(y) \succeq \frac{1}{t} \left(1 - \frac{D^2}{t}\right) I.$$ If $t \ge D^2$, then $-\nabla^2_y \log \rho_{t}(y) \succeq 0$ for all $y \in {\mathbb{R}}^n$, which means $X_t \sim \rho_t$ is log-concave.
We are now ready to prove Theorem \[Thm:EventConv\].
By Lemma \[Lem:BddLC\], $X_t \sim \rho_t$ is log-concave for $t \ge D^2$. By Lemma \[Lem:KJMut\] and \[Lem:DerMut\], this implies mutual information is convex for all $t \ge D^2$.
Proof of Corollary \[Cor:EventConv\]
------------------------------------
Analogous to Lemma \[Lem:BddLC\], we have the following result.
\[Lem:BddConvLC\] If $X_0 \sim \rho_0$ is a convolution of a $D$-bounded and a log-concave distribution, then along the heat flow, $X_t \sim \rho_t$ is log-concave for all $t \ge D^2$.
We write $X_0 = B_0+C$ where $B_0$ is a $D$-bounded random variable and $C$ is a log-concave random variable independent of $B$. Then $X_t = X_0 + \sqrt{t}Z = (B_0 + \sqrt{t}Z) + C = B_t+C$ where $B_t = B_0 + \sqrt{t}Z$ is the heat flow from $B_0$. By Lemma \[Lem:BddLC\], $B_t$ is log-concave for $t \ge D^2$. Then by the Prékopa Leindler inequality, $X_t = B_t+C$ is also log-concave for all $t \ge D^2$.
We are now ready to prove Corollary \[Cor:EventConv\].
By Lemma \[Lem:BddConvLC\], $X_t \sim \rho_t$ is log-concave for $t \ge D^2$. By Lemma \[Lem:KJMut\] and \[Lem:DerMut\], this implies mutual information is convex for all $t \ge D^2$.
Proof of Lemma \[Lem:HesHeat\] {#App:ProofHesHeat}
------------------------------
We use the same setting and notation as in Appendix \[App:ProofEventConv\].
We first establish the following result.
\[Lem:HessCov\] Along the heat flow, for all $x,y \in {\mathbb{R}}^n$, $$\begin{aligned}
\label{Eq:HessCov}
-\nabla^2_y \log \rho_{0|t}(x\,|\,y) = \frac{\operatorname{Cov}(\rho_{0|t}(\cdot\,|\,y))}{t^2}.\end{aligned}$$
We observe that the conditional density $\rho_{0|t}(x\,|\,y)$ can be written as an exponential family distribution over $x$ with parameter $\eta = \frac{y}{t}$: $$\rho_{0|t}(x\,|\,y) = h(x) e^{\langle x,\eta \rangle - A(\eta)}$$ where $h(x) = \rho_0(x) e^{-\frac{\|x\|^2}{2t}}$ is the base measure, and $$A(\eta) = \log \int_{{\mathbb{R}}^n} h(x) e^{\langle x,\eta \rangle} \, dx$$ is the log-partition function, or normalizing constant. Then we have $$-\nabla^2_y \log \rho_{0|t}(x\,|\,y) = \frac{1}{t^2} \nabla^2_\eta A(\eta).$$ By a general identity for exponential family [@WJ08], or simply by differentiating, we have that $$\nabla^2_\eta A(\eta) = \operatorname{Cov}(\rho_{0|t}(\cdot\,|\,y)).$$ Combining the two expressions above yields the result.
We are now ready to prove Lemma \[Lem:HesHeat\].
From the factorization $$\rho_t(y) \rho_{0|t}(x\,|\,y) = \rho_0(x) \rho_{t|0}(y\,|\,x)$$ we have, along the heat flow and by Lemma \[Lem:HessCov\], $$\begin{aligned}
-\nabla^2_y \log \rho_t(y) &= -\nabla^2_y \log \rho_{t|0}(y\,|\,x) + \nabla^2_y \log \rho_{0|t}(x\,|\,y) \\
&= \frac{1}{t} I -\frac{1}{t^2}\operatorname{Cov}(\rho_{0|t}(\cdot\,|\,y)),\end{aligned}$$ as desired.
Proof of Theorem \[Thm:EventConvFI\]
------------------------------------
Let $X_t = X_0 + \sqrt{t} Z$ denote the heat flow. We first establish some results.
\[Lem:KHeat\] Along the heat flow, $$K(X_0;X_t) = \frac{2}{t^3} \operatorname{Var}(X_0\,|\,X_t) - \frac{1}{t^4} {\mathbb{E}}[\|\operatorname{Cov}(\rho_{0|t}(\cdot\,|\,X_t))\|^2_{{\mathrm{HS}}}].$$
Squaring and taking the expectation of the identity in Lemma \[Lem:HesHeat\] gives $$\begin{aligned}
K&(X_t) = \frac{1}{t^2} {\mathbb{E}}\Big[\Big\|I - \frac{1}{t} \operatorname{Cov}(\rho_{0|t}(\cdot\,|\,X_t))\Big\|^2_{{\mathrm{HS}}}\Big] \\
&= \frac{\|I\|^2_{{\mathrm{HS}}}}{t^2} - \frac{2}{t^3} {\mathbb{E}}[\operatorname{Var}( \rho_{0|t}(\cdot\,|\,X_t) )] \\
&~~~~ + \frac{1}{t^4} {\mathbb{E}}[\|\operatorname{Cov}(\rho_{0|t}(\cdot\,|\,X_t))\|^2_{{\mathrm{HS}}}] \\
&= \frac{n}{t^2} - \frac{2}{t^3} \operatorname{Var}(X_0\,|\,X_t) + \frac{1}{t^4} {\mathbb{E}}[\|\operatorname{Cov}(\rho_{0|t}(\cdot\,|\,X_t))\|^2_{{\mathrm{HS}}}].\end{aligned}$$ Since $K(X_t\,|\,X_0) = n/t^2$, this implies the desired result.
\[Lem:VarJ\] Assume $J(X_0) < +\infty$. Along the heat flow, $$\operatorname{Var}(X_0\,|\,X_t) \ge \frac{n^2}{J(X_0) + \frac{n}{t}}.$$
For any random variable $X \sim \rho$ in ${\mathbb{R}}^n$ with a smooth density, recall the uncertainty relationship $$J(X) \operatorname{Var}(X) \ge n^2,$$ which also follows from the Cauchy-Schwarz inequality and integration by parts. Applying this to the conditional densities $\rho_{0|t}(\cdot\,|\,y)$ yields $$\operatorname{Var}(\rho_{0|t}(\cdot\,|\,y)) \ge \frac{n^2}{J(\rho_{0|t}(\cdot\,|\,y))}.$$ Taking expectation over $Y = X_t$ and noting that ${\mathbb{E}}[\frac{1}{J}] \ge \frac{1}{{\mathbb{E}}[J]}$ by Cauchy-Schwarz, we get $$\operatorname{Var}(X_0\,|\,X_t) \ge {\mathbb{E}}\left[\frac{n^2}{J(\rho_{0|t}(\cdot\,|\,X_t))}\right] \ge \frac{n^2}{J(X_0\,|\,X_t)}.$$ Finally, recall from that $J(X_0\,|\,X_t) = J(X_0)+\frac{n}{t}$.
Recall that $M_4(X_0)$ is the fourth moment of $X_0$.
\[Lem:CovM4\] Assume $M_4(X_0) < +\infty$. Along the heat flow, $${\mathbb{E}}[\|\operatorname{Cov}(\rho_{0|t}(\cdot\,|\,X_t))\|^2_{{\mathrm{HS}}}] \le M_4(X_0).$$
Let $\mu_0 = {\mathbb{E}}[X_0]$. For each $y \in {\mathbb{R}}^n$, $$\begin{aligned}
\|\operatorname{Cov}(\rho_{0|t}(\cdot\,|\,y))\|^2_{{\mathrm{HS}}}
&\le (\operatorname{Tr}(\operatorname{Cov}(\rho_{0|t}(\cdot\,|\,y))))^2 \\
&= (\operatorname{Var}(\rho_{0|t}(\cdot\,|\,y)))^2 \\
&\le \left( {\mathbb{E}}_{\rho_{0|t}(\cdot\,|\,y)}[\|X-\mu_0\|^2]\right)^2 \\
&\le {\mathbb{E}}_{\rho_{0|t}(\cdot\,|\,y)}[\|X-\mu_0\|^4].\end{aligned}$$ Taking expectation over $Y=X_t$ and applying the tower property of expectation gives the result.
We are now ready to prove Theorem \[Thm:EventConvFI\].
By Lemma \[Lem:KHeat\], \[Lem:VarJ\], and \[Lem:CovM4\], we have $$K(X_0;X_t) \ge \frac{2n^2}{t^3(J(X_0)+\frac{n}{t})} - \frac{M_4(X_0)}{t^4}.$$ The right-hand side above is nonnegative if $2n^2t^4 \ge t^3M_4(X_0)(J(X_0)+\frac{n}{t})$, or equivalently, if $$2n^2 t^2 - t J(X_0)M_4(X_0) - nM_4(X_0) \ge 0.$$ Therefore, $K(X_0;X_t) \ge 0$ if $t$ is larger than the upper root of the quadratic polynomial above, which is the case when $$t \ge \frac{J(X_0)M_4(X_0)}{4n^2}\left(1+\sqrt{1+\frac{8n}{J(X_0)^2 M_4(X_0)}}\right).$$ Furthermore, by Cauchy-Schwarz and the uncertainty relationship, $$J(X_0)^2 M_4(X_0) \ge J(X_0)^2 \operatorname{Var}(X_0)^2 \ge n^4.$$ Plugging this to the bound above and further using $n \ge 1$, we conclude that $K(X_0;X_t) \ge 0$, and hence mutual information is convex, whenever $$t \ge \frac{J(X_0)M_4(X_0)}{n^2}.$$
Proof of Theorem $\ref{Thm:GenMixt}$
------------------------------------
At each $t > 0$, the density of $X_t \sim \sum_{i=1}^k p_i {\mathcal{N}}(a_i,tI)$ is $$\rho_t(y) = \frac{1}{(2\pi t)^{n/2}} \sum_{i=1}^k p_i e^{-\frac{\|y-a_i\|^2}{2t}}.$$ The entropy of $X_t$ is $$H(X_t)
= \frac{n}{2} \log (2\pi t) - {\mathbb{E}}\left[ \log \left( \sum_{i=1}^k p_i e^{-\frac{\|X_t-a_i\|^2}{2t}} \right) \right].$$ The expectation is over the mixture $X_t \sim \sum_{i=1}^k p_i {\mathcal{N}}(a_i, tI)$, which we split into a sum over $i=1,\dots,k$ of the individual expectations over $Y \sim {\mathcal{N}}(a_i, tI)$. When $Y \sim {\mathcal{N}}(a_i, tI)$, we write $Y = a_i + \sqrt{t} Z$ where $Z \sim {\mathcal{N}}(0,I)$. Then we can write the entropy above as [$$\begin{aligned}
&H(X_t) - \frac{n}{2} \log (2\pi t) \\
&= - \sum_{i=1}^k p_i {\mathbb{E}}\Big[\log\Big(p_i e^{-\frac{\|Z\|^2}{2}} + \sum_{j \neq i} p_j e^{-\frac{\|\sqrt{t}Z+a_i-a_j\|^2}{2t}}\Big)\Big] \\
&= - \sum_{i=1}^k p_i {\mathbb{E}}\Big[ \log p_i - \frac{\|Z\|^2}{2} + \log \Big(1 + \sum_{j \neq i} \frac{p_j}{p_i} e^{\frac{\|Z\|^2}{2}-\frac{\|\sqrt{t}Z+a_i-a_j\|^2}{2t}}\Big)\Big] \\
&= h(p) + \frac{n}{2} - \sum_{i=1}^k p_i {\mathbb{E}}\Big[ \log \Big(1 + \sum_{j \neq i} \frac{p_j}{p_i} e^{\frac{\|Z\|^2}{2}-\frac{\|\sqrt{t}Z+a_i-a_j\|^2}{2t}}\Big)\Big]\end{aligned}$$ ]{} where $h(p) = -\sum_{i=1}^k p_i \log p_i$ is the discrete entropy.
Since $H(X_t\,|\,X_0) = \frac{n}{2} \log (2\pi t e)$, we have for mutual information [$$h(p) - I(X_0;X_t) = \sum_{i=1}^k p_i {\mathbb{E}}\Big[ \log \Big(1 + \sum_{j \neq i} \frac{p_j}{p_i} e^{\frac{\|Z\|^2}{2}-\frac{\|\sqrt{t}Z+a_i-a_j\|^2}{2t}}\Big)\Big].$$ ]{}
Clearly $h(p)-I(X_0;X_t) \ge 0$ since the logarithm on the right-hand side above is positive.
On the other hand, using the inequality $\log(1+\sum_j x_j) \le \sum_j \log(1+x_j)$ for $x_j > 0$, we also have the upper bound [$$h(p)-I(X_0;X_t) \le \sum_{i=1}^k p_i \sum_{j \neq i} {\mathbb{E}}\Big[ \log \Big(1 + \frac{p_j}{p_i} e^{\frac{\|Z\|^2}{2}-\frac{\|\sqrt{t}Z+a_i-a_j\|^2}{2t}}\Big)\Big].$$ ]{} For each $i \neq j$, the exponent on the right-hand side above is $$\frac{\|Z\|^2}{2}-\frac{\|\sqrt{t}Z+a_i-a_j\|^2}{2t} = -\frac{\langle Z,a_i-a_j\rangle}{\sqrt{t}} - \frac{\|a_i-a_j\|^2}{2t},$$ which has the ${\mathcal{N}}(-\frac{\|a_i-a_j\|^2}{2t},\frac{\|a_i-a_j\|^2}{t})$ distribution in ${\mathbb{R}}$, so it has the same distribution as $-\frac{\|a_i-a_j\|^2}{2t} + \frac{\|a_i-a_j\|}{\sqrt{t}} Z_1$ where $Z_1 \sim {\mathcal{N}}(0,1)$ is the standard one-dimensional Gaussian. Thus, we can write the upper bound above as $$h(p) - I(X;Y) \le \sum_{i=1}^k p_i \sum_{j \neq i} {\mathbb{E}}\left[ \log \left(1 + b_{ij} e^{c_{ij} Z_1 - \frac{c_{ij}^2}{2}}\right)\right]$$ where $b_{ij} = \frac{p_j}{p_i}$ and $c_{ij} = \frac{\|a_i-a_j\|}{\sqrt{t}}$, and $Z_1 \sim {\mathcal{N}}(0,1)$ in ${\mathbb{R}}$.
By Lemma \[Lem:Log2\] below, if $c_{ij} \ge \max\{1,\frac{26}{b_{ij}}\}$, then we have $$\begin{aligned}
h(p) - I(X_0;X_t) \le 3 \sum_{i=1}^k p_i \sum_{j \neq i} b_{ij} e^{-0.085c_{ij}^2}.\end{aligned}$$ Note that $b_{ij} = \frac{p_j}{p_i} \le \|p\|_\infty$ and $c_{ij}^2 = \frac{\|a_j-a_i\|^2}{t} \ge \frac{m^2}{t}$, so $$\begin{aligned}
h(p) - I(X_0;X_t) &\le 3 \sum_{i=1}^k p_i \sum_{j \neq i} \|p\|_\infty e^{-0.085 \frac{m^2}{t}} \\
&= 3(k-1) \|p\|_\infty e^{-0.085 \frac{m^2}{t}}.\end{aligned}$$ Now, the condition $c_{ij} \ge \max\{1,\frac{26}{b_{ij}}\}$ is equivalent to $$t \le \frac{\|a_i-a_j\|^2}{\max\{1,\frac{26}{b_{ij}}\}^2}.$$ Since $\|a_i-a_j\|^2 \ge m^2$ and $\frac{1}{b_{ij}} = \frac{p_i}{p_j} \le \|p\|_\infty$, the condition above is satisfied when $$t \le \frac{m^2}{\max\{1,26\|p\|_\infty\}^2} = \frac{m^2}{26^2\|p\|_\infty^2}.$$
Thus, we conclude that if $t \le \frac{m^2}{676\|p\|_\infty^2}$, then $$\begin{aligned}
h(p) - I(X_0;X_t) \le 3 (k-1) \|p\|_\infty e^{-0.085\frac{m^2}{t}}\end{aligned}$$ as desired.
To complete the proof of Theorem \[Thm:GenMixt\], it remains to prove the following estimate.
\[Lem:Log2\] Let $b > 0$, $c \ge \max\{1,\frac{26}{b}\}$, and $Z \sim {\mathcal{N}}(0,1)$. Then $${\mathbb{E}}[\log(1 + be^{cZ-\frac{c^2}{2}})\big] \le 3be^{-0.085 c^2}.$$
We use the standard tail bound $\Pr(Z \ge x) \le \frac{1}{\sqrt{2\pi}} \frac{1}{x} e^{-\frac{x^2}{2}}$ for all $x > 0$, which follows from using the inequality $1 \le \frac{z}{x}$ in the integration. In particular, for $x \ge \frac{1}{\sqrt{2\pi}}$ we have $\Pr(Z \ge x) \le e^{-\frac{x^2}{2}}$.
Let $0 < \eta < 1$. We split the expectation into three parts:
1. For $Z < (1-\eta)\frac{c}{2}$: We have $cZ-\frac{c^2}{2} < -\eta\frac{c^2}{2}$, so $\log(1 + be^{cZ-\frac{c^2}{2}}) \le \log(1+be^{-\eta\frac{c^2}{2}}) \le be^{-\eta\frac{c^2}{2}}$. The contribution to the expectation from this region is at most $be^{-\eta\frac{c^2}{2}} \Pr(Z < (1-\eta)\frac{c}{2}) \le be^{-\eta\frac{c^2}{2}}$.
2. For $(1-\eta)\frac{c}{2} \le Z < \frac{c}{2}$: We have $cZ-\frac{c^2}{2} < 0$, so $\log(1 + be^{cZ-\frac{c^2}{2}}) \le \log(1+b) \le b$. The contribution to the expectation from this region is at most $b \Pr((1-\eta)\frac{c}{2} \le Z < \frac{c}{2}) \le b \Pr(Z \ge (1-\eta)\frac{c}{2}) \le b e^{-(1-\eta)^2\frac{c^2}{8}}$ where the last inequality holds for $c \ge \frac{2}{(1-\eta)\sqrt{2\pi}}$.
3. For $Z \ge \frac{c}{2}$: We have $cZ-\frac{c^2}{2} \ge 0$, so $\log(1 + be^{cZ-\frac{c^2}{2}}) \le \log((1+b)e^{cZ-\frac{c^2}{2}}) = \log(1+b) + cZ - \frac{c^2}{2} \le b + cZ$. The contribution to the expectation from this region is at most $\int_{\frac{c}{2}}^\infty (b+cz) \frac{1}{\sqrt{2\pi}} e^{-\frac{z^2}{2}} dz
= b \Pr(Z \ge \frac{c}{2}) + \frac{c}{\sqrt{2\pi}} e^{-\frac{c^2}{8}}
\le (b+c) e^{-\frac{c^2}{8}}$, where the last inequality holds for $c \ge \frac{2}{\sqrt{2\pi}}$.
Combining the three parts above, we have that for $c \ge \frac{2}{1-\eta}$, $${\mathbb{E}}[\log(1 + be^{cZ-\frac{c^2}{2}})] \le be^{-\eta\frac{c^2}{2}} + b e^{-(1-\eta)^2\frac{c^2}{8}} + (b+c) e^{-\frac{c^2}{8}}.$$ The leading exponent is $\min\{\eta, \frac{(1-\eta)^2}{4}\} \frac{c^2}{2}$, which is maximized by $\eta^\ast = 3-\sqrt{8} \approx 0.1716$. Set $\eta = \eta^\ast$. Note that for $c \ge \frac{2}{(\frac{1}{4}-\eta^\ast)b}$ we have $\frac{c^2}{2}(\frac{1}{4}-\eta^\ast) \ge \frac{c}{b} \ge \log(1+\frac{c}{b})$, so $(b+c)e^{-\frac{c^2}{8}} \le b e^{-\eta^\ast \frac{c^2}{2}}$.
Thus, for $c \ge \max\{\frac{2}{(1-\eta^\ast)\sqrt{2\pi}}, \frac{2}{(\frac{1}{4}-\eta^\ast)b}\}$, we have $${\mathbb{E}}\left[\log\left(1 + be^{cZ-\frac{c^2}{2}}\right)\right] \le 3be^{-\eta^\ast\frac{c^2}{2}}.$$ Since $\frac{\eta^\ast}{2} \approx 0.0858 > 0.085$, $\frac{2}{(1-\eta^\ast)\sqrt{2\pi}} \approx 0.9631 < 1$, and $\frac{2}{\frac{1}{4}-\eta^\ast} \approx 25.5014 < 26$, we can simplify this conclusion by saying that for $c \ge \max\{1,\frac{26}{b}\}$ we have $${\mathbb{E}}\left[\log\left(1 + be^{cZ-\frac{c^2}{2}}\right)\right] \le 3be^{-0.085 c^2},$$ as desired.
Proof of Corollary \[Cor:Last\]
-------------------------------
From Theorem \[Thm:GenMixt\], we have for small $t$, $$\left|\frac{I(X_0;X_t) - h(p)}{t^\ell} \right| \le 3(k-1) \|p\|_\infty \frac{e^{-0.085 \frac{m^2}{t}}}{t^\ell}.$$ Inductively, this implies all derivatives of $I(X_0;X_t)$ tend to $0$ exponentially fast as $t \to 0$.
| ArXiv |
---
abstract: 'Recently we have studied in great detail a model of Hybrid Natural Inflation (HNI) by constructing two simple effective field theories. These two versions of the model allow inflationary energy scales as small as the electroweak scale in one of them or as large as the Grand Unification scale in the other, therefore covering the whole range of possible energy scales. In any case the inflationary sector of the model is of the form $V(\phi)=V_0 \left(1+a \cos(\phi/f)\right)$ where $0\leq a<1$ and the end of inflation is triggered by an independent waterfall field. One interesting characteristic of this model is that the slow-roll parameter $\epsilon(\phi)$ is a non-monotonic function of $\phi$ presenting a [*maximum*]{} close to the inflection point of the potential. Because the scalar spectrum $\mathcal{P}_s(k)$ of density fluctuations when written in terms of the potential is inversely proportional to $\epsilon(\phi)$ we find that $\mathcal{P}_s(k)$ presents a [*minimum*]{} at $\phi_{min}$. The origin of the HNI potential can be traced to a symmetry breaking phenomenon occurring at some energy scale $f$ which gives rise to a (massless) Goldstone boson. Non-perturbative physics at some temperature $T<f$ might occur which provides a potential (and a small mass) to the originally massless boson to become the inflaton (a pseudo-Nambu-Goldstone boson). Thus the inflaton energy scale $\Delta$ is bounded by the symmetry breaking scale, $\Delta\equiv V_H^{1/4} <f.$ To have such a well defined origin and hierarchy of scales in inflationary models is not common. We use this property of HNI to determine bounds for the inflationary energy scale $\Delta$ and for the tensor-to-scalar ratio $r$.'
author:
- |
Gabriel Germán$^{a}
\footnote{Corresponding author: [email protected]}$, Alfredo Herrera-Aguilar$^{b,c}$, Juan Carlos Hidalgo$^{a}$,\
Roberto A. Sussman$^{d}$, José Tapia$^{a,e}$\
\
[*$^a$Instituto de Ciencias Físicas,* ]{}\
[*Universidad Nacional Autónoma de México,*]{}\
[*Apdo. Postal 48-3, C.P. 62251 Cuernavaca, Morelos, México.*]{}\
\
[*$^b$Instituto de Física,* ]{}\
[*Benemérita Universidad Autónoma de Puebla,*]{}\
[*Apdo. Postal J-48, C.P. 72570 Puebla, Puebla, México.*]{}\
\
[*$^c$Instituto de Física y Matemáticas,*]{}\
[*Universidad Michoacana de San Nicolás de Hidalgo,*]{}\
[*Edificio C–3, Ciudad Universitaria, C.P. 58040 Morelia, Michoacán, México.*]{}\
\
[*$^d$Instituto de Ciencias Nucleares,* ]{}\
[*Universidad Nacional Autónoma de México,*]{}\
[*Apdo. Postal 70Ð543, 04510 México D. F., México.*]{}\
\
[*$^e$Centro de Investigación en Ciencias,* ]{}\
[*Universidad Autónoma del Estado de Morelos,*]{}\
[*Avenida Universidad 1001, Cuernavaca, Morelos 62209, México.*]{}
title: General bounds in Hybrid Natural Inflation
---
Introduction {#Intro}
============
In a recent article [@Ross:2016hyb] a model of inflation [@Guth:1980zm], [@Linde:1981mu], [@Albrecht:1982wi], [@Lyth:1998xn] of the hybrid type [@Linde:1994] has been studied with great detail. To show that it is posible within Hybrid Natural Inflation (HNI) [@Ross:2016hyb] to account for inflationary energy scales as small as the electroweak scale, or as large as the Grand Unification scale, two versions of the model have been constructed based on simple effective field theories. The resulting inflationary sector in any case is described by the following potential for the inflaton field $\phi$ $$V(\phi) = V_0\left(1+a\cos \left(\frac{\phi}{f} \right) \right),
\label{pot}$$ where $a$ is a positive constant less than one and $f$ is the scale of (Nambu-Goldstone) symmetry breaking. Here the end of inflation is triggered by an independent sector waterfall field in a rapid phase transition. The potential in Eq.(\[pot\]) is reminiscent of Natural Inflation [@Freese:1990rb], [@Adams:1992bn], [@Freese:2014nla] where $a=1$ sets a vanishing cosmological constant. Here, however, $a$ can take any positive value less than one and as a result the scale $f$ can be sub-Planckian. Once the waterfall field triggers the end of inflation the inflaton fast rolls to a global minimum with vanishing energy.
Hybrid Natural Inflation has also the interesting property that the slow-roll parameter $\epsilon(\phi)$ turns out to be a non-monotonic function of the inflaton [@German:2015qjq]. As a consequence the scalar spectrum of density perturbations develops a [*minimum*]{} (Fig.\[Espectro\]) for a value $\phi_{min}$ close to the inflection point of the potential. We know that inflation in HNI should start before $\phi$ reaches the minimum of the spectrum at $\phi_{min}$ because the spectrum has been observed to be decreasing during at least 8 e-folds of observable inflation. Thus, there should be at least 8 e-folds of inflation from $\phi_H$, at which observable perturbations are produced[^1], to $\phi_{min}$. This minimum amount of inflation with decreasing spectrum should give an upper bound for the tensor-to-scalar ratio $r$ and for the scale of inflation $\Delta$. The remaining $42-52$ e-folds of inflation would occur with an steepening spectrum thus care should be taken to not over-produce primordial black holes (PBH) [@Kohri:2007qn], [@Josan:2009qn], [@Carr:2009jm]. Also the fact that the inflationary energy scale $\Delta$ is bounded by the symmetry breaking scale $f$ imposes [*lower*]{} bounds to these quantities, whenever the minimum of the spectrum is reached after $N_{min}\leq 60$ e«folds of inflation. If all of inflation occurs without $\phi$ reaching $\phi_{min}$ no lower bounds are found.
Our paper is organised as follows: in Section \[slow\] we briefly recall expressions for the slow-roll parameters and observables. We also give an effective field theory derivation of the model we study and the hierarchy of energy scales is discussed. As a warming up exercise we initially study in Section \[NI\] this hierarchy of scales in Natural Inflation (where $a=1$) and Section \[ENI\] deals with “extended” Natural Inflation (ENI) where $a$ is not set to unity from the beginning. This allows us to study the fine tuning of $a$ (to have a vanishing cosmological constant) in terms of the parameters of the model. From here we proceed in Section \[restricted\] to HNI where the hierarchy of energy scales together with the observation that the scalar spectrum is decreasing during $8<N_{min}<60$ e-folds of observable inflation determine bounds for the inflationary energy scale $\Delta$ and for the tensor-to-scalar ratio $r$, this we call the restricted case. In Section \[general\] we obtain general bounds in HNI dropping the previous requirement that $N_{min}$ e-folds of inflation occur with decreasing spectrum. We are able to find general bounds for all the parameters (and observables) of the model and to clearly understand how the scale of inflation in HNI is able to sweep all range of values, from vanishingly small to GUT scales. A brief discussion of constraints coming from Primordial Black Hole (PBH) abundances and considerations regarding low scales of inflation can be found in Section \[PBH\]. Finally Section \[conclusions\] contains our conclusions and a discussion of the main results.
Slow-roll parameters, observables and model construction {#slow}
=========================================================
In slow-roll inflation, the spectral indices are given in terms of the slow-roll parameters of the model, which involve the potential $V(\phi)$ and its derivatives (see e.g. [@Liddle:94], [@Liddle:2000cg]) $$\epsilon \equiv \frac{M^{2}}{2}\left( \frac{V^{\prime }}{V }\right) ^{2},\quad
\eta \equiv M^{2}\frac{V^{\prime \prime }}{V}, \quad
\xi_2 \equiv M^{4}\frac{V^{\prime }V^{\prime \prime \prime }}{V^{2}},\quad
\xi_3 \equiv M^{6}\frac{V^{\prime 2 }V^{\prime \prime \prime \prime }}{V^{3}},
\label{Slowparameters}$$primes denote derivatives with respect to the inflaton $\phi$ and $M$ is the reduced Planck mass $M=2.44\times 10^{18} \,\mathrm{GeV}$. In what follows we set $M=1$. In the slow-roll approximation observables are given by (see e.g. [@Liddle:2000cg]) $$\begin{aligned}
n_{t} &=&-2\epsilon =-\frac{r}{8} , \label{Int} \\
n_{s} &=&1+2\eta -6\epsilon , \label{Ins} \\
n_{sk} &=&\frac{d n_{s}}{d \ln k}=16\epsilon \eta -24\epsilon ^{2}-2\xi_2, \label{Insk} \\
n_{skk} &=&\frac{d^{2} n_{s}}{d \ln k^{2}}=-192\epsilon ^{3}+192\epsilon ^{2}\eta-
32\epsilon \eta^{2} -24\epsilon\xi_2 +2\eta\xi_2 +2\xi_3, \label{Inskk} \\
\mathcal{P}_s(k)&=&\frac{1}{24\pi ^{2}}\frac{V}{\epsilon }=A_s \left( \frac{k}{k_H}\right)^{n_s-1} .
\label{IA} \end{aligned}$$ Here $n_{sk}$ denotes the running of the scalar index and $n_{skk}$ the running of the running, in a self-explanatory notation. All the quantities with a subindex ${}_H$ are evaluated at the scale $\phi_{H}$, at which observable perturbations are produced, some $50-60$ e-folds before the end of inflation. The density perturbation at wave number $k$ is $\mathcal{P}_s(k)$ with amplitude at horizon crossing given by $\mathcal{P}_s(k_H)\approx 2.2\times 10^{-9}$ [@Ade:2015xua], the scale of inflation is $\Delta$ with $\Delta \equiv V_{H}^{1/4}$. The tensor power spectrum parameterised at first order in the SR parameters is $$\mathcal{P}_t(k)=A_t \left( \frac{k}{k_H}\right)^{n_t} ,
\label{PotHNI}$$ it allows to define the tensor-to-scalar ratio as $r\equiv \mathcal{P}_t(k)/\mathcal{P}_s(k)$.
The construction of the model can proceed by initially considering a potential of the form $$V\left(\Phi\right) = V_1\left(\Phi\right)+V_2\left(\Phi\right),
\label {V}$$ where $$V_1\left(\Phi\right)= -m^2|\Phi|^2+\lambda |\Phi|^4+\bar{\Delta}^4.
\label {Phi}$$ is the potential invariant under the $U(1)$ symmetry, $\Phi \rightarrow e^{i\alpha}\Phi$. The origin of the constant term $\bar{\Delta}^4$ above can be traced to terms in the higher energy sector of the theory. For positive mass-square $m^2$, $\Phi$ triggers spontaneous breaking of the $U(1)$ symmetry and the field $\Phi$ gets a vev given by $\tilde{f}$ $$<\Phi_0>\, = \tilde{f}= \frac{m}{\sqrt{2\lambda}}.
\label {Phi}$$ Thus, the potential can be better parameterised by $$\Phi= \frac{1}{\sqrt{2}}\left(\rho+\tilde{f}\right)\, e^{i\frac{\phi}{\tilde{f}}},
\label {Phi}$$ where $\rho$ is a radial field around the minimum of the potential and $\phi$ a Goldstone boson associated with the $U(1)$ symmetry breaking.
The term $V_2\left(\Phi\right)$ in Eq. (\[V\]) explicitly breaks the $U(1)$ symmetry and generates a mass for the Goldstone boson becoming $\phi$ a Pseudo Nambu-Goldstone boson. We can write a very simple form for $V_2$ $$V_2\left(\Phi\right)=\mu^2\left(\Phi^2+ \Phi^{*2}\right) \sim \frac{1}{2}\mu^2\tilde{f}^2\left(e^{i\frac{2\phi}{\tilde{f}}} + e^{-i\frac{2\phi}{\tilde{f}}}\right) \sim \mu^2\tilde{f}^2\cos\left(\frac{2\phi}{\tilde{f}} \right)\ ,
\label {explicit}$$ where $\mu$ is a constant with mass dimensions. It follows that the axion potential is $$V(\phi) =V_0\left[1+a\cos\left(\frac{\phi}{f} \right) \right],
\label {pot1}$$ with $V_0$ defined by $V_0\equiv \bar{\Delta}^4-\frac{m^4}{4\lambda}$, $\tilde{f}=2 f$ and $$a\equiv \frac{4\mu^2 f^2}{V_0}.
\label {apar}$$ The parameter $a$ is bounded as $0\leq a\leq1$. The limiting value $a=0$ reduces the potential to a cosmological constant while $a=1$ gives Natural Inflation.
The origin of the two scales occurring in the potential Eq.(\[pot1\]) is, in principle, well understood and we also know that these scales satisfy a hierarchy: a symmetry breaking phenomenon at some energy scale $f$ gives rise to a (massless) Goldstone boson while non-perturbative physics at temperature $T<f$ provides a potential (and a small mass) to the originally massless boson becoming a pseudo-Nambu-Goldstone boson, the inflaton. In the case of the $QCD$ axion, for example, non-perturbative effects are due to instantons. Thus the inflationary energy scale $\Delta$ is bounded by the symmetry breaking scale $f$ as $\Delta\equiv V_H^{1/4} \approx V_0^{1/4} < f.$ In the following sections we use this hierarchy of scales to extract bounds for the observables.
Natural Inflation {#NI}
==================
As a warming up exercise we beguin with Natural Inflation (NI) [@Freese:1990rb]. The potential for the NI model is given by fixing $a=1$ in Eq.(\[pot1\]) above $$V(\phi) = V_0\left(1+\cos \left(\frac{\phi}{f} \right) \right).
\label{Npot}$$ The coefficient of the $\cos \left(\frac{\phi}{f} \right)$-term has been fine-tuned to one so that the potential vanishes at its minimum. We will see that in NI the hierarchy $\Delta < f$ arises in a very natural way. Defining $c_{\phi}\equiv \cos \left( \frac{\phi }{f}\right)$ the slow-roll parameters relevant for what follows are $$\begin{aligned}
\epsilon &=&\frac{1}{2f^2}\frac{1-c_{\mathrm{\phi}} }{ 1+c_{\mathrm{\phi}} } ,
\label{NIeps}%
\\
\eta &=&-\frac{1}{f^2}\, \frac{c_{\mathrm{\phi}}}{1+c_{\mathrm{\phi}}} ,
\label{NIeta}\end{aligned}$$Defining $\delta_{n_s}\equiv 1-n_{s_H}$, $c_{H}\equiv \cos \left( \frac{\phi _{H}}{f}\right)$ and $c_{e}\equiv \cos \left( \frac{\phi _{e}}{f}\right)$, the functions at the observable scale and at the end of inflation, the equation for the spectral index Eq.(\[Ins\]) at $\phi _H$, can be written as $$\delta_{n_s}= \frac{3 -c_H}{f^2(1+c_{H})},
\label{NIspectral}$$ solving for $c_H$ we get $$c_{H_{NI}}\equiv\frac{3-f^2\delta_{n_s}} {1+f^2\delta_{n_s}},
\label{Nch}$$ where the subindices $NI$ means that $c_H$ has been determined from the NI potential Eq.(\[Npot\]). In the following sections dealing with the Hybrid Natural Inflation (HNI) potential instead of writing $HNI$ subindices, they will be simply dropped. The end of inflation is given by the saturation of the condition $\epsilon = 1$ $$c_{e_{NI}}\equiv\frac{1-2f^2} {1+2f^2}.
\label{Nce}$$ The number of e-folds from $\phi_H$ to the end of inflation at $\phi _e$ is given by $$N\equiv -\int_{\phi _H}^{\phi_e}\frac{V({\phi })}{V^{\prime }({\phi })}{d}{\phi }=f^2\ln \left( \frac{1-c_{e_{NI}}}{1-c_{H_{NI}}}\right)=f^2\ln\left[\frac{2f^2(1+f^2 \delta_{n_s})}{(1+2f^2) (-1+f^2 \delta_{n_s}) } \right],
\label{NN}$$ from where it follows that in NI consistency demands that $$f > \frac{1}{\sqrt{\delta_{n_s}}}.
\label{Nfbound}$$ Assuming the spectral index $n_s$ is known, the number of e-folds $N$ fixes the parameter $f$ through Eq.(\[NN\]) which then fixes $\phi_H$ and $\Delta$. For numerical results we take $n_s=0.965$ [@Ade:2015xua] thus $\delta_{n_s}=0.035$ and $f \gtrsim 5.3$. From Eq.(\[IA\]) the scale of inflation can also be expressed in terms of $f$ $$\Delta=\left(24\pi ^{2}\epsilon(\phi_H)\mathcal{P}_s(k_H)\right)^{1/4} =\left(12\pi^2 \mathcal{P}_s(k_H) \frac{1-c_{H_{NI}} }{f^2\left(1+\,c_{H_{NI}}\right)}\right)^{1/4}=\left(6\,\pi^2 \delta_{n_s}\mathcal{P}_s(k_H)\left(1-\frac{1}{f^2 \delta_{n_s}}\right) \right)^{1/4},
\label{NDelta}$$ from where it follows $$\Delta < \left(6\pi^2\, \delta_{n_s}\mathcal{P}_s(k_H)\right)^{1/4},
\label{NDeltabound}$$ or $\Delta \lesssim 8.3\times 10^{-3} M\approx 2\times 10^{16}\, GeV$. From Eqs.(\[Nfbound\]) and (\[NDeltabound\]) we find that $$\Delta < \left(6\pi^2\delta_{n_s}\mathcal{P}_s(k_H)\right)^{1/4}< \left(6\pi^2\delta_{n_s}^3\mathcal{P}_s(k_H)\right)^{1/4} f\approx 1.6 \times 10^{-3} f < f,
\label{NDeltalessf}$$ thus $\Delta < f$ always. We will see that for this to be the case in HNI the parameter $a$ in the potential has to be bounded from above. The resulting bound for $r$ coming from Eqs.(\[IA\]) and (\[NDeltabound\]) is $$r= \frac{2 \Delta^4}{3 \pi^2 \mathcal{P}_s(k_H)} < 4\delta_{n_s}\approx 0.14.
\label{NDeltalessf}$$ Of course, in NI, once we have determined $f$ by fixing the number of e-folds $r$ follows: for $\delta_{n_s}=0.035$, Eq.(\[NN\]) with $N=60$ requires $f\approx 8.45$ giving a value $r\approx 0.084$. From Eqs.(\[NDelta\]) and Eq.(\[apar\]) we find that tuning $a=1$ is equivalent to tuning the $\mu$ parameter to the value $$\mu^2=\frac{3\,\pi^2}{2 f^2} \left(1-\frac{1}{f^2 \delta_{n_s}}\right) \delta_{n_s}\mathcal{P}_s(k_H),
\label{lambda2}$$ using the numerical values above, Eq.(\[lambda2\]) gives $\mu\approx 3.1 \times 10^{-6} M\approx 7.6 \times 10^{12}GeV$.
Extended Natural Inflation {#ENI}
===========================
We now make a simple extension of the NI model discussed above leaving the parameter $a$ as coefficient of the $\cos \left(\frac{\phi}{f} \right)$-term in the potential Eq.(\[pot1\]) . This parameter is now restricted to the interval $0<a<1$. The potential is thus $$V(\phi) = V_0\left(1+a \cos \left(\frac{\phi}{f} \right) \right).
\label{Hpot}$$ A potential like Eq.(\[Hpot\]) has been studied in the context of Hybrid Natural Inflation (HNI) [@Ross:2016hyb], [@Ross:2009hg], [@Ross:2010fg], where the symmetry breaking scale is sub-Planckian and the end of inflation is triggered by an additional waterfall field. Here we would like to study a simple extension of NI (ENI) allowing (as in NI) super-Planckian values for $f$. This would allow us to study the fine tuning of $a$ in terms of the parameters of the model and see what requirements (if any) the inflationary dynamics impose on them.
We now attempt a similar analysis as in Section\[NI\]. The slow-roll parameters are now given by $$\begin{aligned}
\epsilon &=&\frac{1}{2}\left(\frac{a}{f}\right)^2\frac{1-c_{\mathrm{\phi}} ^{2}}{\left( 1+a\, c_{\mathrm{\phi}} \right)^2} ,
\label{HNIeps}%
\\
\eta &=&-\left( \frac{a}{f^2}\right)\, \frac{c_{\mathrm{\phi}}}{1+a c_{\mathrm{\phi}}} ,
\label{HNIeta}
\\
\xi_2 &=&-\left( \frac{a}{f^2}\right)^2\,\frac{1-c_{\mathrm{\phi}} ^{2}}{\left( 1+a\, c_{\mathrm{\phi}} \right)^2} ,\\
\xi_3 &=& +\left( \frac{a}{f^2}\right)^3\,\frac{1-c_{\mathrm{\phi}} ^{2}}{\left(1+a c_{\mathrm{\phi}}\right)^3} c_{\mathrm{\phi}} .\end{aligned}$$From the expression for the scalar spectral index Eq.(\[Ins\]) we now get $$\delta_{n_s}=\frac{a}{f^{2}}\, \frac{2 c_H+a(3-c_H^2)}{(1+ac_{H})^{2}}. \label{spectral2}$$ At $\phi _{H}$ Eq.(\[spectral2\]) can be solved for $c_H$ [@Ross:2010fg] obtaining the following $$c_{1H}\equiv\frac{1-f^2\delta_{n_s}+\sqrt{1+3a^2-3(1-a^2)f^2\delta_{n_s}}}{a(1+f^2\delta_{n_s})},\text{\ }a\geqslant \frac{1}{3} ,
\label{solution1}$$ and $$c_{2H}\equiv\frac{1-f^2\delta_{n_s}-\sqrt{1+3a^2-3(1-a^2)f^2\delta_{n_s}}}{a(1+f^2\delta_{n_s})},\; a<1.
\label{solution2}$$ The first solution in the limit $a=1$ corresponds to NI. We thus study here the solution $c_{1H}$ only and leave $c_{2H}$ for the following sections. From the constraints $-1<c_{1H}<1$ we get restrictions on the $f$ and $a$ parameters $$\begin{aligned}
\frac{1}{\sqrt{2 \delta_{n_s}}} & < f < &\frac{1}{\sqrt{\delta_{n_s}}}\,,\quad\quad\quad \frac{1}{\sqrt{3}}\left(\frac{3f^2\delta_{n_s}-1} {f^2\delta_{n_s}+1}\right)^{1/2}< a \leq \frac{f^2\delta_{n_s}} {2-f^2\delta_{n_s}} ,
\label{c1Ha}%
\\
\frac{1}{\sqrt{\delta_{n_s}}} & < f\,, &\quad\quad\quad\quad\quad\quad \frac{1}{\sqrt{3}}\left(\frac{3f^2\delta_{n_s}-1} {f^2\delta_{n_s}+1}\right)^{1/2}< a \leq 1,
\label{c1Hb}\end{aligned}$$or using $\delta_{n_s}\approx 0.035$ $$\begin{aligned}
3.78 & < f < &5.35\,,\quad\quad\quad \frac{1}{3}< a \leq 1,
\label{nc1Ha}%
\\
5.35 & < f\,, &\quad\quad\quad\quad\quad\quad \frac{1}{\sqrt{3}} < a \leq 1.
\label{nc1Hb}\end{aligned}$$Thus, we see that the extension of NI corresponds to Eq.(\[solution1\]) together with Eq.(\[c1Hb\]). In NI the end of inflation is given by $\epsilon=1$, here from Eq.(\[HNIeps\]) we find $$\begin{aligned}
c_{e1} & = &-\frac{2f^2-\sqrt{ a^2-2(1-a^2)f^2 }}{a(1+2f^2)},
\label{ce1}%
\\
c_{e2} & = &-\frac{2f^2+\sqrt{ a^2-2(1-a^2)f^2 }}{a(1+2f^2)}.
\label{ce2}\end{aligned}$$Thus, there are two values of $\phi_e$ which can meet the condition $\epsilon=1$ and end inflation. The number of e-folds from $\phi_{{\rm H}}$ to the end of inflation at $\phi _{\mathrm{e}}$ is $$N\equiv -\int_{\phi _H}^{\phi_e}\frac{V({\phi })}{V^{\prime }({\phi })}{d}{\phi }=\frac{f^2}{2a}\left((1+a)\ln \left( \frac{1-c_e}{1-c_H}\right)+(1-a)\ln \left( \frac{1+c_H}{1+c_e}\right)\right),
\label{N}$$ where $c_{e}\equiv \cos \left( \frac{\phi _{e}}{f}\right)$. In ENI we thus have $$N_i=\frac{f^2}{2a}\left((1+a)\ln \left( \frac{1-c_{ei}}{1-c_{1H}}\right)+(1-a)\ln \left( \frac{1+c_{1H}}{1+c_{ei}}\right)\right),\quad\quad i=1,2.
\label{Ni}$$ The function $N_i(f,a)$ defines a two-dimensional sheet tightly folded (see Fig.\[folded\]) as a consequence the number of e-folds $N_1$ is not very different from $N_2$. Thus in what follows we simply talk about the number of e-folds $N$ meaning any of them. In any case the requirement of $N=60$ e-folds of inflation begs for the parameter $a$ to be very close to 1. This is so because the ENI potential esentially differs from NI by a constant term. The inflationary epoch is controled by the derivative of the potential. Thus, a constant term in the potential is not very relevant in the determination of the number of e-folds and hence of the parameters $f$ and $a$; these are very close to the NI values. Not contributing to the cosmological constant problem leads us to impose $a=1$. Thus, a modification of NI to avoid the fine tuning problem requires a more drastic solution. In the following sections we continue our discussion in the context of HNI [@Ross:2016hyb], [@Ross:2009hg], [@Ross:2010fg] where the end of inflation is not due to the saturation of the condition $\epsilon = 1$ but to the destabilisation of the inflaton direction by an extra[ *waterfall*]{} field. This approach liberates the inflationary sector from also ending inflation.
Hybrid Natural Inflation, bounds in the restricted case {#restricted}
=======================================================
The first solution Eq.(\[solution1\]) in the limit $a=1$ corresponds to *Natural Inflation*. However to avoid the possibility of large gravitational corrections to the potential we will concentrate on the case $f<1.$[^2] Thus, the relevant solution is the second one denoted by $c_{2H}$, hereafter $c_H$, given by $$c_H=\frac{1-f^2\delta_{n_s}-\sqrt{1+3a^2-3(1-a^2)f^2\delta_{n_s}}}{a(1+f^2\delta_{n_s})},
\label{ch}$$ corresponds to $c_{2H}$ of Eq.(\[solution2\]). From Eq.(\[ch\]) note that $c_H=1$ when $a=\frac{f^2\delta_{n_s}}{2-f^2\delta_{n_s}}$. Thus, in principle one can have very small $r$ (see Eq.(\[HNIeps\])) and very low-scale of inflation (low $\Delta$, see Eq.(\[IA\])) when $a$ gets close to $\frac{f^2\delta_{n_s}}{2-f^2\delta_{n_s}}$ for any value of $f<1$.
The density perturbation for the HNI potential can be written as $$\mathcal{P}_s(k) =\frac{1}{24\pi ^{2}}\frac{V(\phi)}{\epsilon(\phi)}=\frac{f^2 V_0\left(1+a\cos(\frac{\phi}{f})\right)^3}{12\pi^2 a^2\left(1-\cos(\frac{\phi}{f})^2\right)},
\label{contraste}$$ which presents a minimum (see Fig.\[Espectro\]) for $\phi_{min}$ given by $$c_{{min}}=\cos\left(\frac{\phi_{min}}{f}\right)=\frac{1-\sqrt{1+3a^2}}{a} \approx -\frac{3}{2}a.
\label{cmin}$$ This together with some other properties related with the non-monotonicity of the tensor-to-scalar ratio are studied in [@German:2015qjq]. From the equation for the number of e-folds Eq.(\[N\]) we provisionally take (in what we call the restricted case) the end of the first $8<N_{min}<60$ e-folds of inflation as given by $c_{min}$, this will determine $a$ for a given $f$. The formula for $N_{min}$ e-folds adapted from Eq.(\[N\]) is $$N_{min}=\frac{f^2}{2a}\left((1+a)\ln \left( \frac{1-c_{min}}{1-c_{H}}\right)+(1-a)\ln \left( \frac{1+c_H}{1+c_{min}}\right)\right).
\label{Nmin}$$ We first study this formula analytically for small $a$. The expansion of $c_H$ for small $a$ is given by $$c_H= \frac{1}{2}\left(\frac{f^2 \delta_{n_s}}{a}\right)-\frac{3}{2}\left(1-\frac{5}{12}\left(\frac{f^2 \delta_{n_s}}{a}\right)^2\right)a+\cdot\cdot\cdot \approx \frac{1}{2}\left(\frac{f^2 \delta_{n_s}}{a}\right) +\mathcal{O}(a).
\label{chapp}$$ For small $a$ the term $c_{min}$ is also small and negligible inside the $\log$ of Eq.(\[Nmin\]) thus, $N_{min}$ can be approximated by $$N_{min}\approx\frac{f^2}{2a}\ln \left( \frac{1+c_H}{1-c_{H}}\right)\approx \frac{1}{2 \delta_{n_s}}\left(\frac{f^2 \delta_{n_s}}{a}\right)^2+\frac{1}{24 \delta_{n_s}}\left(\frac{f^2 \delta_{n_s}}{a}\right)^4+\cdot\cdot\cdot,
\label{N8app}$$ from where it follows that $\frac{f^2}{a}$ is approximately constant $$\frac{f^2}{a}\approx \frac{ \sqrt{6}}{\delta_{n_s}} \left(N_s-1\right)^{1/2}.
\label{constant}$$ Here we have defined $N_s\equiv \sqrt{ 1+\frac{2}{3}N_{min}\delta_{n_s} }$ to simplify notation. From $$\Delta=\left(24\pi ^{2}\epsilon(\phi_H)\mathcal{P}_s(k_H)\right)^{1/4} =\left(12\pi^2 \mathcal{P}_s(k_H) \left(\frac{a}{f}\right)^2 \frac{1-c_H^2 }{\left(1+a\,c_H\right)^2}\right)^{1/4},
\label{Delta}$$ we get $$\Delta\approx\left(\pi^2 \delta_{n_s}^2\mathcal{P}_s(k_H)\left(\frac{5-3N_s } {N_s -1}\right)+\mathcal{O}(a) \right)^{1/4} f^{1/2}.
\label{Deltaap}$$ The behavior $\Delta \sim f^{1/2}$ means that $\Delta$ decreases more slowly than $f$. Thus, there is a point where $\Delta=f$ signaling the minimum value of $f$ consistent with $\Delta< f$. Solving $\Delta=f$ to lowest order in $a$ $$f_{min}\approx \pi \delta_{n_s}\mathcal{P}^{1/2}_s(k_H)\left(\frac{5-3N_s} {N_s-1}\right)^{1/2},
\label{fmin}$$ from where it follows that $N_{min}<76$, sufficient for our purpose. From Eqs.(\[constant\]) and (\[Deltaap\]) with $f=f_{min}$ we get the lower limit for $\Delta$ while setting $f=1/\pi$ (for consistency with $\Delta\phi<1$) gives the upper bound $$\Delta_{min}\equiv \pi\delta_{n_s}\,\mathcal{P}^{1/2}_s(k_H)\left(\frac{5-3N_s} {N_s-1}\right)^{1/2} <\Delta< \left(\delta_{n_s}^2\mathcal{P}_s(k_H)\left(\frac{5-3N_s } {N_s -1}\right) \right)^{1/4}\equiv \Delta_{max}.
\label{Deltabounded}$$
The upper limit follows simply from the requirement that $\Delta\phi<1$ and it is not derived from any stronger condition. When there is a fixed number of e-folds $N_{min}<60$ from $\phi_H$ to the minimum at $\phi_{min}$ the scale of inflation as a function of $N_{min}$ is bounded as $\Delta_{min} < \Delta < \Delta_{max}$, see Fig.\[DB\]a. For small $a$ we can also see that the tensor-to-scalar ratio scales with $a$ as follows $$r\approx \frac{2\sqrt{6}}{3}\,\delta_{n_s}\left(N_s -1\right)^{1/2} \left(\frac{5-3N_s}{N_s-1}\right)a+ \mathcal{O}(a^2),
\label{rapp}$$ and so becomes small for small $a$. From Eqs.(\[IA\]) and (\[Deltabounded\]) $r$ is bounded as follows, (see Fig.\[DB\]b), $$r_{min}\equiv \frac{2}{3}\pi^2\,\delta_{n_s}^4\,\mathcal{P}_s(k_H)\left(\frac{5-3N_s}{N_s-1}\right)^2 < r < \frac{2}{3\pi^2}\delta_{n_s}^2\left(\frac{5-3N_s}{N_s-1}\right)\equiv r_{max}.
\label{rbounded}$$ From Eq.(\[IA\]) we see that the scale of inflation decreases much more slowly than the tensor-to-scalar ratio, $\Delta\equiv V_H^{1/4} \sim a^{1/4}$.
In Ref.[@Hebecker:2013zda], HNI was already given a detailed analysis for its feasibility to reach large values of $r$, for sub-Planckian axion decay constant and sub-Planckian field range. There, a constraint on $f$ coming from an embedding of the effective potential of Eq. into a string theory guide the authors to choose the fiducial bound $f\lesssim \frac{\sqrt{3}}{4\pi}$ giving an upper bound $r\simeq 7.6 \times 10^{-4}$ [@Hebecker:2013zda]. Using the same bound for $f$ our upper bound on $r$ changes by a factor of 3/16, from $r\simeq 1.6 \times 10^{-3}$ to $r\simeq 3 \times 10^{-4}$, for $N_{min}=8$ and $n_s=0.965$ as can be seen from Eq. .
The expressions for the spectral indices are $$\begin{aligned}
n_{sk} &\approx&\frac{\delta_{n_s}^2}{6} \left(\frac{5-3N_s} {N_s-1}\right)+ \mathcal{O}(a), \\
n_{skk} &\approx&\frac{\delta_{n_s}^3}{6} \left(\frac{5-3N_s} {N_s-1}\right)+ \mathcal{O}(a), \label{nskkapp}\end{aligned}$$ both are practically constant for small $a$. For numerical values see at the end of Section \[general\].
Hybrid Natural Inflation, bounds in the general case {#general}
====================================================
In the restricted case discussed in section \[restricted\] the number of e-folds $8<N_{min}<60$ is counted from $\phi_H$ to $\phi_{min}$ with the remaining e-folds occurring with increasing spectrum. If all $N=60$ e-folds occur before $\phi$ reaches $\phi_{min}$ we can not use $\phi_{min}$ to count the number of e-folds and we are in the general case where the end of inflation is dictated by the waterfall sector of the theory. From Eq.(\[Delta\]), the equation $\Delta=f$ can actually be solved exactly for $a$ as a function of $f$, we denote this solution by $a_{max}$ $$a_{max}=f^2\left(\frac{\sqrt{3f^8+16f^2\pi^2 \mathcal{P}_s(k_H)-24 f^4\pi^2 \mathcal{P}_s(k_H)\delta_{n_s} +48\pi^4\mathcal{P}_s(k_H)^2\delta_{n_s}^2}}{\sqrt{3}\left(f^6+8\pi^2 \mathcal{P}_s(k_H)-4 f^2\pi^2 \mathcal{P}_s(k_H)\delta_{n_s}\right)} \right),
\label{amax}$$ but contrary to the $N_{min}$ case $a/f^2$ is not approximately constant but clearly depends on $f$. The condition $\Delta < f$ restricts the value of the parameter $a$ to be less than $a_{max}$ for a given $f$. On the other hand the slow-roll condition $a/f^2<1$ (coming from $\eta<1$) restricts $f$ such that the formula for $a_{max}$ is only valid up to the value of $f$, denoted by $f_1$, such that the term in parenthesis in the r.h.s. of Eq.(\[amax\]) equals one, this occurs for $f_{1}\approx 2\pi\sqrt{3 \mathcal{P}_s(k_H)}\approx 5.21 \times 10^{-4}$ (left panel in Fig.\[B2\]). Thus $a_{max}$ is well approximated by $$a_{max}\approx \frac{f^2\delta_{n_s}}{2} \sqrt{1+\frac{f^2}{3\pi^2\delta^2_{n_s}\,\mathcal{P}_s(k_H)}}, \quad\quad f < f_1.
\label{amaxapp}$$ For $f$ larger than $f_{1}$ the condition $\Delta < f$ is always satisfied whenever $a$ is restricted by the stronger condition $a<f^2$. Thus for a given $f < f_{1}$ we get the bounds for $a$ $$a_{min}\equiv \frac{f^2\delta_{n_s}}{2-f^2\delta_{n_s}} < a < a_{max}\,, \quad\quad f < f_1,
\label{abound1}$$ while for $f > f_{1}$ the bounds are $$\frac{f^2\delta_{n_s}}{2-f^2\delta_{n_s}} < a < f^2\,, \quad\quad f > f_1.
\label{abound2}$$ the lower bound in Eqs.(\[abound1\]) and (\[abound2\]) comes from requiring $c_H < 1$ in Eq.(\[ch\]). Thus as we lower the value of $f$ the range of possible $a$ values reduces according to the bounds above. In Fig.\[B2\] (right panel) for any value of $f$ all possible $a$-values define a vertical line in the shaded region.
As a consequence $\Delta$ will be bounded by $f(<f_1)$ whenever $a<a_{max}$ , in this case $r$ is bounded as follows $$0 < r < \frac{2 f^4}{3\pi^2\, \mathcal{P}_s(k_H)}\,, \quad\quad\quad\quad f < f_1,
\label{rGeneralBounded}$$ while for $f > f_{1}$ the scale of inflation can have any value in the interval $0<\Delta <f $ whenever $a<f^2$. Thus, in the general case $\Delta$ is unbounded from below since this was a consequence of fixing the number of e-folds from $\phi_H$ to $\phi_{min}$ to a certain number $N_{min}\leq 60$. In the general case the end of inflation can occur before $\phi$ reaches the minimum of the spectrum at $\phi_{min}$ and no relation beetwen $a$ and $f$ for a fixed $N$ can be found because $\phi_e$ is undetermined. The scale of inflation $\Delta$ is vanishingly small for $a$ approaching $\frac{f^2\delta_{n_s}}{2-f^2\delta_{n_s}}$ because $c_H$ tends to one in that limit. As $a$ goes from $ \frac{f^2\delta_{n_s}}{2-f^2\delta_{n_s}}$ to its upper limit $c_H$ diminishes from 1 and the scale $\Delta$ grows from very small values, this is how the potential is able to cover the whole range of inflationary scales (see Fig.\[B3\]).
Finally the reader would have noticed that we can apply Eq.(\[amax\]) directly to the restricted case, substituting in Eq.(\[Nmin\]) for the number of e-folds from $\phi_H$ up to $\phi_{min}$, extract the values of $f$ which accommodate from 60 to 8 e-folds and evaluate all other quantities of interest. While this is certainly possible and the numerical ranges are given below in Section \[restricted\] we wanted to obtain an approximated [*analytical*]{} expression which can teach us more than a few simple numbers. The corresponding ranges for the [*lower*]{} bounds when the number of e-folds $N_{min}$ go from 8 to 60 are given below (see also Fig. \[B2\]), $$\begin{aligned}
5.75 \times 10^{-6}&< f < & 2.32\times 10^{-5}, \\
6.84\times 10^{-13}&< a_{max} < &2.58\times 10^{-11}, \\
60&>N_{min}>&8, \\
1.40 \times 10^{13}GeV&<\Delta<&5.67\times 10^{13}GeV, \\
3.22 \times 10^{-14}&<r<&8.57 \times 10^{-12}, \\
2.43 \times 10^{-4}&<n_{sk}<&3.97\times 10^{-3}, \\
8.52 \times 10^{-6}&<n_{skk}<&1.39\times 10^{-4}, \label{ranges}\end{aligned}$$ which compare very well with values obtained from the analytical approximations of Section\[restricted\]. Clearly, in the general case we cannot do this because the formula for the number of e-folds Eq.(\[N\]) involves $c_{e}$ which can depend on parameters different from $a$ and $f$.
PBH constraints and low scales of inflation {#PBH}
===========================================
The steepening of the scalar spectrum in HNI (Fig.\[Espectro\]) gives rise to a positive running allowing for the possibility of primordial black hole production during inflation [@Kohri:2007qn], [@Ross:2016hyb]. In terms of the wave number $k$ the scalar power spectrum at first order in the SR parameters is given by $$\mathcal{P}_s(k)=A_s\left( \frac{k}{k_H}\right)^{(n_s-1) + \frac{1}{2}n_{sk} \ln\left(\frac{k}{k_H}\right) +\, \cdot \, \cdot \, \cdot }.\label{power}$$ Due to the constraint coming from the possible over-production of primordial black holes (PBHs) at the end of inflation the Taylor expansion of the power spectrum around its value at horizon crossing, $N_{H}\approx 60$ is bounded by [^3], $$\label{ps:expansion}
C_{PBH}\equiv \ln \left[\frac{\mathcal{P}_s(0)}{\mathcal{P}_s(N_H)}\right] = (n_s- 1) N_H + \frac{1}{2} n_{sk} N_H^2\leq 14,$$ where $\mathcal{P}_s(N= 0)\simeq 10^{-3}$ (see also Refs. [@Josan:2009qn; @Carr:2009jm]) evolves from the initial value $\mathcal{P}_s(N_H) \approx 10^{-9}$. This gives the bound $n_{sk}< 10^{-2}$. For the HNI potential this constraint can be written as $$C_{PBH}=\ln\left[\frac{(1-c_H^2)(1+a\, c_e)^3}{(1-c_e^2)(1+a\, c_H)^3} \right],
\label{pbh}$$ and can be easily satisfied for all cases discussed since here $C_{PBH}<3$. A more stringent bound may be set by PBHs produced during reheating [@Hidalgo:2017dfp], [@Carr:2017edp]. However, this depends on the specific reheating model and its associated equation of state. Such restrictions will be explored elsewhere.
Low scales of inflation $\Delta$ can be obtained when $r$ is very small since, from Eq.(\[IA\]), $\Delta\sim r^{1/4}$. On the other hand, from Eq.(\[HNIeps\]) we see that $r$ is small when $c_H$ is very close to 1. This occurs for $a$ approaching the lower bound in Eqs.(\[abound1\]) and (\[abound2\]). Notice that the parameters $a$ and the scale of symmetry breaking $f$ need not be very small to give small inflationary scales, instead they should be closely related by $a\approx \frac{f^2 \delta_{ns}}{2-f^2 \delta_{ns}}$. In the absence of a mechanism which sets $a$ close to $\frac{f^2 \delta_{ns}}{2-f^2 \delta_{ns}}$ this is fine tuning which is equivalent to starting inflation with $\phi_H$ very close to the origin. In any case the value of $\phi_{H}$ should exceed the quantum fluctuations of the field $\delta\phi\approx\frac{H}{2\pi}\approx\frac{\Delta^2}{2\pi\sqrt{3}}$. From Eqs.(\[HNIeps\]) and (\[IA\]) we get the small $\phi$ behaviour $$r \approx \frac{\delta_{ns}^2}{2} \phi_{H}^2=\frac{2}{3 \pi^2A_s}\Delta^4,
\label{rsmall}$$ from where it follows that $$\phi_{H}\approx\left(\frac{4}{3 \pi^2A_s \delta_{ns}^2}\right)^{1/2}\Delta^2\approx 2.2\times10^5\Delta^2>>\frac{\Delta^2}{2\pi\sqrt{3}}=\delta\phi.
\label{rsmall}$$
Summary and conclusions {#conclusions}
=======================
An interesting characteristic of Hybrid Natural Inflation is that the tensor-to-scalar ratio is a non-monotonic function of $\phi$ with the parameter $\epsilon(\phi)$ developing a [*maximum*]{}. A consequence of this is that the scalar spectrum of density fluctuations develops a [*minimum*]{} for some value $\phi_{min}$ of the inflaton. The value of $\phi_{min}$ is always slightly larger than the value of $\phi$ at the inflection point of the potential at $\phi_{I}=\pi/2$. Since the scalar spectrum has been observed to be decreasing during some 8 e-folds of observable inflation we can determine upper bounds for the scale of inflation and for the tensor-to-scalar ratio. In the [*restricted*]{} case considered in section \[restricted\] we do this by requiring a minimum of $8<N_{min}<60$ e-folds of inflation from the scale $\phi_{H}$, at which observable perturbations are produced, to $\phi_{min}$ where the spectrum stops decreasing. The remaining $0-52$ e-folds of inflation would occur with an steepening spectrum thus care is taken to not over-produce primordial black holes. The condition of having $N_{min}$ e-folds of inflation with a decreasing spectrum fixes the parameter $a$ once the symmetry breaking scale $f$ is determined. A minimum value for $f$ can be obtained by the requirement that the inflationary energy scale $\Delta$ is bounded by $f$. This allows to determine [*lower*]{} bounds for the inflationary energy scale and the tensor-to-scalar ratio. The general case is discussed in section \[general\] where we find upper bounds for $\Delta$ as well as $r$. In the general case the inflationary scale $\Delta$ is unbounded from below and can go all the way to vanishing values. The lower bound of section \[restricted\] is a consequence of fixing the number of e-folds from $\phi_H$ to $\phi_{min}$ to a certain number $N_{min}\leq 60$. In the general case the end of inflation can occur before $\phi$ reaches the minimum of the spectrum at $\phi_{min}$ and no relation beetwen $a$ and $f$ for a fixed $N$ can be found because $\phi_e$ is undetermined. The scale of inflation $\Delta$ is vanishingly small for $a$ approaching $\frac{f^2\delta_{n_s}}{2-f^2\delta_{n_s}}$ because $c_H$ tends to one in that limit. As $a$ goes from $ \frac{f^2\delta_{n_s}}{2-f^2\delta_{n_s}}$ to its upper limit $c_H$ diminishes from 1 and the scale $\Delta$ grows from very small values, this is how the potential is able to sweep the whole range of inflationary scales. By finding lower as well as upper bounds for the parameters $a$ and $f$ we can clearly understand how the scale of inflation in HNI is able to cover the complete range of values, from vanishingly small up to the GUT scale.
Acknowledgements
================
We are grateful to SNI for partial financial support. AHA also acknowledges a VIEP-BUAP-HEAA-EXC17-I research grant. We acknowledge financial support from PAPIIT-UNAM grant IA-103616 [*Observables en cosmología relativista*]{} as well as CONACyT grants 269639 and 269652.
[99]{}
G. G. Ross, G. German and J. A. Vazquez, JHEP [**1605**]{} (2016) 010 doi:10.1007/JHEP05(2016)010 \[arXiv:1601.03221 \[astro-ph.CO\]\]. A. H. Guth, Phys. Rev. D [**23**]{} (1981) 347. doi:10.1103/PhysRevD.23.347 A. D. Linde, Phys. Lett. B [**108**]{} (1982) 389. doi:10.1016/0370-2693(82)91219-9 A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. [**48**]{} (1982) 1220. doi:10.1103/PhysRevLett.48.1220 For reviews see e.g., D. H. Lyth and A. Riotto, Phys. Rept. [**314**]{} (1999) 1 \[hep-ph/9807278\]; D. H. Lyth and A. R. Liddle, Cambridge, UK: Cambridge Univ. Pr. (2009) 497 p; D. Baumann, “TASI Lectures on Inflation,” arXiv:0907.5424 \[hep-th\]. A. Linde, Phys. Rev. D 49 (1994) 748; Other general references for hybrid inflation are: D.H. Lyth, A. Riotto, Phys. Rep. 314 (1999) 1, arXiv:hep-ph/9807278; A.D. Linde, Lecture Notes in Phys. 738 (2008) 1, arXiv:0705.0164 \[hep-th\]; M. Shaposhnikov, J. Phys. Conf. Ser. 171 (2009) 012005.
K. Freese, J. A. Frieman and A. V. Olinto, Phys. Rev. Lett. [**65**]{} (1990) 3233. doi:10.1103/PhysRevLett.65.3233 F. C. Adams, J. R. Bond, K. Freese, J. A. Frieman and A. V. Olinto, Phys. Rev. D [**47**]{} (1993) 426 doi:10.1103/PhysRevD.47.426 \[hep-ph/9207245\]. K. Freese and W. H. Kinney, JCAP [**1503**]{} (2015) 044 doi:10.1088/1475-7516/2015/03/044 \[arXiv:1403.5277 \[astro-ph.CO\]\]. G. German, A. Herrera-Aguilar, J. C. Hidalgo and R. A. Sussman, JCAP [**1605**]{} (2016) no.05, 025 doi:10.1088/1475-7516/2016/05/025 \[arXiv:1512.03105 \[astro-ph.CO\]\]. M. Carrillo-Gonzalez, G. German, A. Herrera-Aguilar, J. C. Hidalgo and R. Sussman, Phys. Lett. B [**734**]{} (2014) 345 doi:10.1016/j.physletb.2014.05.062 \[arXiv:1404.1122 \[astro-ph.CO\]\]. K. Kohri, D. H. Lyth and A. Melchiorri, JCAP [**0804**]{} (2008) 038 doi:10.1088/1475-7516/2008/04/038 \[arXiv:0711.5006 \[hep-ph\]\]. A. S. Josan, A. M. Green and K. A. Malik, Phys. Rev. D [**79**]{} (2009) 103520 \[arXiv:0903.3184 \[astro-ph.CO\]\]. B. J. Carr, K. Kohri, Y. Sendouda and J. Yokoyama, Phys. Rev. D [**81**]{} (2010) 104019 \[arXiv:0912.5297 \[astro-ph.CO\]\]. A. R. Liddle, P. Parsons and J. D. Barrow, Phys. Rev. D [**50**]{} (1994) 7222 \[astro-ph/9408015\]. A. R. Liddle and D. H. Lyth, *Cosmological Inflation and Large-Scale Structure*, Cambridge University Press, (2000).
P. A. R. Ade [*et al.*]{} \[Planck Collaboration\], arXiv:1502.01589 \[astro-ph.CO\]. P. A. R. Ade [*et al.*]{} \[BICEP2 and Planck Collaborations\], “Joint Analysis of BICEP2/Keck Array and Planck Data,” Phys. Rev. Lett. [**114**]{} (2015) 101301 doi:10.1103/PhysRevLett.114.101301 \[arXiv:1502.00612 \[astro-ph.CO\]\].
G. G. Ross and G. German, Phys. Lett. B [**684**]{} (2010) 199 \[arXiv:0902.4676 \[hep-ph\]\]. G. G. Ross and G. German, Phys. Lett. B [**691**]{} (2010) 117 \[arXiv:1002.0029 \[hep-ph\]\].
A. Hebecker, S. C. Kraus and A. Westphal, Phys. Rev. D [**88**]{} (2013) 123506 doi:10.1103/PhysRevD.88.123506 \[arXiv:1305.1947 \[hep-th\]\]. J. C. Hidalgo, J. De-Santiago, G. German, N. Barbosa-Cendejas and W. Ruiz-Luna, arXiv:1705.02308 \[astro-ph.CO\]. B. Carr, T. Tenkanen and V. Vaskonen, arXiv:1706.03746 \[astro-ph.CO\].
[^1]: All quantities with a subindex ${}_H$ are evaluated at the scale $\phi_{H}$, at which observable perturbations are produced, some $50-60$ e-folds before the end of inflation.
[^2]: Recall that $f>\frac{1}{\sqrt{\delta_{n_s}}} \approx 5.3$ in NI.
[^3]: To lowest order in slow-roll $d/dN =- d / d\ln k$. The next order term in the expansion of Eq. , involving the parameter $n_{skk}$, is subdominant.
| ArXiv |
---
abstract: 'Effective bounds for the finite number of surjective holomorphic maps between canonically polarized compact complex manifolds of any dimension with fixed domain are proven. Both the case of a fixed target and the case of varying targets are treated. In the case of varying targets, bounds on the complexity of Chow varieties are used.'
address: |
Ruhr-Universität Bochum\
Fakultät für Mathematik\
D-44780 Bochum\
Germany
author:
- Gordon Heier
title: Effective finiteness theorems for maps between canonically polarized compact complex manifolds
---
Effective bounds for automorphism groups {#autsection}
========================================
Hurwitz proved the following effective finiteness theorem on Riemann surfaces.
\[Hurbound\] Let $X$ be a smooth compact complex curve of genus $g\geq 2$. Then the group $\operatorname{{Aut}}(X)$ of holomorphic automorphisms of $X$ satisfies $$\#\operatorname{{Aut}}(X)\leq84(g-1).$$
For many years after Hurwitz’s proof, this bound has been known to be sharp only for $g=3$ and $g=7$, in which cases there exist, respectively, the classical examples of the Klein quartic in ${{\mathbb{P}}}^2$ given by the homogeneous equation $X^3Y+Y^3Z+Z^3X=0$ and the Fricke curve with automorphism group ${\rm PSL}(2,8)$. Using the theory of finite groups, it was established only in the 1960’s by Macbeath that there are infinitely many $g$ for which the above bound is sharp (see [@Macbeath]).
Xiao was able to establish the following direct (and clearly sharp due to the above) generalization of Hurwitz’s theorem.
Let $X$ be a $2$-dimensional minimal compact complex manifold of general type. Then $$\#\operatorname{{Aut}}(X)\leq(42)^2K_X^2.$$
In arbitrary dimension, the automorphism group of a smooth compact complex manifold of general type is still known to be finite because of the finiteness theorem of Kobayashi-Ochiai ([@KobOch]), which we shall state in the next section. One is tempted to conjecture that in the case of the canonical line bundle being big and nef or even ample, there is an upper bound of the form $C_nK_X^n$. The preprint [@Ts] makes an attempt to prove this conjecture.
In the paper [@Sza], Szabó was able to establish the following effective polynomial upper bound in arbitrary dimension.
\[szabobound\] Let $X$ be an $n$-dimensional compact complex manifold whose canonical line bundle is big and nef. Then the number of birational automorphisms of $X$ is no more than $$(2(n+1)(n+2)!(n+2)K_X^n)^{16n3^n}.$$
The multiple $2(n+1)(n+2)!(n+2)K_X$ is large enough to give a birational morphism from $X$ to projective space. This is proven in [@CaSchn page 8], using results of Demailly [@DBound] and Kollár [@Koeffbase] on effective base point freeness of adjoint line bundles. The goal of [@CaSchn] is to obtain a polynomial bound for the special case of automorphism groups that are abelian.
In arbitrary dimension, effective pluricanonical (birational) embeddings are essential in proving finiteness statements of the type considered in this paper. They enable us to bring the problem into the context of projective varieties and to establish uniform boundedness. In the case of $K_X$ being ample, the following effective theorem on pluricanonical embeddings is available.
\[effpluri\] If $X$ is a compact complex manifold of complex dimension $n$ whose canonical line bundle $K_X$ is ample, then $mK_X$ is very ample for any integer $$\label{pluricaneffbound}
m\geq (e+\frac 1 2)n^\frac 7 3+\frac 1 2 n^\frac 5 3 + (e+\frac 1
2)n^\frac 4 3 + 3n+ \frac 1 2 n^\frac 2 3+5,$$ where $e \approx 2.718$ is Euler’s number.
From now on, we will set $k=k(n)$ to be the round-up of the effective very ampleness bound given in .
To our knowledge, Szabó’s theorem is the one that provides the best bound at this point. However, its proof relies on several methods previously introduced by other authors (e.g., see [@HuckSauer]) and uses the classification of finite simple groups in an essential way. In light of this, the much more straightforward method of Howard and Sommese, which was introduced in [@HoSo], still deserves to be noticed. Their method is actually not only applicable to automorphisms (see next section), and it represents an instance of a proof based entirely on boundedness and rigidity, which, technically speaking, is the main focus of the present paper.
Howard and Sommese prove for the case of a canonically polarized manifold that $\#Aut(X)$ is bounded from above by a number which depends only on the Chern numbers of $X$. Based on their result, we now state the following effective finiteness theorem.
\[HoSoAutbound\] Let $X$ be a compact complex manifold of dimension $n$ whose canonical line bundle is ample. Then $$\#\operatorname{{Aut}}(X) \leq \left((n+1)^2k^nn!2^{n^2}(2k)^{\frac 1 2 n(n+1)}(1+2kn)^nK_X^n\right)^{((k^nK_X^n+n)^2-1} .$$
Before we prove this theorem, we need to prove two auxiliary propositions which make the method of Howard and Sommese entirely effective. The first proposition will be used to bound the dimension of the target projective space for the pluricanonical embedding given by $kK_X$. It is a standard argument.
\[proph0bound\] Let $X$ be an $n$-dimensional compact complex manifold and $L$ a very ample line bundle on $X$. Then
$$h^0(X,L)\leq L^n+n.$$
We proceed by induction.
The case $n=1$ follows immediately from the Riemann-Roch Theorem.
Let $D$ be an effective divisor on $X$ such that $\operatorname{{\mathcal O}}_X(D)=L$. One has the standard short exact sequence $$0\to\operatorname{{\mathcal O}}_X\to\operatorname{{\mathcal O}}_X(L)\to\operatorname{{\mathcal O}}_D(L)\to 0.$$ From this exact sequence, we obtain $$h^0(X,L)\leq h^0(X,\operatorname{{\mathcal O}}_X)+h^0(D,\operatorname{{\mathcal O}}_D(L)).$$ By induction, we find that $$h^0(X,L)\leq 1+(L_{|D})^{n-1}+n-1 = L^n+n.$$
Secondly, we use a result of Demailly, Peternell and Schneider in [@DPS] to compute a bound for the Chern class intersection numbers that occur in the well-known formula for the degree of the ($1$-codimensional) dual of a projective variety. Our effective result is the following.
\[chernintersection\] Let $X$ be a compact complex manifold of dimension $n$ whose canonical line bundle is ample. Let $k$ denote again the round-up of the constant defined in Then the following holds for $i=1,\ldots,n$. $$|c_i(\Omega_X^1).K_X^{n-i}|\leq i!2^{in}(2k)^{\frac 1 2 i(i+1)}(1+2kn)^iK_X^n.$$
Recall that $k$ is such that $kK_X$ is very ample. It follows from the Castelnuovo-Mumford theory of regularity that $\Omega_X^1(2kK_X)$ is generated by global sections and therefore nef. We may thus apply [@DPS Corollary 2.6] to obtain $$\begin{aligned}
0&\leq&c_i(\Omega_X^1(2kK_X))K_X^{n-i}\nonumber\\
&\leq& (c_1(\Omega_X^1(2kK_X)))^iK_X^{n-i}\nonumber\\
&=&(c_1(\Omega_X^1)+2knK_X)^iK_X^{n-i}\nonumber\\
&=&(1+2kn)^iK_X^n\label{B}\end{aligned}$$ for $i=1,\ldots,n$.
In [@Ful page 56], one finds the formula $$\label{chernclassformula}
c_i(\Omega_X^1(2kK_X))=\sum_{\nu=0}^{i}{n-\nu\choose i-\nu}c_\nu(\Omega_X^1)(2kK_X)^{i-\nu},$$ which enables us to prove the Proposition by an induction.
The inequality clearly holds in the case $i=1$.
For $1<i\leq n$, note that it follows from that $$\begin{aligned}
c_i(\Omega_X^1(2kK_X))K_X^{n-i}&=&\left(\sum_{\nu=0}^{i}{n-\nu\choose i-\nu}c_\nu(\Omega_X^1)(2kK_X)^{i-\nu}\right)K_X^{n-i}\\
&=&\sum_{\nu=0}^{i}{n-\nu\choose i-\nu}c_\nu(\Omega_X^1)(2k)^{i-\nu}K_X^{n-\nu}.\end{aligned}$$ Taking absolute values, the triangle inequality yields $$\begin{aligned}
&&|c_i(\Omega_X^1)K_X^{n-i}|\\
&\leq&c_i(\Omega_X^1(2kK_X))K_X^{n-i}+\sum_{\nu=0}^{i-1}{n-\nu\choose i-\nu}|c_\nu(\Omega_X^1)(2k)^{i-\nu}K_X^{n-\nu}|\\
&\stackrel{\eqref{B}}{\leq}&(1+2kn)^iK_X^n+\sum_{\nu=0}^{i-1}{n-\nu\choose i-\nu}(2k)^{i-\nu}|c_\nu(\Omega_X^1)K_X^{n-\nu}|\\
&\stackrel{Ind.}{\leq}&(1+2kn)^iK_X^n+\sum_{\nu=0}^{i-1}{n-\nu\choose i-\nu}(2k)^{i-\nu}\nu! 2^{\nu n} (2k)^{\frac 1 2 \nu(\nu+1)}(1+2kn)^\nu K_X^n\\
&\leq&(1+2kn)^iK_X^n+i2^n(2k)^i(i-1)!2^{(i-1)n}(2k)^{\frac 1 2 i(i-1)}(1+2kn)^{i-1}K_X^{n}\\
&\leq&(1+2kn)^iK_X^n+2^n(2k)^{\frac 1 2 i(i+1)}i!2^{(i-1)n}(1+2kn)^{i-1}K_X^{n}\\
&\leq&(1+2kn)^iK_X^n+(2k)^{\frac 1 2 i(i+1)}i!2^{in}(1+2kn)^{i-1}K_X^{n}\\
&\leq&i!2^{in}(2k)^{\frac 1 2 i(i+1)}(1+2kn)^{i}K_X^{n}\quad \text{q.e.d.}\end{aligned}$$
Now that we have all necessary effective tools at our disposal, we can proceed to the
The proof given in [@HoSo] yields that $$\label{HoSoChern}
\#\operatorname{{Aut}}(X)\leq \left(\sum_{\nu=0}^{n}(-1)^j(n+1-j)(kK_X)^{n-j}c_j(X)\right)^{(h^0(kK_X))^2-1}.$$ Substituting the numerical bounds derived in Propositions \[proph0bound\] and \[chernintersection\] and estimating in an obvious way, we obtain that $$\#\operatorname{{Aut}}(X) \leq\left((n+1)^2k^nn!2^{n^2}(2k)^{\frac 1 2 n(n+1)}(1+2kn)^nK_X^n\right)^{(k^nK_X^n+n)^2-1}.$$
Effective finiteness theorems for maps with a fixed target {#fixed target}
==========================================================
For surjective meromorphic maps between compact complex spaces there is the following finiteness theorem due to Kobayashi-Ochiai.
Let $X$ be any compact complex space and $Y$ a compact complex space of general type. Then the number of surjective meromorphic maps between $X$ and $Y$ is finite.
There are no known effective versions of this theorem due to the fact that there are no effective birational embedding theorems for manifolds with merely big canonical line bundle in higher dimensions. The case of $X$ and $Y$ being smooth compact complex curves has been known for a long time as the Theorem of de Franchis, based on [@deF]. Not surprisingly, there are effective bounds in this case that depend only on the genus $g$ of $X$. However, these bounds are often obtained in the more general case of varying targets (see [@HS], [@Guerra]) or in complete analogy to the higher dimensional case. Somewhat surprisingly, those authors that consider specifically the case of two [*fixed*]{} Riemann surfaces and investigate e.g. the induced homomorphisms on the first homology groups (as e.g. in [@tanabe]) do not seem to be able to do much better numerically than those who consider more general situations. All bounds are exponential in $g$, and the question of the true nature of the dependence on $g$ seems to be completely open. Since maps between fixed Riemann surfaces seem to be closer in spirit to automorphisms than to the case of varying targets, where the bound is not polynomial (see next section), and based on some other preliminary evidence, we venture the following conjecture.
\[deFranchisConj\] There is a polynomial function $B(g)$ with the following property. For two fixed smooth compact complex curves $X$ and $Y$ of genus at least $2$ with the genus of $X$ equal to $g$, and the number of surjective holomorphic maps from $X$ to $Y$ is no more than $B(g)$.
As was already indicated in the previous section, the method of Howard and Sommese for automorphism groups can also be used to obtain a bound for the number of maps between any two fixed canonically polarized manifolds. The details of this straightforward generalization can be found in [@BD]. In fact, the bound one arrives at is the same as the expression we already encountered in . So we simply state the following theorem.
\[efffixedtarget\] Let $X$ and $Y$ be fixed compact complex manifolds with ample canonical line bundles. Let $n$ be the dimension of $X$. Then the number of surjective holomorphic maps between $X$ and $Y$ is no more than $$\left((n+1)^2k^nn!2^{n^2}(2k)^{\frac 1 2 n(n+1)}(1+2kn)^nK_X^n\right)^{(k^nK_X^n+n)^2-1} .$$
As we move on, we remark that a bound for the above theorem can also be obtained by using the Chow variety method discussed in the next section, since the graph of a surjective holomorphic map $X\to Y$ corresponds to an isolated point in a certain Chow variety of $X\times Y$. However, since this leads in fact to a worse bound, we will not discuss this in detail.
Effective finiteness theorems for maps with varying targets
===========================================================
The following theorem is often referred to as the Theorem of de Franchis-Severi. Its statement is obtained from the statement of the de Franchis Theorem by allowing the targets $Y$ to vary among smooth compact complex curves of genus at least $2$.
\[deFSev\] Let $X$ be a smooth compact complex curve. Then the set of all holomorphic maps $f:X\to Y$, where $Y$ is any (variable) smooth compact complex curve of genus at least $2$, is finite.
In [@HS], Howard and Sommese proved that if $X$ is of genus $g$, the number of holomorphic maps in Theorem \[deFSev\] modulo automorphisms of the target spaces is no more than $$\label{HSbound}
\left(\frac 1 2
(2\sqrt{6}(g-1)+1)^{2+2g^2}g^2(g-1)(\sqrt{2})^{g(g-1)}+1\right).$$ We denote this expression by $\mathcal S'(g)$. Since the cardinality of the automorphism group of any one of the targets is at most $\tfrac{1}{2}\cdot 84(g-1)$ due to Hurwitz, one can alternatively say that the number of holomorphic maps in Theorem \[deFSev\] is no more than $$\mathcal S(g):= 42(g-1)\cdot\mathcal S'(g).$$
In their paper, Howard and Sommese apparently overlooked the fact that their technique counts maps only modulo automorphisms of the targets. This fact was observed by Kani in his paper [@Kani]. This oversight can, of course, easily be remedied by adding the factor $42(g-1)$. In Kani’s paper, isomorphism classes of targets are counted instead of maps by means of a “packing argument”. We chose to quote the result of Howard and Sommese because its proof is closer to the point of view taken in the present paper.
It is certainly interesting to note that [@Kani §4] exhibits a relatively straightforward example of a series of Riemann surfaces $X$ that shows that the cardinality of the sets of maps defined in Theorem \[deFSev\] cannot be bounded by a polynomial in $g$. In fact, what is shown is that the number of isomorphism classes of targets in these sets cannot be bounded by a polynomial. Therefore, Kani’s example does not contradict our Conjecture \[deFranchisConj\]. We would like to remark that the statements in [@BD page 802] and [@TsaiIMRN page 110], which say that there is such a contradiction, represent a misinterpretation of Kani’s example.
The following conjecture (which is sometimes referred to as Iitaka-Severi Conjecture) represents a generalization of Theorem \[deFSev\]. As we shall see in the proof of Theorem \[section3thm\], the difficulty in proving it lies in the fact (and only in the fact) that there are no uniform birational embedding theorems for manifolds with big canonical line bundle in higher dimensions (not even ineffective ones).
Let $X$ be a compact complex manifold. Then the number of birational equivalence classes of compact complex manifolds of general type having a member $Y$ for which there exists a dominant rational map from $X$ to $Y$ is finite.
Our result in this section is the following effective generalization of Theorem \[deFSev\].
\[section3thm\] Let $X$ be an $n$-dimensional compact complex manifold whose canonical line bundle $K_X$ is ample. Then the number $\mathcal F(X)$ of surjective holomorphic maps $f:X\to Y$, where $Y$ is any $n$-dimensional compact complex manifold with ample canonical bundle, is no more than $$2^nk^nK_X^n\cdot{(N+1)\cdot 2^nk^nK_X^n\choose
N}^{(N+1)(2^nk^nK_X^n{2^nk^nK_X^n+n-1\choose n}+{2^nk^nK_X^n+n-1\choose
n-1})},$$ where $$N=(k^nK_X^n+n)^2-1,$$ and $k=k(n)$ is the effective very ampleness bound from [Theorem \[effpluri\]]{}.
Although the bound for $\mathcal F(X)$ looks somewhat complicated, its behavior with respect to $n$ and $K_X^n$ is easy to determine using Sterling’s formula. Namely, there exist explicit (exponential) functions $\alpha(n), \beta(n)$ such that $$\mathcal F (X)\leq (\alpha(n)K_X^n)^{\beta(n)(K_X^n)^{n+5}}.$$
In the case of $Y$ being a compact complex surface of general type, effective bounds in the same spirit as our Theorem \[section3thm\] were given by Tsai in [@TsaiJAG] (see also [@TsaiIMRN], [@TsaiCrelle] and also [@Guerra]). Ineffective results related to Theorem \[section3thm\] have also been established in [@MDLM], [@Maehara], [@BD].
It is well known that a result of the type of Theorem \[section3thm\] can be proved by what is commonly referred to as a “boundedness and rigidity argument”. The basic idea is to show that the objects in question can be associated to Chow points in a finite (or even effectively finite) number of Chow varieties and that in any irreducible component the Chow points can correspond to at most one of the objects in question. Then, clearly, the number of the objects in question is no more than the number of relevant irreducible components of the Chow varieties. Keeping this strategy in mind, we now start the
Let $f:X\to Y$ be one of the maps under consideration and let $\Gamma_f$ denote its graph. Let $p_1,p_2$ denote the two canonical projections of $X\times Y$. Let $\phi_f$ denote the isomorphism $X\to \Gamma_f, x \mapsto
(x,f(x))$. The line bundle $p_1^*(kK_X)\otimes p_2^*(kK_Y)$ is very ample on $X\times Y$ and embeds $\Gamma_f\subset X\times Y$ into $$\begin{aligned}
X\times {{\mathbb{P}}}^{h^0(Y,kK_Y)-1}\hookrightarrow{{\mathbb{P}}}^{h^0(X,kK_X)\cdot h^0(Y,kK_Y)-1}.\end{aligned}$$ Due to Proposition \[proph0bound\] and the fact that $K_X^n\geq K_Y^n$ (see below), we can assume $\Gamma_f$ to be embedded into $$X\times{{\mathbb{P}}}^{N_1}\hookrightarrow P^N$$ with $$N_1 := k^nK_X^n+n-1$$ and $$N := (k^nK_X^n+n)^2-1.$$
The degree of $\Gamma_f$ (measured in ${{\mathbb{P}}}^N$) can be estimated as follows: $$\begin{aligned}
\deg(\Gamma_f)&=&\int_{\Gamma_f}(p_1^*c_1(kK_X)+p_2^*c_1(kK_Y))^n\\
&=&\int_X\phi_f^*(p_1^*c_1(kK_X)+p_2^*c_1(kK_Y))^n\\
&=&\int_X (c_1(kK_X)+f^*c_1(kK_Y))^n\\
&\leq&\int_X (2c_1(kK_X))^n\\
&=&2^nk^nK_X^n.\end{aligned}$$ Note that the inequality is due to the fact that $K_X=f^*K_Y+D$, where $D$ is an effective divisor and $K_X$ and $f^*K_Y$ are ample, whence $$\begin{aligned}
K_X^{n-j}\left(f^*K_Y\right)^j&=&K_X^{n-j-1}\left(f^*K_Y+D\right)\left(f^*K_Y\right)^j\\
&=&K_X^{n-j-1}\left(f^*K_Y\right)^{j+1}+K_X^{n-j-1} . D .
\left(f^*K_Y\right)^j\\
&\geq& K_X^{n-j-1}\left(f^*K_Y\right)^{j+1}\end{aligned}$$ for $j=0,\ldots,n-1$. In particular, this computation yields $$K_X^n\geq (f^*K_Y)^n\geq K_Y^n,$$ which we used previously.
We now come to the rigidity part of the proof. In [@TsaiCrelle Corollary 3.2] it is shown that if $\pi:Z\to \Delta$ is a holomorphic family of smooth projective varieties of general type over a disk with $Z_0\cong Y$, there is no surjective holomorphic map $F:X\times\Delta\to Z$ with $F(X\times\{t\})=\pi^{-1}(t)$ unless $Z\cong Y\times \Delta$ and $F(\cdot,t)$ is independent of $t$.
Now take an irreducible component $I$ of $\operatorname{{Chow}}_{n,d}(X\times{{\mathbb{P}}}^{N_1})$ that contains a point corresponding to one of our graphs $\Gamma_f \subset X\times Y\subset X\times{{\mathbb{P}}}^{N_1}$. According to our previous boundedness considerations, we have $d\leq 2^nk^nK_X^n$. To be able to apply the rigidity property stated above, we need the following parametrization statement. For the details of its proof, we refer to [@Maehara Section 3], noting that our situation is essentially the same as the one treated by Maehara.
There is a Zariski-open subset $U\subset I$ such that all Chow points $[\Gamma] \in U$ correspond to surjective holomorphic maps $f_{[\Gamma]}:X\to Y_{[\Gamma]}$ with $Y_{[\Gamma]}\subset{{\mathbb{P}}}^{N_1}$ being an $n$-dimensional projective manifold of general type. Moreover, $U$ contains all Chow points $[\Gamma_f]\in I$ that come from graphs of maps $f:X\to Y$ of the type considered in the statement of the Theorem.
Based on this parametrization statement, the above-mentioned rigidity property implies that for $[\Gamma_1],[\Gamma_2] \in U$, we have $f_{\Gamma_1}=f_{\Gamma_2}$, i.e. the number $\mathcal F(X)$ is no more than the number of relevant irreducible components of $\operatorname{{Chow}}_{n,d}(X\times{{\mathbb{P}}}^{N_1})$ for $d=1,\ldots, 2^nk^nK_X^n$. Clearly, only those components of $\operatorname{{Chow}}_{n,d}(X\times{{\mathbb{P}}}^{N_1})$ are relevant whose general points represent irreducible cycles. However, from [@Kollarbook] (and also [@Guerra]), the following proposition is known.
Let $W\subset {{\mathbb{P}}}^n$ be a projective variety defined by equations of degree no more than $\tilde \delta$. Let $\operatorname{{Chow}}'_{k,\delta}(W)$ denote the union of those irreducible components of $\operatorname{{Chow}}_{k,\delta}(W)$ whose general points represent irreducible cycles. Then the number of irreducible components of $\operatorname{{Chow}}'_{k,\delta}(W)$ is no more than $${(n+1)\max\{\delta,\tilde \delta\}\choose n}^{(n+1)(\delta {\delta+k-1\choose k }+{\delta+k-1 \choose k-1})}.$$
Bounds on the complexity (i.e. the number of irreducible components) of Chow varieties have previously been produced by a number of authors. For example, the problem was extensively studied in Catanese’s [@Ca], and also in the papers of Green-Morrison ([@GM]) and Tsai ([@TsaiIMRN]). A new approach to handling Chow varieties of $1$-dimensional cycles is introduced in [@heiereffshaf].
Since the degrees of the defining equations of $X\times {{\mathbb{P}}}^{N_1}\subset {{\mathbb{P}}}^N$ under the Segre embedding are no more than $k^nK^n_X$, we conclude that our cardinality $\mathcal F(X)$ can be estimated from above by $$\begin{aligned}
&&\sum_{d=1}^{2^nk^nK_X^n} \# \text{ of irreducible components of }\operatorname{{Chow}}'_{n,d}(X\times{{\mathbb{P}}}^{N_1})\\
&\leq&2^nk^nK_X^n\cdot{({N}+1)\cdot 2^nk^nK_X^n\choose
{N}}^{({N}+1)(2^nk^nK_X^n{2^nk^nK_X^n+n-1\choose
n}+{2^nk^nK_X^n+n-1\choose n-1})}.\end{aligned}$$
We remark that the nonequidimensional case (i.e. $\dim X > \dim Y$) can be reduced to our Theorem \[section3thm\] by taking hyperplane sections. We shall express this fact as follows.
\[nonequidimcase\] If we take the targets $Y$ in [Theorem \[section3thm\]]{} to be $n'$-dimensional with $n-n'>0$, then the analogous cardinality $\mathcal F_{n'}(X)$ is no more than the number obtained when replacing $K_X^n$ with $((n-n')k+1)^{n'}k^{(n-n')}K_X^n$ in the bound obtained in [Theorem \[section3thm\]]{}.
We keep the notation from the proof of Theorem \[section3thm\]. For a generic hyperplane section $X\cap H$ of $X$ in ${{\mathbb{P}}}^{N_1}$, the restriction of the maps in question to $X\cap H$ is still surjective (for an easy proof of this fact see [@MDLM]). Therefore, after taking $n-n'$ general hyperplane sections, we obtain an $n'$-dimensional submanifold $\tilde X$ to which we can apply Theorem \[section3thm\].
An $(n-n')$-fold iteration of the adjunction formula yields $$\begin{aligned}
K_{\tilde X}^{n'}&=&(\frac 1 k \mathcal O(1)|_{\tilde X}+(n-n')\mathcal O(1)|_{\tilde X})^{n'}\\
&=&((n-n')+\frac 1 k)^{n'}k^nK_X^n\\
&=&((n-n')k+1)^{n'}k^{(n-n')}K_X^n.\end{aligned}$$
The strategy of an effective boundedness and rigidity proof can be used in a number of similar settings. For example, in the paper [@heiereffshaf], a uniform effective bound is established for the finiteness statement of the Shafarevich Conjecture over function fields (Theorem of Parshin-Arakelov). The arguments in that paper are more delicate due to the more complicated situation (one has to deal with moduli maps instead of maps of the form $f:X\to Y$), but the underlying principle is essentially the same.
It is a great pleasure to thank Professor Yum-Tong Siu for many invaluable discussions on (effective) algebraic geometry in general and finiteness theorems of the type discussed in the present paper in particular. These discussions took place while I enjoyed the generous hospitality of the Mathematics Department of Harvard University and the Institute of Mathematical Research at the University of Hong Kong. It is with sincere gratitude that I acknowledge support through the Schwerpunktprogramm “Globale Methoden in der komplexen Geometrie” of the Deutsche Forschungsgemeinschaft through the chair of Professor Alan Huckleberry at Bochum University.
[MDLM82]{} T. Bandman and G. Dethloff. Estimates of the number of rational mappings from a fixed variety to varieties of general type. , 47(3):801–824, 1997.
F. Catanese. Chow varieties, [H]{}ilbert schemes and moduli spaces of surfaces of general type. , 1(4):561–595, 1992.
F. Catanese and M. Schneider. Polynomial bounds for abelian groups of automorphisms. , 97(1-2):1–15, 1995. Special issue in honour of Frans Oort.
J.-P. Demailly. A numerical criterion for very ample line bundles. , 37(2):323–374, 1993.
M. de Franchis. Un teorema sulle involuzioni irrationali. , 36:368, 1913.
J.-P. Demailly, Th. Peternell, and M. Schneider. Compact complex manifolds with numerically effective tangent bundles. , 3(2):295–345, 1994.
W. Fulton. , volume 2 of [*Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics \[Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics\]*]{}. Springer-Verlag, Berlin, second edition, 1998.
M. Green and I. Morrison. The equations defining [C]{}how varieties. , 53(3):733–747, 1986.
L. Guerra. Complexity of [C]{}how varieties and number of morphisms on surfaces of general type. , 98(1):1–8, 1999.
G. Heier. Effective freeness of adjoint line bundles. , 7:31–42, 2002. (arXiv:math.AG/0108118)
G. Heier. Uniformly effective [S]{}hafarevich [C]{}onjecture on families of hyperbolic curves over a curve with prescribed degeneracy locus. arXiv:math.AG/0311085.
A. Howard and A. J. Sommese. On the orders of the automorphism groups of certain projective manifolds. In [*Manifolds and Lie groups (Notre Dame, Ind., 1980)*]{}, pages 145–158. Birkhäuser Boston, Mass., 1981.
A. Howard and A. J. Sommese. On the theorem of de [F]{}ranchis. , 10(3):429–436, 1983.
A. T. Huckleberry and M. Sauer. On the order of the automorphism group of a surface of general type. , 205(2):321–329, 1990.
A. Hurwitz. ber algebraische [G]{}ebilde mit eindeutigen [T]{}ransformationen in sich. , 41:403–442, 1893.
E. Kani. Bounds on the number of nonrational subfields of a function field. , 85(1):185–198, 1986.
S. Kobayashi and T. Ochiai. Meromorphic mappings onto compact complex spaces of general type. , 31(1):7–16, 1975.
J. Koll[á]{}r. Effective base point freeness. , 296(4):595–605, 1993.
J. Koll[á]{}r. . Springer-Verlag, Berlin, 1996.
A. M. Macbeath. On a theorem of [H]{}urwitz. , 5:90–96, 1961.
K. Maehara. A finiteness property of varieties of general type. , 262(1):101–123, 1983.
M. Martin-Deschamps and R. Lewin-M[é]{}n[é]{}gaux. Surfaces de type général dominées par une variété fixe. , 110(2):127–146, 1982.
E. Szab[ó]{}. Bounding automorphism groups. , 304(4):801–811, 1996.
M. Tanabe. A bound for the theorem of de [F]{}ranchis. , 127(8):2289–2295, 1999.
I-Hsun Tsai. Dominant morphisms on surfaces of general type modulo biholomorphic equivalence. , (3):101–111, 1997.
I-Hsun Tsai. Dominating the varieties of general type. , 483:197–219, 1997.
I-Hsun Tsai. Chow varieties and finiteness theorems for dominant maps. , 7(4):611–625, 1998.
H. Tsuji. Bound of automorphisms of projective varieties of general type. arXiv:math.AG/0004138.
Gang Xiao. Bound of automorphisms of surfaces of general type. [I]{}. , 139(1):51–77, 1994.
Gang Xiao. Bound of automorphisms of surfaces of general type. [II]{}. , 4(4):701–793, 1995.
| ArXiv |
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abstract: 'Given a Riemannian space $N$ of dimension $n$ and a field $D$ of symmetric endomorphisms on $N$, we define the extension $M$ of $N$ by $D$ to be the Riemannian manifold of dimension $n+1$ obtained from $N$ by a construction similar to extending a Lie group by a derivation of its Lie algebra. We find the conditions on $N$ and $D$ which imply that the extension $M$ is Einstein. In particular, we show that in this case, $D$ has constant eigenvalues; moreover, they are all integer (up to scaling) if $\det D \ne 0$. They must satisfy certain arithmetic relations which imply that there are only finitely many eigenvalue types of $D$ in every dimension (a similar result is known for Einstein solvmanifolds). We give the characterisation of Einstein extensions for particular eigenvalue types of $D$, including the complete classification for the case when $D$ has two eigenvalues, one of which is multiplicity free. In the most interesting case, the extension is obtained, by an explicit procedure, from an almost Kähler Ricci flat manifold (in particular, from a Calabi-Yau manifold). We also show that all Einstein extensions of dimension four are Einstein solvmanifolds. A similar result holds valid in the case when $N$ is a Lie group with a left-invariant metric, under some additional assumptions.'
address:
- 'IITP, Russian Academy of Sciences, Moscow 127051, Russia'
- 'Department of Mathematics and Statistics, La Trobe University, Melbourne 3086, Australia'
author:
- 'D.Alekseevsky'
- 'Y.Nikolayevsky'
bibliography:
- 'oneext.bib'
title: Einstein extensions of Riemannian manifolds
---
[^1]
Introduction {#s:intro}
============
The construction and the study of Einstein manifolds is one of the main avenues of Riemannian Geometry. One of the starting points of our paper is the theory of Einstein homogeneous manifolds of negative scalar curvature. Assuming *Alekseevsky Conjecture* (and the fact that the isometry group is linear), such manifolds are necessarily solvmanifolds, solvable Lie groups with a left-invariant Einstein metric. At present, the theory of Einstein solvmanifolds is very well developed [@Lsurv]. The basic construction is as follows. At the level of Lie algebras, one starts with a nilpotent Lie algebra ${\mathfrak{n}}$ with a special *nilsoliton inner product* characterised by the property that its Ricci operator is a linear combination of the identity operator and the *Einstein derivation* $D$. The derivation $D$ is always symmetric and its eigenvalues, up to scaling, are natural numbers (not every nilpotent Lie algebra admits such a derivation and such an inner product; those which do are called *nilsolitons*). The rank one extension of ${\mathfrak{n}}$ by $D$ is a solvable Lie algebra ${\mathfrak{s}}$. Extending the inner product from ${\mathfrak{n}}$ to ${\mathfrak{s}}$ in such a way that the extension is orthogonal (and choosing the correct scaling factor) one obtains a metric Einstein solvable Lie algebra whose solvable Lie group, with the corresponding left-invariant metric, is an Einstein solvmanifold $M$. All rank one Einstein extensions can be obtained in this way and the higher rank extensions can be obtained from rank one extensions by a known procedure [@Heb Theorem 4.18] and [@Lstand]. One can see that the resulting Riemannian metric on $M$ has precisely the form as in the definition below. The main idea of this paper is to drop the homogeneity assumption and to construct rank one Einstein extensions of arbitrary Riemannian manifolds by a field of symmetric endomorphisms $D$, as described below.
\[def:main\] Let $(N, g)$ be a Riemannian manifold of dimension $n > 1$, and $D$ a field of symmetric endomorphisms on $(N, g)$. For $u \in {\mathbb{R}}$, the *$D$-deformation* of the metric $g$ on $N$ is the metric on $N$ given by $g^u:=(\exp(uD))^*g$. The *$D$-extension* is the Riemannian manifold $(M, g^D)$ given by $$(M:={\mathbb{R}}\times N, \; g^D:= du^2 + g^u).$$
When $D$ has eigenvalues $q_1, \dots, q_m$ of constant multiplicities and $V(q_i)$ are the corresponding eigendistributions, the $D$-deformation is given by $g^u = e^{2q_1u} g_1 + \dots + e^{2q_mu} g_m$, and the $D$-extension, by $$g^D = du^2 + g^u = du^2 + e^{2q_1u} g_1 + \dots + e^{2q_mu} g_m,$$ where $g_i:= g_{|V(q_i)}$. Clearly, $D$ remains symmetric with respect to all the metrics $g^u$ on $N$.
This construction, both in the Riemannian and pseudo-Riemannian cases, is known in the literature (see e.g., [@Herv]) and also appears in the theory of Riemannian submersions [@Bes Chapter 9]. It is a generalisations of the warped product metrics; note however that we make no assumptions on the integrability of the eigendistributions of $D$.
\[def:stable\] A manifold $(N, g)$ (and the metric $g$ on the manifold $N$) is called *Ricci $D$-stable* if the Ricci operator ${\operatorname{Ric}}^u := {\operatorname{Ric}}_{g^u}$ does not depend on $u$, and is called *$D$-Einstein* if the extension $(M,g^D)$ is Einstein.
Our main question is, when a metric is $D$-Einstein, or in other words, *when the extension $(M, g^D)$ is Einstein*? As we will see, in many cases, this general construction bears a remarkable resemblance to the homogeneous (the solvmanifold) case, and in some cases (as in Theorem \[th:dim3\]), the Einstein condition even *implies* the homogeneity. Below we present the structure of the paper and the main results.
In Section \[s:ric\], we compute the Ricci tensor of $(M, g^D)$ and prove the following theorem which gives a necessary and sufficient condition for a Riemannian manifold $(N,g)$ to admit an Einstein $D$-extension.
\[th:Dconst\] Let $(M, g^D)$ be the $D$-extension of $(N,g)$. Then $(M, g^D)$ is Einstein if and only if the following two conditions are satisfied:
1. \[it:Dconst1\] The endomorphism $D$ has constant eigenvalues and $$\label{eq:ein1div}
{\operatorname{div}}D = 0,$$ where ${\operatorname{div}}$ is the divergence relative to $g$ *(*so that $({\operatorname{div}}D)X={\operatorname{Tr}}(Y \mapsto (\nabla_Y D)X)$*)*.
2. \[it:Dconst2\] The manifold $(N,g)$ is Ricci $D$-stable and $$\label{eq:ein1Ru}
{\operatorname{Ric}}^u= ({\operatorname{Tr}}D) \, D - {\operatorname{Tr}}(D^2) \, {\mathrm{id}}.$$
The Einstein constant of $g^D$ is $- {\operatorname{Tr}}(D^2)$.
\[ex:id\] A Ricci flat manifold $(N^n, g)$ is ${\mathrm{id}}$-Einstein, i.e. the metric $g^{{\mathrm{id}}} = du^2 + e^{2u}g$ is Einstein with the Einstein constant $-n$. In particular, if $g$ is Euclidean, then $g^{{\mathrm{id}}}$ is a hyperbolic metric written in horospherical coordinates. The converse (“any ${\mathrm{id}}$-Einstein manifold is Ricci flat") follows from .
\[ex:product\] A direct product $(N_1 \times N_2, g_1+g_2)$ of Ricci $D_i$-stable manifolds $(N_i, g_i), \; i = 1, 2$, is Ricci $D$-stable, where $D=D_1 \oplus D_2$. Moreover, it is $D$-Einstein if and only if for $i = 1, 2$, condition of Theorem \[th:Dconst\] is satisfied and ${\operatorname{Ric}}_{g_i} = ({\operatorname{Tr}}D) D_i - {\operatorname{Tr}}(D^2) {\mathrm{id}}_{N_i}$.
\[ex:group\] Let $(N, g)$ be a Lie group with a left-invariant metric and $D$ be defined by a symmetric derivation of the Lie algebra of $N$. Then $(N, g)$ is $D$-stable (see Section \[s:homo\]).
In Section \[s:eigen\] we study the *eigenvalue type of $D$*, the vector $\mathbf{p}=(p_1, \dots, p_n)^t$ of its eigenvalues (recall that all of them must be constant by Theorem \[th:Dconst\]). We call $\mathbf{p}$ *the spectral vector*. We show in Lemma \[l:scalar\] that $D$ is scalar if and only if $(N, g)$ is Ricci flat, and that in all the other cases, the eigenvalues satisfy some restrictions, in particular, some nontrivial relations of the form $p_k=p_i+p_j$.
In the case when all $p_i$ are nonzero, we can be much more specific. In the Euclidean space ${\mathbb{R}}^n$ with an orthonormal basis $f_i, \; i=1, \dots n$, introduce the subset $F=\{f_i+f_j-f_k \, : \, i \ne j, \, k \ne i,j\}$ and consider the vectors $\mathbf{p}=(p_1, \dots, p_n)^t$ and $\mathbf{1}_n=(1,\dots,1)^t$ ($n$ ones). In the (finite) set $F \cap \mathbf{p}^\perp$, choose a maximal linearly independent subset $F_{\mathbf{p}}=\{v_1, \dots, v_m\}$ (any of them, if there are more than one). Let $V$ be an $n \times m$ matrix whose vector columns are the vectors $v_a$ (so that if $v_a=f_i+f_j-f_k \in F_{\mathbf{p}}$, then the $a$-th column of $V$ has a one in the $i$-th and in the $j$-th rows, a minus one in the $k$-th row, and zeros elsewhere). The following theorem is analogous to [@Heb Theorem 4.14, Corollary 4.17] for rank one Einstein solvmanifolds.
\[th:nonzero\] Let $(M,g^D)$ be an Einstein $D$-extension of $(N,g)$ with $\det D \ne 0$.
1. \[it:nonzero1\] Then the projection of $\mathbf{1}_n$ to $\mathbf{p}^\perp$ belongs to the convex cone hull of the set $F \cap \mathbf{p}^\perp$.
2. \[it:nonzero2\] If, in addition, ${\operatorname{Tr}}D \ne 0$, then up to scaling, the spectral vector is given by $$\label{eq:eigenstr}
\mathbf{p}=\mathbf{1}_n-V(V^tV)^{-1}\mathbf{1}_m.$$ In particular, up to scaling, all the $p_i$ are integers. Moreover, for every dimension $n$, there is only a finite number of the eigenvalue types of the operators $D$, with $\det D \ne 0$ and ${\operatorname{Tr}}D \ne 0$.
If the operator $D$ is non-scalar, the simplest possible case to consider is when it has an eigenvalue of multiplicity $n-1$, so that $\mathbf{p}=(\lambda, \dots, \lambda, \nu)^t, \; {\lambda}\ne \nu$. Up to scaling, we can have $({\lambda}, \nu)=(0,1), (1,0)$, or $(1,2)$, where the fact that $\nu/{\lambda}=2$ in the latter case follows from Theorem \[th:nonzero\] (or from Lemma \[l:scalar\] below). These three eigenvalue types are studied in Section \[s:Heis\]. If $\mathbf{p}=(0, \dots, 0, 1)^t$, we have the following theorem.
\[th:01\] Let $(M,g^D)$ be the $D$-extension of $(N,g)$ with the spectral vector $\mathbf{p}=(0, \dots, 0, 1)^t$. If $(M,g^D)$ is Einstein, then $(N,g)$ is locally isometric to the Riemannian product of the real line and an Einstein manifold $N'$ of dimension $n-1$ with the Einstein constant $-1$. The manifold $(M,g^D)$ is locally isometric to the Riemannian product of the hyperbolic plane of curvature $-1$ and $N'$.
When $\mathbf{p}=(1, \dots, 1, 0)^t$ we show that $(M, g^D)$ is a warped product with a two-dimensional base, and that it can be obtained as a “double extension", by two commuting extensions, of a Ricci flat manifold $N'$ (Theorem \[th:10\]).
In the third case (which is probably the most interesting one), we prove the following theorem.
\[th:12\] Let $(M,g^D)$ be the $D$-extension of $(N,g)$ with the spectral vector $\mathbf{p}=(1, \dots, 1, 2)^t$. The the following are equivalent.
1. \[it:12ein\] The extension $(M,g^D)$ is Einstein.
2. \[it:12Keta\] The Riemannian manifold $(N,g)$ is locally K-contact $\eta$-Einstein with ${\operatorname{ric}}= -2 g + (n+1) \eta \otimes \eta$, where $\eta$ is the one-form dual to a unit eigenvector $\xi$ of $D$ with the eigenvalue $2$.
3. \[it:12KahlerRf\] The metric $g$ on $N$ is locally given by $g = g' + (dt+\theta')^2$, where $t \in {\mathbb{R}}$ and $(N',g')$ is an almost Kähler, Ricci flat manifold and $\theta'$ is a $1$-form on $N'$ such that $d\theta'=\omega$, the Kähler form on $N'$.
Under any of the above three conditions, the Einstein metric $g^D$ on $M$ is almost Kähler and is locally given by $g^D=du^2+e^{2u}g'+e^{4u}(dt+\theta')^2$ in the notation of .
We prove Theorem \[th:12\] in Section \[ss:12\]. Recall that a unit Killing vector field $\xi$ on a Riemannian manifold $(N,g)$ defines a *$K$-contact structure* if $\xi^{\perp}$ is a contact distribution with the contact form $\eta = g\circ \xi$ and the restriction $J = \Phi_{|\xi^{\perp}}$ of the endomorphism $\Phi=-\nabla \xi$ to $\xi^{\perp}$ is an almost CR structure, that is, $J^2 = -{\mathrm{id}}$. It is called a *Sasakian structure* if $(\xi^{\perp}, J)$ is a CR structure or equivalently, if $(\nabla_X \Phi)= \xi \otimes g \circ X - X \otimes g \circ \xi$, and it is called *$\eta$-Einstein*, if ${\operatorname{ric}}= ag + b\eta \otimes \eta $ for some constants $a,b \in {\mathbb{R}}$ [@Blair].
\[rem:contact\] Note that in , “locally" can be replaced by “globally" if $N$ is orientable (or otherwise we can replace $N$ by its orientable cover). We also note that the equivalence of and is a known fact [@Blair; @BGbook], which we included for completeness; the almost Kähler, Ricci flat manifold in $(N',g')$ in is locally the “leaf space" of the geodesic foliation on $(N,g)$ defined by $\xi$.
If we additionally assume $N$ to be compact (and orientable), then by the result of [@BG Theorem 7.2], $(N,g)$ is Sasakian, and then $(N',g')$ is Kähler and Ricci flat and hence $(M,g^D)$ is Einstein Kähler. Moreover, if the geodesic foliation on $N$ defined by $\xi$ is *regular* (this means that every point has a neighbourhood through which every geodesic of that foliation passes at most once), one obtains that $(N,g)$ is a circle bundle over a Calabi-Yau manifold $(N',g')$.
On the other hand, starting with an almost Kähler non-Kähler Ricci flat manifold $(N',g')$ (an example is constructed in [@NP]) one gets that $(N,g)$ is K-contact, but is not Sasakian.
\[rem:deform\] It is interesting to compare our construction to the standard cone construction in contact geometry. Let $(N,g, \eta)$ be a contact metric structure (this means that $\eta$ is a contact form such that the associated Reeb vector field $\xi$ is unit and $g^{-1} \circ d \eta_{|{\operatorname{Ker}}\eta}$ is an almost CR structure on ${\operatorname{Ker}}\eta$). Then the cone $C = {\mathbb{R}}_+ \times N$ with the metric $g_C = du^2 + u^2 g, \; u \in {\mathbb{R}}_+$, is an almost Kähler manifold with the Kähler form $\omega = d (\frac12 u^2 \eta)= u du \wedge \eta + \frac12 u^2 d \eta$. It is a Kähler manifold if and only if $(N,g, \eta)$ is a Sasakian manifold.
Our construction can be viewed as the “$\mathcal{D}$-deformation cone". For a contact metric structure $(N,g, \eta)$ and a positive number $r > 0$, the $\mathcal{D}$-deformed structure is given by $\tilde{\eta}=r\eta, \, \tilde{\xi}=r^{-1} \xi$ and $\tilde{g} = rg + r(r - 1) \eta \otimes \eta$; see [@Blair §7.3], [@BGbook §7.3.3]. A $\mathcal{D}$-deformation preserves the property of being K-contact, Sasakian and $\eta$-Einstein. Our metric $(N,g^u)$ is obtained from $(N,g)$ by the $\mathcal{D}$-deformation with $r=e^{2u}$.
In Section \[s:dim3\], we consider the case $n=3$, the first case when our construction produces interesting examples. Remarkably, in that case the Einstein condition forces the homogeneity.
\[th:dim3\] Let $(M,g^D)$ be the $D$-extension of the manifold $(N,g)$ of dimension $3$. If $(M,g^D)$ is Einstein, then both $(N,g)$ and $(M,g^D)$ are locally isometric to Lie groups with left-invariant metrics; $N$ is a nilmanifold or a solvmanifold, $D$ is a derivation, and $(M,g^D)$ is an Einstein solvmanifold. All the possible cases, up to scaling, are listed in Table \[t:d3\].
$p_i$ ${\mathfrak{n}}$ $(M, g^D)$ $ds^2$
--------- ------------------ ------------------------ -------------------------------------------------------------------
$0,0,0$ *abelian* ${\mathbb{R}}^4$ $du^2+(dx^1)^2+(dx^2)^2+(dx^3)^2$
$1,1,1$ *abelian* $H^4(-1)$ $du^2+e^{2u}((dx^1)^2+(dx^2)^2+(dx^3)^2)$
$1,1,2$ ${\mathbb{C}}H^2 (-4)$
$1,p,0$ $du^2+(dx^3)^2+e^{2 (u- p x^3)}(dx^1)^2+e^{2 (p u+ x^3)}(dx^2)^2$
: Four-dimensional Einstein extensions.[]{data-label="t:d3"}
In the first column of Table \[t:d3\], we list the eigenvalues $p_i$ of the derivation $D$ of the Lie algebra ${\mathfrak{n}}$ of $N$, with the corresponding eigenvectors $\overline{e}_i, \, i=1,2,3$. The second column gives the types and the defining relations for ${\mathfrak{n}}$ (note that the relations for the Lie algebra ${\mathfrak g}$ of $M$ are obtained by adding the relations $[e_4,\overline{e}_i]=p_i \overline{e}_i$ to the relations for ${\mathfrak{n}}$). The third and the fourth columns give the homogeneous spaces to which $(M,g^D)$ is locally isometric and the explicit forms of the metric $(M,g^D)$ in local coordinates respectively, where we denote $H^m(c)$ the hyperbolic space of curvature $c$, and ${\mathbb{C}}H^2 (-4)$ the complex hyperbolic space of holomorphic curvature $-4$. Note that in the last row of Table \[t:d3\], $p$ is arbitrary. The fact that the metric $g^D$ on $M$ is indeed the Riemannian product of two hyperbolic planes (of curvature $-(p^2+1)$ each) can be seen by the change of variables $y_1=(p^2+1)^{-1/2}(u-p x^3), \; y_2=(p^2+1)^{-1/2}(pu+x^3)$.
In Section \[s:homo\], we turn our attention to the case when $N$ is a Lie group with a left-invariant metric $g$ and $D$ is left-invariant. Denote ${\mathfrak{n}}$ the Lie algebra of $N$. It would be interesting to know if the condition that $(M,g^D)$ is Einstein forces it to be an Einstein solvmanifold. We answer this question in positive in two cases.
\[th:group\] Suppose the extension $(M,g^D)$ of a Lie group $(N,g)$ by $D$ is Einstein, and that both $g$ and $D$ are left-invariant.
1. \[it:bpos\] If the Killing form of ${\mathfrak{n}}$ is nonnegative, then $D$ is a symmetric derivation of ${\mathfrak{n}}$.
2. \[it:decomphom\] Let ${\mathfrak{n}}={\mathfrak h}\oplus {\mathfrak{m}}$ be a decomposition into an orthogonal sum of an abelian subalgebra ${\mathfrak h}$ and a nilpotent ideal ${\mathfrak{m}}$ and suppose that one of the following conditions is satisfied:
1. \[it:decompD\] either $D$ preserves the decomposition ${\mathfrak{n}}={\mathfrak h}\oplus {\mathfrak{m}}$,
2. \[it:ni1ab\] or ${\mathfrak{m}}$ is abelian.
Then there exists a metric solvable Lie group $(N',g')$ and an isometry $\Phi: (N,g) \to (N',g')$ such that $D'=(d\Phi)D$ is left-invariant relative to $N'$ and $D'(e)$ is a symmetric derivation of the Lie algebra ${\mathfrak{n}}'$ of $N'$.
In both cases, the manifold $(M, g^D)$ is an Einstein solvmanifold.
Ricci curvature of $D$-deformation and $D$-extension. Proof of Theorem \[th:Dconst\] {#s:ric}
====================================================================================
Ricci curvature of the $D$-extension {#ss:ricDext}
------------------------------------
In this subsection, we find the Ricci tensor of the $D$-extension $(M, g^D)$, assuming neither $D$-stability, nor $D$-Einstein property.
Locally, on an open, connected domain $U$ of $(N, g)$ on which the eigenvalues $p_i$ of $D$ have constant multiplicities, we can choose a smooth orthonormal frame $\overline{e}_i, \; i=1, \dots, n$, of eigenvectors of $D$. We extend $\overline{e}_i$ and $D$ to $M':={\mathbb{R}}\times U \subset M$ by the Lie translation along the vector field $\partial_u$. Let $\overline{\theta}^i$ be the one-forms dual to $\overline{e}_i$. The metric $g^D$ on $M'$ is given by $$\label{eq:metricM}
ds^2=du^2+\sum\nolimits_{i=1}^n e^{2up_i}(\overline{\theta}^i)^2.$$
Let $e_a, \; a=0, \dots , n$, be the orthonormal frame on $(M', g^D)$ given by $e_0=\partial_u, \; e_i = e^{-up_i}\overline{e}_i$ for $i >0$, and let $\theta^a$ be the $1$-forms dual to $e_a$, so that $$\theta^0=du, \qquad \theta^i=e^{up_i}\overline{\theta}^i, \; \text{for }i >0.$$
The structure equations for $(M, g^D)$ are given by $$\label{eq:structure}
d \theta^a = - \sum\nolimits_b \psi^a_b \wedge \theta^b,\; \psi^a_b =- \psi^b_a, \qquad {\Omega}^a_b = d \psi^a_b + \sum\nolimits_c \psi^a_c \wedge \psi^c_b,$$ where $\psi_a^b$ and ${\Omega}_a^b$ are the connection and the curvature two-forms respectively. Decomposing them by the basis $\theta^a \wedge \theta^b$ we obtain $$\psi_b^a = \sum\nolimits_c {\Gamma}_{bc}^a \theta^c, \; \; {\Gamma}_{bc}^a = {\langle}{\nabla}_c e_b, e_a{\rangle}, \;\; {\Gamma}_{ac}^b = - {\Gamma}_{bc}^a,
\qquad {\Omega}^a_b = - {\Omega}^b_a = \tfrac 12 \sum\nolimits_{c,d} R_{abcd} \theta^c \wedge \theta^d.$$ We use the same notation with the bar for the corresponding objects relative to the metric $g$ on $N$; the convention for the index ranges is $0 \le a,b,c, \dots \le n$ and $1 \le i,j,k, \dots \le n$.
We have $d\theta^0=0, \; d\theta^i=e^{up_i}(p_i \theta^0 + u \sum\nolimits_j e_j(p_i) \theta^j) \wedge \overline{\theta}^i - e^{up_i}\sum\nolimits_j \overline{\psi}^i_j \wedge \overline{\theta}^j$, so $$\sum\nolimits_{b,c} {\Gamma}^0_{bc} \theta^c \wedge \theta^b=0, \quad
\sum\nolimits_{b,c} {\Gamma}^i_{bc} \theta^c \wedge \theta^b=-p_i \theta^0 \wedge \theta^i- u e^{up_i}\sum\nolimits_{j}\overline{e_j}(p_i) \overline{\theta}^j \wedge \overline{\theta}^i + e^{up_i}\sum\nolimits_{j,k}\overline{{\Gamma}}^i_{jk} \overline{\theta}^k \wedge \overline{\theta}^j.$$ It follows that $$\label{eq:gij0}
{\Gamma}^i_{00}={\Gamma}^0_{i0}=0, \quad {\Gamma}^0_{ii}=p_i, \quad {\Gamma}^i_{j0}={\Gamma}^0_{ij}=0, \quad \text{for } i \ne j,$$ and that $\sum\nolimits_{j,k}(e^{u(p_j+p_k)}{\Gamma}^i_{jk} -e^{up_i}(u \overline{e_j}(p_i) {\delta}_{ik} + \overline{{\Gamma}}^i_{jk}) )\overline{\theta}^j \wedge \overline{\theta}^k= 0$. By the cyclic permutation of $i,j,k$ we obtain $$\label{eq:gijk}
\begin{split}
{\Gamma}^i_{jk}=& \tfrac12 e^{u(p_k-p_i-p_j)}(\overline{{\Gamma}}^k_{ji}-\overline{{\Gamma}}^k_{ij}+u ({\delta}_{ki}\overline{e_j}-{\delta}_{kj}\overline{e_i})(p_k))\\
-&\tfrac12 e^{u(p_i-p_j-p_k)}(\overline{{\Gamma}}^i_{kj}-\overline{{\Gamma}}^i_{jk}+u ({\delta}_{ij}\overline{e_k}-{\delta}_{ik}\overline{e_j})(p_i))\\
-&\tfrac12 e^{u(p_j-p_k-p_i)}(\overline{{\Gamma}}^j_{ik}-\overline{{\Gamma}}^j_{ki}+u ({\delta}_{jk}\overline{e_i}-{\delta}_{ji}\overline{e_k})(p_j)).
\end{split}$$ By the Ricci tensor ${\operatorname{ric}}$ of $(M',g^D)$ is given by $$\label{eq:ricci}
{\operatorname{ric}}_{ab}=\sum\nolimits_c R_{cacb}=\sum\nolimits_c \big(e_c({\Gamma}^c_{ab})-e_b({\Gamma}^c_{ac})+ \sum\nolimits_d({\Gamma}^c_{dc}{\Gamma}^d_{ab}-{\Gamma}^c_{ad}{\Gamma}^d_{bc})\big). $$ Then by $$\label{eq:ric00}
{\operatorname{ric}}_{00}=-\sum\nolimits_i p_i^2=-{\operatorname{Tr}}(D^2),$$ and by (\[eq:gij0\], \[eq:gijk\]), $$\label{eq:ric0i}
\begin{split}
{\operatorname{ric}}_{0i}&=e^{-up_i}\Big(\overline{e}_i({\operatorname{Tr}}D -p_i)+\sum\nolimits_j\overline{{\Gamma}}^i_{jj}(p_i-p_j)+u(\overline{e}_i(\tfrac12{\operatorname{Tr}}(D^2))-p_i\overline{e}_i({\operatorname{Tr}}D))\Big)\\
&=e^{-up_i}\Big(\overline{e}_i({\operatorname{Tr}}D)-\overline{{\langle}\overline{e}_i, \sum\nolimits_j(\overline{{\nabla}}_{\overline{e}_j}D)\overline{e}_j{\rangle}} +u(\overline{e}_i(\tfrac12{\operatorname{Tr}}(D^2))-p_i\overline{e}_i({\operatorname{Tr}}D))\Big).
\end{split}$$ Note that the connection forms of the metric $g^u$ on $N$ relative to the frame $e_i$ are the same as those of $(M,g^D)$ and are given by , so from (or from the Gauss equation) we obtain $$\label{eq:ricij}
{\operatorname{ric}}_{ij}={\operatorname{ric}}^u_{ij}-{\delta}_{ij} p_i {\operatorname{Tr}}D,$$ where ${\operatorname{ric}}^u$ is the Ricci tensor of $(N,g^u)$.
Proof of Theorem \[th:Dconst\] {#ss:pfofDconst}
------------------------------
Suppose $(M, g^D)$ is Einstein. Let $U$ be an open, connected domain of $(N, g)$ on which the eigenvalues $p_i$ of $D$ have constant multiplicities and on which the rank of $D$ is constant. Let $M':={\mathbb{R}}\times U \subset M$. Then by the Einstein constant is $-{\operatorname{Tr}}(D^2)$, so from and at the points of $(M', g^D)$ we obtain: $$\begin{aligned}
&p_i \overline{e}_i ({\operatorname{Tr}}D)=0, \quad \text{for all } i=1, \dots, n, \label{eq:dTr} \\
&\overline{{\operatorname{div}}} D = \overline{{\operatorname{grad}}} ({\operatorname{Tr}}D), \label{eq:divgrad} \\
&{\operatorname{Ric}}^u= ({\operatorname{Tr}}D) \, D - {\operatorname{Tr}}(D^2) \, {\mathrm{id}}, \label{eq:RicuD}\end{aligned}$$ where $\overline{{\operatorname{grad}}}$ is the gradient relative to $g$.
The right-hand side of is independent of $u$, while the left-hand side, by and , is the sum of expressions of the form $f_{\alpha}e^{ul_{\alpha}}, \, f_{\alpha}ue^{ul_{\alpha}}$, and $f_{\alpha}u^2e^{ul_{\alpha}}$, where $l_{\alpha}$ are linear combinations of the $p_i$’s with integer coefficients, and $f_{\alpha}$ are functions on $N$. For any sum $F$ of such expressions, we denote by $[u^2 \exp]F$ the sum of all terms of $F$ of the form $f_{\alpha}u^2e^{ul_{\alpha}}$. Then gives $[u^2 \exp]{\operatorname{ric}}^u_{ij}=0$. To compute ${\operatorname{ric}}^u_{ij}$ we use replacing $a,b,c,d$ with $i,j,k,l$. Then from we obtain $$\begin{split}
0&=[u^2 \exp]{\operatorname{ric}}^u_{ij}=[u^2 \exp](e^{-up_k}\overline{e}_k({\Gamma}^k_{ij})-e^{-up_j}\overline{e}_j({\Gamma}^k_{ik}) -{\Gamma}^k_{lj}{\Gamma}^l_{ik}+{\Gamma}^k_{lk}{\Gamma}^l_{ij}+{\Gamma}^k_{il}{\Gamma}^l_{kj}-{\Gamma}^k_{il}{\Gamma}^l_{jk})\\
&=u^2 e^{-u(p_i+p_j)}\Bigl(\overline{e}_i({\operatorname{Tr}}D)\overline{e}_j(p_i)+\overline{e}_j({\operatorname{Tr}}D)\overline{e}_i(p_j)-2\overline{e}_j(p_i)\overline{e}_i(p_j) -\sum\nolimits_k\overline{e}_j(p_k)\overline{e}_i(p_k)\Bigr)\\ &+{\delta}_{ij}u^2\sum\nolimits_k e^{-2up_k}\overline{e}_k(p_i)\overline{e}_k(2p_k- {\operatorname{Tr}}D).
\end{split}$$ In particular, taking $i=j$ and summing up by $i=1, \dots , n$ we get $$\begin{gathered}
\sum\nolimits_i\bigl(e^{-2up_i}\bigl(2\overline{e}_i({\operatorname{Tr}}D)\overline{e}_i(p_i)-2(\overline{e}_i(p_i))^2 -\sum\nolimits_k(\overline{e}_i(p_k))^2\bigr)\bigr) \\ +\sum\nolimits_k e^{-2up_k}\overline{e}_k({\operatorname{Tr}}D)\overline{e}_k(2p_k- {\operatorname{Tr}}D)=0.\end{gathered}$$ On the domain $U \subset N$, the rank $r: = {\operatorname{rk}}D$ is constant, and by relabelling we can assume that $p_{r+1}=\dots=p_{n}=0$ and that all the functions $p_1, \dots, p_r$ are nonzero on $U$. Then from we get $\overline{e}_i(p_i)\overline{e}_i({\operatorname{Tr}}D)=0$, for all $i=1, \dots, n$, and so the above equation gives $$-\sum\nolimits_i e^{-2up_i}\bigl(2(\overline{e}_i(p_i))^2+\sum\nolimits_k(\overline{e}_i(p_k))^2\bigr)
-\sum\nolimits_k e^{-2up_k}(\overline{e}_k({\operatorname{Tr}}D))^2=0.$$ Thus $\overline{e}_i(p_k)=0$, for all $i,k = 1, \dots, n$, at all the points of $U$. It follows that the coefficients of the characteristic polynomial of $D$ are locally constant on every open, connected domain $U \subset N$ where the eigenvalues of $D$ have constant multiplicities. As the union of such domains is dense in $N$, the eigenvalues of $D$ are constant on the whole manifold $N$. Then (\[eq:divgrad\], \[eq:RicuD\]) imply , as required.
The converse easily follows from and .
Ricci and scalar curvature of the $D$-deformation with constant eigenvalues {#ss:ricuEin}
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In this subsection, we compute the Ricci tensor and the scalar curvature of the $D$-deformation *assuming the eigenvalues of $D$ to be constant*. Note that by Theorem \[th:Dconst\], this condition must be always satisfied if the $D$-extension is Einstein.
From , the connection components of $g^u$ are given by $$\label{eq:gijkc}
\begin{split}
{\Gamma}^i_{jk}&= \tfrac12 e^{u(p_k-p_i-p_j)}(\overline{{\Gamma}}^k_{ji}-\overline{{\Gamma}}^k_{ij})
-\tfrac12 e^{u(p_i-p_j-p_k)}(\overline{{\Gamma}}^i_{kj}-\overline{{\Gamma}}^i_{jk})
-\tfrac12 e^{u(p_j-p_k-p_i)}(\overline{{\Gamma}}^j_{ik}-\overline{{\Gamma}}^j_{ki})\\
&=\tfrac12 \overline{{\Gamma}}^i_{jk}(e^{u(p_i-p_j-p_k)}+e^{u(p_j-p_k-p_i)})\\
&+\tfrac12\overline{{\Gamma}}^k_{ji}(e^{u(p_k-p_i-p_j)}-e^{u(p_j-p_k-p_i)})-\tfrac12 \overline{{\Gamma}}^k_{ij}(e^{u(p_k-p_i-p_j)}-e^{u(p_i-p_j-p_k)}).
\end{split}$$
Introduce the functions $\mu_{ij|k}=\overline{{\langle}[\overline{e}_i,\overline{e}_j], \overline{e}_k{\rangle}}$. Then from we have $$\label{eq:gijklie}
\begin{split}
\mu_{ij|k}&=\overline{{\Gamma}}_{ji}^k-\overline{{\Gamma}}_{ij}^k,\\
{\Gamma}^i_{jk}&= \tfrac12 e^{u(p_k-p_i-p_j)}\mu_{ij|k}-\tfrac12 e^{u(p_i-p_j-p_k)}\mu_{jk|i}-\tfrac12 e^{u(p_j-p_k-p_i)}\mu_{ki|j}.
\end{split}$$ Substituting into we obtain: $$\label{eq:Ricuij}
\begin{split}
{\langle}{\operatorname{Ric}}^u e_i, e_j{\rangle}_u= &-\tfrac12e^{-u(p_i+p_j)}\bigl(\overline{e}_j(\sum\nolimits_{k}\mu_{ki|k}) + \overline{e}_i(\sum\nolimits_{k}\mu_{kj|k}) +\sum\nolimits_{k,l}\mu_{jk|l}\mu_{il|k}\bigr) \\
&+\tfrac12e^{u(p_i-p_j)}\bigl(\sum\nolimits_{k}e^{-2up_k}\overline{e}_k(\mu_{kj|i})+\sum\nolimits_{k,l}e^{-2up_l}\mu_{lj|i}\mu_{kl|k}\bigr) \\
&+\tfrac12e^{u(p_j-p_i)}\bigl(\sum\nolimits_{k}e^{-2up_k}\overline{e}_k(\mu_{ki|j})+\sum\nolimits_{k,l}e^{-2up_l}\mu_{li|j}\mu_{kl|k}\bigr) \\
&+\tfrac14e^{u(p_i+p_j)}\sum\nolimits_{k,l}e^{-2u(p_l+p_k)}\mu_{kl|i}\mu_{kl|j} \\
&-\tfrac12e^{-u(p_i+p_j)}\sum\nolimits_{k,l}e^{2u(p_l-p_k)}\mu_{ik|l}\mu_{jk|l}.
\end{split}$$ In particular, we get $$\label{eq:Ricuii}
\begin{split}
{\langle}{\operatorname{Ric}}^u e_i, e_i{\rangle}_u= &-\tfrac12e^{-2up_i}\bigl(2\overline{e}_i(\sum\nolimits_{k}\mu_{ki|k}) +\sum\nolimits_{k,l}\mu_{ik|l}\mu_{il|k}\bigr) \\
&+\sum\nolimits_{k}e^{-2up_k}\bigl(\overline{e}_k(\mu_{ki|i})+\mu_{ki|i}\sum\nolimits_{l}\mu_{lk|l}\bigr) \\
&+\tfrac14\sum\nolimits_{k,l}e^{2u(p_i-p_l-p_k)}(\mu_{kl|i})^{2} -\tfrac12\sum\nolimits_{k,l}e^{2u(p_l-p_k-p_i)}(\mu_{ik|l})^{2},
\end{split}$$ and for the scalar curvature of the $D$-deformation, $$\label{eq:scalu}
\begin{split}
{\operatorname{scal}}^u =& \sum\nolimits_{k}e^{-2up_k}\Bigl(2\overline{e}_k\Bigl(\sum\nolimits_{i}\mu_{ki|i}\Bigr)-\Bigl(\sum\nolimits_{i}\mu_{ki|i}\Bigr)^2-\tfrac12 \sum\nolimits_{i,l}\mu_{ki|l}\mu_{kl|i}\Bigr) \\
&-\tfrac14\sum\nolimits_{i,k,l}e^{2u(p_i-p_l-p_k)}(\mu_{kl|i})^{2}.
\end{split}$$
If we additionally assume the $D$-extension to be Einstein, then by we obtain $$\begin{aligned}
\label{eq:ein1Rudetailes}
{\langle}{\operatorname{Ric}}^u e_i, e_j{\rangle}_u & = (({\operatorname{Tr}}D) p_i - {\operatorname{Tr}}(D^2)) {\delta}_{ij}, \\
{\operatorname{scal}}^u & = ({\operatorname{Tr}}D)^2 - n {\operatorname{Tr}}(D^2), \label{eq:scaludetailes}\end{aligned}$$ where the left-hand sides are given by and respectively. We also note that can be explicitly written in the equivalent form $$\label{eq:mijj}
\sum\nolimits_j\overline{{\Gamma}}^i_{jj}(p_i-p_j)=\sum\nolimits_j \mu_{ij|j}(p_i-p_j)=0.$$
Summarising the above we can express the conditions of Theorem \[th:Dconst\] explicitly as follows.
\[cor:Einext\] The extension $(M,g^D)$ is Einstein if and only if $D$ has constant eigenvalues and equations *(*with ${\operatorname{Ric}}^u$ given by *)* and are satisfied.
The eigenvalue type of $D$. Proof of Theorem \[th:nonzero\] {#s:eigen}
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By Theorem \[th:Dconst\], the eigenvalues $p_i$ of $D$ are constant. In this section, we show that there are strong algebraic restrictions on $p_i$.
We have the following lemma (note that assertion is well-known, as in that case $g^D$ is a warped product).
\[l:scalar\] Suppose that $(M,g^D)$ is Einstein. Then
1. \[it:sc1\] $D$ is scalar if and only if $N$ is Ricci flat.
2. \[it:sc2\] In all the other cases, there exist $i, j, k$ with $i \ne j$ such that $p_k=p_i+p_j$.
3. \[it:sc3\] If $p_i+p_j-p_k \notin \{0, p_1, \dots, p_n\}$, then $\mu_{ij|k}=0$.
Suppose that $D$ is scalar. Then from ${\operatorname{Ric}}^u=0$. In particular, ${\operatorname{Ric}}^0=0$, as required.
Conversely, if ${\operatorname{Ric}}^0=0$, then by we have $({\operatorname{Tr}}D) \, D = {\operatorname{Tr}}(D^2) \, {\mathrm{id}}$. Taking the traces of both sides we obtain $({\operatorname{Tr}}D)^2 = n{\operatorname{Tr}}(D^2)$, which is only possible when $D$ is scalar, by the Cauchy-Schwarz inequality.
Suppose such a triple $i, j, k$ does not exist. In particular, this means that all the $p_i$ are nonzero. Then the only possible way for the scalar curvature ${\operatorname{scal}}^u$ given by to be constant is when it is identically zero. By this implies that $({\operatorname{Tr}}D)^2 = n{\operatorname{Tr}}(D^2)$, so that $D$ is scalar.
Let $\mathcal{S}$ be the set of all triples $(i,j,k)$, with $i \ne j$, satisfying the assumption of the assertion (note that for $i=j$, the claim is trivial). Then $k \ne i, j$ (as otherwise $p_i+p_j-p_k \in \{p_i,p_j\}$). Then from , ${\operatorname{scal}}^u = \sum\nolimits_{k}e^{-2up_k}(\dots) + (\dots) -\tfrac14\sum\nolimits_{(k,l,i) \in \mathcal{S}}e^{2u(p_i-p_l-p_k)}(\mu_{kl|i})^{2}$, where $(\dots)$ are some expressions not involving $u$. As no terms in the last sum have zero exponents or the same exponents as the terms in the first sum (by the definition of $\mathcal{S}$), and as ${\operatorname{scal}}^u$ is a constant, we get $\sum\nolimits_{(k,l,i) \in \mathcal{S}}e^{2u(p_i-p_l-p_k)}(\mu_{kl|i})^{2}=0$, so $\mu_{kl|i}=0$, for all $(k,l,i) \in \mathcal{S}$.
\[rem:survive\] It follows from Lemma \[l:scalar\] and from and that in the diagonal components $({\operatorname{Ric}}^u)^i_i={\langle}{\operatorname{Ric}}^u e_i, e_i{\rangle}_u$ of the Ricci tensor of $g^u$, the only non-vanishing terms are those with $e^{-2uq}$, where $q \in P:=\{0, p_1, \dots, p_n\}$. Let $P=\{0, q_1, \dots, q_m\}$ (without repetitions). Collecting the similar terms in , we obtain $$\label{eq:collect}
({\operatorname{Ric}}^u)^i_i=\sum\nolimits_{a=1}^m e^{-2uq_a} r_{ia} +r_{i0},$$ where the expressions $r_{ia}, r_{i0}$ do not depend on $u$ (but only on $(N, g)$), and so for all $i=1, \dots, n$, implies $$\label{eq:equate}
r_{ia}=0, \;\text{ for all } a=1, \dots, m, \quad \text{and } r_{i0}=\Big(\sum\nolimits_j p_j\Big) p_i - \Big(\sum\nolimits_j p_j^2\Big).$$
Suppose that all the $p_i$ are nonzero. Then from and , the terms $r_{i0}$ of the diagonal elements $({\operatorname{Ric}}^u)_i^i={\langle}{\operatorname{Ric}}^u e_i, e_i{\rangle}_u$ of ${\operatorname{Ric}}^u$ are given by $$r_{i0}=\tfrac14\sum\nolimits_{k,l: p_i-p_l-p_k=0}(\mu_{kl|i})^{2} -\tfrac12\sum\nolimits_{k,l: p_l-p_k-p_i=0}(\mu_{ik|l})^{2},$$ Note that, as $p_i \ne0$, all the three subscripts $i,k,l$ in each of the above summations are pairwise non-equal. It follows that $$\begin{split}
r_{i0}&=\tfrac14\sum\nolimits_{k,l: f_k+f_l-f_i \in (F \cap {\mathbf{p}}^\perp)}(\mu_{kl|i})^{2} -\tfrac12\sum\nolimits_{k,l: f_i+f_k-f_l \in (F \cap {\mathbf{p}}^\perp)}(\mu_{ik|l})^{2}\\
&=\tfrac12\sum\nolimits_{a: v_a=f_k+f_l-f_i \in (F \cap {\mathbf{p}}^\perp)}(\mu_{kl|i})^{2} -\tfrac12\sum\nolimits_{a: v_a=f_i+f_k-f_l \in (F \cap {\mathbf{p}}^\perp)}(\mu_{ik|l})^{2},
\end{split}$$ where $v_a$ is some labeling of the elements of the set $F \cap {\mathbf{p}}^\perp$. Denote $\mu_a^2=(\mu_{ij|k})^2$ for $v_a=f_i+f_j-f_k \in (F \cap {\mathbf{p}}^\perp)$. Then we obtain $$r_{i0}=\tfrac12\sum\nolimits_{a: v_a \in (F \cap {\mathbf{p}}^\perp), {\langle}v_a,f_i{\rangle}=-1}\mu_a^2 -\tfrac12\sum\nolimits_{a: v_a \in (F \cap {\mathbf{p}}^\perp), {\langle}v_a,f_i{\rangle}=1}\mu_a^2=-\tfrac12\sum\nolimits_{a: v_a \in (F \cap {\mathbf{p}}^\perp)}{\langle}v_a,f_i{\rangle}\mu_a^2.$$ By the second equation of , the latter expression equals ${\langle}{\mathbf{p}},\mathbf{1}_n{\rangle}p_i-\|{\mathbf{p}}\|^2$, so we obtain $$\label{eq:Fpperp}
\sum\nolimits_{a: v_a \in (F \cap {\mathbf{p}}^\perp)} (\tfrac12\mu_a^2) v_a =\|{\mathbf{p}}\|^2\mathbf{1}_n-{\langle}{\mathbf{p}},\mathbf{1}_n{\rangle}{\mathbf{p}}.$$ The right-hand side is the projection of $\mathbf{1}_n$ to ${\mathbf{p}}^\perp$ multiplied by $\|{\mathbf{p}}\|^2$, and the claim follows.
From we obtain that $Vc=\|{\mathbf{p}}\|^2\mathbf{1}_n-{\langle}{\mathbf{p}},\mathbf{1}_n{\rangle}{\mathbf{p}}$, for some vector $c \in {\mathbb{R}}^m$ (note that its components are not necessarily nonnegative). By construction, $V^t \mathbf{1}_n = \mathbf{1}_m, \; {\operatorname{rk}}V = m$ and $V^t{\mathbf{p}}=0$ (in particular, $n > m$, as otherwise ${\mathbf{p}}=0$). Then $V^tVc=\|{\mathbf{p}}\|^2\mathbf{1}_m$, so $c=\|{\mathbf{p}}\|^2 (V^tV)^{-1}\mathbf{1}_m$. Therefore ${\langle}{\mathbf{p}},\mathbf{1}_n{\rangle}{\mathbf{p}}=\|{\mathbf{p}}\|^2(\mathbf{1}_n-V(V^tV)^{-1}\mathbf{1}_m)$, and equation follows, as ${\langle}{\mathbf{p}},\mathbf{1}_n{\rangle}= {\operatorname{Tr}}D$ is assumed to be nonzero.
As all the components of the vector on the right-hand side of are rational, all the $p_i$’s, up to scaling, are integer.
The last claim of the assertion follows from the fact that for every $n$, there is a finite number of possible matrices $V$.
\[rem:34\] The conditions imposed by Theorem \[th:nonzero\] on the eigenvalue type of $D$ are quite restrictive (although somewhat implicit). For example, it follows that if $n=3$ and $\det D \ne 0$, then $D= {\operatorname{diag}}(1,1,2)$ and $D= {\mathrm{id}}$ are the only possible eigenvalue types, up to scaling. Indeed, there can be no more than one relation of the form $p_i+p_j-p_k=0$, with $i \ne j$, between $p_1,p_2,p_3$. It follows that $F \cap {\mathbf{p}}^\perp$ is either empty (then $D$ is scalar by Lemma \[l:scalar\]), or consists of a single element $f_1+f_2-f_3$, up to relabelling. In the latter case, the matrix $V$ is $3 \times 1$, $V=(1,1,-1)^t$, and by , the vector $\mathbf{p}$ is a multiple of $(1,1,2)^t$. If $n=4$ and $\det D \ne 0$, then considering all the possibilities for the matrix $V$ we obtain that all the eigenvalue types, up to scaling, are $$\begin{gathered}
(1,1,1,1)^t,\; (2,2,3,4)^t,\; (3,4,4,7)^t,\; (1,2,3,4)^t,\; (1,1,2,2)^t,\\ (1,1,1,2)^t,\; (1,1,2,3)^t, \; (-1,1,1,2)^t,\; (-1,1,2,3)^t.\end{gathered}$$
\[rem:drop\] In general, the condition $\det D \ne 0$ in Theorem \[th:nonzero\] cannot be dropped, as shows the analysis of the case $n=3$ in Section \[s:dim3\] (for example, in the last row of Table \[t:d3\] in Theorem \[th:dim3\], $p$ can be any real number). One might ask however, if the assumption ${\operatorname{Tr}}D \ne 0$ in Theorem \[th:nonzero\] can be removed. It follows from the proof that the condition ${\operatorname{Tr}}D = 0$ is equivalent to $V(V^tV)^{-1}\mathbf{1}_m=\mathbf{1}_n$ (which geometrically means that the manifold $(N,g)$ is by itself Einstein). The following example shows the necessity of this assumption, at least at the algebraic level. Let $n=6$ and let $\mathbf{p}=(-3,-2,-1,1,2,3)^t$. Then $F \cap {\mathbf{p}}^\perp = \{v_{142}, v_{153}, v_{231}, v_{243}, v_{264}, v_{354},v_{365},v_{456}\}$, where $v_{ijk}=f_i+f_j-f_k$. The projection of $\mathbf{1}_6$ to ${\mathbf{p}}^\perp$ is $\mathbf{1}_6$ which belongs to the convex cone hull of the set $F \cap {\mathbf{p}}^\perp$ as $2 \cdot \mathbf{1}_6 = 3v_{142}+v_{153}+2v_{231}+v_{243}+2v_{264}+v_{354}+v_{365}+v_{456}$, but is not satisfied (taking the inner product of both sides with $\mathbf{p}$ we obtain $\|\mathbf{p}\|^2=0$).
Einstein $D$-extension of the eigenvalue type $\mathbf{p} = (\lambda, \cdots, \lambda, \nu)^t$ {#s:Heis}
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Three cases {#s:threecases}
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In the previous section, we considered the conditions which the Einstein property of $(M,g^D)$ imposes on $D$. Clearly, there are also some conditions on $(N,g)$. For example, the Ricci eigenvalues of $(N,g)$ are constant by , and the scalar curvature is negative, unless $(N,g)$ is Ricci flat, by Lemma \[l:scalar\].
In this section, we give a complete characterisation of $(N,g)$ and of $(M,g^D)$ in the case when $D$ has the “next simplest" eigenvalue type after being a scalar operator, namely when one of the eigenvalues of $D$ has multiplicity $n-1$. As we know from the Introduction, there are only three possibilities, up to scaling; the Ricci operator ${\operatorname{Ric}}^u$ is given by :
1. \[it:01\] $\mathbf{p}=(0, \dots, 0, 1),\quad {\operatorname{Ric}}^u = {\operatorname{diag}}(-1, \dots, -1, 0)$,
2. \[it:10\] $\mathbf{p}=(1, \dots, 1, 0),\quad {\operatorname{Ric}}^u = {\operatorname{diag}}(0, \dots, 0, 1-n)$,
3. \[it:12\] $\mathbf{p}=(1, \dots, 1, 2),\quad {\operatorname{Ric}}^u = {\operatorname{diag}}(-2, \dots, -2, n-1)$.
We consider them in Theorem \[th:01\], Theorem \[th:10\], and Theorem \[th:12\] respectively. We start with the following lemma valid in all three cases. Denote $S_i=\sum_k \mu_{ki|k}$.
\[l:multn-1\] Let $(M,g^D)$ be the $D$-extension of $(N,g)$ with the spectral vector $\mathbf{p} = (\lambda, \cdots, \lambda, \nu)^t$, $\lambda \ne \nu$. Then
1. \[it:multn-1Smu\] ${\operatorname{div}}D = 0$ if and only if $S_n=0$ and $\mu_{in|n}=0$, for all $i < n$.
2. \[it:multn-1sym\] Locally there exists a frame $\overline{e}_1, \dots, \overline{e}_{n-1}$ for the ${\lambda}$-eigendistribution of $D$ such that $\mu_{ni|j}=\mu_{nj|i}$, for all $i,j = 1, \dots, n-1$. Such a frame can be chosen arbitrarily on a hypersurface transversal to $\overline{e}_n$.
follows from .
Let $W = (w_{ij})$ be an $(n-1)\times (n-1)$ orthogonal matrix whose entries are smooth functions on $N$, and define $\overline{e}'_i=\sum_k w_{ik}\overline{e}_k$. Relative to the orthonormal frame $\overline{e}'_1, \dots , \overline{e}'_{n-1},\overline{e}_n$, we have $\mu'_{ni|j} = {\langle}[\overline{e}_n,\overline{e}'_i],\overline{e}'_j{\rangle}= \sum_k \overline{e}_n(w_{ik}) w_{jk}+\sum_{k,s} w_{ik} w_{js} \mu_{nk|s}$. Let $K, K'$ be $(n-1)\times (n-1)$ skew-symmetric matrices defined by $K_{ij}=\mu_{ni|j}-\mu_{nj|i}$ and $K'_{ij}=\mu'_{ni|j}-\mu'_{nj|i}$. Then $K'=WKW^t + \overline{e}_n(W)W^t - W\overline{e}_n(W^t)$ which is equivalent to $W^tK'W=K+2W^t\overline{e}_n(W)$. Solving the equation $\overline{e}_n(W)=-\frac12 WK$ along the integral curves of $\overline{e}_n$, with the initial condition $W={\mathrm{id}}$ on a hypersurface transversal to $\overline{e}_n$, we get $K'=0$, as required.
For the rest of this section, we assume the frame $\overline{e}_i$ to be chosen as in Lemma \[l:multn-1\].
Proof of Theorem \[th:01\] {#ss:01}
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From Lemma \[l:scalar\] we get $\mu_{ij|n}=0$, for all $i,j < n$. Using this fact and Lemma \[l:multn-1\] we find $({\operatorname{Ric}}^u)_n^n= -e^{-2u} \sum\nolimits_{k,l < n}(\mu_{nk|l})^2$ from . But $({\operatorname{Ric}}^u)_n^n = 0$ by , so $\mu_{nk|l}=0$, for all $k,l$. Therefore $\mu_{ij|k}=0$ whenever at least one of the subscripts equals $n$. It follows that the vector field $\overline{e}_n$ is parallel, and so $(N,g)$ is locally isometric to the Riemannian product of the real line and an $(n-1)$-dimensional manifold $N'$. Furthermore, for $i,j<n$ we get $({\operatorname{Ric}}^u)_i^j = -{\delta}_{ij}$ by . It follows that $N'$ is Einstein, with the Einstein constant $-1$. Then the metric $g$ on $N$ is given by $\overline{ds^2}=(dx^n)^2 + ds'^2$, where $ds'{}^2$ is an Einstein metric on $N'$, and hence the metric $g^D$ on $M$ is given by $\overline{ds^2}=du^2+e^{2u}(dx^n)^2 + ds'^2$. Thus $(M,g^D)$ is locally isometric to the Riemannian product of the hyperbolic plane of curvature $-1$ and the Einstein manifold $N'$ with the Einstein constant $-1$.
\[rem:convth01\] Note that the converse to Theorem \[th:01\] is easily verified: starting with the Riemannian product of the real line ${\mathbb{R}}$ and an Einstein manifold $N'$ of dimension $n-1$ with the Einstein constant $-1$ and extending it by the endomorphisms $D$ with the spectral vector $\mathbf{p}=(0, \dots, 0, 1)^t$ whose kernel is $TN'$ we get an Einstein manifold $(M,g^D)$ isometric to the Riemannian product of the hyperbolic plane of curvature $-1$ and $N'$.
Case {#ss:10}
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In the case $\mathbf{p}=(1, \dots,1, 0)^t$, we have the following theorem.
\[th:10\] Let $(M,g^D)$ be the $D$-extension of $(N,g)$ with the spectral vector $\mathbf{p}=(1, \dots, 1, 0)^t$. If $(M,g^D)$ is Einstein, then the $1$-eigendistribution of $D$ is integrable and the manifold $(N,g)$ is locally an extension of a Ricci flat manifold $(N',g')$ of dimension $n-1$ by a field of symmetric endomorphism $D'$ with constant eigenvalues, such that ${\operatorname{Tr}}D' = 0$, ${\operatorname{Tr}}D'^2=n-1$, and ${\operatorname{div}}' D'=0$. The Einstein metric $g^D$ on $M$ is locally given by $ds^2=du^2+dt^2+\sum_{i=1}^{n-1} e^{2u+2tq_i} (\theta'^i)^2$, where $q_i$ are the eigenvalues of $D'$ and $\{\theta'^i\}$ is the coframe dual to an orthonormal frame of eigenvectors of $D'$ on $N'$.
From Lemma \[l:multn-1\] and Lemma \[l:scalar\] we obtain that $S_n=0$ and that $\mu_{ij|n}=0$, for all $i,j$. It follows that the vector field $\overline{e}_n$ is geodesic and that its orthogonal distribution is integrable. Therefore we can define a function $x^n$ locally on $N$ in such a way that $\overline{\theta}^n=dx^n$. Without loss of generality, assume that $N'$ is the level hypersurface of $N$ defined by the equation $x^n=0$. Denote $g'$ the induced metric on $N'$.
From we get $({\operatorname{Ric}}^u)_i^j=0$ for $i,j < n$. Then by , the sum of the terms of $({\operatorname{Ric}}^u)_i^j$ which do not depend on $u$ gives $\overline{e}_n(\mu_{ni|j})=0$, for all $i,j < n$. We can now specify the frame $\overline{e}_i, \; i<n$, further. Note that $\mu_{ni|j}= {\langle}[\overline{e}_n, \overline{e}_i],\overline{e}_j{\rangle}= {\langle}\overline{\nabla}_{\overline{e}_n} \overline{e}_i, \overline{e}_j{\rangle}- {\langle}\overline{\nabla}_{\overline{e}_i}\overline{e}_n, \overline{e}_j{\rangle}= {\langle}\overline{\nabla}_{\overline{e}_n} \overline{e}_i, \overline{e}_j{\rangle}+ {\langle}\overline{\nabla}_{\overline{e}_i}\overline{e}_j, \overline{e}_n{\rangle}$. On the hypersurface $N'$, we have ${\langle}\overline{\nabla}_{\overline{e}_i} \overline{e}_j, \overline{e}_n{\rangle}=h(\overline{e}_i, \overline{e}_j)$, where $h$ is the second fundamental form of $N'$. As $\mu_{ni|j}=\mu_{nj|i}$ we obtain that on $N'$, the expression ${\langle}\overline{\nabla}_{\overline{e}_n} \overline{e}_i, \overline{e}_j{\rangle}$ is symmetric relative to $i,j$, hence ${\langle}\overline{\nabla}_{\overline{e}_n} \overline{e}_i, \overline{e}_j{\rangle}=0$, and so $\mu_{ni|j}=h(\overline{e}_i, \overline{e}_j)$ on $N'$. We can now choose the frame $\overline{e}_i, \; i<n$, on $N'$ consisting of orthonormal eigenvectors of the second fundamental form $h$ and then extend it to $N$ as in Lemma \[l:multn-1\]. Then the symmetric matrix $(\mu_{ni|j})$ restricted to $N'$ is diagonal, and from the fact that $\overline{e}_n(\mu_{ni|j})=0$, for all $i,j < n$, we obtain that $(\mu_{ni|j})$ is diagonal locally on $N$, so that $\mu_{ni|j}=q_i {\delta}_{ij}$ and $\overline{e}_n(q_i)=0$. Then the vector fields $e'_i=e^{-x^nq_i} \overline{e}_i, \; i<n$, are Lie parallel along the integral curves of $\overline{e}_n=\partial_n$, as also are their dual one-forms ${\theta'}^i= e^{x^nq_i} \overline{\theta}^i$. It follows that the metric $g$ on $N$ is locally given by $\overline{ds^2}=\sum\nolimits_{i=1}^n (\overline{\theta}^i)^2 = (dx^n)^2+\sum\nolimits_{i<n} e^{2x^nq_i}({\theta'}^i)^2$. Thus the metric $(M,g^D)$ has the required form and moreover, by , $(N,g)$ is locally an extension of $(N',g')$ by the field of the symmetric endomorphism $D'$ defined by $D'\overline{e}_i=q_i \overline{e}_i$ at the points of $N'$.
We also have ${\operatorname{Tr}}D' = \sum_i \mu_{ni|i}=-S_n=0$ by Lemma \[l:multn-1\]. Furthermore, from we get $({\operatorname{Ric}}^u)_n^n=1-n$, which by gives ${\operatorname{Tr}}D'^2 = \sum_i \mu_{ni|i}^2 = n-1$. Moreover, as $({\operatorname{Ric}}^u)_i^j=0$ for $i,j < n$, we have $({\operatorname{Ric}}^0)_i^j=0$ (note that $g^0=g$), and so applied to the extension of $(N',g')$ by $D'$ implies that $(N',g')$ is Ricci flat. We also have $({\operatorname{Ric}}^0)_n^i=0$ for $i < n$, so by applied to the extension of $(N',g')$ by $D'$ we obtain ${\operatorname{div}}' D'=0$.
The metric $g^D$ is locally obtained as the result of two consecutive extensions from the metric $g'$ on $N'$, first by $D'$ and then by $D$. Note that these two extensions “commute", so we can first extend the Ricci flat metric $(N',g')$ by the identity endomorphism to obtain an Einstein manifold $\tilde N$ with the Einstein constant $-1$ (compare to Lemma \[l:scalar\]), and then extend again by the endomorphism $\tilde D$ which coincides with $D'$ on $TN'$, is zero on $\partial_u$, and is Lie parallel along $\partial_u$, to obtain the Einstein metric $(M, g^D)$. By Theorem \[th:Dconst\], the eigenvalues of $\tilde D$ are constant, and so the eigenvalues of $D'$ are also constant, as claimed.
\[rem:10\] Note that the condition for $(M,g^D)$ to be Einstein given in Theorem \[th:10\] is only necessary. To make it sufficient one has to additionally require that all the $D'$-deformations of the metric $(N',g')$ are Ricci flat, not only the metric $(N',g')$ itself. In general, it may be too difficult to classify Ricci flat deformations of a Ricci flat manifold, even under the additional assumption that ${\operatorname{Tr}}D' = 0$, ${\operatorname{Tr}}D'^2=n-1$, and ${\operatorname{div}}' D'=0$. One simple example is the Riemannian product of Ricci flat manifolds, with the operator $D'$ acting by scaling on each factor (compare to Example \[ex:product\]). When $\dim N' = 2$, this is the only possible case (see the end of the proof of Theorem \[th:dim3\] in the next section).
Proof of Theorem \[th:12\] {#ss:12}
--------------------------
$\Leftrightarrow$ . By Corollary \[cor:Einext\], is equivalent to the fact that equations are satisfied and that ${\operatorname{Ric}}^u$ given by satisfies . By Lemma \[l:multn-1\], is equivalent to the fact that $S_n=0$ and $\mu_{in|n}=0$, for all $i < n$. Furthermore, we can assume that the frame $\overline{e}_1, \dots, \overline{e}_{n-1}$ is chosen in such a way that $\mu_{ni|j}=\mu_{nj|i}$ (and there is still the freedom of choosing it (locally) arbitrarily on a hypersurface transversal to $\overline{e}_n$).
From and by Lemma \[l:multn-1\] we get $$({\operatorname{Ric}}^u)_n^n= -e^{-4u}\sum\nolimits_{k,l < n}(\mu_{nk|l})^2 +\tfrac14\sum\nolimits_{k,l<n}(\mu_{kl|n})^{2}=n-1,$$ which is equivalent to the fact that $$({\operatorname{Ric}}^0)_n^n = n-1 \quad \text{and} \quad \mu_{nk|l}=0,$$ for all $k, l < n$. Then from and we obtain $d\overline{\theta}^i=-\frac12 \sum_{j,k<n} \mu_{jk|i} \overline{\theta}^j \wedge \overline{\theta}^k$. It follows that $d(\sum_{j,k<n} \mu_{jk|i} \overline{\theta}^j \wedge \overline{\theta}^k)=0$, so $\overline{e}_n(\mu_{jk|i})=0$, for all $j,k < n$, and in particular, $\overline{e}_n(S_i)=0$.
Now computing the components $({\operatorname{Ric}}^u)_i^i, \; ({\operatorname{Ric}}^u)_i^n, \; i <n$, by and and using we get $$\begin{gathered}
({\operatorname{Ric}}^u)_i^i = e^{-2u} (({\operatorname{Ric}}^0)_i^i + \tfrac12 \sum\nolimits_{k<n}(\mu_{ik|n})^{2}) - \tfrac12 \sum\nolimits_{k<n}(\mu_{ik|n})^{2} = -2,
\\
({\operatorname{Ric}}^u)_i^n = e^{-u} ({\operatorname{Ric}}^0)_i^n = 0,\end{gathered}$$ which is equivalent to the fact that for all $i < n$, $$({\operatorname{Ric}}^0)_i^i = -2, \; ({\operatorname{Ric}}^0)_i^n = 0, \quad \text{and} \quad \sum\nolimits_{k<n}(\mu_{ik|n})^{2}=4.$$
Now taking $\xi=\overline{e}_n$ we find that $\xi$ is geodesic if and only if $\mu_{in|n}=0$. Furthermore, choosing an orthonormal frame in the distribution $\xi^\perp$ as in Lemma \[l:multn-1\] (so that $\mu_{ni|j}=\mu_{nj|i}$ for $i,j <n$) and defining $J=-\nabla \xi$ we obtain ${\langle}J\overline{e}_i, \overline{e}_j{\rangle}= \mu_{ni|j}+\frac12\mu_{ij|n}$ for $i, j < n$, and so $\xi$ is Killing if and only if $\mu_{ni|j}=0$. Then the condition that the contact structure defined by $\xi$ is K-contact is equivalent to the fact that $\sum\nolimits_{k<n}(\mu_{ik|n})^{2}=4$, for all $i < n$. Finally, the condition that $(N,g)$ is $\eta$-Einstein, with ${\operatorname{ric}}= -2 g + (n+1) \eta \otimes \eta$, where $\eta=\overline{\theta}^n$, is equivalent to the fact that $({\operatorname{Ric}}^0)_i^i = -2, \; ({\operatorname{Ric}}^0)_i^n = 0$ and $({\operatorname{Ric}}^0)_n^n = n-1$, as the orthonormal basis $\overline{e}_i$ at a point can be chosen arbitrarily.
$\Leftrightarrow$ . The foliation defined by $\xi$ on $(N,g)$ is geodesic. Locally take $N'=N/\xi$ and define the metric $g'$ on $N'$ in such a way that the natural projection is a submersion (this is possible as $\xi$ is Killing). Then the restriction of $J$ to $N'$ defines an almost Kähler structure. The fact that $(N',g')$ is Ricci flat follows from [@BG Equation 7.3] or by a direct calculation. The implication $\Leftrightarrow$ is proved by reversing the construction.
Finally, it is easy to see that $(M,g^D)$ is almost Kähler, with the fundamental 2-form $e^{2u}(2du \wedge (dt+\theta') + \omega)$, in the notation of .
Four-dimensional Einstein extensions {#s:dim3}
====================================
In this section we consider the case $n=3$, the lowest dimension when our construction provides interesting examples. Note that in the case $n =2$, there are only two independent connection components, $\overline{{\Gamma}}^1_{21}$ and $\overline{{\Gamma}}^1_{22}$, and implies that $\overline{{\Gamma}}^1_{21}(p_1-p_2) = \overline{{\Gamma}}^1_{22}(p_1-p_2) = 0$, so either $(N,g)$ is flat, or $D$ is scalar, which again implies that $(N,g)$ is flat by Lemma \[l:scalar\]. Then the manifold $(M,g^D)$ is hyperbolic.
We consider all the possible eigenvalue types of $D$.
In the case when $D$ is scalar, the manifold $(N, g)$ is flat by Lemma \[l:scalar\]. We can locally introduce Cartesian coordinates $x^1, x^2,x^3$ on $N$ and set $\overline{e}_i=\partial_i$. Then $N$ is abelian, $D$ is left-invariant, and its value at the identity of $N$ is obviously a derivation of the abelian Lie algebra ${\mathfrak{n}}$ of $N$. We get the first two rows of Table \[t:d3\], up to scaling.
Next suppose that two out of three eigenvalues $p_i$ are zeros. Up to scaling, we can assume that $p_1=p_2=0, \; p_3=1$. By Theorem \[th:01\] we can choose local coordinates on $N$ in such a way that $\overline{ds^2}=(dx^3)^2 + (dx^1)^2 + e^{2x^1}(dx^2)^2$, and the orthonormal frame of eigenvectors of $D$ is $\overline{e}_1=\partial_1, \, \overline{e}_3=\partial_3$, and $\overline{e}_2= e^{-x^1} \partial_2$. Then $N$ is locally a solvable Lie group, with the only nontrivial relation in ${\mathfrak{n}}$ being $[\overline{e}_1,\overline{e}_2]=-\overline{e}_2$. Moreover, $D$ is left-invariant and is a derivation of ${\mathfrak{n}}$. Up to relabelling we obtain the case in the last row of Table \[t:d3\], with $p=0$.
Suppose that $D$ is non-scalar and nonsingular. Up to scaling, we get $D={\operatorname{diag}}(1,1,2)$ by Remark \[rem:34\]. Then by Theorem \[th:12\], we can choose local coordinates on $N$ in such a way that $\overline{ds^2}=ds'^2+(dx^3+\theta')^2$, where $ds'^2=(dx^1)^2+(dx^2)^2$ is a two-dimensional flat metric and $\theta'=x^1 dx^2-x^2 dx^1$. Then $\overline{ds^2}=(dx^1)^2+(dx^2)^2+(dx^3+x^1 dx^2-x^2 dx^1)^2$. An orthonormal frame of eigenvectors of $D$ can be chosen as $\overline{e}_1={\partial}_1+x^2{\partial}_3, \, \overline{e}_2= \partial_2-x^1 \partial_3$, and $\overline{e}_3=-\partial_3$. Then $N$ is locally the Heisenberg Lie group, with the only nontrivial relation in ${\mathfrak{n}}$ being $[\overline{e}_1,\overline{e}_2]=2\overline{e}_3$. The endomorphism field $D$ is left-invariant and its value at the identity is a derivation of ${\mathfrak{n}}$. We obtain the case in the third row of Table \[t:d3\]. The fact that the extension $(M,g^D)$ is locally isometric to ${\mathbb{C}}H^2$ is well-known (see e.g., [@Heb Section 6.5]).
Suppose that one of the eigenvalues of $D$ is zero and the other two are nonzero. Up to relabeling and scaling, we can take $p_1=1, \; p_3=0, \; p_2 =p$, where $|p| \ge 1$.
We have the following lemma.
\[l:pnot1\] If $p \ne 1$, then the only nonzero $\mu_{ij|k}, \; i<j$, are $\mu_{23|2}= 1$ and $\mu_{13|1}=- p$.
As in Section \[s:Heis\], we denote $S_i=\sum_k \mu_{ki|k}$. Consider three cases.
Let $p \ne -1, 1, 2$. Then for pairwise non-equal $i,j,k$, we have $p_i+p_j-p_k \notin \{0,1,p\}$, so $\mu_{ij|k}=0$, by Lemma \[l:scalar\]. From we obtain that $\mu_{21|2}=\frac{1}{p}S_1, \; \mu_{31|3}=\frac{p-1}{p}S_1, \; \mu_{12|1}=pS_2, \; \mu_{32|3}=(1-p)S_2, \; \mu_{13|1}= \frac{p}{p-1}S_3, \; \mu_{23|2}=\frac{1}{1-p}S_3$. Collecting the similar terms of as in , we obtain by : $$\begin{gathered}
\overline{e}_1(S_1)+S_1^2=\overline{e}_1(S_1)+\tfrac{1+(1-p)^2}{p^2} S_1^2=0, \quad \overline{e}_2(S_2)+S_2^2=\overline{e}_2(S_2)+(p^2+(1-p)^2)S_2^2=0, \\ \overline{e}_3(S_3)+S_3^2=(1-p)^2,\quad \overline{e}_3(S_3)+\tfrac{1+p^2}{(1-p)^2}S_3^2=p^2+1,\end{gathered}$$ and so $S_1=S_2=0$ and $S_3=1-p$ (up to changing the sign of $\overline{e}_3$), and the claim follows.
Now suppose $p=2$. Then by Lemma \[l:scalar\], we have $\mu_{12|3}=\mu_{31|2}=0$. From we obtain that $\mu_{21|2}=\mu_{31|3}=\frac12 S_1, \; \mu_{12|1}=2S_2$, $\mu_{32|3}=-S_2, \; \mu_{13|1}=2S_3, \; \mu_{23|2}=-S_3$. From and we get: $$\begin{gathered}
\overline{e}_1(S_1)+S_1^2+\mu_{23|1}^2=\overline{e}_1(S_1)+\tfrac12 S_1^2-\tfrac12\mu_{23|1}^2=0, \quad \overline{e}_2(S_2)+S_2^2=\overline{e}_2(S_2)+5S_2^2=0, \\ \overline{e}_3(S_3)+S_3^2=1,\quad \overline{e}_3(S_3)+5S_3^2=5,\end{gathered}$$ which then implies that $S_1=S_2=\mu_{23|1}=0$ and $S_3=-1$ (up to changing the sign of $\overline{e}_3$), and the claim follows.
The last case is $p=-1$. By Lemma \[l:scalar\], we get $\mu_{23|1}=\mu_{31|2}=0$. From we obtain $\mu_{21|2}=-S_1, \; \mu_{31|3}=2S_1, \; \mu_{12|1}=-S_2, \; \mu_{32|3}=2S_2, \; \mu_{13|1}=\mu_{23|2}=\frac12 S_3$. Then equations and give: $$\begin{gathered}
\overline{e}_1(S_1)+S_1^2=\overline{e}_1(S_1)+5 S_1^2=0, \quad \overline{e}_2(S_2)+S_2^2=\overline{e}_2(S_2)+5S_2^2=0, \\ \overline{e}_3(S_3)+S_3^2+\mu_{12|3}^2=4,\quad 2\overline{e}_3(S_3)+S_3^2-\mu_{12|3}^2=4,\end{gathered}$$ which implies that $S_1=S_2=0$ and $S_3^2+3\mu_{12|3}^2=4, \; \overline{e}_3(S_3)=2\mu_{12|3}^2$. So all the $\mu_{ij|k}, i < j$, except possibly $\mu_{13|1}=\mu_{23|2}=\frac12 S_3$ and $\mu_{12|3}$, vanish. Then $d\overline{\theta}^1 = -\frac12 S_3 \overline{\theta}^1 \wedge \overline{\theta}^3, \; d\overline{\theta}^2 =-\frac12 S_3 \overline{\theta}^2 \wedge \overline{\theta}^3, \; d\overline{\theta}^3=-\mu_{12|3}\overline{\theta}^1 \wedge \overline{\theta}^2$. Differentiating the last equation we get $0=(\overline{e}_3(\mu_{12|3})+\mu_{12|3}S_3)\overline{\theta}^1 \wedge \overline{\theta}^2 \wedge \overline{\theta}^3$, and so $\overline{e}_3(\mu_{12|3})=-\mu_{12|3}S_3$. But then differentiating the equation $S_3^2+3\mu_{12|3}^2=4$ along $\overline{e}_3$ and using the fact that $\overline{e}_3(S_3)=2\mu_{12|3}^2$ we obtain $S_3\mu_{12|3}^2=0$. It follows that $\mu_{12|3}=0, \; S_3=2$ (up to changing the sign of $\overline{e}_3$), and $\mu_{23|2}=\mu_{13|1}=1$.
We return to the proof of the theorem. Suppose $p \ne 1$. Then it follows from Lemma \[l:pnot1\] that $d\overline{\theta}^1 = p \overline{\theta}^1 \wedge \overline{\theta}^3, \; d\overline{\theta}^2 =- \overline{\theta}^2 \wedge \overline{\theta}^3, \; d\overline{\theta}^3=0$, so we can choose local coordinates on $N$ such that $\overline{\theta}^3=dx^3, \overline{\theta}^1=e^{-p x^3}dx^1, \; \overline{\theta}^2=e^{x^3}dx^2$. Then the metric $g$ is locally given by $\overline{ds^2}=e^{-2p x^3}(dx^1)^2+e^{2x^3}(dx^2)^2+(dx^3)^2$. Furthermore, the eigenvectors $\overline{e}_i$ of $D$ satisfy the relations $[\overline{e}_3,\overline{e}_1]= p \overline{e}_1, \; [\overline{e}_3,\overline{e}_2]=- \overline{e}_2, \; [\overline{e}_1,\overline{e}_2]=0$. Hence $N$ is (locally) a solvable, non-nilpotent Lie group, and $D$ is left-invariant and is a derivation of ${\mathfrak{n}}$. The Einstein metric $g^D$ on $M$ is locally given by $ds^2=e^{2(u-p x^3)}(dx^1)^2+e^{2(pu+x^3)}(dx^2)^2 +(dx^3)^2 +du^2$, which is the Riemannian product of two hyperbolic planes of curvature $-(p^2+1)$, which can be seen by the change of variables $y_1=(p^2+1)^{-1/2}(u-px^3), \; y_2=(p^2+1)^{-1/2}(pu+ x^3)$.
The only remaining case to consider is $p=1$, so that the eigenvalues of $D$ are $p_1=p_2=1$, $p_3=0$. By Theorem \[th:10\], $(N,g)$ is the extension of a flat two-dimensional manifold $(N',g')$ by a symmetric endomorphism $D'$ such that ${\operatorname{Tr}}D'=0, {\operatorname{Tr}}D'^2=2$, and ${\operatorname{div}}' D'=0$. Choosing local Cartesian coordinates $x^1, x^2$ on $N'$, we obtain $D'=\left(\begin{smallmatrix}a&b\\b&-a\end{smallmatrix}\right)$, for some functions $a$ and $b$ on $N'$, with $a^2+b^2=2$. The condition ${\operatorname{div}}' D'=0$ gives $\partial_{x^1} a + \partial_{x^2} b =\partial_{x^1} b - \partial_{x^2} a= 0$, so $a - \mathrm{i}b$ is a holomorphic function with a constant module. It follows that both $a$ and $b$ are constants, so choosing $\partial_{x^1}, \partial_{x^2}$ to be the unit eigenvectors of $D'$ we obtain $D'={\operatorname{diag}}(-1,1)$. Then the metric $g$ on $N$ is locally given by $ds^2=(dx^3)^2+e^{-2x^3} (dx^1)^2+e^{2x^3} (dx^2)^2$. Choosing the unit eigenvectors of $D$ as $\overline{e}_1=e^{x^3}\partial_1, \; \overline{e}_2=e^{-x^3}\partial_2, \; \overline{e}_3=\partial_3$ we obtain $[\overline{e}_3,\overline{e}_1]= \overline{e}_1, \; [\overline{e}_3,\overline{e}_2]=- \overline{e}_2$, and $[\overline{e}_1,\overline{e}_2]=0$, so $N$ is a Lie group defined by the corresponding Lie algebra ${\mathfrak{n}}$ and $D$ is a derivation of ${\mathfrak{n}}$. By Theorem \[th:10\], the Einstein metric $g^D$ on $M$ is given by $ds^2=du^2+(dx^3)^2+e^{2(u-x^3)} (dx^1)^2+e^{2(u+x^3)} (dx^2)^2$, as in the last row of Table \[t:d3\], with $p=1$.
Extensions of a Lie group. Proof of Theorem \[th:group\] {#s:homo}
========================================================
Suppose that $N$ is a Lie group, and both the metric $g$ and the endomorphism field $D$ are left-invariant. We will mostly work on the level of Lie algebras. We can take the vector fields $\overline{e}_i$ left-invariant. Then $\mu_{ij|k}=\overline{{\langle}[\overline{e}_i,\overline{e}_j], \overline{e}_k{\rangle}}$ are constants and are the structure constants of the Lie algebra ${\mathfrak{n}}$ of $N$, and we have $\sum\nolimits_{k,l}\mu_{jk|l}\mu_{il|k} = \sum\nolimits_{k,l}{\langle}{\operatorname{ad}}_j\overline{e}_k, \overline{e}_l{\rangle}{\langle}{\operatorname{ad}}_i\overline{e}_l, \overline{e}_k{\rangle}= B(\overline{e}_i,\overline{e}_j)$, where $B$ is the Killing form of ${\mathfrak{n}}$. Moreover, $S_l=\sum_k\mu_{kl|k}=-{\operatorname{Tr}}{\operatorname{ad}}_l$ (where we abbreviate ${\operatorname{ad}}_{\overline{e}_i}$ to ${\operatorname{ad}}_i$). Then equation (which is equivalent to ) takes the form $$\label{eq:mijjhom}
{\operatorname{Tr}}({\operatorname{ad}}_{DX}-{\operatorname{ad}}_X D)=0, \quad \text{for all }X \in {\mathfrak{n}},$$ and equations and give $$\label{eq:Ricuijhom}
\begin{split}
({\operatorname{Ric}}^u)_i^j= &-\tfrac12e^{-u(p_i+p_j)}B(\overline{e}_i,\overline{e}_j) -\tfrac12 \sum\nolimits_{l}(e^{u(p_i-p_j-2p_l)}\mu_{lj|i}+ e^{u(p_j-p_i-2p_l)}\mu_{li|j}){\operatorname{Tr}}{\operatorname{ad}}_l \\
&+\tfrac14e^{u(p_i+p_j)}\sum\nolimits_{k,l}e^{-2u(p_l+p_k)}\mu_{kl|i}\mu_{kl|j} -\tfrac12e^{-u(p_i+p_j)}\sum\nolimits_{k,l}e^{2u(p_l-p_k)}\mu_{ik|l}\mu_{jk|l}\\
= &\; (({\operatorname{Tr}}D) p_i-({\operatorname{Tr}}D^2)){\delta}_{ij}.
\end{split}$$ From (or from ) we also obtain $$\label{eq:scaluhom}
\begin{split}
{\operatorname{scal}}^u & = -\sum\nolimits_{k}e^{-2up_k}(({\operatorname{Tr}}{\operatorname{ad}}_k)^2+\tfrac12B(\overline{e}_k,\overline{e}_k)) -\tfrac14\sum\nolimits_{i,k,l}e^{2u(p_i-p_l-p_k)}(\mu_{kl|i})^{2}\\ & = ({\operatorname{Tr}}D)^2 - n({\operatorname{Tr}}D^2).
\end{split}$$ One can rewrite equation in a different form using the action of the group $\mathrm{GL}({\mathfrak{n}})$ on the Lie bracket of ${\mathfrak{n}}$ as in [@Lsurv; @Lstand; @N] (for a similar approach, with $\mathrm{GL}({\mathfrak{n}})$ acting on the inner product, see [@Heb Section 3]). For the metric Lie algebra ${\mathfrak{n}}$, denote the Lie bracket by $\mu(X,Y):=[X,Y]$, and for $A \in \mathrm{GL}({\mathfrak{n}})$, define the new Lie bracket on the underlying Euclidean space $({\mathbb{R}}^{n},{\overline{\langle \cdot, \cdot \rangle}})$ of ${\mathfrak{n}}$, keeping the inner product fixed, by $A.\mu(X,Y)= A\mu(A^{-1}X,A^{-1}Y)$. The resulting metric Lie algebra is isomorphic (but not, in general, isometric) to $({\mathfrak{n}}, {\overline{\langle \cdot, \cdot \rangle}})$. In our case, taking $A=e^{uD}$ we obtain the metric Lie algebra $({\mathfrak{n}}(u), {\overline{\langle \cdot, \cdot \rangle}})$ with the Lie bracket $e^{uD}.\mu(X,Y)=e^{uD}[e^{-uD}X,e^{-uD}Y]$ whose structure constants $\mu^u_{ij|k}$ are given by $\mu^u_{ij|k}=e^{u(p_k-p_i-p_j)}\mu_{ij|k}$, so equation takes the form ${\operatorname{Ric}}^{{\mathfrak{n}}(u)}=({\operatorname{Tr}}D) D -({\operatorname{Tr}}D^2){\mathrm{id}}$, where ${\operatorname{Ric}}^{{\mathfrak{n}}(u)}$ is the Ricci operator of $({\mathfrak{n}}(u), {\overline{\langle \cdot, \cdot \rangle}})$. Using [@Bes sec 7.38] (or ) we obtain that for the Lie bracket $e^{uD}.\mu$, equation is equivalent to $$\label{eq:Ricgroup}
\begin{split}
\overline{{\langle}{\operatorname{Ric}}^{{\mathfrak{n}}(u)}X ,X{\rangle}} \mkern-7mu & \mkern7mu =-\tfrac12 B(e^{-uD}X,e^{-uD}X)-\overline{{\langle}(e^{uD}{\operatorname{ad}}_{e^{-uD}H}e^{-uD})X,X{\rangle}}\\
&+ \tfrac14 \sum\nolimits_{k,l}\overline{{\langle}[e^{-uD}E_k, e^{-uD}E_l], e^{uD}X{\rangle}}^2 -\tfrac12 {\operatorname{Tr}}({\operatorname{ad}}_{e^{-uD}X}^*e^{2uD} \! {\operatorname{ad}}_{e^{-uD}X}e^{-2uD})\\
&= ({\operatorname{Tr}}D) \overline{{\langle}DX,X{\rangle}}-({\operatorname{Tr}}D^2)\overline{\|X\|}^2,
\end{split}$$ for all $X \in {\mathfrak{n}}$, where $H \in {\mathfrak{n}}$, *the mean curvature vector* of the unimodular ideal, is defined by $\overline{{\langle}H,X{\rangle}}={\operatorname{Tr}}{\operatorname{ad}}_X$, and $\{E_k\}$ is an arbitrary orthonormal basis for $({\mathfrak{n}}, {\overline{\langle \cdot, \cdot \rangle}})$ (not necessarily a basis of eigenvectors of $D$).
From Theorem \[th:Dconst\] we obtain the following.
\[cor:homo\] Suppose $(N,g)$ is a Lie group and both $g$ and $D$ are left-invariant. The extension $(M,g^D)$ is Einstein if and only if equations *(*or equivalently *)* and are satisfied.
An immediate consequence of is the fact that the Ricci tensor of the metric Lie algebra $({\mathfrak{n}}(u), {\overline{\langle \cdot, \cdot \rangle}})$ must be independent of $u$. One obvious case when this happens is when $D$ is a derivation of ${\mathfrak{n}}$, as then $e^{uD}$ is an automorphism, and so $e^{uD}.\mu(X,Y) = e^{uD}[e^{-uD}X,e^{-uD}Y] = \mu(X,Y)$. In that case, the resulting Einstein manifold $(M,g^D)$ is a Lie group with a left-invariant metric. Moreover, assuming Alekseevsky Conjecture, the manifold $(M,g^D)$ must be an Einstein solvmanifold (if $\det D \ne 0$, this follows from the fact that ${\mathfrak{n}}$ is nilpotent [@Jac]).
However, $D$ is not necessarily a derivation. The simplest example is when $D={\mathrm{id}}$. Then $(N,g)$ is Ricci flat by Lemma \[l:scalar\], hence is flat by [@AK], hence ${\mathfrak{n}}={\mathfrak{n}}_1 \ltimes {\mathfrak{n}}_2$, an (orthogonal) semidirect product of the abelian algebras ${\mathfrak{n}}_1$ and ${\mathfrak{n}}_2$, with ${\mathfrak{n}}_2$ acting on ${\mathfrak{n}}_1$ by commuting skew-symmetric endomorphisms [@AK; @BB1]. Therefore the algebra ${\mathfrak{n}}$ is not necessarily abelian, while $D={\mathrm{id}}$ can be a derivation only of an abelian Lie algebra. Note however that $(N,g)$ is isometric to an abelian group and the extension $(M, g^D)$ is a solvable group with the hyperbolic metric. In the proof of Theorem \[th:group\] below, we will see more complicated examples of the same phenomenon. However, we know *no examples of non-homogeneous Einstein extensions of a Lie group with a left-invariant metric by a left-invariant $D$*. Under some additional assumptions on the structure of ${\mathfrak{n}}$, as in Theorem \[th:group\], the fact that the extension $(M, g^D)$ is Einstein forces it to be an Einstein solvmanifold.
For $q \in {\mathbb{R}}$, introduce the sets $\mathcal{S}_q=\{(k,l,i) \, : \, k \ne l, p_l+p_k-p_i=q\}$ and $P_q=\{i \, : \, p_i=q\}$. Let $Q = \{ q \in {\mathbb{R}}\, : \, P_q \cup \mathcal{S}_q \ne \varnothing\}$. Then gives $${\operatorname{scal}}^u = -\sum\nolimits_{q \in Q} e^{-2uq}\bigl(\sum\nolimits_{k \in P_q}(({\operatorname{Tr}}{\operatorname{ad}}_k)^2+\tfrac12B(\overline{e}_k,\overline{e}_k)) +\tfrac14\sum\nolimits_{(k,l,i) \in \mathcal{S}_q} \mu_{kl|i}^{2}\bigr).$$ It follows that for all $q \in Q \setminus \{0\}$, we get $\sum\nolimits_{k \in P_q}(({\operatorname{Tr}}{\operatorname{ad}}_k)^2+\tfrac12B(\overline{e}_k,\overline{e}_k)) +\tfrac14\sum\nolimits_{(k,l,i) \in \mathcal{S}_q} \mu_{kl|i}^{2}=0$. As by assumption $B \ge 0$, all the terms on the left-hand side are zeros. Hence $\mu_{kl|i}=0$, unless $p_l+p_k = p_i$ and $k \ne l$, and ${\operatorname{Tr}}{\operatorname{ad}}_k=B(\overline{e}_k,\overline{e}_k)=0$, unless $p_k=0$. The former fact implies that for all $k,l$ we have $D[\overline{e}_k,\overline{e}_l]-[D\overline{e}_k,\overline{e}_l]-[\overline{e}_k,D\overline{e}_l]=\sum_i (p_i-p_k-p_l) \mu_{kl|i}=0$, so $D$ is a derivation of ${\mathfrak{n}}$. It follows that the extension $(M,g^D)$ is a metric Einstein Lie group, whose Lie algebra ${\mathfrak g}$ is the extension of ${\mathfrak{n}}$ by the derivation $D$.
To see that ${\mathfrak g}$ is solvable, consider the Killing form $B_{\mathfrak g}$ of ${\mathfrak g}$. As for all $X \in {\mathfrak{n}}$ we have ${\operatorname{ad}}_X e_0 \in {\mathfrak{n}}$, it follows that $B_{\mathfrak g}(X,X)=B(X,X) \ge 0$. Moreover, as ${\operatorname{Tr}}{\operatorname{ad}}_k=0$, unless $p_k=0$, we get ${\operatorname{Tr}}{\operatorname{ad}}_{DX}=0$ for all $X \in {\mathfrak{n}}$, and so $B_{\mathfrak g}(X, e_0)=0$ by . As $B_{\mathfrak g}(e_0,e_0) = {\operatorname{Tr}}D^2 \ge 0$, the Killing form $B_{\mathfrak g}$ is nonnegative, hence ${\mathfrak g}$ is solvable [@Heb Remark 4.8(a)].
Denote $\dim {\mathfrak h}=d$ and let ${\mathfrak h}= {\operatorname{Span}}(\overline{e}_a \, : \, a = 1, \dots, d), \; {\mathfrak{m}}= {\operatorname{Span}}(\overline{e}_k \, : \, k = d+1, \dots, n)$. Throughout the proof, the indices $a, b, c$ range from $1$ to $d$, and the indices $k,l,s$, from $d+1$ to $n$. Note that $\mu_{kl|a} = \mu_{ab|k}=0$.
\[l:mukldi\] [ ]{}
1. \[it:mutriv\] $\mu_{ab|c}=\mu_{ab|k}=\mu_{kb|c}=\mu_{kl|c}=\mu_{ak|c}=0$, for all $a,b,c \le d < k, l$.
2. \[it:muakl\] If $\mu_{ak|l} \ne 0$ for $a \le d < k, l$, then
- either $p_a=0$ and $p_k=p_l$,
- or $p_a \ne 0$ and either $p_k=p_l$ and then $\mu_{ak|l}+\mu_{al|k} = 0$, or $p_l-p_k-p_a=0$.
3. \[it:mukls\] For all $k, l, s > d$, we have $\mu_{sl|k}=0$ unless $p_k-p_l-p_s=0$.
4. \[it:muQN\] $\sum\nolimits_{a: p_a = q} \big(\sum\nolimits_{s: p_s = p_k} \mu_{as|k}\mu_{as|l} - \sum\nolimits_{s: p_s = p_l} \mu_{ak|s}\mu_{al|s} \big)=0$, for all $k, l > d$ with $p_l-p_k =q \ne 0$.
is obvious, as ${\mathfrak h}$ is abelian and is orthogonal to the derived algebra of ${\mathfrak g}$.
Denote $Q:=\{p_l - p_k \, : \, k,l =d+1, \dots, n\}$. Take $i = j= a$ in equation (so that $\overline{e}_i = \overline{e}_j \in {\mathfrak h}$). If $p_a = 0$ we get by $$({\operatorname{Ric}}^u)_a^a= -({\operatorname{Tr}}D^2) = -\tfrac12 B(\overline{e}_a,\overline{e}_a) - \tfrac12 \sum\nolimits_{k,l} e^{2u(p_l-p_k)}\mu_{ak|l}^2,$$ and so $\mu_{ak|l}=0$ unless $p_l=p_k$. If $p_a \ne 0$ we obtain $$\begin{split}
({\operatorname{Ric}}^u)_a^a= &\; ({\operatorname{Tr}}D) p_a-({\operatorname{Tr}}D^2) = -\tfrac12e^{-2up_a}\sum\nolimits_{k,l}\mu_{ak|l}\mu_{al|k} -\tfrac12e^{-2up_a}\sum\nolimits_{k,l} e^{2u(p_l-p_k)}\mu_{ak|l}^2\\
= &-\tfrac12 \sum\nolimits_{q \in Q \setminus \{0, p_a\}} e^{2u(q-p_a)}\sum\nolimits_{k,l: p_l-p_k=q}\mu_{ak|l}^2
-\tfrac12 \sum\nolimits_{k,l: p_l-p_k=p_a} \mu_{ak|l}^2 \\
&-\tfrac12e^{-2up_a}\Big(\sum\nolimits_{k,l} \mu_{ak|l} \mu_{al|k} + \sum\nolimits_{k,l: p_l=p_k} \mu_{ak|l}^2\Big).
\end{split}$$ It follows that $\mu_{ak|l} = 0$ unless $p_l-p_k \in \{0, p_a\}$. But then the expression in the last brackets equals $\frac12 \sum\nolimits_{k,l: p_l=p_k} (\mu_{ak|l}+\mu_{al|k})^2$ which implies that $\mu_{ak|l}+\mu_{al|k} = 0$ when $p_l=p_k$.
Take $i = j = k$ (so that $\overline{e}_i = \overline{e}_j \in {\mathfrak{m}}$) in . Using we get $$\begin{split}
({\operatorname{Ric}}^u)_k^k= &\; ({\operatorname{Tr}}D) p_k-({\operatorname{Tr}}D^2) = - \sum\nolimits_{a} e^{-2u p_a} \mu_{ak|k} {\operatorname{Tr}}{\operatorname{ad}}_a \\
&+\tfrac12 e^{2u p_k} \sum\nolimits_{a,l} e^{-2u(p_l+p_a)} \mu_{al|k}^2 - \tfrac12 e^{-2u p_k} \sum\nolimits_{a,l} e^{2u(p_l-p_a)} \mu_{ak|l}^2\\
&+\tfrac14 e^{2u p_k}\sum\nolimits_{s,l} e^{-2u(p_l+p_s)}\mu_{sl|k}^2 - \tfrac12 e^{-2u p_k} \sum\nolimits_{s,l} e^{2u(p_l-p_s)} \mu_{ks|l}^2.\\
\end{split}$$ But the sum of the first three terms on the right-hand side is constant (does not depend on $u$) by , and therefore the sum of the last two terms must also be a constant. Summing up these sums by $k$ we get $-\tfrac14 \sum\nolimits_{k,s,l} e^{2u(p_k-p_l-p_s)} \mu_{sl|k}^2$, and the claim follows.
Let $k,l > d$ with $p_l-p_k=q \ne 0$. Using , and we obtain $$\begin{split}
0=({\operatorname{Ric}}^u)_k^l= & \; \tfrac12e^{u(p_k+p_l)}\sum\nolimits_{a,s}e^{-2u(p_s+p_a)}\mu_{as|k}\mu_{as|l} -\tfrac12e^{-u(p_k+p_l)}\sum\nolimits_{a,s}e^{2u(p_s-p_a)}\mu_{ak|s}\mu_{al|s}\\
= & \; \tfrac12e^{u(p_k+p_l)}\sum\nolimits_{a,s: (p_a, p_s) \in \{(q, p_k), (-q,p_l)\}}e^{-2u(p_s+p_a)}\mu_{as|k}\mu_{as|l} \\
& \; - \tfrac12e^{-u(p_k+p_l)}\sum\nolimits_{a,s: (p_a, p_s) \in \{(-q,p_k),(q,p_l)\}}e^{2u(p_s-p_a)}\mu_{ak|s}\mu_{al|s}\\
= & \; \tfrac12e^{-uq} \sum\nolimits_{a: p_a = q} \Big(\sum\nolimits_{s: p_s = p_k} \mu_{as|k}\mu_{as|l} - \sum\nolimits_{s: p_s = p_l} \mu_{ak|s}\mu_{al|s} \Big)\\
& \; + \tfrac12e^{uq}\sum\nolimits_{a: p_a = -q} \Big(\sum\nolimits_{s: p_s = p_l} \mu_{as|k}\mu_{as|l} - \sum\nolimits_{s: p_s = p_k} \mu_{ak|s}\mu_{al|s}\Big),
\end{split}$$ and the claim follows.
The claim of Lemma \[l:mukldi\] is equivalent to the fact that the restriction of $D$ to ${\mathfrak{m}}$ is a derivation. The restriction of $D$ to ${\mathfrak h}$ is also a derivation, as ${\mathfrak h}$ is abelian. But $D$ may fail to be a derivation of the whole algebra ${\mathfrak{n}}$, as by Lemma \[l:mukldi\], the expression $(p_l-p_k-p_a) \mu_{ak|l}$ is not necessarily zero. To “fix" that we will modify ${\mathfrak{n}}$ by a twisting, but first we will further clarify the action of ${\operatorname{ad}}_{\mathfrak h}$ on ${\mathfrak{m}}$.
Let $\{q_1, \dots, q_m\} = \{p_{d+1}, \dots, p_n\}$ be the eigenvalues of the restriction of $D$ to ${\mathfrak{m}}$ labelled in such a way that $q_1 < \dots < q_m$, and let $d_{\alpha}, \; {\alpha}=1, \dots, m$, be the multiplicity of $q_{\alpha}$. Specify the basis $\overline{e}_k, \; k =d+1, \dots, n$, for ${\mathfrak{m}}$ in such a way that $p_{d+1} = \dots = p_{d+d_1} = q_1, \; p_{d+d_1+1} = \dots = p_{d+d_2} = q_2, \dots, p_{n-d_m+1} = \dots = p_n = q_m$. For $p_a=0$, denote $T_a$ the matrix of the restriction of ${\operatorname{ad}}_{\overline{e}_a}$ to ${\mathfrak{m}}$ relative to the chosen basis for ${\mathfrak{m}}$ (note that ${\operatorname{ad}}_{\overline{e}_a}$ acts trivially on ${\mathfrak h}$). We have $(T_a)_{kl}=\mu_{al|k}$, and so by Lemma \[l:mukldi\], the matrix $T_a$ is block-diagonal with the diagonal blocks having dimensions $d_1 \times d_1, \, d_2 \times d_2, \dots, d_m \times d_m$, in that order (so that $T_a$ commutes with $D_{|{\mathfrak{m}}}$).
If $p_a \ne 0$, then ${\operatorname{ad}}_{\overline{e}_a}$ still acts trivially on ${\mathfrak h}$. For the restriction of ${\operatorname{ad}}_{\overline{e}_a}$ to ${\mathfrak{m}}$, relative to the chosen basis for ${\mathfrak{m}}$, we have $({\operatorname{ad}}_{\overline{e}_a})_{kl} = \mu_{al|k}$, and so by Lemma \[l:mukldi\], $({\operatorname{ad}}_{\overline{e}_a})_{|{\mathfrak{m}}} = Q_a + N_a$, where $Q_a$ is a block-diagonal skew-symmetric matrix whose diagonal blocks have dimensions $d_1 \times d_1, \, d_2 \times d_2, \dots, d_m \times d_m$, in that order (so that $Q_a$ commutes with $D_{|{\mathfrak{m}}}$), and $N_a$ is a strictly upper or lower triangular matrix (depending on the sign of $p_a$) which may only have nonzero entries in the blocks $d_{\alpha}\times d_{\beta}$ such that $q_{\alpha}-q_{\beta}=p_a$ (so that $[D_{|{\mathfrak{m}}}, N_a] = p_a N_a$). In terms of the $\mu$’s, when $p_a \ne 0$, we have $$\label{eq:QNmu}
\begin{gathered}
(Q_a)_{kl}=\mu_{al|k}, \quad Q_a^t = -Q_a, \quad (Q_a)_{kl} \ne 0 \,\Rightarrow \, p_k=p_l, \\
(N_a)_{kl}=\mu_{al|k}, \quad (N_a)_{kl} \ne 0 \,\Rightarrow \, p_k=p_l+p_a.
\end{gathered}$$ Let $d_0 \ge 0$ be the multiplicity of the eigenvalue $0$ of the restriction of $D$ to ${\mathfrak h}$. Relabel the basis $\overline{e}_a, \; a =1, \dots, d$, for ${\mathfrak h}$ in such a way that $p_1=\dots=p_{d_0}=0$ and $p_a \ne 0$ for $d_0 < a \le d$. We have the following lemma.
\[l:allcommute\] All the matrices $T_a, Q_b, N_b$, where $1 \le a \le d_0, \; d_0 < b \le d$, pairwise commute.
As ${\mathfrak h}$ is abelian, the operators ${\operatorname{ad}}_{\overline{e}_a}$ commute. Then $[T_a, T_b] = 0$, for all $a, b \le d_0$. Moreover, for all $a, b$ such that $1 \le a \le d_0 < b \le d$, we have $[T_a, Q_b + N_b] = 0$, and so $[T_a, Q_b] = [T_a, N_b] = 0$, because $T_a$ and $Q_b$ are block-diagonal, but all the nonzero blocks of $N_b$ are outside the diagonal.
Furthermore, for $a, b > d_0$, we have $0 = [Q_a+N_a, Q_b+N_b] = [Q_a, Q_b] + [Q_a, N_b] + [N_a, Q_b] + [N_a, N_b]$. The same argument on the block structure of the $Q_a$’s and $N_a$’s now implies that $[Q_a, Q_b] = [Q_a, N_b] = [Q_b, N_a] = [N_a, N_b] = 0$, with only two possible exceptions:
1. \[it:-\] $p_a = - p_b \ne 0$; then $[N_a, N_b]$ is block-diagonal and so we only get $[Q_a, N_b] = [Q_b, N_a] = 0$ and $[Q_a, Q_b]+[N_a,N_b]=0$.
2. \[it:+\] $p_a = p_b \ne 0$; then $[Q_a, N_b]$ and $[N_a, Q_b]$ have nonzero blocks at the same places and so we only get $[Q_a, Q_b] = [N_a, N_b] = 0$ and $[Q_a, N_b]+[N_a,Q_b]=0$.
Consider case first. Denote $q=p_a = p_b \ne 0$, and let $k, l > d$ be such that $p_l-p_k=q$ (if no such pair $(k,l)$ exists, then $N_a=0$, for all $a$ with $p_a=q$, by Lemma \[l:mukldi\]). Then by Lemma \[l:mukldi\] and from we get $$\begin{split}
0 &=\sum\nolimits_{a: p_a = q} \Big(\sum\nolimits_{s: p_s = p_k} \mu_{as|k}\mu_{as|l} - \sum\nolimits_{s: p_s = p_l} \mu_{ak|s}\mu_{al|s} \Big) \\
&= \sum\nolimits_{a: p_a = q} \Big(\sum\nolimits_{s: p_s = p_k} (Q_a)_{ks} (N_a)_{ls} - \sum\nolimits_{s: p_s = p_l} (Q_a)_{sl} (N_a)_{sk} \Big)\\
&= \sum\nolimits_{a: p_a = q} ((Q_aN_a^t)_{kl} - (N_a^tQ_a)_{kl}),
\end{split}$$ as $(Q_a)_{ks} = 0$ when $p_s \ne p_k$ and $(Q_a)_{sl} = 0$ when $p_s \ne p_l$, by . But then $\sum\nolimits_{a: p_a = q} [Q_a, N_a^t]_{kl} = 0$, and since all the entries $[Q_a, N_a^t]_{kl}$ with $p_l-p_k \ne q$ are zeros from we get $\sum\nolimits_{a: p_a = q} [Q_a, N_a^t] = 0$, that is, $\sum\nolimits_{a: p_a = q} [Q_a, N_a] = 0$, as the matrices $Q_a$ are skew-symmetric.
Now take the commutator of the latter equation with $Q_b$ such that $p_b=q$. As in our case $[Q_a, Q_b] = 0$ and $[Q_a, N_b]=[Q_b,N_a]$ we obtain $0=\sum\nolimits_{a: p_a = q} [Q_b,[Q_a, N_a]] = \sum\nolimits_{a: p_a = q} [Q_a,[Q_b, N_a]] = \sum\nolimits_{a: p_a = q} [Q_a,[Q_a, N_b]]$. Multiplying by $N_b^t$ and taking the trace we obtain $0= \sum\nolimits_{a: p_a = q} {\operatorname{Tr}}([Q_a,[Q_a, N_b]] N_b^t)= -\sum\nolimits_{a: p_a = q} {\operatorname{Tr}}([Q_a, N_b] [Q_a,N_b]^t)$, as the matrices $Q_a$ are skew-symmetric. It follows that $[Q_a, N_b]=0$, for all $a, b$ such that $p_a=p_b \ne 0$.
Now consider case . We have $[Q_a, Q_b]+[N_a,N_b]=0$. Taking the commutator with $Q_b$ and using the fact that $Q_b$ and $N_a$ commute we get $0=[Q_b,[Q_a, Q_b]]+[Q_b,[N_a,N_b]]=[Q_b,[Q_a, Q_b]]+[N_a,[Q_b,N_b]]=[Q_b,[Q_a, Q_b]]$, as $[Q_b,N_b]=0$ from case . But then multiplying by $Q_a$ and taking the trace we obtain ${\operatorname{Tr}}([Q_a, Q_b]^2)=0$, and so $[Q_a, Q_b]=0$.
For $b=d_0+1, \dots, d$ define the operators $\mathcal{Q}_b, \mathcal{N}_b \in {\operatorname{End}}({\mathfrak{n}})$ by $$\mathcal{Q}_b ({\mathfrak h}) = \mathcal{N}_b ({\mathfrak h}) = 0, \qquad \mathcal{Q}_b X = Q_b X, \; \mathcal{N}_b X = N_b X \; \text{for } X \in {\mathfrak{m}}.$$ Note that ${\operatorname{ad}}_{\overline{e}_b} = \mathcal{Q}_b + \mathcal{N}_b$. Moreover, as both $D_{|{\mathfrak{m}}}$ and $({\operatorname{ad}}_{\overline{e}_b})_{|{\mathfrak{m}}} = Q_b + N_b$ are derivations of ${\mathfrak{m}}$ and as $[D_{|{\mathfrak{m}}}, Q_b]=0,\; [D_{|{\mathfrak{m}}}, N_b] = p_b N_b, \; p_b \ne 0$, all the $Q_b$’s and $N_b$’s are derivations of ${\mathfrak{m}}$. It follows from Lemma \[l:allcommute\] that the operators ${\operatorname{ad}}_{\overline{e}_a}, \; a \le d_0$ and $\mathcal{Q}_b, \mathcal{N}_b, \; b > d_0$ are commuting derivations of the whole algebra ${\mathfrak{n}}$.
We now consider the metric solvable Lie algebra ${\mathfrak{n}}'$, defined on the same underlying linear space as ${\mathfrak{n}}$, with the same inner product ${\langle \cdot, \cdot \rangle}'={\overline{\langle \cdot, \cdot \rangle}}$, and with the Lie bracket $[ \cdot, \cdot]'$ defined as follows: $${\operatorname{ad}}_{\overline{e}_k}'={\operatorname{ad}}_{\overline{e}_k}, \; k > d, \qquad {\operatorname{ad}}_{\overline{e}_a}'={\operatorname{ad}}_{\overline{e}_a}, \; a \le d_0, \qquad {\operatorname{ad}}_{\overline{e}_b}'=\mathcal{N}_b, \; d_0 < b \le d.$$ Then ${\mathfrak{n}}'$ is indeed a Lie algebra, and what is more, $D$ is a symmetric derivation of $({\mathfrak{n}}',{\langle \cdot, \cdot \rangle}')$. The algebra ${\mathfrak{n}}'$ is obtained from ${\mathfrak{n}}$ by the twisting $X \mapsto X + \phi(X)$, where $\phi$ is the homomorphism from ${\mathfrak{n}}$ to the Lie algebra of skew-symmetric derivations of ${\mathfrak{n}}$ defined on the basis by $\phi(\overline{e}_b) = -\mathcal{Q}_b$ for $d_0 < b \le d$, and $\phi(\overline{e}_k) = \phi(\overline{e}_a) = 0$ for $a \le d_0$ and $k > d$. It follows from [@Aneg § 2] that the metric Lie groups $(N, g)$ and $(N', g')$ are isometric (${\mathfrak{n}}'$ is a standard modification of ${\mathfrak{n}}$ [@GW]). Furthermore, as the field of endomorphisms $D$ defined on the underlying Riemannian space $(N, g)$ is $N$-left-invariant and as $[D,\phi(X)]=0$, for all $X \in {\mathfrak{n}}$, it is also $N'$-left-invariant.
We will prove that $D$ respects the decomposition ${\mathfrak{n}}={\mathfrak h}\oplus {\mathfrak{m}}$. Then the claim will follow from assertion .
Denote $\dim {\mathfrak h}=d$ and let ${\mathfrak h}= {\operatorname{Span}}(E_a \, : \, a = 1, \dots, d), \; {\mathfrak{m}}= {\operatorname{Span}}(E_k \, : \, k = d+1, \dots, n)$. The index $a$ will range from $1$ to $d$, and the index $l$, from $d+1$ to $n$.
Polarising with $u=0$ we obtain that for all $X \in {\mathfrak{m}}, \; Y \in {\mathfrak h}$, $$\overline{{\langle}{\operatorname{Ric}}^{{\mathfrak{n}}(0)}X,Y{\rangle}} = 0 = ({\operatorname{Tr}}D) \overline{{\langle}DX,Y{\rangle}}.$$ If ${\operatorname{Tr}}D \ne 0$, we are done. Suppose ${\operatorname{Tr}}D = 0$. Then it follows from that for each $u \in {\mathbb{R}}$, the metric Lie algebra $({\mathfrak{n}}(u), {\overline{\langle \cdot, \cdot \rangle}})$ is Einstein, with the Einstein constant $-({\operatorname{Tr}}D^2)$. The same is true for the metric Lie algebra $({\mathfrak{n}}, {\overline{\langle \cdot, \cdot \rangle}}_u)$ which is isometrically isomorphic to $({\mathfrak{n}}(u), {\overline{\langle \cdot, \cdot \rangle}})$. Taking $u=0$ and $X \in {\mathfrak{m}}$ in we obtain $$\begin{split}
\overline{{\langle}{\operatorname{Ric}}^{{\mathfrak{n}}(0)}X ,X{\rangle}} &= -\overline{{\langle}{\operatorname{ad}}_{H}X,X{\rangle}} + \tfrac12 \sum\nolimits_{a,l}\overline{{\langle}[E_a, E_l], X{\rangle}}^2 -\tfrac12 {\operatorname{Tr}}({\operatorname{ad}}_{X}^*{\operatorname{ad}}_{X}) \\
& = -\overline{{\langle}{\operatorname{ad}}_{H}X,X{\rangle}} +\tfrac12 \sum\nolimits_{a,l}\overline{{\langle}{\operatorname{ad}}_{E_a}^* X, E_l{\rangle}}^2 -\tfrac12 \sum\nolimits_{a} \overline{{\langle}{\operatorname{ad}}_{X}^*{\operatorname{ad}}_{X} E_a,E_a{\rangle}} \\
&= -\overline{{\langle}{\operatorname{ad}}_{H}X,X{\rangle}} + \tfrac12 \sum\nolimits_{a}\overline{\|{\operatorname{ad}}_{E_a}^* X\|}^2 -\tfrac12 \sum\nolimits_{a} \overline{\|{\operatorname{ad}}_{E_a}X\|}^2 \\
&= -({\operatorname{Tr}}D^2)\overline{\|X\|}^2,
\end{split}$$ and so $$\label{eq:AH+AH'}
-\tfrac12(A_H+A_H^*) +\tfrac12 \sum\nolimits_{a}[A_{E_a}, A_{E_a}^*]= -({\operatorname{Tr}}D^2) {\mathrm{id}}_{{\mathfrak{m}}},$$ where $A_Y, \; Y \in {\mathfrak h}$, is the restriction of ${\operatorname{ad}}_Y$ to ${\mathfrak{m}}$.
In particular, taking the traces of both sides of we obtain $$\label{eq:H^2}
(n-d){\operatorname{Tr}}(D^2)= {\operatorname{Tr}}A_H = {\operatorname{Tr}}{\operatorname{ad}}_H = \|H\|^2.$$ If ${\mathfrak{n}}$ is unimodular, then $H=0$, and so $D=0$ (which is clearly a derivation). Moreover, $(N,g)$ is flat and $(M,g^D)$ is the Riemannian product of $(N,g)$ and the line. Suppose ${\mathfrak{n}}$ is non-unimodular. Then $({\mathfrak{n}}, {\overline{\langle \cdot, \cdot \rangle}})$ is a standard metric solvable Einstein Lie algebra. In particular, by [@Heb Corollary 4.11], the derived algebra of ${\mathfrak{n}}$ coincides with ${\mathfrak{m}}$, and by [@Heb Theorem 4.10(1)], all the operators ${\operatorname{ad}}_Y, \; Y \in {\mathfrak h}$, are normal. Then equation gives $$\label{eq:AH+AH'new}
\tfrac12(A_H+A_H^*) = ({\operatorname{Tr}}D^2) {\mathrm{id}}_{{\mathfrak{m}}}.$$
Every algebra $({\mathfrak{n}}, {\overline{\langle \cdot, \cdot \rangle}}_u), \; u \in {\mathbb{R}}$, is a metric solvable non-unimodular Lie algebra, with the same Einstein constant $-{\operatorname{Tr}}(D^2)$. As any Einstein solvmanifold is standard by [@Lstand], the ${\overline{\langle \cdot, \cdot \rangle}}_u$-orthogonal complement ${\mathfrak h}_u$ to ${\mathfrak{m}}=[{\mathfrak{n}}, {\mathfrak{n}}]$ must be abelian. We have ${\mathfrak h}_u = e^{-2uD}{\mathfrak h}$, and so $[e^{-2uD}X, e^{-2uD}Y]=0$, for all $X, Y \in {\mathfrak h}$. Differentiating by $u$ at $u=0$ we obtain $$\label{eq:huabelian}
[DX, Y] + [X, DY] = 0, \quad \text{for all} X, Y \in {\mathfrak h}.$$ Furthermore, equation still holds if we replace ${\overline{\langle \cdot, \cdot \rangle}}$ with ${\overline{\langle \cdot, \cdot \rangle}}_u$ and $H$ with $H_u$, the mean curvature vector of $({\mathfrak{n}}, {\overline{\langle \cdot, \cdot \rangle}}_u)$ defined by ${\langle}H_u, X{\rangle}_u = {\operatorname{Tr}}{\operatorname{ad}}_X$, for all $X \in {\mathfrak{n}}$. We have ${\langle}e^{uD}H_u, e^{uD}Y{\rangle}={\langle}H,Y{\rangle}$ and so $H_u=e^{-2uD}H$. Then from we get $(n-d){\operatorname{Tr}}(D^2)=\|H_u\|^2_u={\langle}e^{-2uD}H,H{\rangle}$. Decomposing $H$ by an orthonormal basis of eigenvectors of $D$ we obtain $DH=0$. Then from with $X=H$ we obtain $DY \in {\operatorname{Ker}}{\operatorname{ad}}_H$, for all $Y \in {\mathfrak h}$. Clearly, ${\mathfrak h}\subset {\operatorname{Ker}}{\operatorname{ad}}_H$, and if $X \in {\mathfrak{m}}\cap {\operatorname{Ker}}{\operatorname{ad}}_H$, then by we get $0= \tfrac12{\langle}(A_H+A_H^*)X, X{\rangle}= ({\operatorname{Tr}}D^2) \|X\|^2$, and so $X = 0$, as $D \ne 0$. It follows that $D{\mathfrak h}\subset {\mathfrak h}$, and so $D$ preserves the decomposition ${\mathfrak{n}}={\mathfrak h}\oplus{\mathfrak{m}}$.
[^1]: The authors were partially supported by ARC Discovery Grant DP130103485.
| ArXiv |
---
abstract: 'We calculate QCD corrections to transversely polarized Drell-Yan process at a measured $Q_T$ of the produced lepton pair in the dimensional regularization scheme. The $Q_T$ distribution is discussed resumming soft gluon effects relevant for small $Q_T$.'
address:
- |
Radiation Laboratory, RIKEN\
2-1 Hirosawa, Wako, Saitama 351-0198, JAPAN\
[email protected]
- |
Theory Division, High Energy Accelerator Research Organization (KEK)\
1-1 OHO, Tsukuba 305-0801, JAPAN
- 'Department of Physics, Juntendo University, Inba-gun, Chiba 270-1695, JAPAN'
author:
- HIROYUKI KAWAMURA
- JIRO KODAIRA and HIROTAKA SHIMIZU
- KAZUHIRO TANAKA
title: '$Q_T$ RESUMMATION IN TRANSVERSELY POLARIZED DRELL-YAN PROCESS [^1]'
---
Hard processes with polarized nucleon beams enable us to study spin-dependent dynamics of QCD and the spin structure of nucleon. The helicity distribution $\Delta q(x)$ of quarks within nucleon has been measured in polarized DIS experiments, and $\Delta G(x)$ of gluons has also been estimated from the scaling violations of them. On the other hand, the transversity distribution $\delta q(x)$, i.e. the distribution of transversely polarized quarks inside transversely polarized nucleon, can not be measured in inclusive DIS due to its chiral-odd nature,[@RS:79] and remains as the last unknown distribution at the leading twist. Transversely polarized Drell-Yan (tDY) process is one of the processes where the transversity distribution can be measured, and has been undertaken at RHIC-Spin experiment.
We compute the 1-loop QCD corrections to tDY at a measured $Q_T$ and azimuthal angle $\phi$ of the produced lepton in the dimensional regularization scheme. For this purpose, the phase space integration in $D$-dimension, separating out the relevant transverse degrees of freedom, is required to extract the $\propto \! \cos(2\phi)$ part of the cross section characteristic of the spin asymmetry of tDY.[@RS:79] The calculation is rather cumbersome compared with the corresponding calculation in unpolarized and longitudinally polarized cases, and has not been performed so far. We obtain the NLO ${\cal O}(\alpha_s)$ corrections to the tDY cross section in the $\overline{\rm MS}$ scheme. We also include soft gluon effects by all-order resummation of logarithmically enhanced contributions at small $Q_T$ (“edge regions of the phase space”) up to next-to-leading logarithmic (NLL) accuracy, and obtain the first complete result of the $Q_T$ distribution for all regions of $Q_T$ at NLL level.
We first consider the NLO ${\cal O}(\alpha_s)$ corrections to tDY: $h_1(P_1,s_1)+h_2(P_2,s_2)\rightarrow l(k_1)+\bar{l}(k_2)+X$, where $h_1,h_2$ denote nucleons with momentum $P_1,P_2$ and transverse spin $s_1,s_2$, and $Q=k_1+k_2$ is the 4-momentum of DY pair. The spin dependent cross section $\Delta_T d \sigma \equiv (d \sigma (s_1 , s_2) - d \sigma (s_1 , - s_2))/2$ is given as a convolution $$\Delta_T d\sigma = \int d x_1 d x_2\,
\delta H (x_1 \,,\,x_2 ; \mu_F)\,
\Delta_T d \hat{\sigma} (s_1\,,\,s_2 ; \mu_F),$$ where $\mu_F$ is the factorization scale, and $$\delta H (x_1 \,,\,x_2 ; \mu_F)\, = \sum_i e_i^2
[\delta q_i(x_1 ; \mu_F)\delta \bar{q}_i(x_2 ; \mu_F)
+\delta \bar{q}_i(x_1 ; \mu_F)\delta q_i(x_2 ; \mu_F)]$$ is the product of transversity distributions of the two nucleons, and $\Delta_T d \hat{\sigma}$ is the corresponding partonic cross section. Note that, at the leading twist level, the gluon does not contribute to the transversely polarized process due to its chiral odd nature. We compute the one-loop corrections to $\Delta_T d \hat{\sigma}$, which involve the virtual gluon corrections and the real gluon emission contributions, e.g., $q (p_1 , s_1) + \bar{q} (p_2 , s_2)
\to l (k_1) + \bar{l} (k_2) + g$, with $p_i = x_i P_i$. We regularize the infrared divergence in $D=4 - 2 \epsilon$ dimension, and employ naive anticommuting $\gamma_5$ which is a usual prescription in the transverse spin channel.[@WV:98] In the $\overline{\rm MS}$ scheme, we eventually get,[@KKST; @KKST2] to NLO accuracy, $$\begin{aligned}
\frac{\Delta_T d \sigma}{d Q^2 d Q_T^2 d y d \phi}
= N\, \cos{(2 \phi )}
\left[ X\, (Q_T^2 \,,\, Q^2 \,,\, y)
+ Y\, (Q_T^2 \,,\, Q^2 \,,\, y) \right],
\label{cross section}\end{aligned}$$ where $N = \alpha^2 / (3\, N_c\, S\, Q^2)$ with $S=(P_1 +P_2 )^2$, $y$ is the rapidity of virtual photon, and $\phi$ is the azimuthal angle of one of the leptons with respect to the initial spin axis. For later convenience, we have decomposed the cross section into the two parts: the function $X$ contains all terms that are singular as $Q_T \rightarrow 0$, while $Y$ is of ${\cal O}(\alpha_s)$ and finite at $Q_T=0$. Writing $X = X^{(0)} + X^{(1)}$ as the sum of the LO and NLO contributions, we have[@KKST; @KKST2] $X^{(0)} = \delta H (x_1^0\,,\,x_2^0\,;\, \mu_F )\ \delta (Q_T^2)$, and $$\begin{aligned}
X^{(1)} &=& \frac{\alpha_s}{2 \pi} C_F\
\Biggl\{ \delta H (x_1^0\,,\,x_2^0\,;\, \mu_F )
\left[\, 2\, \left( \frac{\ln Q^2 / Q_T^2}{Q_T^2} \right)_+
- \frac{3}{(Q_T^2)_+}
+ \left(\, - 8 + \pi^2 \right) \delta (Q_T^2) \right]
\nonumber\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!+&& \!\!\!\!\!\!\!\!\left( \frac{1}{(Q_T^2)_+}
+ \delta (Q_T^2) \ln \frac{Q^2}{\mu_F^2} \right)\!\!\!
\left[ \int^1_{x_1^0} \frac{d z}{z}
\delta P_{qq}^{(0)} (z)\
\delta H
\left( \frac{x_1^0}{z}, x_2^0 ;\ \mu_F \right)
+ ( x_1^0 \leftrightarrow x_2^0 ) \right] \Biggr\} ,
\label{eq:x}\end{aligned}$$ where $x_1^0 = \sqrt{\tau}\ e^y , x_2^0 =\sqrt{\tau}\ e^{-y}$ are the relevant scaling variables with $\tau =Q^2/S$, and $\delta P_{qq}^{(0)} (z) = 2 z/(1 - z)_+
+ (3/2)\, \delta (1 - z)$ is the LO transverse splitting function.[@AM:90] In (\[eq:x\]), the terms involving $\delta(Q_T^2 )$ come from the virtual gluon corrections, while the other terms represent the recoil effects due to the real gluon emissions. For the analytic expression of $Y$, see Ref.[@KKST2]. Eq. (\[cross section\]) gives the first NLO result in the $\overline{\rm MS}$ scheme. We note that there has been a similar NLO calculation of tDY cross section in massive gluon scheme.[@VW:93] We also note that, integrating (\[cross section\]) over $Q_T$, our result coincides with the corresponding $Q_T$-integrated cross sections obtained in previous works employing massive gluon scheme[@VW:93] and dimensional reduction scheme,[@CKM:94] via the scheme transformation relation.[@WV:98]
The cross section (\[cross section\]) becomes very large when $Q_T \ll Q$, due to the terms behaving $\sim \alpha_s \ln(Q^2/Q_T^2 )/Q_T^2$ and $\sim \alpha_s /Q_T^2$ in the singular part $X$. It is well-known that, in unpolarized and longitudinally polarized DY, large “recoil logs” of similar nature appear in each order of perturbation theory as $\alpha_s^n \ln^{2n-1}(Q^2/Q_T^2 )/Q_T^2$, $\alpha_s^n \ln^{2n-2}(Q^2/Q_T^2 )/Q_T^2$, and so on, corresponding to LL, NLL, and higher contributions, respectively, and that the resummation of those “double logarithms” to all orders is necessary to obtain a well-defined, finite prediction of the cross section.[@CSS:85] Because the LL and NLL contributions are universal,[@DG:00] we can work out the all-order resummation of the corresponding logarithmically enhanced contributions in (\[cross section\]) up to the NLL accuracy, based on the general formulation[@CSS:85] of the $Q_T$ resummation. This can be conveniently carried out in the impact parameter $b$ space, conjugate to the $Q_T$ space. As a result, the singular part $X$ of (\[cross section\]) is modified into the corresponding resummed part, which is expressed as the Fourier transform, $$\begin{aligned}
X \rightarrow&&
\sum_{i}e_i^2 \int_0^{\infty} d b \frac{b}{2}
J_0 (b Q_T)
e^{\, S (b , Q)}
( C_{qq} \otimes \delta q_i )
\left( x_1^0 , \frac{b_0^2}{b^2} \right)
( C_{\bar{q} \bar{q}} \otimes \delta\bar{q}_i )
\left( x_2^0 , \frac{b_0^2}{b^2} \right)
\nonumber\\
&&+ ( x_1^0 \leftrightarrow x_2^0 ) .
\label{resum}\end{aligned}$$ Here $b_0 = 2e^{-\gamma_E}$, and the large logarithmic corrections are resummed into the Sudakov factor $e^{S (b , Q)}$ with $S(b,Q)=-\int_{b^2_0/b^2}^{Q^2}(d\kappa^2/\kappa^2)
\{ (\ln{\frac{Q^2}{\kappa^2}} )A_q(\alpha_s(\kappa))+
B_q(\alpha_s(\kappa)) \}$. The functions $A_q$, $B_q$ as well as the coefficient functions $C_{qq}, C_{\bar{q}\bar{q}}$ are calculable in perturbation theory, and at the present accuracy of NLL, we get:[@KKST; @KKST2] $A_q(\alpha_s ) =(\alpha_s /\pi) C_F +(\alpha_s /2\pi)^2
2C_F \{(67/18-\pi^2/6)C_G-5N_f /9 \}$, $B_q (\alpha_s )=-3C_F(\alpha_s /2\pi)$, $C_{qq}(z, \alpha_s )=C_{\bar{q} \bar{q}}(z, \alpha_s )
=\delta(1-z)\{1+(\alpha_s /4\pi)C_F(\pi^2-8)\}$. We have utilized a relation[@KT:82] between $A_q$ and the DGLAP kernels in order to obtain the two-loop term of $A_q$. The other contributions have been determined so that the expansion of the above formula (\[resum\]) in powers of $\alpha_s (\mu_F )$ reproduces $X$ of (\[cross section\]), (\[eq:x\]) to the NLO accuracy. Eq. (\[cross section\]) with (\[resum\]) presents the first result of the NLL $Q_T$ resummation formula for tDY. The NLO parton distributions in the $\overline{\rm MS}$ scheme have to be used.
One more step is necessary to make the QCD prediction of tDY. Similarly to other all-order resuumation formula, our result (\[resum\]) is suffered from the IR renormalons due to the Landau pole at $b= (b_0 /Q)e^{(1/2\beta_0 \alpha_s (Q))}$ in the Sudakov factor, and it is necessary to specify a prescription to avoid this singularity. Here we deform the integration contour in (\[resum\]) in the complex $b$ space, following the method introduced in the joint resummation.[@LKSV:01] Obviously prescription to define the $b$ integration is not unique reflecting IR renormalon ambiguity, e.g., “$b_{*}$ prescription” to “freeze” effectively the $b$ integration along the real axis is frequently used.[@CSS:85] The renormalon ambiguity should be eventually compensated in the physical quantity by the power corrections $\sim (b \Lambda_{\rm QCD})^n$ ($n=2,3, \ldots$) due to non-perturbative effects. Correspondingly, we make the replacement $e^{S (b , Q)}\rightarrow e^{S (b , Q)}F^{NP}(b)$ in (\[resum\]) with the “minimal” ansatz for non-perturbative effects, [@CSS:85; @LKSV:01] $F^{NP}(b)=\exp(-g b^2)$ with a non-perturbative parameter $g$. Fig.1 shows the $Q_T$ distribution of tDY at $\sqrt{S}=100$ GeV, $Q=10$ GeV, $y=\phi=0$, and with a model for the transversity $\delta q(x)$ that saturates the Soffer’s inequality at a low scale.[@MSSV:98] Solid line shows the NLO result using (\[cross section\]), and the dashed and dot-dashed lines show the NLL result using (\[cross section\]), (\[resum\]), $F^{NP}(b)=\exp(-g b^2)$, with $g=0.5$ GeV$^2$ and $g=0$, respectively.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank W. Vogelsang for valuable discussions. The work of J.K. was supported by the Grant-in-Aid for Scientific Research No. C-16540255. The work of K.T. was supported by the Grant-in-Aid for Scientific Research No. C-16540266.
[0]{}
J.P.Ralston and D.Soper, [*Nucl. Phys.*]{}[**B152**]{}, 109 (1979).
W.Vogelsang and A.Weber, [*Phys. Rev.*]{}[**D48**]{}, 2073 (1993).
A.P.Contogouris, B.Kamal and Z.Merebashvili, [*Phys. Lett.*]{} [**B337**]{}, 167 (1994).
W.Vogelsang, [*Phys. Rev.*]{} [**D57**]{}, 1886 (1998).
X.Artru and M.Mukhfi, [*Z. Phys.*]{} [**C45**]{}, 669(1990).
H.Kawamura [*et al.*]{},[*Nucl.Phys.Proc.Suppl.*]{} [**135**]{}, 19 (2004).
H.Kawamura, J.Kodaira, H.Shimizu and K.Tanaka, in preparation.
W.Vogelsang, [*Phys. Rev.*]{} [**D57**]{}, 1886 (1998).
J.C.Collins, D.Soper, G.Sterman, [*Nucl. Phys.*]{} [**B250**]{}, 199 (1985).
J. Kodaira and L. Trentadue, [*Phys. Lett.*]{} [**B112**]{} 66 (1982).
D. de Florian and M. Grazzini,[*Phys.Rev. Lett.*]{} [**85**]{} 4678 (2000).
E.Laenen, G.Sterman and W.Vogelsang, [*Phys. Rev.*]{} [**D63**]{}, 114018 (2001); S.Kulesza, G.Sterman and W.Vogelsang, [*ibid.*]{}[**D66**]{},014001 (2002); E.Laenen, in these proceedings.
O. Martin [*et al.*]{}, [*Phys. Rev.*]{} [**D57**]{}, 3084 (1998); [*ibid*]{} [**D60**]{} (1999) 117502.
[^1]: A talk presented by H.Kawamura
| ArXiv |
---
author:
- Chiara Panosetti
- 'Simon B. Anniés'
- Cristina Grosu
- Stefan Seidlmayer
- Christoph Scheurer
bibliography:
- 'references.bib'
title: 'DFTB modelling of lithium intercalated graphite with machine-learned repulsive potential'
---
Abstract
========
Lithium ion batteries have been a central part of consumer electronics for decades. More recently, they have also become critical components in the quickly arising technological fields of electric mobility and intermittent renewable energy storage. However, many fundamental principles and mechanisms are not yet understood to a sufficient extent to fully realize the potential of the incorporated materials. The vast majority of concurrent lithium ion batteries make use of graphite anodes. Their working principle is based on intercalation—the embedding and ordering of (lithium-) ions in the two-dimensional spaces between the graphene sheets. This important process—it yields the upper bound to a battery’s charging speed and plays a decisive role for its longevity—is characterized by multiple phase transitions, ordered and disordered domains, as well as non-equilibrium phenomena, and therefore quite complex. In this work, we provide a simulation framework for the purpose of better understanding lithium intercalated graphite and its behaviour during use in a battery. In order to address the large systems sizes and long time scales required to investigate said effects, we identify the highly efficient, but semi-empirical Density Funtional Tight Binding (DFTB) as a suitable approach and combine particle swarm optimization (PSO) with the machine learning (ML) based Gaussian Process Regression (GPR) to obtain the necessary parameters. Using the resulting parametrization, we are able to reproduce experimental reference structures at a level of accuracy which is in no way inferior to much more costly *ab initio* methods. We finally present structural properties and diffusion barriers for some exemplary system states.
Introduction {#intro}
============
Within the past decade, studies investigating the consequences of man-made climate change [@Sharp2011; @Fisher2012; @Program2018] have become more specific, the predicted time frames shorter and the warnings more urgent. The immediate and radical reduction of carbon dioxide emissions by replacing fossil fuel based energy sources with renewable ones has been found to be the only reasonable approach to at least limit those consequences. [@Anderson2016] While the generation of electric energy from wind and sun is already quite advanced and efficient, its storage and transport are the main factors holding it back compared to coal and oil. Currently, two main approaches are being pursued in order to eliminate these drawbacks. One aims directly at the synthesis of alternative liquid or gas-phase fuels. The other intends to improve upon existing battery technology—especially lithium ion batteries—enough, to make it a serious contender in terms of energy sustenance. In this work, we intend to lay some groundwork for gaining deeper insight into some of the atomistic mechanisms limiting the (dis-)charging speed and lifetime of the most common types of lithium ion batteries, with graphite intercalation anodes. Ever since graphite was ascertained experimentally and theoretically to be an excellent candidate as an anode for Li-ion batteries, numerous attempts were made at fully describing the working system. [@Hennig1959; @Guerard1975; @Hawrylak1984; @Conard1994; @Nitta2015] Most of the electrochemical properties of the anode material itself are well-known. However, in particular transport processes during strongly driven operating conditions, like fast charging, are only poorly understood at a microscopic level. These technologically important macroscopic conditions are accompanied [*e.g.*]{} by temperature variations, leading to a capacity fade during ageing, as well as lithium plating. All of them limit the lifetime of the battery. [@Gallagher2016; @Wandt2018; @Yang2018] Against this background, experiments and theory are pushed quite far to gain insight into the real processes occuring during the electrochemical operation. Depending on the quantities accessible via experiments and theory, two different hypotheses are regularly invoked to explain the findings in the range of 0% (graphite) to 100% (LiC$_6$) state of charge (SOC): the staging and the domain model. The lithium intercalation process shows evidence of multiple phase transitions in the voltage vs. SOC diagram. The corresponding system configurations are termed “stages” I, II and so forth. In the simple staging model, these correspond directly to the numbers of empty galleries (spaces between graphene sheets) between the fully occupied ones (see Figure \[fig:staging\]).
![Sketch of Li-intercalated graphite in stage I to III configurations [@Smith2017]. Violet spheres represent lithium ions, black lines correspond to graphene sheets. Bottom right: illustration of the domain model [@Daumas1969]. The structure has the same nominal stoichiometry as the structure in stage II (top right). \[fig:staging\]](./staging_compact.png "fig:"){width="0.9\linewidth"}\
In the domain model, these motifs are not assumed to range over meso-/macroscopic dimensions but to form regions of finite lateral extent. Consequently, it is quite clear that different SOC with the same nominal stoichiometry LiC$_x$ will not be configurationally homogeneous, making Li-intercalated graphite a profoundly non-trivial system to address.
In order to effectively connect to experimental studies, a theoretical framework for simulating large-scale and long-duration non-equilibrium processes in the graphite anode, based on kinetic Monte Carlo (kMC) [@Andersen2019] simulations is required. The first step towards this goal is gaining the ability to quickly and accurately calculate diffusion barriers on the fly, which is the primary motivation of this work. This requires the ability to reproduce reliably and accurately the layer distances (ideally of all possible configurations, but predominantly of the dilute, low-saturation stages) and the forces affecting the lithium-ions, while the strains within the graphene layers are of lesser importance.
Large-scale atomistic simulations typically pursue force field approaches [@Duin2001] for those systems where energetics and kinetics are well described within the upper end of the SOC range. However, those approaches are limited when it comes to the entire range of different SOC, from extremely diluted stages to fully concentrated ones. Recently, a Gaussian Approximation Potential (GAP) was reported to be able to describe amorphous carbon well. [@Deringer2017] However, when the latter was later extended to model lithium intercalation, [@Fujikake2018] it became apparent that the insertion of lithium into those host structures requires a non-trivial description of the electrostatic interaction. Contrary to most approaches, including the one presented in this work, Fujikake *et al.* did not treat the full Li-C system, but attempted to model the energy and force differences arising from lithium intercalation separately, and then added them to the carbon GAP. More specifically, their machine learning process (ML) is based on fitting the energy and force differences between identical carbon host structures, but with and without an intercalated lithium atom. However due to the fact that the lithium intercalation energies are significantly larger in magnitude than the electrostatic lithium-lithium interaction energies, they were not able to recover the latter from the data to a satisfactory degree and had to manually add an extra correction term (fitted to DFT) in order to account for those contributions. To avoid similar shortcomings, we rather base our approach on Density Functional Tight Binding (DFTB) [@Elstner1998], a semi-empirical—and thus computationally much cheaper—approximation to Density Functional Theory (DFT), [@Kohn1965] which has been the most common technique for high-accuracy electrochemical simulations for many decades [@Koskinen2009]. However, since DFTB’s speedup is achieved by pre-calculating atomic interactions to avoid calculating them at runtime, this comes at the cost—or rather, initial investment—of pairwise parametrization. As of now, no Li-Li and Li-C DFTB parameters are available. In the following, we combine for the first time the recently developed Particle Swarm Optimization [@Shi1998] parametrization approach as first proposed by Chou [*et al.*]{} [@Chou2016] with a more flexible ML repulsive potential [@Engelmann2018], to obtain finely-tuned parameters for this system—taking advantage of its physics, albeit perhaps at the expense of some transferability. Let us however stress that the parametrization procedure employed here remains completely general, as the system specificity lies entirely in the choice of the training set(s).
The electronic part {#electro}
===================
{width="0.24\linewidth"} {width="0.24\linewidth"} {width="0.24\linewidth"} {width="0.24\linewidth"}
In DFTB jargon, the so-called “electronic part” includes the semi-empirical band structure and the Coulombic contributions to the total energy of the system. [@Koskinen2009] These depend parametrically on the diagonal elements $\varepsilon$ of the non-interacting Hamiltonian, the Hubbard-$U$ and a confinement potential which is used to cut off the diffuse tails of the basis orbitals. For the free atom, the first two quantities are tabulated for most elements or can be calculated with DFT. However, using the free atom values is an approximation, and the decision whether it is justified must be made carefully on a case to case basis. The confinement potential, on the other hand, is always treated as a parameter. Quadratic [@Stohr2018] and general power-law functional forms [@Wahiduzzaman2013] are commonly used, as well as the Woods-Saxon potential [@Chou2016] (also employed here) which assures a smoother transition to zero in the orbital tails. Each of these parameters needs to be determined for every chemical species present in the system of interest, typically in a non-linear optimization process. In the PSO, each particle then represents a set of parameters ($\left\{\varepsilon\right\}$, $\left\{U\right\}$, and the confinement constants), with which the DFTB interaction is constructed, so that the parametrization can be improved by minimizing a cost function. The central task is thus the definition of a meaningful cost function. Frequently, one uses the weighted sum of an arbitrary number of contributions $f(\sigma^{DFT}, \sigma^{DFTB})$, each providing a measure of the deviation between DFT and DFTB for some system property $\sigma$. Hereby, as we are optimizing the electronic parameters only, the chosen target properties must not depend on repulsion. For our system, we compare the band structures of metallic lithium, graphene and diamond. Additional details on the definition of the corresponding cost function are provided in the SI. Figure \[fig:bandstructures\] shows our resulting band structures. Overall, we recognize decent agreement for all band structures, while some deviations are expected given the minimal basis in DFTB. For example, the pronounced mismatch in the conduction band at the $H$ point in the lithium band structure as well as the incorrectly direct band gap of diamond can be ascribed to this over-simplification in the DFTB model. For the two carbon systems, we see very good qualitative agreement for most regions of the band structures, but notice a small degree of overall compression towards the Fermi level. Given the systematic nature of this imperfection, we speculate that for further improvement, it would probably be necessary to include $U$ and $\varepsilon$ in our parameter space, which would simultaneously increase the dimensionality of the optimization problem. As an additional validation criterion, we examine the charge population (which DFTB provides by default) for the lithium ions in LiC$_6$. Our parametrization produces a value of $0.853\,e$, in agreement with the value of $0.86\,e$ calculated by Krishnan [@Krishnan2013] with Bader charge analysis [@Bader1990]. Given this excellent agreement and also considering the fact that the repulsion potential is capable of quite effectively correcting small imperfections in the electronic part, we decide not to optimize the latter any further in this work—a decision justified in retrospect by the excellent results we present. However, let us still emphasize the opportunity for improvement here, should it eventually become necessary.
The repulsion potential {#rep}
=======================
It is common practice to assume some analytical form for the repulsive potential and fit the chosen functional parameters as to minimize a set of DFT-DFTB force differences [@Koskinen2009]—a protocol easily implemented also for the PSO approach. However, the main limitation and bias results from the choice of said parametrized functional form. It needs to be sufficiently flexible to cover a large space of systems and bonding situations. This typically yields a high dimensional non-linear optimization problem, which might still be insufficient to capture unexpected subtle, yet extremely relevant physical features. We rather adopt the method recently developed by A. Engelmann [@Engelmann2018], which employs Gaussian Process Regression (GPR) [@Rasmussen2006] to create a flexible functional form “on the fly”, while adapting to the physics captured by the training data set, instead of forcing us to guess it *a priori*. In the SI, we give a short introduction to the method and explain the character and effect of the related hyperparameters, referring the reader to Rasmussen [@Rasmussen2006] for the underlying stochastic theory and to Engelmann [@Engelmann2018] for the application to DFTB repulsive potentials. For the global damping, correlation distance, and data noise hyperparameters, we verified (see SI) that a sizeable subspace of the overall hyperparameter-space is appropriate, and choosing pretty much any combination of values within that subspace will produce very similar, correct results. The same is not necessarily true for the cutoff radii of the C-C and the Li-C repulsion. Since the electronic energy contribution is entirely based on just a sum of non-interacting atomic contributions, the repulsion potential has to account for different chemical environments affecting the same type of atom. In a GPR setting it is therefore of paramount importance to sample a sufficiently large set of training data which covers all interatomic distance ranges and chemical environments relevant for a faithful representation of the system studied. Ideally, it should also be ascertained that the model quality is stable w.r.t. the explicit choice of hyperparameters such as the cutoff radii.
The training data {#training}
-----------------
![Interlayer distances for graphite (grey), LiC$_{12}$ (SOC 50%, grey-purple) and LiC$_6$ (SOC 100%, bright purple) as a function of C-C repulsion cutoff trained. Note that for LiC$_{12}$, there are two different layer distances to consider: one for the empty gallery and one for the full gallery. Here, we plot the average of the two. The dashed lines show the experimental layer distances we aim to reproduce (Sources: Trucano [*et al.*]{} [@Trucano1975] (graphite), Vadlamani [*et al.*]{} [@Vadlamani2014] (LiC$_{12}$ and LiC$_6$). The green coloured area represents the range within which we are satisfied with the performance.[]{data-label="fig:vsCC"}](./all_vs_CC_colors.png){width="0.9\linewidth"}
In terms of DFT functional, our starting point is PBE [@Perdew1996], which has been used by the majority of researchers working on intercalation phenomena and is known to describe LiC$_6$ well. However, it does not reproduce the dispersive interaction between graphene sheets. In order to address this, we finally (see “Set 3” below) combine the reference PBE calculation with a Many Body Dispersion (MBD) treatment and the DFTB model with a computationally cheap Lennard Jones (LJ) [@Zhechkov2005] dispersion correction [@Rappe1992]. The rationale for this choice is that PBE should reproduce galleries containing many lithium atoms correctly and LJ-dispersion should predict empty galleries well, while not interfering too much with the PBE-description of the concentrated ones. However, it is unclear, how this interaction shapes out for intermediate, dilute lithium stoichiometries. During our investigations, we find that this approach works somewhat decently, but needs some controlled adjustments (vide infra) in order to produce truly satisfactory results.
As a first guess, we construct a set of training structures (Set 1) which consists of a balanced mix of Li$_n$C$_{36}$ super-cells ($n \in (0, 1, ..., 6)$), in order to represent the entire range of charging states (exemplary structures are shown in the SI). Additionally, those structures are rattled (each atom randomly displaced), as well as compressed or expanded. This procedure yields a smooth distribution of bond lengths and forces. We then train a GPR repulsion potential by matching DFTB against PBE forces for this structural ensemble, aiming at a first, mostly transferable model. The standard LJ DFTB correction is subsequently applied on top of this parametrized DFTB model. With this approach, we are able to find parametrizations that reproduce all layer distances (of graphite, LiC$_{12}$ and of LiC$_6$) correctly, albeit not for a stable range of all parameters (in particular the Li-C cutoff, see below). As shown in Figure \[fig:vsCC\], the choice of cutoff radius for the C-C repulsion potential does not have a major influence on the layer-distances for quite a large range of values. In fact, the point at which the predictions stop being accurate can be identified as approximately the experimental values for the interlayer distances. Going beyond that with the cutoff radius essentially corresponds to including interlayer interactions in the potential fit, mixing their description with the intralayer covalent bonds. Thus, the restriction of the cutoff radius we find here is physically motivated by the range separation of the interactions that characterize our system: as the 2nd next neighbour distance in a relaxed graphene sheet is around $2.45$ Å and the layer distance is $3.35$ Å, the cutoff range defined by the (smallest) plateau in Figure \[fig:vsCC\] represents a sweet spot where the GPR learns 2nd next neighbour interactions but does not yet (mistakenly) take any interlayer interactions (even in the compressed structures) into account in the repulsion potential. In light of these findings, we select the cutoff value $2.6$ Å for the C-C-repulsion potential. Indeed, we did not encounter any reason to change this selection during the entirety of this work (despite rigorously testing it for each of the training data sets).
However, with this first training set we do not obtain an equally stable plateau as a function of the Li-C repulsive cutoff (see SI). Furthermore, we discover that the quite strongly distorted graphite planes in these structures lead to large forces compared with those acting on the intercalated lithium-ions hindering the performance in lithium-forces prediction. We tackle the second problem first: while the rattled, scaled structures in Set 1 cover a sufficiently large range of bond lengths, they only account for configurations with the lithium-ions sitting over the centre of a graphite ring, *i.e.* in a local energy minimum. We recognize this as the reason for the comparably small lithium-forces. In order to balance out this structural bias, we calculate a number of transition paths for lithium diffusion processes using a Nudged Elastic Band (NEB) method [@Henkelman2000; @Henkelman2000a]. Exemplary structures can be found in the SI. Now, we are able to extract structures from these trajectories, in which the lithium ions are subject to stronger forces commensurable with the graphite-layers. For our second training set (Set 2), we replace around $45\%$ of the rattled and scaled structures with those transition path geometries. By this measure, we are able to improve the accuracy for predicting forces on Li-ions significantly, without sacrificing the description of the graphite layers. However, while we do observe a plateau for the resulting layer distances with respect to the Li-C cutoff, the interlayer distances are not reproduced equally well as in Figure \[fig:vsCC\] for Set 1 (see Figure \[fig:vsLiC\], yellow area), while only the LiC$_{12}$ interlayer distances assume correct values (see Figure \[fig:vsLiC\], green areas), yet outside the plateau.
![Interlayer distances for LiC$_{12}$ (SOC 50%, grey-purple) and LiC$_{6}$ (SOC 100%, bright purple) as a function of Li-C repulsion cutoff, with a fixed C-C cutoff set to 2.6 Å. The repulsion was trained on a set analogous to Set 1 ([*cf.*]{} text), where 45% of the structures were replaced by geometries randomly extracted from intra-layer Li diffusion paths. For LiC$_{12}$, the plotted interlayer distance is the average between the values for the filled and the empty gallery. The dashed lines show the experimental layer distances. The yellow coloured area represents the range within which the results are stable, however at a wrong value.[]{data-label="fig:vsLiC"}](./all_vs_LiC_set2.png){width="0.9\linewidth"}
![Interlayer distances for LiC$_{12}$ (SOC 50%, grey-purple) and LiC$_{6}$ (SOC 100%, bright purple) as a function of Li-C repulsion cutoff, with a fixed C-C cutoff set to 2.6 Å. The repulsion was trained on a set analogous to Set 2 ([*cf.*]{} text), where 55% of the structures were replaced by geometries with MBD-corrected forces. For LiC$_{12}$, the plotted interlayer distance is the average between the values for the filled and the empty gallery. The dashed lines show the experimental layer distances. []{data-label="fig:vsLiC_set3"}](./all_vs_LiC_set3.png){width="0.9\linewidth"}
{width="0.9\linewidth"}
This behaviour suggests that our problem here does not lie in the choice of the training set, but rather in the treatment of long-ranged interactions.
Let us consider the underlying predicament: so far, the DFTB-part of the force residues used for the ML process is calculated without LJ dispersion correction. We then construct the repulsion potential with the purpose of making those DFTB calculations match references based on PBE-DFT, which reliably predicts layer distances for LiC$_6$. By then using LJ (required to obtain the correct empty layer distance in graphite) in our actual DFTB calculations (after the parametrization process), we cause the aforementioned offset for highly lithiated compounds. Using LJ already for the force-residue calculations during the ML seems like the obvious solution to this problem. However, this presents a new issue in the lower-saturation range (Li$_x$C$_6$, $x<0.5$). There, we previously fitted the repulsion to PBE-DFT references, which are not correct in that range without dispersion correction. The resulting DFTB forces are then shifted by LJ towards the correct value (as is indicated by the quite decent results for LiC$_{12}$ with Set 2). But after the modification, we would then fit the *final* DFTB forces (that result after applying the LJ) to the (incorrect) PBE-DFT references, thus improving our performance for highly saturated system states, but ruining it for dilute ones, by effectively double counting dispersive contributions. It becomes apparent that in order to make this approach work, we need to utilize dispersion corrected DFT reference forces which are also correct for low saturation states and, at the same time, compatible with the computationally cheap DFTB-LJ correction.
Our ansatz is that we can essentially—to a degree—encode the difference between the LJ dispersion and the “true” dispersion into the repulsion potential. At this point we stress that *ideally*, both the true, non-local exchange correlation functional in DFT and an ideal repulsion energy in DFTB would already encompass all dispersion effects, and it is solely due to approximations in the derivations, [*e.g.*]{} of GGAs, that they do not in these models. Therefore, rather than mixing our repulsion potential with something fundamentally different (which would be physically questionable), what we do here simply corresponds to partially adding a contribution back in, that should have been there in the first place. To our knowledge, the currently best way to calculate dispersion corrected lithium intercalated graphite, with correct layer distances predicted for the entire saturation range, is the MBD correction [@Tkatchenko2012]. This method is computationally rather expensive, but since we only need to run DFT calculations for our training data set, which is very limited in size, this is not vital to us. We do realize that this approach most likely comes with some cost in terms of transferability. In order to retain as much of it as possible, we choose not to replace *all* force residues, but only $\approx50\%$, which proves sufficient to demonstrate the effectiveness of the presented method in a general way. Nonetheless, further investigating the effect this percentage has on the performance is certainly a task that should be tackled in the future. Of course, alternatively to our approach, it is possible to simply apply the MBD correction scheme directly to our DFTB calculations. However, doing so would cost us one to two orders of magnitude in speed, as MBD then becomes the dominating step in terms of computation time. Exactly as we had hoped for, we have succeeded at shifting the predicted interlayer distances (within the stable Li-C cutoff plateau) into the very close proximity of the experimental reference values for both LiC$_6$ and LiC$_{12}$ (Figure \[fig:vsLiC\_set3\]). Especially the excellent results for the stage II compound LiC$_{12}$ show that our parametrization is now able to handle *both* mainly ionic concentrated *and* mainly dispersive dilute layers to a satisfactory degree. In Figure \[2Dlandscape\], we illustrate the effect our modification has on the repulsion potential landscape for a wide range of Li-C cutoff radii. First (and most notably), we have moved and solidified the local minimum related to the next-neighbour lithium-carbon interaction (see bottom right). For the Set 2 and Set 3 potentials, the minima (blue and black dashed lines) are located at atomic distances of $2.41$ Å and $2.35$ Å respectively, which correspond to LiC$_6$ interlayer distances of $3.83$ Å and $3.67$ Å, the exact values which *do*, in fact, result from the relaxation of those structures, using the two repulsion potentials respectively. The 2D maps (top) show that this behaviour is apparent for an entire range of cutoff radii, thus ruling out the possibility that the fit is only accidentally correct (as it happens, [*e.g.*]{}, for Set 2, see Figure \[fig:vsLiC\]). In bottom left, we can also clearly see the upper ($\sim 5.8$Å) and lower ($\sim 4.5$Å) boundaries for the cutoff radius, beyond which the physicality of the model falls apart. They define exactly the range within which we find the stable cutoff dependency plateau, which is now at the correct numerical value as shown in Figure \[fig:vsLiC\_set3\]. We may identify the upper boundary at $5.8$ Å (as further discussed in the SI), as the distance between a lithium ion and the second closest graphene sheet, which is an intuitively plausible limitation. It is less obvious, though, to assign a clear physical meaning to the lower bound at $4.5$ Å, as it cannot be directly related to any particular structural feature of LiC$_6$. The most likely cause, we believe, is that the cosine-shaped cutoff function employed in the GPR framework starts cutting off physically relevant details from the repulsion potential below that. A physically motivated lower bound may be identified by evaluating the mean absolute forces acting on lithium as a function of Li-C cutoff, shown in Figure \[Fig:maf\].
![Mean absolute forces acting on Li for validation set??? compared to DFT reference calculations (black dashed line)[]{data-label="Fig:maf"}](./rmsd_forces_set3.png){width="0.9\linewidth"}
Overall, we now observe two separate Li-C cutoff plateaus: between approximately 4.3 Å and 5.7 Å, we obtain accurate layer distances (Figure \[fig:vsLiC\_set3\]), while for radii above roughly 5.0 Å, our predictions for forces and transition energies are correct (Figure \[Fig:maf\]). This duality can very simply be explained by the fact that the first property is mostly a z-direction phenomenon (and interactions with the second closest graphene sheet limit the physicality of our model), while the other takes place almost exclusively in the x-y-plane, where no such limitation applies. Given this difference in fundamental nature, it is very plausible to trust both these plateaus. Thus, their overlap (5.0–5.7 Å) defines the region within which any value of the Li-C cutoff radius produces an almost identical parametrization that performs very well, for all our benchmark criteria, in a stable and trustworthy manner.
Results: interlayer distances and diffusion barriers
====================================================
[l l | r r r r r r ]{}\
&compound &experiment &DFTB &DFT in [@Krishnan2013] &filled gallery &empty gallery &barrier\
C$_6$ &– &$3.355$ Å &$+46$ mÅ &$+62$ mÅ &– & $+46$ mÅ\
LiC$_6$ &stage I &$3.687$ Å &$-12$ mÅ &$+56$ mÅ &$-12$ mÅ &– &$351$ meV\
LiC$_{12}$ &stage II &$3.511$ Å &$+16$ mÅ &$-16$ mÅ &$+112$\* mÅ &$-79$\* mÅ &$401$ meV\
LiC$_{18}$ &stage III &$3.470$ Å &$-30$ mÅ&$+173$ mÅ &$+196$\* mÅ &$-50$\* mÅ &$393$ meV\
\
\
Table \[table1\] reports some resulting inter-layer distances and diffusion barriers based on our new DFTB parametrization in Table \[table1\], compared with experimentally determined values, as well as previous theoretical findings.
Furthermore, we draw qualitative conclusions from these results and summarize their implications on the intercalation mechanism. As a quick reminder, stages I, II and III correspond to every, every other and every third gallery being filled (to any degree) with lithium. Additionally, we describe the concentration of the intercalant in a filled gallery as dilute (low) or concentrated (high), thus allowing for a simple classification of fundamentally different compounds.
Here, we take only concentrated stages into consideration. For all calculations, we chose an Li-C cutoff radius of $5.5$ Å, following the findings discussed above. As Table \[table1\] clearly illustrates, we systematically outperform the method by Krishnan *et al.* [@Krishnan2013]—in terms of accuracy—for every structure they provide comparison for. This is especially remarkable considering the fact that they used full GGA-DFT with dispersion corrections in post-processing, which is the current state-of-the-art approach, as well as significantly more computationally expensive than our method.\
Subsequently, we investigate intra-layer next-neighbour diffusion barriers and compare our results to recent experimental findings from [@Umegaki2017] (based on muon spin relaxation spectroscopy). Our calculations yield purely *microscopic* results within 50 meV from each other for all three relevant compounds, as is shown in Table \[table1\]. The deviations between them can be attributed to slight differences in the filled-layer spacing of the different structures.\
In contrast, the experimentally determined *active* barriers of 270 meV for LiC$_6$ and 170 meV for LiC$_{12}$ show a strong dependency on the systems stage. We believe this difference to be caused by correlation effects. Capturing those using kinetic Monte Carlo simulation is something we intend to do in the near future.
Conclusions and outlook
=======================
In this work, we put forward—for the first time combining particle swarm (i.e. PSO) and machine learning [@Engelmann2018] (i.e. GAP) approaches for this task—a well-performing DFTB-parametrization for lithium intercalated graphite which is capable of very accurately reproducing various structural properties and qualitative trends relating to the intercalation mechanism for a wide variety of Li$_x$C$_{36}$ compounds. In the course of this process, we believe to have shown that Density Functional Tight Binding (DFTB) is a superior approach for modelling intercalation compared with force field methods (*e.g.* the GAP by [@Fujikake2018] requires a manual correction term for lithium-lithium interactions which our method does not). Furthermore, we share key details and choices along this process and thus provide guidance for similar endeavours in the future.
| ArXiv |
---
abstract: 'In a two-flavor color superconductor, the $SU(3)_c$ gauge symmetry is spontaneously broken by diquark condensation. The Nambu-Goldstone excitations of the diquark condensate mix with the gluons associated with the broken generators of the original gauge group. It is shown how one can decouple these modes with a particular choice of ’t Hooft gauge. We then explicitly compute the spectral density for transverse and longitudinal gluons of adjoint color 8. The Nambu-Goldstone excitations give rise to a singularity in the real part of the longitudinal gluon self-energy. This leads to a vanishing gluon spectral density for energies and momenta located on the dispersion branch of the Nambu-Goldstone excitations.'
address:
- |
Institut für Theoretische Physik, Johann Wolfgang Goethe-Universität\
Robert-Mayer-Str. 8–10, D-60054 Frankfurt/Main, Germany\
E-mail: [email protected]
- |
School of Physics and Astronomy, University of Minnesota\
116 Church Street S.E., Minneapolis, MN 55455, U.S.A.\
E-mail: [email protected]
author:
- 'Dirk H. Rischke'
- 'Igor A. Shovkovy[^1]'
title: 'Longitudinal gluons and Nambu-Goldstone bosons in a two-flavor color superconductor'
---
Introduction
============
Cold, dense quark matter is a color superconductor [@bailinlove]. For two massless quark flavors (say, up and down), Cooper pairs with total spin zero condense in the color-antitriplet, flavor-singlet channel. In this so-called two-flavor color superconductor, the $SU(3)_c$ gauge symmetry is spontaneously broken to $SU(2)_c$ [@arw]. If we choose to orient the (anti-) color charge of the Cooper pair along the (anti-) blue direction in color space, only red and green quarks form Cooper pairs, while blue quarks remain unpaired. Then, the three generators $T_1,\, T_2,$ and $T_3$ of the original $SU(3)_c$ gauge group form the generators of the residual $SU(2)_c$ symmetry. The remaining five generators $T_4, \ldots, T_8$ are broken. (More precisely, the last broken generator is a combination of $T_8$ and the generator ${\bf 1}$ of the global $U(1)$ symmetry of baryon number conservation, for details see Ref. [@sw2] and below).
According to Goldstone’s theorem, this pattern of symmetry breaking gives rise to five massless bosons, the so-called Nambu-Goldstone bosons, corresponding to the five broken generators of $SU(3)_c$. Physically, these massless bosons correspond to fluctuations of the order parameter, in our case the diquark condensate, in directions in color-flavor space where the effective potential is flat. For gauge theories (where the local gauge symmetry cannot truly be spontaneously broken), these bosons are “eaten” by the gauge bosons corresponding to the broken generators of the original gauge group, [*i.e.*]{}, in our case the gluons with adjoint colors $a= 4, \ldots, 8$. They give rise to a longitudinal degree of freedom for these gauge bosons. The appearance of a longitudinal degree of freedom is commonly a sign that the gauge boson becomes massive.
In a dense (or hot) medium, however, even [*without*]{} spontaneous breaking of the gauge symmetry the gauge bosons already have a longitudinal degree of freedom, the so-called [*plasmon*]{} mode [@LeBellac]. Its appearance is related to the presence of gapless charged quasiparticles. Both transverse and longitudinal modes exhibit a mass gap, [*i.e.*]{}, the gluon energy $p_0 \rightarrow m_g > 0$ for momenta $p \rightarrow 0$. In quark matter with $N_f$ massless quark flavors at zero temperature $T=0$, the gluon mass parameter (squared) is [@LeBellac] $$\label{gluonmass}
m_g^2 = \frac{N_f}{6\, \pi^2} \, g^2 \, \mu^2\,\, ,$$ where $g$ is the QCD coupling constant and $\mu$ is the quark chemical potential.
It is [*a priori*]{} unclear how the Nambu-Goldstone bosons interact with these longitudinal gluon modes. In particular, it is of interest to know whether coupling terms between these modes exist and, if yes, whether these terms can be eliminated by a suitable choice of (’t Hooft) gauge. The aim of the present work is to address these questions. We shall show that the answer to both questions is “yes”. We shall then demonstrate by focussing on the gluon of adjoint color 8, how the Nambu-Goldstone mode affects the spectral density of the longitudinal gluon.
Our work is partially based on and motivated by previous studies of gluons in a two-flavor color superconductor [@carterdiakonov; @dhr2f; @dhrselfenergy]. The gluon self-energy and the resulting spectral properties have been discussed in Ref. [@dhrselfenergy]. In that paper, however, the fluctuations of the diquark condensate have been neglected. Consequently, the longitudinal degrees of freedom of the gluons corresponding to the broken generators of $SU(3)_c$ have not been treated correctly. The gluon polarization tensor was no longer explicitly transverse (a transverse polarization tensor $\Pi^{\mu\nu}$ obeys $P_\mu \, \Pi^{\mu \nu} = \Pi^{\mu \nu}\, P_\nu = 0$), and it did not satisfy the Slavnov-Taylor identity. As a consequence, the plasmon mode exhibited a certain peculiar behavior in the low-momentum limit, which cannot be physical (cf. Fig. 5 (a) of Ref. [@dhrselfenergy]). It was already realized in Ref. [@dhrselfenergy] that the reason for this unphysical behavior is the fact that the mixing of the gluon with the excitations of the condensate was neglected. It was moreover suggested in Ref. [@dhrselfenergy] that proper inclusion of this mixing would amend the shortcomings of the previous analysis. The aim of the present work is to follow this suggestion and thus to correct the results of Ref. [@dhrselfenergy] with respect to the longitudinal gluon. Note that in Ref. [@carterdiakonov] fluctuations of the color-superconducting condensate were taken into account in the calculation of the gluon polarization tensor. As a consequence, the latter is explicitly transverse. However, the analysis was done in the vacuum, at $\mu=0$, not at (asymptotically) large chemical potential.
The outline of the present work is as follows. In Section \[II\] we derive the transverse and longitudinal gluon propagators including fluctuations of the diquark condensate. In Section \[III\] we use the resulting expressions to compute the spectral density for the gluon of adjoint color 8. Section \[IV\] concludes this work with a summary of our results.
Our units are $\hbar=c=k_B=1$. The metric tensor is $g^{\mu \nu}= {\rm diag}\,(+,-,-,-)$. We denote 4-vectors in energy-momentum space by capital letters, $K^{\mu} = (k_0,{\bf k})$. Absolute magnitudes of 3-vectors are denoted as $k \equiv |{\bf k}|$, and the unit vector in the direction of ${\bf k}$ is $\hat{\bf k} \equiv
{\bf k}/k$.
Derivation of the propagator for transverse and longitudinal gluons {#II}
===================================================================
In this section, we derive the gluon propagator taking into account the fluctuations of the diquark condensate. A short version of this derivation can be found in Appendix C of Ref. [@msw] \[see also the original Ref. [@gusyshov]\]. Nevertheless, for the sake of clarity and in order to make our presentation self-contained, we decide to present this once more in greater detail and in the notation of Ref. [@dhrselfenergy]. As this part is rather technical, the reader less interested in the details of the derivation should skip directly to our main result, Eqs. (\[transverse\]), (\[longitudinal\]), and (\[hatPi00aa\]).
We start with the grand partition function of QCD,
\[Z\] $$\label{ZQCD}
{\cal Z} = \int {\cal D} A \; e^{ S_A }
\;{\cal Z}_q[A]\,\, ,$$ where $${\cal Z}_q[A] = \int {\cal D} \bar{\psi} \, {\cal D} \psi\,
\exp \left[ \int_x \bar{\psi} \left(
i \gamma^\mu \partial_\mu + \mu \gamma_0 + g \gamma^\mu A_\mu^a T_a
\right) \psi \right]
\,. \label{Zquarks}$$
is the grand partition function for massless quarks in the presence of a gluon field $A^\mu_a$. In Eq. (\[Z\]), the space-time integration is defined as $\int_x \equiv \int_0^{1/T} d\tau \int_V d^3{\bf x}\,$, where $V$ is the volume of the system, $\gamma^\mu$ are the Dirac matrices, and $T_a= \lambda_a/2$ are the generators of $SU(N_c)$. For QCD, $N_c = 3$, and $\lambda_a$ are the Gell-Mann matrices. The quark fields $\psi$ are $4 N_c N_f$-component spinors, [*i.e.*]{}, they carry Dirac indices $\alpha = 1, \ldots,4$, fundamental color indices $i=1,\ldots,N_c$, and flavor indices $f=1,\ldots,N_f$. The action for the gauge fields consists of three parts, $$\label{L_A}
S_A = S_{F^2} + S_{\rm gf} + S_{\rm FPG}\,\, ,$$ where $$S_{F^2} = - \frac{1}{4} \int_x F^{\mu \nu}_a \, F_{\mu \nu}^a$$ is the gauge field part; here, $F_{\mu\nu}^a = \partial_\mu A_\nu^a
- \partial_\nu A_\mu^a + g f^{abc} A_\mu^b A_\nu^c$ is the field strength tensor. The part corresponding to gauge fixing, $S_{\rm gf}$, and to Fadeev-Popov ghosts, $S_{\rm FPG}$, will be discussed later.
For fermions at finite chemical potential it is advantageous to introduce the charge-conjugate degrees of freedom explicitly. This restores the symmetry of the theory under $\mu \rightarrow - \mu$. Therefore, in Ref. [@dhr2f], a kind of replica method was applied, in which one first artificially increases the number of quark species, and then replaces half of these species of quark fields by charge-conjugate quark fields. More precisely, first replace the quark partition function ${\cal Z}_q[A]$ by ${\cal Z}_M[A] \equiv \left\{ {\cal Z}_q[A] \right\}^M$, $M$ being some large integer number. (Sending $M \rightarrow 1$ at the end of the calculation reproduces the original partition function.) Then, take $M$ to be an even integer number, and replace the quark fields by charge-conjugate quark fields in $M/2$ of the factors ${\cal Z}_q[A]$ in ${\cal Z}_M[A]$. This results in $${\cal Z}_M[A] =
\int \prod_{r=1}^{M/2} {\cal D} \bar{\Psi}_r \, {\cal D} \Psi_r \; \exp
\left\{ \sum_{r=1}^{M/2} \left[
\int_{x,y} \bar{\Psi}_r(x) \,{\cal G}_0^{-1} (x,y)\,
\Psi_r(y) + \int_x g\, \bar{\Psi}_r(x) \, A_\mu^a(x)\,\hat{\Gamma}^\mu_a\,
\Psi_r(x) \right] \right\} \,\, . \label{Zquarks2}$$ Here, $r$ labels the quark species and $\Psi_r$, $\bar{\Psi}_r$ are $8 N_c N_f$-component Nambu-Gor’kov spinors, $$\Psi_r \equiv \left( \begin{array}{c}
\psi_r \\
\psi_{C r}
\end{array}
\right) \,\,\, , \,\,\,\,
\bar{\Psi}_r \equiv ( \bar{\psi}_r \, , \, \bar{\psi}_{C r} )\,\, ,$$ where $\psi_{C r} \equiv C \bar{\psi}_r^T$ is the charge conjugate spinor and $C=i \gamma^2 \gamma_0$ is the charge conjugation matrix. The inverse of the $8 N_c N_f \times 8 N_c N_f$-dimensional Nambu-Gor’kov propagator for non-interacting quarks is defined as $$\label{S0-1}
{\cal G}_0^{-1} \equiv
\left( \begin{array}{cc}
[G_0^+]^{-1} & 0 \\
0 & [G_0^-]^{-1}
\end{array} \right)\,\, ,$$ where $$\label{G0pm-1}
[G_0^\pm]^{-1}(x,y) \equiv -i \left( i \gamma_\mu \partial^\mu_x
\pm \mu \gamma_0 \right) \delta^{(4)}(x-y)$$ is the inverse propagator for non-interacting quarks (upper sign) or charge conjugate quarks (lower sign), respectively. The Nambu-Gor’kov matrix vertex describing the interaction between quarks and gauge fields is defined as follows: $$\label{Gamma}
\hat{\Gamma}^\mu_a \equiv \left( \begin{array}{cc}
\Gamma^\mu_a & 0 \\
0 & \bar{\Gamma}^\mu_a
\end{array} \right)\,\, ,$$ where $\Gamma^\mu_a \equiv \gamma^\mu T_a$ and $\bar{\Gamma}^\mu_a \equiv C (\gamma^\mu)^T C^{-1} T_a^T \equiv -\gamma^\mu
T_a^T$.
Following Ref. [@bailinlove] we now add the term $\int_{x,y} \bar{\psi}_{C r}(x)\, \Delta^+(x,y) \,
\psi_r(y)$ and the corresponding charge-conjugate term $\int_{x,y} \bar{\psi}_r(x)\, \Delta^-(x,y) \, \psi_{C r}(y)$, where $\Delta^- \equiv \gamma_0 \, (\Delta^+)^\dagger \, \gamma_0$, to the argument of the exponent in Eq. (\[Zquarks2\]). This defines the quark (replica) partition function in the presence of the gluon field $A^\mu_a$ [*and*]{} the diquark source fields $\Delta^+$, $\Delta^-$: $${\cal Z}_M[A,\Delta^+,\Delta^-] \equiv
\int \prod_{r=1}^{M/2} {\cal D} \bar{\Psi}_r \, {\cal D} \Psi_r \; \exp
\left\{ \sum_{r=1}^{M/2} \left[
\int_{x,y} \bar{\Psi}_r(x) \,{\cal G}^{-1} (x,y)\,
\Psi_r(y) + \int_x g\, \bar{\Psi}_r(x) \, A_\mu^a(x)\,\hat{\Gamma}^\mu_a\,
\Psi_r(x) \right] \right\} \,\, , \label{Zquarks3}$$ where $$\label{G-1}
{\cal G}^{-1} \equiv
\left( \begin{array}{cc}
[G^+_0]^{-1} & \Delta^- \\
\Delta^+ & [G^-_0]^{-1}
\end{array} \right)$$ is the inverse quasiparticle propagator.
Inserting the partition function (\[Zquarks3\]) into Eq. (\[ZQCD\]), the (replica) QCD partition function is then computed in the presence of the (external) diquark source terms $\Delta^\pm(x,y)$, ${\cal Z} \rightarrow {\cal Z}[\Delta^+,\Delta^-]$. In principle, this is not the physically relevant quantity, from which one derives thermodynamic properties of the color superconductor. The diquark condensate is not an external field, but assumes a nonzero value because of an intrinsic property of the system, namely the attractive gluon interaction in the color-antitriplet channel, which destabilizes the Fermi surface.
The proper functional from which one derives thermodynamic functions is obtained by a Legendre transformation of $\ln {\cal Z} [\Delta^+, \Delta^-]$, in which the functional dependence on the diquark source term is replaced by that on the corresponding canonically conjugate variable, the diquark condensate. The Legendre-transformed functional is the effective action for the diquark condensate. If the latter is [*constant*]{}, the effective action is, up to a factor of $V/T$, identical to the effective potential. The effective potential is simply a function of the diquark condensate. Its explicit form for large-density QCD was derived in Ref. [@eff-pot]. The value of this function at its maximum determines the pressure. The maximum is determined by a Dyson-Schwinger equation for the diquark condensate, which is identical to the standard gap equation for the color-superconducting gap. It has been solved in the mean-field approximation in Refs. [@rdpdhr2; @schaferwilczek; @miransky]. In the mean-field approximation [@rdpdhrscalar], $$\label{mfa}
\Delta^+(x,y) \sim
\langle \, \psi_{C r}(x) \, \bar{\psi}_r(y)\, \rangle
\,\,\,\, , \,\,\,\,\,
\Delta^-(x,y) \sim
\langle \, \psi_r(x) \, \bar{\psi}_{C r}(y) \, \rangle \,\, .$$ In this work, we are interested in the gluon propagator, and the derivation of the pressure via a Legendre transformation of $\ln {\cal Z}[\Delta^+,\Delta^-]$ is of no concern to us. In the following, we shall therefore continue to consider the partition function in the presence of (external) diquark source terms $\Delta^\pm$.
The diquark source terms in the quark (replica) partition function (\[Zquarks3\]) could in principle be chosen differently for each quark species. This could be made explicit by giving $\Delta^\pm$ a subscript $r$, $\Delta^\pm \rightarrow
\Delta^\pm_r$. However, as we take the limit $M \rightarrow 1$ at the end, it is not necessary to do so, as only $\Delta^\pm_1
\equiv \Delta^\pm$ will survive anyway. In other words, we use the [*same*]{} diquark sources for [*all*]{} quark species.
The next step is to explicitly investigate the fluctuations of the diquark condensate around its expectation value. These fluctuations correspond physically to the Nambu-Goldstone excitations (loosely termed “mesons” in the following) in a color superconductor. As mentioned in the introduction, there are five such mesons in a two-flavor color superconductor, corresponding to the generators of $SU(3)_c$ which are broken in the color-superconducting phase. If the condensate is chosen to point in the (anti-) blue direction in fundamental color space, the broken generators are $T_4, \ldots, T_7$ of the original $SU(3)_c$ group and the particular combination $B \equiv ({\bf 1} + \sqrt{3} T_8)/3$ of generators of the global $U(1)_B$ and local $SU(3)_c$ symmetry [@sw2].
The effective action for the diquark condensate and, consequently, for the meson fields as fluctuations of the diquark condensate, is derived via a Legendre transformation of $\ln {\cal Z}[\Delta^+,\Delta^-]$. In this work, we are concerned with the properties of the gluons and thus refrain from computing this effective action explicitly. Consequently, instead of considering the physical meson fields, we consider the variables in ${\cal Z}[\Delta^+,\Delta^-]$, which correspond to these fields. These are the fluctuations of the diquark source terms $\Delta^\pm$. We choose these fluctuations to be complex phase factors multiplying the magnitude of the source terms,
\[DeltaPhi\] $$\begin{aligned}
\Delta^+(x,y) & = &
{\cal V}^* (x)\, \Phi^+(x,y) \, {\cal V}^\dagger(y) \,\, , \\
\Delta^-(x,y) & = &
{\cal V}(x) \, \Phi^-(x,y) \, {\cal V}^T(y) \,\, ,\end{aligned}$$
where $$\label{phase}
{\cal V}(x) \equiv \exp \left[ i \left( \sum_{a=4}^7 \varphi_a(x) T_a
+ \frac{1}{\sqrt{3}}\, \varphi_8(x) B \right) \right]\,\,.$$ The extra factor $1/\sqrt{3}$ in front of $\varphi_8$ as compared to the treatment in Ref. [@msw] is chosen to simplify the notation in the following.
Although the fields $\varphi_a$ are not the meson fields themselves, but external fields which, after a Legendre transformation of $\ln {\cal Z}[\Delta^+,\Delta^-]$, are replaced by the meson fields, we nevertheless (and somewhat imprecisely) refer to them as meson fields in the following. After having explicitly introduced the fluctuations of the diquark source terms in terms of phase factors, the functions $\Phi^\pm$ are only allowed to fluctuate in magnitude. For the sake of completeness, let us mention that one could again have introduced different fields $\varphi_{a r}$ for each replica $r$, but this is not really necessary, as we shall take the limit $M \rightarrow 1$ at the end of the calculation anyway.
It is advantageous to also subject the quark fields $\psi_r$ to a nonlinear transformation, introducing new fields $\chi_r$ via $$\label{chi}
\psi_r = {\cal V}\, \chi_r
\,\,\,\, , \,\,\,\,\,
\bar{\psi}_r = \bar{\chi}_r\, {\cal V}^\dagger\,\, .$$ Since the meson fields are real-valued and the generators $T_4, \ldots, T_7$ and $B$ are hermitian, the (matrix-valued) operator ${\cal V}$ is unitary, ${\cal V}^{-1} = {\cal V}^\dagger$. Therefore, the measure of the Grassmann integration over quark fields in Eq. (\[Zquarks3\]) remains unchanged. From Eq. (\[chi\]), the charge-conjugate fields transform as $$\psi_{C r} = {\cal V}^* \, \chi_{C r}
\,\,\,\, , \,\,\,\,\,
\bar{\psi}_{C r} = \bar{\chi}_{C r} \, {\cal V}^T\,\, ,$$
The advantage of transforming the quark fields is that this preserves the simple structure of the terms coupling the quark fields to the diquark sources, $$\bar{\psi}_{C r}(x)\, \Delta^+(x,y) \, \psi_r(y)
\equiv \bar{\chi}_{C r}(x)\, \Phi^+(x,y) \, \chi_r(y)
\,\,\,\, , \,\,\,\,\,
\bar{\psi}_r(x)\, \Delta^-(x,y) \, \psi_{C r}(y)
\equiv \bar{\chi}_r(x)\, \Phi^-(x,y) \, \chi_{C r}(y) \,\, .$$ In mean-field approximation, the diquark source terms are proportional to $$\label{mfa2}
\Phi^+(x,y)
\sim \langle \, \chi_{C r}(x) \, \bar{\chi}_r(y)\, \rangle
\,\,\,\, , \,\,\,\,\,
\Phi^-(x,y)
\sim \langle \, \chi_r(x) \, \bar{\chi}_{C r}(y) \, \rangle\,\, .$$
The transformation (\[chi\]) has the following effect on the kinetic terms of the quarks and the term coupling quarks to gluons:
$$\begin{aligned}
\bar{\psi}_r \, \left( i \, \gamma^\mu \partial_\mu
+ \mu \, \gamma_0 + g \, \gamma_\mu \, A^\mu_a T_a \right)\,
\psi_r & = & \bar{\chi}_r \, \left( i\, \gamma^\mu \partial_\mu
+ \mu \, \gamma_0 + \gamma_\mu \, \omega^\mu \right) \, \chi_r \,\, , \\
\bar{\psi}_{C r} \, \left( i \, \gamma^\mu \partial_\mu
- \mu \, \gamma_0 - g \, \gamma_\mu \, A^\mu_a T_a^T \right)\,
\psi_{C r} & = & \bar{\chi}_{C r} \, \left( i\, \gamma^\mu \partial_\mu
- \mu \, \gamma_0 + \gamma_\mu \, \omega^\mu_C \right) \, \chi_{C r} \,\, ,\end{aligned}$$
where
\[Maurer1\] $$\omega^\mu \equiv {\cal V}^\dagger \, \left( i \, \partial^\mu
+ g\, A^\mu_a T_a \right) \, {\cal V}$$ is the $N_c N_f \times N_c N_f$-dimensional Maurer-Cartan one-form introduced in Ref. [@sannino] and $$\omega^\mu_C \equiv {\cal V}^T \, \left( i \, \partial^\mu
- g\, A^\mu_a T_a^T \right) \, {\cal V}^*$$
is its charge-conjugate version. Note that the partial derivative acts only on the phase factors ${\cal V}$ and ${\cal V}^*$ on the right.
Introducing the Nambu-Gor’kov spinors $$X_r \equiv \left( \begin{array}{c}
\chi_r \\
\chi_{C r}
\end{array}
\right) \,\,\, , \,\,\,\,
\bar{X}_r \equiv ( \bar{\chi}_r \, , \, \bar{\chi}_{C r} )$$ and the $2 N_c N_f \times 2 N_c N_f$-dimensional Maurer-Cartan one-form $$\label{Maurer2}
\Omega^\mu(x,y) \equiv -i \, \left( \begin{array}{cc}
\omega^\mu(x) & 0 \\
0 & \omega_C^\mu(x)
\end{array}
\right)\, \delta^{(4)}(x-y) \,\, ,$$ the quark (replica) partition function becomes $${\cal Z}_M[\Omega,\Phi^+,\Phi^-] \equiv
\int \prod_{r=1}^{M/2} {\cal D} \bar{X}_r \, {\cal D} X_r \; \exp
\left\{ \sum_{r=1}^{M/2}
\int_{x,y} \bar{X}_r(x) \,\left [\, {\cal S}^{-1} (x,y)
+ \gamma_\mu \Omega^\mu(x,y) \, \right] \, X_r(y)
\right\} \,\, , \label{Zquarks4}$$ where $${\cal S}^{-1} \equiv
\left( \begin{array}{cc}
[G^+_0]^{-1} & \Phi^- \\
\Phi^+ & [G^-_0]^{-1}
\end{array} \right)\,\, .$$
We are interested in the properties of the gluons, and thus may integrate out the fermion fields. This integration can be performed analytically, with the result $${\cal Z}_M[\Omega,\Phi^+,\Phi^-] \equiv
\left[ \,{\rm det} \left( {\cal S}^{-1} + \gamma_\mu \Omega^\mu
\right) \, \right]^{M/2} \,\, . \label{Zquarks5}$$ The determinant is to be taken over Nambu-Gor’kov, color, flavor, spin, and space-time indices. Finally, letting $M \rightarrow 1$, we obtain the QCD partition function (in the presence of meson, $\varphi_a$, and diquark, $\Phi^\pm$, source fields) $$\label{ZQCD2}
{\cal Z}[\varphi,\Phi^+, \Phi^-] =
\int {\cal D} A \; \exp\left[ S_A
+ \frac{1}{2} \,
{\rm Tr} \ln \left({\cal S}^{-1} + \gamma_\mu \Omega^\mu \right) \,
\right]\,\, .$$ Remembering that $\Omega^\mu$ is linear in $A^\mu_a$, cf. Eq. (\[Maurer2\]) with (\[Maurer1\]), in order to derive the gluon propagator it is sufficient to expand the logarithm to second order in $\Omega^\mu$, $$\begin{aligned}
\frac{1}{2}\,
{\rm Tr} \ln \left({\cal S}^{-1} + \gamma_\mu \Omega^\mu \right) \,
& \simeq & \frac{1}{2}\,{\rm Tr} \ln {\cal S}^{-1}
+ \frac{1}{2}\,{\rm Tr} \left( {\cal S}\, \gamma_\mu \Omega^\mu \right)
- \frac{1}{4} {\rm Tr} \left(
{\cal S} \, \gamma_\mu \Omega^\mu \, {\cal S} \, \gamma_\nu
\Omega^\nu \right) \nonumber \\
& \equiv & S_0[\Phi^+,\Phi^-] + S_1[\Omega,\Phi^+,\Phi^-] +
S_2[\Omega,\Phi^+,\Phi^-]\,\,, \label{expandlog}\end{aligned}$$ with obvious definitions for the $S_i$. The quasiparticle propagator is $${\cal S} \equiv
\left( \begin{array}{cc}
G^+ & \Xi^- \\
\Xi^+ & G^-
\end{array} \right)\,\,,$$ with $$G^\pm = \left\{ [G_0^\pm]^{-1} - \Sigma^\pm \right\}^{-1}
\,\,\,\, , \,\,\,\,\,
\Sigma^\pm = \Phi^\mp \, G_0^\mp \, \Phi^\pm
\,\,\,\, ,\,\,\,\,\,
\Xi^\pm = - G_0^\mp \, \Phi^\pm \, G^\pm\,\, .$$
To make further progress, we now expand $\omega^\mu$ and $\omega_C^\mu $ to linear order in the meson fields,
\[linearomega\] $$\begin{aligned}
\omega^\mu & \simeq & g \, A^\mu_a \, T_a - \sum_{a=4}^7
\left( \partial^\mu \varphi_a \right)\, T_a -
\frac{1}{\sqrt{3}}\, \left(\partial^\mu \varphi_8\right)\, B\,\, , \\
\omega_C^\mu & \simeq & - g \, A^\mu_a \, T_a^T + \sum_{a=4}^7
\left( \partial^\mu \varphi_a \right) \, T_a^T +
\frac{1}{\sqrt{3}}\, \left( \partial^\mu \varphi_8\right) \, B^T\,\, .\end{aligned}$$
The term $S_1$ in Eq. (\[expandlog\]) is simply a tadpole source term for the gluon fields. This term does not affect the gluon propagator, and thus can be ignored in the following.
The quadratic term $S_2$ represents the contribution of a fermion loop to the gluon self-energy. Its computation proceeds by first taking the trace over Nambu-Gor’kov space, $$\begin{aligned}
S_2 & = & -\frac{1}{4} \int_{x,y} {\rm Tr}_{c,f,s} \left[
G^+(x,y) \, \gamma_\mu \omega^\mu(y)\, G^+(y,x) \, \gamma_\nu \omega^\nu(x)
+ G^-(x,y) \, \gamma_\mu \omega_C^\mu(y)\, G^-(y,x) \, \gamma_\nu
\omega_C^\nu(x) \right. \nonumber \\
& & \hspace*{2.1cm} + \left.
\Xi^+(x,y) \, \gamma_\mu \omega^\mu(y)\, \Xi^-(y,x) \, \gamma_\nu
\omega_C^\nu(x)
+ \Xi^-(x,y) \, \gamma_\mu \omega_C^\mu(y)\, \Xi^+(y,x) \, \gamma_\nu
\omega^\nu(x) \right] \,\,. \label{S2}\end{aligned}$$ The remaining trace runs only over color, flavor, and spin indices. Using translational invariance, the propagators and fields are now Fourier-transformed as
$$\begin{aligned}
G^\pm (x,y) & = & \frac{T}{V} \sum_K e^{-i K \cdot (x-y)} \, G^\pm(K)\,\, ,\\
\Xi^\pm (x,y) & = & \frac{T}{V} \sum_K e^{-i K \cdot (x-y)} \,
\Xi^\pm(K)\,\, ,\\
\omega^\mu (x) & = & \sum_P e^{-i P \cdot x} \,
\omega^\mu(P)\,\, ,\\
\omega_C^\mu (x) & = & \sum_P e^{-i P \cdot x} \,
\omega_C^\mu(P)\,\, .\end{aligned}$$
Inserting this into Eq. (\[S2\]), we arrive at Eq. (C16) of Ref. [@msw], which in our notation reads $$\begin{aligned}
S_2 & = & -\frac{1}{4} \sum_{K,P} {\rm Tr}_{c,f,s} \left[
G^+(K) \, \gamma_\mu \omega^\mu(P)\, G^+(K-P) \, \gamma_\nu \omega^\nu(-P)
+ G^-(K) \, \gamma_\mu \omega_C^\mu(P)\, G^-(K-P) \, \gamma_\nu
\omega_C^\nu(-P) \right. \nonumber \\
& & \hspace*{1.1cm} + \left.
\Xi^+(K) \, \gamma_\mu \omega^\mu(P)\, \Xi^-(K-P) \, \gamma_\nu
\omega_C^\nu(-P)
+ \Xi^-(K) \, \gamma_\mu \omega_C^\mu(P)\, \Xi^+(K-P) \, \gamma_\nu
\omega^\nu(-P) \right] \,\,. \end{aligned}$$ The remainder of the calculation is straightforward, but somewhat tedious. First, insert the (Fourier-transform of the) linearized version (\[linearomega\]) for the fields $\omega^\mu$ and $\omega_C^\mu$. This produces a plethora of terms which are second order in the gluon and meson fields, with coefficients that are traces over color, flavor, and spin. Next, perform the color and flavor traces in these coefficients. It turns out that some of them are identically zero, preventing the occurrence of terms which mix gluons of adjoint colors 1, 2, and 3 (the unbroken $SU(2)_c$ subgroup) among themselves and with the other gluon and meson fields. Furthermore, there are no terms mixing the meson fields $\varphi_a,\, a=4, \ldots 7,$ with $\varphi_8$. There are mixed terms between gluons and mesons with adjoint color indices $4, \ldots, 7$, and between the gluon field $A_8^\mu$ and the meson field $\varphi_8$.
Some of the mixed terms (those which mix gluons and mesons of adjoint colors 4 and 5, as well as 6 and 7) can be eliminated via a unitary transformation analogous to the one employed in Ref. [@dhr2f], Eq. (80). Introducing the tensors
$$\begin{aligned}
\Pi^{\mu \nu}_{11} (P) & \equiv & \Pi^{\mu \nu}_{22} (P) \equiv
\Pi^{\mu \nu}_{33} (P) = \frac{g^2}{2} \, \frac{T}{V}
\sum_K {\rm Tr}_{s} \left[ \gamma^\mu \, G^+ (K) \, \gamma^\nu
\, G^+(K-P) + \gamma^\mu \, G^- (K) \, \gamma^\nu
\, G^-(K-P) \right. \nonumber \\
& & \left. \hspace*{5.1cm}
+\, \gamma^\mu \, \Xi^- (K) \, \gamma^\nu
\, \Xi^+(K-P) + \gamma^\mu \, \Xi^+ (K) \, \gamma^\nu
\, \Xi^-(K-P) \right] \,\,, \label{Pi11}\end{aligned}$$
cf. Eq. (78a) of Ref. [@dhr2f], $$\begin{aligned}
\Pi^{\mu \nu}_{44} (P) & \equiv & \Pi^{\mu \nu}_{66} (P)
= \frac{g^2}{2} \, \frac{T}{V}
\sum_K {\rm Tr}_{s} \left[ \gamma^\mu \, G_0^+ (K) \, \gamma^\nu
\, G^+(K-P) + \gamma^\mu \, G^- (K) \, \gamma^\nu
\, G_0^-(K-P) \right]\,\, , \label{Pi44diag} \end{aligned}$$ cf. Eq. (83a) of Ref. [@dhr2f], $$\begin{aligned}
\Pi^{\mu \nu}_{55} (P) & \equiv & \Pi^{\mu \nu}_{77} (P)
= \frac{g^2}{2} \, \frac{T}{V}
\sum_K {\rm Tr}_{s} \left[ \gamma^\mu \, G^+ (K) \, \gamma^\nu
\, G_0^+(K-P) + \gamma^\mu \, G_0^- (K) \, \gamma^\nu
\, G^-(K-P) \right]\,\, . \label{Pi55diag}\end{aligned}$$ cf. Eq. (83b) of Ref. [@dhr2f], as well as $$\begin{aligned}
\Pi^{\mu \nu}_{88} (P) & = & \frac{2}{3} \, \Pi_0^{\mu \nu}(P)
+ \frac{1}{3} \, \tilde{\Pi}^{\mu \nu} (P) \,\, , \label{Pi88}\\
\tilde{\Pi}^{\mu \nu} (P) & = & \frac{g^2}{2} \, \frac{T}{V}
\sum_K {\rm Tr}_{s} \left[ \gamma^\mu \, G^+ (K) \, \gamma^\nu
\, G^+(K-P) + \gamma^\mu \, G^- (K) \, \gamma^\nu
\, G^-(K-P) \right. \nonumber \\
& & \left. \hspace*{1.8cm}
-\, \gamma^\mu \, \Xi^- (K) \, \gamma^\nu
\, \Xi^+(K-P) - \gamma^\mu \, \Xi^+ (K) \, \gamma^\nu
\, \Xi^-(K-P) \right] \,\,, \label{Pitilde}\end{aligned}$$ cf. Eq. (78c) of Ref. [@dhr2f], where $\Pi_0^{\mu \nu}$ is the gluon self-energy in a dense, but normal-conducting system, $$\Pi_0^{\mu \nu} (P) = \frac{g^2}{2}\, \frac{T}{V} \sum_K {\rm Tr}_{s}
\left[\gamma^\mu \, G_0^+ (K)\, \gamma^\nu \,
G_0^+(K-P) + \gamma^\mu \,G_0^-(K)\,\gamma^\nu \,G_0^-(K-P) \right]\,\, ,
\label{Pi0}$$
cf. Eq. (27b) of Ref. [@dhr2f], the final result can be written in the compact form (cf. Eq. (C19) of Ref. [@msw]) $$\label{S2final}
S_2 = - \frac{1}{2} \, \frac{V}{T} \, \sum_P \sum_{a=1}^8
\left[A_\mu^a(-P) - \frac{i}{g}\, P_\mu\, \varphi^a(-P)\right]
\, \Pi^{\mu \nu}_{aa}(P) \,
\left[A_\nu^a(P) + \frac{i}{g}\, P_\nu\, \varphi^a(P)\right] \,\, .$$ In deriving Eq. (\[S2final\]), we have made use of the transversality of the polarization tensor in the normal-conducting phase, $\Pi_0^{\mu \nu}(P) \, P_\nu = P_\mu \, \Pi_0^{\mu \nu}(P)=0$. Note that the tensors $\Pi^{\mu \nu}_{aa}$ for $a= 1, \, 2,$ and 3 are also transverse, but those for $a=4,\ldots,8$ are not. This can be seen explicitly from the expressions given in Ref. [@dhrselfenergy]. The compact notation of Eq. (\[S2final\]) is made possible by the fact that $\varphi^a \equiv 0$ for $a = 1,2,3$, and because we introduced the extra factor $1/\sqrt{3}$ in Eq. (\[phase\]) as compared to Ref. [@msw].
To make further progress, it is advantageous to tensor-decompose $\Pi^{\mu \nu}_{aa}$. Various ways to do this are possible [@msw]; here we follow the notation of Ref. [@LeBellac]. First, define a projector onto the subspace parallel to $P^\mu$, $$\label{E}
{\rm E}^{\mu \nu} = \frac{P^\mu \, P^\nu}{P^2}\,\, .$$ Then choose a vector orthogonal to $P^\mu$, for instance $$N^\mu \equiv \left( \frac{p_0\, p^2}{P^2}, \frac{p_0^2\, {\bf p}}{P^2}
\right) \equiv \left(g^{\mu \nu} - {\rm E}^{\mu \nu}\right)\, f_\nu\,\, ,$$ with $f^\mu = (0,{\bf p})$. Note that $N^2 = -p_0^2\,p^2/P^2$. Now define the projectors $$\label{BCA}
{\rm B}^{\mu \nu} = \frac{N^\mu\, N^\nu}{N^2}\,\,\,\, , \,\,\,\,\,
{\rm C}^{\mu \nu} = N^\mu \, P^\nu + P^\mu\, N^\nu \,\,\,\, , \,\,\,\,\,
{\rm A}^{\mu \nu} = g^{\mu \nu} - {\rm B}^{\mu \nu} - {\rm E}^{\mu \nu} \,\, .$$ Using the explicit form of $N^\mu$, one convinces oneself that the tensor ${\rm A}^{\mu \nu}$ projects onto the spatially transverse subspace orthogonal to $P^\mu$, $${\rm A}^{00} = {\rm A}^{0i}=0\,\,\,\, , \,\,\,\,\,
{\rm A}^{ij} = - \left(\delta^{ij} - \hat{p}^i \, \hat{p}^j \right)\,\, .$$ (Reference [@LeBellac] also uses the notation $P_T^{\mu \nu}$ for ${\rm A}^{\mu \nu}$.) Consequently, the tensor ${\rm B}^{\mu \nu}$ projects onto the spatially longitudinal subspace orthogonal to $P^\mu$, $${\rm B}^{00} = - \frac{p^2}{P^2} \,\,\,\, , \,\,\,\,\,
{\rm B}^{0i} = - \frac{p_0\, p^i}{P^2}\,\,\,\, ,\,\,\,\,
{\rm B}^{ij} = - \frac{p_0^2}{P^2}\,\hat{p}^i\,\hat{p}^j\,\, .$$ (Reference [@LeBellac] also employs the notation $P_L^{\mu \nu}$ for ${\rm B}^{\mu \nu}$.) With these tensors, the gluon self-energy can be written in the form $$\label{tensor}
\Pi^{\mu \nu}_{aa}(P) = \Pi^{\rm a}_{aa}(P) \, {\rm A}^{\mu \nu}
+ \Pi^{\rm b}_{aa}(P) \, {\rm B}^{\mu \nu} + \Pi^{\rm c}_{aa}(P)\,
{\rm C}^{\mu \nu} + \Pi^{\rm e}_{aa}(P)\, {\rm E}^{\mu \nu}\,\, .$$ The polarization functions $\Pi^{\rm a}_{aa},\, \Pi^{\rm b}_{aa}, \,
\Pi^{\rm c}_{aa},$ and $\Pi^{\rm e}_{aa}$ can be computed by projecting the tensor $\Pi^{\mu \nu}_{aa}$ onto the respective subspaces of the projectors (\[E\]) and (\[BCA\]). Introducing the abbreviations $$\Pi^t_{aa}(P) \equiv \frac{1}{2} \,
\left( \delta^{ij} - \hat{p}^i\, \hat{p}^j \right) \, \Pi^{ij}_{aa}(P)
\,\,\,\, ,\,\,\,\,\,
\Pi^\ell_{aa}(P) \equiv \hat{p}_i \, \Pi^{ij}_{aa}(P)\, \hat{p}_j \,\, .$$ these functions read
\[Pifunctions\] $$\begin{aligned}
\Pi^{\rm a}_{aa}(P) & = & \frac{1}{2}\, \Pi^{\mu \nu}_{aa}(P)\,
{\rm A}_{\mu \nu} = - \Pi^t_{aa}(P) \,\, , \label{Pia} \\
\Pi^{\rm b}_{aa}(P) & = & \Pi^{\mu \nu}_{aa}(P)\, {\rm B}_{\mu \nu}
= - \frac{p^2}{P^2} \, \left[ \Pi^{00}_{aa}(P)
+ 2\, \frac{p_0}{p}\, \Pi^{0i}_{aa}(P)\,\hat{p}_i
+ \frac{p_0^2}{p^2}\, \Pi^\ell_{aa}(P) \right] \,\, , \\
\Pi^{\rm c}_{aa}(P) & = & \frac{1}{2\, N^2 \, P^2}\, \Pi^{\mu \nu}_{aa}(P)\,
{\rm C}_{\mu \nu}
= -\frac{1}{P^2}\, \left[ \Pi^{00}_{aa}(P)
+ \frac{p_0^2+p^2}{p_0\,p}\, \Pi^{0i}_{aa}(P)\,\hat{p}_i
+ \Pi^\ell_{aa}(P) \right] \,\, , \\
\Pi^{\rm e}_{aa}(P) & = & \Pi^{\mu \nu}_{aa}(P)\, {\rm E}_{\mu \nu}
= \frac{1}{P^2}\, \left[ p_0^2 \, \Pi^{00}_{aa}(P)
+ 2\,p_0\,p \, \Pi^{0i}_{aa}(P) \, \hat{p}_i
+ p^2 \, \Pi^\ell_{aa}(P) \right] \,\, .\end{aligned}$$
For the explicitly transverse tensor $\Pi^{\mu \nu}_{11}$, the functions $\Pi^{\rm c}_{11} = \Pi^{\rm e}_{11} \equiv 0$. The same holds for the HDL polarization tensor $\Pi_0^{\mu \nu}$. For the other gluon colors $a=4, \ldots, 8$, the functions $\Pi^{\rm c}_{aa}$ and $\Pi^{\rm e}_{aa}$ do not vanish. Note that the dimensions of $\Pi^{\rm a}_{aa},\, \Pi^{\rm b}_{aa},$ and $\Pi^{\rm e}_{aa}$ are $[{\rm MeV}^2]$, while $\Pi^{\rm c}_{aa}$ is dimensionless.
Now let us define the functions $$\label{functions}
A_{\perp\, \mu}^a(P) = {{\rm A}_\mu}^\nu\, A^a_\nu(P)
\,\,\,\, , \,\,\,\,\,
A_\parallel^a(P) = \frac{ P^\mu \, A^a_\mu(P)}{P^2}
\,\,\,\, , \,\,\,\,\,
A_N^a(P) = \frac{ N^\mu \, A^a_\mu(P)}{N^2}\,\, .$$ Note that $A_\parallel^a(-P) = - P^\mu \, A^a_\mu(-P)/P^2$, and $A_N^a(-P) = - N^\mu \, A^a_\mu(-P)/N^2$, since $N^\mu$ is odd under $P \rightarrow -P$. The fields $A_\parallel^a(P)$ and $A_N^a(P)$ are dimensionless. With the tensor decomposition (\[tensor\]) and the functions (\[functions\]), Eq. (\[S2final\]) becomes $$\begin{aligned}
S_2 & = & -\frac{1}{2}\, \frac{V}{T} \sum_P \sum_{a=1}^8
\left\{ \frac{}{} A_{\perp\, \mu}^a(-P)\, \Pi^{\rm a}_{aa}(P)\,
{\rm A}^{\mu \nu}\, A_{\perp\,\nu}^a(P)
- A_N^a(-P) \, \Pi^{\rm b}_{aa}(P)\, N^2 \, A_N^a(P)
\right.
\nonumber \\
& & - \left[A_\parallel^a(-P) + \frac{i}{g}\, \varphi^a(-P)\right]\,
\Pi^{\rm c}_{aa}(P) \,N^2 P^2 \, A_N^a(P)
- A_N^a(-P) \, \Pi^{\rm c}_{aa}(P) \,N^2 P^2 \
\left[A_\parallel^a(P) + \frac{i}{g}\, \varphi^a(P)\right]
\nonumber \\
& & - \left. \left[A_\parallel^a(-P) + \frac{i}{g}\, \varphi^a(-P)\right]\,
\Pi^{\rm e}_{aa}(P) \, P^2 \,
\left[A_\parallel^a(P) + \frac{i}{g}\, \varphi^a(P)\right] \right\}\,\,.
\label{decompose}\end{aligned}$$
In any spontaneously broken gauge theory, the excitations of the condensate mix with the gauge fields corresponding to the broken generators of the underlying gauge group. The mixing occurs in the components orthogonal to the spatially transverse degrees of freedom, [*i.e.*]{}, for the spatially longitudinal fields, $A_N^a$, and the fields parallel to $P^\mu$, $A_\parallel^a$. For the two-flavor color superconductor, these components mix with the meson fields for gluon colors $4, \ldots, 8$. The mixing is particularly evident in Eq. (\[decompose\]).
The terms mixing mesons and gauge fields can be eliminated by a suitable choice of gauge. The gauge to accomplish this goal is the ’t Hooft gauge. The “unmixing” procedure of mesons and gauge fields consists of two steps. First, we eliminate the terms in Eq. (\[decompose\]) which mix $A_N^a$ and $A_\parallel^a$. This is achieved by substituting $$\label{sub}
\hat{A}_\parallel^a(P) = A_\parallel^a(P) +
\frac{\Pi^{\rm c}_{aa}(P)\, N^2}{\Pi^{\rm e}_{aa}(P)} \, A_N^a(P)\,\, .$$ (We do not perform this substitution for $a=1,2,3$; for these gluon colors $\Pi^{\rm c}_{aa}\equiv 0$, such that there are no terms in Eq. (\[decompose\]) which mix $A_\parallel^a$ and $A_N^a$). This shift of the gauge field component $A_\parallel^a$ is completely innocuous for the following reasons. First, the Jacobian $\partial(\hat{A}_\parallel, A_N)/\partial(A_\parallel, A_N)$ is unity, so the measure of the functional integral over gauge fields is not affected. Second, the only other term in the gauge field action, which is quadratic in the gauge fields and thus relevant for the derivation of the gluon propagator, is the free field action $$\label{S0}
S_{F^2}^{(0)}
\equiv - \frac{1}{2} \, \frac{V}{T} \sum_P \sum_{a=1}^8 A_\mu^a(-P) \,
\left(P^2\, g^{\mu \nu} - P^{\mu}\, P^{\nu} \right) \, A_\nu^a(P)
\equiv - \frac{1}{2} \, \frac{V}{T} \sum_P \sum_{a=1}^8 A_\mu^a(-P) \, P^2 \,
\left({\rm A}^{\mu \nu} + {\rm B}^{\mu \nu} \right) \, A_\nu^a(P)\,\, ,$$ and it does not contain the parallel components $A_\parallel^a(P)$. It is therefore also not affected by the shift of variables (\[sub\]).
After renaming $\hat{A}_\parallel^a \rightarrow A_\parallel^a$, the final result for $S_2$ reads: $$\begin{aligned}
S_2 & = & -\frac{1}{2}\, \frac{V}{T} \sum_P \sum_{a=1}^8
\left\{ \frac{}{} A_{\perp\, \mu}^a(-P)\, \Pi^{\rm a}_{aa}(P)\, A^{\mu \nu}\,
A_{\perp\,\nu}^a(P)
- A_N^a(-P) \,\hat{\Pi}^{\rm b}_{aa}(P) \, N^2 \, A_N^a(P) \right.
\nonumber \\
& & \hspace*{1.85cm}
- \left. \left[A_\parallel^a(-P) + \frac{i}{g}\, \varphi^a(-P)\right]\,
\Pi^{\rm e}_{aa}(P) \, P^2 \,
\left[A_\parallel^a(P) + \frac{i}{g}\, \varphi^a(P)\right] \right\}\,\,,
\label{S2finalunmix}\end{aligned}$$ where we introduced $$\label{hatPib}
\hat{\Pi}^{\rm b}_{aa}(P) \equiv \Pi^{\rm b}_{aa}(P)
- \frac{\left[\Pi^{\rm c}_{aa}(P)\right]^2 N^2 P^2}{\Pi^{\rm e}_{aa}(P)} \,\, .$$ The ’t Hooft gauge fixing term is now chosen to eliminate the mixing between $A_\parallel^a$ and $\varphi^a$: $$S_{\rm gf} = \frac{1}{2 \, \lambda} \,\frac{V}{T}
\sum_P \sum_{a=1}^8
\left[ P^2\, A_\parallel^a(-P) -
\lambda\, \frac{i}{g} \, \Pi^{\rm e}_{aa}(P)\, \varphi^a(-P) \right] \,
\left[ P^2\, A_\parallel^a(P) -
\lambda\, \frac{i}{g}\, \Pi^{\rm e}_{aa}(P)\, \varphi^a(P) \right] \,\, .
\label{L_gf}$$ This gauge condition is non-local in coordinate space, which seems peculiar, but poses no problem in momentum space. Note that $P^2\, A_\parallel^a(P) \equiv P^\mu \, A_\mu^a(P)$. Therefore, in various limits the choice of gauge (\[L\_gf\]) corresponds to covariant gauge, $$S_{\rm cg} = \frac{1}{2 \, \lambda} \,\frac{V}{T}
\sum_P \sum_{a=1}^8 A_\mu^a(-P)\, P^\mu\, P^\nu \, A_\nu^a(P) \,\, .
\label{L_cg}$$ The first limit we consider is $T, \mu \rightarrow 0$, [*i.e.*]{} the vacuum. Then, $\Pi^{\rm e}_{aa} \equiv 0$, and Eq. (\[L\_gf\]) becomes (\[L\_cg\]). The second case is the limit of large 4-momenta, $P \rightarrow \infty$. As shown in Ref. [@dhrselfenergy], in this region of phase space the effects from a color-superconducting condensate on the gluon polarization tensor are negligible. In other words, the gluon polarization tensor approaches the HDL limit. The physical reason is that gluons with large momenta do not see quark Cooper pairs as composite objects, but resolve the individual color charges inside the pair. Consequently, $\Pi^{\rm e}_{aa}(P) \, P^2 \rightarrow P_\mu \, \Pi^{\mu \nu}_0(P)\,
P_\nu \equiv 0$ for $P \rightarrow \infty$ and, for large $P$, the individual terms in the sum over $P$ in Eqs. (\[L\_gf\]) and (\[L\_cg\]) agree. Finally, for gluon colors $a=1,2,3$, $\Pi^{\rm e}_{aa} \equiv 0$, since the self-energy $\Pi^{\mu \nu}_{11}$ is transverse. Thus, for $a=1,2,3$ the terms in Eqs. (\[L\_gf\]) and (\[L\_cg\]) are identical.
The decoupling of mesons and gluon degrees of freedom becomes obvious once we add (\[L\_gf\]) to (\[S2finalunmix\]) and (\[S0\]), $$\begin{aligned}
S_{F^2}^{(0)} + S_2 + S_{\rm gf}
& = & -\frac{1}{2} \, \frac{V}{T} \sum_P \sum_{a=1}^8
\left\{ \frac{}{} A_{\perp\, \mu}^a(-P)\,
\left[ P^2 + \Pi^{\rm a}_{aa}(P) \right]\,
{\rm A}^{\mu \nu}\, A_{\perp\,\nu}^a(P) \right. \nonumber \\
& & \hspace*{2.1cm} - \; A_N^a(-P) \, \left[ P^2 + \hat{\Pi}^{\rm b}_{aa}(P)
\right] \, N^2 \, A_N^a(P) \nonumber \\
& & \hspace*{2.1cm} - \;
A_\parallel^a(-P) \, \left[ \frac{1}{\lambda}\, P^2
+ \Pi^{\rm e}_{aa}(P) \right]\, P^2 \, A_\parallel^a(P) \nonumber \\
& & \hspace*{2.1cm} + \left. \frac{\lambda}{g^2}\, \varphi^a(-P)\,
\left[ \frac{1}{\lambda}\, P^2 + \Pi^{\rm e}_{aa}(P) \right] \,
\Pi^{\rm e}_{aa}(P)\, \varphi^a(P) \right\} \,\, .
\label{SgfS2}\end{aligned}$$ Consequently, the inverse gluon propagator is $${\Delta^{-1}}^{\mu \nu}_{aa}(P) =
\left[ P^2 + \Pi^{\rm a}_{aa}(P) \right]\, {\rm A}^{\mu \nu}
+ \left[ P^2 + \hat{\Pi}^{\rm b}_{aa}(P) \right] \, {\rm B}^{\mu \nu} +
\left[ \frac{1}{\lambda}\, P^2 + \Pi^{\rm e}_{aa}(P) \right] \,
{\rm E}^{\mu \nu}\,\, .$$ Inverting this as discussed in Ref. [@LeBellac], one obtains the gluon propagator for gluons of color $a$, $$\label{glueprop}
\Delta^{\mu \nu}_{aa}(P) = \frac{1}{P^2 + \Pi^{\rm a}_{aa}(P)}\,
{\rm A}^{\mu \nu} +
\frac{1}{P^2 + \hat{\Pi}^{\rm b}_{aa}(P)}\, {\rm B}^{\mu \nu}
+ \frac{\lambda}{P^2 + \lambda\, \Pi^{\rm e}_{aa}(P)} \,
{\rm E}^{\mu \nu}\,\, .$$ For any $\lambda \neq 0$, the gluon propagator contains unphysical contributions parallel to $P^\mu$, which have to be cancelled by the corresponding Faddeev-Popov ghosts when computing physical observables. Only for $\lambda = 0$ these contributions vanish and the gluon propagator is explicitly transverse, [*i.e.*]{}, $P_\mu\, \Delta^{\mu \nu}_{aa}(P) =
\Delta^{\mu \nu}_{aa}(P)\,P_\nu = 0$. Also, in this case the ghost propagator is independent of the chemical potential $\mu$. The contribution of Fadeev-Popov ghosts to the gluon polarization tensor is then $\sim g^2\,T^2$ and thus negligible at $T=0$. We shall therefore focus on this particular choice for the gauge parameter in the following. Note that for $\lambda = 0$, the inverse meson field propagator is $$\label{NGbosons}
D^{-1}_{aa}(P) \equiv \Pi^{\rm e}_{aa}(P)\, P^2
= P_\mu \, \Pi^{\mu \nu}_{aa}(P)\, P_\nu \,\, ,$$ and the dispersion relation for the mesons follows from the condition $D^{-1}_{aa}(P)=0$, as demonstrated in Ref. [@zarembo] for a three-flavor color superconductor in the color-flavor-locked phase.
The gluon propagator for transverse and longitudinal modes can now be read off Eq. (\[glueprop\]) as coefficients of the corresponding tensors ${\rm A}^{\mu \nu}$ (the projector onto the spatially transverse subspace orthogonal to $P^\mu$) and ${\rm B}^{\mu \nu}$ (the projector onto the spatially longitudinal subspace orthogonal to $P^\mu$). For the transverse modes one has [@LeBellac] $$\label{transverse}
\Delta^t_{aa}(P) \equiv \frac{1}{P^2 + \Pi^{\rm a}_{aa}(P)}
= \frac{1}{P^2 - \Pi^t_{aa}(P)}\,\, ,$$ where we used Eq. (\[Pia\]). Multiplying the coefficient of ${\rm B}^{\mu \nu}$ in Eq. (\[glueprop\]) with the standard factor $-P^2/p^2$ [@LeBellac], one obtains for the longitudinal modes $$\label{longitudinal}
\hat{\Delta}^{00}_{aa}(P) \equiv - \frac{P^2}{p^2}
\, \frac{1}{P^2 + \hat{\Pi}^{\rm b}_{aa}(P)}
= - \frac{1}{p^2 - \hat{\Pi}^{00}_{aa}(P)}\,\, ,$$ where the longitudinal gluon self-energy $$\label{hatPi00aa}
\hat{\Pi}^{00}_{aa}(P) \equiv p^2 \,
\frac{\Pi^{00}_{aa}(P)\,\Pi^\ell_{aa}(P) -
\left[ \Pi^{0i}_{aa}(P) \, \hat{p}_i \right]^2 }{
p_0^2 \, \Pi^{00}_{aa}(P) + 2\, p_0\,p\, \Pi^{0i}_{aa}(P) \, \hat{p}_i
+ p^2 \, \Pi^\ell_{aa}(P) }$$ follows from the definition of $\hat{\Pi}^{\rm b}_{aa}$, Eq.(\[hatPib\]), and the relations (\[Pifunctions\]). The longitudinal gluon propagator $\hat{\Delta}^{00}_{aa}$ [*must not be confused*]{} with the the $00$-component of $\Delta^{\mu \nu}_{aa}$. We deliberately use this (slightly ambiguous) notation to facilitate the comparison of our new and correct results with those of Ref. [@dhrselfenergy], which were partially incorrect. The results of that paper were derived in Coulomb gauge, where the $00$-component of the propagator is indeed [*identical*]{} to the longitudinal propagator (\[longitudinal\]). We were not able to find a ’t Hooft gauge that converged to the Coulomb gauge in the various limits discussed above, and consequently had to base our discussion on the covariant gauge (\[L\_cg\]) as limiting case of Eq. (\[L\_gf\]).
To summarize this section, we have computed the gluon propagator for gluons in a two-flavor color superconductor. Due to condensation of quark Cooper pairs, the $SU(3)_c$ gauge symmetry is spontaneously broken to $SU(2)_c$, leading to the appearance of five Nambu-Goldstone bosons. In general, these bosons mix with some components of the gauge fields corresponding to the broken generators. To “unmix” them we have used a form of ’t Hooft gauge which smoothly converges to covariant gauge in the vacuum, as well as for large gluon momenta, and when the gluon polarization tensor is explicitly transverse. Finally, choosing the gauge fixing parameter $\lambda=0$ we derived the gluon propagator for transverse, Eq. (\[transverse\]), and longitudinal modes, Eq. (\[longitudinal\]) with (\[hatPi00aa\]).
Spectral properties of the eighth gluon {#III}
=======================================
In this section, we explicitly compute the spectral properties of the eighth gluon. We shall confirm the results of Ref. [@dhrselfenergy] for the transverse mode and amend those for the longitudinal mode, which have not been correctly computed in Ref. [@dhrselfenergy]. In particular, we shall show that the plasmon dispersion relation now has the correct behavior $p_0 \rightarrow m_g$ as $p \rightarrow 0$. Furthermore, the longitudinal spectral density vanishes for gluon energies and momenta located on the dispersion branch of the Nambu-Goldstone bosons, [*i.e.*]{}, for energies and momenta given by the roots of Eq. (\[NGbosons\]). For the eighth gluon, this condition can be written in the form $P_\mu\, \tilde{\Pi}^{\mu \nu}(P)\, P_\nu = 0$ [@gusyshov; @zarembo], since the HDL self-energy is transverse, $P_\mu\, \Pi^{\mu \nu}_0(P)\, P_\nu \equiv 0$.
Polarization tensor
-------------------
We first compute the polarization tensor for the transverse and longitudinal components of the eighth gluon. To this end, it is convenient to rewrite the longitudinal gluon self-energy (\[hatPi00aa\]) in the form $$\begin{aligned}
\label{Pi0088}
\hat{\Pi}^{00}_{88}(P) & \equiv &
\frac{2}{3} \, \Pi_0^{00}(P) + \frac{1}{3}\, \hat{\Pi}^{00}(P) \,\, , \\
\hat{\Pi}^{00}(P) & \equiv & p^2 \,
\frac{\tilde{\Pi}^{00}(P)\,\tilde{\Pi}^\ell(P) -
\left[ \tilde{\Pi}^{0i}(P) \, \hat{p}_i \right]^2 }{
p_0^2 \, \tilde{\Pi}^{00}(P) + 2\, p_0\,p\, \tilde{\Pi}^{0i}(P) \, \hat{p}_i
+ p^2 \, \tilde{\Pi}^\ell(P) } \,\, , \label{hatPi00}\end{aligned}$$ with $\tilde{\Pi}^\ell (P) \equiv \hat{p}_i \, \tilde{\Pi}^{ij}(P)\,
\hat{p}_j$.
Let us now explicitly compute the polarization functions. As in Ref. [@dhrselfenergy] we take $T=0$, and we shall use the identity $$\label{ident}
\frac{1}{x+i \eta} \equiv {\cal P}\, \frac{1}{x} - i \pi \, \delta(x)\,\, ,$$ where ${\cal P}$ stands for the principal value description, in order to decompose the polarization tensor into real and imaginary parts. The imaginary parts can then be straightforwardly computed, while the real parts are computed from the dispersion integral
$$\label{dispint}
{\rm Re} \, \Pi(p_0,{\bf p}) \equiv \frac{1}{\pi} \, {\cal P}
\int_{- \infty}^{\infty} d \omega\, \frac{{\rm Im}\, \Pi(\omega,{\bf p})}{
\omega - p_0} + C\,\, ,$$
where $C$ is a (subtraction) constant. If ${\rm Im}\, \Pi(\omega, {\bf p})$ is an odd function of $\omega$, ${\rm Im}\, \Pi(-\omega, {\bf p}) = - {\rm Im}\, \Pi(\omega, {\bf p})$, Eq. (\[dispint\]) becomes Eq. (39) of Ref. [@dhrselfenergy], $$\label{odd}
{\rm Re} \, \Pi(p_0,{\bf p}) \equiv \frac{1}{\pi} \, {\cal P}
\int_0^{\infty} d \omega\, {\rm Im}\, \Pi_{\rm odd}(\omega,{\bf p}) \,
\left(\frac{1}{\omega+p_0} + \frac{1}{\omega - p_0} \right) + C\,\, ,$$ and if it is an even function of $\omega$, ${\rm Im}\, \Pi(-\omega, {\bf p}) = {\rm Im}\, \Pi(\omega, {\bf p})$, we have instead $$\label{even}
{\rm Re} \, \Pi(p_0,{\bf p}) \equiv \frac{1}{\pi} \, {\cal P}
\int_0^{\infty} d \omega\, {\rm Im}\, \Pi_{\rm even}(\omega,{\bf p}) \,
\left(\frac{1}{\omega-p_0} - \frac{1}{\omega + p_0} \right) + C\,\, ,$$
Since the polarization tensor for the transverse gluon modes, $\Pi^t_{88} \equiv \frac{2}{3}\, \Pi_0^t + \frac{1}{3}\, \tilde{\Pi}^t$, has already been computed in Ref. [@dhrselfenergy], we just cite the results. The imaginary part of the transverse HDL polarization function reads (cf. Eq. (22b) of Ref. [@dhrselfenergy])
$${\rm Im}\, \Pi^t_0(P) = - \pi\, \frac{3}{4}\, m_g^2 \,
\frac{p_0}{p}\, \left(1- \frac{p_0^2}{p^2} \right) \, \theta(p-p_0)\,\, .$$
The corresponding real part is computed from Eq. (\[odd\]), with the result (cf. Eqs. (40b) and (41) of Ref. [@dhrselfenergy]) $${\rm Re}\, \Pi^t_0(P) = \frac{3}{2}\, m_g^2\, \left[
\frac{p_0^2}{p^2} + \left( 1- \frac{p_0^2}{p^2} \right) \,
\frac{p_0}{2\, p} \, \ln \left| \frac{p_0 + p}{p_0 - p} \right| \,
\right]\,\, .$$
We have used the fact that the value of the subtraction constant is $C^t_0=m_g^2$, which can be derived from comparing a direct calculation of ${\rm Re}\, \Pi^t_0$ using Eq. (19b) of Ref. [@dhrselfenergy] with the above computation via the dispersion formula (\[odd\]).
The imaginary part of the tensor $\tilde{\Pi}^t$ is given by (cf. Eq. (36) of Ref. [@dhrselfenergy]) $$\begin{aligned}
\lefteqn{{\rm Im}\, \tilde{\Pi}^t(P) =
- \pi\, \frac{3}{4}\, m_g^2 \, \theta(p_0 - 2\, \phi)\, \frac{p_0}{p}
\left\{ \frac{}{}
\theta(E_p - p_0) \, \left[ \left( 1 - \frac{p_0^2}{p^2}\, (1+s^2)
\right) \, {\bf E}(t) - s^2 \,\left( 1- 2\, \frac{p_0^2}{p^2} \right) \,
{\bf K}(t) \right]
\right.} \nonumber \\
& + & \left. \theta(p_0 - E_p) \left[ \left( 1 - \frac{p_0^2}{p^2}\, (1+s^2)
\right) \, E(\alpha,t) - \left( 1- \frac{p_0^2}{p^2} \right)\,
\frac{p}{p_0}\, \sqrt{1 - \frac{4\, \phi^2}{p_0^2 - p^2}}
- s^2 \,\left( 1- 2\, \frac{p_0^2}{p^2} \right)\, F(\alpha,t) \right]
\right\}\,\, ,\end{aligned}$$ where $\phi$ is the value of the color-superconducting gap, $E_p = \sqrt{p^2 + 4 \phi^2}$, $t = \sqrt{1-4\phi^2/p_0^2}$, $s^2 = 1 - t^2$, $\alpha = \arcsin [p/(t p_0)]$, and $F(\alpha,t)$, $E(\alpha,t)$ are elliptic integrals of the first and second kind, while ${\bf K}(t) \equiv F( \pi/2, t)$ and ${\bf E}(t) \equiv E( \pi/2,t)$ are the corresponding complete elliptic integrals. The real part is again computed from Eq. (\[odd\]). The integral has to be done numerically, see Appendix A of Ref. [@dhrselfenergy] for details. The subtraction constant is, for reasons discussed at length in Ref. [@dhrselfenergy], identical to the one in the HDL limit, $C^t \equiv C^t_0 = m_g^2$. Finally, taking the linear combination $\Pi^t_{88} \equiv \frac{2}{3}\, \Pi_0^t + \frac{1}{3}\, \tilde{\Pi}^t$ completes the calculation of the transverse polarization function $\Pi^t_{88}$.
In order to compute the polarization function for the longitudinal gluon, $\hat{\Pi}^{00}_{88}$, we have to know the functions $\Pi_0^{00}(P)$, $\tilde{\Pi}^{00}(P)$, $\tilde{\Pi}^{0i}(P)\, \hat{p}_i$, and $\tilde{\Pi}^\ell(P)$. The first two functions, $\Pi_0^{00}(P)$ and $\tilde{\Pi}^{00}(P)$ have also been computed in Ref. [@dhrselfenergy]. The imaginary part of the longitudinal HDL polarization function is (cf. Eq. (22a) of Ref. [@dhrselfenergy])
$${\rm Im}\, \Pi_0^{00}(P) = - \pi\, \frac{3}{2}\, m_g^2 \,
\frac{p_0}{p} \, \theta(p-p_0)\,\, .$$
The real part is computed from Eq. (\[odd\]), with the result (cf. Eqs. (40a) and (41) of Ref. [@dhrselfenergy]) $${\rm Re}\, \Pi^{00}_0(P) = - 3\, m_g^2\, \left( 1-
\frac{p_0}{2\, p} \, \ln \left| \frac{p_0 + p}{p_0 - p} \right| \,
\right)\,\, . \label{RePi000}$$
Here, the subtraction constant is $C^{00}_0 = 0$.
The imaginary part of the function $\tilde{\Pi}^{00}$ is (cf. Eq. (35) of Ref. [@dhrselfenergy]) $${\rm Im}\, \tilde{\Pi}^{00}(P) =
- \pi\, \frac{3}{2}\, m_g^2 \, \theta(p_0 - 2\, \phi)\, \frac{p_0}{p}
\left\{ \frac{}{}
\theta(E_p - p_0) \, {\bf E}(t) +
\theta(p_0 - E_p) \left[ E(\alpha,t) -
\frac{p}{p_0}\, \sqrt{1 - \frac{4\, \phi^2}{p_0^2 - p^2}}
\right] \right\}\,\, .$$ The real part is computed from Eq. (\[odd\]), with the subtraction constant $C^{00} \equiv C^{00}_0 = 0$. Again, the integral has to be done numerically.
It remains to compute the functions $\tilde{\Pi}^{0i}(P)\, \hat{p}_i$ and $\tilde{\Pi}^\ell(P)$. First, one performs the spin traces in Eq. (\[Pitilde\]) to obtain Eqs. (102b) and (102c) of Ref. [@dhr2f]. Then, taking $T=0$,
\[Pi0iPil\] $$\begin{aligned}
\tilde{\Pi}^{0i}(P)\, \hat{p}_i & = &
\frac{g^2}{2} \int \frac{d^3 {\bf k}}{(2 \pi)^3} \,
\sum_{e_1,e_2= \pm} \left( e_1\, \hat{\bf k}_1 \cdot {\bf p}
+ e_2\, \hat{\bf k}_2 \cdot {\bf p} \right)\,
\left( \frac{\xi_2}{2 \epsilon_2} - \frac{\xi_1}{2 \epsilon_1} \right)
\nonumber \\
& & \hspace*{2.6cm} \times \,
\left( \frac{1}{p_0 + \epsilon_1 + \epsilon_2 + i \eta}
+ \frac{1}{p_0 - \epsilon_1 - \epsilon_2+ i \eta} \right) \,\, , \\
\tilde{\Pi}^\ell(P) & = &
- \frac{g^2}{2} \int \frac{d^3 {\bf k}}{(2 \pi)^3} \,
\sum_{e_1,e_2= \pm} \left[ \left(1- e_1e_2\, \hat{\bf k}_1 \cdot {\bf k}_2
\right) + 2\, e_1 e_2\, \hat{\bf k}_1 \cdot {\bf p}\;
\hat{\bf k}_2 \cdot {\bf p} \right]\,
\frac{\epsilon_1 \epsilon_2 - \xi_1 \xi_2 - \phi_1 \phi_2}{2 \,
\epsilon_1 \epsilon_2}
\nonumber \\
& & \hspace*{2.6cm} \times \,
\left( \frac{1}{p_0 + \epsilon_1 + \epsilon_2 + i \eta}
- \frac{1}{p_0 - \epsilon_1 - \epsilon_2+ i \eta} \right) \,\, , \end{aligned}$$
where ${\bf k}_{1,2} = {\bf k} \pm {\bf p}/2$, $\phi_i \equiv \phi^{e_i}_{{\bf k}_i}$ is the gap function for quasiparticles ($e_i = +1$) or quasi-antiparticles ($e_i = -1$) with momentum ${\bf k}_i$, $\xi_i \equiv e_ik_i - \mu$, and $\epsilon_i \equiv \sqrt{\xi_i^2
+ \phi_i^2}$.
One now repeats the steps discussed in detail in Section II.A of Ref. [@dhrselfenergy] to obtain (for $p_0 \geq 0$)
\[Pi0iPil2\] $$\begin{aligned}
{\rm Im}\, \tilde{\Pi}^{0i}(P)\, \hat{p}_i & = &
\pi\, \frac{3}{2}\, m_g^2 \, \theta(p_0 - 2\, \phi)\, \frac{p_0^2}{p^2}
\left\{ \frac{}{}
\theta(E_p - p_0) \, \left[ {\bf E}(t) - s^2 \, {\bf K}(t) \right]
\right. \nonumber \\
& & \hspace*{3.2cm}
+ \left. \theta(p_0 - E_p) \left[ E(\alpha,t) - \frac{p}{p_0}\, \sqrt{1
- \frac{4\, \phi^2}{p_0^2 - p^2}} - s^2 \, F(\alpha,t) \right]
\right\}\,\, , \\
{\rm Im}\, \tilde{\Pi}^\ell(P) & = &
- \pi\, \frac{3}{2}\, m_g^2 \, \theta(p_0 - 2\, \phi)\, \frac{p_0^3}{p^3}
\left\{ \frac{}{}
\theta(E_p - p_0) \, \left[ (1+s^2)\, {\bf E}(t)
- 2\, s^2 \, {\bf K}(t) \right]
\right. \nonumber \\
& & \hspace*{2.5cm}
+ \left. \theta(p_0 - E_p) \left[ (1+s^2)\, E(\alpha,t) -
\frac{p}{p_0}\, \sqrt{1 - \frac{4\, \phi^2}{p_0^2 - p^2}}
- 2\, s^2 \, F(\alpha,t) \right]
\right\}\,\, .\end{aligned}$$
One observes that in the limit $\phi \rightarrow 0$, the functions (\[Pi0iPil2\]) approach the HDL result
$$\begin{aligned}
{\rm Im}\, \Pi_0^{0i}(P)\, \hat{p}_i & = &
\pi\, \frac{3}{2}\, m_g^2 \, \frac{p_0^2}{p^2} \, \theta(p-p_0)\,\, , \\
{\rm Im}\, \Pi_0^\ell(P) & = &
- \pi\, \frac{3}{2}\, m_g^2 \, \frac{p_0^3}{p^3} \, \theta(p-p_0)\,\,.\end{aligned}$$
Applying Eq. (\[ident\]) to Eqs. (\[Pi0iPil\]) we immediately see that the imaginary part of $\tilde{\Pi}^{0i}(P)\, \hat{p}_i$ is [*even*]{}, while that of $\tilde{\Pi}^\ell(P)$ is [*odd*]{}. Thus, in order to compute the real part of $\tilde{\Pi}^{0i}(P)\, \hat{p}_i$, we have to use Eq. (\[even\]), while the real part of $\tilde{\Pi}^\ell(P)$ has to be computed from Eq. (\[odd\]). When implementing the numerical procedure discussed in Appendix A of Ref. [@dhrselfenergy] for the integral in Eq. (\[even\]), one has to modify Eq. (A1) of Ref. [@dhrselfenergy] appropriately.
Finally, one has to determine the values of the subtraction constants $C^{0i}$ and $C^\ell$. We again use the fact that $C^{0i} \equiv C^{0i}_0$ and $C^\ell \equiv C^\ell_0$, where the index “0” refers to the HDL limit. The corresponding constants are determined by first computing ${\rm Re}\, \Pi_0^{0i}(P)\,
\hat{p}_i$ and ${\rm Re}\, \Pi_0^\ell(P)$ from the dispersion formulas (\[odd\]) and (\[even\]). The result of this calculation is then compared to that of a direct computation using, for instance, the result (\[RePi000\]) for ${\rm Re}\, \Pi_0^{00}(P)$ and then inferring ${\rm Re}\, \Pi_0^{0i}(P) \, \hat{p}_i$ and ${\rm Re}\, \Pi_0^\ell(P)$ from the transversality of $\Pi_0^{\mu \nu}$. The result is $C^{0i} \equiv C^{0i}_0 = 0$ and $C^{\ell}\equiv C^\ell_0 = m_g^2$.
At this point, we have determined all functions entering the transverse and longitudinal polarization functions for the eighth gluon. In Fig. \[fig1\] we show the imaginary parts and in Fig. \[fig2\] the real parts, for a fixed gluon momentum $p= 4\, \phi$, as a function of gluon energy $p_0$ (in units of $2\, \phi$). The units for the imaginary parts are $-3 \, m_g^2/2$, and for the real parts $+ 3\, m_g^2/2$. For comparison, in parts (a) and (g) of these figures, we show the results from Ref. [@dhrselfenergy] for the longitudinal and transverse polarization function of the gluon with adjoint color 1. In parts (d), (e), and (f) the functions $\tilde{\Pi}^{00}$, $-\tilde{\Pi}^{0i}\, \hat{p}_i$, and $\tilde{\Pi}^\ell$ are shown. According to Eq. (\[hatPi00\]) these are required to determine $\hat{\Pi}^{00}$, shown in part (b). Using Eq. (\[Pi0088\]), this result is then combined with the HDL polarization function $\Pi^{00}_0$ to compute $\hat{\Pi}^{00}_{88}$, shown in part (c). Finally, the transverse polarization function for gluons of color 8 is shown in part (i). This function is given by the linear combination $\Pi^t_{88} = \frac{2}{3}\, \Pi^t_0 + \frac{1}{3}\, \tilde{\Pi}^t$ of the transverse HDL polarization function $\Pi^t_0$ and the function $\tilde{\Pi}^t$, both of which are shown in part (h). In all figures, the results for the two-flavor color superconductor are drawn as solid lines, while the dotted lines correspond to those in a normal conductor, $\phi \rightarrow 0$ (the HDL limit).
Note that parts (a), (d), (g), (h), and (i) of Figs. \[fig1\] and \[fig2\] agree with parts (a), (b), (d), (e), and (f) of Figs. 2 and 3 of Ref. [@dhrselfenergy]. The new results are parts (e) and (f) of Figs. \[fig1\] and \[fig2\], which are used to determine the functions in parts (b) and (c), the latter showing the correct longitudinal polarization function for the eighth gluon. In Ref. [@dhrselfenergy], this function was not computed correctly, as the effect from the fluctuations of the condensate on the polarization tensor of the gluons was not taken into account.
The singularity around a gluon energy somewhat smaller than $p_0 = 2\, \phi$ visible in Figs. \[fig2\] (b) and (c) seems peculiar. It turns out that it arises due to a zero in the denominator of $\hat{\Pi}^{00}$ in Eq. (\[hatPi00\]), [*i.e.*]{}, when $P_\mu\, \tilde{\Pi}^{\mu \nu}(P)\, P_\nu = 0$. As discussed above, this condition defines the dispersion branch of the Nambu-Goldstone excitations [@zarembo]. Therefore, the singularity is tied to the existence of the Nambu-Goldstone excitations of the diquark condensate.
Spectral densities
------------------
Let us now determine the spectral densities for longitudinal and transverse modes, defined by (cf. Eq. (45) of Ref. [@dhrselfenergy]) $$\rho^{00}_{88}(p_0, {\bf p}) \equiv \frac{1}{\pi}\,
{\rm Im}\, \hat{\Delta}^{00}_{88} (p_0 + i \eta, {\bf p})
\,\,\,\, , \,\,\,\,\,
\rho^t_{88}(p_0, {\bf p}) \equiv \frac{1}{\pi}\,
{\rm Im}\, \Delta^t_{88} (p_0 + i \eta, {\bf p})$$ The longitudinal and transverse spectral densities for gluons of color 8 are shown in Figs. \[fig3\] (c) and (d), for fixed gluon momentum $p = m_g/2$ and $m_g = 8\, \phi$. For comparison, the corresponding spectral densities for gluons of color 1 are shown in parts (a) and (b). Parts (a), (b), and (d) are identical to those of Fig. 6 of Ref. [@dhrselfenergy], part (c) is new and replaces Fig. 6 (c) of Ref. [@dhrselfenergy]. One observes a peak in the spectral density around $p_0 = m_g$. This peak corresponds to the ordinary longitudinal gluon mode (the plasmon) present in a dense (or hot) medium.
Note that the longitudinal spectral density for gluons of color 8 vanishes at an energy somewhat smaller than $p_0 = m_g/4$. The reason is the singularity of the real part of the gluon self-energy seen in Figs. \[fig2\] (b) and (c). The location of this point is where $P_\mu \, \tilde{\Pi}^{\mu \nu}(P)\, P_\nu =0$, [*i.e.*]{}, on the dispersion branch of the Nambu-Goldstone excitations.
Finally, we show in Fig. \[fig4\] the dispersion relations for all excitations, defined by the roots of
\[disprel\] $$p^2 - {\rm Re}\, \hat{\Pi}^{00}_{88}(p_0, {\bf p}) = 0$$ for longitudinal gluons (cf. Eq. (47a) of Ref. [@dhrselfenergy]), and by the roots of $$p_0^2 - p^2 - {\rm Re}\, \Pi^t_{88}(p_0, {\bf p}) = 0$$
for transverse gluons (cf. Eq. (47b) of Ref. [@dhrselfenergy]). Let us mention that not all excitations found via Eqs. (\[disprel\]) correspond to truly stable quasiparticles, [*i.e.*]{}, the imaginary parts of the self-energies do not always vanish along the dispersion curves. Nevertheless, in that case Eqs. (\[disprel\]) can still be used to identify peaks in the spectral densities, which correspond to [*unstable*]{} modes (which decay with a rate proportional to the width of the peak). As long as the width of the peak (the decay rate of the quasiparticles) is small compared to its height, it makes sense to refer to these modes as quasiparticles.
Fig. \[fig4\] corresponds to Fig. 5 of Ref. [@dhrselfenergy]. In fact, part (b) is identical in both figures. Fig. \[fig4\] (a) differs from Fig. 5 (a) of Ref. [@dhrselfenergy], reflecting our new and correct results for the longitudinal gluon self-energy. In Fig. 5 (a) of Ref. [@dhrselfenergy], the dispersion curve for the longitudinal gluon of color 8 was seen to diverge for small gluon momenta. In Ref. [@dhrselfenergy] it was argued that this behavior was due to neglecting the mesonic fluctuations of the diquark condensate. Indeed, properly accounting for these modes, we obtain a reasonable dispersion curve, approaching $p_0 = m_g$ as the momentum goes to zero. In Fig. \[fig4\] (a) we also show the dispersion branch for the Nambu-Goldstone excitations (dash-dotted). This is strictly speaking not given by a root of Eq. (\[disprel\]), but by the singularity of the real part of the longitudinal gluon self-energy. However, because this singularity involves a change of sign, a normal root-finding algorithm applied to Eq. (\[disprel\]) will also locate this singularity. As expected [@zarembo], the dispersion branch is linear, $$p_0 \simeq \frac{1}{\sqrt{3}}\, p \,\, ,$$ for small gluon momenta, and approaches the value $p_0 = 2\, \phi$ for $p \rightarrow \infty$.
Conclusions {#IV}
===========
In cold, dense quark matter with $N_f=2$ massless quark flavors, condensation of quark Cooper pairs spontaneously breaks the $SU(3)_c$ gauge symmetry to $SU(2)_c$. This results in five Nambu-Goldstone excitations which mix with some of the components of the gluon fields corresponding to the broken generators. We have shown how to decouple them by a particular choice of ’t Hooft gauge. The unphysical degrees of freedom in the gluon propagator can be eliminated by fixing the ’t Hooft gauge parameter $\lambda = 0$. In this way, we derived the propagator for transverse and longitudinal gluon modes in a two-flavor color superconductor accounting for the effect of the Nambu-Goldstone excitations.
We then proceeded to explicitly compute the spectral properties of transverse and longitudinal gluons of adjoint color 8. The spectral density of the longitudinal mode now exhibits a well-behaved plasmon branch with the correct low-momentum limit $p_0 \rightarrow m_g$. Moreover, the spectral density vanishes for gluon energies and momenta corresponding to the dispersion relation for Nambu-Goldstone excitations. We have thus amended and corrected previous results presented in Ref. [@dhrselfenergy].
Our results pose one final question: using the correct expression for the longitudinal self-energy of adjoint colors $4,\ldots,8$, do the values of the Debye masses derived in Ref. [@dhr2f] change? The answer is “no”. In the limit $p_0 = 0,\,
p \rightarrow 0$, application of Eqs. (120), (124), and (129) of Ref. [@dhr2f] to Eq. (\[hatPi00aa\]) yields $\hat{\Pi}^{00}_{aa}(0) \equiv \Pi^{00}_{aa}(0)$, and the results of Ref. [@dhr2f] for the Debye masses remain valid.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank G. Carter, D. Diakonov, and R.D. Pisarski for discussions. We thank R.D. Pisarski in particular for a critical reading of the manuscript and for the suggestion to use ’t Hooft gauge to decouple meson and gluon modes. D.H.R. thanks the Nuclear Theory groups at BNL and Columbia University for their hospitality during a visit where part of this work was done. He also gratefully acknowledges continuing access to the computing facilities of Columbia University’s Nuclear Theory group. I.A.S. would like to thank the members of the Institut für Theoretische Physik at the Johann Wolfgang Goethe-Universität for their hospitality, where part of this work was done. The work of I.A.S. was supported by the U.S.Department of Energy Grant No. DE-FG02-87ER40328.
[99]{}
D. Bailin and A. Love, Phys. Rept. [**107**]{}, 325 (1984). M. Alford, K. Rajagopal, and F. Wilczek, Phys. Lett. B [**422**]{}, 247 (1998); R. Rapp, T. Schäfer, E.V. Shuryak, and M. Velkovsky, Phys. Rev. Lett. [**81**]{}, 53 (1998). T. Schäfer and F. Wilczek, Phys. Rev. D [**60**]{}, 074014 (1999). M. Le Bellac, [*Thermal Field Theory*]{} (Cambridge, Cambridge University Press, 1996).
G.W. Carter and D. Diakonov, Nucl. Phys. B [**582**]{}, 571 (2000). D.H. Rischke, Phys. Rev. D [**62**]{}, 034007 (2000).
D.H. Rischke, Phys. Rev. D [**64**]{}, 094003 (2001). V.A. Miransky, I.A. Shovkovy, and L.C.R. Wijewardhana, Phys. Rev. D [**64**]{}, 096002 (2001). V.P. Gusynin and I.A. Shovkovy, Nucl. Phys. A [**700**]{}, 577 (2002). V.A. Miransky, I.A. Shovkovy, and L.C.R. Wijewardhana, Phys. Lett. B [**468**]{}, 270 (1999). R.D. Pisarski and D.H. Rischke, Phys. Rev. D [**61**]{}, 074017 (2000). T. Schäfer and F. Wilczek, Phys. Rev. D [**60**]{}, 114033 (1999). D.K. Hong, V.A. Miransky, I.A. Shovkovy, and L.C.R. Wijewardhana, Phys. Rev. D [**61**]{}, 056001 (2000) \[Erratum-ibid. D [**62**]{}, 059903 (2000)\]. R.D. Pisarski and D.H. Rischke, Phys. Rev. D [**60**]{}, 094013 (1999). R. Casalbuoni, Z. Duan, and F. Sannino, Phys. Rev. D [**62**]{}, 094004 (2000). K. Zarembo, Phys. Rev. D [**62**]{}, 054003 (2000).
[^1]: On leave of absence from Bogolyubov Institute for Theoretical Physics, 252143 Kiev, Ukraine.
| ArXiv |
---
abstract: |
While the individuals chosen for a genome-wide association study (GWAS) may not be closely related to each other, there can be distant (cryptic) relationships that confound the evidence of disease association. These cryptic relationships violate the GWAS assumption regarding the independence of the subjects’ genomes, and failure to account for these relationships results in both false positives and false negatives.
This paper presents a method to correct for these cryptic relationships. We accurately detect distant relationships using an expectation maximization (EM) algorithm for finding the identity coefficients from genotype data with know prior knowledge of relationships. From the identity coefficients we compute the kinship coefficients and employ a kinship-corrected association test.
We assess the accuracy of our EM kinship estimation algorithm. We show that on genomes simulated from a Wright-Fisher pedigree, our method converges quickly and requires only a relatively small number of sites to be accurate. We also demonstrate that our kinship coefficient estimates outperform state-of-the-art covariance-based approaches and PLINK’s kinship estimate.
To assess the kinship-corrected association test, we simulated individuals from deep pedigrees and drew one site to recessively determine the disease status. We applied our EM algorithm to estimate the kinship coefficients and ran a kinship-adjusted association test. Our approach compares favorably with the state-of-the-art and far out-performs a naïve association test.
We advocate use of our method to detect cryptic relationships and for correcting association tests. Not only is our model easy to interpret due to the use of identity states as latent variables, but also inference provides state-of-the-art accuracy.
cryptic relatedness, identity states, kinship coefficients, expectation maximization
author:
- 'Bonnie Kirkpatrick and Alexandre Bouchard-Côté'
bibliography:
- 'pedbib.bib'
title: 'Correcting for Cryptic Relatedness in Genome-Wide Association Studies'
---
Introduction
============
Accuracy in disease association studies is heavily influenced by cryptic relatedness and population substructure, [@Astle2009]. Both false positives and false negatives result from these influences, because of false assumptions of independence between individuals that are actually related. Disease associations are primarily detected using the genome-wide association study (GWAS) which is typically a case-control or cohort association study implemented using a test for correlation between the disease and the genotypes, [@Risch1996]. While GWAS typically assumes independence between the individuals, a growing number of methods are designed to detect relationships in the form of statistical dependence between the genomes of individuals and correct the correlation calculation for these dependencies. This paper presents a novel method for detecting relationships and correcting the association analysis.
Methods exist that correct association test for relationships that are either known or unknown. These methods often use a two-step approach. First, pair-wise relationships in the form of identity states or kinship coefficients must be inferred either from a known pedigree structure, [@thompson1985], or from data, [@Purcell2007; @Browning2011; @Milligan2003; @Sun2014]. Second, the inferred relationships are used to correct the test for association, [@Yu2005; @Thornton2007; @Rakovski2009; @Kang2010; @Cordell2014].
In this work, we focus on inferring pair-wise relationships, for which existing methods can be split into four categories. First, there are exact methods that compute the kinship coefficients from pedigree structures in quadratic time in the number of individuals in the pedigree, [@Kirkpatrick2016a; @Abney2009; @thompson1985; @Karigl81]. Second, there are ways to estimate the probabilities of the condensed identity coefficients, [@Jacquard1972], from data using a maximum likelihood estimator, [@Milligan2003]. Third, there are ways to estimate the probabilities of the outbred condensed identity coefficients, [@Purcell2007; @Browning2011][^1], but such methods work only for outbred pedigrees and fail to capture more complicated relationships, as we see later in this paper. Forth, there has been work on inferring kinship coefficients from admixed individuals, [@Thornton2012]. Our EM method has a running time, for a pair of individuals, that is linear in the number of sites. To estimate a kinship coefficient matrix for $n$ individuals, our method is quadratic in $n$, due to the number of pairs of individual. Our method avoids the assumption of having a known pedigree, taking only the genotypes as an input. We introduce one latent variable for each pair of individuals and each site, which encodes our uncertainty on the identity by descent states. The distribution over these states for each pair of individuals is learned using a simple expectation maximization (EM) algorithm and provides an informative, yet concise and learnable, summary of the relationship of each pair of individuals. We show that this EM algorithm quickly converges to an accurate estimate. We assess the accuracy in two ways. First, we measure the accuracy of the reconstructed latent variables using simulated pedigree data (the true value of the latent variable can be computed efficiently from the held-out pedigree structure). We asses the kinship estimates compared with PLINK and with a covariance-based estimate. Second, we demonstrate that our method can be used to correct GWAS for cryptic relatedness without assuming the knowledge of the pedigree structure.
Background {#sec:background}
==========
A pedigree is an annotated directed acyclic graph where the nodes are diploid individuals and the edges are directed from parent to child (Figure \[fig:ped-ibd\]A). The graph is acyclic, because no individual can be their own ancestor. The pedigree graph is typically annotated with the gender (male or female), affection status, and the genotype status of the individuals. The gender is used to enforce on the graph the marriage constraint that each individual with parents in the graph has at most one parent of each gender. The *founders* are the individuals who do not have parents in the pedigree while *non-founders* are all the individuals having parent(s) in the pedigree graph. The *affection status* indicates whether an individual has a disease or phenotype. The genotype status indicates whether the individual has available genotypes.
The pedigree graph depicts familial relationships between individuals. Each diploid individual inherits one copy of each chromosome from each parent, and at different sites (loci) in the genome, an individual may inherit from different copies of the parent’s chromosome. In representing inheritance, we consider a single site in the genome and the binary inheritance decision made along each parent-child edge in the pedigree graph. We represent this with an *inheritance path* (Figure \[fig:ped-ibd\]B). For a particular genomic site, the nodes of the inheritance path represent the alleles in all the individuals in the pedigree graph (i.e. for individual $a$ the allele nodes are $a_1$ and $a_2$). The edges of the inheritance path graph proceed from a parent allele to a child allele and indicate that the parent allele is copied with fidelity (i.e. without mutation) to the child allele. For each pedigree graph with $n$ edges there are $2^n$ possible inheritance path graphs.
In an inheritance path, if two alleles are in the same connected component, we say those alleles are *identical by descent (IBD)*. For example, the paternal alleles from two siblings are IBD if and only if the two children inherit from the same allele of the father’s two alleles. The two alleles from the same individual whose parents are related to each other may appear in the same connected component due to inbreeding.
When we focus on two individuals, $a$ and $b$, who are genotyped, a natural question is: which subsets of the four alleles of those two individuals are IBD? For a given inheritance path, the *identity state* captures the answer to this question (Figure \[fig:ped-ibd\]C). The identity state is a graph with four nodes, one for each of the four alleles $\{a_1,a_2,b_1,b_2\}$, and with an edge between two alleles if and only if these two alleles are IBD. Therefore, there is an edge between two alleles in the identity state if and only if they appear in the same connected component of the inheritance path graph.
The identity state is related to the genotype data in the following way. Each of these two individuals may be either heterozygous (two different alleles, i.e. $a_1 \ne a_2$) or homozygous (two copies of the same allele, $a_1 = a_2$) at each site assayed. Certainly if alleles $x,y$ appear together in a connected component of the identity state, they are IBD and must be identical alleles. However, if alleles $x,y$ appear in different connected components of the identity state, then they are not IBD and they may or may not be identical alleles.
For any given pedigree, there are 15 possible identity states, [@Jacquard1972]. For a pedigree with $n$ edges and a pair of individuals $a$ and $b$, the identity state is a random variable, $S$, having a corresponding distribution called the *identity coefficients*, $\bbP[S=s]$, which is the fraction of the $2^n$ inheritance paths for the pedigree that exhibit identity state $s$.
![[**Example pedigree, inheritance path, and identity state.**]{} [**A)**]{} An example of a pedigree dag is shown at the left with the edges directed implicitly downwards, the circles representing females, and the boxes representing males. The two individuals with known genotypes are labeled $a$ and $b$. [**B)**]{} An example of an inheritance path graph for this pedigree is shown in the center with the dots being nodes representing alleles. Although the allele nodes are unlabeled, here, for each individual the inheritance path edges are confined to the options permitted by the pedigree graph. Notice that while there are cycles in graph (A), the inheritance path graph, which has alleles (dots) as nodes, has no cycles (such a graph is called a forest, as it is not connected in general). There are three trees in this inheritance path forest. [**C)**]{} An example of an identity state is shown on the right. This identity state is the one produced from the connected components of the inheritance path graph in the center (B).[]{data-label="fig:ped-ibd"}](ped-path-is.eps){width="5in"}
While the identity state describes the possible IBD between two genotyped individuals, the *condensed identity state*, [@Jacquard1972], more directly relates the IBD to the genotype data. In the identity state the two alleles of one individual are distinguishable, however in the genotype, due to our inability to measure which allele is on which chromosome, the alleles are exchangeable. The condensed identity states are a grouping or partition of the identity states such that two identity states appearing in the same group are isomorphic when the labels on the nodes belonging to individual $a$ are permuted and the labels on the nodes belonging to the individual $b$ are permuted (any valid permutation is permitted, including the permutation that does not change the node labels of the individual). According to this definition, there are nine condensed identity states. Let $c_i$ be the number of condensed identity states which contain $i$ distinct identity states. Then the 9 condensed identity states group the 15 identity states into groups $c_1=5, c_2=3, c_3=0, c_4=1$ (see Figure \[fig:identstates\] for drawings of all the identity state graphs grouped into the condensed identity states). Similar to the identity coefficient, each of these condensed identity states has a corresponding *condensed identity coefficient* which is the fraction of the $2^n$ inheritance paths for the pedigree with $n$ edges that exhibit the condensed identity state.
![[**Identity States.**]{} The 15 identity states are grouped so that each row corresponds to one of the 9 condensed identity states. The number of founder alleles for each identity state is listed, along with whether the identity state is outbred. []{data-label="fig:identstates"}](identitystates.eps){width="3in"}
We need one last concept, still defined with respect to a pair of individuals in a pedigree. The *kinship coefficient* for that pair of individuals is defined as the probability of IBD when choosing one allele from each individual uniformly at random. The exact kinship coefficients for every pair of individuals are typically computed from known relationships–a pedigree graph–using a recursive computation, [@Kirkpatrick2016a; @Abney2009; @Karigl81; @thompson1985].
Methods {#sec:methods}
=======
\[em\]
We focus on estimating the identity coefficients. [^2] We do not directly estimate the kinship coefficients from the genotypes, because we have no model by which the kinship coefficients can generate the genotypes. On the other hand, if we know the identity states, their coefficients, and the allele frequencies in the founders, there is a generative model for producing the genotype. Since the kinship coefficients can be expressed as an expectation with respect to the identity coefficients, they contain a degenerate version of the information contained in the identity coefficients and do not provide a convenient generative model for the genotype.
The estimated identity coefficients are then transformed into estimates of the kinship using equations developed by [@Kirkpatrick2011xxxx; @Kirkpatrick2016a], which we repeat for convenience. For an identity state, let $t \in \{aa, ab, bb\}$ denote an *edge type*, for example, $ab$ indicates any edge between the alleles in two different individuals $a$ and $b$ (i.e., and edge between one of the nodes $\{a_1, a_2\}$ and one of the nodes $\{b_1,b_2\}$). Let ${{\mathcal S}}$ be the set of all the identity states for a pedigree and pair of individuals. Let $e(s,t)$ for $s \in {{\mathcal S}}$ be an indicator function that is one if and only if the identity state has an edge of type $t$. Equations (\[eq:phi\_ab\]) and (\[eq:phi\_aa\]) give the kinship in terms of the identity state distribution. $$\label{eq:phi_ab}
\Phi_{a,b} = \sum_{s\in{{\mathcal S}}} \frac{e(s,ab)}{4} \; \bbP[S=s].$$ where $s$ is the identity state. Rather than the kinship coefficient for the diagonal, we consider the inbreeding coefficient (see [@Kirkpatrick2016a] for details of how these quantities are related): $$\label{eq:phi_aa}
\Phi_{a,a} = \sum_{s\in{{\mathcal S}}} e(s,aa) \; \bbP[S=s],$$ where $e(s,aa)$ indicates whether the single possible edge between nodes $a_1$ and $a_2$ exists.
In this section, we first introduce a model with the identity states as latent variables. Second, we discuss our EM algorithm for doing inference with this model. Accuracy discussions appear in the Results section.
Model
-----
Let the identity states for the $m$ sites be represented as $S = (S_1, \dots,S_m)$. Let the pair genotype data be represented as $G = (G_1, \dots,G_m)$ where each $G_j$ is a tuple of two genotypes $G_j=(G^1_j,G^2_j)$, one genotype for each individual of interest. For each genotype, the values $G^i_j$ takes values in $\{0,1,2\}$, the count of the minor allele. The vector $p = (p_1, \dots, p_m)$ contains the founder allele frequencies for each site.
In our model, the likelihood is a function of the allele frequencies and is defined as a marginal probability where the identity states are marginalized out $$\bbP[G=g|p] = \sum_{s\in{{\mathcal S}}} \bbP[G=g,S=s|p].$$ The joint probability of the data and the identity states is defined as the product of independent sites:[^3] $$\begin{aligned}
\bbP[G=g,S=s|p] &=& \prod_j \bbP[G_j=g_j,S_j=s_j|p_j] \\
&=& \prod_j \bbP[G_j=g_j|S_j=s_j,p_j]\; \bbP[S_j=s_j].\end{aligned}$$ Since all of the inheritance paths are drawn from the same pedigree, the distribution on the identity states is the same for all sites: $\bbP[S_j=s] = \bbP[S_{j'}=s]$ for all $s$. We denote the parameters of this shared categorical distribution by $(d_s : s\in{{\mathcal S}})$, i.e. $d_s = \bbP[S_j=s]$.
A key component of our model is the conditional probability $\bbP[G_j=g|S_j=s,p_j]$. To write an expression for this conditional probability, we introduce the *allele assignment* which must be consistent with both the genotype and the identity state. Recall that an identity state $s$ has vertices $V = \{a_1,a_2,b_1,b_2\}$. Let an allele assignment be given by a map $a:V \to \{0,1\}$. We say that an allele assignment $a$ with $1$ as the minor allele is consistent with the genotypes for two individuals $g=(g^1,g^2)$ if and only if $a(a_1)+a(a_2) = g^1$ and $a(b_1)+a(b_2) = g^2$. Indicate this genotype consistency by the indicator function $C$ (i.e., $C(a,g) = 1$, otherwise $C(a,g) = 0$). Further let $CC(s)$ be the partition of $V$ into connected components extracted from the identity state graph of $s$. Then we say that an allele assignment $a$ is legal with respect to the identity state if and only if the function $a$ is constant on each connected component (i.e., for each connected component $c \in CC(s)$, $a(x) = a(y)$ whenever $x,y \in c$). We represent a legal allele assignment with the indicator $L(a,s)=1$ and $L(a,s)=0$ otherwise. Now, let $$A(s,g) = \{a | C(a,g)=1 \textrm{ and } L(a,s)=1\}$$ be the set of legal and consistent assignments of the genotype alleles to the allele nodes of the identity state.
We can now write an expression for the conditional probability $\bbP[G_j=g|S_j=s,p_j]$ which is: $$\bbP[G_j=g|S_j=s,p_j] \propto \sum_{a \in A(s,g)} p_j^{n_0(a)} (1-p_j)^{N_{cc}-n_0(a)},$$ where $A(s,g)$ is the set of legal assignments of the genotype alleles to the allele nodes of the identity state, $N_{cc}$ is the number of connected components in the identity state, $n_0(a)$ is the number of connected components of the identity state that are labeled with the minor allele by assignment $a$.
Inference
---------
For inference, we use the EM algorithm below. This algorithm takes as input an estimate of the allele frequencies $\hat{p}$ together with the genotypes for a pair of individuals and predicts the identity state coefficients for those individuals. First, we will describe a method to estimate $\hat{p}$ from many genotypes, and second, we will give the details of the EM estimator.
For simplicity and computational efficiency, we estimate the allele frequencies $\hat{p}$ using the Laplace estimator based on the genotypes of independent individuals.[^4] Since our simulations produce pairs of related individuals, we consider the genotypes of one individual per simulated pair, but pool the data from many pairs. Later, in our simulations, this same allele frequency estimate $\hat{p}$ is then given to all the methods that infer kinship, ours and others. To estimate the identity state coefficient, we use an EM algorithm that iteratively produces successive estimates $d_s^{(t)}$ of the probability distribution $\bbP[S_j=s]$ from above. These estimates are obtained using simple and efficient update rules. For the [**E-step**]{} we update our estimate of $N^{(t)}_s$ which is the expected number of times that identity state $s$ occurred $$\begin{aligned}
\label{eq:ml-eq}
N^{(t)}_s = \sum_{j} \frac{d_s^{(t-1)} \times \bbP[G_j=g_j|S_j=s,\hat{p}]}{\sum_{s'} d_{s'}^{(t-1)} \times \bbP[G_j=g_j|S_j=s',\hat{p}]}\end{aligned}$$ while the [**M-step**]{} consists of updating our estimate of the identity coefficients $$d_s^{(t)} = \frac{N^{(t)}_s}{m}.$$ Iterative application of the E-step and M-step yields a sequence of estimates: $$(N^{(t)}_s,d_s^{(t)})~~\textrm{for } t=1,2,\dots~.$$ As we show in the Supplement, this algorithm is efficient, requiring a small number of iterations before convergence.
Inferred relationships can be used to correct association tests for cryptic relatedness thereby reducing spurious, or false-positive, associations. This combined algorithm of our estimates of kinship coefficients informing MQLS, we call *pedigree-free MQLS (PFMQLS)*.
Results {#sec:results}
=======
There are two categories of results, estimate accuracy and corrections for spurious associations, and simulations that go along with each category. We simulated pedigree replicates from the Wright-Fisher (WF) model with parameters: $N$, the number of male and the number of female individuals per generation (meaning there are $2N$ individuals per generation), and $G$, the number of generations.
The data was simulated by holding the pedigree fixed, and drawing an inheritance path for each site from the uniform distribution over inheritance paths. For each site and inheritance path, the allelic data was drawn using the parameter, $p$—the vector of minor allele frequencies, one for each site. The founder alleles for site $j$, with possible values in $\{0,1\}$, were each drawn from the Bernoulli distribution with parameter $p_j$. Once the founder alleles were selected, they were copied along the inheritance path without mutation, so that a descendant inherited a copy of a founding allele from their founding ancestor if there was an undirected path in the inheritance path graph between the descendant and the founder allele. This simulated data consisted of haplotypes, since it is known which allele was inherited from each parent.
We discuss accuracy in three ways. Each accuracy measure uses variations on how the genotyped individuals are sampled from the simulation.
Improved Estimation of Kinship Coefficients
-------------------------------------------
To asses the accuracy of the kinship estimates, we simulated pedigree replicates. For each pedigree, we sampled two extant individuals of interest for which to simulate allelic data. Recall that the data simulation provides haplotypes. To obtain the genotypes, the record of each allele’s parent was discarded. The genotype data of the two extant individuals, and not the pedigree, was given to the estimation algorithms.
To estimate the allele frequencies, since they vary from site to site, our method requires a set of individuals all simulated using the same allele frequencies. Therefore, our full simulation produces a number of pairs of individuals, where each pair is simulated from a single pedigree, and all the pedigrees share the same parameter $p$. Recall that we obtain $p$ using the Laplace estimator applied to independent individuals, one individual from each pedigree.
To assess estimation accuracy, we use a gold standard estimates that are computed during pedigree and inheritance path simulation. Once the pedigree has been drawn, we apply the algorithm for computing the exact kinship coefficients and the inbreeding coefficients, [@Kirkpatrick2016a]. Once the inheritance paths have been drawn, we can compute the empirical identity state distribution represented in the data[^5]. These quantities 1) the kinship and inbreeding coefficients and 2) the identity state distribution, are used as the gold-standard for estimates. Table \[table:kinship\] shows the results of two estimation methods: the EM algorithm introduced in the Methods section and the covariance-based kinship coefficient estimator introduced by [@Astle2009]. We found that REAP, which infers kinship from admixed individuals, [@Thornton2012], does not apply to our setting even when the parameters are set for no admixture, as it found unreasonable kinship and inbreeding coefficients.
Comparison with PLINK
---------------------
Essentially the worst case of population structure would be if a sample of individuals was from a family, yet they were thought to be unrelated individuals. Suppose also that the founders of the family are potentially inbred. Even if the pedigree were known, the pedigree relationships would under-represent the inbreeding since some of the inbreeding occurred chronologically before the known relationships.
In order to compare our method to PLINK, [@Purcell2007], which estimates the kinship coefficients, we simulate just such a scenario. We use the pedigree Wright-Fisher simulation to produce the founder haplotypes from an inbred population with $N=8$ individuals per generation and $2,4,6,8,...,40$ generations all with $m=500$ sites. From the most recent generation of the Wright-Fisher pedigree, we draw $4$ founders for an outbred $12$-individual pedigree, see Figure \[fig:pedigree\]. We simulated the recent pedigree genotypes with recombination and considered $6$ of the individuals to have observed genotypes from which we estimated the kinship coefficients.
To compute the accuracy of the kinship estimates, we found the actual kinship coefficients of the $12$ person pedigree. This required using a new method of computing kinship coefficients from known founder kinship coefficients, due to [@Kirkpatrick2016a]. Both methods, ours and PLINK provide estimates of the kinship coefficients. We computed the sum of the absolute value of the differences between the matrix entries of the estimates and the actual kinship coefficients. This sum is the $L_1$ accuracy. In all cases, our method has accuracy far superior to that of PLINK, see Figure \[fig:plink\].
![ [**Outbred Pedigree.**]{} This outbred pedigree was used to simulate genotypes from inbred founder haplotypes. The shaded individuals had genotypes that were typed and used to estimate kinship coefficients with PLINK and the EM method in this paper. []{data-label="fig:pedigree"}](pedigree.pdf){width="3in"}
![ [**Accuracy of Kinship Estimates.**]{} Comparing kinship estimates of PLINK and our EM method using the $L_1$ accuracy demonstrates that our method has superior accuracy to that of PLINK. Our method’s accuracy margin improves as the amount of inbreeding in the founders increases. []{data-label="fig:plink"}](cmpsim-paper.pdf){width="3in"}
Mathematically, we compare the outbred inference method used by PLINK, [@Purcell2007], to our identity state approach which considers both outbred and inbred identity states. Notice that PLINK’s approach is limited to considering the 7 outbred identity states which have a “Yes” in the first column of Figure \[fig:identstates\]. Our approach considers both inbred and outbred identity states—all states shown in Figure \[fig:identstates\]—in a structured learning setting where the identity state for each site is selected along with the frequency for that state.
From the bar plot in Figure \[fig:overfitting\] we can compute the average number of parameters for the outbreeding condensed identity states ($9/3 = 3.00$) and for both inbreeding and outbreeding condensed identity states ($13/9 \simeq 1.44$). This shows that on average, PLINK’s outbred model will over-fit as compared with our method which selects the best model from both inbred and outbred identity states.
![[**Over-fitting.**]{} The number of founder alleles are on the x-axis, and the y-axis shows the count of the number of condensed identity states with the given number of founder alleles. The left, blue bars count only the outbred condensed identity states, while the right, red bars count all the condensed identity states (outbred and inbred). Thus, the red bars are at least the height of the blue bars. From this bar plot we can compute the average number of parameters for the outbreeding condensed identity states ($9/3 = 3.00$) and for both inbreeding and outbreeding condensed identity states ($13/9 \simeq 1.44$).[]{data-label="fig:overfitting"}](overfitting.pdf){width="3in"}
This over-fitting by PLINK is what we see, when PLINK’s $\hat{\pi}$ estimate of kinship nearly recapitulates the kinship computed from an outbred pedigree. This happens because PLINK explains away the excess homozygosity in the data by outbreeding which introduces many independent founder alleles—the founder alleles all appear in distinct connected components of the identity states. On the other hand our approach discovers that outbreeding explains the excess homozygosity less well than inbreeding, because the outbreeding explanation has a lower likelihood than an explanation involving inbreeding. Therefore our approach estimates, from the data, kinship coefficients that deviate from that predicted by the (outbred) pedigree structure, precisely because inbreeding among the founders provides a better model. Indeed, under the setting with a lot of inbreeding among the founders of a pedigree, an outbred-only model like that used by PLINK might have a significantly lower likelihood than a model that allows inbreeding as an explanation for the observed excess homozygosity.
Fewer Spurious Associations
---------------------------
Similar to the simulations in the previous section, we simulated pedigree replicates from the Wright-Fisher (WF) model. Unlike the previous simulations, we sampled $k$ extant individuals and discard the pedigree graph. The genotypes at site $j$ were simulated as before by drawing the founder alleles uniformly at random from the population distribution which is Bernoulli with parameter $p_j$, and then inheriting those alleles along the edges indicated by the inheritance path.The case-control simulation then involves two pedigree replicates which share the same founder allele frequencies. The presence of the two sets of family relationships confounds most association tests and results in very low power.
Each site was independently taken to be the disease site, resulting in $m$ true positive tests per simulated pair of pedigrees. The affection status of each individual was computed from the genotype assuming an almost recessive trait and the minor allele to be the disease allele. Assuming the minor allele is $0$, the penetrance probabilities for the disease given the genotype of person $i$ were $\bbP(D|G^i=0)=0.95$ and $\bbP(D|G^i=1)=\bbP(D|G^i=2)=0.05$. Several association tests were applied to the simulated genotypes of the two pedigrees: the Cochran-Armitage trend test, the ROADTRIPS RM test, [@Thornton2010], and the PFMQLS test which is the MQLS test given the kinship coefficients estimated by the EM algorithm. The simulations where conducted with the following parameters: $N=50$ number of male/females per generation, $G=25$ number of generations, $k=10$ individuals sampled, and $m=400$ sites. We suggest using the Bonferroni threshold.[^6] Overall, we find that the Bonferroni threshold can favor the PFMQLS test, see Supplement.
We summarized the data in a receiver operating characteristic (ROC) plot. The $(x,y)$ points for the ROC plot are the false positive rate (FPR) and true positive rate (TPR) for a particular p-value threshold for the test. By considering multiple thresholds, we can look across all the simulations to find the FPR for all the non-disease site and to find the TPR for all the disease sites, see Figure \[fig:roc\]. While there is a slight difference between the performance of the PFMQLS and ROADTRIPS, the number of simulations suggests that this difference may not be significant. The performance of the Cochran-Armitage trend test (CATT) is very poor due to many false positives. Both PFMQLS and ROADTRIPS avoid the spurious false positives. We conclude that PFMQLS is as good as the state-of-the-art represented by ROADTRIPS while providing an intuitive and interpretable model of relatedness.
![[**ROC plot.**]{} The x-axis is the false positive rate and the y-axis is the true positive rate. There are several tests shown: the Cochran-Armitage trend test (CATT), the pedigree-free MQLS test (PFMQLS), and the ROADTRIPS RM test. The performances of RM and PFMQLS are almost indistinguishable and far superior to that of CATT. []{data-label="fig:roc"}](roc-105.eps){width="3in"}
Conclusion
==========
We present a method for inferring the kinship coefficients that relies on identity states—a more detailed description of the pedigree than used by most kinship inference methods. Our method is an EM algorithm that infers the identity state distribution without assuming a known pedigree. The accuracy of our method depends on the number of sites and is reliable with as few as 64 independent sites as input[^7]. Our results show that our kinship estimates out-perform the covariance kinship method and other recent methods for kinship estimation by a large margin. Our EM kinship estimates can also correct an association test to produce state-of-the-art accuracy. Constructing the kinship matrix with our method requires a pair-wise comparison of individuals’ genomes. Such an approach can be computationally intensive, and future work includes a method avoiding this quadratic cost. A potential drawback of using an EM algorithm is that it finds local optima: our current results each use a single run of EM, and so random restarts could potentially improve the results.
Other areas of future work involve simulating complex diseases which produce the disease trait by interaction of multiple sites in the genome. The results presented here were for a simple nearly recessive disease which probably accounts for the high accuracy of PFMQLS and ROADTRIPS as seen in the area under the curve for the ROC plot. In addition to simulating complex diseases, future work involves tailoring a test to the setting of using kinship corrections to detect epistasis.
Author Disclosure Statement. {#author-disclosure-statement. .unnumbered}
============================
BK is the owner of Intrepid Net Computing.
Supplement: Correcting for Cryptic Relatedness in Genome-Wide Association Studies {#supplement-correcting-for-cryptic-relatedness-in-genome-wide-association-studies .unnumbered}
=================================================================================
Reducing Spurious Associations
------------------------------
Inferred relationships can be used to correct association tests for cryptic relatedness thereby reducing spurious, or false-positive, associations. Notably, the MQLS, [@Thornton2007] test relies on kinship coefficients calculated from a known pedigree to correct for the dependencies caused by relatedness that would confound tests that assume independence, such as the $\chi^2$ test. We propose to reduce spurious associations in data sets having an unknown pedigree by using our EM algorithm for estimating the kinship coefficients for every pair of individuals. Recall that for every pair of individuals, we obtain estimates of the inbreeding coefficients of each individual and the kinship coefficient between them. For each of the $N$ individuals, we will have $N-1$ estimates of the inbreeding coefficient which we average to obtain a single estimate. This leaves us with a matrix of estimates with the off-diagonals being estimates of the kinship coefficients and the diagonal being the estimates of the inbreeding coefficients.
This combined algorithm of our estimates of kinship coefficients informing MQLS, we call *pedigree-free MQLS (PFMQLS)*. While it is possible to tailor-design a test based on these EM kinship coefficients, this approach of running MQLS on our EM results allows us to judge whether the kinship coefficients estimated by the EM algorithm can successfully reduce spurious associations.
Results: Improved Estimation of Kinship Coefficients
----------------------------------------------------
Figure \[fig:sites\] shows the effect of the number of sites on the accuracy of the estimates, both in terms of the kinship coefficient estimates and of the identity state distribution estimate. The simulations were performed exactly as they were for the table (see paper) with the actual allele frequencies generated uniformly and the estimated allele frequencies being obtained empirically from independent individuals in the simulation. The estimated allele frequencies were then used to estimate the identity state distribution and the kinship coefficients. We show that it is possible to use only a few EM iterations to obtain a stable solution, Figure \[fig:l1\].
![[**The number of sites and the accuracy of the estimates.**]{} As the number of sites increase, log-scale on the x-axis, the accuracy, on the y-axis of both the kinship and the identity state estimates, improves. The maximum number of sites shown here is $2^6 = 64$.[]{data-label="fig:sites"}](mse-kinship.eps "fig:"){width="3in"} ![[**The number of sites and the accuracy of the estimates.**]{} As the number of sites increase, log-scale on the x-axis, the accuracy, on the y-axis of both the kinship and the identity state estimates, improves. The maximum number of sites shown here is $2^6 = 64$.[]{data-label="fig:sites"}](mse-is-dist.eps "fig:"){width="3in"}
![[**$L_1$ distances between the EM solutions of successive iterations.**]{} For a single pair of individuals, the figure records the $L_1$ distance between every pair of solutions for successive iterations. As the iterations, x-axis, increase the $L_1$ distances, y-axis, decrease rapidly towards zero.[]{data-label="fig:l1"}](em_convergence.eps){width="3in"}
Results: Fewer Spurious Associations
------------------------------------
We ran a simulation as described in the paper with $m=400$ sites. Figure \[fig:tppval\] shows the true positive p-values while Figure \[fig:fppval\] shows the false positive p-values.[^8] Overall, these figures illustrate that the Bonferroni threshold can favor the PFMQLS test.
![[**True positives.**]{} On the x-axis are the sites, and the y-axis the negative log of the p-value. We show here the negative log p-value for the pedigree-free MQLS test (PFMQLS), the Cochran-Armitage trend test (CATT), and the ROADTRIPS RM test. The horizontal line in the figure gives the Bonferroni-corrected threshold for a site-specific significance of $0.05$. Any spikes that protrude above the line indicate sites that are significant and true positive.[]{data-label="fig:tppval"}](plot-tp-105.pdf){width="5in"}
![[**False positives.**]{} On the x-axis are the sites, and the y-axis the negative log of the p-value. We show here the negative log p-value for the pedigree-free MQLS test (PFMQLS), the Cochran-Armitage trend test (CATT), and the ROADTRIPS RM test. The horizontal line in the figure gives the Bonferroni-corrected threshold for a site-specific significance of $0.05$. Any spikes that protrude above the line indicate sites that are significant and false positive. []{data-label="fig:fppval"}](plot-fp-105.pdf){width="5in"}
[^1]: The probabilities of having zero, one, or two alleles identical-by-descent (IBD) are considered.
[^2]: One could re-express our method in terms of the condensed identity coefficients (since the data do not provide information to distinguish between identity states that fall in the same condensed identity states), but the presentation is simpler in the non-condensed setting.
[^3]: In practice, sites are clearly dependent because of recombination, but we assume sites have been sufficiently subsampled to approximate independence. We show in the next section that our method requires relatively few sites, so this is a reasonable approximation in practice.
[^4]: Smoothing of $\hat{p}$ is required to avoid degeneracies where our method fails due to divisions by zero in Equation (\[eq:ml-eq\]).
[^5]: Unlike the kinship coefficients, the exact algorithm for obtaining the identity state distribution is exponential, so we use a Monte Carlo approximation similar to [@Sun2014ugrad].
[^6]: Our association results are demonstrated with p-values, but our method is equally amenable to the use of false discovery rates and the computation of q-values.
[^7]: From the whole genome of correlated sites, it is feasible to extract many more than 64 independent sites for input to our method.
[^8]: Our association results are demonstrated with p-values, but our method is equally amenable to the use of false discovery rates and the computation of q-values.
| ArXiv |
---
abstract: 'This paper explores the fundamental properties of distributed minimization of a sum of functions with each function only known to one node, and a pre-specified level of node knowledge and computational capacity. We define the optimization information each node receives from its objective function, the neighboring information each node receives from its neighbors, and the computational capacity each node can take advantage of in controlling its state. It is proven that there exist a neighboring information way and a control law that guarantee global optimal consensus if and only if the solution sets of the local objective functions admit a nonempty intersection set for fixed strongly connected graphs. Then we show that for any tolerated error, we can find a control law that guarantees global optimal consensus within this error for fixed, bidirectional, and connected graphs under mild conditions. For time-varying graphs, we show that optimal consensus can always be achieved as long as the graph is uniformly jointly strongly connected and the nonempty intersection condition holds. The results illustrate that nonempty intersection for the local optimal solution sets is a critical condition for successful distributed optimization for a large class of algorithms.'
author:
- 'Guodong Shi, Alexandre Proutiere and Karl Henrik Johansson[^1]'
title: |
**Distributed Optimization: Convergence Conditions\
from a Dynamical System Perspective[^2]**
---
[**Keywords:**]{} Distributed optimization, Dynamical Systems, Multi-agent systems, Optimal consensus
Introduction
============
Motivation
----------
Distributed optimization is on finding a global optimum using local information exchange and cooperative computation over a network. In such problems, there is a global objective function to be minimized, say, and each node in the network can only observe part of the objective. The update dynamics is executed through an update equation implemented in each node of the network, based on the information received from the local objective and the neighbors.
The literature has not to sufficient extent studied the real meaning of “distributed" optimization, or the [*level*]{} of distribution possible for convergence. Some algorithms converge faster than others, while they depend on more information exchange and a more complex iteration rule. For a precise study of the level of distribution for optimization methods, the way nodes share information, and the computational capacity of each node should be specified. Thus, an interesting question arises: fixing the knowledge set and the computational capacity, what is the best performance of any distributed algorithm? In this paper, we investigate the fundamental performance limits of distributed algorithms when the constraints on how nodes exchange information and on their computational capacity are fixed. We address these limits from a dynamical system point of view and characterize some fundamental conditions on the global objective function for a distributed solution to exist.
Related Works
-------------
Distributed optimization is a classical topic in applied mathematics with several excellent textbooks, e.g., [@book1; @book2; @book3].
Assuming that some estimate of the subgradient for each component of the overall objective function can be passed over the network from one node to another via deterministic or randomized iteration, a class of subgradient-based incremental algorithms was investigated in [@solodov; @rabbat; @nedic01; @bjsiam; @ram]. A series of results were established combining consensus and subgradient computation. This idea can be traced back to 1980s to the pioneering work [@tsi]. A subgradient method for fixed undirected topology was given in [@bj08]. Then in [@nedic09], convergence bounds for time-varying graphs with various connectivity assumptions were shown. This work was then extended to a constrained optimization case in [@nedic10], where each agent is assumed to always lie in a particular convex set. Consensus and optimization were shown to be guaranteed when each node makes a projection onto its own set at each step. Following the ideas of [@nedic10], a randomized discrete-time algorithm and a deterministic continuous-time algorithm were presented for optimal consensus in [@shirandom] and [@shitac], respectively, where in both cases the goal is to form a consensus within the intersection of the optimal solution sets of the local objective functions. An augmented Lagrangian algorithm was presented for constrained optimization with directed gossip communication in [@jmf]. An alternative approach was presented in [@lu1], where the nodes keep their gradient sum equal to zero during the iteration by utilizing gossiping.
Dynamical system solutions to distributed optimization problem have been considered for more than fifty years. The Arrow-Hurwicz-Uzawa flow was shown to converge to the set of saddle points for a constrained convex optimization problem [@ahu]. In [@brockett], a simple and elegant continuous-time protocol was presented to solve linear programming problems. More recently, in [@elia], a continuous-time solution having second-order node dynamics was proposed for solving distributed optimization problems for fixed bidirectional graphs. In [@eben], a smooth vector field was shown to be able to drive the system trajectory to converge to the saddle point of the Lagrangian of a convex and constrained optimization problem. In [@shitac], a network of first-order dynamical system was proposed to solve convex intersection computation problems with directed time-varying communication graphs. Besides optimization, a continuous-time interpretation to discrete-time algorithms was discussed for recursive stochastic algorithms in [@ljung].
Consensus algorithms have been proven to be useful in the design of distributed optimization methods [@nedic09; @nedic10; @shirandom; @shitac; @elia; @lu1]. Consensus methods have also been extensively studied for both discrete-time and continuous-time models in the past decade, some references related to the current paper include [@tsi; @jad03; @julien2; @lwang; @lin07; @ren05; @mar; @caoming1; @mor; @nedic08; @shi09; @shi11].
Main Contribution
-----------------
This paper considers the following distributed optimization model. The network consists of $N$ nodes with directed communication. Each node $i$ has a convex objective function $f_i: \mathds{R}^m\rightarrow \mathds{R}$. The goal of the network is to reach consensus meanwhile minimizing the function $\sum_{i=1}^N f_i$. At any time $t$, each node $i$ observes the gradient of $f_i$ at its current state $g_i(t)$ and the neighboring information $n_i(t)$ from its neighbors. The map $n_i(t)$ is zero when the nodes state is equal to all its neighbors’ state. The evolution of the nodes’ states is given by a first-order integrator with right-hand side being a control law $\mathcal{J}(n_i,g_i)$ taking feedback from $g_i(t)$ and $n_i(t)$. We assume $\mathcal{J}(n_i,g_i)$ to be injective in $g_i$ when $n_i$ takes value zero.
The main results we obtain are stated as follows:
- We prove that there exists a neighboring information rule $n_i$ and a control law $\mathcal{J}$ guaranteeing global optimal consensus if and only if the intersection of the solution sets of $f_i,i=1,\dots,N$, is nonempty intersection set for fixed strongly connected graphs.
- We show that given any $\epsilon>0$, there exists a control law $\mathcal{J}$ that guarantees global optimal consensus with error no larger than $\epsilon$ for fixed, bidirectional, and connected graphs under mild conditions.
- We show that optimal consensus can always be achieved for time-varying graphs as long as the graph is uniformly jointly strongly connected and the nonempty intersection condition above holds.
We conclude that the nonempty intersection of the solution sets of the local objectives seems to be a fundamental condition for distributed optimization.
Paper Organization
------------------
In Section 2, some preliminary mathematical concepts and lemmas are introduced. In Section 3, we formulate the considered optimization model, node dynamics, and define the problem of interest. Section 4 focuses on fixed graphs. A necessary and sufficient condition is presented for the exact solution of optimal consensus, and then approximate solutions are investigated as $\epsilon$-optimal consensus. Section 5 is on time-varying graphs, and we show optimal consensus under uniformly jointly strongly connected graphs. Finally, in Section 6 some concluding remarks are given.
Preliminaries
=============
In this section, we introduce some notations and provide preliminary results that will be used in the rest of the paper.
Directed Graphs
---------------
A directed graph (digraph) $\mathcal
{G}=(\mathcal {V}, \mathcal {E})$ consists of a finite set $\mathcal{V}$ of nodes and an arc set $\mathcal {E}$, where an arc is an ordered pair of distinct nodes of $\mathcal {V}$ [@god]. An element $(i,j)\in\mathcal {E}$ describes an arc which leaves $i$ and enters $j$. A [*walk*]{} in $\mathcal
{G}$ is an alternating sequence $\mathcal
{W}:
i_{1}e_{1}i_{2}e_{2}\dots e_{m-1}i_{m}$ of nodes $i_{\kappa}$ and arcs $e_{\kappa}=(i_{\kappa},i_{\kappa+1})\in\mathcal {E}$ for $\kappa=1,2,\dots,m-1$. A walk is called a [*path*]{} if the nodes of the walk are distinct, and a path from $i$ to $j$ is denoted as $i\rightarrow j$. $\mathcal
{G}$ is said to be [*strongly connected*]{} if it contains path $i\rightarrow j$ and $j\rightarrow i$ for every pair of nodes $i$ and $j$. A digraph $\mathcal {G}$ is called [*bidirectional*]{} when for any two nodes $i$ and $j$, $(i,j)\in\mathcal{E}$ if and only if $(j,i)\in\mathcal{E}$. Ignoring the direction of the arcs, the connectivity of a bidirectional digraph is transformed to that of the corresponding undirected graph. A time-varying graph is defined as $\mathcal
{G}_{\sigma(t)}=(\mathcal {V},\mathcal {E}_{\sigma(t)})$ where $\sigma:[0,+\infty)\rightarrow \mathcal {Q}$ denotes a piecewise constant function, where $\mathcal {Q}$ is a finite set containing all possible graphs with node set $\mathcal{V}$. Moreover, the joint graph of $\mathcal
{G}_{\sigma(t)}$ in time interval $[t_1,t_2)$ with $t_1<t_2\leq +\infty$ is denoted as $\mathcal {G}([t_1,t_2))= \cup_{t\in[t_1,t_2)}
\mathcal {G}(t)=(\mathcal {V},\cup_{t\in[t_1,t_2)}\mathcal
{E}_{\sigma(t)})$.
Dini Derivatives
----------------
The upper [*Dini derivative*]{} of a continuous function $h: (a,b)\to \mathds{R}$ ($-\infty\leq a<b\leq \infty$) at $t$ is defined as $$D^+h(t)=\limsup_{s\to 0^+} \frac{h(t+s)-h(t)}{s}.$$ When $h$ is continuous on $(a,b)$, $h$ is non-increasing on $(a,b)$ if and only if $ D^+h(t)\leq 0$ for any $t\in (a,b)$. The next result is convenient for the calculation of the Dini derivative [@dan; @lin07].
\[lemdini\] Let $V_i(t,x): \mathds{R}\times \mathds{R}^d \to \mathds{R}\;(i=1,\dots,n)$ be $C^1$ and $V(t,x)=\max_{i=1,\dots,n}V_i(t,x)$. If $
\mathcal{I}(t)=\{i\in \{1,2,\dots,n\}\,:\,V(t,x(t))=V_i(t,x(t))\}$ is the set of indices where the maximum is reached at $t$, then $
D^+V(t,x(t))=\max_{i\in\mathcal{ I}(t)}\dot{V}_i(t,x(t)).
$
Limit Sets
----------
Consider the following autonomous system $$\label{i1} \dot{x}=f(x),$$ where $f:\mathds{R}^d\rightarrow \mathds{R}^d$ is a continuous function. Let $x(t)$ be a solution of (\[i1\]) with initial condition $x(t_0)=x^0$. Then $\Omega_0\subset \mathds{R}^d$ is called a [*positively invariant set*]{} of (\[i1\]) if, for any $t_0\in\mathds{R}$ and any $x^0\in\Omega_0$, we have $x(t)\in\Omega_0$, $t\geq t_0$, along every solution $x(t)$ of (\[i1\]).
We call $y$ a $\omega$-limit point of $x(t)$ if there exists a sequence $\{t_k\}$ with $\lim_{k\rightarrow \infty}t_k=\infty$ such that $$\lim_{k\rightarrow \infty}x(t_k)=y.$$ The set of all $\omega$-limit points of $x(t)$ is called the $\omega$-limit set of $x(t)$, and is denoted as $\Lambda^+\big(x(t)\big)$. The following lemma is well-known [@rou].
\[leminvariant\] Let $x(t)$ be a solution of (\[i1\]). Then $\Lambda^+\big(x(t)\big)$ is positively invariant. Moreover, if $x(t)$ is contained in a compact set, then $\Lambda^+\big(x(t)\big)\neq \emptyset$.
Convex Analysis
---------------
A set $K\subset \mathds{R}^d$ is said to be [*convex*]{} if $(1-\lambda)x+\lambda
y\in K$ whenever $x\in K,y\in K$ and $0\leq\lambda \leq1$. For any set $S\subset \mathds{R}^d$, the intersection of all convex sets containing $S$ is called the [*convex hull*]{} of $S$, denoted by $co(S)$.
Let $K$ be a closed convex subset in $\mathds{R}^d$ and denote $|x|_K\doteq\inf_{y\in K}| x-y |$ as the distance between $x\in \mathds{R}^d$ and $K$, where $|\cdot|$ is the Euclidean norm. There is a unique element ${P}_{K}(x)\in K$ satisfying $|x-{P}_{K}(x)|=|x|_K$ associated to any $x\in \mathds{R}^d$ [@aubin]. The map ${P}_{K}$ is called the [*projector*]{} onto $K$. The following lemma holds [@aubin].
\[lemconvex\] (i). $\langle {P}_{K}(x)-x,{P}_{K}(x)-y\rangle\leq 0,\quad \forall y\in
K$.
(ii). $|{P}_{K}(x)-{P}_{K}(y)|\leq|x-y|, x,y\in \mathds{R}^d$.
\(iii) $|x|_K^2$ is continuously differentiable at $x$ with $\nabla |x|_K^2=2\big(x-{P}_{K}(x)\big)$.
Let $f: \mathds{R}^d\rightarrow \mathds{R}$ be a real-valued function. We call $f$ a convex function if for any $x,y\in\mathds{R}^d$ and $0\leq\lambda \leq1$, it holds that $f\big((1-\lambda)x+\lambda y\big)\leq (1-\lambda)f(x)+\lambda f(y)$. The following lemma states some well-known properties for convex functions.
\[lemfunction\] Let $f:\mathds{R}^d\rightarrow \mathds{R}\in C^1$ be a convex function.
(i). $f(x)\geq f(y)+\big\langle x-y, \nabla f(y)\big\rangle$.
(ii). Any local minimum is a global minimum, i.e., $\arg \min f=\big\{z: \nabla f(z)=0 \big\}$.
Problem Definition
==================
Objective
---------
Consider a network with node set $\mathcal
{V}=\{1,2,\dots,N\}$ modeled in general as a directed graph $\mathcal{G}=(\mathcal {V}, \mathcal {E})$. A node $j$ is said to be a [*neighbor*]{} of $i$ at time $t$ when there is an arc $(j, i)\in \mathcal
{E}$, and we denote $\mathcal{N}_i$ the set of neighbors for node $i$.
Node $i$ is associated with a cost function $f_i: \mathds{R}^m\rightarrow \mathds{R}, m>0$ which is observed by node $i$ only. The objective for the network is to cooperatively solve the optimization problem $$\label{1}
\begin{array}{cl}
\mathop{\rm minimize}\ & \sum_{i=1}^N f_i(z) \\
\textrm{subject to} & z\in \mathds{R}^m.
\end{array}$$
We impose the following assumption on the functions $f_i, i=1,\dots,N$.
[**A1.**]{} For all $i=1,\dots,N$, we have (i) $f_i\in C^1$; (ii) $f_i$ is a convex function; (iii) $\arg \min f_i\neq \emptyset$.
Problem (\[1\]) is equivalent with the following problem: $$\label{16}
\begin{array}{cl}
\mathop{\rm minimize}\ & \sum_{i=1}^N f_i(z_i) \\
\textrm{subject to} & z_i\in \mathds{R}^m \\
& z_1=\dots=z_N.
\end{array}$$ From (\[16\]) we see that consensus algorithms are a natural mean for solving the optimization problem (\[1\]).
Information Flow
----------------
The state of node $i$ at time $t$ is denoted as $x_i(t)\in \mathds{R}^m$. We define the information flow for node $i$ as follows.
- The local optimization information $g_i(t)$ node $i$ receives from its objective $f_i$ at time $t$ is the gradient of $f_i$ at its current state, i.e., $$\begin{aligned}
g_i(t)\doteq \nabla f_i\big(x_i(t)\big).\end{aligned}$$
- The neighboring information $n_i(t)$ node $i$ receives from its neighbors at time $t$ is $$\begin{aligned}
n_i(t)\doteq \hbar_i\big(x_i(t), x_j(t): j \in
\mathcal{N}_i \big),\end{aligned}$$ where $\hbar_i: \mathds{R}^m\times \mathds{R}^{m|\mathcal{N}_i|}\rightarrow \mathds{R}^l$ is a continuous function, $|\mathcal{N}_i|$ denotes the number of elements in $\mathcal{N}_i$, and $l$ is a given integer indicating the dimension of the neighboring information.
Let $\hbar= \hbar_1 \otimes \dots \otimes \hbar_N: \mathds{R}^{m(1+|\mathcal{N}_1|)} \times \dots \times \mathds{R}^{m(1+|\mathcal{N}_N|)} \rightarrow \mathds{R}^{Nl}$ denote the direct sum of $\hbar_i, i=1,\dots,N$. Then $\hbar$ represents the rule of all neighboring information flow over the whole network. We impose the following assumption.
[**A2.**]{} $\hbar\in \mathscr{R} \doteq \Big\{ h_1 \otimes \dots \otimes h_N$: $h_i$: $ \mathds{R}^{m(1+|\mathcal{N}_i|)} \mapsto\mathds{R}^{l}$ and $h_i\equiv0$ within the local consensus manifold $\big\{x_i=x_{j}: j \in
\mathcal{N}_i \big\}$ for all $i\in\mathcal{V}\Big\}$.
Assumption A2 is to say that the neighboring information a node receives from its neighbors becomes trivial when the node is in the same state as all its neighbors. This is a quite natural assumption in the literature on distributed averaging and optimization algorithms [@jad03; @mor; @saber04; @nedic09; @nedic10].
Computational Capacity
----------------------
We adopt a dynamical system model to define the way nodes update their respective states. The evolution of the nodes’ states is restricted to be a first-order integrator: $$\label{2}
\dot{x}_i=u_i, \quad i=1,\dots,N,$$ where the right-hand side $u_i$ is interpreted as a control input and the control law is characterized as $$\begin{aligned}
\label{5}
u_i= \mathcal {J} \big(n_i,g_i\big),\ i=1,\dots,N\end{aligned}$$ with $\mathcal{J}:
\mathds{R}^l\times \mathds{R}^m\rightarrow \mathds{R}^m$.
For the control law $\mathcal{J}$, we impose the following assumption.
[**A3.**]{} $\mathcal{J} \in \mathscr{C}\doteq \big\{\mathcal{F}(\cdot,\cdot)\in C^0:\
\mathds{R}^l\times \mathds{R}^m\rightarrow \mathds{R}^m,\ \mathcal{F}(0,\cdot)\ \mbox{is injective}\big\}$.
Assumption A3 indicates that the control law applied in each node should have the same structure, irrespectively of individual local optimization information or neighboring information. Note that our network model is homogeneous because one cannot tell the difference from one node to another. We assume that the control law $\mathcal{J}(0,\cdot)$ is injective, so each node takes different response to different gradient information on the local consensus manifold. Again, Assumption A3 is widely applied in the literature [@jad03; @mor; @saber04; @nedic09; @nedic10].
Problem
-------
Let $x(t)=(x_1^T(t),\dots,x_N^T(t))^T\in \mathds{R}^{mN}$ be the trajectory of system (\[2\]) with control law (\[5\]) for initial condition $x^0=x(t_0)$. Denote $F(z)=\sum_{i=1}^N f_i(z)$. We introduce the following definition.
Global [*optimal consensus*]{} of (\[2\])–(\[5\]) is achieved if for all $x^0\in \mathds{R}^{mN}$, we have $$\label{3}
\limsup_{t\rightarrow +\infty} F\big(x_i(t)\big)= \min_{z\in \mathds{R}^m} F(z)$$ and $$\label{4}
\lim_{t\rightarrow +\infty} \big |x_i(t)-x_j(t)\big|=0,\quad i,j=1,\dots,N.$$
The problem considered in this paper is to characterize conditions on the control law $\mathcal{J}$ under which global optimal consensus is achieved. In Section 4 this is done for fixed graphs and in Section 5 for time-varying graphs.
Fixed Graphs
============
In this section, we consider the possibility of solving optimal consensus using control law (\[5\]) under fixed communication graphs. We first discuss whether exact optimal consensus can be reached for directed graphs. Then we show the existence of an approximate solution for optimal consensus over bidirectional graphs.
Exact Solution
--------------
We make an assumption on the solution set of $F=\sum_{i=1}^N f_i$.
[**A4.** ]{} $\arg \min F(z)\neq\emptyset$ is a bounded set.
The main result on the existence of a control law solving optimal consensus is stated as follows.
\[thm1\]Assume that A1 and A4 hold. Let the communication graph $\mathcal{G}$ be fixed and strongly connected. There exist a neighboring information rule $\hbar \in \mathscr{R}$ and a control law $\mathcal{J}\in \mathscr{C}$ such that global optimal consensus is achieved if and only if $$\begin{aligned}
\label{intersection}
\bigcap_{i=1}^N \arg \min f_i(z)\neq \emptyset.\end{aligned}$$
According to Theorem \[thm1\], the optimal solution sets of $f_i$, $i=1,\dots,N$, having nonempty intersection is a critical condition for the existence of a control law (\[5\]) that solves the optimal consensus problem. Condition (\[intersection\]) is obviously a strong constraint which in general does not hold. Therefore, basically Theorem \[thm1\] suggests that exact solution of optimal consensus is seldom possible for the given model.
It follows from the proof below that the necessity statement of Theorem \[thm1\] relies only on the fact that the limit set of an autonomous system is invariant. It is straightforward to verify that for a discrete-time autonomous dynamical system defined by $$\begin{aligned}
y_{k+1}=f(y_k)\end{aligned}$$ with $f$ a continuous function, its limit set is invariant. Therefore, if we consider a model with discrete-time update as $$\begin{aligned}
\label{203}
x_i(k+1)=x_i(k)+u_i(k)\end{aligned}$$ with $$\begin{aligned}
\label{204}
u_i(k)=\mathcal{J}\big(n_i(k),g_i(k)\big),\end{aligned}$$ where $n_i$, $g_i$, and $\mathcal{J}$ agree with the definitions above, the necessity statement of Theorem \[thm1\] still holds. However, the sufficiency statement of Theorem \[thm1\] may in general not hold for discrete-time updates since even for the centralized optimization problem, there is not always an algorithm with constant step size which can solve the problem exactly, cf., [@bertsekas].
In [@nedic09], a discrete-time algorithm was provided for solving (\[1\]), where the structure of the nodes’ update is the sum of a consensus term averaging the neighbors’ states, and a subgradient term of the local objective function with a fixed step size. It is easy to see that the algorithm in [@nedic09] can be rewritten as (\[203\]) and (\[204\]) as long as the graph is fixed and the step size is constant. All the properties we impose on the information flow and update dynamics are kept. Convergence bounds were established for the case with constant step size in [@nedic09]. Theorem \[thm1\] shows that proposing a convergence bound is in general the best we can do for algorithms like the one developed in [@nedic09], and the result also explains why a time-varying step size may be necessary in distributed optimization algorithms, as in [@nedic10].
In the rest of this subsection, we first give the proof of the necessity claim of Theorem \[thm1\], and then we present a simple proof for the sufficiency part with bidirectional graphs. The sufficiency part of Theorem \[thm1\] in fact follows from the upcoming conclusion, Theorem \[thm4\], which does not rely on Assumption A4.
### Necessity
We now prove the necessity statement in Theorem \[thm1\] by a contradiction argument. Suppose $\bigcap_{i=1}^N \arg \min f_i(z)= \emptyset$ and there exists a distributed control in the form of (\[5\]), say $\mathcal {J}_0 \big(n_i,g_i\big)$, under which global optimal consensus is reached for certain neighboring information flow $n_i$ satisfying Assumption A2. Let $x(t)$ be a trajectory of system (\[2\]) with control $\mathcal {J}_0 \big(n_i,g_i\big)$ and $\Lambda^+(x(t))$ be its $\omega$-limit set. The definition of optimal consensus leads to that $x(t)$ converges to the bounded set $\Big( \arg \min F(z)\Big)^N\bigcap \mathcal {M}$, where $\Big( \arg \min F(z)\Big)^N$ denotes the $N$’th power set of $ \arg \min F(z)$ and $\mathcal{M}$ denotes the consensus manifold, defined by $$\begin{aligned}
\mathcal{M}\doteq \big\{x=(x_1^T\dots x_N^T)^T: \ x_1=\dots=x_N;\ x_i\in \mathds{R}^m, i=1,\dots,N\big\}.\end{aligned}$$ Therefore, each trajectory $x(t)$ is contained in a compact set.
Based on Lemma \[leminvariant\], we conclude that $\Lambda^+(x(t))\neq \emptyset$ and $$\begin{aligned}
\label{6}
\Lambda^+(x(t))\subseteq \Big( \arg \min F(z)\Big)^N\bigcap \mathcal {M},\end{aligned}$$ Moreover, $\Lambda^+(x(t))$ is positively invariant since system (\[2\]) is autonomous under control $\mathcal {J}_0 \big(n_i,g_i\big)$ when the communication graph is fixed. This is to say, any trajectory of system (\[2\]) under control $\mathcal {J}_0 \big(n_i,g_i\big)$ must stay within $\Lambda^+(x(t))$ for any initial value in $\Lambda^+(x(t))$.
Now we take $y\in\Lambda^+(x(t))$. Then we have $y\in \Big( \arg \min F(z)\Big)^N\bigcap \mathcal {M}$ according to (\[6\]), and thus $y=(z_\ast^T \dots z_\ast^T)^T$ for some $z_\ast\in \arg \min F(z)$. With Assumption A1, the convexity of the $f_i$’s implies that $$\begin{aligned}
\arg \min F(z)=\big\{z\in\mathds{R}^m:\ \sum_{i=1}^N \nabla f_i(z)=0 \big\}.\end{aligned}$$ On the other hand, we have $$\bigcap_{i=1}^N \arg \min f_i(z)= \bigcap_{i=1}^N \big\{z\in\mathds{R}^m:\ \nabla f_i(z)=0 \big\}= \emptyset.$$ Therefore, there exists two indices $i_1,i_2\in \{1,\dots,N\}$ with $i_1\neq i_2$ such that $$\begin{aligned}
\nabla f_{i_1}(z_\ast)\neq \nabla f_{i_2}(z_\ast).\end{aligned}$$
Consider the solution of (\[2\]) under control $\mathcal {J}_0 \big(n_i,g_i\big)$ for initial time $t_0$ and initial value $y$. The fact that $y$ belongs to the consensus manifold guarantees $$\begin{aligned}
n_{i_1}(t_0)=n_{i_2}(t_0)=0.\end{aligned}$$ With Assumption A4, we have $$\begin{aligned}
\mathcal {J}_0\big(n_{i_1}(t_0), g_{i_1}(t_0)\big)= \mathcal {J}_0\big(0, \nabla f_{i_1}(z_\ast)\big) \neq
\mathcal {J}_0\big(0, \nabla f_{i_2}(z_\ast)\big)= \mathcal {J}_0\big(n_{i_2}(t_0), g_{i_2}(t_0)\big).\end{aligned}$$ This implies $\dot{x}_{i_1}(t_0)\neq \dot{x}_{i_2}(t_0)$. As a result, there exists a constant $\varepsilon>0$ such that $x_{i_1}(t)\neq x_{i_2}(t)$ for $t\in (t_0,t_0+\varepsilon)$. In other word, the trajectory will leave the set $$\Big( \arg \min F(z)\Big)^N\bigcap \mathcal {M}$$ for $(t_0,t_0+\varepsilon)$, and therefore will also leave the set $\Lambda^+(x(t))$. This contradicts the fact that $\Lambda^+(x(t))$ is positively invariant. The necessity part of Theorem \[thm1\] has been proved.
### Sufficiency: Bidirectional Case
We now provide an alternative proof of sufficiency for bidirectional graphs, which is based on some geometrical intuition of the vector field. Note that compared to the proof of Theorem \[thm4\] on directed graphs, this proof uses completely different arguments which indeed cannot be applied to directed graphs. Therefore, we believe the proof given in the following is interesting at its own right, because it reveals some fundamental difference between directed and bidirectional graphs.
Let $a_{ij}>0$ be a constant marking the weight of arc $(j,i)$. We will show that the particular neighboring information flow $$n_i=\sum\limits_{j \in
\mathcal{N}_i}a_{ij}\big(x_j-x_i\big)$$ and control law $$\begin{aligned}
\label{7}
\mathcal{J}_\star(n_i,g_i)=n_i-g_i=\sum\limits_{j \in
\mathcal{N}_i}a_{ij}\big(x_j-x_i\big)-\nabla f_i\big(x_i\big)\end{aligned}$$ ensure global optimal consensus for system (\[2\]). Note that (\[7\]) is indeed a continuous-time version of the algorithm proposed in [@nedic09].
We suppose $\mathcal{G}$ is bidirectional. In this case, we have $a_{ij}=a_{ji}$ for all $i$ and $j$, and we use unordered pair $\{i,j\}$ to denote the edge between node $i$ and $j$.
Noticing that $$\begin{aligned}
\mathcal{J}_\star(n_i,g_i)=\sum\limits_{j \in
\mathcal{N}_i}a_{ij}\big(x_j-x_i\big)-\nabla f_i\big(x_i\big)=-\nabla_{x_i} \Big(\frac{1}{2}\sum\limits_{j \in
\mathcal{N}_i}a_{ij}\big|x_j-x_i\big|^2+f_i(x_i)\Big),\end{aligned}$$ we have that $(\ref{7})$ indeed solves the following convex problem $$\begin{aligned}
\label{a1}
\begin{array}{cl}
\mathop{\rm minimize}\ & F_{\mathcal{G}}(x)\doteq\sum_{i=1}^N f_i(x_i)+ \frac{1}{2}\sum_{\{j,i\}\in\mathcal{E}}a_{ij}\big|x_j-x_i\big|^2 \\
\textrm{subject to} & x_i\in \mathds{R}^m,\ i=1,\dots,N.
\end{array}\end{aligned}$$ We establish the following lemma relating the solution sets of problems (\[1\]) and (\[a1\]).
\[lem2\] Suppose $\bigcap_{i=1}^N \arg \min f_i(z)\neq \emptyset$. Suppose also the communication graph $\mathcal{G}$ is fixed, bidirectional, and connected. Then we have $$\begin{aligned}
\arg \min F_{\mathcal{G}}(x)=\Big( \bigcap_{i=1}^N \arg \min f_i(z)\Big)^N\bigcap \mathcal {M}=\Big( \arg \min F(z)\Big)^N\bigcap \mathcal {M}.\end{aligned}$$
[*Proof.*]{} When $\bigcap_{i=1}^N \arg \min f_i(z)\neq \emptyset$, it is straightforward to see that $$\arg \min F(z)= \bigcap_{i=1}^N \arg \min f_i(z).$$
Now take $x_\ast=(p_\ast^T \dots p_\ast^T)^T\in \Big( \bigcap_{i=1}^N \arg \min f_i(z)\Big)^N\bigcap \mathcal {M}$, where $p_\ast\in \bigcap_{i=1}^N \arg \min f_i(z)$. First we have $x_\ast\in \arg \min_x \sum_{i=1}^N f_i(x_i) $. Second we have $x_\ast\in \arg \min_x \frac{1}{2}\sum_{\{j,i\}\in\mathcal{E}}a_{ij}\big|x_j-x_i\big|^2$. Therefore, we conclude that $x_\ast\in \arg \min F_{\mathcal{G}}(x) $. This gives $$\begin{aligned}
\label{11}
\arg \min F_{\mathcal{G}}(x) \supseteq\Big( \bigcap_{i=1}^N \arg \min f_i(z)\Big)^N\bigcap \mathcal {M}.\end{aligned}$$
On the other hand, convexity gives $$\begin{aligned}
\arg \min F_{\mathcal{G}}(x)=\bigg\{x:\ -(L\otimes I_m) x= \Big( \big(\nabla f_1(x_1)\big)^T \dots \big(\nabla f_N(x_N)\big)^T \Big)^T\bigg\},\end{aligned}$$ where $\otimes$ represents the Kronecker product, $I_m$ is the identity matrix in $\mathds{R}^m$, and $L=D-A$ is the Laplacian of the graph $\mathcal{G}$ with $A=[a_{ij}]$ and $D={\rm diag}(d_1,\dots,d_N)$, where $d_i=\sum_{j=1}^n a_{ij}$. Noticing that $$(\mathbf{1}_N^T\otimes I_m) (L\otimes I_m) =\mathbf{1}_N^TL\otimes I_m=0,$$ where $\mathbf{1}_N=(1\dots 1)^T \in \mathds{R}^N$, we have $$\begin{aligned}
\label{8}
\Big(\mathbf{1}_N^T \otimes I_m\Big) \Big( \big(\nabla f_1(x_1)\big)^T \dots \big(\nabla f_N(x_N)\big)^T \Big)^T=\sum_{i=1}^N \nabla f_i(x_i)=0\end{aligned}$$ for any $x\in\arg \min F_{\mathcal{G}}(x) $.
Now take $x^\ast=(q_1^T \dots q_N^T)^T\in\arg \min F_{\mathcal{G}}(x) $. Suppose there exist two indices $i_\ast$ and $j_\ast$ such that $$\nabla f_{i_\ast}(q_{i_\ast})\neq \nabla f_{j_\ast}(q_{j_\ast}).$$ Then at least one of $\nabla f_{i_\ast}(q_{i_\ast})$ and $\nabla f_{j_\ast}(q_{j_\ast})$ must be nonzero. Taking $\hat{p}\in \bigcap_{i=1}^N \arg \min f_i(z)$, we have $$\sum_{i=1}^N f_i(q_i)>\sum_{i=1}^Nf_i(\hat{p})$$ because for $x=(x_1^T \dots x_N^T)^T\in\arg \min \sum_{i=1}^N f_i(x_i)$, we have $\nabla f_i(x_i)=0, i=1,\dots,N$. Consequently, for $w_\ast=(\hat{p}^T \dots \hat{p}^T)^T$, we have $$F_{\mathcal{G}}(x^\ast)>F_{\mathcal{G}}(w_\ast),$$ which is impossible according to the definition of $x^\ast$ so that such $i_\ast$ and $j_\ast$ cannot exist. In light of (\[8\]), this immediately implies $$\nabla f_i(q_i)=0,\ i=1,\dots,N,$$ or equivalently $$\begin{aligned}
\label{10}
q_i\in \arg \min f_i(z),\ i=1,\dots,N\end{aligned}$$ for all $x^\ast=(q_1^T\dots q_N^T)^T\in\arg \min F_{\mathcal{G}}(x)$.
Therefore, we conclude from (\[10\]) that $$\sum_{i=1}^N f_i(q_i)=\sum_{i=1}^Nf_i(p_\ast),$$ and this implies $$\sum_{\{j,i\}\in\mathcal{E}}a_{ij}\big|q_j-q_i\big|^2=0$$ as long as $x^\ast=(q_1^T \dots q_N^T)^T\in\arg \min F_{\mathcal{G}}(x)$. The connectivity of the communication graph thus further guarantees that $q_1=\dots=q_N$, so we have proved that $
x^\ast\in \Big( \bigcap_{i=1}^N \arg \min f_i(z)\Big)^N\bigcap \mathcal {M}.$ Consequently, we obtain $$\begin{aligned}
\label{12}
\arg \min F_{\mathcal{G}}(x) \subseteq\Big( \bigcap_{i=1}^N \arg \min f_i(z)\Big)^N\bigcap \mathcal {M}.\end{aligned}$$
The desired lemma holds from (\[11\]) and (\[12\]). $\square$
Now since $F_{\mathcal{G}}(x)$ is a convex function and we have $\dot{x}=\nabla F_{\mathcal{G}}(x)$ for system (\[2\]) with control (\[7\]), we conclude that $$\lim_{t\rightarrow \infty}{\rm dist}\big(x(t),{\arg \min F_{\mathcal{G}}(x) }\big)=0.$$ Lemma \[lem2\] ensures $$\lim_{t\rightarrow \infty}{\rm dist}\bigg(x(t),\Big( \bigcap_{i=1}^N \arg \min f_i(z)\Big)^N\bigcap \mathcal {M}\bigg)=0$$ if $\mathcal{G}$ is bidirectional and connected. Equivalently, global optimal consensus is reached.
We see from the proof above that the construction of $F_{\mathcal{G}}(x)$ is critical because the convergence argument is based on the fact that the gradient of $F_{\mathcal{G}}(x)$ is consistent with the communication graph. It can be easily verified that finding such a function is in general impossible for directed graphs.
Approximate Solution
--------------------
Theorem \[thm1\] indicates that optimal consensus is impossible no matter how the control law $\mathcal{J}$ is chosen from $\mathscr{C}$ as long as the nonempty intersection condition (\[intersection\]) is not fulfilled. In this subsection, we discuss the approximate solution of the optimal consensus problem in the absence of (\[intersection\]). We introduce the following definition.
Global [*$\epsilon$-optimal consensus*]{} is achieved if for all $x^0\in \mathds{R}^{mN}$, we have $$\label{3a}
\limsup_{t\rightarrow +\infty} F\big(x_i(t)\big)\leq \min_{z\in \mathds{R}^m} F(z)+\epsilon$$ and $$\label{4a}
\lim_{t\rightarrow +\infty} \big |x_i(t)-x_j(t)\big|\leq \epsilon,\quad i,j=1,\dots,N.$$
Denoting $F_{\mathcal{G}}(x;K)=\sum_{i=1}^N f_i(x_i)+ \frac{K}{2}\sum_{\{j,i\}\in\mathcal{E}}a_{ij}\big|x_j-x_i\big|^2$, we impose the following assumption.
[**A5.** ]{} (i) $\arg \min F(z)\neq\emptyset$; (ii) $\arg \min F_{\mathcal{G}}(x;K)\neq\emptyset$ for all $K\geq 0$; (iii) $\bigcup_{K\geq 0} \arg \min F_{\mathcal{G}}(x;K)$ is bounded.
For $\epsilon$-optimal consensus, we present the following result.
\[thm2\] Assume that A1 and A5 hold. Let the communication graph $\mathcal{G}$ be fixed, bidirectional, and connected. Then for any $\epsilon>0$, there exist a neighboring information rule $\hbar \in \mathscr{R}$ and a control law $\mathcal{J}\in \mathscr{C}$ such that global $\epsilon$-optimal consensus is achieved.
[*Proof.* ]{} Again, let $a_{ij}>0$ be any constant marking the weight of arc $(j,i)$ and $a_{ij}=a_{ji}$ for all $(i,j)\in \mathcal{E}$. Fix $\epsilon$. We will show that under neighboring information flow $$n_i=\sum\limits_{j \in
\mathcal{N}_i}a_{ij}\big(x_j-x_i\big),$$ there exists a constant $K_\epsilon>0$ such that the control law $$\begin{aligned}
\label{50}
u_i=\mathcal{J}_{K_\epsilon}(n_i,g_i)\doteq K_\epsilon n_i-g_i\end{aligned}$$ guarantees global $\epsilon$-optimal consensus.
It is straightforward to see that $$\begin{aligned}
\mathcal{J}_{K}(n_i,g_i)=K\sum\limits_{j \in
\mathcal{N}_i}a_{ij}\big(x_j-x_i\big)-\nabla f_i\big(x_i\big)=-\nabla_{x_i} \Big(\frac{K}{2}\sum\limits_{j \in
\mathcal{N}_i}a_{ij}\big|x_j-x_i\big|^2+f_i(x_i)\Big).\end{aligned}$$ System (\[2\]) with control law $u_i=\mathcal{J}_K(n_i,g_i)$ can be written into the following compact form $$\begin{aligned}
\label{90}
\dot{x}=-\nabla F_{\mathcal{G}}(x;K),\ \ x=(x_1^T \dots x_N^T)^T\in \mathds{R}^{mN}.\end{aligned}$$ Then the convexity of $F_{\mathcal{G}}(x;K)$ ensures that control law $\mathcal{J}_{K}(n_i,g_i)$ asymptotically solves the convex optimization problem $$\begin{aligned}
\label{ka1}
\begin{array}{cl}
\mathop{\rm minimize}\ & F_{\mathcal{G}}(x;K)=\sum_{i=1}^N f_i(x_i)+ \frac{K}{2}\sum_{\{j,i\}\in\mathcal{E}}a_{ij}\big|x_j-x_i\big|^2 \\
\textrm{subject to} & x_i\in \mathds{R}^m,\ i=1,\dots,N.
\end{array}\end{aligned}$$
Convexity gives $$\begin{aligned}
\label{92}
\arg \min F_{\mathcal{G}}(x;K)=\bigg\{x:\ -K(L\otimes I_m) x= \Big( \big(\nabla f_1(x_1)\big)^T \dots \big(\nabla f_N(x_N)\big)^T \Big)^T\bigg\}.\end{aligned}$$ Under Assumptions A1 and A5, we have that $$\begin{aligned}
L_0\doteq \sup \Big\{ \big|\nabla \tilde{F}(x)\big|:\ x\in \bigcup_{K\geq0} \arg \min F_{\mathcal{G}}(x;K)\Big\}\end{aligned}$$ is a finite number, where $\tilde{F}(x)=\sum_{i=1}^N f_i(x_i)$. We also define $$\begin{aligned}
D_0\doteq \sup \Big\{ \big|z_\ast-x_i\big|:\ i=1,\dots,N,\ x\in \bigcup_{K\geq 0} \arg \min F_{\mathcal{G}}(x;K)\Big\},\end{aligned}$$ where $z_\ast\in \arg \min F$ is an arbitrarily chosen point.
Let $p=(p_1^T \dots p_N^T)^T \in \arg \min F_{\mathcal{G}}(x;K)$ with $p_i\in \mathds{R}^m, i=1, \dots,N$. Since the graph is bidirectional and connected, we can sort the eigenvalues of the Laplacian $L\otimes I_m$ as $$0=\lambda_1=\dots=\lambda_{m} <\lambda_{m+1}\leq \dots \leq\lambda_{mN}.$$ Let $l_1\dots,l_{mN}$ be the orthonormal basis of $\mathds{R}^{mN}$ formed by the right eigenvectors of $L\otimes I_m$, where $l_1,\dots,l_m$ are eigenvectors corresponding to the zero eigenvalue. Suppose $p=\sum_{k=1}^{mN}c_k l_{k}$ with $c_k\in\mathds{R}, k=1,\dots,mN$.
According to (\[92\]), we have $$\begin{aligned}
\Big|K(L\otimes I_m)p\Big|^2= K^2\Big|\sum_{k=m+1}^{mN} c_k \lambda_k l_k\Big|^2=K^2\sum_{k=m+1}^{mN} c_k^2 \lambda_k^2 \leq L_0^2,\end{aligned}$$ which yields $$\begin{aligned}
\label{94}
\sum_{k=m+1}^{mN} c_k^2 \leq \Big(\frac{L_0}{K\lambda_2^\ast}\Big)^2,\end{aligned}$$ where $\lambda_2^\ast>0$ denotes the second smallest eigenvalue of $L$.
Now recall that $$\begin{aligned}
\mathcal{M}\doteq \big\{x=(x_1^T\dots x_N^T)^T: \ x_1=\dots=x_N;\ x_i\in \mathds{R}^m, i=1,\dots,N\big\}.\end{aligned}$$ is the consensus manifold. Noticing that $\mathcal{M}={\rm span} \{l_1,\dots, l_m\}$, we conclude from (\[94\]) that $$\begin{aligned}
\label{100}
\sum_{k=m+1}^{mN} c_k^2 =\Big| \sum_{k=m+1}^{mN} c_k l_k \Big|^2=| p |_{\mathcal{M}}^2=\sum_{i=1}^N \Big|p_i-\frac{\sum_{i=1}^{N} p_i}{N}\Big|^2\leq \Big(\frac{L_0}{K\lambda_2^\ast}\Big)^2.\end{aligned}$$ The last equality in (\[100\]) is due to the fact that $\mathbf{1}_N \otimes\Big( \frac{\sum_{i=1}^{N} p_i}{N}\Big)$ is the projection of $p$ on to $\mathcal{M}$. Thus, for any $\varsigma >0$, there is $K_1(\varsigma)>0$ such that when $K\geq K_1(\varsigma)$, $$\begin{aligned}
\label{101}
\Big|p_i-p_{\rm ave}\Big|\leq \varsigma,\ i=1,\dots,N\end{aligned}$$ and $$\begin{aligned}
|F(p_i)-F(p_{\rm ave})\Big|\leq \varsigma,\ i=1,\dots,N,\end{aligned}$$ where $p_{\rm ave}=\frac{\sum_{i=1}^N p_i}{N}$.
On the other hand, with (\[92\]), we have $$\begin{aligned}
\label{102}
\sum_{i=1}^N \nabla f_i(p_i)=\sum_{i=1}^N \nabla f_i(p_{\rm ave}+\hat{p}_i)=0,\end{aligned}$$ where $\hat{p}_i=p_i-p_{\rm ave}$. Now according to (\[101\]) and (\[102\]), since each $f_i\in C^1$, for any $\varsigma >0$, there is $K_2(\varsigma)>0$ such that when $K\geq K_2(\varsigma)$, $$\begin{aligned}
\Big|\sum_{i=1}^N \nabla f_i(p_{\rm ave}) \Big|\leq \frac{\varsigma}{D_0}.\end{aligned}$$ This implies $$\begin{aligned}
F(p_{\rm ave})\leq F(z_\ast)+|z_\ast-p_{\rm ave}|\times \Big|\sum_{i=1}^N \nabla f_i(p_{\rm ave}) \Big|\leq F(z_\ast)+\varsigma.\end{aligned}$$
Therefore, for any $\epsilon>0$, we can take $K_0=\max\{K_1(\epsilon/2), K_2(\epsilon/2) \}$. Then when $K\geq K_0$, we have $$\begin{aligned}
|p_i-p_j|\leq \epsilon;\ \ F(p_i)\leq \min_z F(z)+\epsilon\end{aligned}$$ for all $i$ and $j$. Now that $F_{\mathcal{G}}(x;K)$ is a convex function and observing (\[90\]), every limit point of system (\[2\]) with control law $\mathcal{J}_K(n_i,g_i)$ is contained in the set $\arg \min F_{\mathcal{G}}(x;K)$. Noting that $p$ is arbitrarily chosen from $\arg \min F_{\mathcal{G}}(x;K)$, $\epsilon$-optimal consensus is achieved as long as we choose $K_\epsilon\geq K_0$. This completes the proof. $\square$
Theorem \[thm2\] can be compared to the results given in [@solodov], where a discrete-time incremental algorithm with constant step size was shown to be able to reach an $\epsilon$-approximate solution of (\[1\]). Incremental algorithms relies on global iteration along each local objective function alternatively [@solodov; @ram; @bjsiam]. They are therefore fundamentally different with the model we discuss.
For the discrete-time algorithm proposed in [@nedic09], a bound of the convergence error was expressed explicitly as a function of the fixed step size. However, this bound will not vanish as the fixed step size tends to zero or infinity [@nedic09]. Note that the parameter $K$ in the control law $\mathcal{J}_{K}(n_i,g_i)$ can be viewed as a step size. As shown in Theorem \[thm2\], the convergence error vanishes as $K$ tends to infinity, which is essentially different with the discrete-time case in [@nedic09].
From Theorems \[thm1\] and \[thm2\], we conclude that even though without the nonempty intersection condition (\[intersection\]), it is impossible to reach exact optimal consensus via control law of the form of (\[5\]), it is still possible to find a control law that guarantees approximate optimal consensus with arbitrary accuracy.
Discussion: Global vs. Local
----------------------------
A fundamental question in distributed optimization is whether global optimization can be obtained by neighboring information flow and cooperative computation. We have the following observation.
- Note that in this paper, to determine a proper $K$ in (\[50\]) for a given $\epsilon$ relies on knowledge of the structure of the network, and the information of all $f_i, i=1,\dots,N$. Finding a proper control law for $\epsilon$-optimal consensus requires thus [*global*]{} knowledge of the network. Apparently also the nonempty intersection condition in Theorem \[thm1\] is a [*global*]{} constraint.
- Incremental algorithms with constant step size have been shown to be able to reach $\epsilon$-optimal solution for any error bound $\epsilon$ as long as the step size is sufficiently small, e.g., [@solodov; @nedic01; @bjsiam]. In an incremental algorithm, iteration is carried out by only one node alternatively on each local objective function, which is is equivalent to the fact that the $N$ nodes perform the iteration, but any node can access the states of all other nodes. Therefore, it means that the underlying graph is indeed complete, which is certainly a [*global*]{} constraint.
- One can also use time-varying step size. In [@nedic10], it was shown that global optimization can be achieved by a algorithm combining consensus algorithm and subgradient computation with a time-varying step size. However, this time-varying step size must be applied to all nodes homogeneously, which makes it a [*global*]{} parameter.
From the above observations, we can conclude that in general for distributed optimization methods, some [ global]{} information (or constraint) is somehow inevitable to guarantee a global (exact or $\epsilon$-approximate) convergence. This reveals some fundamental limit of distributed information collection and algorithm design.
Assumption Feasibility
----------------------
This subsection discusses the feasibility of Assumptions A4 and A5 and shows that some mild conditions are enough to ensure A4 and A5.
Let A1 hold. If $\tilde{F}(x)=\sum_{i=1}^N f_i(x_i)$ is coercive, i.e., $\tilde{F}(x)\rightarrow \infty$ as long as $|x|\rightarrow \infty$, then A4 and A5 hold.
[*Proof.*]{} Assume that A1 holds.
a). Since $\tilde{F}(x)=\sum_{i=1}^N f_i(x_i)$ is coercive, it follows straightforwardly that $F(z)=\sum_{i=1}^Nf_i(z)$ is also coercive. As a result, $\arg \min F(z)\neq \emptyset$ is a bounded set. Thus, A4 and A5.(i) hold. b). Observing that $\frac{K}{2}\sum_{\{j,i\}\in\mathcal{E}}a_{ij}\big|x_j-x_i\big|^2\geq 0$ for all $x=(x_1^T \dots x_N^T)^T\in \mathds{R}^{mN}$ and that $\tilde{F}(x)=\sum_{i=1}^N f_i(x_i)$ is coercive, we obtain that $\arg \min F_{\mathcal{G}}(x;K)\neq\emptyset$ for all $K\geq 0$. Thus, A5.(ii) holds.
c). Based on a), we can denote $F_\ast=\min_z F(z)=F(z_\ast)$. Since $\sum_{i=1}^Nf_i(x_i)$ is coercive, there exists a constant $M(F_\ast)>0$ such that $\sum_{i=1}^N f_i(x_i)> F_\ast$ for all $|x|>M$. This implies $$\begin{aligned}
F_{\mathcal{G}}(x;K)> F_{\mathcal{G}}(\mathbf{1}_N\otimes z_\ast;K)=F_\ast\end{aligned}$$ for all $|x|> M$. That is to say, the global minimum of $F_{\mathcal{G}}(x;K)$ is reached within the set $\{|x|\leq M\}$ for all $K>0$. Therefore, we have $$\begin{aligned}
\bigcup_{K\geq 0} \arg \min F_{\mathcal{G}}(x;K)\subseteq \big\{|x|\leq M\big\}.\end{aligned}$$ This proves A5.(iii). $\square$
Next, we propose another case when A4 and A5 hold.
Let A1 hold. Suppose each $\arg \min f_i$ is bounded and the argument space for each $f_i$ is $\mathds{R}$, i.e., $m=1$. Then A4 and A5 holds.
[*Proof.*]{} Assume that A1 holds.
a). Let $x_i^\ast \in \arg \min f_i$. Denote $y_\ast=\min \{x_1^\ast,\dots,x_N^\ast\}$. Then for any $i=1,\dots,N$, we have $$\begin{aligned}
0\geq f_i(x_i^\ast)-f_i(y_\ast)\geq (x_i^\ast -y_\ast)\nabla f_i(y_\ast)\end{aligned}$$ according to inequality (i) of Lemma \[lemfunction\]. This immediately yields $\nabla f_i(y_\ast)\leq 0$ for all $i=1,\dots,N$.
Thus, for any $y<y_\ast$, we have $$\begin{aligned}
F(y)-F(y_\ast)\geq (y -y_\ast)\nabla F(y_\ast)=\sum_{i=1}^N (y -y_\ast) \nabla f_i(y_\ast)\geq 0,\end{aligned}$$ which implies $F(y)\geq F(y_\ast)$ for all $y<y_\ast$.
A symmetric analysis leads to that $F(y)\geq F(y^\ast)$ for all $y>y^\ast$ with $y^\ast=\max \{x_1^\ast,\dots,x_N^\ast\}$. Therefore, we obtain $ F(y)\geq \min\{F(y_\ast), F(y^\ast)\}$ for all $y\neq [y_\ast, y^\ast]$. This implies that a global minimum is reached within the interval $[y_\ast, y^\ast]={\rm co}\{x_1^\ast,\dots,x_N^\ast\}$ and A5.(i) thus follows.
If $\arg \min f_i$ is bounded for $i=1,\dots,N$, there exist $b_i\leq d_i, i=1,\dots, N$ such that $\arg \min f_i=[b_i, d_i]$. Define $b_\ast =\min\{b_1,\dots,b_N\}$ and $d^\ast= \max\{d_1,\dots,d_N\}$. Following a similar argument we have $\arg \min F \subseteq [b_\ast, d^\ast]$. Thus A4 holds.
b). Introduce the following cube in $\mathds{R}^N$: $$\mathcal{C}_\ast^\eta\doteq \Big\{x=(x_1^T \dots x_N^T)^T: \ x_i \in [y_\ast-\eta, y^\ast+\eta],i=1,\dots,N\Big\},$$ where $\eta>0$ is a given constant.
[*Claim.*]{} For any $K\geq 0$, $\mathcal{C}_\ast^\eta$ is an invariant set of system (\[2\]) under control law $\mathcal{J}_K(n_i,g_i)$.
Define $\Psi(x(t))=\max_{i\in\mathcal{V}} x_i(t)$. Then based on Lemma \[lemdini\], we have $$\begin{aligned}
D^+\Psi(x(t))&=\max_{i\in \mathcal{I}_0(t)} \frac{d}{dt}x_i(t)\nonumber\\
&=\max_{i\in \mathcal{I}_0(t)} \sum\limits_{j \in
\mathcal{N}_i}a_{ij}\big(x_j-x_i\big)-\nabla f_i\big(x_i\big) \nonumber\\
&\leq \max_{i\in \mathcal{I}_0(t)} \Big[-\nabla f_i\big(x_i\big) \Big],\end{aligned}$$ where $\mathcal{I}_0(t)$ denotes the index set which contains all the nodes reaching the maximum for $\Psi(x(t))$.
Since $$\begin{aligned}
0\geq f_i(x_i^\ast)-f_i(y_\ast+\eta)\geq (x_i^\ast -y_\ast-\eta)\nabla f_i(y_\ast+\eta),\ i=1,\dots,N\end{aligned}$$ we have $\nabla f_i( y^\ast+\eta)\geq 0$ for all $i=1,\dots,N$. As a result, we obtain $$\begin{aligned}
D^+\Psi(x(t))\Big|_{\Psi(x(t))=y^\ast+\eta}\leq 0,\end{aligned}$$ which implies $\Psi(x(t))\leq y^\ast+\eta$ for all $t\geq t_0$ under initial condition $\Psi(x(t_0))\leq y^\ast+\eta$. Similar analysis ensures that $\min_{i\in\mathcal{V}} x_i(t)\geq y^\ast-\eta$ for all $t\geq t_0$ as long as $\min_{i\in\mathcal{V}} x_i(t_0)\geq y^\ast-\eta$. This proves the claim.
Note that every trajectory of system (\[2\]) under control law $\mathcal{J}_K(n_i,g_i)$ asymptotically solves (\[ka1\]). This immediately leads to that $ F_{\mathcal{G}}(x;K)$ reaches its minimum within $\mathcal{C}_\ast^\eta$ for any $K\geq 0$ since $\mathcal{C}_\ast^\eta$ is an invariant set. Then A5.(ii) holds straightforwardly.
c). Since $\arg \min f_i$ is bounded for $i=1,\dots,N$, there exist $b_i\leq d_i, i=1,\dots, N$ such that $\arg \min f_i=[b_i, d_i]$. Define $b_\ast =\min\{b_1,\dots,b_N\}$ and $d^\ast= \max\{d_1,\dots,d_N\}$. We will prove the conclusion by showing $\arg \min F_{\mathcal{G}}(x;K) \subseteq \mathcal{C}_\ast$ for all $K\geq 0$, where $$\mathcal{C}_\ast\doteq \Big\{x=(x_1^T \dots x_N^T)^T: \ x_i \in [b_\ast, d^\ast],i=1,\dots,N\Big\}.$$
Let $z=(z_1 \dots,z_N)^T\in \arg \min F_{\mathcal{G}}(x;K)$. First we show $\max\{z_1,\dots,z_N\} \leq d^\ast$ by a contradiction argument. Suppose $\max\{z_1,\dots,z_N\} > d^\ast$.
Now let $i_1,\dots, i_k$ be the nodes reaching the maximum state, i.e., $z_{i_1}=\dots=z_{i_k}=\max\{z_1,\dots,z_N\}$. There will be two cases.
- Let $k=N$. We have $z_1=\dots=z_N=y$ in this case. Then for all $i$ and $x_i^\ast \in \arg \min f_i$, we have $$\begin{aligned}
0>f_i(x_i^\ast)-f_i(y)\geq (x_i^\ast -y) \nabla f_i(y)\end{aligned}$$ which yields $ \nabla f_i(y)>0, i=1,\dots,N$ since $y>d^\ast$. This immediately leads to $$\begin{aligned}
F_{\mathcal{G}}(z;K)=F(y)>\min F \geq \min F_{\mathcal{G}}(z;K),\end{aligned}$$ which contradicts the fact that $z\in \arg \min F_{\mathcal{G}}(x;K)$.
- Let $k<N$. Then we denote $s_\ast=\max\big \{ z_i: i\notin\{i_1,\dots,i_k\}, i=1,\dots,N \big\}$, which is actually the second largest value in $\{z_1,\dots,z_N\}$. We define a new point $\hat{z}=(\hat{z}_1 \dots,\hat{z}_N)^T$ by $\hat{z}_i=z_i, i\notin\{i_1,\dots,i_k\}$ and $$\begin{aligned}
\hat{z}_i=\begin{cases}
d^\ast, & \mbox{if $s_\ast<d^\ast$}\\
s_\ast, & \mbox{otherwise}
\end{cases}\end{aligned}$$ for $i\in \{i_1,\dots,i_k\}$. Then it is easy to obtain that $F_{\mathcal{G}}(z;K)>F_{\mathcal{G}}(\hat{z};K)$, which again contradicts the choice of $z$.
Therefore, we have proved that $\max\{z_1,\dots,z_N\} \leq d^\ast$. Based on a symmetric analysis we also have $\min\{z_1,\dots,z_N\} \geq b_\ast$. Therefore, we obtain $\arg \min F_{\mathcal{G}}(x;K) \subseteq \mathcal{C}_\ast$ for all $K\geq 0$ and A5.(iii) follows. $\square$
Time-varying Graphs
===================
Now we consider time-varying graphs. The communication in the multi-agent network is modeled as $\mathcal {G}_{\sigma(t)}=(\mathcal {V},\mathcal
{E}_{\sigma(t)})$ with $\sigma:[0,+\infty)\rightarrow \mathcal {Q}$ being a piecewise constant function, where $\mathcal {Q}$ is a finite set indicating all possible graphs. In this case the neighbor set for each node is time-varying, and we let $\mathcal{N}_i(\sigma(t))$ represent the set of agent $i$’s neighbors at time $t$. As usual in the literature [@jad03; @lin07; @shi09], an assumption is given to how fast $\mathcal {G}_{\sigma(t)}$ can vary.
[**A6.**]{} [*(Dwell Time)*]{} There is a lower bound $\tau_D>0$ between two consecutive switching time instants of $\sigma(t)$.
We have the following definition.
\(i) $\mathcal
{G}_{\sigma(t)}$ is said to be [*uniformly jointly strongly connected*]{} if there exists a constant $T>0$ such that $\mathcal {G}([t,t+T))$ is strongly connected for any $t\geq0$.
\(ii) $\mathcal
{G}_{\sigma(t)}$ is said to be [*uniformly jointly quasi-strongly connected*]{} if there exists a constant $T>0$ such that $\mathcal {G}([t,t+T))$ has a spanning tree for any $t\geq0$.
With time-varying graphs, $$\begin{aligned}
n_i(t)\doteq \hbar_i\big(x_i(t), x_j(t): j \in
\mathcal{N}_i(\sigma(t)) \big).\end{aligned}$$ where $\hbar_i: \mathds{R}^m\times \mathds{R}^{m|\mathcal{N}_i(\sigma(t))|}\rightarrow \mathds{R}^l$ is now piecewise defined. As a result, assumption A2 is transformed to the following piecewise version.
[**A7.**]{} $\hbar\in \mathscr{R}_\ast \doteq \Big\{ h_1 \otimes \dots \otimes h_N$: $h_i$ maps $ \mathds{R}^{m(1+|\mathcal{N}_i(\sigma(t))|)}$ to $\mathds{R}^{l}$ on each time interval when $\sigma(t)$ is constant, and $h_i\equiv0$ within the time-varying local consensus manifold $\big\{x_i=x_{j}: j \in
\mathcal{N}_i (\sigma(t))\big\}$ for all $i\in\mathcal{V}\Big\}$.
For optimal consensus with time-varying graphs, we present the following result.
\[thm3\] Suppose A1 and A6 hold and $\mathcal
{G}_{\sigma(t)}$ is uniformly jointly strongly connected. Suppose $\bigcap_{i=1}^N \arg \min f_i\neq \emptyset$ contains at least one interior point. Then there exist a neighboring information rule $\hbar \in \mathscr{R}_\ast$ and a control law $\mathcal{J}\in \mathscr{C}$ such that global optimal consensus is achieved and $$\begin{aligned}
\label{17}
\lim_{t\rightarrow\infty}x_i(t)=x_\ast.
\end{aligned}$$ for some $x_\ast\in \bigcap_{i=1}^N \arg \min f_i$.
Note that (\[17\]) is indeed a stronger conclusion than our definition of optimal consensus as Theorem \[thm3\] guarantees that all the node states converge to a common point in the global solution set of $F(z)$. We will see from the proof of Theorem \[thm3\] that this state convergence highly relies on the existence of an interior point of $\bigcap_{i=1}^N \arg \min f_i$. In the absence of such an interior point condition, it turns out that optimal consensus still stands. We present another theorem stating the fact.
\[thm4\] Suppose A1 and A6 hold and $\mathcal
{G}_{\sigma(t)}$ is uniformly jointly strongly connected. Suppose also $\bigcap_{i=1}^N \arg \min f_i\neq \emptyset$. Then there exist a neighboring information rule $\hbar \in \mathscr{R}_\ast$ and a control law $\mathcal{J}\in \mathscr{C}$ such that global optimal consensus is achieved.
The proofs of Theorems \[thm3\] and \[thm4\] rely on the following neighboring information flow $$\begin{aligned}
\label{201}
n_i=\sum\limits_{j \in
\mathcal{N}_i(\sigma(t))}a_{ij}(t)\big(x_j-x_i\big),\end{aligned}$$ where $a_{ij}(t)>0$ is any weight function associated with arc $(j,i)$. The resulting control law is $$\begin{aligned}
\mathcal{J}_\star(n_i,g_i)=n_i-g_i.\end{aligned}$$ An assumption is made on each $a_{ij}(t),i,j=1,2,...,N$.
[**A8.**]{} [*(Weights Rule)*]{} (i) Each $a_{ij}(t)$ is piece-wise continuous and $a_{ij}(t)\geq0$ for all $i$ and $j$.
(ii). There are $a^\ast>0$ and $a_\ast>0$ such that $
a_\ast\leq a_{ij}(t)\leq a^\ast,\quad
t\in \mathds{R}^+.$
Preliminary Lemmas
------------------
We establish three useful lemmas in this subsection.
Suppose $\bigcap_{i=1}^N \arg \min f_i\neq \emptyset$ and take $z_\ast\in \bigcap_{i=1}^N \arg \min f_i$. We define $$\begin{aligned}
V_i(t)=\big|x_i(t)-z_\ast\big|^2,\ i=1,\dots,N,\end{aligned}$$ and $$\begin{aligned}
V(t)=\max_{i=1,\dots,N} V_i(t).\end{aligned}$$ The following lemma holds with the proof in Appendix A.1.
\[lemmono\] Let A1 and A8 hold. Suppose $\bigcap_{i=1}^N \arg \min f_i\neq \emptyset$. Then along any trajectory of system (\[2\]) with neighboring information (\[201\]) and control law $\mathcal{J}_\star(n_i,g_i)$, we have $D^+V(t)\leq 0$ for all $t\in \mathds{R}^+$.
A direct consequence of Lemma \[lemmono\] is that when $\bigcap_{i=1}^N \arg \min f_i\neq \emptyset$, we have $$\begin{aligned}
\lim_{t\rightarrow \infty} V(t)=d_\ast^2\end{aligned}$$ for some $d_\ast\geq 0$ along any trajectory of system (\[2\]) with control law $\mathcal{J}_\star(n_i,g_i)$. However, it is still unclear whether $V_i(t)$ converges or not. We establish another lemma indicating that with proper connectivity condition for the communication graph, all $V_i(t)$’s have the same limit $d_\ast^2$. The proof can be found in Appendix A.2.
\[lemlimit\] Let A1, A6, and A8 hold. Suppose $\bigcap_{i=1}^N \arg \min f_i\neq \emptyset$ and $\mathcal
{G}_{\sigma(t)}$ is uniformly jointly strongly connected. Then along any trajectory of system (\[2\]) with neighboring information (\[201\]) and control law $\mathcal{J}_\star(n_i,g_i)$, we have $\lim_{t\rightarrow \infty} V_i(t)=d_\ast^2 $ for all $i$.
The next lemma shows that each node will reach its own optimum along the trajectories of system (\[2\]) under control law $\mathcal{J}_\star(n_i,g_i)$. The proof is in Appendix A.3.
\[lemnodeoptimum\] Let A1, A6, and A8 hold. Suppose $\bigcap_{i=1}^N \arg \min f_i\neq \emptyset$ and $\mathcal
{G}_{\sigma(t)}$ is uniformly jointly strongly connected. Then along any trajectory of system (\[2\]) with control law $\mathcal{J}_\star(n_i,g_i)$, we have $\limsup_{t\rightarrow \infty} \big| x_i(t)\big|_{\arg\min f_i}=0$ for all $i$.
Proof of Theorem \[thm3\]
-------------------------
The proof of Theorem \[thm3\] relies on the following lemma.
\[lemunique\] Let $z_1,\dots,z_{m+1}\in\mathds{R}^m$ and $d_1,\dots,d_{m+1}\in\mathds{R}^+$. Suppose there exist solutions to equations (with variable $y$) $$\label{23}
\begin{cases}
|y-z_1|^2 =d_1;\\
\ \ \ \ \ \ \vdots\\
|y-z_{m+1}|^2 =d_{m+1}.
\end{cases}$$ Then the solution is unique if ${\rm rank}\big(z_2-z_1, \dots, z_{m+1}-z_1\big)=m$.
[*Proof.*]{} Take $j>1$ and let $y$ be a solution to the equations. Noticing that $$\langle y-z_1,y-z_1\rangle=d_1; \quad \langle y-z_j,y-z_j\rangle=d_j$$ we obtain $$\begin{aligned}
\langle y,z_j-z_1\rangle= \frac{1}{2}\Big(d_1-d_j+|z_j|^2-|z_1|^2\Big), \ j=2,\dots,m+1.\end{aligned}$$ The desired conclusion follows immediately. $\square$
We now prove Theorem \[thm3\]. Let $r_\star=(r_1^T \dots r_N^T)^T$ be a limit point of a trajectory of system (\[2\]) with control law $\mathcal{J}_\star(n_i,g_i)$.
We first show consensus. Based on Lemma \[lemlimit\], we have $\lim_{t\rightarrow \infty} V_i(t)=d_\ast$ for all $z_\ast\in\bigcap_{i=1}^N \arg \min f_i$. This is to say, $|r_i-z_\ast|=d_\ast$ for all $i$ and $z_\ast\in\bigcap_{i=1}^N \arg \min f_i$. Since $\bigcap_{i=1}^N \arg \min f_i\neq \emptyset$ contains at least one interior point, it is obvious to see that we can find $z_1,\dots,z_{m+1}\in\bigcap_{i=1}^N \arg \min f_i$ with ${\rm rank}\big(z_2-z_1, \dots, z_{m+1}-z_1\big)=m$ and $d_1,\dots,d_{m+1}\in\mathds{R}^+$, such that each $r_i, i=1,\dots,N$ is a solution of equations (\[23\]). Then based on Lemma \[lemunique\], we conclude that $r_1=\dots=r_N$. Next, with Lemma \[lemnodeoptimum\], we have $| r_i|_{\arg\min f_i}=0$. This implies that $r_1=\dots=r_N\in\bigcap_{i=1}^N \arg \min f_i$, i.e., optimal consensus is achieved.
We turn to state convergence. We only need to show that $r_\star$ is unique along any trajectory of system (\[2\]) with neighboring information (\[201\]) and control law $\mathcal{J}_\star(n_i,g_i)$. Now suppose $r_\star^1=\mathbf{1}_N\otimes r^1$ and $r_\star^2=\mathbf{1}_N\otimes r^2$ are two different limit points with $r^1\neq r^2 \in \bigcap_{i=1}^N \arg \min f_i$. According to the definition of a limit point, we have that for any $\varepsilon>0$, there exists a time instant $t_\varepsilon$ such that $|x_i(t_\varepsilon)-r^1|\leq \varepsilon$ for all $i$. Note that Lemma \[lemmono\] indicates that the disc $B(r^1,\varepsilon)=\{y: |y-r^1|\leq \varepsilon\}$ is an invariant set for initial time $t_\varepsilon$. While taking $\varepsilon={|r^1-r^2|}/{4}$, we see that $r^2\notin B(r^1,|r^1-r^2|/{4})$. Thus, $r^2$ cannot be a limit point.
Now since the limit point is unique, we denote it as $\mathbf{1}_N\otimes x_\ast$ with $x_\ast\in \bigcap_{i=1}^N \arg \min f_i$. Then we have $\lim_{t\rightarrow\infty}x_i(t)=x_\ast$ for all $i=1,\dots,N$. This completes the proof.
Proof of Theorem \[thm4\]
-------------------------
In this subsection, we prove Theorem \[thm4\]. We need the following lemma on robust consensus, which can be found in [@shicdc].
\[lemrobust\] Consider a network with node set $\mathcal{V}=\{1,\dots,N\}$ with time-varying communication graph $\mathcal{G}_{\sigma(t)}$. Let the dynamics of node $i$ be $$\begin{aligned}
\dot{x}_i=\sum\limits_{j \in
\mathcal{N}_i(\sigma(t))}a_{ij}(t)\big(x_j-x_i\big)+w_i(t),\end{aligned}$$ where $w_i(t)$ is a piecewise continuous function. Suppose A6 and A8 hold and $\mathcal{G}_{\sigma(t)}$ is uniformly jointly quasi-strongly connected. Then we have $$\lim_{t\rightarrow +\infty} \big |x_i(t)-x_j(t)\big|=0,\quad i,j=1,\dots,N$$ if $\lim_{t\rightarrow \infty}w_i(t)=0$ for all $i$.
Lemma \[lemnodeoptimum\] indicates that $\limsup_{t\rightarrow \infty} \big| x_i(t)\big|_{\arg\min f_i}=0$ for all $i$, which yields $$\begin{aligned}
\lim_{t\rightarrow \infty}\nabla f_i\big(x_i(t)\big)=0\end{aligned}$$ for all $i$ according to Assumption A1. Then the consensus part in the definition of optimal consensus follows immediately from Lemma \[lemrobust\]. Again by Lemma \[lemnodeoptimum\], we further conclude that $\limsup_{t\rightarrow \infty} {\rm dist}\big( x_i(t), \bigcap_{i=1}^N \arg \min f_i\big)=0$. The desired conclusion thus follows.
Conclusions
===========
Various algorithms have been proposed in the literature for the distributed minimization of $\sum_{i=1}^N f_i$ with $f_i$ only known to node $i$. This paper explored some fundamental properties for distributed methods given a certain level of node knowledge, computational capacity, and information flow. It was proven that there exists a control law that ensures global optimal consensus if and only if $\arg \min f_i, i=1,\dots,N$, admit a nonempty intersection set for fixed strongly connected graphs. We also showed that for any error bound, we can find a control law which guarantees global optimal consensus within this bound for fixed, bidirectional, and connected graphs under some mild conditions such as that $f_i$ is coercive for some $i$. For time-varying graphs, it was proven that optimal consensus can always be achieved as long as the graph is uniformly jointly strongly connected and the nonempty intersection condition holds. It was then concluded that nonempty intersection for the local optimal solution sets is a critical condition for distributed optimization using consensus processing.
More challenges lie in exploring the corresponding limit of performance for high-order schemes, the optimal structure of the underlying communication graph for distributed optimization, and the fundamental communication complexity required for global convergence.
Acknowledgment {#acknowledgment .unnumbered}
===============
The authors would like to thank Prof. Angelia Nedić for helpful discussions and for pointing out relevant literature.
Appendix {#appendix .unnumbered}
========
**A.1 Proof of Lemma \[lemmono\]** {#a.1-proof-of-lemma-lemmono .unnumbered}
-------------------------------------
Based on Lemma \[lemdini\], we have $$\begin{aligned}
\label{20}
D^+V(t)&=\max_{i\in \mathcal{I}(t)} \frac{d}{dt}V_i(t)\nonumber\\
&=\max_{i\in \mathcal{I}(t)} 2\Big\langle x_i(t)-z_\ast, \sum\limits_{j \in
\mathcal{N}_i(\sigma(t))}a_{ij}(t)\big(x_j-x_i\big)-\nabla f_i\big(x_i\big)\Big\rangle,\end{aligned}$$ where $\mathcal{I}(t)$ denotes the index set which contains all the nodes reaching the maximum for $V(t)$.
Let $m\in\mathcal{I}(t)$. Denote $$Z_t=\big\{z:\ |z-z_\ast|\leq \sqrt{V(t)} \big\}$$ as the disk centered at $z_\ast$ with radius $\sqrt{V(t)}$. Take $y=x_m(t)+(x_m(t)-z_\ast)$. Then from some simple Euclidean geometry it is obvious to see that $P_{Z_t}(y)=x_m(t)$, where $P_{Z_t}$ is the [*projector*]{} onto $Z_t$. Thus, for all $j\in\mathcal{N}_{m}(\sigma(t))$, we obtain $$\begin{aligned}
\label{18}
\big\langle x_m(t)-z_\ast,x_j(t)-x_m(t)\big\rangle&=\big\langle y-x_m(t),x_j(t)-x_m(t)\big\rangle\nonumber\\
&=\big\langle y-P_{Z_t}(y),x_j(t)-P_{Z_t}(y)\big\rangle\nonumber\\
&\leq 0\end{aligned}$$ according to inequality (i) in Lemma \[lemconvex\] since $x_j(t)\in Z_t$. On the other hand, based on inequality (i) in Lemma \[lemfunction\], we also have $$\begin{aligned}
\label{19}
\big\langle x_m(t)-z_\ast,-\nabla f_m\big(x_m(t)\big)\big\rangle\leq f_m(z_\ast)-f_m\big(x_m(t)\big) \leq 0\end{aligned}$$ in light of the definition of $z_\ast$.
With (\[20\]), (\[18\]) and (\[19\]), we conclude that $$\begin{aligned}
D^+ V(t) =\max_{i\in \mathcal{I}(t)} 2\big\langle x_i(t)-z_\ast, \sum\limits_{j \in
\mathcal{N}_i(\sigma(t))}a_{ij}(t)\big(x_j-x_i\big)-\nabla f_i\big(x_i\big)\big\rangle\leq 0,\end{aligned}$$ which completes the proof. $\square$
**A.2 Proof of Lemma \[lemlimit\]** {#a.2-proof-of-lemma-lemlimit .unnumbered}
--------------------------------------
In order to prove the desired conclusion, we just need to show $\liminf_{t\rightarrow \infty} V_i(t)=d_\ast^2$ for all $i$.
With Lemma \[lemmono\], we conclude that $\forall\varepsilon>0, \exists M(\varepsilon)>0$, s.t., $$\begin{aligned}
\label{24}
\sqrt{V_i(t)}\leq d_\ast+\varepsilon\end{aligned}$$ for all $i$ and $t\geq M$.
For all $t\geq M$ and all $i,j\in\mathcal{V}$, we have $$\begin{aligned}
\label{25}
\big\langle x_i(t)-z_\ast,x_j(t)-x_i(t) \big\rangle\leq-V_i(t)+(d_\ast+\varepsilon)\sqrt{V_i(t)}.\end{aligned}$$
If $x_i(t)=z_\ast$ (\[25\]) follows trivially from (\[24\]). Otherwise we take $y_\ast= z_\ast+ (d_\ast+\varepsilon)\frac{x_i(t)-z_\ast}{|x_i(t)-z_\ast|}$ and $B_t=\big\{z: |z-z_\ast|\leq d_\ast+\varepsilon\big \}$. Here $B_t$ is the disk centered at $z_\ast$ with radius $d_\ast+\varepsilon$, and $y_\ast$ is a point within the boundary of $B_t$ and falls the same line with $z_\ast$ and $x_{i_0}(t)$. Take also $q_\ast=y_\ast+x_i(t)-z_\ast$. Then we have $$\begin{aligned}
\big\langle x_i(t)-z_\ast,x_j(t)-y_\ast \big\rangle&=\big\langle q_\ast-y_\ast,x_j(t)-y_\ast \big\rangle\nonumber\\
&=\big\langle q_\ast-P_{B_t}(q_\ast),x_j(t)-P_{B_t}(q_\ast) \big\rangle\nonumber\\
&\leq 0\end{aligned}$$ according to inequality (i) in Lemma \[lemconvex\], which leads to $$\begin{aligned}
\big\langle x_i(t)-z_\ast,x_j(t)-x_i(t) \big\rangle&=\big\langle x_i(t)-z_\ast,x_j(t)-y_\ast \big\rangle+\big\langle x_i(t)-z_\ast,y_\ast-x_i(t) \big\rangle\nonumber\\
&\leq\big\langle x_i(t)-z_\ast,y_\ast-x_i(t)\big\rangle\nonumber\\
&=-V_i(t)+(d_\ast+\varepsilon)\sqrt{V_i(t)}.\end{aligned}$$ This proves the claim.
Now suppose there exists $i_0\in\mathcal{V}$ with $\liminf_{t\rightarrow \infty} V_i(t)=\theta_{i_0}^2<d_\ast^2$. Then we can find a time sequence $\{t_k\}_1^\infty$ with $\lim_{k\rightarrow \infty}t_k =\infty$ such that $$\begin{aligned}
\label{28}
\sqrt{V_{i_0}(t_k)}\leq \frac{\theta_{i_0}+d_\ast}{2}.\end{aligned}$$ We divide the rest of the proof into three steps.
[*Step 1.*]{} Take $t_{k_0}>M$. We bound $V_{i_0}(t)$ in this step.
With the weights rule A8, (\[25\]) and inequality (i) in Lemma \[lemfunction\], we see that $$\begin{aligned}
\label{45}
\frac{d}{dt} V_{i_0}(t)&=2\Big \langle x_{i_0}(t)-z_\ast, \sum\limits_{j \in
\mathcal{N}_{i_0}(\sigma(t))}a_{i_0j}(t)\big(x_j-x_{i_0}\big)-\nabla f_{i_0}\big(x_{i_0}(t)\big) \Big\rangle\nonumber\\
&\leq 2\sum\limits_{j \in
\mathcal{N}_{i_0}(\sigma(t))}a_{i_0j}(t) \Big \langle x_{i_0}(t)-z_\ast,x_j(t)-x_{i_0}(t) \Big\rangle+f_{i_0}\big(z_\ast\big)-f_{i_0}\big(x_{i_0}(t)\big)\nonumber\\
&\leq 2(N-1)a^\ast\Big(-V_{i_0}(t)+(d_\ast+\varepsilon)\sqrt{V_{i_0}(t)}\Big),\end{aligned}$$ for all $t\geq t_{k_0}$, which implies $$\begin{aligned}
\label{26}
\frac{d}{dt}\sqrt{V_{i_0}(t)} \leq -(N-1)a^\ast\Big(\sqrt{V_{i_0}(t)}-(d_\ast+\varepsilon)\Big),\ \ t\geq t_{k_0}.\end{aligned}$$
In light of Grönwall’s inequality, (\[28\]) and (\[26\]) yield $$\begin{aligned}
\label{30}
\sqrt{V_{i_0}(t)} &\leq e^{-(N-1)^2a^\ast T_D}\sqrt{V_{i_0}(t_{k_0})}+\Big(1-e^{-(N-1)^2a^\ast T_D}\Big)(d_\ast+\varepsilon)\nonumber\\
&\leq \frac{e^{-(N-1)^2a^\ast T_D}}{2} \theta_{i_0}+\Big(1-\frac{e^{-(N-1)^2a^\ast T_D}}{2}\Big)(d_\ast+\varepsilon)\nonumber\\
&\doteq \Lambda_\ast.\end{aligned}$$ for all $t\in[t_{k_0}, t_{k_0}+(N-1)T_D]$ with $T_D=T+\tau_D$, where $T$ comes from the definition of uniformly jointly strongly connected graphs and $\tau_D$ represents the dwell time.
[*Step 2.*]{} Since the graph is uniformly jointly strongly connected, we can find an instant $\hat{t}\in[t_{k_0},t_{k_0}+T]$ and another node $i_1\in\mathcal{V}$ such that $(i_0,i_1)\in\mathcal{G}_{\sigma(t)}$ for $t\in[\hat{t}, \hat{t}+\tau_D]$. In this step, we continue to bound $V_{i_1}(t)$.
Similar to (\[25\]), for all $t\geq M$ and all $i,j\in\mathcal{V}$, we also have $$\begin{aligned}
\label{29}
\big\langle x_i(t)-z_\ast,x_j(t)-x_i(t) \big\rangle\leq-\sqrt{V_i(t)}\Big(\sqrt{V_i(t)}-\sqrt{V_j(t)}\Big)\end{aligned}$$ when $V_j(t)\leq V_i(t)$. Then based on (\[25\]), (\[30\]), and (\[29\]), we obtain $$\begin{aligned}
\label{31}
\frac{d}{dt} V_{i_1}(t)
&\leq 2\sum\limits_{j \in
\mathcal{N}_{i_1}(\sigma(t))}a_{i_1j}(t) \Big \langle x_{i_1}(t)-z_\ast,x_j(t)-x_{i_1}(t) \Big\rangle\nonumber\\
&= 2\sum\limits_{j \in
\mathcal{N}_{i_1}(\sigma(t))\setminus \{i_0\}}a_{i_1j}(t) \Big \langle x_{i_1}(t)-z_\ast,x_j(t)-x_{i_1}(t) \Big\rangle+2a_{i_1i_0}(t) \Big \langle x_{i_1}(t)-z_\ast,x_{i_0}(t)-x_{i_1}(t) \Big\rangle\nonumber\\
&\leq 2(N-2)a^\ast\Big(-V_{i_1}(t)+(d_\ast+\varepsilon)\sqrt{V_{i_1}(t)}\Big)-2a_\ast\sqrt{V_{i_1}(t)}\Big(\sqrt{V_{i_1}(t)}-\sqrt{V_{i_0}(t)}\Big)\nonumber\\
&\leq- 2\Big((N-2)a^\ast+a_\ast\Big)V_{i_1}(t) +2\sqrt{V_{i_1}(t)} \Big((N-2)a^\ast(d_\ast+\varepsilon)+\Lambda_\ast a_\ast\Big)\end{aligned}$$ for $t\in[\hat{t},\hat{t}+\tau_D]$, where without loss of generality we assume $V_{i_1}(t)\geq V_{i_0}(t)$ during all $t\in[\hat{t},\hat{t}+\tau_D]$.
Then (\[31\]) gives $$\begin{aligned}
\frac{d}{dt} \sqrt{V_{i_1}(t)}
&\leq- \Big((N-2)a^\ast+a_\ast\Big)\sqrt{V_{i_1}(t)} + \Big((N-2)a^\ast(d_\ast+\varepsilon)+\Lambda_\ast a_\ast\Big), t\in[\hat{t},\hat{t}+\tau_D]\end{aligned}$$ which yields $$\begin{aligned}
\sqrt{V_{i_1}(\hat{t}+\tau_D)}&\leq e^{- \big((N-2)a^\ast+a_\ast\big)\tau_D}(d_\ast+\varepsilon)+\Big(1-e^{- \big((N-2)a^\ast+a_\ast\big)\tau_D}\Big)\frac{(N-2)a^\ast(d_\ast+\varepsilon)+\Lambda_\ast a_\ast}{(N-2)a^\ast+a_\ast}\nonumber\\
&=\frac{ a_\ast\big(1-e^{- ((N-2)a^\ast+a_\ast)\tau_D}\big)}{(N-2)a^\ast+a_\ast}\times\frac{e^{-(N-1)^2a^\ast T_D}}{2} \theta_{i_0}\nonumber\\
&\ \ \ \ \ \ \ \ \ \ \ +\Big(1-\frac{ a_\ast\big(1-e^{- ((N-2)a^\ast+a_\ast)\tau_D}\big)}{(N-2)a^\ast+a_\ast}\times\frac{e^{-(N-1)^2a^\ast T_D}}{2}\Big)(d_\ast+\varepsilon)\end{aligned}$$ again by Grönwall’s inequality and some simple algebra.
Next, applying the estimate of node $i_0$ in step 1 on $i_1$ during time interval $[\hat{t}+\tau_D,t_{k_0}+(N-1)T_D]$, we arrive at $$\begin{aligned}
\sqrt{V_{i_1}(t)}&\leq \frac{ a_\ast\big(1-e^{- ((N-2)a^\ast+a_\ast)\tau_D}\big)}{(N-2)a^\ast+a_\ast}\times\frac{e^{-2(N-1)^2a^\ast T_D}}{2} \theta_{i_0}\nonumber\\
&\ \ \ \ \ \ \ \ \ \ \ +\Big(1-\frac{ a_\ast\big(1-e^{- ((N-2)a^\ast+a_\ast)\tau_D}\big)}{(N-2)a^\ast+a_\ast}\times\frac{e^{-2(N-1)^2a^\ast T_D}}{2}\Big)(d_\ast+\varepsilon)\end{aligned}$$ for all $t\in[t_{k_0}+T_D, t_{k_0}+(N-1)T_D]$.
[*Step 3.*]{} Noticing that the graph is uniformly jointly strongly connected, the analysis of steps 1 and 2 can be repeatedly applied to nodes $i_3,\dots,i_{N-1}$, and eventually we have that for all $i_0,\dots,i_{N-1}$, $$\begin{aligned}
\sqrt{V_{i_m}\big( t_{k_0}+(N-1)T_D\big)}&\leq \Big(\frac{ a_\ast\big(1-e^{- ((N-2)a^\ast+a_\ast)\tau_D}\big)}{(N-2)a^\ast+a_\ast}\Big)^{N-2}\times\frac{e^{-(N-1)^3a^\ast T_D}}{2} \theta_{i_0}\nonumber\\
&\ \ \ \ \ \ \ \ \ \ \ +\Bigg(1-\Big(\frac{ a_\ast\big(1-e^{- ((N-2)a^\ast+a_\ast)\tau_D}\big)}{(N-2)a^\ast+a_\ast}\Big)^{N-2}\times\frac{e^{-(N-1)^3a^\ast T_D}}{2} \Bigg)(d_\ast+\varepsilon)\nonumber\\
&<d_\ast\end{aligned}$$ for sufficiently small $\varepsilon$ because $\theta_{i_0}<d_\ast$ and $$\Big(\frac{ a_\ast\big(1-e^{- ((N-2)a^\ast+a_\ast)\tau_D}\big)}{(N-2)a^\ast+a_\ast}\Big)^{N-2}\times\frac{e^{-(N-1)^3a^\ast T_D}}{2} <1$$ is a constant. This immediately leads to that $$\begin{aligned}
V\big(t_{k_0}+(N-1)T_D\big)<d_\ast,\end{aligned}$$ which contradicts the definition of $d_\ast$.
This completes the proof.
**A.3 Proof of Lemma \[lemnodeoptimum\]** {#a.3-proof-of-lemma-lemnodeoptimum .unnumbered}
--------------------------------------------
With Lemma \[lemlimit\], we have that $\lim_{t\rightarrow \infty} V_i(t)=d_\ast^2$ for all $i\in\mathcal{V}$. Thus, $\forall\varepsilon>0, \exists M(\varepsilon)>0$, s.t., $$\begin{aligned}
\label{40}
d_\ast\leq \sqrt{V_i(t)}\leq d_\ast+\varepsilon\end{aligned}$$ for all $i$ and $t\geq M$. If $d_\ast=0$, the desired conclusion follows straightforwardly. Now we suppose $d_\ast>0$.
Assume that there exists a node $i_0$ satisfying $\limsup_{t\rightarrow \infty} \big| x_{i_0}(t)\big|_{\arg\min f_{i_0}}>0$. Then we can find a time sequence $\{t_k\}_1^\infty$ with $\lim_{k\rightarrow \infty}t_k =\infty$ and a constant $\delta$ such that $$\begin{aligned}
\label{42}
\big| x_{i_0}(t_k)\big|_{\arg\min f_{i_0}}\geq\delta, \ k=1,\dots.\end{aligned}$$ Denote also $B_1\doteq\big\{z: |z-z_\ast|\leq d_\ast+1\big \}$ and $G_1=\max\big\{ \nabla f_{i_0}(y):\ y\in B_1\big\}$. Assumption A1 ensures that $G_1$ is a finite number since $B_1$ is compact. By taking $\varepsilon=1$ in (\[40\]), we see that $x_i(t)\in B_1$ for all $i$ and $t\geq M(1)$. As a result, we have $$\begin{aligned}
\label{41}
\Big|\frac{d}{dt}{x}_{i_0}(t)\Big|=\Big|\sum_{j\in\mathcal{N}_{i_0}(\sigma(t))} a_{i_0 j}(t)(x_j-x_{i_0})+\nabla f_{i_0}(x_{i_0})\Big|\leq 2(n-1) a^\ast (d_\ast+1)+G_1.\end{aligned}$$
Combining (\[42\]) and (\[41\]), we conclude that $$\begin{aligned}
\label{43}
\big| x_{i_0}(t)\big|_{\arg\min f_{i_0}}\geq \frac{\delta}{2}, \ t\in[t_k,t_k+\tau],\end{aligned}$$ for all $k=1,\dots$, where by definition $\tau=\frac{\delta}{2\big( 2(n-1) a^\ast (d_\ast+1)+G_1\big)}$.
Now we introduce $$D_\delta\doteq \min \Big\{f_{i_0}(y)-f_{i_0}(z_\ast):\ \big| x_{i_0}(t)\big|_{\arg\min f_{i_0}}\geq \frac{\delta}{2}\ {\rm and}\ y\in B_1\Big\}.$$ Then we know $D_\delta >0$ again by the continuity of $f_{i_0}$. According to (\[45\]), (\[40\]), and (\[43\]), we obtain $$\begin{aligned}
\frac{d}{dt} V_{i_0}(t)
&\leq 2(N-1)a^\ast\Big(-V_{i_0}(t)+(d_\ast+\varepsilon)\sqrt{V_{i_0}(t)}\Big)+f_{i_0}\big(z_\ast\big)-f_{i_0}\big(x_{i_0}(t)\big)\nonumber\\
&\leq 2(N-1)a^\ast (d_\ast+\varepsilon)\varepsilon -D_\delta,\end{aligned}$$ for $t\in [t_k,t_k+\tau]$, $k=1,\dots$. This leads to $$\begin{aligned}
\label{46}
V_{i_0}(t_k+\tau)&\leq V_{i_0}(t_k)+\Big(2(N-1)a^\ast (d_\ast+\varepsilon)\varepsilon -D_\delta\Big)\tau \nonumber\\
&\leq d_\ast+\varepsilon+\Big(2(N-1)a^\ast (d_\ast+\varepsilon)\varepsilon -D_\delta\Big)\tau\nonumber\\
&< d_\ast\end{aligned}$$ as long as $\varepsilon$ is sufficiently small so that $$\varepsilon \Big(1+2(N-1)a^\ast (d_\ast+\varepsilon)\Big) <D_\delta \tau.$$ We see that (\[46\]) contradicts (\[40\]). The desired conclusion thus follows.
[99]{}
S. Boyd and L. Vandenberghe. [*Convex Optimization*]{}. New York, NY: Cambridge University Press, 2004.
D. P. Bertsekas. [*Nonlinear Programming*]{}. Second Edition, Athena Scientific, Belmont, Massachusetts, 1999.
E. K. P. Chong and S. H. Zak. [*An Introduction to Optimization.*]{} 2nd Edition, Wiley, 2001.
R. T. Rockafellar. [*Convex Analysis.*]{} New Jersey: Princeton University Press, 1972.
J. Aubin and A. Cellina. [*Differential Inclusions*]{}. Berlin: Speringer-Verlag, 1984.
M. Yokoo. [*Distributed Constraint Satisfaction: Foundations of Cooperation in Multi-agent Systems*]{}, Springer, 2001.
B. Faltings. [*Distributed Constraint Programming.*]{} Elsevier, 2006.
C. Godsil and G. Royle, New York: Springer-Verlag, 2001.
N. Rouche, P. Habets, and M. Laloy, . New York: Springer-Verlag, 1977.
J. Danskin, “The theory of max-min, with applications," [*SIAM J. Appl. Math.*]{}, vol. 14, 641-664, 1966.
F. Clarke, Yu.S. Ledyaev, R. Stern, and P. Wolenski. [*Nonsmooth Analysis and Control Theory*]{}. Berlin: Speringer-Verlag, 1998.
S. Martinez, J. Cortes, and F. Bullo, “ Motion coordination with distributed information," [*IEEE Control Syst. Mag.*]{}, vol. 27, no. 4, pp. 75-88, 2007.
J. Cortés, “Discontinuous dynamical systems–A tutorial on solutions, nonsmooth analysis, and stability," [*IEEE Control Systems Magazine*]{}, vol. 28, no. 3, 36-73, 2008.
R. Olfati-Saber, “Flocking for multi-agent dynamic systems: algorithms and theory," [*IEEE Trans. Autom. Control*]{}, vol. 51, no. 3, pp. 401-420, 2006.
M. Cao, A. S. Morse and B. D. O. Anderson, “Reaching a consensus in a dynamically changing environment: a graphical approach," , vol. 47, no. 2, 575-600, 2008.
J. M. Hendrickx, A. Olshevsky, and J. N. Tsitsiklis, “Distributed anonymous discrete function computation," [*IEEE Trans. Autom. Control*]{}, vol. 56, no. 10, pp. 2276-2289, 2011.
R. Olfati-Saber and R. Murray, “Consensus problems in the networks of agents with switching topology and time delays," [*IEEE Trans. Autom. Control*]{}, vol. 49, no. 9, pp. 1520-1533, 2004.
J. Fax and R. Murray, “Information flow and cooperative control of vehicle formations," , vol. 49, no. 9, 1465-1476, 2004.
H. G. Tanner, A. Jadbabaie, G. J. Pappas, “Flocking in fixed and switching networks," [*IEEE Trans. Autom. Control*]{}, vol. 52, no.5, pp. 863-868, 2007.
A. Jadbabaie, J. Lin, and A. S. Morse, “Coordination of groups of mobile autonomous agents using nearest neighbor rules," [*IEEE Trans. Autom. Control*]{}, vol. 48, no. 6, pp. 988-1001, 2003.
J. Tsitsiklis, D. Bertsekas, and M. Athans, “Distributed asynchronous deterministic and stochastic gradient optimization algorithms," [*IEEE Trans. Autom. Control*]{}, vol. 31, no. 9, pp. 803-812, 1986.
W. Ren and R. Beard. [*Distributed Consensus in Multi-vehicle Cooperative Control*]{}, Springer-Verlag, London, 2008.
W. Ren and R. Beard, “Consensus seeking in multi-agent systems under dynamically changing interaction topologies," [*IEEE Trans. Autom. Control*]{}, vol. 50, no. 5, pp. 655-661, 2005.
F. Xiao and L. Wang, “Asynchronous consensus in continuous-time multi-agent systems with switching topology and time-varying delays," [*IEEE Trans. Automatic Control*]{}, vol. 53, no. 8, 1804-1816, 2008.
G. Shi and Y. Hong, “Global target aggregation and state agreement of nonlinear multi-agent systems with switching topologies," [*Automatica*]{}, vol. 45, no. 5, pp. 1165-1175, 2009.
G. Shi, Y. Hong and K. H. Johansson, “Connectivity and set tracking of multi-agent systems guided by multiple moving leaders," [*IEEE Trans. Autom. Control*]{}, vol. 57, no. 3, pp. 663-676, 2012.
G. Shi and K. H. Johansson,“Multi-agent robust consensus-Part I: convergence analysis," IEEE CDC, pp. 5744-5749, 2011.
Z. Lin, B. Francis, and M. Maggiore, “State agreement for continuous-time coupled nonlinear systems," [*SIAM J. Control Optim.*]{}, vol. 46, no. 1, pp. 288-307, 2007.
F. Xiao and L. Wang, “Asynchronous consensus in continuous-time multi-agent systems with switching topology and time-varying delays," [*IEEE Trans. Automatic Control*]{}, vol. 53, no. 8, 1804-1816, 2008.
L. Moreau, “Stability of multiagent systems with time-dependent communication links," [*IEEE Trans. Autom. Control*]{}, vol. 50, no. 2, pp. 169-182, 2005.
A. Nedić, A. Olshevsky, A. Ozdaglar, and J. N. Tsitsiklis, “Distributed subgradient methods and quantization effects," in [*Proc. IEEE Conf. on Decision and Control, Cancun, Mexico*]{}, pp. 4177-4184, 2008.
A. Nedić and A. Ozdaglar, “Distributed subgradient methods for multi-agent optimization," [*IEEE Trans. Autom. Control*]{}, vol. 54, no. 1, pp. 48-61, 2009.
A. Nedić, A. Ozdaglar and P. A. Parrilo, “Constrained consensus and optimization in multi-agent networks," [*IEEE Trans. Autom. Control*]{}, vol. 55, no. 4, pp. 922-938, 2010.
G. Shi and K. H. Johansson, “Randomized optimal consensus of multi-agent systems," [*Automatica*]{}, to appear, arxiv version available from [http://arxiv.org/abs/1108.3223]{}.
M. Rabbat and R. Nowak, “Distributed optimization in sensor networks," in [*IPSN’04*]{}, pp. 20-27, 2004.
B. Johansson, T. Keviczky, M. Johansson, and K. H. Johansson, “Subgradient methods and consensus algorithms for solving convex optimization problems," in [*Proc. IEEE Conf. on Decision and Control*]{}, Cancun, Mexico, pp. 4185-4190, 2008.
S. S. Ram, A. Nedić, and V. V. Veeravalli, “Stochastic incremental gradient descent for estimation in sensor networks," in [*Proc. Asilomar Conference on Signals, Systems, and Computers*]{}, Pacific Grove, pp. 582-586, 2007.
J. Lu, C. Y. Tang, P. R. Regier, and T. D. Bow, “Gossip algorithms for convex consensus optimization over networks," [*IEEE Trans. Autom. Control*]{}, vol. 56, pp. 2917-2923, 2011.
J. Lu and C. Y. Tang, “Zero-gradient-sum algorithms for distributed convex optimization: the continuous-time case," [*IEEE Trans. Autom. Control*]{}, vol. 59, no. 7, pp. 2348-2354, 2012.
M. V. Solodov, “Incremental gradient algorithms with stepsizes bounded away from zero," [*Computational Optimization and Applications*]{}, vol. 11, pp. 23-35, 1998.
A. Nedić and D.P. Bertsekas, “Incremental subgradient methods for nondifferentiable optimization," [*SIAM J. on Optimization*]{}, vol. 12, pp. 109-138, 2001.
S. S. Ram, A. Nedić, and V. V. Veeravalli, “Incremental stochastic subgradient algorithms for convex optimization," [*SIAM Journal on Optimization*]{}, vol. 20, no. 2, 691-717, 2009.
B. Johansson, M. Rabi and M. Johansson. “A randomized incremental subgradient method for distributed optimization in networked systems," [*SIAM Journal on Optimization*]{}, vol. 20, no. 3, pp. 1157-1170, 2009.
D. Jakovetić, J. Xavier and J. M. F. Moura, “Cooperative convex optimization in networked systems: augmented lagrangian algorithms with directed gossip communication," http://arxiv.org/abs/1007.3706, 2011.
K. Arrow, L. Hurwicz and H. Uzawa. [*Studies in linear and non-linear programming*]{}. Stanford University Press, 1958.
R. W. Brockett, “Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems," [*Linear Algebra and its Applications*]{}, vol. 146, pp. 79-91, 1991.
L. Ljung, “Analysis of recursive stochastic algorithms," [*IEEE Trans. Autom. Control*]{}, vol. 22, no. 4, pp. 551-575, 1977.
J. Wang and N. Elia, “Control approach to distributed optimization," [*The 48th Annual Allerton Conference*]{}, pp. 557-561, 2010.
H.-B. Dürr and C. Ebenbauer, “A smooth vector field for saddle point problems," [*The 50th IEEE Conference on Decision and Control and European Control Conference*]{}, pp. 4654-4660, 2011.
G. Shi, K. H. Johansson and Y. Hong, “Reaching an optimal consensus: dynamical systems that compute intersections of convex sets," [*IEEE Transactions on Automatic Control*]{}, to appear, arxiv version available from [http://arxiv.org/abs/1112.1333]{}.
[^1]: The authors are with ACCESS Linnaeus Centre, School of Electrical Engineering, Royal Institute of Technology, Stockholm 10044, Sweden. Email: [[email protected], [email protected], [email protected]]{}
[^2]: This work has been supported in part by the Knut and Alice Wallenberg Foundation, the Swedish Research Council, and KTH SRA TNG.
| ArXiv |
---
abstract: 'The particle unbound $^{26}$O nucleus is located outside the neutron drip line, and spontaneously decays by emitting two neutrons with a relatively long life time due to the centrifugal barrier. We study the decay of this nucleus with a three-body model assuming an inert $^{24}$O core and two valence neutrons. We first point out the importance of the neutron-neutron final state interaction in the observed decay energy spectrum. We also show that the energy and and angular distributions for the two emitted neutrons manifest a clear evidence for the strong neutron-neutron correlation in the three-body resonance state. In particular, we find an enhancement of two-neutron emission in back-to-back directions. This is interpreted as a consequence of [*dineutron correlation*]{}, with which the two neutrons are spatially localized before the emission.'
author:
- 'K. Hagino'
- 'H. Sagawa'
title: ' Correlated two-neutron emission in the decay of unbound nucleus $^{26}$O'
---
Correlations among particles lead to a variety of rich phenomena in many-fermion systems, such as superconductivity and superfluidity. The spatial distribution of particles is also affected by the correlations. For many-electron systems, the Coulomb repulsion between electrons yields the so called Coulomb hole, in which the distribution of the second electron is largely suppressed in the vicinity of the first electron [@CN61; @RRB78]. In atomic nuclei, in contrast, an attractive nuclear force leads to the dineutron and diproton correlations, with which two nucleons are spatially localized in the surface region of nuclei[@BBR67; @CIMV84]. These nuclear correlations have attracted lots of attention recently [@BE91; @Zhukov93; @HS05; @MMS05; @PSS07], in connection to physics of weakly bound nuclei.
In order to probe the inter-particle correlation, it has been a standard way in atomic physics to measure a double ionization with strong laser fields[@WSD94; @WGW00; @BKJ12; @BLHE12]. It has been observed that the ionization rate is significantly enhanced due to the electronic correlation, and moreover, there is a strong momentum correlation between the two emitted electrons. The corresponding experiment in nuclear physics is the Coulomb breakup of the Borromean nuclei $^{11}$Li and $^6$He, in which those nuclei are broken up to the core nuclei, $^9$Li and $^4$He, and two neutrons in the Coulomb field of a target nucleus [@N06; @A99; @NK12]. The observed breakup probabilities, especially those for the $^{11}$Li nucleus, show a sharp peak in the low-energy region, which can be accounted for only by taking into account the neutron-neutron correlations. Furthermore, from the observed strength distribution, the opening angle between the valence neutrons in the ground state of the Borromean nuclei has been inferred employing the cluster sum rule [@N06; @HS07; @BH07]. For both $^{11}$Li and $^6$He, the extracted opening angles were significantly smaller than the value for the independent neutrons, that is, 90 degrees, and clearly indicate the existence of the dineutron correlation.
A small drawback with the cluster sum rule approach is that it yields only an expectation value of the opening angle and a detailed angular distribution cannot be studied with this method. For this reason, the energy and the angular distributions of the emitted neutrons from the Coulomb breakup have been investigated[@EB92]. However, it has been concluded that those distributions are largely determined by the properties of the neutron-core system, and thus it is difficult to acquire detailed information on the neutron-neutron correlations from the Coulomb breakup measurement [@HSNS09; @KKM10].
It is therefore desirable to seek for other probes for the nucleonic correlation. Among them, the two-proton radioactivity, that is, the spontaneous emission of two protons of proton-unbound nuclei, has been considered to be a good candidate for that purpose [@PKGR12]. An attractive feature of this phenomenon is that the two valence protons are emitted without an influence of disturbance of nuclei due to an external field. Very recently, the ground state [*two-neutron*]{} emission was discovered for $^{16}$Be[@SKB12]. Earlier measurements on the two-neutron emission include those for $^{10}$He [@J10] and $^{13}$Li [@J10; @A08]. These are a counter part of the two-proton emission of proton-rich nuclei, corresponding to a penetration of two neutrons over a centrifugal barrier. Subsequently, the two-neutron emission was discovered also for $^{26}$O[@LDK12; @CSA12] and $^{13}$Li [@KLD13]. So far, the experimental data have been analyzed only with a schematic dineutron model [@SKB12; @KLD13] (see also Ref. [@MOA12]). Although such schematic model appears to reproduce the data, realistic three-body model calculations with configuration mixings and full neutron-neutron correlations have been clearly urged.
In this paper, we apply the three-body model with a density-dependent contact interaction between the valence neutrons to the decay problem of $^{26}$O, assuming $^{24}$O to be an inert core. This model has been successfully applied to describe the ground state properties and the [Coulomb break-up]{} of neutron-rich nuclei[@BE91; @HS05; @HSNS09; @EBH97]. In order to describe the decay of neutron-unbound nucleus, we shall take into account the couplings to continuum by the Green’s function technique, which was invented in Ref. [@EB92] in order to describe the continuum dipole excitations of $^{11}$Li. We shall discuss the role of neutron-neutron correlation in the decay probability, as well as in the energy and the angular distributions of the emitted neutrons.
In the experiment of Ref. [@LDK12], the $^{26}$O nucleus was produced in the single proton-knockout reaction from a secondary $^{27}$F beam. We therefore first construct the ground state of $^{27}$F with a three-body model, assuming the $^{25}$F+$n$+$n$ structure. We then assume a sudden proton removal, that is, the $^{25}$F core changes to $^{24}$O keeping the configuration for the $n$+$n$ subsystem of $^{26}$O to be the same as in the ground state of $^{27}$F. This initial state, $\Psi_i$, is then evolved with the Hamiltonian for the three-body $^{24}$O+$n$+$n$ system for the two-neutron decay.
We therefore consider two three-body Hamiltonians, one for the initial state $^{25}$F+$n$+$n$ and the other for the final state $^{24}$O+$n$+$n$. For both the systems, we use similar Hamiltonians as that in Refs. [@HS05; @EBH97], $$H=\hat{h}_{nC}(1)+\hat{h}_{nC}(2)+v(1,2)
+\frac{{\mbox{\boldmath $p$}}_1\cdot{\mbox{\boldmath $p$}}_2}{A_cm},
\label{3bh}$$ where $A_c$ is the mass number of the core nucleus, $m$ is the nucleon mass, and $\hat{h}_{nC}$ is the single-particle (s.p.) Hamiltonian for a valence neutron interacting with the core. The last term in Eq. (\[3bh\]) is the two-body part of the recoil kinetic energy of the core nucleus [@EBH97], while the one-body part is included in $\hat{h}_{nC}$. We use a contact interaction between the valence neutrons, $v$, given as[@BE91; @HS05; @EBH97], $$v({\mbox{\boldmath $r$}}_1,{\mbox{\boldmath $r$}}_2)=\delta({\mbox{\boldmath $r$}}_1-{\mbox{\boldmath $r$}}_2)
\left(v_0+\frac{v_\rho}{1+\exp[(r_1-R_\rho)/a_\rho]}\right).
\label{vnn}$$ Here, the strength $v_0$ is determined to be $-$857.2 MeV$\cdot$fm$^{3}$ from the scattering length for the $nn$ scattering together with the cutoff energy, which we take $E_{\rm cut}=30$ MeV. See Refs.[@HS05; @EBH97] for the details. The second term in Eq. (\[vnn\]) simulates the density dependence of the interaction. Taking $R_\rho=1.34\times A_c^{1/3}$ fm and $a_\rho$=0.72 fm, we adjust the value of $v_\rho$ to be 952.3 MeV$\cdot$fm$^{3}$ so as to reproduce the experimental two-neutron separation energy of $^{27}$F, $S_{\rm 2n}$=2.80(18) MeV[@JSM07].
We employ a Woods-Saxon form for the s.p. potential in $\hat{h}_{nC}$. For the $^{24}$O+$n+n$ system, we take $a=0.72$ fm and $R_0=1.25A_c^{1/3}$ fm with $A_c=24$, and determine the values of $V_0=-44.1$ MeV and $V_{ls}$=45.87 MeV$\cdot$fm$^2$ in order to reproduce the single-particle energies of $\epsilon_{2s_{1/2}}=-4.09(13)$ MeV and $\epsilon_{1d_{3/2}}=770^{+20}_{-10}$ keV [@H08]. This potential yields the width for the 1$d_{3/2}$ state of $\Gamma_{1d_{3/2}}=92.9$ keV, which is compared with the empirical value, $\Gamma_{1d_{3/2}}=172(30)$ keV [@H08]. For the $^{25}$F+$n+n$ system, one has to modify the Woods-Saxon potential in order to take into account the presence of the valence proton in the core nucleus. The important effect comes from the tensor force between the valence proton and neutrons [@O05], which primarily modifies the spin-orbit potential in the mean-field approximation[@SBF77; @CSFB07; @LBB07]. We thus use the same Woods-Saxon potential for $^{25}$F+$n+n$ system as that for the $^{24}$O+$n+n$ system except for the spin-orbit potential, whose strength is weakened to $V_{ls}$=33.50 MeV$\cdot$fm$^2$ in order to reproduce the energy of $1d_{3/2}$ state in $^{27}$F, $\epsilon_{1d_{3/2}}=-0.811$ MeV.
With the initial wave function thus obtained, the decay energy spectrum can be computed as [@EB92], $$\begin{aligned}
\frac{dP}{dE}&=&\frac{1}{\pi}\Im \langle \Psi_i|G_0(E)|\Psi_i\rangle \nonumber \\
&&-\frac{1}{\pi}
\Im\langle \Psi_i|G_0(E)v(1+G_0(E)v)^{-1}G_0(E)|\Psi_i\rangle.
\label{decayenergy}\end{aligned}$$ where $\Im$ denotes the imaginary part. In Eq. , $G_0(E)$ is the unperturbed Green’s function given by, $$G_0(E)=\sum_{\rm 1,2}\frac{|(j_1j_2)^{(0^+)}\rangle\langle(j_1j_2)^{(0^+)}|}
{e_1+e_2-E-i\eta},
\label{green0}$$ where $\eta$ is an infinitesimal number and the sum includes all independent two-particle states coupled to the total angular momentum of $J=0$ with the positive parity, described by the three-body Hamiltonian for $^{24}$O+$n+n$. As in our previous study for the continuum E1 excitations of the $^{11}$Li nucleus [@HSNS09], we have neglected the two-body part of the recoil kinetic energy in order to derive Eq. (\[decayenergy\]), while we keep all the recoil terms in constructing the initial state wave function.
![(Color online) The decay energy spectrum for the two-neutron emission decay of $^{26}$O. The solid line denotes the result with the full inclusion of the final state neutron-neutron ($nn$) interaction, while the dashed line shows the result without the final state $nn$ interaction. The theoretical curves are drawn with a finite width of 0.21 MeV, which is the same as the experimental energy resolution. The experimental data, normalized to the unit area, are taken from Ref. [@LDK12]. ](fig1)
Figure 1 shows the decay energy spectrum obtained with Eq. . The solid line shows the correlated spectrum, in which the final state $nn$ interaction is fully taken into account, while the dashed line shows the result without the final state $nn$ interaction. The latter corresponds to the first term in Eq. (\[decayenergy\]). Since the width of the three-body resonance state is extremely small, which is experimentally the order of 10$^{-10}$ MeV [@K13], we have introduced a finite width for a presentation purpose. That is, in evaluating the unperturbed Green’s function, Eq. (\[green0\]), we set $\eta$ =0.21 MeV, that is to be the same as the experimental energy resolution. Without the final state $nn$ interaction, the two valence neutrons in $^{26}$O occupy the s.p. resonance state of 1$d_{3/2}$ at 770 keV, and the peak in the decay energy spectrum appears at twice this energy. When the final state $nn$ interaction is taken into account, the peak is largely shifted towards a lower energy and appears at 0.14 MeV, in a good agreement with the experimental data.
\
The energy distribution of the two emitted neutrons is shown in Fig. 2, in which a decay amplitude is calculated to a specific two-particle final state [@EB92], $$\begin{aligned}
M_{j,l,k_1,k_2}
&=&\langle (jj)^{(00)}|1-vG_0+vG_0vG_0-\cdots|\Psi_i\rangle,
\label{amplitude1}
\\
&=&\langle (jj)^{(00)}|(1+vG_0)^{-1}|\Psi_i\rangle.
\label{amplitude2}\end{aligned}$$ The unperturbed Green’s function, $G_0$, is evaluated at $E=e_1+e_2$. Notice that a series of $-vG_0+vG_0vG_0-\cdots$ in Eq. (\[amplitude1\]) describes the multiple rescattering effect of the two neutrons during the emission due to the final state $nn$ interaction, which is included to the all orders in Eq. (\[amplitude2\]). In contrast to the case of decay energy spectra shown in Fig. 1, we take $\eta$ in Eq. (\[green0\]) to be an infinitesimal number in evaluating the unperturbed Green’s function and use the Gauss-Legendre integration technique for Eq. as described in Ref. [@EB92]. The energy spectrum is then computed as, $$\frac{d^2P}{de_1de_2}=\sum_{j,l}|M_{j,l,k_1,k_2}|^2\,\frac{dk_1}{de_1}
\frac{dk_2}{de_2},$$ where the factors $dk/de$ are due to the normalization of the continuum single-particle wave functions, for which we follow Ref. [@EB92].
Figure 2(a) shows the energy distribution obtained by switching off the final state $nn$ interaction. The energy distribution is dominated by the single-particle $d_{3/2}$ resonance state at 0.77 MeV. A ridge appears as in the energy distribution for dipole excitations of Borromean nuclei [@EB92; @HSNS09]. The energy distribution with the $nn$ final state interaction is shown in Fig. 2(b). The energy distribution is drastically changed, being highly concentrated along the line of $e_1+e_2\sim$ 0.14 MeV with an extremely small width. The variation with $e_1$ is weak along this line, although the maximum still appears at $e_1=e_2$. This is a clear manifestation of a three-body resonance, and is in marked contrast to the continuum dipole excitations, in which the final state $nn$ interaction does not affect much the shape of the energy distribution [@HSNS09].
The angular distribution of the emitted neutrons can be also calculated using the decay amplitude, Eq. (\[amplitude1\]). The amplitude for emitting the two neutrons with spin components of $s_1$ and $s_2$ and momenta ${\mbox{\boldmath $k$}}_1$ and ${\mbox{\boldmath $k$}}_2$ reads [@EB92], $$\begin{aligned}
f_{s_1s_2}({\mbox{\boldmath $k$}}_1,{\mbox{\boldmath $k$}}_2)&=&
\sum_{j,l}e^{-il\pi}e^{i(\delta_1+\delta_2)}\,
M_{j,l,k_1,k_2} \nonumber \\
&&\times \langle [{\cal Y}_{jl}(\hat{{\mbox{\boldmath $k$}}}_1)
{\cal Y}_{jl}(\hat{{\mbox{\boldmath $k$}}}_2)]^{(00)}|\chi_{s_1}\chi_{s_2}\rangle,
\label{angularamplitude}\end{aligned}$$ where ${\cal Y}_{jlm}$ is the spin-spherical harmonics, $\chi_s$ is the spin wave function, and $\delta$ is the nuclear phase shift. The angular distribution is then obtained as $$\frac{dP}{d\theta_{12}}=4\pi\sum_{s_1,s_2}
\int dk_1dk_2\, |f_{s_1s_2}(k_1,\hat{{\mbox{\boldmath $k$}}}_1=0,k_2,
\hat{{\mbox{\boldmath $k$}}}_{2}=\theta_{12})|^2,
\label{angular}$$ where we have set $z$-axis to be parallel to ${\mbox{\boldmath $k$}}_1$ and evaluated the angular distribution as a function of the opening angle, $\theta_{12}$, of the two emitted neutrons.
The angular distribution obtained without including the final state $nn$ interaction is shown by the dotted line in Fig. 3. The main component in the initial wave function, $\Psi_i$, is the $d_{3/2}$ configuration, and the angular distribution is almost symmetric around $\theta_{12}=\pi/2$. In the presence of the final state $nn$ interaction, the angular distribution becomes highly asymmetric, in which the emission of two neutrons in the opposite direction (that is, $\theta_{12}=\pi$) is enhanced[@GMZ13], as is shown by the solid line. Notice that we have obtained the correlated distribution by evaluating Eq. (\[angular\]) only at $e_1+e_2=0.14$ and then normalize it, since it is hard to carry out the integrations in Eq. (\[angular\]) when the resonance width is extremely small. We do not expect that this procedure causes any significant error in evaluating the angular distribution. The asymmetric angular distribution for the correlated case originates from the interference between opposite party components, as in the dineutron correlation in the density distribution [@CIMV84]. For the $^{26}$O nucleus, it is due to the interference between the $l=0$ and $l=1$ components. The dot-dashed line in Fig. 3 shows the result obtained by including only $l=0$ in Eq. (\[angularamplitude\]), while the dashed line shows the result with $l$=0 and 1. One can see that the angular distribution is almost exhausted by these two angular momenta and they contribute with almost equal amplitudes. For higher partial waves $l\geq2$, the scattering wave functions in Eq. (\[amplitude2\]) are highly damped inside the centrifugal barrier since the energy is quite low ($e_1\sim e_2 \sim$ 0.07 MeV). In other words, the two neutrons are rescattered into $s$-wave and $p$-wave states by multistep process due to the interaction $v$ (see Eq. (\[amplitude1\])) and these low $l$ components uniquely enhance the penetrability, even though the main component in the initial wave function is the $d$-wave state. This picture is consistent with what Grigorenko [*et al.*]{} have argued in Ref.[@GMZ13].
The enhancement of angular distribution at backward angles for $^{26}$O has also been seen theoretically in the dipole excitations of $^{11}$Li [@EB92] and both theoretically and experimentally in the two-proton emission decay of $^6$Be [@G09]. This reflects the spatial correlation of the three-body resonance state of $^{26}$O. Figure 4 shows the two-particle density for a resonance state of $^{26}$O obtained with the box boundary condition as a function of $r_1=r_2=r$ and the opening angle between the two neutron, $\theta_{12}$. One finds that the density distribution is well localized in the small $\theta_{12}$ region, which is clear manifestation of the dineutron correlation [@HS05]. It has been well known that the configurations with opposite parity have to contribute coherently in order to form the dineutron correlation [@CIMV84; @PSS07; @HVPBS11]. In the angular distribution in Fig. 3, a phase factor, $e^{-il\pi}$, in the amplitude in Eq. (\[angularamplitude\]) alters the sign of the contributions of odd partial waves, leading to the opposite tendency from the density, that is, the preference of emission of two-neutrons in the back-to-back angles. The nuclear phase shifts, $\delta_1+\delta_2$, plays a minor role in the decay of $^{26}$O, partly because the decay energy is extremely small. Evidently, the back-to-back emission of two neutrons in the momentum space from the decay of $^{26}$O is another manifestation of the strong dineutron correlation in the coordinate space of ground state density distribution.
For $^{16}$Be and $^{13}$Li, the experimental angular distributions show an enhancement of emission with relatively small opening angles[@SKB12; @KLD13]. It has yet to be clarified why these nuclei show different angular distributions from $^{26}$O (and from $^{6}$Be and $^{11}$Li). One possible reason is that the nuclear phase shift might play a more important role in these nuclei so that the phase factor $e^{-il\pi}$ is canceled out. Another reason may be the core excitation, with which the $nn$ configuration with coupled angular momenta of $J\neq0$ is largely admixed in the ground state wave function. In order to confirm these points, three-body model calculations for these nuclei with the core excitations are clearly needed, but we leave them as a future work.
In summary, we have used the three-body model with a contact neutron-neutron interaction in order to analyze the two-neutron emission decay of the unbound neutron-rich nucleus $^{26}$O. Using the Green’s function technique, we have analyzed the decay energy spectrum, the energy and the angular distributions of the two emitted neutrons. We have pointed out that the final state n-n interaction plays a crucial role to reproduce the strong low energy peak of the experimental decay energy spectrum. We have also argued that the energy distribution is a clear manifestation of a three-body resonance state and its density distribution is strongly reflected in the angular distribution of the emitted neutrons. In particular, the angular distribution clearly prefers the emission of the two neutrons in the back-to-back angles, that can be interpreted as a clear evidence for the dineutron correlation. So far, the energy and the angular distributions for the two-neutron decay of $^{26}$O have not yet been measured experimentally. It would be extremely intriguing if they will be measured at new generation RI beam facilities, such as the SAMURAI facility at RIBF at RIKEN [@AN13].
We thank Z. Kohley, T. Nakamura, A. Navin, Y. Kondo and T. Aumann for useful discussions. This work was supported by JSPS KAKENHI Grant Numbers 22540262 and 25105503.
[99]{}
C.A. Coulson and A.H. Neilson, Proc. Phys. Soc. [**78**]{}, 831 (1961).
P. Rehmus, C.C.J. Roothaan, and R.S. Berry, Chem. Phys. Lett. [**58**]{}, 321 (1978).
G.F. Bertsch, R.A. Broglia, and C. Riedel, Nucl. Phys. A[**91**]{}, 123 (1967).
F. Catara, A. Insolia, E. Maglione, and A. Vitturi, Phys. Rev. C[**28**]{}, 1091 (1984).
G.F. Bertsch and H. Esbensen, Ann. Phys. (N.Y.) [**209**]{}, 327 (1991).
M.V. Zhukov [*et. al.*]{}, Phys. Rep. [**231**]{}, 151 (1993).
K. Hagino and H. Sagawa, Phys. Rev. C[**72**]{}, 044321 (2005).
M. Matsuo, K. Mizuyama, and Y. Serizawa, Phys. Rev. C[**71**]{}, 064326 (2005).
N. Pillet, N. Sandulescu, and P. Schuck, Phys. Rev. C[**76**]{}, 024310 (2007).
B. Walker [*et al.*]{}, Phys. Rev. Lett. [**73**]{}, 1227 (1994).
Th. Weber [*et al.*]{}, Nature [**405**]{}, 658 (2000).
B. Bergues [*et al.*]{}, Nature Commun. [**3**]{}, 813 (2012).
W. Becker, X.J. Liu, P.J. Ho, and J.H. Eberly, Rev. Mod. Phys. [**84**]{}, 1011 (2012).
T. Nakamura [*et al.*]{}, Phys. Rev. Lett. [**96**]{}, 252502 (2006).
T. Aumann [*et al.*]{}, Phys. Rev. C[**59**]{}, 1252 (1999).
T. Nakamura and Y. Kondo, Lec. Notes in Phys. [**848**]{}, 67 (2012).
K. Hagino and H. Sagawa, Phys. Rev. C[**76**]{}, 047302 (2007).
C.A. Bertulani and M.S. Hussein, Phys. Rev. C[**76**]{}, 051602 (2007).
H. Esbensen and G.F. Bertsch, Nucl. Phys. [**A542**]{}, 310 (1992).
K. Hagino, H. Sagawa, T. Nakamura, and S. Shimoura, Phys. Rev. C[**80**]{}, 031301(R) (2009).
Y. Kikuchi [*et al.*]{}, Phys. Rev. C[**81**]{}, 044308 (2010).
M. Pfützner, M. Karny, L.V. Grigorenko, and K. Riisager, Rev. Mode. Phys. [**84**]{}, 567 (2012), and references therein.
A. Spyrou [*et al.*]{}, Phys. Rev. Lett. [**108**]{}, 102501 (2012).
H.T. Johansson [*et al.*]{}, Nucl. Phys. [**A842**]{}, 15 (2010); Nucl. Phys. [**A847**]{}, 66 (2010).
Yu. Aksyutina [*et al.*]{}, Phys. Lett. [**B666**]{}, 430 (2008).
E. Lunderberg [*et al.*]{}, Phys. Rev. Lett. [**108**]{}, 142503 (21012).
C. Caesar [*et al.*]{}, arXiv:1209.0156.
Z. Kohley [*et al.*]{}, Phys. Rev. C[**87**]{}, 011304(R) (2013).
F.M. Marqués [*et al.*]{}, Phys. Rev. Lett. [**109**]{}, 239201 (2012).
H. Esbensen, G.F. Bertsch, and K. Hencken, Phys. Rev. C[**56**]{}, 3054(1997).
B. Jurado [*et al.*]{}, Phys. Lett. [**B649**]{}, 43 (2007).
C.R. Hoffman [*et al.*]{}, Phys. Rev. Lett. [**100**]{}, 152502 (2008).
T. Otsuka [*et al.*]{}, Phys. Rev. Lett. [**95**]{}, 232502 (2005); Phys. Rev. Lett. [**105**]{}, 032501 (2010).
Fl. Stancu, D.M. Brink, and H. Flocard, Phys. Lett. [**68B**]{}, 108 (1977).
G. Colo, H. Sagawa, S. Fracasso, and P.F. Bortignon, Phys. Lett. [**B646**]{}, 227 (2007).
T. Lessinski [*et al.*]{}, Phys. Rev. C[**76**]{}, 014312 (2007).
Z. Kohley [*et al.*]{}, Phys. Rev. Lett. [**110**]{}, 152501 (2013).
L.V. Grigorenko, I.G. Mukha, and M.V. Zhukov, Phys. Rev. Lett. [**111**]{}, 042501 (2013).
L.V. Grigorenko [*et al.*]{}, Phys. Rev. C[**80**]{}, 034602 (2009).
K. Hagino, A. Vitturi, F. Perez-Bernal, and H. Sagawa, J. of Phys. G[**38**]{}, 015105 (2011).
T. Aumann and T. Nakamura, Phys. Scr. [**T152**]{}, 014012 (2013).
| ArXiv |
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abstract: 'A relatively simple model problem where a single electron moves in two relativistically-strong obliquely intersecting plane wave-packets is studied using a number of different numerical solvers. It is shown that, in general, even the most advanced solvers are unable to obtain converged solutions for more than about 100 fs in contrast to the single plane-wave problem, and that some basic metrics of the orbit show enormous sensitivity to the initial conditions. At a bare minimum this indicates an unusual degree of non-linearity, and may well indicate that the dynamics of this system are chaotic.'
author:
- 'A.P.L.Robinson'
- 'K.Tangtartharakul'
- 'K.Weichman'
- 'A.V.Arefiev'
title: Extreme Nonlinear Dynamics in Vacuum Laser Acceleration with a Crossed Beam Configuration
---
Introduction
============
Since the development of Chirped Pulse Amplification lasers [@danson_vulcan_2004; @hernandez-gomez_vulcan_2010; @tang_optical_2008; @hooker_improving_2011], the field of ultra-intense laser-matter interactions has grown considerably. Initially this technology allowed the development of TW-scale lasers that breached the 10$^{18}$Wcm$^{-2}\mu$m$^2$, but subsequent progress has lead to the construction of 10PW scale systems [@hernandez-gomez_vulcan_2010], with 100PW systems under development. The field now spans a large number of sub-topics including laser wakefield acceleration of electrons [@mangles_monoenergetic_2004], laser-driven ion acceleration [@gaillard_increased_2011], laser- driven x-ray [@kneip_observation_2008] and neutron sources, advanced inertial fusion concepts such as Fast Ignition [@m.tabak_review_2005], studies of both Warm Dense Matter and Hot Dense Matter [@d.j.hoarty_observations_2013], radiation reaction studies, and even probing QED physics [@heinzl_observation_2006; @heinzl_exploring_2009]. It is likely that the latter topics in that list will become more dominant as multi-PW facilities become fully operational in the following years. Numerical simulation codes, particularly Particle-in-Cell (PIC) codes [@birdsall_plasma_2018; @pukhov_strong_2003], have been instrumental in driving the field forward, both in terms of interpreting experiments and in making predictions that have motivated crucial experimental work. Perhaps the best known example of PIC’s predictive capabilities is that of Pukhov and Meyer-ter-Vehn’s prediction [@pukhov_laser_2002] of the ‘bubble regime’ of laser wakefield acceleration, which was later validated by three different research groups [@mangles_monoenergetic_2004].
The PIC algorithm is itself dependent on a number of algorithms, some of which were developed separately, such as the Yee FDTD method [@kane_yee_numerical_1966] for numerical electromagnetics. Importantly this includes a ‘particle-pusher’ algorithm which advances the macroparticles position and momentum. The quality and capability of any individual PIC code will depend on the set of algorithms chosen for these different components. A common choice for the particle-pusher is the Boris method [@birdsall_plasma_2018]. The Boris method is a second order accurate leapfrog-type method that is centred in time. It is a method that has enjoyed considerable success, and which has been employed in a number of different PIC codes including EPOCH [@arber_contemporary_2015]. Developing higher order versions of the Boris method is a non-trivial proposition, and it is has been questioned whether or this endeavour would actually yield any serious benefits to laser-plasma or accelerator science [@higuera_structure-preserving_2017; @londrillo_2010].
In the past few years however it has been recognized that the Boris method has at least one serious defect, namely that constant motion is not maintained in the case of uniform crossed ${\bf E} \ne 0$ and ${\bf B} \ne 0$ fields (for the choice of particle velocity for which a force-free scenario is obtained). This was first identified by Vay [@vay_simulation_2008], who proposed a variation on the Boris pusher that resolved this issue. Later Higuera and Cary [@higuera_structure-preserving_2017] proposed an algorithm that both solved the issue of the ${\bf E} \times {\bf B}$ velocity and which also preserved phase-space volume (unlike Vay’s method). Alongside these developments, Arefiev also showed that considerable care needs to be taken in setting the time step when integrating the orbits of an electron in a relativistic laser pulse. Altogether these developments underline how the particle-pusher problem needs careful study to ensure that particle-pusher algorithms can be trusted when employed to study the strongly relativistic and highly complex configurations encountered in ultra-intense laser-matter problems.
Despite these developments the methods of Vay and Higuerra-Cary are still only second order accurate methods. For problems where the overall behaviour of the system is quite ’regular’ this means that they will be quite adequate in the majority of cases. What has not been given so much consideration is whether the dynamics can always be assumed to be sufficiently ‘regular’. Some researchers have pointed out that some laser interaction problems will have a ‘stochastic’ nature [@z.-m.sheng_stochastic_2002; @sheng_efficient_2004; @meyer-ter-vehn_electron_1999], this terminology appears to actually mean that the dynamics are [*chaotic*]{}[@strogatz_nonlinear_2015; @ott_coping_1994; @handfinch]. If the Lyapunov time is larger than the time-scale of interest then this is not a problem for numerical simulation. However if the Lyapunov time becomes much shorter than the time-scale of interest then the ability to predict future states of the system will be highly limited even with very sophisticated numerical solvers.
In this paper we present a relatively simple test problem for a single electron : two plane EM Gaussian wave-packets that cross at an oblique angle and which are $\pi$ out of phase. The electron is initially at rest and which sits ‘off-axis’ by a fraction of the vacuum wavelength. To the best of the authors’ knowledge this problem does not have an analytic solution. We have studied the ability of a number of leapfrog pushers, RK4 method, and more sophisticated adaptive algorithms to solve the electron orbits in this problem. We have found that, in general, all of these solvers are only able to obtain converged orbits for a fraction ($<$20%) of the total problem duration (100-200 fs out of 1 ps). Complete converged orbits are only obtained in a few cases, and usually only the RK4 method (or better) is able to do this. A survey of the sensitivity to initial conditions was carried out, and it was found that there are regions of parameter space which exhibit extreme sensitivity to initial conditions. This indicates that this problem, however simple it may seem, in fact is [*chaotic*]{} in nature, as expected given earlier studies [@z.-m.sheng_stochastic_2002; @sheng_efficient_2004; @meyer-ter-vehn_electron_1999], however in this case it would appear that the chaotic dynamics is severly problematic for numerical integration. We suggest that this may have important ramifications for both Vacuum Laser Acceleration (VLA) [@hartemann_chirped-pulse_1999; @thevenet_vacuum_2016; @plettner_proof--principle_2005; @troha_vacuum_1999; @robinson_interaction_2018] and Direct Laser Acceleration (DLA) [@pukhov_particle_1999; @naseri_channeling_2012; @arefiev_beyond_2016; @robinson_breaking_2017; @arefiev_spontaneous_2016; @huang_characteristics_2016; @zhang_synergistic_2015; @willingale_surface_2013; @robinson_generating_2013].
Description of Model Problem {#model}
============================
We consider a problem where two relativisitically-strong plane EM wave-packets intersect obliquely. We want to study the relativistic motion of an electron that is initially at rest. This can be described by the following formulae for the electric fields of the incident waves:
$$\begin{aligned}
&&{\bf E} = {\bf E}_1 + {\bf E}_2, \\
&&{\bf E}_1 = E\cos\psi_1f_{env,1}\left[ -\sin(\theta_{cb}/2),\cos(\theta_{cb}/2),0\right], \\
&&{\bf E}_2 = E\cos\psi_2f_{env,2}\left[ \sin(\theta_{cb}/2),\cos(\theta_{cb}/2),0\right],\end{aligned}$$
![Schematic of the simulation set-up showing key parameters.[]{data-label="sketch"}](sketch5.png){width="\columnwidth"}
where $\psi_1 = {\bf k_1.r} - \omega_L{t} + \phi_1$, $\psi_2 = {\bf k_2.r} - \omega_L{t}+\phi_2$, ${\bf k}_1 = [\cos(\theta_{cb}/2),\sin(\theta_{cb}/2),0]$, ${\bf k}_2 = [\cos(\theta_{cb}/2),-\sin(\theta_{cb}/2),0]$. For the envelope functions, we use $f_{env} = \exp(-(\psi/k_L + 5c\tau_L)^2/(2c\tau_L))$. There are corresponding magnetic fields in the z-direction. This corresponds to two intersecting plane wave-packets that are aligned obliquely to the $x$-axis with the E-field polarized in the $xy$-plane in each case. The angle between the wavevectors of the two wave-packets is $\theta_{cb}$. For our baseline problem we consider the case where $E = 5\omega_Lm_ec/e$ (i.e. each plane wave-packet has $a_0 =$ 5), $\theta_{cb}= $40$^\circ$, $\lambda_L =$1 $\mu$m, and $\tau_L = $20 fs. The two wavepackets are $\pi$ out of phase, i.e. $\phi_1 =$ 0, $\phi_2 = \pi$. The electron is initially at rest at $x =$0,$z = $0, and $y = y_0$. A schematic of the problem is shown in fig.\[sketch\].
Since this problem is quite close to that considered previously [@z.-m.sheng_stochastic_2002; @sheng_efficient_2004; @meyer-ter-vehn_electron_1999], we should expect that chaotic dynamics are likely to be encountered. A very significant difference with earlier studies is that the value of the normalized vector potential in this case is significantly larger ($a >$ 5 here). However since Mendonca’s [@mendonca_1983] criterion is $a_1a_2 >$1/16 we expect that chaotic dynamics will only be more prevalent in this problem.
Analysis with Standard Algorithms {#standard}
=================================
In the first part of our study we have used the following solvers : (i) the Boris pusher [@birdsall_plasma_2018], (ii) the Vay pusher[@vay_simulation_2008], (iii) the Higuera-Cary pusher[@higuera_structure-preserving_2017], and (iv) the 4th order Runge-Kutta (RK4) algorithm [@numrecipes], to study this problem. Note that (i)–(iii) are formally 2nd order algorithms (although they differ in their treatment of the ${\bf E} \times {\bf B}$ velocity) , and only (iv) is formally 4th order. These were applied to study the baseline case (case 1). We shall not re-state the details of these here, and we refer the reader to the given references for further details. We have tested and checked our implementations, in particular by testing that they reproduce the motion in a single plane wave-packet. The baseline numerical integration is carried out over 18000 time steps with $\Delta{t} = $0.05 fs. To examine convergence, the time step is multiplied by a factor $1/M$, and the total number of time steps by $M$ in order to keep the total duration of the integration constant. In general, we regard two trajectories as being converged if the variables in question are within 5% of one another. All of these solvers reproduce the analytic prediction for the single plane-wave problem with $M$ = 1 and the solutions of each solver are practically identical.
For each solver we obtained solutions of $M =$1,2,4, and 8. The results for the Boris pusher, in terms of $p_y$ are shown in fig.\[fig:figure1\]. By following sequence of cases, we can see that the solution is not converging.
![\[fig:figure1\]The results from the Boris Pusher for the baseline case. Value of $M$ for each line is shown in the legend. Solution shows no sign of convergence with increasing $M$.](figure1.pdf){width="\columnwidth"}
The behaviour of the Boris pusher is in sharp contrast with the RK4 algorithm. The results of the RK4 algorithm, also in terms of $p_y$, are shown in fig.\[fig:figure2\]. Here the four curves almost perfectly overlap, showing clearly that there has been very good convergence, and that it has happened very rapidly.
![\[fig:figure2\]The results from the RK4 algorithm for the baseline case. Value of $M$ for each line is shown in the legend. All four curves overlap almost perfectly, indicating extremely rapid convergence.](figure4.pdf){width="\columnwidth"}
The behaviour of both the Vay and Higuera-Cary pushers are shown in fig.s \[fig:figure3\] and \[fig:figure4\]. By comparing fig.s \[fig:figure3\] and \[fig:figure4\] to fig. \[fig:figure2\] we can see that, when $M =$8, both the Vay and the Higuera-Cary pushers come very close to the solution obtained by the RK4 algorithm. This should lead to confidence in the solution obtained by the RK4 solver. It is clearly good that both the Vay and Higuera-Cary pushers are able to eventually reach this solution, however the rate of convergence is rather slow, and it requires that one adopts a very small time ($M =$8) time step. In figure \[fig:figure5\] we directly compare the Vay, Higuera-Cary, and RK4 solutions for $M =$8. As can be seen they all lie extremely close to one another, showing that the Vay and Higuera-Cary solvers are able to approach the RK4 solution, whereas the Boris solver cannot for $M \le 8$.
![\[fig:figure3\]The results from the Vay Pusher for the baseline case. Value of $M$ for each line is shown in the legend. ](figure2.pdf){width="\columnwidth"}
![\[fig:figure4\]The results from the Higuera-Cary pusher for the baseline case. Value of $M$ for each line is shown in the legend. ](figure3.pdf){width="\columnwidth"}
![\[fig:figure5\]Comparison of the solutions from the Vay, Higuera-Cary, and RK4 pushers for $M =$8, showing that, in the $M =$8 case, convergence is obtained. ](figure5.pdf){width="\columnwidth"}
In the second part of our study we extended this to multiple cases to see if these findings reflected a general trend. As is evident from fig.s \[fig:figure1\]–\[fig:figure4\], even when convergence is not obtained over the entire 900 fs, convergence in fact can occur over a time period that is a fraction of the total duration of the problem. When extending the study we instead looked at the fraction of the problem duration over which convergence was obtained (instead of whether or not [*total*]{} convergence was obtained). The results are summarized in table \[table1\], which shows the convergence obtained for each case as a percentage of the total problem duration (900 fs), and for each solver tried. The special cases of the convergence obtained by the Vay and Higuera-Cary pushers in the baseline case are noted by an asterisk.
Case Boris Vay Hig.-Cary RK4
------------------------------------------------------------- ------- ------- ----------- -------
1.$a_0 =$5, $y_0 = \lambda/4$, $\theta_{cb} = $ 40$^\circ$ 14.7 100\* 100\* 100
2.$a_0 =$5, $y_0 = \lambda/2$, $\theta_{cb} = $ 40$^\circ$ 11.1 11.0 9.1 12.1
3.$a_0 =$10, $y_0 = \lambda/2$, $\theta_{cb} = $ 40$^\circ$ 12.8 13.3 12.9 100.0
4.$a_0 =$10, $y_0 = \lambda/4$, $\theta_{cb} = $ 40$^\circ$ 10.2 10.2 11.3 17.8
5.$a_0 =$5, $y_0 = \lambda/8$, $\theta_{cb} = $ 40$^\circ$ 10.9 11.6 11.6 17.6
6.$a_0 =$10, $y_0 = \lambda/8$, $\theta_{cb}= $ 40$^\circ$ 10.8 11.9 11.9 17.1
7.$a_0 =$5, $y_0 = \lambda/4$, $\theta_{cb} = $ 60$^\circ$ 11.6 11.9 11.7 14.0
8.$a_0 =$5, $y_0 = \lambda/4$, $\theta_{cb}= $ 80$^\circ$ 9.3 9.3 9.8 16.6
9.$a_0 =$5, $y_0 = \lambda/4$, $\theta_{cb}= $ 20$^\circ$ 14.8 14.8 14.8 41.3
10.$a_0 =$5, $y_0 = \lambda/4$, $\theta_{cb} = $ 10$^\circ$ 48.7 48.3 49.0 62.5
: \[table1\]Summary of results for different cases. Shown in the percentage of the total problem duration for which a given pusher is able to obtain convergence for $M \le$8. The special cases of the Vay and Higuera-Cary pushers in the baseline case are noted by an asterisk.
From Table \[table1\] we find that the baseline case unfortunately represents a rather optimistic one from the point of view of numerically solving this problem. In general we found that even the RK4 pusher was unable to produce converged solutions for more than 18% of the problem duration. Converged solutions over the full duration were only obtained by the RK4 solver in a couple of cases. Also as the approach angle, $\theta_{cb}$, becomes very small, it is much easier to obtain convergence.
All the leapfrog solvers perform less well than the RK4 pusher. The differences between the three are usually rather small (again suggesting that the baseline case, happens to be a special case). It therefore appears that, in general, the enhanced leapfrog solvers are not substantially better at the crossed beam problem than the Boris pusher.
We have also examined the effect that the different solvers have on distributions arising from an ensemble of different initial conditions. This was done for 10000 different particles initialized at rest with the initial $x$-position spanning -0.5 to +0.5$\lambda_L$ ($y_{init} = \lambda_L/4$). The problem was run up to 450 fs with $M = $1. Otherwise the problem corresponds to the baseline case. We compared the distributions that arose from using the Boris and the RK4 solvers, which are shown in fig.s \[fig:figure9\] and \[fig:figure10\] respectively.
![\[fig:figure9\]Distribution at 450 fs of ensemble calculation (see text) for the case of the Boris solver. ](crossed_borisphase.jpeg){width="\columnwidth"}
![\[fig:figure10\]Distribution at 450 fs of ensemble calculation (see text) for the case of the RK4 solver. ](crossed_rk4phase.jpeg){width="\columnwidth"}
In the case of the RK4 solver we see that there is a very strong spike at high energy, denoted as ’B’ in fig. \[fig:figure10\]. This feature is absent in fig. \[fig:figure9\], and instead we see a different feature denoated as ’A’ in this figure. Given that the strongest accumulations of particles are completely different for different solvers applied to the same ensemble/problem, we can conclude that the issues observed with single trajectories will lead to significant differences in particles distributions as well.
It therefore appears to be the case that the crossed beam problem presented here represents a far harder test than the single plane wave of single electron trajectory calculation. To the best of the author’s knowledge this is currently the hardest test case, at least specifically for laser-plasma studies, as the conventional particle pushers tested here are known to capable of producing fully converged solutions (for $M \le$ 8) over the full duration. This is certainly the case for the single plane wave problem. More importantly the results presented in Table \[table1\] already indicate the most likely reason as to why this problem is so challenging : namely that the dynamics has become chaotic. We see that, in the general case, a converged solution can only be obtained for a short period of time. We also see that there are strange isolated cases where a full converged solution can be obtained. The observation of these features motivated a more detailed study of the problem.
Parameter Scans with Advanced Algorithms {#advanced}
========================================
In the second phase of this study, another class of solvers was used, namely the [MATLAB]{} suite of ODE solvers. In broad terms, applying these solvers to the problem lead to results similar to those presented in Sec. \[standard\], with convergence only obtained over a limited period of time and for a small angle between the beams. Out of the entire suite, ODE113 performed the best. This solver is a variable-step, variable-order (VSVO) Adams-Bashforth-Moulton Predictor-Corrector solver of order 1 to 13. It was found that convergence was reliably obtained when the angle between the beams was limited to no more than $\theta_{cb} =$ 30$^\circ$. We have cross-checked the results obtained with ODE113 against the RK4 algorithm, and found good agreement between the two.
In order to study the sensitivity to the initial conditions, parameters scans were then carried out by varying $\theta_{cb}$, $\phi_1$, and $\phi_2$. For each set of initial conditions a calculation was run up to 200 fs. Two outputs were recorded : (a) the ratio of the final displacement in $y$ to that in $x$ ($r_y/r_x$), and (b) the time at which the maximum $\gamma$ occurred ($\tau_{\gamma,{max}}$, normalized to the laser period). Two types of scans were carried out, [*coarse*]{} and [*fine*]{}. For the coarse scans, 100 points were used for each parameter over a large range : $\pm \pi$ for phases and 10–30$^\circ$ for $\theta_{cb}$. For the fine scans, a fraction of each range was used and 200 points were then used for each parameter. In all other respects, the calculations are the same as the baseline calculation described in Sec. \[model\]. By moving from the analysis of Sec. \[standard\] where we looked at 10 cases to these parameter scans where we look at 10000-40000 cases per scan, we can obtain a much clearer idea of how sensitive this problem can be to the initial conditions.
In Fig.s \[fig:figure6\] and \[fig:figure7\] we show the results from a coarse parameter scan of $\theta_{cb}$ and $\phi_2$ with $\phi_1$ held fixed at 0$^{\circ}$. The sub-figures show plots of fine parameter scans in the regions indicated.
![\[fig:figure6\]Results from parameter scan over $\phi_2$ and $\theta_{cb}$ in terms of $r_y/r_x$. Main plot is a coarse scan, with fine scans as sub-plots a and b. ](AngleDeltaPhi_Slope_All.png){width="\columnwidth"}
![\[fig:figure7\]Results from parameter scan over $\phi_2$ and $\theta_{cb}$ in terms of $\tau_{\gamma,{max}}$. Main plot is a coarse scan, with fine scans as sub-plots a and b. ](AngleDeltaPhi_TimeMaxGamma_All.png){width="\columnwidth"}
What we observe from these extensive parameters scans is that the parameter space appears to consist of two types of regions. There are regions where the results of the calculations vary (relatively) slowly and smoothly as the initial conditions are changed. Examples of these are shown in the ‘a’ sub-figures in both fig. \[fig:figure6\] and fig. \[fig:figure7\]. There are also regions where small variations of the initial conditions leads to gross changes in the results including rapid changes in sign. Examples of these sub-regions are shown in the ‘b’ sub-figures in both fig. \[fig:figure6\] and fig. \[fig:figure7\].
As we were observing strong point-to-point changes in fig. \[fig:figure7\](b) along $\theta_{cb} = $ 28$^\circ$ we repeated this set of calculations at twice the resolution in $\phi_2$ (i.e. now with 400 points in $\phi_2$ across the ‘fine’ range). The results are shown in fig. \[fig:figure8\]. It can be seen that there is no improvement in terms of being able to ‘resolve’ the detail in this region.
![\[fig:figure8\]Line-out of Fig.\[fig:figure7\](b) along $\theta_{cb} =$ 28$^\circ$](LineOut_TimeGammaMax2.png){width="\columnwidth"}
We can summarize the results from this second phase of the study as follows : (i) we have done an extensive parameter scan of the initial conditions / problem parameters using an advanced ODE solver, (ii) this reveals regions in parameter space that are very sensitive to the initial conditions / problem parameters, (iii) we are not able to ‘resolve’ this sensitivity by successively refining the set of points over which we scan. These observations suggest that we are actually looking at a system that exhibits chaotic dynamics, as we expected from earlier studies.
Conclusions
===========
In this paper we have examined an apparently simple model problem in relativistic single electron motion relevant to ultra-intense laser-plasma interactions, involving two obliquely intersecting plane wave-packets. The findings for this model problem, which are presented herein can be summarized as follows:
1. [Under a wide range of conditions converged solutions cannot be obtained for a 1 ps period using a wide range of different solvers including the Boris method, 4th order Runge-Kutta, and the [MATLAB]{} suite of ODE solvers.]{}
2. [Converged solutions appear to occur in isolated ranges of problem parameters.]{}
3. [Converged solutions can, in general, only be obtained over quite short durations, especially compared to benchmarks such as the single plane-wave problem where this is not an issue.]{}
4. [When extensive parameter scans are carried out across initial conditions / problem parameters, it is found that regions in parameter space exist where there is a very high degree of sensitivity to these initial conditions (or problem parameters).]{}
5. [Progressively increasing the resolution of these sensitive regions does not lead to any improved resolution of the highly sensitive region.]{}
Our findings have, in the authors’ view, two main consequences. Firstly, great care needs to be taken when using PIC codes to study laser-plasma interactions. Prior to this study it was generally assumed that algorithms such as the Boris pusher would produce reasonably accurate results irrespective of the field configuration under consideration. In light of this study, we no longer think this can be assumed. We suggest that PIC simulations are accompanied by complementary studies of the single particle motion to ensure that converged orbits can be obtained.
Secondly, these findings suggest that the root cause of both the issues of convergence and the sensitivity to initial conditions is at the very least indicative of extreme nonlinearity, but it quite strongly suggests that the dynamics of this problem are [*chaotic*]{}. This is entirely consistent with earlier studies [@z.-m.sheng_stochastic_2002; @sheng_efficient_2004; @meyer-ter-vehn_electron_1999], however these results now indicate that it is quite easy for the Lyapunov time to become sufficiently short that numerical integration is inhibited. This would explain the very limited ability of nearly all methods to obtain converged solutions, and it also explains the very high sensitivity to initial conditions. We do not claim to provide any rigorous proof that the dynamics of this system are chaotic, only to submit the results of numerical calculations that show that this might be the case, and that further investigation should be carried out. We do however draw the attention of the reader to earlier studies where such detailed analysis was carried out [@mendonca_1983]. If this simple model problem is indeed shown to have chaotic dynamics then this could have quite profound implications for the field of ultra-intense laser-plasma interactions, as it would then imply that a number of laser-target configurations where there are interfering laser fields would have the potential for chaotic dynamics.
Acknowledgements {#acknowledgements .unnumbered}
================
The work of K.T. and A.V.A was supported by the National Science Foundation (PHY 1632777). K.W. was supported in part by the DOE Office of Science under Grant No. DE-SC0018312 and in part by the DOE Computational Science Graduate Fellowship under Grant No. DE-FG02-97ER25308.
| ArXiv |
---
abstract: 'This paper introduces and analyses the new grid-based tensor approach for approximate solution of the eigenvalue problem for linearized Hartree-Fock equation applied to the 3D lattice-structured and periodic systems. The set of localized basis functions over spatial $(L_1,L_2,L_3)$ lattice in a bounding box (or supercell) is assembled by multiple replicas of those from the unit cell. All basis functions and operators are discretized on a global 3D tensor grid in the bounding box which enables rather general basis sets. In the periodic case, the Galerkin Fock matrix is shown to have the three-level block circulant structure, that allows the FFT-based diagonalization. The proposed tensor techniques manifest the twofold benefits: (a) the entries of the Fock matrix are computed by 1D operations using low-rank tensors represented on a 3D grid, (b) the low-rank tensor structure in the diagonal blocks of the Fock matrix in the Fourier space reduces the conventional 3D FFT to the product of 1D FFTs. We describe fast numerical algorithms for the block circulant representation of the core Hamiltonian in the periodic setting based on low-rank tensor representation of arising multidimensional functions. Lattice type systems in a box with open boundary conditions are treated by our previous tensor solver for single molecules, which makes possible calculations on large $(L_1,L_2,L_3)$ lattices due to reduced numerical cost for 3D problems. The numerical simulations for box/periodic $(L,1,1)$ lattice systems in a 3D rectangular “tube” with $L$ up to several hundred confirm the theoretical complexity bounds for the tensor-structured eigenvalue solvers in the limit of large $L$.'
author:
- 'V. KHOROMSKAIA, [^1]'
- 'B. N. KHOROMSKIJ [^2]'
title: 'Tensor Numerical Approach to Linearized Hartree-Fock Equation for Lattice-type and Periodic Systems'
---
*AMS Subject Classification:* ** 65F30, 65F50, 65N35, 65F10
*Key words:* Hartree-Fock equation, tensor-structured numerical methods, 3D grid-based tensor approximation, Fock operator, core Hamiltonian, periodic systems, lattice summation, block circulant matrix, Fourier transform.
Introduction {#sec:introduct}
============
The efficient numerical simulation of periodic and perturbed periodic systems is one of the most challenging computational tasks in quantum chemistry calculations of crystalline, metallic and polymer-type compounds. The reformulation of the nonlinear Hartree-Fock equation for periodic molecular systems based on the Bloch theory [@Bloch:1925] has been addressed in the literature for more than forty years ago, and nowadays there are several implementations mostly relying on the analytic treatment of arising integral operators [@CRYSTAL:2000; @CRYSCOR:12; @GAUSS:09]. The mathematical analysis of spectral problems for PDEs with the periodic-type coefficients was an attractive topic in the recent decade, see [@CancesDeLe:08; @CanEhrMad:2012; @Ortn:ArX] and the references therein. However, the systematic developments and optimization of the basic numerical algorithms in the Hartree-Fock calculations for large lattice structured compounds still are largely unexplored.
Grid-based approaches for single molecules and moderate size systems based on the locally adaptive grids and multiresolution techniques have been discussed (see [@HaFaYaBeyl:04; @SaadRev:10; @Frediani:13; @CanEhrMad:2012; @Ortn:ArX; @BiVale:11; @RahOsel:13] and references therein).
In this paper, we consider the Hartree-Fock equation for extended systems composed of atoms or molecules, determined by means of an $(L_1, L_2, L_3)$ lattice in a box, both for open boundary conditions and in the periodic setting (supercell). The grid-based tensor-structured method is applied (see [@KhKhFl_Hart:09; @VKH_solver:13; @KhorSurv:10; @VeBoKh:Ewald:14] and references therein) to calculate the core Hamiltonian in the localized Gaussian-type basis sets living on a box/periodic spatial lattice. To perform numerical integration by using low-rank tensor formats we represent all basis functions on the fine global grid covering the whole computational box (supercell). The Hartree-Fock equation for periodic systems is reformulated as the eigenvalue problem for large block circulant matrices which are diagonalizable in the Fourier space, that allows efficient computations on large lattices of size $L=\max\{L_1,L_2,L_3\}$. In the following we consider the model problem for the Fock operator confined to the core Hamiltonian part.
One of the severe difficulties in the Hartree-Fock calculations for lattice-structured periodic or box-restricted systems is the computation of 3D lattice sums of a large number of long-distance Coulomb interaction potentials. This problem is traditionally treated by the so-called Ewald-type summation techniques [@Ewald:27] combined with the fast multipole expansion or/and FFT methods. Notice that the traditional approaches for lattice summation by the Ewald-type methods scale as $O(L^3 \log L)$ at least, for both periodic and box-type lattice sums. We apply the recent lattice summation method [@VeBoKh:Ewald:14] by assembled rank-structured tensor decomposition, which reduces the asymptotic cost at this computational step to linear scaling in $L$, i.e. $O(L)$.
In the presented approach the Fock matrix is calculated directly by 3D grid-based tensor numerical methods in the basis set of localized Gaussian-type-orbitals (GTO) specified by $m_0$ elements in the 3D unit cell [@VeKh_Diss:10; @VKH_solver:13]. Hence, we do not impose explicitly the periodicity-like features of the solution by means of the approximation ansatz that is normally the case in the Bloch formalism. Instead, the periodic properties of the considered system appear implicitly through the block structure in the Fock matrix. In periodic case this matrix is proved to inherit the three-level symmetric block circulant form, that allows its efficient diagonalization in the Fourier basis [@KaiSay_book:99; @Davis]. In the case of $d$-dimensional lattice ($d=1,2,3$), the weak overlap between lattice translated basis functions improves the block sparsity thus reducing the storage cost to $O(m_0^2 L)$, while the FFT-based diagonalization procedure amounts to $O(m_0^2 L^d \log L)$ operations. Introducing the low-rank tensor structure into the diagonal blocks of the Fock matrix represented in the Fourier space, and using the initial block-circulant structure it becomes possible to further reduce the numerical costs to linear scaling in $L$, $O(m_0^2 L \log L)$. We present numerical tests in the case of a rectangular 3D “tube” composed of $(L, 1, 1)$ cells with $L$ up to several hundred.
In the new approach one can potentially benefit from the additional flexibility that allows to treat slightly perturbed periodic systems in a straightforward way. Such situations may arise, for example, in the case of finite extended systems in a box (open boundary conditions) also considered in this paper, or for slightly perturbed periodic compounds, say for quasi-periodic systems with vacancies [@BGKh:12]. The proposed numerical scheme can be applied in the framework of self-consistent Hartree-Fock calculations, in particular, in the reduced Hartree-Fock model [@CancesDeLe:08], where the similar block-structure in the Fock matrix can be observed. The Wannier-type basis constructed by the lattice translation of the initial localized molecular orbitals precomputed on the reference unit cell, can be also adapted to our framework.
Furthermore, the arising block-structured matrix representing the stiffness matrix $H$ of the core Hamiltonian, as well as some auxiliary function-related tensors, can be shown to be well suited for further optimization by imposing the low-rank tensor formats, and in particular, the quantics-TT (QTT) tensor approximation [@KhQuant:09] of long vectors, which especially benefits in the limiting case of large $L$-periodic systems. In the QTT approach the algebraic operations on the 3D $n\times n\times n$ Cartesian grid can be implemented with logarithmic cost $O(\log n)$. Literature surveys on tensor algebra and rank-structured tensor methods for multi-dimensional PDEs can be found in [@Kolda; @KhorSurv:10; @GraKresTo:13], see also [@HaKhSaTy:08; @DoKhSavOs_mEIG:13].
The rest of the paper is organized as follows. Section \[sec\_MLBlock-circ\] recalls the main properties of the multilevel block circulant matrices with special focus on their diagonalization by FFT. Section \[sec:core\_H\] includes the main results on the analysis of core Hamiltonian on lattice structured compounds. In particular, section \[Core\_Hamil\] describes the tensor-structured calculation of the core Hamiltonian for large lattice-type molecular/atomic systems. We recall tensor-structured calculation of the Laplace operator and fast summation of lattice potentials by assembled canonical tensors. The complexity reduction due to low-rank tensor structures in the matrix blocks is discussed (see Proposition \[prop:low\_rank\_coef\]). Section \[sec:Core\_Ham\_period\_FFT\] discusses in detail the block circulant structure of the core Hamiltonian and presents numerical illustrations for $(L,1,1)$ lattice systems. Appendix recalls the classical results on the properties of block circulant/Toeplitz matrices.
Diagonalizing multilevel block circulant matrices {#sec_MLBlock-circ}
=================================================
The direct Hartree-Fock calculations for lattice structured systems in the localized GTO-type basis lead to the symmetric block circulant/Toeplitz matrices (see Appendix \[sec\_Append:block-circ\]), where the first-level blocks, $A_0,...,A_{L-1}$, may have further block structures. In particular, the Galerkin approximation of the 3D Hartree-Fock core Hamiltonian in periodic setting leads to the symmetric, three-level block circulant matrix.
Multilevel block circulant/Toeplitz matrices {#ssec:MLblock-circ}
--------------------------------------------
In this section we consider the extension of (one-level) block circulant matrices described in Appendix. First, we recall the main notions of multilevel block circulant (MBC) matrices with the particular focus on the three-level case. Given a multi-index ${\bf L}=(L_1, L_2, L_3)$, we denote $|{\bf L}|=(L_1,\, L_2,\, L_3)$. A matrix class ${\cal BC} (d,{\bf L},m_0)$ ($d=1,2,3$) of $d$-level block circulant matrices can be introduced by the following recursion.
\[def:Bcirc\] For $d=1$, define a class of one-level block circulant matrices by ${\cal BC} (1,{\bf L},m)\equiv {\cal BC} (L_1,m)$ (see Appendix), where ${\bf L}=(L_1,1,1)$. For $d=2$, we say that a matrix $A\in \mathbb{R}^{|{\bf L}|m_0 \times |{\bf L}| m_0}$ belongs to a class ${\cal BC} (d,{\bf L},m_0)$ if $$A = \operatorname{bcirc}(A_1,...,A_{L_1})\quad \mbox{with}\quad
A_j\in {\cal BC}(d-1,{\bf L}_{[1]},m_0),\; j=1,...,L_1,$$ where ${\bf L}_{[1]}=(L_2,L_3)\in \mathbb{N}^{d-1} $. Similar recursion applies to the case $d=3$.
Likewise to the case of one-level BC matrices (see Appendix), it can be seen that a matrix $A \in {\cal BC} (d,{\bf L},m_0)$, $d=1,2,3$, of size $|{\bf L}| m_0 \times |{\bf L}| m_0$ is completely defined (parametrized) by a $d$th order matrix-valued tensor ${\bf A}=[A_{k_1 ... k_d}]$ of size $L_1\times ... \times L_d $, ($k_\ell=1,...,L_\ell$, $\ell=1,...,d$), with $m_0\times m_0$ matrix entries $A_{k_1 ... k_d}$, obtained by folding of the generating first column vector in $A$. A diagonalization of a MBC matrix is based on representation via a sequence of cycling permutation matrices $\pi_{L_1}, ...,\pi_{L_d}$, $d=1,2,3$. Recall that the $q$-dimensional Fourier transform (FT) can be defined via the Kronecker product of the univariate FT matrices (rank-$1$ operator), $$F_{\bf L}=F_{L_1}\otimes \cdots \otimes F_{L_d}.$$ The block-diagonal form of a MBC matrix is well known in the literature. Here we prove the diagonal representation in a form that is useful for the description of numerical algorithms. To that end we generalize the notations ${\cal T}_L$ and $\widehat{A}$ (see Section \[sec\_Append:block-circ\]) to the class of multilevel matrices. We denote by $\widehat{A}\in \mathbb{R}^{|{\bf L}|m_0\times m_0}$ the first block column of a matrix $A\in {\cal BC} (d,{\bf L},m_0)$, with a shorthand notation $$\widehat{A}=[A_0,A_1,...,A_{L-1}]^T,$$ so that a $|{\bf L}|\times m_0 \times m_0$ tensor ${\cal T}_{\bf L} \widehat{A}$ represents slice-wise all generating $m_0\times m_0$ matrix blocks. Notice that in the case $m_0=1$, $\widehat{A}\in \mathbb{R}^{|{\bf L}|}$ represents the first column of the matrix $A$. Now the Fourier transform $F_{\bf L}$ applies to ${\cal T}_{\bf L} \widehat{A}$ columnwise, and the backward reshaping of the resultant tensor, ${\cal T}_{\bf L}'$, returns an $|{\bf L}|m_0 \times m_0$ block matrix column.
\[lem:DiagMLCirc\] A matrix $A\in {\cal BC} (d,{\bf L},m_0)$, is block-diagonalysed by the Fourier transform, $$\label{eqn:DiagMLcirc}
A= (F_{\bf L}^\ast \otimes I_{m_0}) \operatorname{bdiag} \{ \bar{A}_{\bf 0}, \bar{A}_{\bf 1},\ldots ,
\bar{A}_{\bf L-1}\}(F_{\bf L} \otimes I_{m_0}),$$ where $$\left[ \bar{A}_{\bf 0}, \bar{A}_{\bf 1},\ldots , \bar{A}_{\bf L-1}\right]^T =
{\cal T}_{\bf L}'(F_{\bf L} ({\cal T}_{\bf L} \widehat{A})).$$
First, we confine ourself to the case of three-level matrices, i.e. $d=3$, and proceed $$\begin{aligned}
A & = \sum\limits^{L_1 -1}_{k_1=0} \pi_{L_1}^{k_1} \otimes {\bf A}_{k_1} \\ \nonumber
& = \sum\limits^{L_1 -1}_{k_1=0} \pi_{L_1}^{k_1}\otimes
(\sum\limits^{L_2 -1}_{k_2=0} \pi_{L_2}^{k_2}\otimes {\bf A}_{k_1 k_2} )=
\sum\limits^{L_1 -1}_{k_1=0}\sum\limits^{L_2 -1}_{k_2=0}
\pi_{L_1}^{k_1}\otimes \pi_{L_2}^{k_2}\otimes {\bf A}_{k_1 k_2} \\ \nonumber
& = \sum\limits^{L_1 -1}_{k_1=0}\sum\limits^{L_2 -1}_{k_2=0}\sum\limits^{L_3 -1}_{k_3=0}
\pi_{L_1}^{k_1}\otimes \pi_{L_2}^{k_2}\otimes \pi_{L_3}^{k_3}\otimes A_{k_1 k_2 k_3}, \nonumber\end{aligned}$$ where ${\bf A}_{k_1}\in \mathbb{R}^{L_2\times L_3 \times m_0\times m_0}$, ${\bf A}_{k_1 k_2} \in \mathbb{R}^{L_3 \times m_0\times m_0}$ and $A_{k_1 k_2 k_3}\in \mathbb{R}^{m_0\times m_0}$.
Diagonalizing the periodic shift matrices $\pi_{L_1}^{k_1}, \pi_{L_2}^{k_2}$, and $\pi_{L_3}^{k_3}$ via the Fourier transform (see Appendix), we arrive at $$\begin{aligned}
\label{eqn:MLbcircDiag2}
A & = (F_{\bf L}^\ast \otimes I_{m_0}) \left[\sum\limits^{L_1 -1}_{k_1=0}\sum\limits^{L_2 -1}_{k_2=0}
\sum\limits^{L_3 -1}_{k_3=0}
D_{L_1}^{k_1}\otimes D_{L_2}^{k_2}\otimes D_{L_3}^{k_3}\otimes A_{k_1 k_2 k_3} \right]
(F_{\bf L}\otimes I_{m_0})
\\
& = (F_{\bf L}^\ast \otimes I_{m_0})
\mbox{bdiag}_{m_0\times m_0} \{{\cal T}_{\bf L}'(F_{\bf L} ({\cal T}_{\bf L} \widehat{A}))\}
(F_{\bf L}\otimes I_{m_0}),\nonumber
\end{aligned}$$ with the monomials of diagonal matrices $D_{L_\ell}^{k_\ell}\in \mathbb{R}^{L_\ell \times L_\ell}$, $\ell=1,2,3$ are defined by (\[eqn:diagshift\]).
The generalization to the case $d >3$ can be proven by the similar argument.
Taking into account representation (\[eqn:symBc\]), the multilevel symmetric block circulant matrix can be described in form (\[eqn:DiagMLcirc\]), such that all real-valued diagonal blocks remain symmetric.
Similar to Definition \[def:Bcirc\], a matrix class ${\cal BT}_s (d,{\bf L},m_0)$ of symmetric $d$-level block Toeplitz matrices can be introduced by the following recursion.
\[def:BToepl\] For $d=1$, ${\cal BT}_s (1,{\bf L},m_0)\equiv {\cal BT}_s (L_1,m_0)$ is the class of one-level symmetric block circulant matrices with ${\bf L}=(L_1,1,1)$. For $d=2$ we say that a matrix $A\in \mathbb{R}^{|{\bf L}|m \times |{\bf L}| m_0}$ belongs to a class ${\cal BT}_s (d,{\bf L},m_0)$ if $$A= \operatorname{btoepl}_s(A_1,...,A_{L_1})\quad
\mbox{with}\quad A_j\in {\cal BT}_s(d-1,{\bf L_{[1]}},m_0),\; j=1,...,L_1.$$ Similar recursion applies to the case $d=3$.
The following remark compares the properties of circulant and Toeplitz matrices.
\[rem:BToepl\] A block Toeplitz matrix does not allow diaginalization by FT as it was the case for block circulant matrices. However, it is well known that a block Toeplitz matrix can be extended to the double-size (at each level) block circulant that makes it possible the efficient matrix-vector multiplication, and, in particular, the efficient application of power method for finding the senior eigenvalues.
Low-rank tensor structure in matrix blocks {#ssec:Tensor_bcirc}
------------------------------------------
In the case $d=3$, the general block-diagonal representation (\[eqn:DiagMLcirc\]) - (\[eqn:MLbcircDiag2\]) takes form $$A= (F_{\bf L}^\ast \otimes I_{m_0}) (\sum\limits^{L_1 -1}_{k_1=0}\sum\limits^{L_2 -1}_{k_2=0}
\sum\limits^{L_3 -1}_{k_3=0}
D_{L_1}^{k_1}\otimes D_{L_2}^{k_2}\otimes D_{L_3}^{k_3}\otimes A_{k_1 k_2 k_3} )
(F_{\bf L}\otimes I_{m_0}),$$ that allows the reduced storage costs of order $O(|{\bf L}| m_0^2)$, where $|{\bf L}|=L^3$. For large $L$ the numerical cost may become prohibitive. However, the above representation indicates that the further storage and complexity reduction becomes possible if the third-order coefficients tensor ${\bf A}= [A_{k_1 k_2 k_3}]$, $k_\ell=0,...,L_\ell-1$, with the matrix entries $A_{k_1 k_2 k_3}\in \mathbb{R}^{m_0\times m_0}$, allows some low-rank tensor representation (approximation) in the multiindex ${\bf k}$ described by a small number of parameters.
To fix the idea, let us assume the existence of rank-$1$ separable matrix factorization, $$A_{k_1 k_2 k_3} = A_{k_1}^{(1)}\odot A_{k_2}^{(2)} \odot A_{k_3}^{(3)},
\quad A_{k_1}^{(1)}, A_{k_2}^{(2)},A_{k_3}^{(3)} \in \mathbb{R}^{m_0\times m_0},
\quad \mbox{for} \quad k_\ell=0,...,L_\ell-1,$$ where $\odot$ denotes the Hadamard (pointwise) product of matrices. The latted representation can be written in the factorized tensor-product form $$\begin{aligned}
\label{eqn:bcirc_R1}
& D_{L_1}^{k_1} \otimes D_{L_2}^{k_2}\otimes D_{L_3}^{k_3}\otimes A_{k_1 k_2 k_3} \\
= &
((D_{L_1}^{k_1}\otimes A_{k_1}^{(1)}) \otimes I_{L_2}\otimes I_{L_3} )\odot
(I_{L_1} \otimes (D_{L_2}^{k_2} \otimes A_{k_2}^{(2)})\otimes I_{L_3} ) \odot
(I_{L_1} \otimes I_{L_2}\otimes (D_{L_3}^{k_3} \otimes A_{k_3}^{(3)})). \end{aligned}$$ Given $\ell \in \{1,...,d\}$ and a matrix $A\in \mathbb{R}^{L_\ell \times L_\ell}$, define the [*tensor prolongation*]{} mapping, ${\cal P}_\ell: \mathbb{R}^{L_\ell\times L_\ell}\to \mathbb{R}^{|{\bf L}|\times |{\bf L}|}$, by $$\label{eqn:Tensor_prolong}
{\cal P}_\ell(A):= \bigotimes_{i=1}^{\ell-1}I_{L_i}\otimes A \bigotimes_{i=\ell+1}^{d}I_{L_i}.$$ This leads to the powerful matrix factorization $$\begin{aligned}
A=& (F_{\bf n}^\ast \otimes I_m)
\left[\sum\limits^{L_1 -1}_{k_1=0} {\cal P}_1(D_{L_1}^{k_1}\otimes A_{k_1}^{(1)}) \odot
\sum\limits^{L_2 -1}_{k_2=0} {\cal P}_2(D_{L_2}^{k_2} \otimes A_{k_2}^{(2)}) \odot
\sum\limits^{L_3 -1}_{k_3=0} {\cal P}_3(D_{L_3}^{k_3} \otimes A_{k_3}^{(3)})\right]
(F_{\bf L}\otimes I_{m_0}),\\
=& (F_{\bf L}^\ast \otimes I_m)
\left[ {\cal P}_1(\sum\limits^{L_1-1 -1}_{k_1=0} D_{L_1}^{k_1}\otimes A_{k_1}^{(1)}) \odot
{\cal P}_2(\sum\limits^{L_2 -1}_{k_2=0} D_{L_2}^{k_2} \otimes A_{k_2}^{(2)}) \odot
{\cal P}_3(\sum\limits^{L_3 -1}_{k_3=0} D_{L_3}^{k_3} \otimes A_{k_3}^{(3)})\right]
(F_{\bf n}\otimes I_{m_0}),\\
= & (F_{\bf L}^\ast \otimes I_{m_0})
\left[ {\cal P}_1(\mbox{bdiag} F_{L_1} A^{(1)}) \odot
{\cal P}_2(\mbox{bdiag} F_{L_2} A^{(2)})\odot
{\cal P}_3(\mbox{bdiag} F_{L_3} \otimes A^{(3)})\right]
(F_{\bf L}\otimes I_{m_0}),\end{aligned}$$ where the tensor $A^{(\ell)}\in \mathbb{R}^{L_\ell\times m_0 \times m_0}$ is defined by concatenation $A^{(\ell)}=[A_{0}^{(\ell)},...,A_{L_\ell-1}^{(\ell)}]^T$, and the tensor prolongation ${\cal P}_\ell$ is defined by (\[eqn:Tensor\_prolong\]). This representation requires only 1D Fourier transforms thus reducing the numerical cost to $$O(m_0^2 {\sum}_{\ell=1}^d L_\ell \log L_\ell).$$ Moreover, and it is even more important, that the eigenvalue problem for the large matrix $A$ now reduces to only $L_1+L_2+L_3 \ll L_1 L_2 L_3$ small $m_0 \times m_0$ matrix eigenvalue problems.
The above block-diagonal representation for $d=3$ generalizes easily to the case of arbitrary dimension $d$. Finally, we prove the following general result.
\[thm:tens\_FFT\] Introduce the notation $D_{\bf L}^{\bf k}= D_{L_1}^{k_1}\otimes D_{L_2}^{k_2}\otimes \cdots \otimes D_{L_d}^{k_d}$, then we have $$A= (F_{\bf L}^\ast \otimes I_{m_0})(\sum\limits^{\bf L -1}_{\bf k=0}
D_{\bf L}^{\bf k}\otimes A_{\bf k}) (F_{\bf L} \otimes I_{m_0}).$$ Assume the separability of a tensor $[A_{\bf k}]$ in the ${\bf k}$ space, we arrive at the factorized block-diagonal form of $A$ $$A= (F_{\bf L}^\ast \otimes I_{m_0})
\left[ {\cal P}_1(F_{L_1} A^{(1)}) \odot
{\cal P}_2(F_{L_2} A^{(2)})\odot \dots \odot
{\cal P}_d (F_{L_d} A^{(d)})\right]
(F_{\bf L}\otimes I_{m_0}).$$
The rank-$1$ decomposition was considered for the ease of exposition only. The above low-rank representations can be easily generalized to the case of canonical or Tucker formats in ${\bf k}$ space (see Proposition \[prop:low\_rank\_coef\] below).
Notice that in the practically interesting 3D case the use of MPS/TT type factorizations does not take the advantage over the Tucker format since the Tucker and MPS ranks in 3D appear to be close to each other. Indeed, the HOSVD for a tensor of order $3$ leads to the same rank estimates for both the Tucker and MPS/TT tensor formats.
Core Hamiltonian for lattice structured compounds {#sec:core_H}
=================================================
In this section we analyze the structure of the Galerkin matrix for the core Hamiltonian part in the Fock operator with respect to the localized GTO basis replicated over a lattice, $\{g_m (x) \}_{1\leq m \leq N_b}, x \in {\mathbb{R}^3}$ in a box, or in a supercell with the priodic boundary conditions.
The core Hamiltonian in a GTO basis set {#Core_Hamil}
---------------------------------------
The nonlinear Fock operator ${\cal F}$ in the governing Hartree-Fock eigenvalue problem, describing the ground state energy for $2N_b$-electron system, is defined by $$\left[-\frac{1}{2} \Delta - v_c(x) +
\int_{\mathbb{R}^3} \frac{\rho({ y})}{\|{ x}-{ y}\|}\, d{ y}\right] \varphi_i({ x})
- \int_{\mathbb{R}^3} \; \frac{\tau({ x}, { y})}{\|{x} - { y}\|}\, \varphi_i({ y}) d{ y}
= \lambda_i \, \varphi_i({ x}), \quad x\in \mathbb{R}^3,$$ where $i =1,...,N_{orb}$. The linear part in the Fock operator is presented by the core Hamiltonian $$\label{eqn:HFcore}
{\cal H}=-\frac{1}{2} \Delta - v_c,$$ while the nonlinear Hartree potential and exchange operators, depend on the unknown eigenfunctions (molecular orbitals) comprising the electron density, $\rho({ y})= 2 \tau(y,y)$, and the density matrix, $
\tau(x,y) =\sum\limits^{N_{orb}}_{i=1} \varphi_i (x)\varphi_i (y),\quad x,y\in \mathbb{R}^3,
$ respectively. The electrostatic potential in the core Hamiltonian is defined by a sum $$\label{eqn:ElectrostPot}
v_c(x)= \sum_{\nu=1}^{M}\frac{Z_\nu}{\|{x} -a_\nu \|},\quad
Z_\nu >0, \;\; x,a_\nu\in \mathbb{R}^3,$$ where $M$ is the total number of nuclei in the system, $a_\nu$, $Z_\nu$, represent their Cartesian coordinates and the respective charge numbers. Here $\|\cdot \|$ means the distance function in $\mathbb{R}^3$. Given a general set of localized GTO basis functions $\{g_\mu\}$ ($\mu=1,...,N_b$), the occupied molecular orbitals $\psi_i$ are approximated by $$\label{expand}
\psi_i=\sum\limits_{\mu=1}^{N_b} C_{\mu i} g_\mu, \quad i=1,...,N_{orb},$$ with the unknown coefficients matrix $C=\{C_{ \mu i} \}\in \mathbb{R}^{N_b \times N_{orb}}$ obtained as the solution of the discretized Hartree-Fock equation with respect to $\{g_\mu\}$, and described by $N_b\times N_b$ Fock matrix. Since the number of basis functions scales cubically in $L$, $N_b = m_0 L^3$, the calculation of the Fock matrix may become prohibitive as $L$ increases ($m_0$ is the number of basis functions in the unit cell).
In what follows we describe the grid-based tensor method for the block-structured representation of the core Hamiltonian in the Fock matrix in a box and in a supercell subject to the periodic boundary conditions. The stiffness matrix $H=\{h_{\mu \nu}\}$ of the core Hamiltonian (\[eqn:HFcore\]) is represented by the single-electron integrals, $$\label{eqn:Core_Ham}
h_{\mu \nu}= \frac{1}{2} \int_{\mathbb{R}^3}\nabla g_\mu \cdot \nabla g_\nu dx -
\int_{\mathbb{R}^3} v_c(x) g_\mu g_\nu dx, \quad 1\leq \mu, \nu \leq N_b,$$ such that the resulting $N_b\times N_b$ Galerkin system of equations governed by the reduced Fock matrix $ H$ reads as follows $$\begin{aligned}
\label{eqn:HF discr}
H C &= SC \Lambda, \quad \Lambda= diag(\lambda_1,...,\lambda_{N_{orb}}), \\
C^T SC &= I_N, \nonumber\end{aligned}$$ where the mass (overlap) matrix $S=\{s_{\mu \nu} \}_{1\leq \mu, \nu \leq N_b}$, is given by $
s_{\mu \nu}=\int_{\mathbb{R}^3} g_\mu g_\nu dx.
$
The numerically extensive part in (\[eqn:Core\_Ham\]) is related to the integration with the large sum of lattice translated Newton kernels. Indeed, let $M_0$ be the number of nuclei in the unit cell, then the expensive calculations are due to the summation over $M_0 L^3$ Newton kernels, and further spacial integration of this sum with the large set of localized atomic orbitals $\{g_\mu\}$, ($\mu=1,...,N_b$), where $N_b$ is of order $m_0 L^3$.
The present approach solves this problem by using the fast and accurate grid-based tensor method for evaluation of the electrostatic potential $v_c$ defined by the lattice sum in (\[eqn:ElectrostPot\]), see [@VeBoKh:Ewald:14], and subsequent efficient computation and structural representation of the stiffness matrix $V_c$, $$V_c=[V_{\mu \nu}]:\quad V_{\mu \nu}= \int_{\mathbb{R}^3} v_c(x) g_\mu g_\nu dx,
\quad 1\leq \mu, \nu \leq N_b,$$ by numerical integration by using the low-rank tensor representation on the grid of all functions involved.
This approach is applicable to the large $L\times L \times L$ lattice. In the next sections, we show that in the periodic case the resultant stiffness matrix $H=\{h_{\mu \nu}\}$ of the core Hamiltonian can be parametrized in the form of a symmetric, three-level block circulant matrix. In the case of lattice system in a box the block structure of $H$ is a small perturbation of the block Toeplitz matrix.
Low-rank tensor form of the nuclear potential in a box {#ssec:nuclear}
------------------------------------------------------
We consider the nuclear (core) potential operator describing the Coulomb interaction of the electrons with the nuclei, see (\[eqn:ElectrostPot\]). In the scaled unit cell $\Omega=[-b/2,b/2]^3$, we introduce the uniform $n \times n \times n$ rectangular Cartesian grid $\Omega_{n}$ with the mesh size $h=b/n$. Let $\{ \psi_\textbf{i}\}$ be the set of tensor-product piecewise constant basis functions, $ \psi_\textbf{i}(\textbf{x})=\prod_{\ell=1}^d \psi_{i_\ell}^{(\ell)}(x_\ell)$ for ${\bf i}=(i_1,i_2,i_3)\in I \times I \times I $, $i_\ell \in I=\{1,...,n\}$. The Newton kernel is discretized by the projection/collocation method in the form of a third order tensor of size $n\times n \times n$, defined point-wise as $$\begin{aligned}
\mathbf{P}:=[p_{\bf i}] \in \mathbb{R}^{n\times n \times n}, \quad
p_{\bf i} =
\int_{\mathbb{R}^3} \frac{\psi_{{\bf i}}({x})}{\|{x}\|} \,\, \mathrm{d}{x},
\label{galten}\end{aligned}$$ see [@KhKhFl_Hart:09; @BeHaKh:08; @VeKh_Diss:10; @VKH_solver:13]. Our low-rank canonical decomposition of the $3$rd order tensor $\mathbf{P}$ is based on using exponentially convergent $\operatorname*{sinc}$-quadratures for approximation of the Laplace-Gauss transform, see [@Stenger; @GHK:05; @HaKhtens:04I], $$\frac{1}{z}= \frac{2}{\sqrt{\pi}}\int_{\mathbb{R}_+} e^{- z^2 t^2 } dt,$$ which can be adapted to the Newton kernel by substitution $z=\sqrt{x_1^2 + x_2^2 + x_3^2}$. Rational type approximation by exponential sums have been addressed in [@Braess:95; @Braess:BookApTh]. We denote the resultant $R$-term canonical representation by $$\label{eqn:sinc_general}
\mathbf{P} \approx \mathbf{P}_R
= \sum\limits_{q=1}^{R} {\bf p}^{(1)}_q \otimes {\bf p}^{(2)}_q \otimes {\bf p}^{(3)}_q
\in \mathbb{R}^{n\times n \times n}.
$$ In a similar way, we also introduce the “master tensor”, $\widetilde{\bf P}_R \in \mathbb{R}^{\widetilde{n}\times \widetilde{n} \times \widetilde{n}}$, approximating the Newton kernel in the extended (accompanying) domain $\widetilde{\Omega} \supset \Omega$, and associated with the grid parameter $\widetilde{n}=n_0+n$ (say, $n_0=n$), $$\label{eqn:master_pot}
\widetilde{\bf P}_R=
\sum\limits_{q=1}^{R} \widetilde{\bf p}^{(1)}_q \otimes
\widetilde{\bf p}^{(2)}_q \otimes \widetilde{\bf p}^{(3)}_q
\in \mathbb{R}^{\widetilde{n}\times \widetilde{n} \times \widetilde{n}}.$$
The core potential for the molecule is approximated by the canonical tensor $${\bf P}_{c} = \sum_{\nu=1}^{M_0} Z_\nu {\bf P}_{{c},\nu}\approx \widehat{\bf P}_{c}
\in \mathbb{R}^{n\times n \times n},$$ with the rank bound $rank({\bf P}_{c})\leq M_0 R$, where the rank-$R$ tensor ${\bf P}_{{c},\nu}$ represents the single Coulomb potential shifted according to coordinates of the corresponding nuclei, [@VeBoKh:Ewald:14], $$\label{eqn:core_tens}
{\bf P}_{c,\nu} = {\cal W}_{\nu} \widetilde{\bf P}_R =
\sum\limits_{q=1}^{R} {\cal W}_{\nu}^{(1)} \widetilde{\bf p}^{(1)}_q \otimes
{\cal W}_{\nu}^{(2)} \widetilde{\bf p}^{(2)}_q
\otimes {\cal W}_{\nu}^{(3)} \widetilde{\bf p}^{(3)}_q\in \mathbb{R}^{n\times n \times n},
$$ such that every rank-$R$ canonical tensor ${\cal W}_{\nu} \widetilde{\bf P}_R \in \mathbb{R}^{n\times n \times n}$ is thought as a sub-tensor of the master tensor obtained by a shift and restriction (windowing) of $\widetilde{\bf P}_R$ onto the $n \times n \times n$ grid $\Omega_{n}$ in the unit cell $\Omega$, $\Omega_{n} \subset \Omega_{\widetilde{n}}$. A shift from the origin is specified according to the coordinates of the corresponding nuclei, $a_\nu$, counted in the $h$-units. Here $\widehat{\bf P}_{c}$ is the rank-$R_c$ ($R_c\leq M_0 R$, actually $R_c \approx R$) canonical tensor obtained from ${\bf P}_{c}$ by the rank optimization procedure (see [@VeBoKh:Ewald:14], Remark 2.2).
For the tensor representation of the Newton potentials, ${\bf P}_{{c},\nu}$, we make use of the piecewise constant discretization on the equidistant tensor grid, where, in general, the univariate grid size $n$ can be noticeably smaller than that used for the piecewise linear discretization applied to the Laplace operator. Indeed, since we use the global basis functions for the Galerkin approximation to the eigenvalue problem, the grid-based representation of these basis functions can be different in the calculation of the kinetic and potential parts in the Fock operator. The corresponding choice is the only controlled by the respective approximation error and by the numerical efficiency depending on the separation rank parameters. The error $\varepsilon >0$ arising due to the separable approximation of the nuclear potential is controlled by the rank parameter $R_{P}= rank({\bf P}_{c})$. Now letting $rank({\bf G}_m) = R_m$ implies that each matrix element is to be computed with linear complexity in $n$, $O(R_kR_m R_{P} \, n)$. The almost exponential convergence of the rank approximation in $R_{P}$ allows us the choice $R_{P}=O(|\log \varepsilon |)$.
Let us discuss the lattice structured systems. Low-rank tensor decomposition of the Coulomb interaction defined by the large lattice sum is proposed in [@VeBoKh:Ewald:14]. Given the potential sum $v_c$ in the scaled unit cell $\Omega=[-b/2,b/2]^3$, of size $b\times b \times b$, we consider an interaction potential in a symmetric box (supercell) $$\Omega_L =B_1\times B_2 \times B_3,
$$ consisting of a union of $L_1 \times L_2 \times L_3$ unit cells $\Omega_{\bf k}$, obtained from $\Omega$ by a shift proportional to $ b$ in each variable, and specified by the lattice vector $b {\bf k}$, where ${\bf k}=(k_1,k_2,k_3)\in \mathbb{Z}^3$, $-(L_\ell-1)/2 \leq k_\ell\leq (L_\ell-1)/2 $, ($\ell=1,2,3$), such that, without loss of generality, we assume $L_\ell= 2 p_\ell +1, p_\ell\in \mathbb{N}$. Hence, we have $$B_\ell = \frac{b}{2}[- L_\ell ,L_\ell ], \quad \mbox{for} \quad
L_\ell \in \mathbb{N},$$ where $L_\ell=1$ corresponds to one-layer systems in the respective variable. Recall that $b=n h$, where $h$ is the spacial grid size that is the same for all spacial variables. To simplify the discussion, we often consider the case $L_\ell = L$. We also introduce the accompanying domain $\widetilde{\Omega}_L$. In the case of extended system in a box, further called case (B), the summation problem for the total potential $v_{c_L}$ is formulated in the box $\Omega_L= \bigcup_{k_1,k_2,k_3=-(L-1)/2}^{(L-1)/2} \Omega_{\bf k}$ as well as in the accompanying domain $\widetilde{\Omega}_L$. On each $\Omega_{\bf k}\subset \Omega_L$, the potential sum of interest, $v_{\bf k}(x)=(v_{c_L})_{|\Omega_{\bf k}}$, is obtained by summation over all unit cells in $\Omega_L$, $$\label{eqn:EwaldSumE}
v_{\bf k}(x)= \sum_{\nu=1}^{M_0} \sum\limits_{k_1,k_2,k_3=-(L-1)/2}^{(L-1)/2}
\frac{Z_\nu}{\|{x} -a_\nu (k_1,k_2,k_3)\|}, \quad x\in \Omega_{\bf k},
$$ where $a_\nu (k_1,k_2,k_3)=a_\nu + b {\bf k}$. This calculation is performed at each of $L^3$ elementary cells $\Omega_{\bf k}\subset \Omega_L$, which is implemented by the tensor summation method described in [@VeBoKh:Ewald:14]. The resultant lattice sum is presented by the canonical tensor ${\bf P}_{c_L}$ with the rank $R_0 \leq M_0 R$, $$\label{eqn:EwaldTensorGl}
{\bf P}_{c_L}= \sum\limits_{\nu=1}^{M_0} Z_\nu \sum\limits_{q=1}^{R}
(\sum\limits_{k_1=0}^{L-1}{\cal W}_{\nu({k_1})} \widetilde{\bf p}^{(1)}_{q}) \otimes
(\sum\limits_{k_2=0}^{L-1} {\cal W}_{\nu({k_2})} \widetilde{\bf p}^{(2)}_{q}) \otimes
(\sum\limits_{k_3=0}^{L-1}{\cal W}_{\nu({k_3})} \widetilde{\bf p}^{(3)}_{q}).$$ The numerical cost and storage size are bounded by $O(M_0 R L N_L )$, and $O(M_0 R N_L)$, respectively (see [@VeBoKh:Ewald:14], Theorem 3.1), where $N_L= nL$. The lattice sum is also computed in the accompanying domain $\widetilde{\Omega}_L$, $\widetilde{\bf P}_{c_L}$, where the grid size is equal to $N_L +2 n_0$.
The lattice sum in (\[eqn:EwaldTensorGl\]) converges only conditionally as $L\to \infty$. This aspect will be addressed in Section (\[ssec:Complexity\_EigPr\]) following the approach introduced in [@VeBoKh:Ewald:14].
Nuclear potential operator in a box {#ssec:Core_Ham_gener}
-----------------------------------
First, consider the case of a single molecule in the unit cell. Given the GTO-type basis set $\{{g}_k\}$, $k=1,...,m_0$, i.e. $N_b=m_0$, associated with the scaled unit cell and extended to the local bounding box $\widetilde{\Omega}$. The corresponding rank-$1$ coefficients tensors ${\bf G}_k={\bf g}_k^{(1)}\otimes{\bf g}_k^{(2)} \otimes{\bf g}_k^{(3)}$ representing their piecewise constant approximations $\{\overline{g}_k\}$ on the fine $\widetilde{n}\times \widetilde{n}\times \widetilde{n}$ grid. Then the entries of the respective Galerkin matrix for the core potential operator $v_c$ in (\[eqn:ElectrostPot\]), ${V}_c=\{{V}_{km}\}$, are represented (approximately) by the following tensor operations, $$\label{eqn:nuc_pot}
{V}_{km} \approx \int_{\widetilde{\Omega}_L} V_c(x) \overline{g}_k(x) \overline{g}_m(x) dx \approx
\langle {\bf G}_k \odot {\bf G}_m , {\bf P}_{c}\rangle =: {v}_{km} ,
\quad 1\leq k, m \leq m_0.$$ In the case of lattice syastem in a box we define the basis set on a supercell $\Omega_{L}$ (and on $\widetilde{\Omega}_L$) by translation of the generating basis by the lattice vector $\delta {\bf k}$, i.e., $\{g_{\mu}({x})\} \mapsto \{g_{\mu}({x+\delta {\bf k} })\}$, where ${\bf k}=(k_1,k_2,k_3)$, $0 \leq k_\ell\leq L_\ell -1$, ($\ell=1,2,3$), assuming zero extension of $\{g_{\mu}({x+\delta {\bf k} })\}$ beyond each local bounding box $\widetilde{\Omega}_{\bf k}$. In this construction the total number of basis functions is equal to $N_b=m_0 L_1 L_2 L_3$. In practically interesting case of localized atomic orbital basis functions, the matrix $V_{c_L}$ exhibits the special block sparsity pattern since the effective support of localized atomic orbitals associated with every unit cell $\Omega_{\bf k} \subset \widetilde{\Omega}_{\bf k}$ overlaps only fixed (small) number of neighboring cells.
In the following, the matrix block entries will be numbered by a pair of multi-indicies, $V_{c_L}=\{V_{{\bf k}{\bf m}}\}$, ${\bf k}=(k_1,k_2,k_3)$, where the $m_0\times m_0$ matrix block $V_{{\bf k}{\bf m}}$ is defined by $$\label{eqn:nuc_MatrSparsP}
V_{{\bf k}{\bf m}} = \langle {\bf G}_{\bf k} \odot {\bf G}_{\bf m} , {\bf P}_{c_L}\rangle,
\quad - L/2 \leq k_\ell, m_\ell \leq L/2,\quad \ell=1,2,3,$$ where the canonical tensors ${\bf G}_{\bf k}$ inherit the same block numbering.
We denote by $L_0$ the number of cells measuring the overlap in basis functions in each spacial direction (overlap constant).
\[lem:SparseCaseE\] Assume that the number of overlapping cells in each spacial direction does not exceed $L_0$, then in case (B): (a) the number of non-zero blocks in each block row (column) of the symmetric Galerkin matrix $V_{c_L}$ does not exceed $(2 L_0 + 1)^3$, (b) the required storage is bounded by $m_0^2 [(L_0 + 1)L]^3$.
In case (B), the matrix representation $V_{c_L}=\{v_{km}\}\in \mathbb{R}^{N_b\times N_b}$ of the tensor as in (\[eqn:nuc\_pot\]) is obtained elementwise by the following tensor operations $$\label{nuc_potMatrTot}
\overline{v}_{km}= \int_{\mathbb{R}^3} v_c(x) \overline{g}_k(x) \overline{g}_m(x) dx
\approx \langle {\bf G}_k \odot {\bf G}_m , {\bf P}_{c_L}\rangle =: v_{km},
\quad 1\leq k, m \leq N_b,$$ where $\{\overline{g}_k\}$ denotes the piecewise constant representations to the respective Galerkin basis functions. This leads to the final expression $$\begin{split}
{v}_{km} & = \sum\limits_{\nu=1}^{M_0} Z_\nu \sum\limits_{q=1}^{R_{\cal N}}
\langle {\bf G}_k \odot {\bf G}_m ,
(\sum\limits_{k_1=0}^{L_1-1} {\cal W}_{\nu({k_1})} \widetilde{\bf p}^{(1)}_{q}) \otimes
(\sum\limits_{k_2=0}^{L_2-1} {\cal W}_{\nu({k_2})} \widetilde{\bf p}^{(2)}_{q}) \otimes
(\sum\limits_{k_3=0}^{L_3-1} {\cal W}_{\nu({k_3})} \widetilde{\bf p}^{(3)}_{q}) \rangle \\
&=
\sum\limits_{\nu=1}^{M_0} Z_\nu \sum\limits_{q=1}^{R_{\cal N}}
\prod\limits_{\ell=1}^3
\langle {\bf g}_k^{(\ell)} \odot { \bf g}_m^{(\ell)},
\sum\limits_{k_\ell=1}^{L_\ell} {\cal W}_{\nu({k_\ell})} \widetilde{\bf p}^{(\ell)}_{q} \rangle.
\end{split}$$ Taking into account the block representation (\[eqn:nuc\_MatrSparsP\]) and the overlapping property $$\label{eqn:Overlap_Basis}
{\bf G}_{\bf k} \odot {\bf G}_{\bf m}=0 \quad \mbox{if} \quad | k_\ell - m_\ell| \geq L_0,$$ we analyze the block sparsity pattern in the Galerkin matrix $V_{c_L}$. Given $3 M_0 R_{\cal N} $ vectors $\sum\limits_{k_\ell=1}^{L_\ell} {\cal W}_{\nu({k_\ell})} \widetilde{\bf p}^{(\ell)}_{q}\in \mathbb{R}^{N_L}$, where $N_L$ denotes the total number of grid points in $\Omega_L$ in each space variable. Now the numerical cost to compute ${v}_{km}$ for every fixed index $(k,m)$ is estimated by $O(M_0 R_{\cal N} N_L)$ indicating linear scaling in the large grid parameter $N_L$ (but not cubic).
Fixed the row index in $(k,m)$, then (b) follows from the bound on the total number of cells $\Omega_{\bf k}$ in the effective integration domain in (\[nuc\_potMatrTot\]), that is $(2 L_0 + 1)^3$, and the symmetry of $V_{c_L}$.
Figure \[fig:3DCorePerScellErr\] illustrates the sparsity pattern of the nuclear potentail operator $V_{c_L}$ in the matrix $H$, for $(L, 1, 1)$ lattice in a supercell with $L=32$ and $m_0=4$, corresponding to the overlapping parameter $L_0=3$. One can observe the nearly-boundary effects due to the non-equalized contributions from the left and from the right (supercell in a box).
Figure \[fig:3DCorePerScellErr\] shows the difference between matrices $V_{c_L}$ in periodic (see §\[ssec:Core\_Ham\_period\] for more details) and non-periodic cases. The relative norm of the difference is vanishing if $L\to \infty$.
![ Matrix $V_{c_L}$ in a supercell for $L_0=3, L=32$ (left). Difference between matrices $V_{c_L}$ in periodic and single-box cases (middle). Block-sparsity in the matrix $V_{c_L}$ in periodic case (right).[]{data-label="fig:3DCorePerScellErr"}](NM_Lx64_1_1.eps "fig:"){width="5.0cm"}![ Matrix $V_{c_L}$ in a supercell for $L_0=3, L=32$ (left). Difference between matrices $V_{c_L}$ in periodic and single-box cases (middle). Block-sparsity in the matrix $V_{c_L}$ in periodic case (right).[]{data-label="fig:3DCorePerScellErr"}](NM_err_Lx64_1_1.eps "fig:"){width="5.0cm"} ![ Matrix $V_{c_L}$ in a supercell for $L_0=3, L=32$ (left). Difference between matrices $V_{c_L}$ in periodic and single-box cases (middle). Block-sparsity in the matrix $V_{c_L}$ in periodic case (right).[]{data-label="fig:3DCorePerScellErr"}](VH_P_64_1_1_matr.eps "fig:"){width="5.0cm"}
Notice that the quantized approximation of canonical vectors involved in ${\bf G}_k$ and ${\bf P}_{c_L}$ reduces this cost to the logarithmic scale, $O(M_0 R_{\cal N} \log N_L)$, that is important in the case of large $L$ in view of $N_L=O(L)$.
The block $L_0$-diagonal structure of the matrix $V_{c_L}=\{V_{{\bf k}{\bf m}}\}$, ${\bf k}\in \mathbb{Z}^3$ ($- L/2 \leq {k}_\ell L/2$) described by Lemma \[lem:SparseCaseE\] allows the essential saving in the storage costs.
However, the polynomial complexity scaling in $L$ leads to severe limitations on the number of unit cells. These limitations can be relax if we look more precisely on the defect between matrix ${V}_{c_L}$ and its block-circulant version corresponding to the periodic boundary conditions (see §\[ssec:Core\_Ham\_period\]). This defect can be split into two components with respect to their local and non-local features:
1. Non-local effect due to the asymmetry in the interaction potential sum on the lattice in a box.
2. The near boundary (local) defect that effects only those blocks in $V_{c_L}=\{V_{{\bf k}{\bf m}}\}$ lying in the $L_0$-width of $\partial\Omega_L$, $$L_0+1 -(L-1)/2\leq k_\ell, m_\ell \leq (L-1)/2-1 -L_0.$$
Item (A) is related to a slight modification of the core potential to the shift invariant Toeplitz-type form $V_{{\bf k}{\bf m}} = V_{|{\bf k}-{\bf m}|}$ by replication of the central block corresponding to $k=0$, as considered in the next section. In this way the overlap condition (\[eqn:Overlap\_Basis\]) for the tensor ${\bf G}_{\bf k}$ will impose the block sparsity.
The boundary effect in item (B) becomes relatively small for large number of cells so that the block-circulant part of the matrix $V_{c_L}$ is getting dominating as $L\to \infty$.
The full diagonalization for above mentioned matrices is impossible. However the efficient storage and fast matrix-times-vector algorithms can be applied in the framework of iterative methods for calculation of a small subset of eigenvalues.
Discrete Laplacian and the mass matrix {#ssec:Lap_Op}
--------------------------------------
The Laplace operator part included in eigenvalue problem for a single molecule is posed in the unit cell $ \Omega=[-b/2,b/2]^3 \in \mathbb{R}^3 $, subject to the homogeneous Dirichlet boundary conditions on $\partial \Omega$. In periodic case they should be substituted by the periodic boundary conditions. For given discretization parameter $\overline{n} \in \mathbb{N}$, we use the equidistant $\overline{n}\times \overline{n} \times \overline{n}$ tensor grid $\omega_{{\bf 3},\overline{n}}=\{x_{\bf i}\} $, ${\bf i} \in {\cal I} :=\{1,...,\overline{n}\}^3 $, with the mesh-size $h=2b/(\overline{n} + 1)$, which might be different from the grid $\omega_{{\bf 3},n} $ introduced for representation of the interaction potential (usually, $n\leq \overline{n}$).
Define a set of piecewise linear basis functions $\overline{g}_k := {\bf I}_1 g_k $, $k=1,...,N_b$, by linear tensor-product interpolation via the set of product hat functions, $\{\xi_{\bf i}\}= \xi_{i_1} (x_1) \xi_{i_2} (x_2)\xi_{i_3} (x_3)$, ${\bf i} \in {\cal I}$, associated with the respective grid-cells in $\omega_{{\bf 3},N}$. Here the linear interpolant ${\bf I}_1= {I}_1\times {I}_1 \times {I}_1$ is a product of 1D interpolation operators, $\overline{g}_k^{(\ell)}= {I}_1 {g}_k^{(\ell)}$, $\ell=1,...,3$, where ${I}_1:C^0([-b,b])\to W_h:=span\{\xi_i\}_{i=1}^{\overline{n}}$ is defined over the set of piecewise linear basis functions by $$({I}_1 \, w)(x_\ell):=\sum_{i=1}^{\tilde{n}} w(x_{i_\ell})\xi_{i}(x_\ell),
\quad x_{\bf i} \in \omega_{{\bf 3},\tilde{n}}.$$
With these definitions, the rank-$3$ tensor representation of the standard FEM Galerkin stiffness matrix for the Laplacian, $A_3$, in the tensor basis $\{\xi_i(x_1) \xi_j (x_2)\xi_k (x_3) \} $, $i,j,k = 1,\ldots \overline{n}$, is given by $$A_3 := A^{(1)} \otimes S^{(2)} \otimes S^{(3)} + S^{(1)} \otimes A^{(2)} \otimes S^{(3)}
+ S^{(1)} \otimes S^{(2)} \otimes A^{(3)}\in
\mathbb{R}^{\overline{n}^{\otimes 3}\times \overline{n}^{\otimes 3}},$$ where the 1D stiffness and mass matrices $A^{(\ell)}, S^{(\ell)} \in \mathbb{R}^{\overline{n}\times \overline{n}}$, $\ell=1,\,2,\,3$, are represented by $$A^{(\ell)} := \{ \langle \frac{d}{d x_\ell} \xi_i(x_\ell) , \frac{d}{d x_\ell} \xi_j(x_\ell)
\rangle \}^{\overline{n}}_{i,j=1} = \frac{1}{h} \mbox{tridiag} \{-1,2,-1\},$$ $$S^{(\ell)}=\{ \langle \xi_i ,\xi_j\rangle \}^{\overline{n}}_{i,j=1} = \frac{h}{6}\;
\mbox{tridiag} \{1,4,1\},$$ respectively.
This leads to the separable grid-based approximation of the initial basis functions $g_k(x)$, $$\label{eq. Gaus pwl}
g_k (x) \approx \overline{g}_k (x)=\prod^3_{\ell=1}
\overline{g}_k^{(\ell)} (x_{\ell})=\prod^3_{\ell=1}
\sum\limits^{\overline{n}}_{i=1} g_{k}^{(\ell)}(x_{i_\ell}) \xi_i (x_{\ell}),$$ where the rank-$1$ coefficients tensor ${\bf G}_k$ is given by ${\bf G}_k= {\bf g}_k^{(1)} \otimes {\bf g}_k^{(2)} \otimes {\bf g}_k^{(3)}$, with the canonical vectors ${\bf g}_k^{(\ell)}=\{g_{k_i}^{(\ell)}\}\equiv \{g_{k}^{(\ell)}(x_{i_\ell})\}$. Let us agglomerate the rank-$1$ tensors ${\bf G}_k\in \mathbb{R}^{{\overline{n}}^{\otimes 3}}$, ($k=1,...,N_b$) in a tensor-valued matrix $G\in \mathbb{R}^{N^{\otimes 3}\times N_b}$, the Galerkin matrix in the basis set ${\bf G}_k $ can be written in a matrix form $$A_G= G^T A_3 G\in \mathbb{R}^{N_b \times N_b},$$ corresponding to the standard matrix-matrix transform under the change of basis. The matrix entries in $A_G=\{a_{k m}\}$ can be represented by $$a_{k m}= \langle A_3 {\bf G}_k, {\bf G}_m\rangle, \quad k,m = 1,...,N_b.$$ Likewise, for the entries of the stiffness matrix we have $s_{k m}= \langle {\bf G}_k, {\bf G}_m\rangle$.
It is easily seen that in the periodic case both matrices, $A_G$ and $S$, take the multilevel block circulant structure.
Linearized spectral problem by FFT-diagonalization {#sec:Core_Ham_period_FFT}
==================================================
There are two possibilities for mathematical modeling of the $L$-periodic molecular systems, composed, of $(L, L, L)$ elementary unit cells. In the first approach, the system is supposed to contain an infinite set of equivalent atoms that map identically into itself under any translation by $L$ units in each spacial direction. The other model is based on the ring-type periodic structures consisting of $L$ identical units in each spacial direction, where every unit cell of the periodic compound will be mapped to itself by applying a rotational transform from the corresponding rotational group symmetry [@SzOst:1996].
The main difference between these two concepts is in the treatment of the lattice sum of Coulomb interactions, thought, in the limit of $L\to \infty$ both models approach each other. In this paper we mainly follow the first approach with the particular focus on the asymptotic complexity optimization for large lattice parameter $L$. The second concept is useful for understanding the block structure of the Galerkin matrices for Laplacian and the identity operators.
Block circulant structure of core Hamiltonian in periodic case {#ssec:Core_Ham_period}
--------------------------------------------------------------
Inthis section we consider the periodic case, further called case (P), and derive the more refined sparsity pattern of the matrix $V_{c_L}$ using the $d$-level ($d=1,2,3$) tensor structure in this matrix. The matrix block entries are numbered by a pair of multi-indices, $V_{c_L}=\{V_{{\bf k}{\bf m}}\}$, ${\bf k}=(k_1,k_2,k_3)$, where the $m_0\times m_0$ matrix block $V_{{\bf k}{\bf m}}$ is defined by (\[eqn:nuc\_MatrSparsP\]). Figure \[fig:3DPeriodStruct\] illustrates an example of 3D lattice-type structure of size $(4, 4, 2)$.
![Example of the 3D lattice-type structure of size $(4, 4, 2)$. []{data-label="fig:3DPeriodStruct"}](periodic42.eps){width="7.0cm"}
Following [@VeBoKh:Ewald:14] we introduce the periodic cell ${\cal R}= \mathbb{Z}^d$, $d=1,2,3$ for the $\bf k$ index, and consider a 3D $B$-periodic supercell $\Omega_L= B\times B\times B$, with $B= \frac{b}{2}[-L,L]$. The total electrostatic potential in the supercell $\Omega_L$ is obtained by, first, the lattice summation of the Coulomb potentials over $\Omega_L$ for (rather large) $L$, but restricted to the central unit cell $\Omega_0$, and then by replication of the resultant function to the whole supercell. Hence, that the total potential sum $v_{c_L}(x)$ is designated at each elementary unit-cell in $\Omega_L$ by the same value (${\bf k}$-translation invariant). The effect of the conditional convergence of the lattice summation can be treated by using the extrapolation to the limit (regularization) on a sequence of different lattice parameters $L$ as described in [@VeBoKh:Ewald:14].
The electrostatic potential in any of $B$-periods can be obtained by copying the respective data from $\Omega_L$. The basis set in $\Omega_L$ is constructed by replication from the the master unit cell $\Omega_0$ over the whole periodic lattice.
Consider the case $d=3$ in more detail. Recall that the reference value $v_{c_L}(x)$ will be computed at the central cell $\Omega_0$, indexed by $(0,0,0)$, by summation over all contributions from $L^3$ elementary sub-cells in $\Omega_L$. For technical reasons here and in the following we vary the summation index in $k_\ell=0,..., L-1$ and obtain $$\label{eqn:EwaldSumP}
v_0(x)= \sum_{\nu=1}^{M_0} \sum\limits_{k_1,k_2,k_3=0}^{L-1}
\frac{Z_\nu}{\|{x} -a_\nu (k_1,k_2,k_3)\|},
\quad x\in \Omega_0.$$ The local lattice sum on the index set $n\times n \times n$ corresponding to $\Omega_0$, is represented by $${\bf P}_{\Omega_0} = \sum_{\nu=1}^{M_0} Z_\nu \sum\limits_{k_1,k_2,k_3=0}^{L-1} \sum\limits_{q=1}^{R_{\cal N}}
{\cal W}_{\nu({\bf k})} \widetilde{\bf p}^{(1)}_{q}
\otimes \widetilde{\bf p}^{(2)}_{q} \otimes \widetilde{\bf p}^{(3)}_{q}
\in \mathbb{R}^{n\times n \times n},$$ for the corresponding local projected tensor of small size $n\times n \times n$. Here the $\Omega$-windowing operator, $
{\cal W}_{\nu({\bf k})}={\cal W}_{\nu(k_1)}^{(1)}\otimes {\cal W}_{\nu(k_2)}^{(2)}
\otimes {\cal W}_{\nu(k_3)}^{(3)},
$ restricts onto the small $n\times n \times n$ unit cell by shifting by the lattice vector ${\bf k}=(k_1,k_2,k_3)$. This reduces both the computational and storage costs by factor $L$.
In the 3D case, we set $q=3$ in the notation for multilevel BC matrix. Similar to the case of one-level BC matrices, we notice that a matrix $A\in {\cal BC} (3,{\bf L},m)$ of size $|{\bf L}| m \times |{\bf L}| m$ is completely defined by a $3$-rd order coefficients tensor ${\bf A}=[A_{k_1 k_2 k_3}]$ of size $L_1 \times L_2 \times L_3 $, ($k_\ell=0,...,L_\ell-1$, $\ell=1,2,3$), with $m\times m$ block-matrix entries, obtained by folding of the generating first column vector in $A$.
\[lem:SparseCaseP\] Assume that in case (P) the number of overlapping unit cells (in the sense of supports of basis functions) in each spatial direction does not exceed $L_0$. Then the Galerkin matrix $V_{c_L}$ exhibits the symmetric, three-level block circulant Kronecker tensor-product form. i.e. $V_{c_L} \in {\cal BC} (3,{\bf L},m_0)$, (${\bf L}=(L_1,L_2,L_3)$) $$\label{eqn:BC-Core}
V_{c_L}= \sum\limits_{k_1=0}^{L_1-1} \sum\limits_{k_2=0}^{L_2-1} \sum\limits_{k_3=0}^{L_3-1}
\pi_{L_1}^{k_1}\otimes \pi_{L_2}^{k_2}\otimes \pi_{L_3}^{k_3}\otimes A_{k_1 k_2 k_3},
\quad A_{k_1 k_2 k_3}\in \mathbb{R}^{m_0\times m_0},$$ where the number non-zero matrix blocks $A_{k_1 k_2 k_3}$ does not exceed $(L_0+1)^3$.
The required storage is bounded by $m_0^2 [(L_0 + 1)]^3$ independent of $L$. The set of non-zero generating matrix blocks $\{ A_{k_1 k_2 k_3}\}$ can be calculated in $O(m_0^2 [(L_0 + 1)]^3 n)$ operations.
Furthermore, assume that the QTT ranks of the assembled canonical vectors do not exceed $r_0$. Then the numerical cost can be reduced to the logarithmic scale, $O( m_0^2 [(L_0 + 1)]^3\log n)$.
First, we notice that the shift invariance property in the matrix $V_{c_L}=\{V_{{\bf k}{\bf m}}\}$ is a consequence of the translation invariance in the canonical tensor ${\bf P}_{c_L}$ (periodic case), and in the basis-tensor ${\bf G}_{\bf k}$ (by construction), $$\label{eqn:Basis_shift}
{\bf G}_{\bf k m}:= {\bf G}_{\bf k} \odot {\bf G}_{\bf m}= {\bf G}_{|{\bf k} -{\bf m}|} \quad
\mbox{for} \quad | k_\ell|, |m_\ell| \leq L-1,$$ so that we have $$\label{nuc_BCirculantP}
V_{{\bf k}{\bf m}} = V_{|{\bf k}-{\bf m}|}, \quad 0\leq k_\ell, m_\ell \leq L-1.
$$ This ensures the perfect three-level block-Toeplitz structure of $V_{c_L}$ (compare with the case of a box). Now the block circulant pattern in ${\cal BC} (3,{\bf L},m_0)$ is imposed by the periodicity of a lattice-structured basis set.
To prove the complexity bounds we observe that a matrix $V_{c_L} \in {\cal BC} (3,{\bf L},m_0)$ can be represented in the Kronecker tensor product form (\[eqn:BC-Core\]), obtained by an easy generalization of (\[eqn:bcircPol\]). In fact, we apply (\[eqn:bcircPol\]) by successive slice-wise and fiber-wise splitting to obtain $$\begin{split}
V_{c_L}
& = \sum\limits_{k_1=0}^{L_1-1}\pi_{L_1}^{k_1}\otimes {\bf A}_{k_1} \\
&= \sum\limits_{k_1=0}^{L_1-1}\pi_{L_1}^{k_1}\otimes
\left( \sum\limits_{n_2=0}^{L_2-1} \pi_{L_2}^{k_2}\otimes {\bf A}_{k_1 k_2} \right)\\
&=\sum\limits_{k_1=0}^{L_1-1}\pi_{L_1}^{k_1}\otimes
\left( \sum\limits_{k_2=0}^{L_2-1} \pi_{L_2}^{k_2}\otimes
\left(\sum\limits_{k_3=0}^{L_3-1} \pi_{L_3}^{k_3}\otimes {A}_{k_1 k_2 k_3} \right) \right),
\end{split}$$ where ${\bf A}_{k_1}\in \mathbb{R}^{L_2\times L_3 \times m_0\times m_0}$, ${\bf A}_{k_1 k_2} \in \mathbb{R}^{L_3 \times m_0\times m_0}$, and $A_{k_1 k_2 k_3}\in \mathbb{R}^{m_0\times m_0}$. Now the overlapping assumption ensures that the number of non-zero matrix blocks $A_{k_1 k_2 k_3}$ does exceed $(L_0+1)^3$.
Furthermore, the symmetric mass matrix, $S_{c_L}=\{{s}_{\mu \nu} \}\in \mathbb{R}^{N_b\times N_b}$, for the Galerkin representation of the identity operator reads as follows, $$\label{Ident_pot}
{s}_{\mu \nu}= \langle {\bf G}_\mu , {\bf G}_\nu \rangle
=\langle S^{(1)}{\bf g}_\mu^{(1)},{\bf g}_\nu^{(1)} \rangle
\langle S^{(2)} {\bf g}_\mu^{(2)},{\bf g}_\nu^{(2)} \rangle
\langle S^{(3)} {\bf g}_\mu^{(3)},{\bf g}_\nu^{(3)} \rangle,
\quad 1\leq \mu, \nu \leq N_b,$$ where $N_b=m_0 L^3$. It can be seen that in the periodic case the block structure in the basis-tensor ${\bf G}_{\bf k} $ imposes the three-level block circulant structure in the mass matrix $S_{c_L}$ $$\label{eqn:BC-mass}
S_{c_L}= \sum\limits_{k_1=0}^{L_1-1} \sum\limits_{k_2=0}^{L_2-1} \sum\limits_{k_3=0}^{L_3-1}
\pi_{L_1}^{k_1}\otimes \pi_{L_2}^{k_2}\otimes \pi_{L_3}^{k_3}\otimes S_{k_1 k_2 k_3},
\quad S_{k_1 k_2 k_3}\in \mathbb{R}^{m_0\times m_0}.$$ By the previous arguments we conclude that $S_{k_1 k_2 k_3}=S^{(1)}_{k_1} S^{(2)}_{k_2} S^{(3)}_{k_3}$ implying the rank-$1$ separable representation in (\[eqn:BC-mass\]).
Likewise, it is easy to see that the stiffness matrix representing the (local) Laplace operator in the periodic setting has the similar block circulant structure, $$\label{eqn:BC-Laplace}
\Delta_{c_L}= \sum\limits_{k_1=0}^{L_1-1} \sum\limits_{k_2=0}^{L_2-1} \sum\limits_{k_3=0}^{L_3-1}
\pi_{L_1}^{k_1}\otimes \pi_{L_2}^{k_2}\otimes \pi_{L_3}^{k_3}\otimes B_{k_1 k_2 k_3},
\quad B_{k_1 k_2 k_3}\in \mathbb{R}^{m_0\times m_0},$$ where the number non-zero matrix blocks $B_{k_1 k_2 k_3}$ does not exceed $(L_0+1)^3$. In this case the matrix block $B_{k_1 k_2 k_3}$ admits a rank-$3$ product factorization.
This proves the sparsity pattern of our tensor approximation to $H$.
In the Hartree-Fock calculations for lattice structured systems we deal with the multilevel, symmetric block circulant/Toeplitz matrices, where the first-level blocks, $A_0,...,A_{L_1-1}$, may have further block structures. In particular, Lemma \[lem:SparseCaseP\] shows that the Galerkin approximation of the 3D Hartree-Fock core Hamiltonian in periodic setting leads to the symmetric, three-level block circulant matrix.
Figure \[fig:3DCoreHamPer\] represents the block-sparsity in the core Hamiltonian matrix in a box for $L=8$ (left), and the rotated matrix profile (right).
![Block-sparsity in the core Hamiltonian matrix in a box for $L=8$ (left); Rotated matrix profile (right).[]{data-label="fig:3DCoreHamPer"}](Core_4_128_CHd.eps "fig:"){width="6.0cm"}![Block-sparsity in the core Hamiltonian matrix in a box for $L=8$ (left); Rotated matrix profile (right).[]{data-label="fig:3DCoreHamPer"}](Core_4_128_CH_rotd.eps "fig:"){width="6.0cm"}
In the next section we discuss computational details of the FFT-based eigenvalue solver on the example of 3D linear chain of molecules.
Regularized spectral problem and complexity analysis {#ssec:Complexity_EigPr}
----------------------------------------------------
Combining the block circulant representations (\[eqn:BC-Core\]), (\[eqn:BC-Laplace\]) and (\[eqn:BC-mass\]), we are able to represent the eigenvalue problem for the Fock matrix in the Fourier space as follows $$\label{eqn:HF-FSpace}
\sum\limits^{L_1 -1}_{k_1=0}\sum\limits^{L_2 -1}_{k_2=0}\sum\limits^{L_3 -1}_{k_3=0}
D_{L_1}^{k_1}\otimes D_{L_2}^{k_2}\otimes D_{L_3}^{k_3}\otimes
(B_{k_1 k_2 k_3} + A_{k_1 k_2 k_3}) U
= \lambda
\sum\limits^{L_1 -1}_{k_1=0}\sum\limits^{L_2 -1}_{k_2=0}\sum\limits^{L_3 -1}_{k_3=0}
D_{L_1}^{k_1}\otimes D_{L_2}^{k_2}\otimes D_{L_3}^{k_3} S_{k_1 k_2 k_3} U,$$ with the diagonal matrices $D_{L_\ell}^{k_\ell}\in \mathbb{R}^{L_\ell \times L_\ell}$, $\ell=1,2,3$, where $U=F_{\bf L}\otimes I_m C$. The equivalent block-diagonal form reads $$\label{eqn:HF-FSpace-Block}
\mbox{bdiag}_{m_0\times m_0}
\{{\cal T}_{\bf L}'[F_{\bf L} ({\cal T}_{\bf L}\widehat{B})
+ F_{\bf L} ({\cal T}_{\bf L}\widehat{A})] -
\lambda {\cal T}_{\bf L}'(F_{\bf L} [{\cal T}_{\bf L} \widehat{S})] \} U =0.
$$ The block structure specified by Lemma \[lem:SparseCaseP\] allows to apply the efficient eigenvalue solvers via FFT based diagonalization in the framework of Hartree-Fock calculations, in general, with the numerical cost $O(m_0^2 L^d \log L)$.
![Molecular orbitals, i.e. the eigenvectors represented in GTO basis: the $4$th orbital (left), the $8$th orbital (right). []{data-label="fig:3DCoreEigVect"}](Vec_128_3_4.eps "fig:"){width="6.0cm"} ![Molecular orbitals, i.e. the eigenvectors represented in GTO basis: the $4$th orbital (left), the $8$th orbital (right). []{data-label="fig:3DCoreEigVect"}](Vec_128_3_8.eps "fig:"){width="6.0cm"}
\[prop:low\_rank\_coef\] The low-rank structure in the coefficients tensor mentioned above (see Section \[ssec:Tensor\_bcirc\]) allows to reduce the factor $L^d \log L$ to $L \log L$ for $d=2,3$. It was already observed in the proof of Lemma \[lem:SparseCaseP\] that the respective coefficients in the overlap and Laplacian Galerkin matrices can be treated as the rank-$1$ and rank-$3$ tensors, respectively. Clearly, the factorization rank for the nuclear part of the Hamiltonian does not exceed $R_{\cal N}$. Hence, Theorem \[thm:tens\_FFT\] can be applied in generalized form.
Figure \[fig:3DCoreEigVect\] visualizes molecular orbitals on fine spatial grid with $n=2^{14}$: the $4$th orbital (left), the $8$th orbital (right). The eigenvectors are computed in GTO basis for $(L,1,1)$ system with $L=128$ and $m_0=4$.
Table \[Table\_Times\_SupSvsPer\] compares CPU times in sec. (Matlab) for the full eigenvalue solver on a 3D $(L, 1, 1)$ lattice in a box, and for the FTT-based diagonalization in the periodic supercell, all computed for $m_0=4$, $L=2^p$ ($p=7,8,...,15$). The number of basis function (problem size) is given by $N_b=m_0 L$.
![Spectrum of the core Hamiltonian.[]{data-label="fig:3DCoreEig"}](Spectr_H4_128_1_1.eps "fig:"){width="6.0cm"} ![Spectrum of the core Hamiltonian.[]{data-label="fig:3DCoreEig"}](Spectr_H4_256_1_1.eps "fig:"){width="6.0cm"}
\[c\][|r|r|r|r|r|r|r|r|r|r|]{}Problem size $N_b=n_0 L $ & $512$ & $1024$ & $2048$ & $4096$ & $8192$ & $16384$ & $32768$ & $65536$ & $131072$\
Full EIG-solver & $0.67$ & $5.49$ & $48.6 $ & $497.4$ & $--$ & $--$ & $--$ & $--$ & $--$\
FFT diagonalization & $0.10$ & $0.09$ & $0.08 $ & $0.14$ & $0.44$ & $1.5$ & $5.6$ & $22.9$ & $89.4$\
Figure \[fig:3DCoreEig\] represents the spectrum of the core Hamiltonian in a box vs. those in a periodic supercell for different number of cells $L=128,256$, where $m_0=4$. The systematic difference between the eigenvalues in both cases can be observed even for very large $L$. This spectral pollution effects have been discussed and theoretically analyzed in [@CancesDeLe:08].
![Spectrum of the core Hamiltonian for a $(L,1,1)$ lattice with $L=256$, and $m_0=4$, in a box (left) and for periodic case (right).[]{data-label="fig:3DSpectBand"}](Spectrum_EVP_512_b.eps "fig:"){width="6.0cm"} ![Spectrum of the core Hamiltonian for a $(L,1,1)$ lattice with $L=256$, and $m_0=4$, in a box (left) and for periodic case (right).[]{data-label="fig:3DSpectBand"}](Spectrum_EVP_512_p.eps "fig:"){width="6.0cm"}
Figure \[fig:3DSpectBand\] presents the spectral bands for a $(L,1,1)$ lattice system in a box and in the periodic setting, for $L=256$, and $m_0=4$.
![Average energy per unit cell vs. $L$ for a $(L,1,1)$ lattice in a 3D rectangular “tube“.[]{data-label="fig:3DAveEig"}](Average_Lambda_512.eps){width="7.0cm"}
Figure \[fig:3DAveEig\] demonstrates the relaxation of the average energy per unit cell with $m_0=4$, for a $(L,1 ,1)$ lattice structure in a 3D rectangular “tube‘ up to $L=512$, for both periodic and open boundary conditions.
Conclusions {#sec:Conclusions}
===========
We have introduced and analyzed the grid-based tensor product approach to discretization and solution of the Hartree-Fock equation in ab initio modeling of the lattice-structured molecular systems. In this presentation we consider the case of core Hamiltonian. All methods and algorithms developed in this paper are implemented and tested in Matlab.
The proposed tensor techniques manifest the twofold benefits: (a) the entries of the Fock matrix are computed by 1D operations using low-rank tensors represented on a 3D grid, (b) the low-rank tensor structure in the diagonal blocks of the Fock matrix in the Fourier space reduces the conventional 3D FFT to the product of 1D Fourier transforms. The main contributions include:
- Fast computation of Fock matrix by 1D matrix-vector operations using low-rank tensors represented on a 3D spacial grid.
- Analysis and numerical implementation of the multilevel block circulant representation of the Fock matrix in the periodic setting.
- Investigation of the low-rank tensor structure in the diagonal blocks of the Fock matrix represented in the Fourier space, that allows to reduce the conventional 3D FFT to the product of 1D FFTs.
- Numerical tests illustrating the computational efficiency of the tensor-structured methods applied to the reduced Hartree-Fock equation for lattice-type and periodic systems. Numerical experiments on verification of the theoretical results on the asymptotic complexity estimated of the presented algorithms.
Here we confine ourself to the case of core Hamiltonian part in the full Fock matrix (linear part in the Fock operator). The rigorous study of the fully nonlinear self-consistent Hartree-Fock eigenvalue problem for periodic and lattice-structured systems in a box is a matter of future research.
Appendix: Overview on block circulant matrices {#sec_Append:block-circ}
==============================================
We recall that a one-level block circulant matrix $A\in {\cal BC} (L,m_0)$ is defined by [@Davis], $$\label{eqn:block_c}
A=\operatorname{bcirc}\{A_0,A_1,...,A_{L-1}\}=
\begin{bmatrix}
A_0 & A_{L-1} & \cdots & A_{2} & A_{1} \\
A_{1} & A_0 & \cdots & \vdots & A_{2} \\
\vdots & \vdots & \ddots & A_0 & \vdots \\
A_{L-1} & A_{L-2} & \cdots & A_{1} & A_0 \\
\end{bmatrix}
\in \mathbb{R}^{L m_0\times L m_0},$$ where $A_k \in \mathbb{R}^{m_0\times m_0}$ for $k=0,1, \ldots ,L-1$, are matrices of general structure. The equivalent Kronecker product representation is defined by the associated matrix polynomial, $$\label{eqn:bcircPol}
A= \sum\limits^{L -1}_{k=0} \pi^k \otimes A_k =:p_A(\pi),$$ where $\pi=\pi_L\in \mathbb{R}^{L \times L}$ is the periodic downward shift (cycling permutation) matrix, $$\pi_L:=
\begin{bmatrix}
0 & 0 & \cdots & 0 & 1 \\
1 & 0 & \cdots & 0 & 0 \\
\vdots &\vdots & \ddots & \vdots & \vdots \\
0 & \cdots & 1 & 0 & 0 \\
0 & 0 & \cdots & 1 & 0 \\
\end{bmatrix}
,$$ and $\otimes$ denotes the Kronecker product of matrices.
In the case $m_0=1$ a matrix $A\in {\cal BC} (L,1)$ defines a circulant matrix generated by its first column vector $\widehat{a}=(a_0,...,a_{L-1})^T$. The associated scalar polynomial then reads $$p_A(z):= a_0 + a_1 z + ... +a_{L-1} z^{L-1},$$ so that (\[eqn:bcircPol\]) simplifies to $$A=p_A(\pi_L).$$ Let $\omega= \omega_L= \exp(-\frac{2\pi i}{L})$, we denote by $$F_L=\{f_{k\ell}\}\in \mathbb{R}^{L\times L}, \quad \mbox{with} \quad
f_{k\ell}=\frac{1}{\sqrt{L}}\omega_L^{(k-1)(\ell-1)},\quad
k,l=1,...,L,$$ the unitary matrix of Fourier transform. Since the shift matrix $\pi_L$ is diagonalizable in the Fourier basis, $$\label{eqn:diagshift}
\pi_L=F_L^\ast D_L F_L,\quad D_L= \mbox{diag}\{1,\omega,...,\omega^{L-1} \},$$ the same holds for any circulant matrix, $$\label{eqn:circDiag}
A = p_A(\pi_L) = F_L^\ast p_A(D_L) F_L,$$ where $$p_A(D_L)=\mbox{diag}\{p_A(1),p_A(\omega),...,p_A(\omega^{L-1})\}= \mbox{diag}\{F_L a\}.$$
Conventionally, we denote by $\mbox{diag}\{x\}$ a diagonal matrix generated by a vector $x$. Let $X$ be an $L m_0\times m_0$ matrix obtained by concatenation of $m_0\times m_0$ matrices $X_k$, $k=0,...,L-1$, $X=\operatorname{conc}(X_0,...,X_{L-1})=[X_0,...,X_{L-1}]^T$. For example, the first block column in (\[eqn:block\_c\]) has the form $\operatorname{conc}(A_0,...,A_{L-1})$. We denote by $\mbox{bdiag}\{X\}$ the $L m_0\times L m_0$ block-diagonal matrix of block size $L$ generated by $m_0\times m_0$ blocks $X_k$.
It is known that similarly to the case of circulant matrices (\[eqn:circDiag\]), block circulant matrix in ${\cal BC} (L,m_0)$ is unitary equivalent to the block diagonal one by means of Fourier transform via representation (\[eqn:bcircPol\]), see [@Davis]. In the following, we describe the block-diagonal representation of a matrix $A\in {\cal BC} (L,m_0)$ in the form that is convenient for generalization to the multi-level block circulants as well as for the description of FFT based implementational schemes. To that end, let us introduce the reshaping (folding) transform ${\cal T}_L$ that maps a $L m_0\times m_0$ matrix $X$ (i.e., the first block column in $A$) to $L\times m_0\times m_0$ tensor $B={\cal T}_L X$ by plugging the $i$th $m_0\times m_0$ block in $X$ into a slice $B(i,:,:)$. The respective unfolding returns the initial matrix $X={\cal T}_L' B$. We denote by $\widehat{A}\in \mathbb{R}^{L m_0\times m_0}$ the first block column of a matrix $A\in {\cal BC} (L,m_0)$, with a shorthand notation $$\widehat{A}=[A_0,A_1,...,A_{L-1}]^T,$$ so that the $L\times m_0\times m_0$ tensor ${\cal T}_L \widehat{A}$ represents slice-wise all generating $m_0\times m_0$ matrix blocks.
\[prop:eig\_bcmatr\] For $A\in {\cal BC} (L,m_0)$ we have $$\label{eqn:bcircDiag}
A= (F_L^\ast \otimes I_{m_0}) \operatorname{bdiag}
\{ \bar{A}_0, \bar{A}_1,\ldots , \bar{A}_{L-1}\}
(F_L \otimes I_{m_0}),$$ where $$\bar{A}_j = \sum\limits^{L -1}_{k=0} \omega_L^{jk} A_k \in \mathbb{C}^{m_0 \times m_0},$$ can be recognized as the $j$-th $m_0\times m_0$ matrix block in block column ${\cal T}_L'(F_L ({\cal T}_L \widehat{A}))$, such that $$\left[ \bar{A}_0, \bar{A}_1,\ldots , \bar{A}_{L-1}\right]^T =
{\cal T}_L'(F_L ({\cal T}_L \widehat{A})).$$ A set of eigenvalues $\lambda$ of $A$ is then given by $$\label{eqn:lambdAbc}
\{\lambda | Ax = \lambda x, \; x\in \mathbb{C}^{L m_0}\}=
\bigcup\limits_{j=0}^{L-1} \{ \lambda |\bar{A}_j u = \lambda u, \; u \in \mathbb{C}^{m_0} \}.$$ The eigenvectors corresponding to the spectral sets $$\Sigma_j= \{\lambda_{j, m} |\bar{A}_j u_{j,m} = \lambda_{j,m} u_{j,m},
\; u_{j,m} \in \mathbb{C}^{m_0}\},
\quad j= 0,1,\ldots , L-1,\quad m=1,...,m_0,$$ can be represented in the form $$\label{eqn:eigvecA}
U_{j,m}=(F_L^\ast \otimes I_{m}) \bar{U}_{j,m},\quad \mbox{where} \quad
\bar{U}_{j,m}= E_{[j]} \operatorname{vec}\, [u_{0,m},u_{1,m},...,u_{L-1,m}],$$ with $E_{[j]}=\operatorname{diag}\{e_j\}\otimes I_{m_0} \in \mathbb{R}^{L m_0\times L m_0} $, and $e_j\in \mathbb{R}^{L}$ being the $j$th Euclidean basis vector.
We combine representations (\[eqn:bcircPol\]) and (\[eqn:diagshift\]) to obtain $$\begin{aligned}
\label{eqn:Bcircdiag}
A & = \sum\limits^{L -1}_{k=0} \pi^k \otimes A_k =
\sum\limits^{L -1}_{k=0} (F_L^\ast D^k F_L) \otimes A_k \\ \nonumber
& = (F_L^\ast \otimes I_{m_0}) (\sum\limits^{L -1}_{k=0} D^k
\otimes A_k)(F_L \otimes I_{m_0}) \\ \nonumber
& = (F_n^\ast \otimes I_m)(\sum\limits^{L -1}_{k=0}
\mbox{bdiag}\{A_k,\omega_L^k A_k,...,\omega_L^{k(L-1)}A_{k} \} )
(F_L \otimes I_{m_0})\\ \nonumber
& = (F_L^\ast \otimes I_{m_0})
\mbox{bdiag}\{\sum\limits^{L -1}_{k=0} A_k,\sum\limits^{L -1}_{k=0} \omega_L^k A_k,...,
\sum\limits^{L -1}_{k=0} \omega_L^{k(L-1)}A_{k} \} (F_L \otimes I_{m_0})\\ \nonumber
& = (F_L^\ast \otimes I_{m_0}) \mbox{bdiag}_{m_0 \times m_0}
\{{\cal T}_L'(F_L ({\cal T}_L \widehat{A}))\} (F_L \otimes I_{m_0}), \nonumber\end{aligned}$$ where the final step follows by the definition of FT matrix and by the construction of ${\cal T}_L$. The structure of eigenvalues and eigenfunctions then follows by simple calculations with block-diagonal matrices.
The next statement describes the block-diagonal form for a class of symmetric BC matrices, ${\cal BC}_s (L,m_0)$, that is a simple corollary of [@Davis], Proposition \[prop:eig\_bcmatr\]. In this case we have $A_0=A_0^T$, and $A_k^T=A_{L-k}$, $k=1,...,L-1$.
\[cor:eig\_symbcmatr\] Let $A\in {\cal BC}_s (L,m_0)$ be symmetric, then $A$ is unitary similar to a Hermitian block-diagonal matrix, i.e., $A$ is of the form $$\label{eqn:F_bc}
A= (F_L \otimes I_{m_0}) \operatorname{bdiag} (\tilde{A}_0, \tilde{A}_1,\ldots , \tilde{A}_{L-1})
(F_L^\ast \otimes I_{m_0}),$$ where $I_{m_0}$ is the $m_0\times m_0$ identity matrix. The matrices $\tilde{A}_j \in \mathbb{C}^{m_0\times m_0}$, $j= 0,1,\ldots , L-1$, are defined for even $n\geq 2$ as $$\label{eqn:symBc}
\tilde{A}_j =A_0 + \sum\limits^{L/2-1}_{k=1} (\omega^{kj}_L A_k +
\overline{\omega}^{kj}_L A^T_k) + (-1)^j A_{L/2}.$$
Corollary \[cor:eig\_symbcmatr\] combined with Proposition \[prop:eig\_bcmatr\] describes a simplified structure of eigendata in the symmetric case. Notice that the above representation imposes the symmetry of each real-valued diagonal blocks $\tilde{A}_j \in \mathbb{R}^{m_0\times m_0}, \; j= 0,1,\ldots , L-1$, in (\[eqn:F\_bc\]).
Finally, we recall that a one-level symmetric block Toeplitz matrix $A\in {\cal BT}_s (L,m_0)$ is defined by [@Davis], $$\label{eqn:block_SToepl}
A=\operatorname{BToepl}_s\{A_0,A_1,...,A_{L-1}\}=
\begin{bmatrix}
A_0 & A_{1}^T & \cdots & A_{L-2}^T & A_{L-1}^T \\
A_{1} & A_0 & \cdots & \vdots & A_{L-2}^T \\
\vdots & \vdots & \ddots & A_0 & \vdots \\
A_{L-1} & A_{L-2} & \cdots & A_{1} & A_0 \\
\end{bmatrix}
\in \mathbb{R}^{L m_0\times L m_0},$$ where $A_k \in \mathbb{R}^{m_0\times m_0}$ for $k=0,1, \ldots ,L-1$, are matrices of a general structure.
[99]{}
Bloch, Andr[é]{}, “Les théorèmes de M.Valiron sur les fonctions entières et la théorie de l’uniformisation”. Annales de la faculté des sciences de l’Université de Toulouse 17 (3): 1-22 (1925). ISSN 0240-2963.
F.A. Bischoff and E.F. Valeev. *Low-order tensor approximations for electronic wave functions: Hartree-Fock method with guaranteed precision.* J. of Chem. Phys., 134, 104104-1-10 (2011).
. *Low-rank quadrature-based tensor approximation of the Galerkin projected Newton/Yukawa kernels.* Comp. Phys. Comm. 183(4) (2012) 904–912.
D. Braess. *Nonlinear approximation theory*. Springer-Verlag, Berlin, 1986.
D. Braess. *Asymptotics for the Approximation of Wave Functions by Exponential-Sums*. J. Approx. Theory, 83: 93-103, (1995).
E. Cancés, A. Deleurence and M. Lewin, *A new approach to the modeling of local defects in crystals: The reduced Hartree-Fock case*. Comm. Math. Phys. 281 (2008) 129-177.
E. Canc[é]{}s, V. Ehrlacher, and Y. Maday. *Periodic Schrödinger operator with local defects and spectral pollution.* SIAM J. Numer. Anal. v. 50, No. 6 (2012), pp. 3016-3035.
J. P. Davis. Circulant matrices. New York. John Wiley & Sons, 1979.
S. Dolgov, B.N. Khoromskij, D. Savostyanov, and I. Oseledets. *Computation of extreme eigenvalues in higher dimensions using block tensor train format.* Comp. Phys. Communications, Volume 185, Issue 4, April 2014, 1207-1216. doi:10.1016/j.cpc.2013.12.017.
L. Frediani, E. Fossgaard, T. Flå and K. Ruud. *Fully adaptive algorithms for multivariate integral equations using the non-standard form and multiwavelets with applications to the Poisson and bound-state Helmholtz kernels in three dimensions.* Molecular Physics, v. 111, 9-11, 2013.
Y. Saad, J. R. Chelikowsky, and S. M. Shontz. *Numerical Methods for Electronic Structure Calculations of Materials.* SIAM Review, v. 52 (1), 2010, 3-54.
R. Dovesi, R. Orlando, C. Roetti, C. Pisani, and V.R. Sauders. *The Periodic Hartree-Fock Method and its Implementation in the CRYSTAL Code.* Phys. Stat. Sol. (b) [**217**]{}, 63 (2000).
V. Ehrlacher, C. Ortner, and A. V. Shapeev. *Analysis of boundary conditions for crystal defect atomistic simulations.* e-prints ArXiv:1306.5334, 2013.
Ewald P.P. *Die Berechnung optische und elektrostatischer Gitterpotentiale.* Ann. Phys [**64**]{}, 253 (1921).
I.P. Gavrilyuk, W. Hackbusch and B.N. Khoromskij. *Hierarchical Tensor-Product Approximation to the Inverse and Related Operators in High-Dimensional Elliptic Problems.* Computing 74 (2005), 131-157.
L. Grasedyck, D. Kressner and C. Tobler. *A literature survey of low-rank tensor approximation techniques.* arXiv:1302.7121v1, 2013.
B.N. Khoromskij and V. Khoromskaia. *Multigrid tensor approximation of function related multi-dimensional arrays.* SIAM J. Sci. Comp. 31(4) (2009) 3002-3026.
B.N. Khoromskij, V. Khoromskaia, and H.-J. Flad. *Numerical Solution of the Hartree-Fock Equation in Multilevel Tensor-structured Format.* SIAM J. Sci. Comp. 33(1) (2011) 45-65.
V. Khoromskaia. *Numerical Solution of the Hartree-Fock Equation by Multilevel Tensor-structured methods.* PhD thesis, TU Berlin, 2010.
V. Khoromskaia, B.N. Khoromskij, and R. Schneider. *Tensor-structured calculation of two-electron integrals in a general basis*. SIAM J. Sci. Comput., [35]{}(2), 2013, A987-A1010.
R.J. Harrison, G.I. Fann, T. Yanai, Z. Gan, and G. Beylkin. *Multiresolution quantum chemistry: Basic theory and initial applications*. J. of Chem. Phys. 121(23) (2004) 11587-11598.
D. R. Hartree. *The Calculation of Atomic Structure*, Wiley, New York, 1957.
Ch. [Froese Fischer]{}. *The Hartree-Fock Method for Atoms – A Numerical Approach*, Wiley, New York, 1977.
M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G.A. Petersson et al., *Gaussian development version Revision H.1*, Gaussian Inc., Wallingford, CT, 2009.
E. A. McCullough, Jr. *Numerical Hartree-Fock methods for molecules*, in P. v. R. Schleyer et al. (eds.), Encyclopedia of Computational Chemistry (Vol. 3), Wiley, Chichester, (1998) 1941–1947.
T. Kailath, and A. Sayed. *Fast reliable algorithms for matrices with structure.* SIAM Publication, Philadelphia, 1999.
V. Khoromskaia and B. N. Khoromskij. *Grid-based lattice summation of electrostatic potentials by assembled rank-structured tensor approximation.* arXiv:1405.2270. CPC 2014, to appear.
B.N. Khoromskij. *Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances.* Chemometr. Intell. Lab. Syst. 110 (2012), 1-19.
T. G. Kolda and B. W. Bader. *Tensor Decompositions and Applications.* SIAM Rev. 51(3) (2009) 455–500.
. *Use of tensor formats in elliptic eigenvalue problems.* Numer. Lin. Alg. Appl., v. 19(1), 2012, 133-151.
B.N. Khoromskij. [*$O(d\log N)$-Quantics Approximation of $N$-$d$ Tensors in High-Dimensional Numerical Modeling.*]{} Constr. Approx. 34 (2011) 257–280.
H.-J. Werner, P.J. Knowles, et al. <span style="font-variant:small-caps;">Molpro</span> version 2010.1, A Package of Ab-Initio Programs for Electronic Structure Calculations.
T. Helgaker, P. J[ø]{}rgensen, and J. Olsen. *Molecular Electronic-Structure Theory*. Wiley, New York, 1999.
V. Khoromskaia. *Computation of the Hartree-Fock Exchange in the Tensor-structured Format.* Comp. Meth. App. Math., 10(2) (2010) 204–218.
V. Khoromskaia. *Black-box Hartree-Fock solver by tensor numerical methods*. Comp. Meth. in Applied Math., vol. 14(2014) No. 1, pp.89-111. T. H. Dunning, Jr. *Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen*, J. Chem. Phys. 90 (1989) 1007–1023.
Venera Khoromskaia. *Numerical Solution of the Hartree-Fock Equation by Multilevel Tensor-structured methods.* PhD thesis, TU Berlin, 2010.\
http://opus4.kobv.de/opus4-tuberlin/frontdoor/index/index/docId/2780.
A. Szabo, and N. Ostlund. *Modern Quantum Chemistry.* Dover Publication, New York, 1996.
C. Pisani, M. Sch[ü]{}tz, S. Casassa, D. Usvyat, L. Maschio, M. Lorenz, and A. Erba. *CRYSCOR: a program for the post-Hartree-Fock treatment of periodic systems*, Phys. Chem. Chem. Phys., 2012, [**14**]{}, 7615-7628.
M. V. Rakhuba, I. V. Oseledets. *Fast multidimensional convolution in low-rank formats via cross approximation.* arXiv:1402.5649, 2013.
S. A. Losilla, D. Sundholm, J. Juselius *The direct approach to gravitation and electrostatics method for periodic systems*. J. Chem. Phys. 132 (2) (2010) 024102.
T. Blesgen, V. Gavini, V. Khoromskaia. *Approximation of the electron density of Aluminium clusters in tensor-product format.* J. Comp. Phys. 231(6) (2012) 2551–2564.
F. Stenger. *Numerical methods based on Sinc and analytic functions.* Springer-Verlag, 1993.
[^1]: Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22-26, D-04103 Leipzig, Germany ([[email protected]]{}).
[^2]: Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22-26, D-04103 Leipzig, Germany ([[email protected]]{}).
| ArXiv |
---
abstract: |
Using a coupling for the weighted sum of independent random variables and the explicit expression of the transition semigroup of Ornstein-Uhlenbeck processes driven by compound Poisson processes, we establish the existence of a successful coupling and the Liouville theorem for general Ornstein-Uhlenbeck processes. Then we present the explicit coupling property of Ornstein-Uhlenbeck processes directly from the behaviour of the corresponding symbol or characteristic exponent. This approach allows us to derive gradient estimates for Ornstein-Uhlenbeck processes via the symbol.
**Keywords:** Ornstein-Uhlenbeck processes; coupling property; Liouville theorem; gradient estimates.
**MSC 2010:** 60J25; 60J75.
author:
- 'René L. Schilling Jian Wang'
title: '**On the Coupling Property and the Liouville Theorem for Ornstein-Uhlenbeck Processes**'
---
[^1] [^2]
Main Results {#section1}
============
Let $(X^x_t)_{t{\geqslant}0}$ be an $n$-dimensional Ornstein-Uhlenbeck process, which is defined as the unique strong solution of the following stochastic differential equation $$\label{ou1}
dX_t = AX_t\,dt + B\,dZ_t,\qquad X_0=x\in{\mathds R}^n.$$ Here $A$ is a real $n\times n$ matrix, $B$ is a real $n\times d$ matrix and $Z_t$ is a Lévy process in ${\mathds R}^d$; note that we allow $Z_t$ to take values in a proper subspace of ${\mathds R}^d$. It is well known that $$X_t^x
=e^{tA}x + \int_0^t e^{(t-s)A}B\,dZ_s.$$ The characteristic exponent or symbol $\Phi$ of $Z_t$, defined by $${\mathds E}\bigl(e^{i{\langle\xi,Z_t\rangle}}\bigr)
=e^{-t\Phi(\xi)},\quad \xi\in{\mathds R}^d,$$ enjoys the following Lévy-Khintchine representation: $$\label{ou2}
\Phi(\xi)
=\frac{1}{2}{\langleQ\xi,\xi\rangle} +i{\langleb,\xi\rangle} +\int_{z\neq 0} \Bigl(1-e^{i{\langle\xi,z\rangle}}+i{\langle\xi,z\rangle}{\mathds 1}_{B(0,1)}(z)\Bigr)\nu(dz),$$ where $Q=(q_{j,k})_{j,k=1}^d$ is a positive semi-definite matrix, $b\in{{\mathds R^d}}$ is the drift vector and $\nu$ is the Lévy measure, i.e.a $\sigma$-finite measure on ${\mathds R}^d\setminus\{0\}$ such that $\int_{z\neq 0}(1\wedge |z|^2)\,\nu(dz)<\infty$. For every $\varepsilon>0$, define ${\nu}_\varepsilon$ on ${\mathds R}^d$ as follows: $${\nu}_\varepsilon(C)
=
\begin{cases}
\nu(C), & \text{if\ \ } \nu({\mathds R}^d)<\infty;\\
\nu(C\setminus \{z: |z|<\varepsilon\}), & \text{if\ \ } \nu({\mathds R}^d)=\infty.
\end{cases}$$
Let $(Y_t)_{t{\geqslant}0}$ be a Markov process on ${\mathds R}^n$ with transition function $P_t(x,\cdot)$. Then, according to [@Li; @T; @SW], we say that $(Y_t)_{t{\geqslant}0}$ admits a *successful coupling* (also: enjoys the *coupling property*) if for any $x,y\in{\mathds R}^n$, $$\label{prex1}
\lim_{t\rightarrow\infty}\|P_t(x,\cdot)-P_t(y,\cdot)\|_{{\mathrm{Var}}}=0,$$ where $\|\cdot\|_{{\mathrm{Var}}}$ stands for the total variation norm. If a Markov process admits a successful coupling, then it also has the Liouville property, i.e. every bounded harmonic function is constant; in this context a function $f$ is harmonic, if $Lf=0$ where $L$ is the generator of the Markov process. See [@CG; @CW] and the references therein for this result and more details on the coupling property.
Let $A$ be an $n\times n$ matrix. We say that an eigenvalue $\lambda$ of $A$ is *semisimple* if the dimension of the corresponding eigenspace is equal to the algebraic multiplicity of $\lambda$ as a root of characteristic polynomial of $A$. Note that for symmetric matrices $A$ all eigenvalues are real and semisimple. Recall that for any two bounded measures $\mu$ and $\nu$ on $({\mathds R}^d,{\mathscr{B}}({\mathds R}^d))$, $\mu\wedge\nu:=\mu-(\mu-\nu)^+$, where $(\mu-\nu)^{\pm}$ refers to the Jordan-Hahn decomposition of the signed measure $\mu-\nu$. In particular, $\mu\wedge\nu=\nu\wedge\mu$, and $\mu\wedge
\nu\,({\mathds R}^d)=\frac{1}{2}\big[\mu({\mathds R}^d)+\nu({\mathds R}^d)-\|\mu-\nu\|_{{\mathrm{Var}}}\big].$
One of our main results is the following
\[th1\] Let $P_t(x,\cdot)$ be the transition probability of the Ornstein-Uhlenbeck process $\{X_t^x\}_{t{\geqslant}0}$ given by . Assume that ${\operatorname{Rank}}(B)=n$ (which implies $n{\leqslant}d$), and that there exist $\varepsilon,\delta>0$ such that $$\label{th2233}
\inf_{z\in{\mathds R}^d,|z|{\leqslant}\delta}\nu _\varepsilon\wedge (\delta_z*\nu_\varepsilon)({\mathds R}^d)>0.$$
If the real parts of all eigenvalues of $A$ are non-positive and if all purely imaginary eigenvalues are semisimple, then there exists a constant $C=C(\varepsilon,\delta,\nu,A,B)>0 $ such that for all $x,y\in{\mathds R}^n$ and $t>0$, $$\label{th21}
\|P_t(x,\cdot)-P_t(y,\cdot)\|_{{\mathrm{Var}}}
{\leqslant}\frac{C(1+|x-y|)}{\sqrt{t}}\wedge2.$$
As a consequence of Theorem \[th1\], we immediately obtain the following result which partly answers the following question about Liouville theorems for non-local operators from [@PZ page 458]: *A challenging task would be to apply other probabilistic techniques, based on ... coupling to non-local operators*.
Under the conditions of Theorem \[th1\], the Ornstein-Uhlenbeck process $\{X_t^x\}_{t{\geqslant}0}$ admits a successful coupling and has the Liouville property.
\[remarkth1\] (1) If $A=0$, $d=n$ and $B={\operatorname{id}}_{{\mathds R}^n}$, then $X_t$ is just a Lévy process on ${\mathds R}^n$. The condition is one possibility to guarantee sufficient jump activity such that the Lévy process $X_t$ admits a successful coupling. To see that is sharp, we can use the example in [@SW Remark 1.2].
\(2) Let $Z_t$ be a (rotationally symmetric) $\alpha$-stable Lévy process $Z_t$, $0<\alpha<2$, and denote by $X_t$ the $n$-dimensional Ornstein-Uhlenbeck process driven by $Z_t$, i.e.$$dX_t = AX_t\,dt + dZ_t.$$ If at least one eigenvalue of $A$ has positive real part, then $X_t$ does not have the coupling property. Indeed, according to [@PZ Example 3.4 and Theorem 3.5], we know that $X_t$ does not have the Liouville property, i.e. there exists a bounded harmonic function which is not constant. According to [@Li Theorem 21.12] or [@CG Theorem 1 and its second remark], $X_t$ does not have the coupling property. This example indicates that the non-positivity of the real parts of the eigenvalues of $A$ is also necessary.
In [@SW Theorem 4.1 and Corollary 4.2] we show that Lévy processes which have the strong Feller property admit the coupling property. A similar conclusion, however, does not hold for general Ornstein-Uhlenbeck processes. Consider, for instance, the one-dimensional Ornstein-Uhlenbeck process given by $$dX_t=X_t\,dt+dZ_t,\qquad X_0=x\in{\mathds R},$$ where $Z_t$ is an $\alpha$-stable Lévy process $Z_t$ on ${\mathds R}$. According to [@PZ2 Theorem 1.1] (or [@NS Theorem A]) and [@PZ2 Proposition 2.1], we know that $X_t$ has the strong Feller property. However, the argument used in Remark \[remarkth1\] shows that this process fails to have the coupling property.
Recently, F.-Y. Wang [@wang1] has studied the coupling property of an Ornstein-Uhlenbeck process $X_t$ defined by . Assume that ${\operatorname{Rank}}(B)=n$ and ${\langleAx,x\rangle}{\leqslant}0$ holds for $x\in{\mathds R}^n$. In [@wang1 Theorem 3.1] it is proved that is satisfied for some constant $C>0$, whenever the Lévy measure of $Z_t$ satisfies $\nu(dz){\geqslant}\rho_0(z)dz$ such that $$\label{wang22}
\int_{\{|z-z_0|{\leqslant}\varepsilon\}}
\frac{dz}{\rho_0(z)}<\infty$$ holds for some $z_0\in{\mathds R}^d$ and some $\varepsilon>0$.
Let us compare F.-Y. Wang’s result with our Theorem \[th1\].
\[improvement\] Assume that holds for some $\rho_0\in L^1_{\mathrm{loc}}({\mathds R}^d\setminus\{0\})$, some $z_0\in{\mathds R}^d$ and some $\varepsilon>0$. Then, there exist a closed subset $F\subset \overline{B(z_0,\varepsilon)}=\{z\in{\mathds R}^d: |z-z_0|{\leqslant}\varepsilon\}$ and a constant $\delta>0$ such that $$\inf_{x\in{\mathds R}^d, |x|{\leqslant}\delta}\int_F\big(\rho_0(z)\wedge\rho_0(z-x)\big)\,dz>0.$$
We postpone the technical proof of Proposition \[improvement\] to Section \[subsec-appendix2\] in the appendix. Proposition \[improvement\] shows that Theorem \[th1\] improves [@wang1 Theorem 3.1], even if the Lévy measure $\nu$ of $Z_t$ has an absolutely continuous component as we will see in the following example.
Let $C_{3/4}$ be a Smith-Volterra-Cantor set in $[0,1]$ with Lebesgue measure ${\operatorname{Leb}}(C_{3/4})=3/4$, i.e. $C_{3/4}$ is a perfect set with empty interior, see e.g. [@AB Chapter 3, Section 18]. Consider the following one-dimensional Ornstein-Uhlenbeck process $$dX_t=-X_t\,dt + dZ_t,\qquad X_0=x\in{\mathds R},$$ where $Z_t$ is a real-valued Lévy process with Lévy measure $\nu(dz)={\mathds 1}_{C_{3/4}}(z)\,dz$. We will see that we can use Theorem \[th1\] to show the coupling property of the process $X_t$ while the criterion from [@wang1 Theorem 3.1] fails.
Let $\delta \in (0, 1/8)$ and $z\in [-\delta, \delta]$. Then $$\begin{aligned}
\nu _\varepsilon\wedge (\delta_z*\nu_\varepsilon)({\mathds R})
&=\int\Bigl({\mathds 1}_{C_{3/4}}(x)\wedge{\mathds 1}_{C_{3/4}}(x+z)\Bigr)dx\\
&={\operatorname{Leb}}\bigl( C_{3/4}\cap (C_{3/4}-z)\bigr)\\
&={\operatorname{Leb}}(C_{3/4}) + {\operatorname{Leb}}(C_{3/4}-z) - {\operatorname{Leb}}\bigl( C_{3/4}\cup (C_{3/4}-z)\bigr)\\
&{\geqslant}\frac 64 - {\operatorname{Leb}}[-|z|,1+|z|]{\geqslant}\frac 14.
\end{aligned}$$ This shows that the conditions of Theorem \[th1\] are satisfied.
On the other hand, since $C_{3/4}$ contains no intervals, we see that for all $z_0\in{\mathds R}$ and $\varepsilon>0$, $$\int_{\{|z-z_0|{\leqslant}\varepsilon\}}\frac{dz}{{\mathds 1}_{C_{3/4}}(z)}=\infty$$ (here we use the convention $\frac 10 = +\infty$). This means that does not hold.
Now we are going to estimate $\|P_t(x,\cdot)-P_t(y,\cdot)\|_{{\mathrm{Var}}}$ for large values of $t$ with the help of the characteristic exponent $\Phi(\xi)$ of the Lévy process $Z_t$. We restrict ourselves to the case where $Q=0$ in , i.e. to Lévy process $(Z_t)_{t{\geqslant}0}$ without a Gaussian part. For $t, \rho>0$, define $$\varphi_t(\rho)
:=\sup_{|\xi|{\leqslant}\rho}\int_0^t{\ensuremath{\operatorname{Re}}}\Phi\big(B^\top e^{sA^\top}\xi\big)\,ds,$$ where $M^\top$ denotes the transpose of the matrix $M$.
\[coup\] Let $P_t(x,\cdot)$ be the transition function of the Ornstein-Uhlenbeck process $\{X_t^x\}_{t{\geqslant}0}$ on ${\mathds R}^n$ given by . Assume that there exists some $t_0 > 0$ such that $$\label{coup1}
\liminf\limits_{|\xi|\rightarrow\infty}\frac{\int_0^{t_0}{\ensuremath{\operatorname{Re}}}\Phi\big(B^\top e^{sA^\top}\xi\big)\,ds}{\log (1+|\xi|)} >2n+2.$$ If $$\label{coup2}
\int \exp\left(-\int_0^t{\ensuremath{\operatorname{Re}}}\Phi\big(B^\top e^{sA^\top}\xi\big)ds\right) |\xi|^{n+2} \,d\xi
= \mathsf{O}\left(\varphi_t^{-1}(1)^{2n+2}\right)
\quad\text{as\ \ } t\to\infty,$$ then there exist $t_1,C>0$ such that for any $x,y\in{\mathds R}^n$ and $t{\geqslant}t_1$, $$\label{coup3}
\|P_t(x,\cdot)-P_t(y,\cdot)\|_{{\mathrm{Var}}}
{\leqslant}C|e^{tA}(x-y)|\,\varphi^{-1}_t(1).$$ In particular, when $$\label{coup4}\xi\mapsto\int_0^\infty{\ensuremath{\operatorname{Re}}}\Phi\big(B^\top
e^{sA^\top}\xi\big)\,ds \quad\textrm{ is locally bounded},$$ we only need the condition to get .
Note that is, e.g. satisfied, if the real parts of all eigenvalues of $A$ are negative and $$\limsup_{|\xi|\rightarrow0}\frac{{\ensuremath{\operatorname{Re}}}\Phi\big(B^\top
\xi\big)}{|\xi|^\kappa}<\infty$$ for some constant $\kappa>0$.
The remaining part of this paper is organized as follows. In Section \[section2\] we first present the proof of Theorem \[th1\], where a coupling for the weighted sum of independent random variables and the explicit expression of the transition semigroup of Ornstein-Uhlenbeck processes driven by a compound Poisson process are used. Then, we follow the approach of our recent paper [@SSW] to prove Theorem \[coup\]. As a byproduct, we also derive explicit gradient estimates for Ornstein-Uhlenbeck processes, cf. the Appendix \[subsec-appendix1\].
Proofs of Theorems {#section2}
==================
We begin with the proof of Theorem \[th1\].
The proof is split into six steps. *Step 1*. For any $\varepsilon>0$, let $(Z_t^\varepsilon)_{t{\geqslant}0}$ be a compound Poisson process on ${\mathds R}^d$ whose Lévy measure is $\nu_\varepsilon$. Then, $(Z_t^\varepsilon)_{t{\geqslant}0}$ and $(Z_t-Z_t^\varepsilon)_{t{\geqslant}0}$ are independent Lévy processes. It follows, in particular, that the random variables $$X_t^{\varepsilon,x}:=e^{tA}x+\int_0^t e^{(t-s)A}B\,dZ_s^\varepsilon$$ and $$X_t^x-X_t^{\varepsilon,x}:=\int_0^t e^{(t-s)A}B\,d(Z_s-Z_s^\varepsilon)$$ are independent for any $\varepsilon>0$ and $t{\geqslant}0$.
*Step 2*. Denote by $\mu_{\varepsilon,t}$ the law of random variable $$X_t^{\varepsilon,0} := X_t^{\varepsilon,x}-e^{tA}x = \int_0^te^{(t-s)A}B\,dZ_s^\varepsilon.$$ We will compute $\mu_{\varepsilon,t}$, which coincides with the law of $\int_0^te^{sA}B\,dZ_s^\varepsilon$, cf. Lemma \[lemmaproofth11\] below. Our argument follows the proof of [@PZ2 Theorem 1.1], which is motivated by [@SA Theorem 27.7].
The law of the compound poisson process $Z_t^\varepsilon$ is given by $$e^{-C_\varepsilon t}\bigg[\delta_0+\sum_{k=1}^\infty \frac{(C_\varepsilon t)^k}{k!}\,\bar{\nu}_\varepsilon^{*k}\bigg],$$ where $C_\varepsilon=\nu_\varepsilon({\mathds R}^d)$, $\bar{\nu}_\varepsilon=\nu_\varepsilon/C_\varepsilon$ and $\bar{\nu}_\varepsilon^{*k}$ is the $k$-fold convolution of $\bar{\nu}_\varepsilon$.
Construct a sequence $(\xi_i)_{i{\geqslant}1}$ of iid random variables which are exponentially distributed with intensity $C_\varepsilon$, and introduce a further sequence $(U_i)_{i{\geqslant}1}$ of iid random variables on ${\mathds R}^d$ with law $\bar{\nu}_\varepsilon$. We will assume that the random variables $(U_i)_{i{\geqslant}1}$ are independent of the sequence $(\xi_i)_{i{\geqslant}1}$. It is not difficult to check that the random variable $$\label{proofs0}
0\cdot{\mathds 1}_{\{\xi_1>t\}} +\sum_{k=1}^\infty{\mathds 1}_{\{\xi_1+\cdots+\xi_k{\leqslant}t<\xi_1+\cdots+\xi_{k+1}\}} \Big(e^{\xi_1A}BU_1+\cdots+e^{(\xi_1+\cdots+\xi_k)A}BU_k\Big)$$ also has the probability distribution $\mu_{\varepsilon,t}$.
Using we find for any $f\in B_b({\mathds R}^n)$, $$\label{proofs1}
{\mathds E}f\bigl(X_t^{\varepsilon,x}\bigr)
=\int f\bigl(e^{tA }x+z\bigr)\,\mu_{\varepsilon,t}(dz)
=f\bigl(e^{tA}x\bigr)\,e^{-C_\varepsilon t}+Hf(x),
$$ where $$\begin{aligned}
&Hf(x)\\
&:={\mathds E}f\left(\sum_{k=1}^\infty{\mathds 1}_{\{\xi_1+\cdots+\xi_k{\leqslant}t<\xi_1+\cdots+\xi_{k+1}\}} \Bigl(e^{tA}x+e^{\xi_1A}BU_1+\cdots+e^{(\xi_1+\cdots+\xi_k)A}BU_k\Bigr)\right)\\
&=\sum_{k=1}^\infty{\mathds E}f\left({\mathds 1}_{\{\xi_1+\cdots+\xi_k{\leqslant}t<\xi_1+\cdots+\xi_{k+1}\}} \Bigl(e^{tA}x+e^{\xi_1A}BU_1+\cdots+e^{(\xi_1+\cdots+\xi_k)A}BU_k\Bigr)\right)\\
&=\sum_{k=1}^\infty \mathop{\int\cdots\int}\limits_{t_1+\cdots+t_k{\leqslant}t<t_1+\cdots+t_{k+1}} C_\varepsilon^{k+1}e^{-C_\varepsilon(t_1+\cdots+t_{k+1})}\,dt_1\cdots dt_{k+1}{\;\;\times}\\
&\qquad\quad{\times}\;\int\limits_{{\mathds R}^d}\cdots\int\limits_{{\mathds R}^d} f\bigl(e^{tA}x+e^{t_1A}By_1+\cdots+e^{(t_1+\cdots+t_k)A}By_k\bigr) \,\bar{\nu}_\varepsilon(dy_1)\cdots \bar{\nu}_\varepsilon(dy_k)\\
&=\sum_{k=1}^\infty \mathop{\int\cdots\int}\limits_{t_1+\cdots+t_k{\leqslant}t<t_1+\cdots+t_{k+1}} C_\varepsilon^{k+1}e^{-C_\varepsilon(t_1+\cdots+t_{k+1})}\,dt_1\cdots dt_{k+1}\;\;\times\\
&\qquad\quad\times\;\int\limits_{{\mathds R}^n} f\bigl(e^{tA}x+z\bigr)\,\mu_{t_1,\cdots,t_k}(dz).\end{aligned}$$ Here $\mu_{t_1,\cdots,t_k}$ is the probability measure on ${\mathds R}^n$ which is the image of the $k$-fold product measure $\bar{\nu}_\varepsilon\times \cdots\times \bar{\nu}_\varepsilon$ under the linear transformation $J_{t_1,\ldots,t_k}$ (independent of $\varepsilon$) acting from $({\mathds R}^{d})^k$ into ${\mathds R}^n$: $$J_{t_1,\ldots,t_k}(y_1,\ldots,y_k)
=e^{t_1A}B y_1 + \cdots +e^{(t_1+\cdots+t_k)A}B y_k,$$ for $y_i\in{\mathds R}^d$ and $i=1,\cdots,k$.
*Step 3*. Let $P_t(x,\cdot)$ and $P_t$ be the transition function and the transition semigroup of the Ornstein-Uhlenbeck process $(X^x_t)_{t{\geqslant}0}$. Similarly, we denote by $P^\varepsilon_t(x,\cdot)$ and $P^\varepsilon_t$ the transition function and the transition semigroup of $(X_t^{\varepsilon,x})_{t{\geqslant}0}$, and by $Q^\varepsilon_t(x,\cdot)$ and $Q^\varepsilon_t$ the transition function and the transition semigroup of $(X_t^x-X_t^{\varepsilon,x})_{t{\geqslant}0}$. By the independence of the processes $(X_t^{\varepsilon,x})_{t{\geqslant}0}$ and $(X_t^x-X_t^{\varepsilon,x})_{t{\geqslant}0}$, we get $$\label{proofs2}\begin{aligned}
\|P_t(x,\cdot)-P_t(y,\cdot)\|_{{\mathrm{Var}}}
&= \sup_{\|f\|_\infty{\leqslant}1}\big|P_tf(x)-P_tf(y)\big|\\
&=\sup_{\|f\|_\infty{\leqslant}1} \big|P_t^\varepsilon Q_t^\varepsilon f(x)-P_t^\varepsilon Q_t^\varepsilon f(y)\big|\\
&{\leqslant}\sup_{\|h\|_\infty{\leqslant}1} \big|P^\varepsilon_th(x)-P^\varepsilon_th(y)\big|.
\end{aligned}$$ Furthermore, it follows from that $$\label{proofs3}\begin{aligned}
&\sup_{\|h\|_\infty{\leqslant}1} \big|P^\varepsilon_th(x)-P^\varepsilon_th(y)\big|\\
&{\leqslant}2e^{-C_\varepsilon t}+\sum_{k=1}^\infty \mathop{\int\cdots\int}\limits_{t_1+\cdots+t_k{\leqslant}t<t_1+\cdots+t_{k+1}} C_\varepsilon^{k+1}e^{-C_\varepsilon(t_1+\cdots+t_{k+1})}\,dt_1\cdots dt_{k+1}\;\;\times\\
&\qquad\times\sup_{\|h\|_\infty{\leqslant}1} \bigg|\int\limits_{{\mathds R}^n} h\big(e^{tA}x+z\big)\,\mu_{t_1,\cdots,t_k}(dz) - \int\limits_{{\mathds R}^n} h\big(e^{tA}y+z\big)\,\mu_{t_1,\cdots,t_k}(dz)\biggr|\\
&= 2e^{-C_\varepsilon t}+\sum_{k=1}^\infty \mathop{\int\cdots\int}\limits_{t_1+\cdots+t_k{\leqslant}t<t_1+\cdots+t_{k+1}} C_\varepsilon^{k+1}e^{-C_\varepsilon(t_1+\cdots+t_{k+1})}\,dt_1\cdots dt_{k+1}\;\;\times\\
&\qquad\times\sup_{\|h\|_\infty{\leqslant}1} \bigg|\int\limits_{{\mathds R}^n} h\big(e^{tA}(x-y)+z\big)\,\mu_{t_1,\cdots,t_k}(dz) - \int\limits_{{\mathds R}^n} h(z)\,\mu_{t_1,\cdots,t_k}(dz)\bigg|\\
&{\leqslant}2e^{-C_\varepsilon t}+\sum_{k=1}^\infty\mathop{\int\cdots\int}\limits_{t_1+\cdots+t_k{\leqslant}t<t_1+\cdots+t_{k+1}} C_\varepsilon^{k+1}e^{-C_\varepsilon(t_1+\cdots+t_{k+1})}\,dt_1\cdots dt_{k+1}\;\;\times\\
&\qquad\times\|\delta_{e^{tA}(x-y)}*\mu_{t_1,\cdots,t_k}-\mu_{t_1,\cdots,t_k}\|_{{\mathrm{Var}}}.
\end{aligned}$$
*Step 4*. For any $a\in{\mathds R}^n$, $a\neq 0$, let $R_a$ be the non-degenerate rotation such that $R_a a=|a|e_1$. Then, by [@SW Lemma 3.2], $$\begin{aligned}
\big\|\delta_{e^{tA}(x-y)}*\mu_{t_1,\cdots,t_k} &-\mu_{t_1,\cdots,t_k}\big\|_{{\mathrm{Var}}}\\
&=\big\|\delta_{|e^{tA}(x-y)|e_1}*\big(\mu_{t_1,\cdots,t_k}\circ R_{e^{tA}(x-y)}^{-1}\big)-\mu_{t_1,\cdots,t_k}\circ R_{e^{tA}(x-y)}^{-1}\big\|_{{\mathrm{Var}}}.\end{aligned}$$ Since $\mu_{t_1,\cdots,t_k}$ is the law of the random variable $$\sum_{i=1}^ke^{(t_1+\cdots+t_i)A}BU_i,$$ $\mu_{t_1,\cdots,t_k}\circ R_{e^{tA}(x-y)}^{-1}$ is the law of the random variable $$\sum_{i=1}^kR_{e^{tA}(x-y)}\big(e^{(t_1+\cdots+t_i)A}BU_i\big).$$
To estimate $\big\|\delta_{|e^{tA}(x-y)|e_1}*\big(\mu_{t_1,\cdots,t_k}\circ R_{e^{tA}(x-y)}^{-1}\big)-\mu_{t_1,\cdots,t_k}\circ R_{e^{tA}(x-y)}^{-1}\big\|_{{\mathrm{Var}}}$, we will use the Mineka and Lindvall-Rogers couplings for random walks. The remainder of this part is based on the proof of [@SW Proposition 3.3]. In order to ease notations, we set ${\mathsf{n}}:=\bar{\nu}_\varepsilon$ and ${\mathsf{n}}^{a}:=\delta_a*\bar\nu_\varepsilon$ for any $a\in{\mathds R}^d$.
Since ${\operatorname{Rank}}(B)=n$, there exists a real $d\times n$ matrix $\bar{B}$ such that $B\bar{B}={\operatorname{id}}_{{\mathds R}^n}$, see e.g. [@Bern Theorem 2.6.1, Page 35]. For any $i{\geqslant}1$, let $(U_i,\Delta U_i)\in {\mathds R}^d \times {\mathds R}^d$ be a pair of random variables with the following distribution $${\mathds P}\big((U_i,\Delta U_i)\in C\times D\big)
=
\begin{cases}
\qquad \frac 12 ({\mathsf{n}}\wedge{\mathsf{n}}^{-a_i})(C), & \text{if\ \ } D=\{a_i\};\\
\qquad \frac 12 ({\mathsf{n}}\wedge{\mathsf{n}}^{a_i})(C), & \text{if\ \ } D= \{-a_i\};\\
\big({\mathsf{n}}- \frac 12 ({\mathsf{n}}\wedge{\mathsf{n}}^{-a_i}+{\mathsf{n}}\wedge{\mathsf{n}}^{a_i})\big)(C), & \text{if\ \ } D=\{0\};
\end{cases}$$ where $C\in{\mathscr{B}}({\mathds R}^d)$, $a_i=\bar B\, e^{(t-(t_1+\cdots+t_i))A}\,(x-y)$ and $D$ is any of the following three sets: $\{-a_i\}$, $\{0\}$ or $\{a_i\}$. Again by [@SW Lemma 3.2], $$\begin{aligned}
{\mathds P}\big(\Delta U_i=-a_i\big)
&=\frac{1}{2}\big({\mathsf{n}}\wedge\big(\delta_{a_i}*{\mathsf{n}})\big)({\mathds R}^d)\\
&=\frac{1}{2}\big({\mathsf{n}}\wedge\big(\delta_{-a_i}*{\mathsf{n}})\big)({\mathds R}^d)\\
&={\mathds P}(\Delta U_i=a_i).\end{aligned}$$ It is clear that the distribution of $U_i$ is ${\mathsf{n}}$. Let $U_i'=U_i+\Delta U_i$. We claim that the distribution of $U_i'$ is also ${\mathsf{n}}$. Indeed, for any $C\in\mathscr{B}({\mathds R}^d)$, $$\begin{aligned}
&{\mathds P}(U_i'\in C)\\
&={\mathds P}(U_i-a_i\in C, \Delta U_i=-a_i)
+ {\mathds P}(U_i+a_i\in C, \Delta U_i=a_i)
+{\mathds P}(U_i \in A, \Delta U_i=0)\\
&= \frac 12\left(\delta_{-a_i}*({\mathsf{n}}\wedge{\mathsf{n}}^{a_i})\right)(C)
+\! \frac12\left(\delta_{a_i}*({\mathsf{n}}\wedge{\mathsf{n}}^{-a_i})\right)(C)
\! +\! \left(\!\!{\mathsf{n}}-\!\! \frac 12\,\big({\mathsf{n}}\wedge{\mathsf{n}}^{-a_i}+{\mathsf{n}}\wedge{\mathsf{n}}^{a_i}\big)\!\!\right)(C)\\
&={\mathsf{n}}(C),\end{aligned}$$ where we have used that $$\delta_{a_i}*({\mathsf{n}}\wedge{\mathsf{n}}^{-a_i})={\mathsf{n}}\wedge{\mathsf{n}}^{a_i}\quad\textrm{ and }\quad\delta_{-a_i}*({\mathsf{n}}\wedge{\mathsf{n}}^{a_i})={\mathsf{n}}\wedge{\mathsf{n}}^{- a_i}.$$
Without loss of generality, we can assume that the pairs $(U_i,U_i')$ are independent for all $i{\geqslant}1$. Now we construct the coupling $$(S_k,S_k')_{k{\geqslant}1}
=\left(\sum_{i=1}^kR_{e^{tA}(x-y)}\big(e^{(t_1+\cdots+t_i)A}BU_i\big),
\sum_{i=1}^kR_{e^{tA}(x-y)}\big(e^{(t_1+\cdots+t_i)A}BU'_i\big)\right)_{k{\geqslant}1}$$ of $$S_k:=\sum_{i=1}^kR_{e^{tA}(x-y)}\big(e^{(t_1+\cdots+t_i)A}BU_i\big).$$ Since $U'_i-U_i=\Delta U_i$ is either $\pm a_i$ or $0$, we know that $$\begin{aligned}
(S_k-& S_k')_{k{\geqslant}1}\\
&=\left(\sum_{i=1}^kR_{e^{tA}(x-y)}\big(e^{(t_1+\cdots+t_i)A}BU'_i\big) -\sum_{i=1}^kR_{e^{tA}(x-y)}\big(e^{(t_1+\cdots+t_i)A}BU_i\big)\right)_{k{\geqslant}1}\\
&=\left(\sum_{i=1}^kR_{e^{tA}(x-y)}\big(e^{(t_1+\cdots+t_i)A}B(U'_i-U_i)\big)\right)_{k{\geqslant}1}\end{aligned}$$ is a random walk on ${\mathds R}^n$ whose steps are symmetrically (but not necessarily identically) distributed and take only the values $\pm |e^{tA}(x-y)| e_1$ and $0$.
Set $S^j_{k}=\sum_{i=1}^k\eta^j_{i}$ and $S^{j\,\prime}_{k}=\sum_{i=1}^k\eta^{j\,\prime}_{i}$ for $1{\leqslant}j{\leqslant}n$, where $$(\eta^{1}_{i},\ldots,\eta^{n}_{i})=R_{e^{tA}(x-y)}\big(e^{(t_1+\cdots+t_i)A}BU_i\big)$$ and $$(\eta_{i}^{1\,\prime},\ldots,\eta^{n\,\prime}_{i})=R_{e^{tA}(x-y)}\big(e^{(t_1+\cdots+t_i)A}BU_i^{\prime}\big).$$ Then $(S^1_k-S^{1\,\prime}_k)_{k{\geqslant}1}$ is a random walk on ${\mathds R}$ whose steps are independent and attain the values $-|e^{tA}(x-y)|$, $0$ and $|e^{tA}(x-y)|$ with probabilities $\frac 12(1-p_i)$, $p_i$ and $\frac 12(1-p_i)$, respectively; the values of the $p_i$ are given by $$\begin{aligned}
p_i
:&={\mathds P}(\eta^{1\,\prime}_i-\eta^1_{i}=0)\\
&= \left({\mathsf{n}}- \tfrac 12 ({\mathsf{n}}\wedge{\mathsf{n}}^{-a_i}+{\mathsf{n}}\wedge{\mathsf{n}}^{a_i})\right)({\mathds R}^d)\\
&= 1-{\mathsf{n}}\wedge{\mathsf{n}}^{-a_i}({\mathds R}^d).\end{aligned}$$ Since $S^j_{k}=S^{j\,\prime}_{k}$ for $2{\leqslant}j{\leqslant}n$, we get $$\label{proofs6}
\|\delta_{e^{tA}(x-y)}*\mu_{t_1,\cdots,t_k}-\mu_{t_1,\cdots,t_k}\|_{{\mathrm{Var}}}
{\leqslant}2\,{\mathds P}(T^S>k),$$ where $$T^S=\inf\{i{\geqslant}1\::\: S^1_{i}=S^{1\,\prime}_{i}+|e^{tA}(x-y)|\}.$$
*Step 5*. Since the real parts of all eigenvalues of $A$ are non-positive and since all purely imaginary eigenvalues are semisimple, we know from [@Bern Proposition 11.7.2, Page 438] that $C_A:=\sup_{t{\geqslant}0}\|e^{tA}\|<\infty$. In particular, when $t{\geqslant}t_1+\cdots+t_i$, $$\big|e^{(t-(t_1+\cdots+t_i))A}(x-y)\big|{\leqslant}C_A|x-y|.$$ From we get that for all $i{\geqslant}1$ and $x, y\in{\mathds R}^n$ with $|x-y|{\leqslant}\delta(C_A\|\bar{B}\|)^{-1}$, $$\label{proofcon}\begin{aligned}
\frac 12(1-p_i)
&= \frac{1}{2}\big({\mathsf{n}}\wedge\big(\delta_{-a_i}*{\mathsf{n}})\big)({\mathds R}^d)\\
&{\geqslant}\frac{1}{2}\inf_{|a|{\leqslant}C_A\|\bar{B}\||x-y|}{\mathsf{n}}\wedge (\delta_a*{\mathsf{n}})({\mathds R}^d)\\
&{\geqslant}\frac{1}{2}\inf_{|a|{\leqslant}\delta}{\mathsf{n}}\wedge (\delta_a*{\mathsf{n}})({\mathds R}^d)\\
&=:\frac{1}{2}\,\gamma(\delta)>0.
\end{aligned}$$
We will now estimate ${\mathds P}(T^S>k)$. Let $V_i$, $i{\geqslant}1$, be independent symmetric random variables on ${\mathds R}$, whose distributions are given by $${\mathds P}(V_i=z)
= \begin{cases}
\frac 12(1-p_i), &\text{if\ \ } z=-|e^{tA}(x-y)|;\\
\frac 12(1-p_i), &\text{if\ \ } z=|e^{tA}(x-y)|;\\
\qquad p_i, &\text{if\ \ } z=0.
\end{cases}$$ Set $Z_k:=\sum_{i=1}^k V_i$. We have seen earlier that $$T^S=\inf\{k{\geqslant}1\::\: Z_k=|e^{tA}(x-y)|\}.$$
For any $k{\geqslant}1$, let $$\eta=\eta(k):=\#\big\{i\::\: i{\leqslant}k\textrm{ and }V_i\neq 0\big\}$$ and set $\tilde{Z}_k :=\sum_{i=1}^k\tilde{V}_i$, where $\tilde{V}_i$ denotes the $i$th $V_j$ such that $V_j\neq 0$. Then, $\tilde{Z}_k$ is a symmetric random walk with iid steps which are either $-|e^{tA}(x-y)|$ or $|e^{tA}(x-y)|$ with probability $1/2$. Define $$T^{\tilde{Z}}:=\inf\{k{\geqslant}1\::\: \tilde{Z}_k=|e^{tA}(x-y)|\}.$$ By , $$\label{lll1}\begin{aligned}
{\mathds P}(T^S>k)
&={\mathds P}\left(T^S>k,\; \eta{\geqslant}\frac12\,\gamma(\delta)k\right)
+{\mathds P}\left(T^S>k,\, \eta{\leqslant}\frac12\, \gamma(\delta)k\right)\\
&{\leqslant}{\mathds P}\left(T^{\tilde{Z}}> \frac12\,\gamma(\delta)k\right)
+{\mathds P}\bigg(\eta{\leqslant}\frac{1}{2}\sum_{i=1}^k(1-p_i)\bigg)\\
&{\leqslant}{\mathds P}\left(T^{\tilde{Z}}>\frac12\,\gamma(\delta)k\right)
+{\mathds P}\bigg(\Big|\eta-\sum_{i=1}^k(1-p_i)\Big|{\geqslant}\frac{1}{2}\sum_{i=1}^k(1-p_i)\bigg).
\end{aligned}$$
Note that $$\eta=\eta(k)=\sum_{i=1}^k\zeta_i,$$ where $\zeta_i = {\mathds 1}_{\{V_i\neq 0\}}$, ${1{\leqslant}i{\leqslant}k}$, are independent random variables with ${\mathds P}(\zeta_i = 0) = p_i$ and ${\mathds P}(\zeta_i=1)=1-p_i$. Chebyshev’s inequality shows that $$\label{lll2}\begin{aligned}
{\mathds P}\bigg(\Big|\eta-\sum_{i=1}^k(1-p_i)\Big|{\geqslant}\frac{1}{2}\sum_{i=1}^k(1-p_i)\bigg)
&{\leqslant}\frac{4{\mathrm{Var}}(\eta)}{\Big(\sum_{i=1}^k(1-p_i)\Big)^2}\\
&=\frac{4\sum_{i=1}^kp_i(1-p_i)}{\Big(\sum_{i=1}^k(1-p_i)\Big)^2}\\
&{\leqslant}\frac{4(1-\gamma(\delta))\sum_{i=1}^k(1-p_i)}{\Big(\sum_{i=1}^k(1-p_i)\Big)^2}\\
&{\leqslant}\frac{4(1-\gamma(\delta))}{\gamma(\delta)k}.
\end{aligned}$$ For the second and the last inequality we have used .
On the other hand, by Lemma \[lemmaproofth13\] below, $$\begin{aligned}
{\mathds P}\bigg(T^{\tilde{Z}}>\frac{\gamma(\delta)k}{2}\bigg)
&={\mathds P}\bigg(\max_{i{\leqslant}\big[\frac{\gamma(\delta)k}{2}\big]}\tilde{Z}_i< |e^{tA}(x-y)|\bigg)\\
&{\leqslant}2\,{\mathds P}\left(0{\leqslant}\tilde{Z}_{\big[\frac{\gamma(\delta)k}{2}\big]} {\leqslant}|e^{tA}(x-y)|\right).\end{aligned}$$ From the construction above, we know that $(\tilde{Z}_k)_{k{\geqslant}1}$ is a symmetric random walk with iid steps with values $\pm |e^{tA}(x-y)|$. Using the central limit theorem we find for sufficiently large values of $k{\geqslant}k_0$ and some constant $C=C(k_0)$ $$\label{lll3}\begin{aligned}
{\mathds P}\left(T^S>\frac12\,\gamma(\delta)k\right)
&= 2\,{\mathds P}\left(0{\leqslant}\frac{Z_k}{|e^{tA}(x-y)|\sqrt{\big[\frac{\gamma(\delta)k}{2}\big]}} {\leqslant}{\left[\frac{\gamma(\delta)k}{2}\right]}^{-1/2}\right)\\
&{\leqslant}\frac{C}{\sqrt{2\pi}} \int_{0}^{ {\left[\frac{\gamma(\delta)k}{2}\right]}^{-1/2}} e^{-u^2/2}\,du\\
&{\leqslant}\frac{C_{\gamma(\delta)}}{\sqrt k}.
\end{aligned}$$
Combining , and gives for all $x,y\in{\mathds R}^n$ with $|x-y| {\leqslant}\delta(C_A\|\bar{B}\|)^{-1}$, $t{\geqslant}t_1+\cdots+t_k$ and $k{\geqslant}k_0$ that $${\mathds P}\big(T^S>k\big)
{\leqslant}\frac{C_{\gamma(\delta)}}{\sqrt k}+\frac{4(1-\gamma(\delta))}{\gamma(\delta)k}.$$ Finally, yields for all $x,y\in{\mathds R}^n$ with $|x-y|{\leqslant}\delta(C_A\|\bar{B}\|)^{-1}$, $t{\geqslant}t_1+\cdots+t_k$ and $k{\geqslant}1$, that $$\label{proofs7}
\|\delta_{e^{tA}(x-y)}*\mu_{t_1,\cdots,t_k}-\mu_{t_1,\cdots,t_k}\|_{{\mathrm{Var}}}
{\leqslant}\frac{C_{1,\delta,{\mathsf{n}}}}{\sqrt k}.$$
*Step 6*. If we combine , and , we obtain that for all $x, y\in{\mathds R}^n$ with $|x-y|{\leqslant}\delta(C_A\|\bar{B}\|)^{-1}$, $$\begin{aligned}\label{proofs8}
\|&P_t(x,\cdot)-P_t(y,\cdot)\|_{{\mathrm{Var}}}\\
&{\leqslant}2e^{-C_\varepsilon t}
+C_{1,\delta,{\mathsf{n}}}\sum_{k=1}^\infty\frac{1}{\sqrt{k}} \mathop{\int\cdots\int}\limits_{t_1+\cdots+t_k{\leqslant}t<t_1+\cdots+t_{k+1}} \!\!\!\!\!\!\!\!C_\varepsilon^{k+1}e^{-C_\varepsilon(t_1+\cdots+t_{k+1})}\,dt_1\cdots dt_{k+1}\\
&{\leqslant}2e^{-C_\varepsilon t}+C_{1,\delta,{\mathsf{n}}}e^{-C_\varepsilon t}\sum_{k=1}^\infty \frac{C_\varepsilon^{k+1}}{\sqrt{k}} \mathop{\int\cdots\int}\limits_{t_1+\cdots+t_k{\leqslant}t}\!\!dt_1\cdots dt_{k}\\
&{\leqslant}2e^{-C_\varepsilon t}+C_{1,\delta,{\mathsf{n}}}C_\varepsilon \sum_{k=1}^\infty\frac{C_\varepsilon^k t^k}{\sqrt{k}\,k!}e^{-C_\varepsilon t}\\
&{\leqslant}2e^{-C_\varepsilon t}+\frac{\sqrt{2}C_{1,\delta,{\mathsf{n}}} C_\varepsilon (1-e^{-C_\varepsilon t})}{\sqrt{C_\varepsilon t}}\\
&{\leqslant}\frac{C_{2,\epsilon,\delta,{\mathsf{n}}}}{\sqrt{t}},
\end{aligned}$$ where the penultimate inequality follows as in [@SW Proposition 2.2].
For any $x,$ $y\in{\mathds R}^n$, set $k=\left[\frac{C_A\|\bar{B}\||x-y|}{\delta}\right]+1$. Pick $x_0, x_1, \ldots, x_k\in{\mathds R}^n$ such that $x_0=x$, $x_k=y$ and $|x_i-x_{i-1}|{\leqslant}\delta(C_A\|\bar{B}\|)^{-1}$ for $1{\leqslant}i{\leqslant}k$. By , $$\begin{aligned}
\|P_t(x,\cdot)-P_t(y,\cdot)\|_{{\mathrm{Var}}}
&{\leqslant}\sum_{i=1}^k\|P_t(x_i,\cdot)-P_t(x_{i-1},\cdot)\|_{{\mathrm{Var}}}\\
&{\leqslant}\frac{C_{\epsilon,\delta,{\mathsf{n}},A,B}(1+|x-y|)}{\sqrt{t}},\end{aligned}$$ which finishes the proof of .
The following two lemmas have been used in the proof of Theorem \[th1\] above. For the sake of completeness we include their proofs.
\[lemmaproofth12\] Let $B\in{\mathds R}^{n\times d}$ and $(Z_t)_{t{\geqslant}0}$ be a $d$-dimensional Lévy process with characteristic exponent $\Phi$ as in . Then, $(Z^B_t)_{t{\geqslant}0}:=(BZ_t)_{t{\geqslant}0}$ is a Lévy process on (a subspace of) ${\mathds R}^n$, and the corresponding characteristic exponent is $${\mathds R}^n\ni\xi \mapsto \Phi_B(\xi):=\Phi(B^\top\xi).$$ The Lévy triplet $(Q_B,b_B,\nu_B)$ of $(Z^B_t)_{t{\geqslant}0}$ is given by $ Q_B=BQB^\top$, $\nu_B(C) = \nu\{y : By\in C\}$ and $$b_B=Bb+\int_{x\neq 0} Bx\,\big({\mathds 1}_{\{z\in{\mathds R}^d:|z|{\leqslant}1\}}(Bx) - {\mathds 1}_{\{z\in{\mathds R}^d:|z|{\leqslant}1\}}(x)\big)\, \nu(dx).$$
For all $\xi\in{\mathds R}^n$ and $t{\geqslant}0$, we have $${\mathds E}(e^{i{\langle\xi,Z^B_t\rangle}})
={\mathds E}(e^{i{\langle\xi,BZ_t\rangle}})
={\mathds E}(e^{i{\langleB^\top\xi,Z_t\rangle}})
=e^{-t\Phi(B^\top\xi)}.$$ The assertion follows from and some straightforward calculations.
\[lemmaproofth11\] Let $A\in{\mathds R}^{n\times n}$, $B\in{\mathds R}^{n\times d}$ and $(Z_t)_{t{\geqslant}0}$ be a $d$-dimensional Lévy process with the characteristic exponent $\Phi$ as in . For all $t>0$ the random variables $\int_0^t e^{(t-s)A}B\,dZ_s$ and $\int_0^t e^{sA}B\,dZ_s$ have the same probability distribution. Furthermore, both random variables are infinitely divisible, and the characteristic exponent (log-characteristic function) is given by $${\mathds R}^n\ni\xi \mapsto \Phi_t(\xi):=\int_0^t\Phi\big(B^\top e^{sA^\top}\xi\big)\,ds.$$
We first assume that $n=d$ and $B={\operatorname{id}}_{{\mathds R}^d}$. For any $t>0$, we can use Lemma \[lemmaproofth12\] and follow the proof of [@SA (17.3)] to deduce $${\mathds E}\left[\exp\left(i{\left\langle\xi,\int_0^t e^{(t-s)A}\,dZ_s\right\rangle}\right)\right]
=\exp\left[-\int_0^t\Phi(e^{(t-s)A^\top}\xi)\,ds\right]$$ for all $\xi\in{\mathds R}^d$. Similarly, for every $\xi\in{\mathds R}^d$, $${\mathds E}\left[\exp\left(i{\left\langle\xi,\int_0^t e^{sA}\,dZ_s\right\rangle}\right)\right]
=\exp\left[-\int_0^t\Phi(e^{sA^\top}\xi)\,ds\right].$$ Since $$\exp\left[-\int_0^t\Phi(e^{(t-s)A^\top}\xi)\,ds\right]
=\exp\left[-\int_0^t\Phi(e^{sA^\top}\xi)\,ds\right],$$ it follows that $\int_0^t e^{(t-s)A}B\,dZ_s$ and $\int_0^t e^{sA}B\,dZ_s$ have the same law.
Now replace in the preceding calculations $A$ with $\frac 1k\,A$, $k{\geqslant}1$, and set $Y_k := \int_0^t e^{s\,\frac 1k\, A}\,dZ_s$. Denote by $Y_k^{(j)}$, $1{\leqslant}j{\leqslant}k$, independent copies of $Y_k$. It is straightforward to see that $\sum_{j=1}^k Y_k^{(j)}$ and $\int_0^t e^{sA}\,dZ_s$ have the same law. This proves the infinite divisibility.
If $n\neq d$, we consider, as in Lemma \[lemmaproofth12\], the Lévy process $(Z^B_t)_{t{\geqslant}0}:=(BZ_t)_{t{\geqslant}0}$ on (a subspace of) ${\mathds R}^n$. Then, for any $\xi\in{\mathds R}^n$, $${\mathds E}\left[\exp\left(i{\left\langle\xi,\int_0^t e^{(t-s)A}B\,dZ_s\right\rangle}\right)\right]
={\mathds E}\left[\exp\left(i{\left\langle\xi,\int_0^t
e^{(t-s)A}\,dZ^B_s\right\rangle}\right)\right],$$ and the claim follows from the first part of our proof.
The following result presents the upper estimate for the distribution of the maximum of a symmetric random walk, by using the reflection principle. Since we could not find a precise reference in the literature, we include the complete proof for the readers’ convenience.
\[lemmaproofth13\] Consider a random walk $(S_i)_{i{\geqslant}1}$ on ${\mathds Z}$ with iid steps, which attain the values $-1$, $1$ and $0$ with probabilities $(1-r)/2$, $(1-r)/2$ and $r$ $(0{\leqslant}r<1)$, respectively. Then for any positive integers $a$ and $k$, we have $$\label{lemmaproofth1311}
2{\mathds P}(S_k> a)
{\leqslant}{\mathds P}\Big(\max_{i{\leqslant}k}S_i{\geqslant}a\Big)
{\leqslant}2{\mathds P}(S_k{\geqslant}a)$$ and $$2{\mathds P}\big(0<S_k<a\big)
{\leqslant}{\mathds P}\Big(\max_{1{\leqslant}i{\leqslant}k}S_i < a\Big){\leqslant}2{\mathds P}\big(0{\leqslant}S_k{\leqslant}a\big).$$
Fix any positive integer $a$ and define $\tau :=\tau_a := \inf\{i{\geqslant}1\::\: S_i=a\}$. Since the random walk has iid steps, it is obvious that $(S_{i+\tau}-S_\tau)_{i{\geqslant}0}$ and $(S_i)_{i{\geqslant}0}$ are independent random walks having the same law. Observing that $S_\tau = a$ and $\left\{\max_{i{\leqslant}k}S_i{\geqslant}a\right\} = \{\tau {\leqslant}k\}$ we find, therefore, $$\begin{aligned}
{\mathds P}\Bigl(\max_{i{\leqslant}k}S_i{\geqslant}a\Bigr)
&={\mathds P}\Bigl(\max_{i{\leqslant}k}S_i{\geqslant}a,\; S_k{\geqslant}a\Bigr) + {\mathds P}\Bigl(\max_{i{\leqslant}k}S_i{\geqslant}a,\; S_k < a\Bigr)\\
&={\mathds P}(S_k{\geqslant}a)+{\mathds P}(\tau{\leqslant}k,S_k<S_\tau)\\
&={\mathds P}(S_k{\geqslant}a)+{\mathds P}(\tau{\leqslant}k,S_k>S_\tau)\\
&={\mathds P}(S_k{\geqslant}a)+{\mathds P}(S_k>a).
\end{aligned}$$ From this we conclude that $$2{\mathds P}(S_k{\geqslant}a)
{\geqslant}{\mathds P}\Bigl(\max_{i{\leqslant}k}S_i{\geqslant}a\Bigr)
{\geqslant}2{\mathds P}(S_k>a).$$ Since ${\mathds P}(S_k{\geqslant}0)={\mathds P}(S_k{\leqslant}0){\geqslant}1/2$, we see $$\begin{aligned}
{\mathds P}\Bigl(\max_{i{\leqslant}k}S_i< a\Bigr)
&=1-{\mathds P}\Bigl(\max_{i{\leqslant}k}S_i{\geqslant}a\Bigr)\\
&{\leqslant}1-2{\mathds P}(S_k>a)\\
&{\leqslant}2\big({\mathds P}(S_k{\geqslant}0)-{\mathds P}(S_k>a)\big)\\
&=2{\mathds P}(0{\leqslant}S_k{\leqslant}a);
\end{aligned}$$ the other inequality follows similarly if we use ${\mathds P}(S_k>0)={\mathds P}(S_k<0){\leqslant}1/2$.
Next, we turn to the proof of Theorem \[coup\].
*Step 1*. As in the proof of Lemma \[lemmaproofth11\] we may, without loss of generality, assume that $n=d$ and $B={\operatorname{id}}_{{\mathds R}^d}$. For $t>0$, denote by $\mu_t$ the law of $X_t^0:=\int_0^t e^{(t-s)A}\,dZ_s$. According to Lemma \[lemmaproofth11\], the law $\mu_t$ is an infinitely divisible probability distribution, and the characteristic exponent of $\mu_t$ is given by $$\Phi_t(\xi):=\int_0^t\Phi\big(e^{sA^\top}\xi\big)\,ds.$$ Since the driving Lévy process $(Z_t)_{t{\geqslant}0}$ has no Gaussian part, the Lévy [triplet]{} $(0,b_t,\nu_t)$ of $\Phi_t$ is given by, cf. [@SAT Theorem 3.1], $$\begin{gathered}
\nu_t(C)= \int_0^t\nu(e^{-sA}C)\,ds,\qquad C\in\mathscr{B}({\mathds R}^d\setminus\{0\}),\\
b_t= \int_0^t e^{sA}b\,ds + \int_{z\neq 0} \int_0^t e^{sA}z\Big({\mathds 1}_{\{|z|{\leqslant}1\}} \big(e^{sA}z\big) - {\mathds 1}_{\{|z|{\leqslant}1\}}(z) \Big)\,ds\,\nu(dz).
\end{gathered}$$ For every $r>0$, let $\{\mu_t^r, t{\geqslant}0\}$ be the family of infinitely divisible probability measures on ${\mathds R}^d$ whose Fourier transform is of the form $\widehat{\mu}^r_t(\xi)=\exp(-\Phi_{t,r}(\xi))$, where $$\Phi_{t,r}(\xi) = \int_{|z|{\leqslant}r} \left(1-e^{i{\langle\xi,z\rangle}}+i{\langle\xi,z\rangle}\right)\,\nu_t(dz)$$ with $\nu_t$ as above.
Set $h(t):=1\big/\varphi_t^{-1}(1)$. Following the proof of [@SSW Propostion 2.2], the conditions and ensure that there exists $t_1>0$ such that for all $t{\geqslant}t_1$, the measure $\mu^{h(t)}_t$ has a density $p^{h(t)}_t\in C^{n+2}_b({\mathds R}^d)$; moreover, $$\label{proofcoup1}
\big|\nabla p^{h(t)}_t(y)\big|
{\leqslant}c(n,\Phi)\,h(t)^{-(n+1)}\big(1+h(t)^{-1}|y|\big)^{-(n+1)}$$ holds for all $y\in{\mathds R}^d$.
*Step 2*. For $r>0$ and $\xi\in{\mathds R}^d$, define $$\Psi_{t,r}(\xi)
:= \Phi_t(\xi)-\Phi_{t,r}(\xi)
= \int_{|z|>r}\left(1-e^{i{\langle\xi,z\rangle}}\right)\,\nu_t(dz) - i{\left\langle\xi,\int_{1<|z|{\leqslant}r}z\,\nu_t(dz)-b_t\right\rangle}.$$ Since $\Psi_{t,r}$ is given by a Lévy-Khintchine formula, it is the characteristic exponent of some $d$-dimensional infinitely divisible random variable. Let $\{{\mathsf{\pi}}_t^{r}, t{\geqslant}0\}$ be the family of infinitely divisible measures whose Fourier transforms are of the form $\widehat{{\mathsf{\pi}}}^{r}_t(\xi)=\exp(-\Psi_{t,r}(\xi))$. Clearly, $\mu_t=\mu_t^r*{\mathsf{\pi}}_t^{r}$ for all $t,r>0$.
Let $P_t(x,\cdot)$ and $P_t$ be the transition function and the transition semigroup of the Ornstein-Uhlenbeck process $\{X_t^x\}_{t{\geqslant}0}$ given by . For all $f\in B_b({\mathds R}^d)$ we have $$\begin{aligned}
P_tf(x)
&=\int f\bigl(e^{tA}x+z\bigr)\,\mu_t(dz)\\
&=\int f\bigl(e^{tA}x+z\bigr)\,\mu^r_t*{\mathsf{\pi}}_t^{r}(dz)\\
&=\iint f\bigl(e^{tA}x+z_1+z_2\bigr)\,{\mathsf{\pi}}_t^{r}(dz_1)\,\mu^r_t(dz_2).
\end{aligned}$$
Taking $r=h(t)$ we get, using the conclusions of step 1, that for all $t{\geqslant}t_1$ and $x\in{\mathds R}^d$, $$\begin{aligned}
P_tf(x)
&= \int p^{h(t)}_t(z_2)\,dz_2 \int f\bigl(e^{tA}x+z_1+z_2\bigr)\,{\mathsf{\pi}}_t^{h(t)}(dz_1)\\
&= \int p^{h(t)}_t\bigl(z_2-e^{tA}x\bigr)\,dz_2 \int f(z_1+z_2)\,{\mathsf{\pi}}_t^{h(t)}(dz_1).
\end{aligned}$$ If $\|f\|_\infty{\leqslant}1$, then $$\bigg\|\int f(z_1+\cdot)\,{\mathsf{\pi}}_t^{h(t)}(dz_1)\bigg\|_\infty
{\leqslant}\|f\|_\infty\, {\mathsf{\pi}}_t^{h(t)}({\mathds R}^d) {\leqslant}1.$$
*Step 3*. For all $x,y\in{\mathds R}^d$, $$\label{proofcoup2}\begin{aligned}
\|P_t(x,&\cdot)-P_t(y,\cdot)\|_{{\mathrm{Var}}}\\
&= \sup_{\|f\|_\infty{\leqslant}1}\big|P_tf(x)-P_tf(y)\big|\\
&= \sup_{\|f\|_\infty{\leqslant}1} \bigg|\int p^{h(t)}_t\bigl(z_2-e^{tA}x\bigr)\,dz_2 \int f(z_1+z_2)\,{\mathsf{\pi}}_t^{h(t)}(dz_1)\\
&\qquad\qquad\quad\mbox{}-\int p^{h(t)}_t\bigl(z_2-e^{tA}y\bigr)\,dz_2 \int f(z_1+z_2)\,{\mathsf{\pi}}_t^{h(t)}(dz_1)\bigg|\\
&{\leqslant}\sup_{\|g\|_\infty{\leqslant}1} \bigg|\int g(z)p^{h(t)}_t\bigl(z-e^{tA}x\bigr)\,dz
- \int g(z)p^{h(t)}_t\bigl(z-e^{tA}y\bigr)\,dz\bigg|\\
&= \sup_{\|g\|_\infty{\leqslant}1} \bigg|\int g(z)\Big(p^{h(t)}_t\bigl(z-e^{tA}x\bigr)-p^{h(t)}_t\bigl(z-e^{tA}y\bigr)\Big)\, dz\bigg|\\
&= \int \Big|p^{h(t)}_t\bigl(z-e^{tA}x\bigr)-p^{h(t)}_t\bigl(z-e^{tA}y\bigr)\Big|\,dz.
\end{aligned}$$
With the argument used in the proof of [@SSW Theorem 3.1], follows from and .
*Step 4*. By assumption , $$\varphi_\infty(\rho):=\sup_{|\xi|{\leqslant}\rho}\int_0^\infty{\ensuremath{\operatorname{Re}}}\Phi\big(B^\top e^{sA^\top}\xi\big)\,ds$$ is finite on $(0,\infty)$; in particular, $\varphi_\infty^{-1}(1)\in(0,\infty]$. On the other hand, for any $t{\geqslant}t_0$, according to , $$\begin{aligned}
\int \exp\left(-\int_0^t{\ensuremath{\operatorname{Re}}}\Phi\big(B^\top e^{sA^\top}\xi\big)\,ds\right)& |\xi|^{n+2} \,d\xi\\
&{\leqslant}\int \exp\left(-\int_0^{t_0}{\ensuremath{\operatorname{Re}}}\Phi\big(B^\top e^{sA^\top}\xi\big)\,ds\right) |\xi|^{n+2} \,d\xi\\
&=:C(t_0)<\infty. \end{aligned}$$ Since the function $t\mapsto \varphi^{-1}_t(1)$ is decreasing on $(0,\infty]$, holds. This finishes the proof.
Appendix {#sec-appendix}
========
Gradient Estimates for Ornstein-Uhlenbeck Processes {#subsec-appendix1}
---------------------------------------------------
Motivated by [@SSW Theorem 1.3], we have the following results for gradient estimates of an Ornstein-Uhlenbeck process. This is the counterpart of Theorem \[coup\]. For $t, \rho>0$, define $$\varphi(\rho)
:=\sup_{|\xi|{\leqslant}\rho}{\ensuremath{\operatorname{Re}}}\Phi\bigl(B^\top \xi\bigr)\quad\textrm{ and }\quad \varphi_t(\rho)
:=\sup_{|\xi|{\leqslant}\rho}\int_0^t{\ensuremath{\operatorname{Re}}}\Phi\big(B^\top e^{sA^\top}\xi\big)\,ds,$$ where $\Phi$ is the characteristic exponent of the driving Lévy process $(Z_t)_{t{\geqslant}0}$ from .
\[strong\] Let $P_t(x,\cdot)$ be the transition function of the $n$-dimensional Ornstein-Uhlenbeck process $\{X_t^x\}_{t{\geqslant}0}$ given by . Assume that $$\label{strong1}
\liminf_{|\xi|\rightarrow\infty} \frac{{\ensuremath{\operatorname{Re}}}\Phi\bigl(B^\top \xi\bigr)}{\log (1+|\xi|)} =\infty.$$ If for any $C>0$, $$\label{strong2}
\int \exp\left[-C t{\ensuremath{\operatorname{Re}}}\Phi\bigl(B^\top\xi\bigr)\right] |\xi|^{n+2} \,d\xi
= \mathsf{O}\left(\varphi^{-1}\Big(\frac{1}{t}\Big)^{2n+2}\right) \qquad\text{as\ \ } t\to 0,$$ then there exists $c>0$ such that for all $t>0$ and $f\in{B}_b({\mathds R}^n)$, $$\label{strong3}
\|\nabla P_t f\|_\infty{\leqslant}c\|f\|_\infty \,\varphi^{-1}\Big(\frac{1}{t\wedge1}\Big).$$ If, in addition, $$\xi\mapsto\int_0^\infty{\ensuremath{\operatorname{Re}}}\Phi\big(B^\top e^{sA^\top}\xi\big)\,ds\quad \textrm{ is locally bounded},$$ then there exist $t_1,c>0$ such that for $t{\geqslant}t_1$ and $f\in{B}_b({\mathds R}^n)$, $$\label{strong4}
\|\nabla P_t f\|_\infty
{\leqslant}c\,\|f\|_\infty \bigg[\|e^{tA}\|\,\varphi_t^{-1}(1)\bigg],$$ where $\|M\|=\sup_{|x|{\leqslant}1}|Mx|$ denotes the norm of the matrix of $M$.
To illustrate the power of Theorem \[strong\], we consider
\[examplegradient\] Let $\mu$ be a finite nonnegative measure on the unit sphere ${\mathds{S}}\subset {\mathds R}^n$ and assume that $\mu$ is nondegenerate in the sense that its support is not contained in any proper linear subspace of ${\mathds R}^n$. Let $\alpha\in(0,2)$, $\beta\in (0,\infty]$ and assume that the Lévy measure $\nu$ satisfies $$\nu(C)
{\geqslant}\int_0^{r_0}\int_{{\mathds{S}}}{\mathds 1}_C(s\theta)s^{-1-\alpha} \,ds\, \mu(d\theta)
+ \int_{r_0}^\infty\int_{{\mathds{S}}}{\mathds 1}_C(s\theta)s^{-1-\beta}\,ds\, \mu(d\theta)$$ for some constant $r_0>0$ and all $C\in {\mathscr{B}}({\mathds R}^n\setminus\{0\})$. Consider the following Ornstein-Uhlenbeck process $X_t$ on ${\mathds R}^n$ given by $$dX_t=AX_t\,dt+dZ_t,$$ where $(Z_t)_{t{\geqslant}0}$ is a Lévy process on ${\mathds R}^n$ with the Lévy measure $\nu$. By Theorem \[strong\] there exists a constant $c>0$ such that for all $t>0$ and $f\in B_b({\mathds R}^n)$, $$\|\nabla P_tf\|_\infty
{\leqslant}c\,\|f\|_\infty \, (t\wedge 1)^{-1/\alpha}.$$ Furthermore, if the real parts of all eigenvalues of $A$ are negative, then there exists a constant $c>0$ such that for all $t>0$ and $f\in B_b({\mathds R}^n)$, $$\|\nabla P_tf\|_\infty
{\leqslant}c\,\|f\|_\infty \,\frac{\|e^{tA}\|\quad}{(t\wedge 1)^{1/\alpha}}.$$
Recently, F.-Y. Wang [@W2 Theorem 1.1] has presented explicit gradient estimates for Ornstein-Uhlenbeck processes, by assuming that the corresponding Lévy measure has absolutely continuous (*with respect to Lebesgue measure*) lower bounds. Since lower bounds of Lévy measure in Example \[examplegradient\] could be much irregular, Theorem \[strong\] is more applicable than [@W2 Theorem 1.1].
Assuming the conditions and , we can mimic the proof of [@SSW Theorem 3.2] to show that there exist $t_1, C>0$ such that for all $x,y\in{\mathds R}^n$ and $t{\leqslant}t_1$, $$\label{proofstrong}
\|P_t(x,\cdot)-P_t(y,\cdot)\|_{{\mathrm{Var}}}
{\leqslant}C\,|e^{tA}(x-y)|\,\varphi^{-1}\Big(\frac{1}{t}\Big).$$ Thus we can apply to find for all $f\in B_b({\mathds R}^n)$ with $\|f\|_\infty= 1$, $$\label{proofstrong}\begin{aligned}
|\nabla P_tf(x)|
&{\leqslant}\limsup_{y\to x}\frac{|P_tf(x)-P_tf(y)|}{|y-x|}\\
&{\leqslant}\limsup_{y\to x}\frac{\sup_{\|w\|_\infty{\leqslant}1}|P_tw(x)-P_tw(y)|}{|y-x|}\\
&{\leqslant}\limsup_{y\to x}\frac{\|P_t(x,\cdot)-P_t(y,\cdot)\|_{{\mathrm{Var}}}}{|y-x|}\\
&{\leqslant}C\,\|e^{tA}\|\,\varphi^{-1}\Big(\frac{1}{t}\Big)\\
&{\leqslant}\Big[C\,\sup_{s{\leqslant}t_1}\|e^{sA}\|\Big]\,\varphi^{-1}\Big(\frac{1}{t}\Big).
\end{aligned}$$ Because of the Markov property of the semigroup $P_t$, the function $$t\mapsto \sup_{f\in B_b({\mathds R}^n),\, \|f\|_\infty =1}{ \|\nabla P_t f\|_\infty}$$ is deceasing. Combining this and yields .
The assertion follows if we combine the above argument with : there exist $t_2,C>0$ such that for all $x,y\in{\mathds R}^n$ and $t{\geqslant}t_2$, $$\begin{gathered}
\|P_t(x,\cdot)-P_t(y,\cdot)\|_{{\mathrm{Var}}}{\leqslant}C\,|e^{tA}(x-y)|\,\varphi_t^{-1}(1).
\qedhere\end{gathered}$$
Proof of Proposition \[improvement\] {#subsec-appendix2}
------------------------------------
Because of , we can choose a closed subset $F\subset \overline{B(z_0, \varepsilon)}$ such that $0\notin F$ and $$\int_F \frac{dz}{\rho_0(z)} < \infty.$$ By the Cauchy-Schwarz inequality, we have $$\left(\int_F \rho_0(z)\,dz\right)^{-1}
{\leqslant}\frac{1}{{\operatorname{Leb}}(F)^2} \int_F \frac{dz}{\rho_0(z)} < \infty.$$ Hence, $$K:=\int_F\rho_0(z)\,dz>0.$$
Since $F$ is a compact set and $0\notin F$, there exists some $\delta_0>0$ such that $
0\notin F+\overline{B(0,\delta_0)},
$ where $F+\overline{B(0,\delta_0)}:=\{a+b:a\in F, |b|{\leqslant}\delta_0\}.$ Since $\rho_0$ is locally integrable, we know that $$K{\leqslant}\int_{F+\overline{B(0,\delta_0)}}\rho_0(z)\,dz<\infty.$$ The remainder of the proof is now similar to the argument which shows that the shift $x\mapsto \|f(\cdot - x)-f\|_{L^1}$, $f\in L^1({\mathds R}^d,{\operatorname{Leb}})$, is continuous, see e.g. [@STR Lemma 6.3.5] or [@RSCC Theorem 14.8]: choose $\chi\in C_c^\infty ({\mathds R}^d)$ such that ${\operatorname{supp}}\chi\subset F+\overline{B(0,\delta_0)}$ and $$\int_{F+\overline{B(0,\delta_0)}}|\rho_0(z)-\chi(z)|\, dz{\leqslant}\frac{K}{4}.$$ Therefore, for any $x\in{\mathds R}^d$ with $|x|{\leqslant}\delta_0$, we obtain $$\begin{aligned}
\int_F & |\rho_0(z)-\rho_0(z-x)|\,dz\\
&{\leqslant}\int_F|\rho_0(z)-\chi(z)|\,dz
+\int_F|\chi(z)-\chi(z-x)|\,dz
+\int_F|\rho_0(z-x)-\chi(z-x)|\,dz\\
&= \int_F|\rho_0(z)-\chi(z)|\,dz
+\int_F|\chi(z)-\chi(z-x)|\,dz
+\int_{F+x}|\rho_0(z)-\chi(z)|\,dz\\
&{\leqslant}2\int_{F+\overline{B(0,\delta_0)}}|\rho_0(z)-\chi(z)|\,dz
+\int_F|\chi(z)-\chi(z-x)|\,dz\\
&{\leqslant}\frac{K}{2}+\int_F|\chi(z)-\chi(z-x)|\,dz.
\end{aligned}$$ By the dominated convergence theorem we see that $$x\mapsto\int_F|\chi(z)-\chi(z-x)|\,dz$$ is continuous on ${\mathds R}^d$. Therefore, there exists $0<\delta{\leqslant}\delta_0$ such that $$\sup_{x\in{\mathds R}^d, |x|{\leqslant}\delta}\int_F|\chi(z)-\chi(z-x)|\,dz{\leqslant}\frac{K}{4}$$ and, in particular, $$\sup_{x\in{\mathds R}^d, |x|{\leqslant}\delta}\int_F|\rho_0(z)-\rho_0(z-x)|\,dz{\leqslant}\frac{3K}{4}.$$ Using $2(a\wedge b)=a+b-|a-b|$ for all $a,b{\geqslant}0$, we get $$\begin{aligned}
\inf_{x\in{\mathds R}^d, |x|{\leqslant}\delta} & \int_F\big(\rho_0(z)\wedge\rho_0(z-x)\big)\,dz\\
&=\frac{1}{2}\inf_{x\in{\mathds R}^d, |x|{\leqslant}\delta} \bigg[\int_F\big(\rho_0(z)+\rho_0(z-x)\big)\,dz
- \int_F\big|\rho_0(z)-\rho_0(z-x)\big|\,dz\bigg]\\
&{\geqslant}\frac{1}{2}\int_F\rho_0(z)\,dz
- \frac{1}{2}\sup_{x\in{\mathds R}^d, |x|{\leqslant}\delta}\int_F\big|\rho_0(z)-\rho_0(z-x)\big|\,dz\\
&{\geqslant}\frac{K}{8}>0.
\end{aligned}$$ This finishes the proof.
Financial support through DFG (grant Schi 419/5-1) and DAAD (PPP Kroatien) (for René L. Schilling) and the Alexander-von-Humboldt Foundation and the Natural Science Foundation of Fujian $($No. 2010J05002$)$ (for Jian Wang) is gratefully acknowledged.
[99]{} Aliprantis, C.D. and Burkinshaw, O.: *Principles of Real Analysis (3rd ed.)*, Academic Press, San Diego, California 1998,
Bernstein, D.S.: *Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory*, Princeton University Press, Princeton 2005.
Cranston, M. and Greven, A.: Coupling and harmonic functions in the case of continuous time Markov processes, *Stoch.Proc. Appl.* **60** (1995), 261–286.
Cranston, M. and Wang, F.-Y.: A condition for the equivalence of coupling and shift-coupling, *Ann. Probab.* **28** (2000), 1666–1679.
Lindvall, T.: *Lectures on the Coupling Method*, Wiley, New York 1992.
Nourdin, I. and Simon, T.: On the absolute continuity of Lévy processes with drift, *Ann. Probab.* **34** (2006), 1035–1051.
Priola, E. and Zabczyk, J.: Liouville theorems for non-local operators, *J. Funct. Anal.* **216** (2004), 455–490.
Priola, E. and Zabczyk, J.: Densities for Ornstein-Uhlenbeck processes with jumps, *Bull. London Math. Soc.* **41** (2009), 41–50.
Sato, K. and Yamazato, M.: Operator-self-decomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type, *Stoch. Proc. Appl.* **17** (1984), 73–100.
Sato, K.: *Lévy processes and Infinitely Divisible Distributions*, Cambridge University Press, Cambridge 1999.
Schilling, R.L.: *Measures, Integrals and Martingales*, Cambridge University Press, Cambridge 2005.
Schilling, R.L., Sztonyk, P. and Wang, J.: Coupling property and gradient estimates of Lévy processes via symbol, to appear in *Bernoulli*, 2011. See also arXiv 1011.1067
Schilling, R.L. and Wang, J.: On the coupling property of Lévy processes, to appear in *Ann. Inst. Henri Poincaré: Probab. Stat.*, 2010. See also arXiv 1006.5288
Stroock, D.W.: *A Concise Introducation to the Theory of Integration (2nd ed.)*, Birkhäuser, Boston 1994.
Thorisson, H.: *Coupling, Stationarity and Regeneration*, Springer, New York 2000.
Wang, F.-Y.: Coupling for Ornstein-Uhlenbeck jump processes, to appear in *Bernoulli*, 2010. See also arXiv:1002.2890v5
Wang, F.-Y.: Gradient estimate for Ornstein-Uhlenbeck jump processes, *Stoch. Proc. Appl.* **121** (2011), 466–478.
[^1]: *R. Schilling:* TU Dresden, Institut für Mathematische Stochastik, 01062 Dresden, Germany. `[email protected]`
[^2]: *J. Wang:* School of Mathematics and Computer Science, Fujian Normal University, 350007, Fuzhou, P.R. China *and* TU Dresden, Institut für Mathematische Stochastik, 01062 Dresden, Germany. `[email protected]`
| ArXiv |
---
author:
- |
JLQCD Collaboration: $^a$[^1], G. Cossu$^{a}$, S. Hashimoto$^{a,b}$, T. Kaneko$^{a,b}$, J. Noaki$^{a}$, M. Tomii$^{b}$\
High Energy Accelerator Research Organization (KEK), Ibaraki 305-0801, Japan Address\
Department of Particle and Nuclear Science, SOKENDAI (The Graduate University for Advanced Studies), Ibaraki 305-0801, Japan
bibliography:
- 'lattice\_2015\_fahy.bib'
title: |
(0,0)(0,0) (300,75)[(0,0)\[l\][[KEK-CP-340]{.nodecor} ]{}]{}
Decay constants and spectroscopy of mesons in lattice QCD using domain-wall fermions
---
Introduction
============
The JLQCD collaboration has recently produced ensembles of lattice configurations with fine lattice spacings and good chiral symmetry. Lattice simulations of 2+1-flavor QCD were performed using the Möbius domain-wall fermions [@mobius] with tree-level Symanzik gauge action. Table \[tab:lattices\] shows the $15$ gauge ensembles generated [@finelattice]. These lattices have lattice spacings of $1/a \approx 2.4$, $3.6$, and $4.5\text{ GeV}$ with pion masses between $230$ MeV and $500$ MeV. For each ensemble, $10,000$ molecular dynamics (MD) times were run after thermalization.
Using domain-wall fermions the Ginsparg-Wilson relation is only approximate. The violation of the Ginsparg-Wilson relation is given by the residual mass. The Möbius representation as well as using stout link-smearing [@stout] make the residual mass small of $\mathcal{O}(1\text{ MeV})$ on the coarsest ($\beta=4.17$) lattices and $<0.2\text{ MeV}$ on the finer lattices [@residual]. Good chiral symmetry enables simpler renormalization such as $Z_V = Z_A$, and simplifies the calculation of the pseudo-scalar decay constants directly utilizing the PCAC relation.
With chiral symmetry preserved, observables can be used to compare lattice QCD results to those of Chiral Perturbation Theory (ChPT). In this work we present results of masses and decay constants of light and charmed pseudo-scalar mesons. These measurements are used to determine the low energy constants (LEC) in ChPT. Additionally the fine lattice spacing enables exploration of charm physics with manageable cutoff effects.
In this work the scale for these lattices was determined from Wilson flow using $t_0$. We use the value from [@scale] as an input. The determination was done using a linear extrapolation to the physical point in $m_\pi^2$ as well as an interpolation of the strange quark mass to reproduce the physical $(M^\text{phys}_{s\bar{s}})^2 = 2M_K^2 - M_\pi^2$. The fitted parameters to describe the mass dependence for the two smaller $\beta$ values were then used to determine the scale on our finest lattice. The results for $a^{-1}$ on each $\beta$ are listed in Table \[tab:lattices\].
Lattice Spacing $L^3\times T$ $L_5$ $a m_{ud}$ $a m_s$ $ m_\pi \text{ [MeV]} $ $ m_{\pi}L $
--------------------------------- ----------------------------------------- ------- ------------ --------- -------------------------- --------------
$\beta = 4.17,$ $ 32^3\times64$ $(L=2.6 \text{ fm})$ 12 0.0035 0.040 230 3.0
$a^{-1}=2.453(4)\text{ GeV}$ 0.0070 0.030 310 4.0
0.0070 0.040 310 4.0
0.0120 0.030 400 5.2
0.0120 0.040 400 5.2
0.0190 0.030 500 6.5
0.0190 0.040 500 6.5
$48^3\times96 $ $ (L=3.9 \text{ fm})$ 12 0.0035 0.040 230 4.4
$\beta= 4.35,$ $48^3\times 96 $ $(L=2.6 \text{ fm})$ 8 0.0042 0.018 300 3.9
$a^{-1}=3.610(9)\text{ GeV} $ 0.0042 0.025 300 3.9
0.0080 0.018 410 5.4
0.0080 0.025 410 5.4
0.0120 0.018 500 6.6
0.0120 0.025 500 6.6
$\beta = 4.47,$ $64^3\times128 $ $(L=2.7 \text{ fm}) $ 8 0.0030 0.015 280 4.0
$a^{-1} = 4.496(9) \text{ GeV}$
: Parameters of the JLQCD gauge ensembles used in this work. Pion masses are rounded to the nearest $10$ MeV. The ensemble with $m_\pi L \approx 3.0$ is excluded in all analysis below to avoid possible finite volume effects. \[tab:lattices\]
Computation of observables
==========================
Pseudo-scalar correlators were produced utilizing our QCD software package Iroiro++ [@iroiro]. These correlators were computed on $200$ gauge configurations separated by $50$ MD times and from two source locations, producing $400$ measurements of the light correlators and $300$ measurements of heavy correlators for each ensemble, except for the $\beta=4.17$ ensemble on the larger volume, which has $600$ light and $400$ heavy measurements. Correlators were produced with unsmeared point sources as well as smeared sources using Gaussian smearing, and the same point and smeared operators are used also for the sinks. Gaussian smearing is defined by the operator $(1-(\alpha/N) \Delta)^N$ where $\Delta$ as the Laplacian and in this work the parameters $\alpha= 20.0$ and $N=200$ were used.
The amplitudes of the unsmeared local operators are required to compute the decay constants. Two-point correlation functions of the form ${\langle P^L(x){P^G}^\dagger(0)\rangle}$ were fit simultaneously with correlators ${\langle P^G(x){P^G}^\dagger(0)\rangle}$ where $L$ indicates an unsmeared local operator while $G$ denotes Gaussian smeared operators. The two-point correlation functions were fit to the functional form $$\begin{aligned}
\label{eq:amp}
C = \underbrace{\frac{1}{2m_\pi} {\langle 0\vert}P {\vert \pi\rangle}{\langle \pi\vert}P^\dagger{\vert 0\rangle}}_{A_{PP}} {\left( e^{-m_\pi t}+e^{-m_\pi (N_t-t)} \right)}\end{aligned}$$ for large $t$ to determine the masses and amplitudes where $P$ is either $P^L$ or $P^G$. The matrix element of ${\langle 0\vert}P{\vert \pi\rangle}$ of the unsmeared operator $P^L$ can be reconstructed from the simultaneous fit of ${\langle P^L(x)P^G(0)\rangle}$ and ${\langle P^G(x)P^G(0)\rangle}$. The decay constants are calculated by utilizing the axial Ward-Takahashi identity ${Z_A\partial_\mu A_\mu=(m_{q_1}+m_{q_2})P}$, where $A_\mu$ is the lattice axial current, and $m_q$’s are the quark masses of the pseudo-scalar meson of interest. This leads to the formula for $f_P$, $$\begin{aligned}
\label{eq:fpi}
f_P = (m_{q_1} + m_{q_2})\sqrt{\frac{2A_{PP}}{m_\pi^3}},\end{aligned}$$ which does not rely on the renormalization constant $Z_A$. The mass $m_q$ used is the bare quark mass plus the residual mass. We use the convention $F_\pi = f_\pi/\sqrt{2}$.
Pion masses and decay constants
===============================
Our measurements of the pion masses and decay constants for ensembles at different bare light quark masses allow us to investigate the consistency with $SU(2)$ ChPT. The quark mass dependence of $M_\pi$ and $F_\pi$ at next-to-next-to-leading order [@PhysRevD.90.114504] is $$\begin{aligned}
\frac{M_\pi^2}{\bar{m}_q } &= 2B{\left[ 1-\frac{1}{2}x \ln\frac{\Lambda_3^2}{M^2} + \frac{17}{8} x^2{\left( \ln\frac{\Lambda_M^2}{M^2} \right)}^2 + k_M x^2 +\mathcal{O}{\left( x^3 \right)} \right]}, \label{mx-expand}\\
F_\pi &= F{\left[ 1 + x \ln\frac{\Lambda_4^2}{M^2}-\frac{5}{4} x^2{\left( \ln\frac{\Lambda_F^2}{M^2} \right)}^2 + k_F x^2 +\mathcal{O}{\left( x^3 \right)} \right]}.
\label{fx-expand}\end{aligned}$$ These are expanded using the parameter $x = M^2/(4\pi F)^2$ where $M^2 = B (\bar{m}_q +\bar{m}_q ) = 2\bar{m}_q \Sigma/F^2$. $\bar{m}_q$ is the appropriately renormalized quark mass, where the renormalization factor is discussed in [@renormalization]. The parameters $\Lambda_3$ and $\Lambda_4$ are related to the effective coupling constants of ChPT through $\bar{\ell}_n = \ln
{\Lambda_n^2/M_\pi^2}$. $\Lambda_M$ and $\Lambda_F$ are linear combinations of different $\Lambda_n$’s [@PhysRevD.90.114504].
The chiral expansions above are fit to the data for $F_\pi$ and $M_\pi^2/\bar{m}_q $ simultaneously at both NLO and NNLO. At NLO only terms up to $\mathcal{O}{{\left( x^2 \right)}}$ in (\[mx-expand\]) and (\[fx-expand\]) are included leaving the free parameters $F, B,\Lambda_3$, and $\Lambda_4$. For NNLO there are the additional free parameters $k_M$, and $k_F$, while the values of $\Lambda_{1}$ and $\Lambda_{2}$ were fixed to the phenomenological value from [@colangelo].
To account for the strange-quark mass dependence the fit function was corrected by a term proportional to $M_{s\bar{s}}^2 = 2M_K^2 - M_\pi^2$. Combining with a lattice spacing dependence, all fits were performed with a prefactor $(1+ \gamma_{a} a^2 + \gamma_{s}(M_{s\bar{s}} -
M_{s\bar{s}}^{\text{phys}}) )$. At NLO the fits have $\chi^2$ less than $1.5$ if including only the ensembles with pion masses $M_\pi < 450 \text{ MeV}$, so the other ensembles were excluded for the NLO fits. For NNLO fits the ensembles of all pion masses were included. The results of the NLO and NNLO fits in the continuum and physical strange quark mass limits are shown in Figure \[fig:Fpi\_x\] by dashed lines.
![Plots of $M_\pi^2/\bar{m}_q $ (left panel) and $F_\pi$ (right panel), both vs. $x=2\bar{m}_q B/(4\pi F)^2$. Fit lines show the best NLO (blue) and NNLO (green) fits in the continuum and physical strange quark mass limits. The NLO fits only include the ensembles for $M_\pi < 450 \text{ MeV}$[]{data-label="fig:Fpi_x"}](combined_mpisqrbymq_x_NNLO_all_comb.eps "fig:"){width="49.00000%"} ![Plots of $M_\pi^2/\bar{m}_q $ (left panel) and $F_\pi$ (right panel), both vs. $x=2\bar{m}_q B/(4\pi F)^2$. Fit lines show the best NLO (blue) and NNLO (green) fits in the continuum and physical strange quark mass limits. The NLO fits only include the ensembles for $M_\pi < 450 \text{ MeV}$[]{data-label="fig:Fpi_x"}](combined_Fpi_NNLO_all_x.eps "fig:"){width="49.00000%"}
![Same as Figure \[fig:Fpi\_x\] except plotted vs. $\xi = M_\pi^2 / (4\pi F_\pi)^2$.[]{data-label="fig:Fpi_xi"}](mpisqrbymq_XI_NNLO_comb.eps "fig:"){width="49.00000%"} ![Same as Figure \[fig:Fpi\_x\] except plotted vs. $\xi = M_\pi^2 / (4\pi F_\pi)^2$.[]{data-label="fig:Fpi_xi"}](Fpi_NNLO_xi_inverse_comb.eps "fig:"){width="49.00000%"}
Alternatively the ChPT expansions can be reorganized using the parameter $\xi = M_\pi^2 /(4\pi F_\pi)^2$. The expansions are $$\begin{aligned}
\frac{M_\pi^2}{\bar{m}_q } &= 2B / {\left[ 1+\frac{1}{2} \xi \ln\frac{\Lambda_3}{M_\pi^2} - \frac{5}{8} \xi^2{\left( \ln\frac{\Omega_M^2}{M_\pi^2} \right)}^2 + c_M \xi^2 +\mathcal{O}{\left( \xi^3 \right)} \right]},\\
F_\pi &= F / {\left[ 1 - \xi \ln\frac{\Lambda_4^2}{M_\pi^2}-\frac{1}{4} \xi^2{\left( \ln\frac{\Omega_F^2}{M_\pi^2} \right)}^2 + c_F \xi^2 +\mathcal{O}{\left( \xi^3 \right)} \right]},
\label{xi-expand}\end{aligned}$$ where similarly the values $\Omega_M$ and $\Omega_F$ are combinations of other LEC’s [@PhysRevD.90.114504]. The pion masses and decay constants are plotted against $\xi$ in Figure \[fig:Fpi\_xi\]. The curves represent the fits of NLO and NNLO.
Our preliminary results for the LEC’s from the NLO fits expanded in $x$ are $F = 83.2(6.3)\text{ MeV}$, $\Sigma^{1/3}[2\text{ MeV}] = 287.9(3.7)\text{ MeV}$, $\bar{\ell}_3 = 3.11(44)$ and $\bar{\ell}_3 = 4.37(22)$. The chiral condensate, $\Sigma$, is renormalized to the one in the $\overline{\text{MS}}$ scheme at $\mu = 2 \text{ GeV}$ using the renormalization factor calculated in [@renormalization]. The values obtained with the two expansion parameters as well as those from NLO and NNLO fits are all consistent within statistical error though the NNLO results have slightly larger uncertainty. $F$ is the decay constant in the chiral limit but at the physical pion mass value we obtain $F_\pi = 88.9(5.2)\text{ MeV}$.
Charmed mesons
==============
The lattice spacings of the JLQCD ensembles were chosen to treat heavy physics with minimal cutoff effects. We produced charmed correlators using domain-wall heavy quarks at three masses close to the charm mass. All results shown are first interpolated to the charm mass using the spin averaged $c\bar{c}$ masses. Charmonium correlators are also used in the analysis of their time-moments to determine the charm quark mass $m_c$ and strong coupling constant $\alpha_s$ [@charm].
Figure \[fig:charmedmasses\] shows the masses of the $D$ and $D_s$ mesons as well as linear fits in $M_\pi^2$ accounting for a dependence on the lattice spacing $a^2$ and interpolated in $m_s$ using $2M_K^2 - M_\pi^2$. The raw data for $M_{D_s}$ appear scattered because the data points with different input strange quark masses are plotted together. After interpolating in $2M_K^2 - M_\pi^2$, the data at different $\beta$ are more consistent with each other. The results after extrapolation are $M_D =1867.7(9.5)\text{ MeV}$ and $M_{D_s} = 1964.2(5.0)\text{ MeV}$. Their experimental values are $M_D^{\text{exp}} = 1864.8 \text{ MeV}$ and $M_{D_s}^{\text{exp}} = 1968.3 \text{ MeV}$. The dependence upon the lattice cutoff $a$ turned out to be minimal with a difference of $\mathcal{O}(1\%)$ between the fitted value at $\beta=4.17$ and the continuum limit.
![Masses of the $D$ meson (left) and $D_s$ meson (right) vs. $M_\pi^2$. These were fit linearly in $M_\pi^2$ accounting for a dependence on the lattice spacing $a^2$ and interpolated in the strange quark mass using $2M_K^2 - M_\pi^2$. The blue dashed line indicated the linear fit extrapolated to the continuum limit while the green dashed line shows the linear fit for the value of $a$ corresponding to our coarsest lattice $\beta=4.17$.[]{data-label="fig:charmedmasses"}](mD_s0.eps "fig:"){width="49.00000%"} ![Masses of the $D$ meson (left) and $D_s$ meson (right) vs. $M_\pi^2$. These were fit linearly in $M_\pi^2$ accounting for a dependence on the lattice spacing $a^2$ and interpolated in the strange quark mass using $2M_K^2 - M_\pi^2$. The blue dashed line indicated the linear fit extrapolated to the continuum limit while the green dashed line shows the linear fit for the value of $a$ corresponding to our coarsest lattice $\beta=4.17$.[]{data-label="fig:charmedmasses"}](mDs_s0.eps "fig:"){width="49.00000%"}
The decay constants of the charmed mesons are also computed using the same process for the pion using the pseudo-scalar current and the appropriate quark masses. These results can be seen in Figure \[fig:charmeddecay\]. The fitted values after linear extrapolation in $M_\pi^2$ and $a^2$ are $f_D = 209.6(5.2)\text{ MeV}$ and $f_{D_s} = 244.4(4.1)\text{ MeV}$ with the dependence on the lattice spacing turning out to be negligible. For the decay constant of the $D$ meson we attempted to analyze with the ChPT fit at NLO [@Grinstein1992369] as well as a linear fit. It favors a smaller value for $f_D$ because of the chiral logarithm, but more precise data would be necessary to confirm, especially because the current fit is strongly influenced by the lightest data point, which has a relatively large error.
![Charmed meson decay constants $f_D$ (left panel) and $f_{D_s}$ (right panel) vs. $M_\pi^2$. On both plots the blue dashed line indicate a linear fit in $M_\pi^2$ accounting for a dependence on the lattice spacing $a^2$ and interpolated in strange quark mass using $2M_K^2 -
M_\pi^2$. The left plot of $f_D$ includes a simple linear fit as well as the chiral NLO fit for the ensembles with $M_\pi < 450 \text{MeV}$.[]{data-label="fig:charmeddecay"}](fD_s0_both.eps "fig:"){width="49.00000%"} ![Charmed meson decay constants $f_D$ (left panel) and $f_{D_s}$ (right panel) vs. $M_\pi^2$. On both plots the blue dashed line indicate a linear fit in $M_\pi^2$ accounting for a dependence on the lattice spacing $a^2$ and interpolated in strange quark mass using $2M_K^2 -
M_\pi^2$. The left plot of $f_D$ includes a simple linear fit as well as the chiral NLO fit for the ensembles with $M_\pi < 450 \text{MeV}$.[]{data-label="fig:charmeddecay"}](fDs_s0.eps "fig:"){width="49.00000%"}
Summary {#sec:sum}
=======
We have shown first results from the recently generated JLQCD lattices. The good chiral properties of these lattices enable successful fits of quantities to NLO and NNLO ChPT. Measurements are still in progress and the precision of the decay constants and LECs should be improved with better statistics and the use of stochastic noise sources.
The fine lattice spacings allow us to compute charmed masses and decay constants with small dependence upon $a$. We plan to produce results with quarks heavier than the charm mass to investigate the lattice spacing dependence for heavy domain wall quarks. If the dependence continues to remain small it may be possible to extrapolate to $B$ physics.
Numerical simulations are performed on the IBM System Blue Gene Solution at High Energy Accelerator Research Organization (KEK) under a support of its Large Scale Simulation Program (No. 13/14-04, 14/15-10). We thank P. Boyle for helping in the optimization of the code for BGQ. This work is supported in part by the Grant-in-Aid of the Japanese Ministry of Education (No. 26400259, 26247043, and 15K05065) and the SPIRE (Strategic Program for Innovative Research) Field5 project.
[^1]: Email:[email protected]
| ArXiv |
---
abstract: 'Pseudoconvexity of a domain in $\Bbb C^n$ is described in terms of the existence of a locally defined plurisubharmonic/holomorphic function near any boundary point that is unbounded at the point.'
address:
- |
Institute of Mathematics and Informatics\
Bulgarian Academy of Sciences\
Acad. G. Bonchev 8, 1113 Sofia, Bulgaria
- |
Carl von Ossietzky Universität Oldenburg\
Institut für Mathematik\
Postfach 2503\
D-26111 Oldenburg, Germany
- |
Institut de Mathématiques\
Université Paul Sabatier, 118 Route de Narbonne, 31062 Toulouse Cedex 9, France
- 'Instytut Matematyki, Uniwersytet Jagielloński, Reymonta 4, 30-059 Kraków, Poland'
author:
- 'Nikolai Nikolov, Peter Pflug, Pascal J. Thomas, Wlodzimierz Zwonek'
title: On a local characterization of pseudoconvex domains
---
[This paper was written during the stay of the third named author at the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences supported by a CNRS grant and the stay of the fourth named author at the Universität Oldenburg supported by a DFG grant No 436 POL 113/106/0-2 (July 2008).]{}
Introduction and results
========================
It is well-known that a domain $D\subset\Bbb C^n$ is pseudoconvex if and only if any of the following conditions holds:
\(i) there is a smooth strictly plurisubharmonic function $u$ on $D$ with $\lim_{z\to\partial D}u(z)=\infty;$
\(ii) for any $a\in\partial D$ there is a $u_a\in\PSH(D)$ with $\lim_{z\to a}u_a(z)=\infty;$
\(iii) there is an $f\in\mathcal O(D)$ such that for any $a\in\partial D$ and any neighborhood $U_a$ of $a$ one has that $\limsup_{G\ni z\to a}|f(z)|=\infty$ for any connected component $G$ of $D\cap U_a$ with $a\in\partial G;$
\(iv) for any $a\in\partial D$ there is a neighborhood $U_a$ of $a$ and an $f_a\in\mathcal O(D\cap U_a)$ such that for any neighborhood $V_a\subset U_a$ of $a$ and any connected component $G$ of $D\cap V_a$ with $a\in\partial G$ one has $\limsup_{G\ni
z\to a}|f_a(z)|\\=\infty$ (see Corollary 4.1.26 in [@Hor]).
If $D$ is $C^1$-smooth, we may assume that $D\cap U_a$ is connected in (iii) and (iv).
Our first aim is to see that in (i) in general ’lim’ cannot be weakened by ’limsup’ even if $D$ is $C^1$-smooth.
\[1\] For any $\varepsilon\in(0,1)$ there is a non-pseudoconvex bounded domain $D\subset\Bbb C^2$ with $C^{1,1-\varepsilon}$-smooth boundary and a negative function $u\in\PSH(D)$ with $\limsup_{z\to a}u(z)=0$ for any $a\in\partial
D.$
In particular, $v:=-\log(-u)\in\PSH(D)$ with $\limsup_{z\to
a}v(z)=\infty$ for any $a\in\partial D.$
If we do not require smoothness of $D,$ following the idea presented in the proof, we may just take $D=\{z\in\Bbb
C^n:\min\{||z||,||z-a||\}<1\},$ $0<||a||<2,$ $n\ge 2.$
On the other, this cannot happen if $D$ is $C^2$-smooth.
\[2\] Let $D\subset\Bbb C^n$ be a $C^2$-smooth domain with the following property: for any boundary point $a\in\partial D$ there is a neighborhood $U_a$ of $a$ and a function $u_a\in\PSH(D\cap U_a)$ such that $\limsup_{z\to a}u_a(z)=\infty.$ Then $D$ is pseudoconvex.
However, if we replace ’limsup’ by ’lim’, we may remove the hypothesis about smoothness of the boundary.
\[3\] Let $D\subset\Bbb C^n$ be a domain with the following property: for any boundary point $a\in\partial
D$ there is a neighborhood $U_a$ of $a$ and a function $u_a\in\PSH(D\cap U_a)$ such that $\lim_{z\to a}u_a(z)=\infty.$ Then $D$ is pseudoconvex.
Note that the assumption in Proposition \[3\] is formally weaker that to assume that $D$ is locally pseudoconvex.
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[*Remark.*]{} The three propositions above have real analogues replacing (non)pseudoconvex domains by (non)convex domains and plurisubharmonic functions by convex functions (for the analogue of Proposition 3 use e.g. Theorem 2.1.27 in [@Hor] which implies that if $D$ is a nonconvex domain in $\Bbb R^n,$ then there exists a segment $[a,b]$ such that $c=\frac{a+b}{2}\in\partial D$ but $[a,b]\setminus\{c\}\subset D$). The details are left to the reader.
Recall now that a domain $D\subset\Bbb C^n$ is called [*locally weakly linearly convex*]{} if for any boundary point $a\in\partial D$ there is a complex hyperplane $H_a$ through $a$ and a neighborhood $U_a$ of $a$ such that $H_a\cap D\cap U_a=\varnothing.$ D. Jacquet asked whether a locally weakly linearly convex domain is already pseudoconvex (see [@Jac], page 58). The answer to this question is affirmative by Proposition \[3\]. The next proposition shows that such a domain has to be even taut[^1] if it is bounded.
\[4\] Let $D\subset\Bbb C^n$ be a bounded domain with the following property: for any boundary point $a\in\partial D$ there is a neighborhood $U_a$ of $a$ and a function $f_a\in\mathcal O(D\cap
U_a)$ such that $\lim_{z\to a}|f_a(z)|=\infty.$ Then $D$ is taut.
Let $D\subset\Bbb C^n$ be a domain and let $K_D(z)$ denote the Bergman kernel of the diagonal. It is well-known that $\log
K_D\in\PSH(D).$ Recall that
\(v) if $D$ is bounded and pseudoconvex, and $\limsup_{z\to
a}K_D(z)=\infty$ for any $a\in\partial D,$ then $D$ is an $L_h^2$-domain of holomorphy ($L_h^2(D):=L^2(D)\cap\mathcal O(D)$) (see [@Pfl-Zwo]).
We show that the assumption of pseudoconvexity is essential.
\[10\] There is a non-pseudoconvex bounded domain $D\subset\Bbb C^2$ such that $\limsup_{z\to a}K_D(z)=\infty$ for any $a\in\partial D$.
Note that the domain $D$ with $u=\log K_D$ presents a similar kind of example as that in Proposition \[1\] (however, the domain has weaker regularity properties).
The example given in Proposition \[10\] is a domain with non-schlicht envelope of holomorphy. This is not accidental as the following result shows.
\[11\] Let $D\subset\Bbb C^n$ be a domain such that $\limsup_{z\to a}K_D(z)=\infty$ for any $a\in\partial D$. Assume that one of the following conditions is satisfied:
– the envelope of holomorphy $\hat D$ of $D$ is a domain in $\CC^n$,
– for any $a\in\partial D$ and for any neighborhood $U_a$ of $a$ there is a neighborhood $V_a\subset U_a$ of $a$ such that $V_a\cap
D$ is connected (this is the case when e.g. $D$ is a $C^1$-smooth domain).
Then $D$ is pseudoconvex.
[*Remark.*]{} Note that the domain in the example is not fat. We do not know what will happen if $D$ is assumed to be fat.
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Making use of the reasoning in [@Irg2] we shall see how Proposition \[10\] implies that the domain from this proposition admits a function $f\in L_h^2(D)$ satisfying the property $\limsup_{z\to a}|f(z)|=\infty$ for any $a\in\partial D.$
\[12\] Let $D$ be the domain from Proposition \[10\]. Then there is a function $f\in L_h^2(D)$ such that $\limsup_{z\to a}|f(z)|=\infty$ for any $a\in\partial D.$
Proof of Proposition 1
======================
First, we shall prove two lemmas.
\[5\] For any $\eps \in (0,1)$ and $C_1$, $C_2>0$, there exists an $F\in
\mathcal C^{1, 1-\eps}(\RR)$ such that:
[(i)]{} $\supp F\subset [-1,+1]$, $0\le F(x)\le C_1$ for all $x
\in \RR$;
[(ii)]{} there is a dense open set $\mathcal U \subset [-1,+1]$ such that $F''(x)$ exists and $F''(x) \le -C_2<0$ for all $x \in
\mathcal U$;
[(iii)]{} $F$ vanishes on a Cantor subset of $[-1,+1].$
An elementary construction yields an even non-negative smooth function $b$ supported on $[-3/4,+3/4]$, decreasing on $[0,
3/4]$, such that $b(x) = 1-4x^2$ for $|x|\le 1/4$, $|b'(x)| \le
C_3$, $-8 \le b''(x) \le C_4$ for all $x\in\Bbb R,$ where $C_3,C_4>0.$
For any $a,p >0$, we set $b_{a,p}(x):= ab(x/p)$, $x\in\RR$.
We shall construct two decreasing sequences of positive numbers $(a_n)_{n\geq 0}$ and $(p_n)_{n\geq 0}$, and intervals $\{
I_{n,i}, J_{n,i}, n \ge 0, 1\le i \le 2^n\}$.
Set $I_{0,1}:= (-1,+1)$ and $J_{0,1}:=[-p_0/4,p_0/4]$, where $p_0<1$. Then $I_{1,1}:= (-1,-p_0)$ and $I_{1,2}:= (p_0,1)$.
In general, if the intervals of the n-th “generation” $ I_{n,i}$ are known, we require $$\label{condpn} p_{n} < \frac{| I_{n,i}|}{2},$$ where $|J|$ denotes the length of an interval $J$. Denote by $c_{n,i}$ the center of $ I_{n,i}$ and put $J_{n,i}:=
[c_{n,i}-p_{n}/4, c_{n,i}+p_{n}/4]$. Denote respectively by $I_{n+1,
2i-1}$ and $I_{n+1, 2i}$ the first and second component of $I_{n,i}
\setminus J_{n,i}$.
Now we write $$f_n (x) := \sum_{i=1}^{2^n} b_{a_n,p_n}(x-c_{n,i}),\ x\in\Bbb R,
\quad F_n:= \sum_{m=0}^n f_m.$$ Note that the terms in the sum defining $f_n$ have disjoint supports contained in $[c_{n,i}-3p_{n}/4, c_{n,i}+3p_{n}/4]\subset
I_{n,i},$ ($J_{n,i}$ does not contain the support of the corresponding term in $f_n;$ it is only a place, where that term coincides with a quadratical polynomial) so that $|f'_n(x)| \le
C_3a_n/p_n$. The function $F=\lim_{n\to\infty} F_n$ will be of class $\mathcal C^1$ if $$\label{condc1} \sum_{n=0}^\infty\frac{a_n}{p_n} < \infty.$$ Also, note that $$|F''_n(x)|\le |F''_{n-1}(x)| + C_4 \frac{a_n}{p_n^2}\mbox{, so } \sup
|F''_n| \le C_4 \sum_{m=1}^n \frac{a_m}{p_m^2}.$$
From now on we choose $$\label{geom} \frac{a_n}{p_n^2} = BA^n\mbox{, for some }A>1, B>0
\mbox{ to be determined.}$$ We then have $\sup |F''_n| \le C_4 B A^{n+1}/(A-1)$.
All the successive terms $f_m, m>n,$ are supported on intervals of the form $I_{m,j}$, thus vanish on the interval $J_{n,i}$, so on those intervals $F$ is a smooth function and $$F''=F''_n=F''_{n-1}-8\frac{a_n}{p_n^2}\le C_4\frac{BA^{n}}{A-1}-8 B A^n;$$ therefore, if we choose $$\label{condA} A>1+\frac{C_4}4,$$ we have $F''(x)\le -4B A^n$ for all $x\in J_{n,i}$, and $1\le i
\le 2^n$.
Set $\mathcal U := \bigcup_{n,i} J_{n,i}^\circ$. We have seen that $| I_{n+1,i}|<|I_{n,j}|/2$ (and those quantities do not depend on $i$ or $j$), so that the complement of $\mathcal U$ has empty interior. This proves claim (ii), by choosing $B=C_2/4$. The other claims are clear from the form of the function $F$, once we provide the sequences $(a_n)$ and $(p_n)$ satisfying , , , and .
Let $a_n:= a_0 \gamma^n$, $p_n=p_0\delta^n$. Then is satisfied by construction and $a_0 = Bp_0^2$. Fix $\delta,p_0\in(0,1/2).$ It follows that $p_{n}<| I_{n,i}|/4$ for all $n$ (by an easy induction). Hence, holds.
By our explicit form, means that $\gamma \delta^{-2}
> 1+\frac{C_4}4$, while means $\gamma \delta^{-1} <
1$, so with $\delta^{-1} > 1+\frac{C_4}4$, it is easy to choose $\gamma$. Finally $\|F\|_\infty \le a_0 (1-\gamma)^{-1} < C_1$ for $a_0$ small enough, which can be achieved by decreasing $p_0$ further.
Given any $\varepsilon >0$, we can modify the choices of $\delta$ and $\gamma$ to obtain that $F' \in \Lambda_{1-\eps}$ (the Hölder class of order $1-\eps$). Given any two points $x,y \in [-1,+1]$ and any integer $n \ge 1$, $$|F'(x) - F'(y)| \le |x-y| \| F''_n \|_\infty + 2 \sum_{m\ge n} \|
f'_m\|_\infty$$ $$\le C \left( (\gamma \delta^{-2})^n |x-y| +
(\gamma \delta^{-1})^n \right),$$ where $C>0$ is a positive constant depending on the parameters we have chosen. Take $n$ such that $\delta |x-y| \le \delta^n \le
|x-y| $. Then $$\frac{|F'(x) - F'(y)|}{|x-y|^{1-\eps}} \le C' (\gamma
\delta^{-2+\eps})^n,$$ and it will be enough to choose $\delta$ and $\gamma$ so that $\gamma \delta^{-2+\eps}\le 1$ and $\gamma \delta^{-2} >
1+\frac{C_4}4$, which can be achieved once we pick $\delta$ small enough. The rest of the parameters are then chosen as above.
[*Remark.*]{} It is clear that $F$ cannot be of class $\mathcal C^2(\Bbb R)$. We do not know if our argument can be pushed to get $F\in \mathcal C^{1,1}(\Bbb R).$
\[6\] For any $\eps\in(0,1)$ there exists a non-pseudoconvex bounded $\mathcal C^{1, 1-\eps}$-smooth domain $D\subset \CC^2$ boundary such that $\partial D$ contains a dense subset of points of strict pseudoconvexity.
We start with the unit ball and cave it in somewhat at the North Pole to get an open set of points of strict pseudoconcavity on the boundary. Let $r_0 < 1/3$ and for $x \in [0,1)$, $$\psi_0 (x) = \min\{\log (1-x^2), x^2-r_0^2\}. \footnote{Note that
the graphs of both functions cut inside the interval
$(r_0/2,r_0).$ Indeed, $x^2-r_0^2>\log(1-x^2)$ for $x\ge r_0^2$
and $x^2-r_0^2<\log(1-x^2)$ for $x\le r_0^2/2.$}$$ We take $\psi$ a $\mathcal C^\infty$ regularization of $\psi_0$ such that $\psi=\psi_0$ outside of $(r_0/2,r_0)$. Consider the Hartogs domain $$D_0 := \left\{ (z,w) \in \CC^2 : |z|<1, \log |w| < \frac12 \psi
(|z|) \right\}.$$ Notice that $D_0 \setminus \{ |z| \le r_0 \} = \BB_2 \setminus \{
|z| \le r_0 \}$, so that $\partial D$ is smooth near $|z|=1$.
Now define $\Phi(z)=\Phi (x+iy) = F(x/r_0) \chi (y/r_0)$, where $F$ is the function obtained in Lemma \[5\], and $\chi$ is a smooth, even cut-off function on $\RR$ such that $0\le \chi \le 1$, $\mbox{supp}\, \chi \subset (-2,2)$, and $\chi\equiv 1$ on $[-1,1]$. We define $$D:= \left\{ (z,w) \in \CC^2 : |z|<1, \log |w| < \frac12 \psi (|z|)
+ \Phi (z) \right\}.$$ Recall that for a Hartogs domain $\{\log|w|<\phi(z), |z|<1\},$ if $\phi$ is of class $\mathcal C^2$ at $z_0$, a boundary point $(z_0,w_0)$ with $|z_0|<1$ is strictly pseudoconvex (respectively, strictly pseudoconcave) if and only if $\Delta \phi (z_0) <0$ (respectively, $\Delta \phi (z_0)>0$). Choosing an appropriate regularization (convolution by a smooth positive kernel of small enough support), we may get that:
- $\Delta \psi (|z|)\le-4$ for $|z|\ge r_0$,
- $\Delta \psi (|z|)=4$ for $|z|\le r_0/2$, and is always $\le 4$.
We consider points $z_0=x+iy$. If $|x|>r_0$, $\Phi (z_0)=0$ and we have pseudoconvex points (the boundary is a portion of the boundary of the ball).
On the other hand, when $x \in r_0 \mathcal U$ (where $\mathcal U$ is the dense open set defined in Lemma \[5\]), $$\Delta \Phi (z_0) = \frac1{r_0^2} \Bigl( F''(x/r_0) \chi (y/r_0) +
F(x/r_0) \chi'' (y/r_0) \Bigr).$$ The only values of $z_0$ for which $F(x/r_0) \chi'' (y/r_0) \neq
0$ or $\chi (y/r_0) <1$ verify $|z_0|>r_0$, and at those points we have, using the fact that $F''(x/r_0) <0$, $$\frac12 \Delta \psi (|z_0|) + \Delta \Phi (z_0) \le -4 +
\frac1{r_0^2} C_1 \|\chi''\|_\infty \le -1$$ if we choose $C_1$ small enough. Hence we have strict pseudoconvexity again.
So we may restrict attention to $|y|\le r_0$ and $\Delta \Phi (z_0)
= F''(x/r_0) /r_0^2.$ Therefore $$\frac12 \Delta \psi (|z_0|) + \Delta \Phi (z_0) \le 2 - C_2/r_0^2
< - 2$$ for a $C_2$ chosen large enough.
Finally, notice that points $(z_0,w_0)$ with $|z_0|<r_0/2$ and $F(x)=0$ verify $(z_0,w_0) \in \partial D_0 \cap
\partial D$, $ D_0 \subset D$, and $D_0$ is strictly pseudoconcave at $(z_0,w_0)$, so $D$ is as well.
[*Proof of Proposition \[1\].*]{} Let $D$ be the domain from Lemma \[6\]. We may choose a dense countable subset $(a_j)\subset\partial D$ of points of strict pseudoconvexity. For any $j,$ there is a negative function $u_j\in\PSH(D)$ with $\lim_{z\to a_j}u_j(z)=0.$ If $(D_j)$ is an exhaustion of $D$ such that $D_j\Subset
D_{j+1}$ and $m_j=-\sup_{D_j}{u_j},$ then it is enough to take $u$ to be the upper semicontinuous regularization of $\sup_j u_j/m_j.$
Proofs of Propositions 2, 3 and 4
=================================
[*Proof of Proposition \[2\].*]{} We may assume that $D$ has a global defining function $r:U\to\RR$ with $U=U(\partial
D)$, $r\in\mathcal{C}^2(U)$, and $\grad r\neq 0$ on $U$, such that $D\cap U=\{z\in U:r(z)<0\}$.
Now assume the contrary. Then we may find a point $z^0\in\partial
D$ such that the Levi form of $r$ at $z^0$ is not positive semidefinite on the complex tangent hyperplane to $\partial D$ at $z_0.$ Therefore, there is a complex tangent vector $a$ with $\L
r(z_0,a)\le-2c<0$, where $\L r(z_0,a)$ denotes its Levi form at $z^0$ in direction of $a$. Moreover, we may assume that $|\frac{\partial r}{\partial z_1}(z_0)|\geq 2c.$
Now choose $V=V(z^0)\subset U$ and $u\in\PSH(D\cap V)$ with $$\limsup_{D\cap V\owns z\to z_0}u(z)=\infty;$$ in particular, there is a sequence of points $D\cap V\ni b^j\to z_0$ such that $u(b^j)\to\infty$.
By the $\mathcal{C}^2$-smooth assumption, there is an $\eps_0>0$ such that for all $z\in\BB(z_0,\eps_0)\subset V$ and all $\tilde
a\in\BB(a,\eps_0)$ we have
$$\L r(z,\tilde a)\leq -c,\quad |\frac{\partial r}{\partial
z_1}(z)|\geq c.$$
Now fix an arbitrary boundary point $z\in\partial
D\cap\BB(z_0,\eps_0)$. Define $$a(z):=a+(-\frac{\sum_{j=1}^n a_j\frac{\partial r}{\partial
z_j}(z)}{\frac{\partial r}{\partial z_1}(z)},0,\dots,0).$$ Observe that this vector is a complex tangent vector at $z$ and $a(z)\in\BB(a,\eps_0)$ if $z\in\BB(z_0,\eps_1)$ for a sufficiently small $\eps_1<\eps_0$.
Now, let $z\in\partial D\cap\BB(z_0,\eps_1)$. Put $$b_1(z):=\frac{\L r(z,a(z))}{2\frac{\partial r}{\partial z_1}(z)}$$ and $$\phi_z(\lambda)=z+\lambda a+(\lambda a_1(z)+\lambda^2
b_1(z),0,\dots,0),\quad \lambda\in\CC.$$
Moreover, if $\eps_1$ is sufficiently small, we may find $\delta,
t_0>0$ such that for all $z\in\partial D\cap\BB(z_0,\eps_1)$ we have $$\overline D\cap \BB(z,\delta)-t\nu(z)\subset D,\quad 0<t\leq t_0,$$ where $\nu(z)$ denotes the outer unit normal vector of $D$ at $z$.
Next using the Taylor expansion of $\phi_z$, $z\in\partial
D\cap\BB(z_0,\eps_1)$, $\eps_1$ sufficiently small, we get $$r\circ\phi_z(\lambda)=|\lambda|^2\Bigl(\L
r(z,a(z))+\eps(z,\lambda\Bigr),$$ where $|\eps(z,\lambda)|\leq \eps(\lambda)\to 0$ if $\lambda\to
0$.
In particular, $\phi_z(\lambda)\in \BB(z,\delta)\cap D\subset
V\cap D$ when $0<|\lambda|\leq \delta_0$ for a certain positive $\delta_0$ and $r\circ\phi_z(\lambda)\leq -\delta_0^2c/2$ when $|\lambda|=\delta_0$.
Hence, $K:=\bigcup_{z\in\partial D\cap \BB(z_0,\eps_1),
|\lambda|=\delta_0}\phi_z(\lambda)\Subset D\cup V$. Choose an open set $W=W(K)\Subset D\cap V$. Then $u\leq M$ on $W$ for a positive $M$.
Finally, choose a $j_0$ such that $b^j=z^j-t_j\nu(z^j)$, $j\geq
j_0$, where $z^j\in\partial D\cap\BB(z_0,\eps_1)$, $0<t_j\leq
t_0$, and $\phi_{z^j}(\lambda)\in W$ when $|\lambda|=\delta_0$. Therefore, by construction, $u(b^j)\leq M$, which contradicts the assumption.
[*Proof of Proposition \[3\].*]{} Assume that $D$ is not pseudoconvex. Then, by Corollary 4.1.26 in [@Hor], there is $\varphi\in\mathcal O(\Bbb D, D)$ such $\dist(\varphi(0),\partial D)<\dist(\varphi(\zeta),\partial D)$ for any $\zeta\in\Bbb D_\ast.$ To get a contradiction, it remains to use similar arguments as in the previous proof and we skip the details.
[*Proof of Proposition \[4\].*]{} It is enough to show that if $\mathcal O(\Bbb D,D)\ni\psi_j\to\psi$ and $\psi(\zeta)\in\partial D$ for some $\zeta\in\Bbb D,$ then $\psi(\Bbb D)\subset\partial D.$ Suppose the contrary. Then it is easy to find points $\eta_k\to\eta\in\Bbb D$ such that $\psi(\eta_k)\in D$ but $a=\psi(\eta)\in\partial D.$ We may assume that $\eta=0$ and $g_a=\frac{1}{f_a}$ is bounded on $D\cap U_a.$ Let $r\in(0,1)$ be such that $\psi(r\Bbb D)\Subset U_a.$ Then $\psi_j(r\Bbb D)\subset U_a$ for any $j\ge j_0.$ Hence $|g_a\circ\psi_j|<1$ and we may assume that $g_a\circ\psi_j\to
h_a\in\mathcal O(r\Bbb D,\Bbb C).$ Since $h_a(\eta)=0,$ it follows by the Hurwitz theorem that $h_a=0.$ This contradicts the fact that $h_a(\eta_k)=g_a\circ\psi(\eta_k)\neq 0$ for $|\eta_k|<r.$
Proofs of Propositions 5, 6 and 7
=================================
[*Proof of Proposition \[10\].*]{} Our aim is to construct a non-pseudoconvex bounded domain $D\subset\CC^2$ such that $\limsup_{z\to a}K_D(z)=\infty$ for any $a\in\partial D$.
Let us start with the domain $P\times\Bbb D$, where $P=\{\lambda\in\Bbb C:\frac{1}{2}<|\lambda|<\frac{3}{2}\}$. Let $$S:=\{(z_1,z_2)=(x_1+iy_1,z_2)\in
P\times\DD:(x_1-1)^2+\frac{1+|z_2|^2}{1-|z_2|^2}y_1^2=\frac{1}{4},y_1>0\}.$$ Define $D:=(P\times\DD)\setminus S$. Note that $D$ is a domain. Its envelope of holomorphy is non-schlicht and consists of the union of $D$ and one additional ’copy’ of the set $$D_1:=\{(z_1,z_2)\in
P\times\DD:(x_1-1)^2+\frac{1+|z_2|^2}{1-|z_2|^2}y_1^2\leq\frac{1}{4},y_1>0\}.$$ In particular, $D$ is not pseudoconvex. Note that convexity of the the interior $D^0$ of $D_1$ implies that $\lim_{z\to\partial
D_1}K_{D^0}(z)=\infty$. Therefore, it follows from the localization result for the Bergman kernel due to Diederich-Fornaess-Herbort formulated for Riemann domains in the paper [@Irg1] that for all $a\in S\subset\partial D_1$ the following property holds: $\lim_{D\cap D_1\owns z \to a}K_D(z)=\infty$ (on the other hand while tending to the points from $S$ from the ’other side’ of the domain $D$ the Bergman kernel is bounded from above). Obviously $P\times\DD$ is Bergman exhaustive, so for any $a\in\partial(P\times\DD)$ the following equality holds $\lim_{z\to
a}K_D(z)=\infty$.
[*Proof of Proposition \[11\].*]{} Recall the following facts that follow from [@Bre].
If the envelope of holomorphy $\hat D$ of the domain $D$ is a domain in $\Bbb C^n$ (is schlicht) then the Bergman kernel $K_D$ extends to a real analytic function $\tilde K_D$ defined on $\hat D$.
Let $\emptyset\neq P_0\subset D$, $P_0\subset P$, $P\setminus
D\neq\emptyset$ and $\bar P_0\cap (\Bbb C^n\setminus
D)\neq\emptyset$, where $P_0,P$ are polydiscs, and the following property is satisfied: for any $f\in\O(D)$ there is a function $\tilde f\in\O(P)$ such that $f=\tilde f$ on $P_0$. Then the Bergman kernel $K_D$ extends to a real analytic function on $P$. More precisely, there is a real analytic function $\tilde K_D$ defined on $P$ such that $\tilde K_D(z)=K_D(z)$, $z\in P_0$.
Both facts above complete the proof of Proposition \[11\].
The proof of Proposition \[12\] is essentially contained in [@Irg2]. However, this PhD Thesis is not publically accessible. Therefore we repeat it here. The idea is the following: if $\limsup_{z\to a}K_D(z)=\infty$ for some $a\in\partial D,$ then there is an $f\in L_h^2(D)$ such that $\limsup_{z\to
a}|f(z)|=\infty$.
[*Proof of Proposition \[12\].*]{} In view of Proposition 5, $\limsup_{z\to a}K_D(z)=\infty$ for any $a\in\partial D$.
Let $a\in\partial D$. We claim that there is an $L_h^2(D)$-function $h$ which is unbounded near $a$.
Assume the contrary. Hence for any $f\in L_h^2(D)$ there exists a neighborhood $U_f$ of $a$ and a number $M_f$ such that $|f|\leq M_f$ on $D\cap U_f.$
Denote by $L$ the unit ball in $L_h^2(D)$ and by $c=\pi^n$.
Let $K_1:=\{z\in D:\dist(z,\partial D)\geq 1\}$ (if this is empty take a smaller number than $1$). By the meanvalue inequality we have for any $f\in L$ that $|f|\leq c$ on $K_1$. By assumption, there are $z_1\in D$ and $f_1\in L$ such that $|z_1-a|<1$ and $|f_1(z_1)|>c$.
Set $g_1:=f_1/c$. Then $g\in L$ and therefore there are a neighborhood $U_1$ of $a$ and number $M_1>1$ such that $|g_1|\leq
M_1$ on $D\cap U_1$.
Set $K_2:=\{z\in D:\dist(z,\partial D)\geq\dist(z_1,\partial D)\}$ and $d=c\dist(z_1,\partial D).$ Then $K_1\subset K_2$. Choose $z_2\in U_1\cap D$, $z_2\notin K_2,$ $|z_2-a|<1/2,$ and $f_2\in L$ with $|f_2(z_2)|\geq d(1^3+1^2M_1)$. Moreover, $|f_2|\leq d$ on $K_2$. Put $g_2:=f_2/d$. Then $g_2\in L$. Choose now a neighborhood $U_2$ of $a$ and a number $M_2$ such that $|g_2|\leq M_2$ on $D\cap
U_2$.
Then we continue this process.
So we have points $z_k\in K_{k-1}$, $z_k\notin K_{k-1}$, $|z_k-a|<1/k,$ and functions $f_k\in L$ with $$|f_k(z_k)|\geq c\dist(z_{k-1},\partial
D)^n(k^3+k^2\sum_{j=1}^{k-1}M_j).$$
Setting $g_k:=f_k/d$ and $h:=\sum_{j=1}^\infty g_j/j^2,$ it is clear that $h\in L_h^2(D)$.
Fix now $k\ge 2$. Then $$|h(z_k)|\geq
\frac{|g_k(z_k)|}{k^2}-\sum_{j=1}^{k-1}\frac{|g_j(z)|}{j^2}-
\sum_{j=k+1}^\infty\frac{|g_j(z)|}{j^2}$$ $$\ge
k+\sum_{j=1}^{k-1}M_j-\sum_{j=1}^{k-1}\frac{M_j}{j^2}-\sum_{j=k+1}^\infty
\frac{1}{j^2}>k-\frac{1}{6}.$$ In particular, $h$ is unbounded at $a$ which is a contradiction.
It remains to choose a dense countable sequence $(a_j)\subset\partial D$ such that any term repeats infinitely many times and to copy the proof of the Cartan-Thullen theorem.
H. J. Bremermann, [*Holomorphic continuation of the kernel function and the Bergman metric in several complex variables*]{} in “Lectures on functions of several complex variables”, Univ. of Mich. Press, 1955, pp. 349–383.
L. Hörmander, [*Notions of convexity*]{}, Birkhäuser, Basel–Boston–Berlin, 1994.
M. Irgens, [*Extension properties of square integrable holomorphic functions*]{}, PhD Thesis, Michigan University, 2002.
M. Irgens, [*Continuation of $L^2$-holomorphic functions*]{}, Math. Z. 247 (2004), 611–617.
D. Jacquet, [*On complex convexity,*]{} Doctoral Dissertation, University of Stockholm, 2008.
P. Pflug, W. Zwonek, [*$L\sp 2\sb h$-domains of holomorphy and the Bergman kernel*]{}, Studia Math. 151 (2002), 99–108.
[^1]: This means that $\mathcal O(\Bbb D,D)$ is a normal family, where $\Bbb D\subset\Bbb C$ is the open unit disc. Note that any taut domain is pseudoconvex and any bounded pseudoconvex domain with $C^1$-smooth boundary is taut.
| ArXiv |
---
abstract: |
We investigate the redshift dependence of X-ray cluster scaling relations drawn from three hydrodynamic simulations of the $\Lambda$CDM cosmology: a [*Radiative*]{} model that incorporates radiative cooling of the gas, a [*Preheating*]{} model that additionally heats the gas uniformly at high redshift, and a [*Feedback*]{} model that self-consistently heats cold gas in proportion to its local star-formation rate. While all three models are capable of reproducing the observed local - relation, they predict substantially different results at high redshift (to $z=1.5$), with the [*Radiative*]{}, [*Preheating*]{} and [*Feedback*]{} models predicting strongly positive, mildly positive and mildly negative evolution, respectively.
The physical explanation for these differences lies in the structure of the intracluster medium. All three models predict significant temperature fluctuations at any given radius due to the presence of cool subclumps and, in the case of the [*Feedback*]{} simulation, reheated gas. The mean gas temperature lies above th e dynamical temperature of the halo for all models at $z=0$, but differs between models at higher redshift with the [*Radiative*]{} model having the lowest mean gaswos temperature at $z=1.5$.
We have not attempted to model the scaling relations in a manner that mimics the observational selection effects, nor has a consistent observational picture yet emerged. Nevertheless, evolution of the scaling relations promises to be a powerful probe of the physics of entropy generation in clusters. First indications are that early, widespread heating is favored over an extended period of heating that is associated with galaxy formation.
author:
- 'Orrarujee Muanwong, Scott T. Kay and Peter A. Thomas'
title: 'Evolution of X-ray cluster scaling relations in simulations with radiative cooling and non-gravitational heating'
---
Introduction {#sec:introduction}
============
X-ray scaling relations of galaxy clusters, namely the temperature–mass, -$M$, relation and the luminosity–temperature, -, relation, play a pivotal role when using the abundance of clusters to constrain cosmological parameters [@HA91; @WEF93; @Eke96; @VL96; @VL99; @Henry97; @Henry00; @Borgani01; @Pierpaoli01; @Seljak02; @Pierpaoli03; @Viana03; @Allen03; @Henry04]. It is well known, however, that accurate calibration of scaling relations is crucial to avoid a major source of systematic error. For example, the $-M$ relation is widely used by many of these authors to constrain the amplitude of mass fluctuations, conventionally defined using the parameter, $\sigma_8$. Systematic deviations in the normalization of the $-M$ relation, particularly due to how cluster mass is estimated (e.g. see @HMS99) is amplified by the steep slope of the temperature function, leading to large variations in $\sigma_8$ (see @Henry04 for a discussion of recent results).
As far as the - relation is concerned, the discrepancies are more prominent as is highly sensitive to the thermodynamics of the of the inner intracluster medium (ICM), and can yield different values for both normalizations and slopes [@EdS91; @WJF97; @AlF98; @Mar98; @XuW00]. The situation is further complicated by the fact that clusters do not scale self-similarly, as would be the case (approximately) if the only source of heating was via gravitational infall [@Kaiser86]. This makes the problem more difficult to investigate theoretically, although it allows studies of cluster scaling relations to reveal more information on the physics governing the structure of the intracluster medium.
The departure from self-similarity can be attributed to an increase in the [*entropy*]{} of the gas that particularly affects low-mass systems [@EH91; @Kaiser91; @Bower97; @TN01; @PCN99; @VB01; @Voit02; @Voit03]. Many theoretical studies have been performed to investigate the effects of various physical processes that can raise the entropy of the gas, based on models involving heating [@ME94; @Balogh99; @KY00; @Low00; @WFN00; @Bower01; @Borgani02] , radiative cooling [@KP97; @Pearce00; @Bryan00; @Muanwong01; @Muanwong02; @DKW02; @WX02], and a combination of the two [@Muanwong02; @KTT03; @Tornatore03; @Valdarnini03; @Borgani04; @Kay04; @McCarthy04].
Measurements of how cluster scaling relations evolve with redshift allow even tighter constraints to be placed on cosmological parameters (and entropy generation models), and observations of cluster properties at high redshift are now starting to become available, owing primarily to the high sensitivity of [*Chandra*]{} and [*XMM–Newton*]{}. From a theoretical point of view, this is an exciting phase as we can now fully exploit the availability of our simulated distant clusters and compare their X-ray properties with real observations. It is therefore timely to investigate further the effects of entropy generation on the evolution of clu ster scaling relations as the available data for high-redshift systems accumulates.
In this paper, we will use cosmological hydrodynamical simulations described in @Muanwong02, hereafter MTKP02, and in @Kay04, hereafter KTJP04, to trace the evolution of the cluster population to high redshift ($z=1.5$). Our results will primarily focus on three ([*Radiative*]{}, [*Preheating*]{} and [*Feedback*]{}) models, all able to reproduce the local - relation. The aims of this paper are to determine how the scaling relations evolve with redshift in the three models and to discover what the evolution of scaling relations can teach us about non-gravitational processes occurring in clusters.
The rest of this paper is outlined as follows. In Section \[sec:srel\] we introduce the X-ray scaling relations and summarize our present observational knowledge of these quantities. Details of our simulated cluster populations are presented in Section \[sec:sims\]. In Section \[sec:results\] we present our main results, first at $z=0$, where the models are in good agreement with each other and the observations, then as a function of redshift, where the models predict widely different results. We discuss the implications of these differences in Section \[sec:discuss\] and demonstrate that the degree of X-ray evolution is driven by the supply of cold, low entropy gas. Finally, we summarize our conclusions in Section \[sec:conclude\].
X-ray cluster scaling relations {#sec:srel}
===============================
@Kaiser86 derived the following relations for temperature $${\mbox{$T_{\rm X}$}}\propto M^{2 \over 3} \, (1+z),
\label{eqn:tmrel}$$ and luminosity $$\begin{aligned}
{\mbox{$L_{\rm X}$}}&\propto& M^{4 \over 3} \, (1+z)^{7 \over 2} \label{eqn:lmrel} \\
&\propto& {\mbox{$T_{\rm X}$}}^{2} \, (1+z)^{3 \over 2} \label{eqn:ltrel},\end{aligned}$$ assuming the distribution of gas and dark matter in clusters is perfectly self-similar and the X-ray emission is primarily thermal bremsstrahlung radiation. Observed clusters do not form a self-similar population but it is nevertheless convenient to describe their behavior using a generalized power-law form $$Y = C_0(z) \, X^\alpha = Y_0 \, X^\alpha \, (1+z)^{A},
\label{eqn:powerlaw}$$ where $C_0(z)$ and $Y_0$ determine the normalization, $\alpha$ is the slope of the relation (in log-space) and $A$ determines how the relation evolves with redshift. Our main results will focus on the determination of $A$.
Observationally, attempts to measure the $-M$ relation at high redshift are currently in their infancy, as they require temperature profiles to be measured so that their mass can be estimated, but initial results are consistent with self-similar evolution ($A \sim
1$, @Maughan05 [@Kotov05]).
Measuring the - relation at higher redshift is a somewhat simpler prospect, and has been attempted by many authors [@MS97; @Fairley00; @Holden02; @Novicki02; @Arnaud02; @Vikhlinin02; @Lumb04; @Ettori04; @Maughan05; @Kotov05]. We summarize recent results that adopt a low-density flat cosmology in Figure \[fig:ltevolobs\], attempting to include in the size of the error bars the uncertainty in $A$ due to the choice of local relation (when quoted by the authors). Although the present situation is by no means clear, taking all results at face value generally favors positive evolution ($0 {\mathrel{{{\hbox to 0pt{\lower 3pt\hbox{$\sim$}\hss}}}
\raise 2.0pt\hbox{$<$}}}A {\mathrel{{{\hbox to 0pt{\lower 3pt\hbox{$\sim$}\hss}}}
\raise 2.0pt\hbox{$<$}}}2$) with the latest results being consistent with sel f-similar evolution ($A=3/2$). Larger samples of high redshift clusters (such as that expected from the [*XMM-Newton*]{} Cluster Survey, @Romer01) will be crucial to accurately constrain the degree of evolution in the - relation.
Simulated cluster populations {#sec:sims}
=============================
Our results are drawn from three similarly-sized $N$-body/SPH simulations of the $\Lambda$CDM cosmology, which have already been published in MTKP02 and KTJP04. The simulation box in MTKP02 has a comoving side of $100{\mbox{$\, h^{-1}{{\rm {Mpc}}}$}}$ with $160^3$ particles each of gas and dark matter, whose particle masses are set to $2.6\times 10^9$ and $2.1\times 10^{10}{\, h^{-1} {{\, \mbox{M$_\odot$}}}}$, respectively. The box used in KTJP04 is bigger with a side of $120{\mbox{$\, h^{-1}{{\rm {Mpc}}}$}}$ using $256^3$ particles each of gas and dark matter, whose particle masses are $1.3\times 10^9$ and $7.3\times 10^9{\, h^{-1} {{\, \mbox{M$_\odot$}}}}$, respectively. Full details can be found in the articles. The key difference between the simulations is the model used to raise the entropy of the intracluster gas, summarized as follows:
1. A [*Radiative*]{} model where the excess entropy originated from the removal of low entropy gas to form stars, causing higher entropy gas to flow adiabatically into the core from larger radii (MTKP02).
2. A [*Preheating*]{} model where entropy was generated impulsively by uniformly heating the gas by 1.5 keV per particle at $z=4$ (MTKP02).
3. A [*Feedback*]{} model where the entropy of (on average) 10 per cent of cooled gas in high density regions was raised by 1000 keV cm$^2$, mimicking the effects of heating due to stars and active galactic nuclei (KTJP04).
These three models differ in the timing and distribution of entropy generation in the intracluster medium. The [*Radiative*]{} model has no explicit feedback of energy but relies on the removal of low-entropy gas via cooling; as such it represents a minimal heating model. The [*Preheating*]{} model contains distributed heating at high redshift such as might occur if entropy generation occurs mainly in low-mass galaxies. By contrast heating in the [*Feedback*]{} model occurs solely in high-density regions. In all our models, there is very little star formation before a redshift of z=4 after which the star-formation rate (sfr) begins to rise rapidly. In the [*Preheating*]{} simulation the sfr is then strongly suppressed, whereas in the other two simulations it peaks at a redshift of z=2 and then declines back down to low values by the present day with a time-variation that matches that of the star-formation history of the Universe. The global baryon fraction in stars (and cold gas) at z=0 is 0.002, 0.076 and 0.127 in the [*Preheating*]{}, [*Radiative*]{} and [*Feedback*]{} simulations, respectively. The largest of these corresponds to a stellar mass density of $\Omega_*$=0.006; thus none of the models has excessive star-formation. These models are far from exhaustive and their precise details should not be taken too seriously. The purpose of this paper is not to examine particular models but to illustrate that the evolution of the X-ray scaling relations can provide a powerful discriminant between different classes of model.
Cluster identification and properties
-------------------------------------
Clusters were selected at four redshifts ($z=$0, 0.5, 1 & 1.5) using the procedure outlined in MTKP02. They are defined to be spheres of matter, centered on the dark matter density maximum, with total mass $$M_{\Delta} = {4 \over 3} \pi R_{\Delta}^{3} \, \Delta \, \rho_{\rm c0}
\, (1+z)^{3},
\label{eqn:mass}$$ where $\rho_{\rm c0}=3H_0^2/8\pi G$ is the critical density at $z=0$. We set $\Delta=500$ as it corresponds to a sufficiently large radius such that the results are not dominated by the core, as well as corresponding approximately to the extent of current X-ray observations. Furthermore, as was shown by @Rowley04, the X-ray properties of simulated clusters within an overdensity of 500 exhibit less scatter than within the virial radius. Our choice of scaling with redshift[^1] is independent of cosmology and would allow the simple power-law scalings to be recovered (equations \[eqn:tmrel\],\[eqn:lmrel\] & \[eqn:ltrel\]) if the clusters were structurally self-similar.
We consider scaling relations involving mass, three measures of temperature, and luminosity, for particle properties averaged within $R_{500}$. The mass, $$M_{500}=\sum_i m_i,$$ where the sum runs over all particles, of mass $m_i$. The dynamical temperature, $$kT_{\rm dyn} = \, { \sum_{i,{\rm gas}} m_i k T_i \, + \, \alpha
\sum_i {1 \over 2} m_i v_i^2 \over \sum_i m_i},
\label{eqn:tdyn}$$ where $\alpha=(2/3)\mu m_{\rm H} \sim 4.2\times 10^{-16}$ keV for a fully ionized primordial plasma, assuming the ratio of specific heats for a monatomic ideal gas, $\gamma=5/3$, and the mean atomic weight of a zero metalicity gas, $\mu m_{\rm H}=10^{-24} {\rm g}$. The first sum in the numerator runs over all gas particles, of temperature, $T_i$, whereas the second sum runs over particles of all types, of speed $v_i$ as measured in the center of momentum frame of the cluster.
We also consider the mass-weighted temperature of hot ($T>10^{5}$K) gas, $$kT_{\rm gas} = { \sum_{i,{\rm hot}} m_i k T_i \over
\sum_{i,{\rm hot}} m_i},
\label{eqn:tgas}$$ and we approximate the X-ray temperature of a cluster using the bolometric emission-weighted temperature, $$kT_{\rm bol} = { \sum_{i,{\rm hot}} m_i \rho_i \Lambda_{\rm
bol}(T_i,Z) T_i \over \sum_{i,{\rm hot}} m_i \rho_i
\Lambda_{\rm bol}(T_i,Z) },
\label{eqn:tbol}$$ where $\rho_i$ is the density and $\Lambda_{\rm bol}$ is the bolometric cooling function used in our simulations [@SD93]; for the [*Radiative*]{} and [*Preheating*]{} runs, $Z=0.3(t/t_0)Z_{\odot}$ (MTKP02), and for the [*Feedback*]{} run, $Z=0.3Z_{\odot}$ (KTJP04). Finally, the X-ray luminosity is approximated by the bolometric emission-weighted luminosity $$L_{\rm bol} = \sum_{i,{\rm hot}}
{m_i \rho_i \Lambda_{\rm bol}(T_i,Z)
\over (\mu m_{\rm H})^2 }.$$
It has been shown recently that the emission-weighted temperature is not an accurate diagnostic of cluster temperature, overpredicting the [*spectroscopic*]{} temperature by $\sim 20-30$ per cent when the emission is predominantly thermal bremsstrahlung [@Mazzotta04; @Rasia05]. At lower temperatures ($kT<3$keV), line emission from heavy elements makes the problem significantly more complicated [@Vikhlinin05]. The volume sampled by our simulations ($\sim 100{\mbox{$\, h^{-1}{{\rm {Mpc}}}$}}$) means that we have very few clusters with $T>3$keV, and so a more accurate measure of the cluster temperature would require significantly more effort than applying a simple formula to our data. We therefore leave such improvements to future work, when larger samples of simulated clusters are available. It would not affect the conclusions of this paper.
Cluster catalogues
------------------
------------------ --------------------------- ----- ----- ----- -----
Model Relation 0.0 0.5 1.0 1.5
[*Radiative*]{} Total 340 190 85 31
$T_{\rm dyn}-M_{500}$ 330 186 84 31
$T_{\rm gas}-M_{500}$ 332 186 82 31
$T_{\rm bol}-M_{500}$ 319 151 64 24
$L_{\rm bol}-M_{500}$ 317 186 85 31
$L_{\rm bol}-T_{\rm bol}$ 256 95 34 14
[*Preheating*]{} Total 283 147 59 22
$T_{\rm dyn}-M_{500}$ 273 143 56 22
$T_{\rm gas}-M_{500}$ 271 143 56 22
$T_{\rm bol}-M_{500}$ 264 134 53 22
$L_{\rm bol}-M_{500}$ 269 143 59 22
$L_{\rm bol}-T_{\rm bol}$ 190 92 48 14
[*Feedback*]{} Total 342 98 45 13
$T_{\rm dyn}-M_{500}$ 328 96 43 12
$T_{\rm gas}-M_{500}$ 327 89 41 11
$T_{\rm bol}-M_{500}$ 305 90 39 10
$L_{\rm bol}-M_{500}$ 339 98 45 13
$L_{\rm bol}-T_{\rm bol}$ 269 67 32 12
------------------ --------------------------- ----- ----- ----- -----
: Numbers of clusters at various redshifts[]{data-label="tab:totnumbers"}
\[tab:clusters\]
Table \[tab:clusters\] lists the numbers of clusters in our catalogues for each of the simulations at all 4 redshifts. The first row for each model gives the total number of clusters in our catalogues, down to a minimum mass, $M_{\rm 500}=1.2\times
10^{13}{\, h^{-1} {{\, \mbox{M$_\odot$}}}}$, corresponding to $\sim 500$ dark matter particles in the [*Radiative*]{} and [*Preheating*]{} simulations, and $\sim 1400$ dark matter particles in the (higher resolution) [*Feedback*]{} simulation. At $z=0$, each model contains around 300 clusters above our mass limit, decreasing by around an order of magnitude by $z=1.5$.
We also made a number of additional cuts to the catalogues, specific to each scaling relation. Firstly, we noted a small number of systems that were significantly offset from the mean relation. On inspection, such objects were found to be erroneous as they were subclumps falling into neighbouring clusters. Thus, for each relation, we discarded all objects with $\Delta \log (Y)>0.1$, larger than intrinsic scatter in the $T_{\rm dyn}-M_{\rm 500}$, $T_{\rm gas}-M_{\rm 500}$ and ${T_{\rm bol}-M_{\rm 500}}$ relations; and $\Delta \log (Y)>0.5$ in the $L_{\rm bol}-M_{500}$ and $L_{\rm bol}-T_{\rm bol}$ relations, respectively. Secondly, for the $L_{\rm bol}-T_{\rm
bol}$ relation, we made an additional cut in temperature, such that the catalogues were complete in $T_{\rm bol}$ (excluding those clusters classed as outliers). For the [*Radiative*]{} model, the minimum temperatures are $kT_{\rm
bol,min}=[0.74,1.0,1.25,1.35]$ keV; for the [*Preheating*]{} model, $kT_{\rm bol,min}=[0.70,0.96,1.1,1.37]$ keV; and for the [*Feedback*]{} model, $kT_{\rm bol,min}=[0.59,1.12,1.31,1.58]$ keV, for $z=[0,0.5,1,1.5]$. The numbers of clusters remaining in each of the relations after these cuts are also listed in Table \[tab:clusters\].
Results {#sec:results}
=======
Scaling relations at redshift zero
----------------------------------
Relation Model $\alpha$ $\log C_0(0)$ $\log Y_0$ $A$
--------------------------- ------------------ ---------- --------------- ------------ ------
$T_{\rm dyn}-M_{\rm 500}$ [*Radiative*]{} 0.70 0.34 0.34 1.1
[*Preheating*]{} 0.70 0.33 0.33 1.1
[*Feedback*]{} 0.69 0.33 0.33 1.2
$T_{\rm gas}-M_{\rm 500}$ [*Radiative*]{} 0.61 0.33 0.33 0.9
[*Preheating*]{} 0.61 0.35 0.35 0.9
[*Feedback*]{} 0.61 0.35 0.35 1.1
$T_{\rm bol}-M_{\rm 500}$ [*Radiative*]{} 0.59 0.38 0.37 0.5
[*Preheating*]{} 0.61 0.35 0.35 0.8
[*Feedback*]{} 0.64 0.33 0.33 1.2
$L_{\rm bol}-M_{\rm 500}$ [*Radiative*]{} 1.82 1.36 1.36 3.9
[*Preheating*]{} 1.92 1.40 1.39 3.1
[*Feedback*]{} 2.10 1.40 1.40 3.2
$L_{\rm bol}-T_{\rm bol}$ [*Radiative*]{} 3.06 0.19 0.20 1.9
[*Preheating*]{} 3.05 0.26 0.24 0.7
[*Feedback*]{} 3.13 0.28 0.28 -0.6
: Best-fit scaling relations[]{data-label="tab:evoparams"}
\[tab:bestfit\]
We first present the scaling relations at $z=0$ as they will form the basis for measuring evolution in the cluster properties with redshift. The parameters $\alpha$ and $C_0(0)$ listed in Table \[tab:bestfit\] are determined from the best least-squares fit to the relation $$\log Y=\log C_0(0) + \alpha\log X,$$ where $X$ and $Y$ represent the appropriate data sets in units of $10^{14}
{\, h^{-1} {{\, \mbox{M$_\odot$}}}}$, 1keV and $10^{42}\,h^{-2}\,{\rm erg}\,{\rm s^{-1}}$ for mass, temperature and luminosity, respectively. We will consider each relation in turn.
Figure \[fig:tdynmz0\] illustrates the $T_{\rm dyn}-M_{500}$ relation for each of the three simulations at $z=0$, with best-fit relations overplotted as straight lines. The dynamical temperature is dominated by the contribution from the more massive dark matter particles, and so the resulting three relations are almost identical. The measured slope of the relation is $\alpha \sim 0.7$ (Table \[tab:bestfit\]), close to, but slightly larger than the self-similar value ($\alpha=2/3$); this deviation is due to the variation of concentration with cluster mass. When the mass-weighted temperature of hot gas is used instead, the relation becomes flatter than the self-similar prediction, with $\alpha \sim 0.6$. This is expected as the excess entropy generation due to cooling and heating is more effective in lower mass clusters (MTKP02).
Shown in Figure \[fig:tbolmz0\] is the $T_{\rm bol}-M_{500}$ relation for each of the 3 models. Cool, dense gas dominates $T_{\rm bol}$ and so this temperature is more susceptible to fluctuations caused by merging substructure, leading to an increase in the scatter when compared to Figure \[fig:tdynmz0\]. Again, the slope is flatter than the self-similar prediction, due to the effects of excess entropy. Differences between the models are larger than for the dynamical temperature but are less than the intrinsic scatter.
Finally, we consider relations involving the bolometric luminosity of the cluster. Fitting the relation between luminosity and mass, we find a slope in the range $\alpha \sim 1.8-2.1$, significantly steeper than the self-similar prediction ($\alpha=4/3$). The departure from self-similarity is exacerbated when we plot bolometric luminosity against temperature (Figure \[fig:lbolt0\]). Here, $\alpha \sim 3.1$ in all models, compared to $\alpha=2$ for the self-similar case. The $L_{\rm bol}-T_{\rm bol}$ relations from the three simulations are in reasonable agreement with one another and in good agreement with the observed luminosity–temperature relation (see MTKP02,KTJP04).
In summary, all three models successfully generate excess entropy in order to break self-similarity at the level required by the observations at low redshift ($z \sim 0$). Thus, based on the local scaling relations alone, we cannot easily discriminate between the source of the entropy excess in clusters: whether it is mainly due to radiative cooling, additional uniform heating at high redshift (prior to cluster formation) or localized heating from galaxy formation at all redshifts.
Evolution of scaling relations with redshift
--------------------------------------------
We now examine whether this degeneracy between models in the scaling relations at $z=0$ can be broken by examining the cluster population at higher redshifts ($z=0.5, 1, 1.5$). None of the relations require a significant variation in $\alpha$ with redshift. To make our results easier to interpret, therefore, we use simple power-law relations of the form given in equation \[eqn:powerlaw\] with $\alpha$ fixed at the $z=0$ values given in Table \[tab:bestfit\].
To find the evolution of each relation, we first determine the normalizations, $C_0$, and their corresponding error bars, at each redshift in the same manner as described for redshift zero in Section 4.1 above. We then minimize the $\chi-$squared to obtain parameters $Y_0$ and $A$ as listed in Table \[tab:bestfit\] to fit the relation $$\log C_0 = \log Y_0 + A\log(1+z).
\label{eqn:normz}$$
### Temperature-Mass Evolution
In Figure \[fig:tmevol\], we present values of $\log(C_0)$ versus redshift for the three temperature–mass relations, with the best-fit straight line overplotted. For the $T_{\rm dyn}-M_{500}$ relation (upper panel), we find similar evolution parameters for the three models, $A=1.1-1.2$, confirming that including the effects of baryonic physics does not significantly affect cluster dynamics. The slight excess over the self-similar value of $A=1$ is consistent with the changing cluster concentrations.
However, both the mass-weighted temperature (middle panel) and especially the emission-weighted temperature (lower panel) show significant variation between the three models. In each case the [*Feedback*]{} simulation approximately follows the scaling found for the dynamical temperature, with the [*Preheating*]{} and the [*Radiative*]{} simulations showing progressively larger deviations below the expected normalization as the redshift increases. The explanation for this lies in the variation of temperature of gas particles within each cluster and how this changes with redshift in the different models. We shall explore this further in Section \[sec:discuss\].
### Luminosity-Mass Evolution
Figure \[fig:lmevol\] illustrates the normalization of the $L_{\rm
bol}-M_{500}$ relation versus redshift for all three models. The [*Preheating*]{} and [*Feedback*]{} models evolve almost identically with redshift ($A \sim 3$), but the [*Radiative*]{} run evolves more strongly ($A \sim 4$). These bracket the self-similar value, $A=3.5$, however, this agreement is somewhat coincidental because the slope of the relation at fixed redshift is much steeper than expected ($\alpha\sim1.8$–2.1 rather than 1.3). The reason why the [*Radiative*]{} simulation has steeper evolution is because of enhanced emission from cool gas at high redshift relative that that at low redshift—see discussion in Section \[sec:discuss\].
### Luminosity-Temperature Evolution
Finally, we consider the evolution of the $L_{\rm bol}-T_{\rm bol}$ relation, with the relations at each redshift shown explicitly for each model in Figure \[fig:lbolt4z\], and the variation of normalization with redshift illustrated in Figure \[fig:ltevol\]. It is interesting to note that the values of $A$ are significantly different between all three models: the [*Feedback*]{} model predicts mildly negative evolution ($A=-0.6$), the [*Preheating*]{} mildly positive evolution ($A=0.7$) and the [*Radiative*]{} strongly positive evolution ($A=1.9$). The latter two models straddle the self-similar value ($A=1.5$).
The difference in slopes between the [*Feedback*]{} and [*Preheating*]{}runs is driven by the differences in their temperature. The further difference between the [*Preheating*]{} and [*Radiative*]{} runs comes roughly equally from the temperature and luminosity evolution.
Discussion {#sec:discuss}
==========
In this paper, we have focused on the evolution of cluster scaling relations in three simulations, each adopting a different model for non-gravitational processes that affect the intracluster gas. In the first, [*Radiative*]{} model, the gas could cool radiatively and a significant fraction cooled down to low temperatures and formed stars (MTKP02). In the [*Preheating*]{} model, the same was true, although the gas was additionally heated uniformly and impulsively by 1.5 keV per particle at $z=4$, before cluster formation. In the third, [*Feedback*]{} model, the heating rate was local and quasi-continuous, in proportion to the star-formation rate from cooled gas. All three models are able to generate the required level of excess core entropy in order to reproduce the $L_{\rm X}-T_{\rm X}$ relation at $z=0$ (MTKP02, KTJP04).
The most striking result presented in this paper is that the three models predict widely different $L_{\rm X}-T_{\rm X}$ relations at high redshifts. The [*Radiative*]{} model predicts strongly positive evolution, the [*Preheating*]{} model mildly positive evolution and the [*Feedback*]{} model, mildly negative evolution. At this point, it should be stressed that the values of $A$ presented in Table \[tab:bestfit\] should not be taken too seriously. No attempt has been made to convert the bolometric, emission-weighted fluxes used in this paper to observable X-ray fluxes in different instrumental bands. Also, the volume of our simulation boxes is such that we only get a modest number of relatively poor clusters at high redshift. Nevertheless, the qualitative difference between the models is very encouraging and suggests that evolution of X-ray properties may act as a strong discriminant between models in the future.
Previous work [e.g. @Pearce00; @Muanwong01] has made great play of the fact that radiative cooling can remove low-entropy material and lead to a raising of the gas temperature above the virial temperature of the host halo. That effect is reproduced by the [*Radiative*]{} simulation in this paper, but it is interesting to note that the bolometric X-ray temperature exceeds the dynamical temperature of the clusters only at very low redshift, $z{\mathrel{{{\hbox to 0pt{\lower 3pt\hbox{$\sim$}\hss}}}
\raise 2.0pt\hbox{$<$}}}0.1$. At higher redshifts it falls below the dynamical temperature and is a factor of 1.6 lower by $z=1.5$. This departure from self-similarity is a consequence of the changing density parameter, $\Omega$, in the concordance cosmology: at high redshift $\Omega$ is close to unity and structures grow freely; at lower redshifts $\Omega$ falls well below unity and the rate of growth of cosmic structures declines.
The behavior of the [*Preheating*]{} simulation is similar, although the relative decline in the ratio of the gas to the dynamical temperature is smaller. The effect cannot, therefore, be due to the cooling of intracluster gas in the cluster cores between a redshift of 1.5 and the present—in the [*Preheating*]{} run very little gas cools below a redshift of 4 and so that cool core gas would still be there today. Instead, we attribute the presence of cool gas to the accretion of low temperature subclumps. Such accretion is a ubiquitous feature of clusters [e.g. @Rowley04]. Cool gas is seen in maps of clusters at low redshift [@Onuora03] and would be expected to be much more prevalent in clusters at high redshift in the cosmology.
To test this hypothesis, we measured the temperature variation within each cluster as follows. First we averaged properties within 20 spherical annuli out to $R_{500}$, to create smoothed dynamical, $\bar{T}_{\rm dyn}$, and gas temperature, $\bar{T}_{\rm gas}$, profiles. Then we measured the mean deviation (in log space) of the gas temperature from the local dynamical temperature $$\tau={1\over N}\,\sum_i\left(\log_{10}T_i-\log_{10}\bar{T}_{\rm dyn}\right),$$ and the root-mean square deviation of the temperature, $\sigma_T$, from the mean, $$\sigma_T^2={1\over N}\,\sum_i\left(\log_{10}T_i-\log_{10}\bar{T}_{\rm
gas}\right)^2,$$ where $N=\sum_i$ and the sum runs over all hot gas particles ($T_i>10^5$K) within $R_{500}$.
As an example, Figure \[fig:delt6720.1002\] shows the values of $\tau$ (crosses) and $\sigma_T$ (circles) for each cluster at $z=1$ in the [*Radiative*]{} simulation. This particular example has been chosen simply because the two properties are well-separated and easy to distinguish on the plot. As can be seen, the mean gas temperature of the more massive clusters typically lies below the local dynamical temperature, by as much as 15 per cent at this redshift. However, the dispersion in temperature is much larger, typically a factor of 1.5, so there will be fluctuations both above and below the dynamical temperature.
Figure \[fig:delt\] demonstrates visually the evolution of $\tau$ and $\sigma_T$ with redshift. At each redshift, the plot shoes the average value of $\tau$ over all clusters with masses greater than 1.2$\times10^{13}h^{-1}$[[ ]{}]{}. The half-width of the shaded regions represent the average values of $\sigma_T$, divided by 10 for clarity. Concentrating first on the [*Radiative*]{} simulation, it can be seen that the mean cluster temperature increases relative to the virial temperature over time and that the dispersion in temperature decreases. This is consistent with a decreasing amount of substructure within the clusters at lower redshift, although it should be noted that part of the effect is due to the narrower range of cluster masses resolved by the simulations at high redshift, as the average value of $\tau$ decreases with increasing cluster mass.
The behavior of the [*Preheating*]{} simulation mimics that of the [*Radiative*]{} one, but with a bias to higher mean temperatures. The [*Feedback*]{} simulation, however, is quite different. It shows a much larger dispersion than the other two, but no bias to low temperatures at high redshift. That is because gas is free to cool down to low temperatures but some of that gas is then heated back up high temperatures by the feedback. At low redshift cooling becomes less important and the [*Feedback*]{} run then shows a slight rise in $\tau$, matching that seen in the other two runs.
Our results have two very important implications for observations of high redshift clusters. Firstly, the behavior of the $L_{\rm
X}-T_{\rm X}$ relation at high redshift will determine the number of high redshift clusters to be found in surveys such as the ongoing [*XMM-Newton*]{} Cluster Survey [@Romer01] and this will have a significant impact upon their use in probing cosmological parameters. A positive evolution such as that shown by the [*Radiative*]{} simulation will yield many more observable high-redshift clusters than the negative evolution of the [*Feedback*]{} model. As discussed in Section \[sec:srel\] and summarized in Figure \[fig:ltevolobs\], the observational situation is far from clear but does seem to indicate positive evolution.
Turning this argument around, our results suggest that observational constraints on the degree of evolution of the $L_{\rm X}-T_{\rm X}$ relation will allow interesting constraints to be placed on the source of entropy generation in clusters, in particular the relative role of cooling and heating and whether most of the heating of the intracluster gas occurred at high redshift (as in the [*Preheating*]{} model) or was a continuous function of redshift (as in the [*Feedback*]{} model). Taking our results at face value with recent observations would suggest that our [*Feedback*]{} model is generating too much excess entropy at $z<1.5$ and that the bulk of the heating must have occurred at higher redshift. However, we stress once again that this result is very tentative.
Conclusions {#sec:conclude}
===========
The evolution of X-ray cluster scaling relations are a crucial component when constraining cosmological parameters with clusters. Observational studies at low redshift have already shown that the scaling relations deviate from self-similar expectations, attributed to non-gravitational heating and cooling processes, but their redshift dependence is only starting to be explored. In this paper we have investigated the sensitivity of the X-ray scaling relations to the nature of heating processes, using three numerical simulations of the ${{\mbox{$\Lambda$CDM}}}$ cosmology with different heating models. While all three simulations reproduce more or less the same scaling relations at $z=0$ (as they were designed to produce the correct level of excess entropy), they predict significantly different results for the evolution of the - relation to $z=1.5$.
In conclusion, our findings strongly suggest that the relative abundance of high and low redshift clusters will place interesting constraints on the nature of non-gravitational entropy generation in clusters. First indications are that an early and widespread preheating of the ICM is to be preferred to an extended period of preheating that is associated with galaxy formation. However much more detailed modeling is required and the observational picture is as yet unclear.
The simulations described in this paper were carried out on the Cray-T3E at the Edinburgh Parallel Computing Centre and the COSmology MAchine in Durham as part of the Virgo Consortium investigations into the formation of structure in the Universe. OM is grateful for the hospitality and support of the Astronomy Centre at the University of Sussex where much of work in this paper was carried out, and for financial support from Khon Kaen University. She and PAT also acknowledge support from the Thailand Research Fund and the Commission on Higher Education grant MRG4680129.
Allen S. W., Fabian A. C., 1998, , 297, L57
Allen S. W., Schmidt R. W., Fabian A. C., Ebeling H., 2003, , 342, 287 non, D. B., & Navarro, J. F., 1999, Nature, 397, 135
Rasia E., Mazzotta P., Borgani S., Moscardini L., Dolag K., Tormen G., Diaferio A., Murante G., 2005, , 618, L1
Arnaud M., Aghanim N., Neumann D. M., 2002, , 389, 1
Balogh M. L., Babul A., Patton D. R., 1999, , 307, 463
Borgani, S., Rosati, P., Tozzi, P., Stanford, S. A., Eisenhardt, P. R., Lidman, C., Holden, B., Della Ceca, R., Norman, C., & Squires, G., 2001, , 561, 13
Borgani S., Governato F., Wadsley J., Menci N., Tozzi P., Quinn T., Stadel J., Lake G., 2002, , 336, 409
Borgani, S., Murante, G., Springel, V., Diaferio, A., Dolag, K., Moscardini, L., Tormen, G., Tornatore, L., Tozzi, P., 2004, , 348, 1078
Bower R. G., 1997, , 288, 355
Bower R. G., Benson A. J., Lacey C. G., Baugh C. M., Cole S., Frenk C. S., 2001, , 325, 497
Bryan G. L., 2000, ApJ, 544, L1
non, D. B., & Navarro, J. F., 1999, Nature, 397, 135
Rasia E., Mazzotta P., Borgani S., Moscardini L., Dolag K., Tormen G., Diaferio A., Murante G., 2005, , 618, L1
Davé R., Katz N., Weinberg D. H., 2002, , 579, 23
Edge A. C., Stewart G. C., 1991, , 252, 414
Eke V. R., Cole, S., & Frenk, C. S., 1996, , 282, 263 non, D. B., & Navarro, J. F., 1999, Nature, 397, 135
Rasia E., Mazzotta P., Borgani S., Moscardini L., Dolag K., Tormen G., Diaferio A., Murante G., 2005, , 618, L1
Eke V. R., Navarro J. F. N., Frenk C. S., 1998, , 503, 569
Ettori S., Tozzi P., Borgani S., Rosati P., 2004, , 417, 13
Evrard A. E., Henry J. P., 1991, , 383, 95
Fairley, B. W., Jones, L. R., Scharf, C., Ebeling, H., Perlman, E., Horner, D., Wegner, G., & Malkan, M., 2000, , 315, 669
Henry, J. P., 1997, , 489, L1 non, D. B., & Navarro, J. F., 1999, Nature, 397, 135
Rasia E., Mazzotta P., Borgani S., Moscardini L., Dolag K., Tormen G., Diaferio A., Murante G., 2005, , 618, L1
Henry, J. P., 2000, , 534, 565
Henry, J. P., 2004, , 609, 603
Henry,non, D. B., & Navarro, J. F., 1999, Nature, 397, 135
Rasia E., Mazzotta P., Borgani S., Moscardini L., Dolag K., Tormen G., Diaferio A., Murante G., 2005, , 618, L1 J. P., & Arnaud, K. A., 1991, , 372, 410
Holden B. P., Stanford S. A., Squires G. K., Rosati P., Tozzi P., Eisenhardt P., Spinrad H., 2002, , 124, 33
Horner, D. J., Mushotzky, R. F., & Scharf, C. A., 1999, , 520, 78
Kaiser, N., 1986, , 222, 323
Kaiser, N., 1991, , 383, 104
Kay S. T., Thomas P. A., Theuns T., 2003, , 343, 608non, D. B., & Navarro, J. F., 1999, Nature, 397, 135
Rasia E., Mazzotta P., Borgani S., Moscardini L., Dolag K., Tormen G., Diaferio A., Murante G., 2005, , 618, L1
Kay S. T., Thomas P. A., Jenkins A., Pearce F. R., 2004, , 355, 1091 (KTJP04)
Knight P. A., Ponman T. J., 1997, , 289, 955
Kotov O., Vikhlinin A., 2005, , submitted (astro-ph/0504233)
Kravtsov A. V., Yepes G., 2000, , 318, 227
Loewenstein M., 2000, , 532, 17
Lumb D. H., Bartlett J. G., Romer A. K., Blanchard A., Burke D. J., Collins C. A., Nichol R. C., Giard M., Marty P., Nevalainen J., Sadat R., Vauclair S. C., 2004, , 420, 853
Markevitch M., 1998, , 504, 27
Maughan B. J., Jones L. R., Ebeling H., Scharf C., 2005, , submitted (astro-ph/0503455)
Mazzotta P., Rasia E., Moscardini L., Tormen G., 2004, , 354, 10
McCarthy, I. G., Balogh, M. L., Babul, A., Poole, G. B., Horner, D. J., 2004, , 613, 811
Metzler C. A., Evrard A. E., 1994, , 437, 564
Muanwong, O., Thomas, P. A., Kay, S. T., Pearce, F. R., & Couchman, H. M. P., 2001, , 552, L27
Muanwong, O., Thomas, P. A., Kay, S. T., & Pearce, F. R., 2002, , 336, 527 (MTKP02)
Mushotzky, R. F., & Scharf, C. A., 1997, , 482, L13
Novicki M. C., Manuela S., Henry J. P., 2002, , 124, 2413
Onuora L., Kay S. T., Thomas P. A., 2003, MNRAS, 341, 1241
Pearce F. R., Thomas P. A., Couchman H. M. P., Edge A. C., 2000, , 317, 1029
Pierpaoli, E., Scott, D., & White, M., 2001, , 325, 77
Pierpaoli, E., Borgani S., Scott D., White M., 2003, , 342, 163
Ponman, T. J, Cannon, D. B., & Navarro, J. F., 1999, Nature, 397, 135
Rasia E., Mazzotta P., Borgani S., Moscardini L., Dolag K., Tormen G., Diaferio A., Murante G., 2005, , 618, L1
Romer A. K., Viana P. T. P., Liddle A. R., Mann R. G., 2001, , 547, 594
Rowley D. R., Thomas P. A., Kay S. T., 2004, , 352, 508
Seljak, U., 2002, , 337, 769
Sutherland R. S., Dopita M. A., 1993, , 88, 253
Thomas P. A., Couchman H. M. P., 1992, , 257, 11
Thomas P. A., Muanwong O., Pearce F. R., Couchman H. M. P., Edge A. C., Jenkins A., Onuora L., 2001, , 324, 450
Tornatore L., Borgani S., Springel V., Matteucci F., Menci N., Murante G., 2003, , 342, 1025
Tozzi P., Norman C., 1998, , 546, 63
Valdarnini R., 2003, , 339, 1117
Viana, P. T. P., & Liddle, A. R., 1996, , 281, 323
Viana, P. T. P., & Liddle, A. R., 1999, , 303, 535
Viana P. T. P., Kay S. T., Liddle A. R., Muanwong O., Thomas P. A., 2003, , 346, 319
Vikhlinin A., VanSpeybroeck L., Markevitch M., Forman W. R., Grego L., 2002, , 578, L107
Vikhlinin A., 2005, , submitted (astro-ph/0504098)
Voit G. M., Bryan G. L., 2001, , 414, 425
Voit G. M., Bryan G. L., Balogh M. L., Bower R. G., 2002, , 576, 601
Voit G. M., Balogh M. L., Bower R. G., Lacey C. G., Bryan G. L., 2003, , 593, 272
White S. D. M., Efstathiou G., Frenk C. S., 1993, , 262, 1023
White D. A., Jones C., Forman W., 1997, , 292, 419
Wu K. K. S., Fabian A. C., Nulsen P. E. J., 2000, , 318, 889
Xue Y.-J., Wu X.-P., 2000, , 538, 65
Wu X.-P., Xue Y.-J., 2002, , 569, 112
[^1]: Many authors prefer to adopt the redshift scaling of the critical density, $E(z)^2=\Omega_{\rm m}(1+z)^{3} +
\Omega_{\Lambda}$ (for a flat universe), rather than the background density, $(1+z)^3$.
| ArXiv |
---
address:
- |
Mathematical Institute\
University of Oslo\
P. O. Box 1053\
N–0316 Oslo, Norway
- 'Université Paris 7, UFR de Mathématiques et Institut de Mathématiques de Jussieu, Case Postale 7012, 2, Place Jussieu, F–75251 Paris Cedex 05'
- |
Mathematical Institute\
University of Bergen\
Allég 55\
N–5007 Bergen, Norway
author:
- Geir Ellingsrud
- Joseph Le Potier
- 'Stein A. Str[ø]{}mme'
title: ' Some Donaldson invariants of $\CC\PP^2$'
---
*In memory of the victims of the Kobe earthquake*
Introduction {#introduction .unnumbered}
============
For an integer $n\ge2$, let $q_{4n-3}$ be the coefficient of the Donaldson polynomial of degree $4n-3$ of $P=\CC\PP^2$. An interpretation of $q_{4n-3}$ in an algebro-geometric context is the following. Let $M_n$ denote the Gieseker-Maruyama moduli space of semistable coherent sheaves on $P$ with rank 2 and Chern classes $c_1=0$ and $c_2=n$. For such a sheaf $F$, the Grauert-Mülich theorem implies that the restriction of $F$ to a general line $L\sub P$ splits as $F_L \iso \OO_L\dsum\OO_L$, and that the exceptional lines form a curve $J(F)$ of degree $n$ in the dual projective plane $P\v$. The association $F\mapsto J(F)$ is induced from a morphism of algebraic varieties, called the Barth map, $f_n\: M_n \to P_n$. Here $P_n=\PP^{n(n+3)/2}$ is the linear system parameterizing all curves of degree $n$ in $P\v$. Let $H\in\Pic(P_n)$ be the hyperplane class and let $\alpha = f_n^*H$. The interpretation of the Donaldson invariant is: $$q_{4n-3} = \int_{M_n} \alpha^{4n-3}.$$ Thus $q_{4n-3}$ is the degree of $f_n$ times the degree of its image. From [@Bart-2] it follows that $f_n$ is generically finite for all $n\ge2$, that $f_2$ is an isomorphism and $q_5=1$, and that $f_3$ is of degree 3 and $q_9=3$. Le Potier [@LePo] proved that $f_4$ is birational onto its image and that $q_{13}=54$. The value of $q_{13}$ has also been computed independently by Tikhomirov and Tyurin [@Tyur-1 prop. 4.1] and by Li and Qin [@Li-Qin thm. 6.29].
The main result in the present note is the following
\[thm1\] $q_{17}=2540$ and $q_{21}=233208$.
The proof consists of two parts. The first part, treated in this note, is to express $q_{4n-3}$ in terms of certain classes on the Hilbert scheme of length-$(n+1)$ subschemes of $P$. This is theorems \[thm2\] and \[thm3\] below. The second part is to evaluate these classes numerically. This has been carried out in [@Elli-Stro-5 prop. 4.2].
Let $H_{n+1}=\Hilb^{n+1}_P$ denote the Hilbert scheme parameterizing closed subschemes of $P$ of length $n+1$. There is a universal closed subscheme $\Z\sub H_{n+1}\x P$. Consider the vector bundles $$\E = R^1{p_1}_* (\I_{\Z}\*{p_2}^* \OO_{P}(-1))\text{ and }
\G = R^1{p_1}_* \I_{\Z}$$ on $H_{n+1}$ of ranks $n+1$ and $n$, respectively, and the linebundle $$\L = \det(\G) \* \det(\E)\i.$$
\[thm2\] Let the notation be as above. Then $$q_{17} = \int_{H_6} s_{12}(\E\*\L) \quad\text{and}\quad
q_{21} = \dfrac25 \int_{H_7} s_{14}(\E\*\L).$$
This result was obtained both by Tikhomirov and Tyurin [@Tyur-Tikh], using the method of “geometric approximation procedure” and by Le Potier [@LePo-3], using “coherent systems”. We present in this note what we believe is a considerably simplified proof, which is strongly hinted at on the last few pages of [@Tyur-Tikh].
The formula for $q_{17}$ is a special case of the following formula:
\[thm3\] For $2\le n\le 5$, we have $$q_{4n-3} =
\dfrac1{2^{5-n}}\int_{H_{n+1}} c_1(\L)^{5-n} s_{3n-3}(\E\*\L).$$
With this it is also easy to recompute $q_5$, $q_9$, and $q_{13}$ using similar techniques as in [@Elli-Stro-5].
We let $h$, $h\v$, and $H$ be the hyperplane classes in $P$, $P\v$, and $P_n$, respectively. In general, if $\omega$ is a divisor class, we denote by $\OO(\omega)$ the corresponding linebundle and its natural pullbacks.
This work is heavily inspired by conversations with A. Tyurin, and we thank him for generously sharing his ideas. We would also like to express our gratitude towards the Taniguchi Foundation.
Hulsbergen sheaves
==================
Barth [@Bart-2] used the term Hulsbergen bundle to denote a stable rank-2 vector bundle $F$ on $P$ with $c_1(F)=0$ and $H^0(P,F(1))\ne0$. We modify this definition a little as follows:
A *Hulsbergen sheaf* is a coherent sheaf $F$ on $P$ which admits a non-split short exact sequence (*Hulsbergen sequence*) $$\label{Hulsbergen}
0 \to \OO_P \to F(1) \to \I_Z(2) \to 0,$$ where $Z\sub P$ is a closed subscheme of finite length (equal to $c_2(F)+1$).
Note that a Hulsbergen sheaf is not necessarily semistable or locally free. However:
\[GM\] Let $F$ be a Hulsbergen sheaf with $c_2(F)=n>0$. Then the set $J(F)\sub P\v$ of exceptional lines for $F$ is a curve of degree $n$, defined by the determinant of the bundle map $$m\: H^1(P,F(-2))\*\OO_{P\v}(-1) \to H^1(P,F(-1))\*\OO_{P\v}$$ induced by multiplication with a variable linear form.
First note from the Hulsbergen sequence that the two cohomology groups have dimension $n$. It is easy to see that any Hulsbergen sheaf is slope semistable, in the sense that it does not contain any rank-1 subsheaf with positive first Chern class. Thus by [@Bart-1 thm. 1], $F_L \iso
\OO_L \dsum \OO_L$ for a general line $L$. On the other hand, it is clear that a line $L$ is exceptional if and only if $m$ is not an isomorphism at the point $[L]\in P\v$.
It is straightforward to construct a moduli space for Hulsbergen sequences. For any length-$(n+1)$ subscheme $Z\sub P$, the isomorphism classes of extensions are parameterized by $\PP(\Ext^1_P(\I_Z(2),\OO_P)\v)$. By Serre duality, $$\Ext^1_P(\I_Z(2),\OO_P)\v \iso H^1(P,\I_Z(-1)).$$ For varying $Z$, these vector spaces glue together to form the vector bundle $\E$ over $H_{n+1}$, hence $D_n=\PP(\E)$ is the natural parameter space for Hulsbergen sequences. Let $\OO(\tau)$ be the associated tautological quotient linebundle. For later use, note that for any divisor class $\omega$ on $H_{n+1}$, we have $\pi_*(\tau+\pi^*\omega)^{k+n} = s_k(\E(\omega))$, where $\pi\: D_n \to H_{n+1}$ is the natural map [@IT].
The tautological quotient $\pi^*\E \to \OO(\tau)$ gives rise to a short exact sequence on $D_n\x P$: $$0 \to \OO(\tau) \to \F(h) \to (\pi\x1)^*\I_{\Z}(2h) \to 0$$ which defines a complete family $\F$ of Hulsbergen sheaves.
As we noted earlier, a Hulsbergen sheaf is not necessarily semistable. On the other hand, the *generic* Hulsbergen sheaf is stable if $n\ge 2$. It follows that the family $\F$ induces a *rational* map $g_n\:
D_n \to M_n$. By [lemma \[GM\]]{} above, there is also a Barth map $b_n\: D_n \to P_n$, defined everywhere, and by construction, the following diagram commutes: $$\begin{CD}
D_n @>b_n>> P_n \\
@V{g_n}VV @VV{||}V \\
M_n @>>f_n> P_n
\end{CD}$$
Put $\lambda=c_1(\pi^*\L)$. Then $b_n^*H = \tau+\lambda$.
Let $L\sub P$ be a line. Twist the universal Hulsbergen sequence by $-2h$ and $-3h$ respectively. Multiplication by an equation for $L$ gives rise to the vertical arrows in a commutative diagram with exact rows on $D_n\x P$: $$\begin{CD}
0 @>>> \OO(\tau-3h) @>>> \F(-2h) @>>> (\pi\x1)^*\I_Z(-h)
@>>>0 \\
@. @VVV @VVV @VVV @.\\
0 @>>> \OO(\tau-2h) @>>> \F(-h) @>>> (\pi\x1)^*\I_Z @>>>0
\end{CD}$$ Pushing this down via the first projection, we get the following exact diagram on $D_n$: $$\begin{CD}
0@>>>R^1{p_1}_* \F(-2h) @>>> \pi^*\E @>>> \OO(\tau)@>>> 0\\
@. @Vm_LVV @VVV @VVV\\
0@>>> R^1{p_1}_* \F(-h) @>\iso>> \pi^*\G @>>> 0
\end{CD}$$ Here the last map of the top row is nothing but the tautological quotient map on $\PP(\E)$. Let $A(L)\sub D_n$ be the set of Hulsbergen sequences where $L$ is an exceptional line for the middle term. Clearly, $A(L)$ is the degeneration locus of the left vertical map $m_L$ above. Hence the divisor class of $A(L)$ is $$\begin{aligned}
[A(L)]&= c_1(R^1{p_1}_* \F(-h)) - c_1(R^1{p_1}_* \F(-2h)) \\
&= \pi^*c_1(\G) - \pi^*c_1(\E) + \tau \\
&= \tau+\lambda.
\end{aligned}$$ On the other hand, $A(L)$ is the inverse image of a hyperplane in $P_n$ under $b_n$, so its divisor class is $b_n^*H$.
The case $n\le 5$
=================
For $2\le n\le 5$, the rational map $g_n$ is dominating, and the general fiber is isomorphic to $\PP^{n-5}$. For $n\ge 5$, the map $g_n$ is generically injective with image of codimension $n-5$. In particular, $g_5$ is birational.
Everything follows from the observation that the fiber over a point $[F]\in M_n$ in the image of $g_n$ is the projectivization of $H^0(P,F(1))$, and that for general such $F$, this vector space has dimension $h^0(F(1))=\max(1,6-n)$, which is easily seen from . The assertion about the codimension follows from a dimension count: $\dim(M_n)=4n-3$ and $\dim(D_n)=3n+2$.
The first half of [theorem \[thm2\]]{} now follows: First of all, since $g_5$ is birational, the two morphisms $f_5$ and $b_5$ have the same image and the same degree. Therefore $q_{17}$ can be computed as $$q_{17} = \int_{D_5} H^{17} =\int_{D_5} (\tau+\lambda)^{17}
= \int_{H_6} s_{12}(\E\*\L).$$
For [theorem \[thm3\]]{}, let $L_1,\dots,L_{5-n}$ be general lines in $P$, and let $B_n\sub D_n$ be the locus of Hulsbergen sequences where the closed subscheme $Z$ meets all these $5-n$ lines. The cohomology class of $B_n$ in $H^*(D_n)$ is $\lambda^{5-n}$.
\[cover\] Let $2\le n\le5$. The general nonempty fiber of $g_n$ meets $B_n$ in $2^{5-n}$ points, hence the rational map $g_n|_{B_n}\: B_n \to M_n$ is dominating and generically finite, of degree $2^{5-n}$.
The general nonempty fiber is of the form $\PP(H^0(P,F(1))\v)$. It suffices to show that the restriction of $\L$ to this fiber has degree 2 (if $n<5$). For this, it suffices to consider a linear pencil in the fiber. So let $\sigma_0$ and $\sigma_1$ be two independent global sections of $F(1)$, and consider the pencil they span. Now $\sigma_0\wedge \sigma_1 \in
H^0(P,\wedge^2F)=H^0(P,\OO_P(2))$ is the equation of a conic $C\sub P$ which contains the zero scheme $V(t_0\sigma_0 + t_1\sigma_1)$ of each section in the pencil, $(t_0,t_1)\in\PP^1$. Since $C$ meets a general line in two points, it follows that there are exactly two members of the pencil whose zero set meets a general line.
To complete the proof of [theorem \[thm3\]]{}, by [lemma \[cover\]]{} we now have for $2\le n\le5$: $$\begin{aligned}
2^{5-n}\,q_{4n-3} &= 2^{5-n}\int_{M_n} H^{4n-3} \\
&=\int_{B_n} (\tau+\lambda)^{4n-3} \\
&=\int_{D_n}
\lambda^{5-n}\,(\tau+\lambda)^{4n-3} \\
&=\int_{H_{n+1}}c_1(\L)^{5-n}
\,s_{3n-3}(\E\*\L).
\end{aligned}$$ This completes the proof of the theorems for $n\le 5$.
The case $n=6$
==============
For $n\ge6$ the techniques above will say something about the restriction of the Barth map to the Brill-Noether locus $B\sub M_n$ of semistable sheaves whose first twist admit a global section. For general $n$ this locus is too small to carry enough information about $M_n$, but in the special case $n=6$, it is actually a divisor, whose divisor class $\beta=[B]$ we can determine. Now $\Pic(M_n)\*\QQ$ has rank 2, generated by $\alpha$ and $\delta=[\Delta]$, the class of the locus $\Delta\sub M_n$ corresponding to non-locally free sheaves [@LePo-1].
In $\Pic(M_6)\*\QQ$, the following relation holds: $$\beta = \frac52 \,\alpha - \frac12\,\delta.$$
Let $\xi\:X\to M_6$ be a morphism induced by a flat family $\F$ of semistable sheaves on $P$, parameterized by some variety $X$. For certain divisor classes $a$ and $d$ on $X$, the second and third Chern classes of $\F$ can be written in the form $$c_2(\F) = a\,h+6\,h^2, \quad c_3(\F) = d\,h^2$$ modulo higher codimension classes on $X$. The Grothendieck Riemann-Roch theorem for the projection $p\: X\x P \to X$ easily gives (for example using [@schubert]) that $$-c_1(p_!\F(h)) = \frac52\, a- \frac12\, d.$$ The locus $\xi\i B\sub X$ is set-theoretically the support of $R^1 p_*\F(h)$. It is not hard to see that one can take the family $X$ in such a way that the 0-th Fitting ideal of $R^1 p_*\F(h)$ is actually reduced. Therefore the left hand side of the equation above is $\xi^*\beta$. On the other hand, $a=\xi^*\alpha$ by the usual definition of the $\mu$ map of Donaldson [@Dona-1], and $d=\xi^*\delta$. Since the family $\F/X$ was arbitrary, the required relation is actually universal, and so holds also in $\Pic(M_6)\*\QQ$. (It suffices to take a family with the properties that (i) $\xi^*\:\Pic_\QQ(M_6)
\to \Pic_\QQ(X)$ is injective, (ii) the Fitting ideal above is reduced, and (iii) the general non-locally free sheaf in the family has colength 1 in its double dual.)
With this, we complete the proof of the second part of [theorem \[thm2\]]{} in the following way. The general fiber of $f_6$ restricted to $\Delta$ has dimension 1, so $f_6(\Delta)$ has dimension 19, see e.g. [@Stro-1]. Therefore we get $$\begin{aligned}
\int_{H_7} s_{14}(\E\*\L) &= \int_{D_6}(\lambda+\tau)^{20} \\
&= \int_{M_6} \beta\, \alpha^{20} \\
&= \int_{M_6} (\frac52\, \alpha - \frac12\,\delta)\,\alpha^{20} \\
&= \frac52\int_{M_6} \alpha^{21} -\frac12\int_{\Delta}\alpha^{20}
= \frac52\, q_{21}.
\end{aligned}$$
A geometric interpretation
==========================
A *Darboux configuration* in $P\v$ consists of a pair $(\Pi,C)$ where $\Pi\sub P\v$ is the union of $n+1$ distinct lines, no three concurrent, and $C\sub P\v$ is a curve of degree $n$ passing through all the nodes of $\Pi$.
If we let $Z\sub P$ consist of the $n+1$ points dual to the components of $\Pi$, we have by Hulsbergen’s theorem [@Bart-2 thm. 4] a natural 1-1 correspondence between Hulsbergen sequences and Darboux configurations $(\Pi,C)$, by letting $C=J(F)$. Therefore $D_n$ can be used as a compactification of the set of Darboux configurations, and the intersection number $$\int_{D_n} \lambda^i (\tau+\lambda)^{3n+2-i} =
\int_{H_{n+1}} c_1(\L)^i s_{2n+2-i}(\E\*\L)$$ can be interpreted as the number of Darboux configurations $(\Pi,C)$ where $\Pi$ passes through $i$ given points and $C$ passes through $3n+2-i$ given points.
It is not known whether the Barth map has degree 1 for $n\ge5$. A related question is the following: Let $(\Pi,C)$ be a general Darboux configuration ($n\ge 5$). Is the inscribed polygon $\Pi$ uniquely determined by $C$?
[10]{}
W. Barth. Moduli of vector bundles on the projective plane. , 42:63–91, 1977.
W. Barth. Some properties of stable rank-2 vector bundles on [$\PP_n$]{}. , 226:125–150, 1977.
S. K. Donaldson. Polynomial invariants for smooth 4-manifolds. , 29:257–315, 1990.
G. Ellingsrud and S. A. Str[ø]{}mme. Bott’s formula and enumerative geometry. To appear in Journal of the AMS.
W. Fulton. . Number 2 in Ergebnisse der Mathematik und ihrer Grenz-Gebiete. Springer-Verlag, Berlin-Heidelberg-New York, 1984.
S. Katz and S. A. Str[ø]{}mme. , a [Maple]{} package for intersection theory and enumerative geometry. Software and documentation available from the authors or by anonymous ftp from ftp.math.okstate.edu or linus.mi.uib.no, 1992.
J. Le Potier. Systèmes cohérent et polynômes de [Donaldson]{}. Preprint.
J. Le Potier. Sur le groupe [Picard]{} de l’espace de modules des fibrés stables sur [$\PP_2$]{}. , 13:141–155, 1981.
J. Le Potier. Fibrés stables sur le plan projectif et quartiques de [Lüroth]{}. Preprint, Oct 1989.
W.-P. Li and Z. Qin. Lower-degree [Donaldson]{} polynomial invariants of rational surfaces. , 2:413–442, 1993.
S. A. Str[ø]{}mme. Ample divisors on fine moduli spaces on the projective plane. , 187:405–523, 1984.
A. Tikhomirov and A. N. Tyurin. Application of geometric approximation procedure to computing the [Donaldson’s]{} polynomials for [$\CC\PP^2$]{}. , 12:1–71, 1994.
A. N. Tyurin. The moduli spaces of vector bundles on threefolds, surfaces and curves [I]{}. Erlangen preprint, 1990.
| ArXiv |
---
abstract: 'We present here an overview of recent work in the subject of astrophysical manifestations of super-massive black hole (SMBH) mergers. This is a field that has been traditionally driven by theoretical work, but in recent years has also generated a great deal of interest and excitement in the observational astronomy community. In particular, the electromagnetic (EM) counterparts to SMBH mergers provide the means to detect and characterize these highly energetic events at cosmological distances, even in the absence of a space-based gravitational-wave observatory. In addition to providing a mechanism for observing SMBH mergers, EM counterparts also give important information about the environments in which these remarkable events take place, thus teaching us about the mechanisms through which galaxies form and evolve symbiotically with their central black holes.'
address: '$^1$ NASA Goddard Space Flight Center, Greenbelt, MD 20771'
author:
- 'Jeremy D. Schnittman$^{1}$'
title: 'Astrophysics of Super-massive Black Hole Mergers'
---
INTRODUCTION {#intro}
============
Following numerical relativity’s [*annus mirabilis*]{} of 2006, a deluge of work has explored the astrophysical manifestations of black hole mergers, from both the theoretical and observational perspectives. While the field has traditionally been dominated by applications to the direct detection of gravitational waves (GWs), much of the recent focus of numerical simulations has been on predicting potentially observable electromagnetic (EM) signatures. Of course, the greatest science yield will come from coincident detection of both the GW and EM signature, giving a myriad of observables such as the black hole mass, spins, redshift, and host environment, all with high precision [@bloom:09]. Yet even in the absence of a direct GW detection (and this indeed is the likely state of affairs for at least the next decade), the EM signal alone may be sufficiently strong to detect with wide-field surveys, and also unique enough to identify unambiguously as a SMBH merger.
In this article, we review the brief history and astrophysical principles that govern the observable signatures of SMBH mergers. To date, the field has largely been driven by theory, but we also provide a summary of the observational techniques and surveys that have been utilized, including recent claims of potential detections of both SMBH binaries and also post-merger recoiling black holes.
While the first public use of the term “black hole” is generally attributed to John Wheeler in 1967, as early as 1964 Edwin Saltpeter proposed that gas accretion onto super-massive black holes provided the tremendous energy source necessary to power the highly luminous quasi-stellar objects (quasars) seen in the centers of some galaxies [@saltpeter:64]. Even earlier than that, black holes were understood to be formal mathematical solutions to Einstein’s field equations [@schwarzschild:16], although considered by many to be simply mathematical oddities, as opposed to objects that might actually exist in nature (perhaps most famously, Eddington’s stubborn opposition to the possibility of astrophysical black holes probably delayed significant progress in their understanding for decades) [@thorne:94].
In 1969, Lynden-Bell outlined the foundations for black hole accretion as the basis for quasar power [@lynden_bell:69]. The steady-state thin disks of Shakura and Sunyaev [@shakura:73], along with the relativistic modifications given by Novikov and Thorne [@novikov:73], are still used as the standard models for accretion disks today. In the following decade, a combination of theoretical work and multi-wavelength observations led to a richer understanding of the wide variety of accretion phenomena in active galactic nuclei (AGN) [@rees:84]. In addition to the well-understood thermal disk emission predicted by [@shakura:73; @novikov:73], numerous non-thermal radiative processes such as synchrotron and inverse-Compton are also clearly present in a large fraction of AGN [@oda:71; @elvis:78].
Peters and Mathews [@peters:63] derived the leading-order gravitational wave emission from two point masses more than a decade before Thorne and Braginsky [@thorne:76] suggested that one of the most promising sources for such a GW signal would be the collapse and formation of a SMBH, or the (near head-on) collision of two such objects in the center of an active galaxy. In that same paper, Thorne and Braginsky build on earlier work by Estabrook and Wahlquist [@estabrook:75] and explore the prospects for a space-based method for direct detection of these GWs via Doppler tracking of inertial spacecraft. They also attempted to estimate event rates for these generic bursts, and arrived at quite a broad range of possibilities, from $\lesssim 0.01$ to $\gtrsim 50$ events per year, numbers that at least bracket our current best-estimates for SMBH mergers [@sesana:07].
However it is not apparent that Thorne and Braginsky considered the hierarchical merger of galaxies as the driving force behind these SMBH mergers, a concept that was only just emerging at the time [@ostriker:75; @ostriker:77]. Within the galactic merger context, the seminal paper by Begelman, Blandford, and Rees (BBR) [@begelman:80] outlines the major stages of the SMBH merger: first the nuclear star clusters merge via dynamical friction on the galactic dynamical time $t_{\rm gal} \sim 10^8$ yr; then the SMBHs sink to the center of the new stellar cluster on the stellar dynamical friction time scale $t_{\rm df} \sim 10^6$ yr; the two SMBHs form a binary that is initially only loosely bound, and hardens via scattering with the nuclear stars until the loss cone is depleted; further hardening is limited by the diffusive replenishing of the loss cone, until the binary becomes “hard,” i.e., the binary’s orbital velocity is comparable to the local stellar orbital velocity, at which point the evolutionary time scale is $t_{\rm hard} \sim N_{\rm inf} t_{\rm
df}$, with $N_{\rm inf}$ stars within the influence radius. This is typically much longer than the Hubble time, effectively stalling the binary merger before it can reach the point where gravitational radiation begins to dominate the evolution. Since $r_{\rm hard} \sim 1$ pc, and gravitational waves don’t take over until $r_{\rm GW} \sim 0.01$ pc, this loss cone depletion has become known as the “final parsec problem” [@merritt:05]. BBR thus propose that there should be a large cosmological population of stalled SMBH binaries with separation of order a parsec, and orbital periods of years to centuries. Yet to date not a single binary system with these sub-parsec separations has even been unambiguously identified.
In the decades since BBR, numerous astrophysical mechanisms have been suggested as the solution to the final parsec problem [@merritt:05]. Yet the very fact that so many different solutions have been proposed and continue to be proposed is indicative of the prevailing opinion that it is still a real impediment to the efficient merger of SMBHs following a galaxy merger. However, the incontrovertible evidence that galaxies regularly undergo minor and major mergers during their lifetimes, coupled with a distinct lack of binary SMBH candidates, strongly suggest that nature has found its own solution to the final parsec problem. Or, as Einstein put it, “God does not care about mathematical difficulties; He integrates empirically.”
For incontrovertible evidence of a SMBH binary, nothing can compare with the direct detection of gravitational waves from space. The great irony of gravitational-wave astronomy is that, despite the fact that the peak GW luminosity generated by black hole mergers outshines the [*entire observable universe*]{}, the extremely weak coupling to matter makes both direct and indirect detection exceedingly difficult. For GWs with frequencies less than $\sim 1$ Hz, the leading instrumental concept for nearly 25 years now has been a long-baseline laser interferometer with three free-falling test masses housed in drag-free spacecraft [@faller:89]. Despite the flurry of recent political and budgetary constraints that have resulted in a number of alternative, less capable designs, we take as our fiducial detector the classic LISA (Laser Interferometer Space Antenna) baseline design [@yellowbook:11].
For SMBHs with masses of $10^6
M_\odot$ at a redshift of $z=1$, LISA should be able to identify the location of the source on the sky within $\sim 10$ deg$^2$ a month before merger, and better than $\sim 0.1$ deg$^2$ with the entire waveform, including merger and ringdown [@kocsis:06; @lang:06; @lang:08; @kocsis:08a; @lang:09; @thorpe:09; @mcwilliams:10]. This should almost certainly be sufficient to identify EM counterparts with wide-field surveys such as LSST [@lsst:09], WFIRST [@spergel:13], or WFXT [@wfxt:12]. Like the cosmological beacons of gamma-ray bursts and quasars, merging SMBHs can teach us about relativity, high-energy astrophysics, radiation hydrodynamics, dark energy, galaxy formation and evolution, and how they all interact.
A large variety of potential EM signatures have recently been proposed, almost all of which require some significant amount of gas in the near vicinity of the merging black holes [@schnittman:11]. Thus we must begin with the question of whether or not there [*is*]{} any gas present, and if so, what are its properties. Only then can we begin to simulate realistic spectra and light curves, and hope to identify unique observational signatures that will allow us to distinguish these objects from the myriad of other high-energy transients throughout the universe.
CIRCUMBINARY DISKS {#disk_theory}
==================
If there is gas present in the vicinity of a SMBH binary, it is likely in the form of an accretion disk, as least at some point in the system’s history. Disks are omnipresent in the universe for the simple reason that it is easy to lose energy through dissipative processes, but much more difficult to lose angular momentum. At larger separations, before the SMBHs form a bound binary system, massive gas disks can be quite efficient at bringing the two black holes together [@escala:05; @dotti:07]. As these massive gas disks are typically self-gravitating, their dynamics can be particularly complicated, and require high-resolution 3D simulations, which will be discussed in more detail in section \[MHD\_simulations\].
Here we focus on the properties of non-self-gravitating circumbinary accretion disks which have traditionally employed the same alpha prescription for pressure-viscous stress scaling as in [@shakura:73]. Much of the early work on this subject was applied to protoplanetary disks around binary stars, or stars with massive planets embedded in their surrounding disks. The classical work on this subject is Pringle (1991) [@pringle:91], who considered the evolution of a 1D thin disk with an additional torque term added to the inner disk. This source of angular momentum leads to a net outflow of matter, thus giving these systems their common names of “excretion” or “decretion” disks. Pringle considered two inner boundary conditions: one for the inflow velocity $v^r(R_{\rm in})\to 0$ and one for the surface density $\Sigma(R_{\rm in}) \to 0$. For the former case, the torque is applied at a single radius at the inner edge, leading to a surface density profile that increases steadily inwards towards $R_{\rm in}$. In the latter case, the torque is applied over a finite region in the inner disk, which leads to a relatively large evacuated gap out to $\gtrsim 6 R_{\rm in}$. In both cases, the angular momentum is transferred from the binary outwards through the gas disk, leading to a shrinking of the binary orbit.
In [@artymowicz:91], SPH simulations were utilized to understand in better detail the torquing mechanism between the gas and disk. They find that, in agreement with the linear theory of [@goldreich:79], the vast majority of the binary torque is transmitted to the gas through the $(l,m)=(1,2)$ outer Lindblad resonance (for more on resonant excitation of spiral density waves, see [@takeuchi:96]). The resonant interaction between the gas and eccentric binary ($e=0.1$ for the system in [@artymowicz:91]) pumps energy and angular momentum into the gas, which gets pulled after the more rapidly rotating interior point mass. This leads to a nearly evacuated disk inside of $r\approx 2a$, where $a$ is the binary’s semi-major axis. The interaction with the circumbinary disk not only removes energy and angular momentum from the binary, but it can also increase its eccentricity, and cause the binary pericenter to precess on a similar timescale, all of which could lead to potentially observable effects in GW observations [@armitage:05; @roedig:11; @roedig:12].
In [@artymowicz:94; @artymowicz:96], Artymowicz & Lubow expand upon [@artymowicz:91] and provide a comprehensive study of the effects of varying the eccentricity, mass ratio, and disk thickness on the behavior of the circumbinary disk and its interaction with the binary. Not surprisingly, they find that the disk truncation radius moves outward with binary eccentricity. Similarly, the mini accretion disks around each of the stars has an outer truncation radius that decreases with binary eccentricity. On the other hand, the location of the inner edge of the circumbinary disk appears to be largely insensitive to the binary mass ratio [@artymowicz:94]. For relatively thin, cold disks with aspect ratios $H/R\approx 0.03$, the binary torque is quite effective at preventing accretion, much as in the decretion disks of Pringle [@pringle:91]. In that case, the gas accretion rate across the inner gap is as much at $10-100\times$ smaller than that seen in a single disk, but the authors acknowledge that the low resolution of the SPH simulation makes these estimates inconclusive [@artymowicz:94].
When increasing the disk thickness to $H/R\approx 0.1$, the gas has a much easier time jumping the gap and streaming onto one of the two stars, typically the smaller one. For $H/R\approx 0.1$, the gas accretion rate is within a factor of two of the single-disk case [@artymowicz:96]. The accretion rate across the gap is strongly modulated at the binary orbital period, although the accretion onto the individual masses can be out of phase with each other. The modulated accretion rate suggests a promising avenue for producing a modulated EM signal in the pre-merger phase, and the very fact that a significant amount of gas can in fact cross the gap is important for setting up a potential prompt signal at the time of merger.
To adequately resolve the spiral density waves in a thin disk, 2D grid-based calculations are preferable to the inherently noisy and diffusive SPH methods. Armitage and Natarayan [@armitage:02] take a hybrid approach to the problem, and use a 2D ZEUS [@stone:92] hydrodynamics calculation to normalize the torque term in the 1D radial structure equation. Unlike [@artymowicz:91], they find almost no leakage across the gap, even for a moderate $H/R=0.07$. However, they do identify a new effect that is particularly important for binary black holes, as opposed to protoplanetary disks. For a mass ratio of $q\equiv m_2/m_1=0.01$, when a small accretion disk is formed around the primary, the evolution of the secondary due to gravitational radiation can shrink the binary on such short time scales that it plows into the inner accretion disk, building up gas and increasing the mass accretion rate and thus luminosity immediately preceding merger [@armitage:02]. If robust, this obviously provides a very promising method for generating bright EM counterparts to SMBH mergers. However, recent 2D simulations by [@baruteau:12] suggest that the gas in the inner disk could actually flow across the gap back to the outer disk, like snow flying over the plow. The reverse of this effect, gas piling up in the outer disk before leaking into the inner disk, has recently been explored by [@kocsis:12a; @kocsis:12b].
In the context of T Tau stars, [@gunther:02; @gunther:04] developed a sophisticated simulation tool that combines a polar grid for the outer disk with a Cartesian grid around the binary to best resolve the flow across the gap. They are able to form inner accretion disks around each star, fed by persistent streams from the circumbinary disk. As a test, they compare the inner region to an SPH simulation and find good agreement, but only when the inner disks are artificially fed by some outer source, itself not adequately resolved by the SPH calculation [@gunther:04]. They also see strong periodic modulation in the accretion rate, due to a relatively large binary eccentricity of $e=0.5$.
![\[fig:macfadyen\] ([*left*]{}) Surface density and spiral density wave structure of circumbinary disk with equal-mass BHs on a circular orbit, shown after the disk evolved for 4000 binary periods. The dimensions of the box are $x=[-5a,5a]$ and $y=[-5a,5a]$. ([*right*]{}) Time-dependent accretion rate across the inner edge of the simulation domain ($r_{\rm in}=a$), normalized by the initial surface density scale $\Sigma_0$. \[reproduced from MacFadyen & Milosavljevic (2008), ApJ [**672**]{}, 83\]](macfadyen_Sigma.epsi "fig:"){width="40.00000%"} ![\[fig:macfadyen\] ([*left*]{}) Surface density and spiral density wave structure of circumbinary disk with equal-mass BHs on a circular orbit, shown after the disk evolved for 4000 binary periods. The dimensions of the box are $x=[-5a,5a]$ and $y=[-5a,5a]$. ([*right*]{}) Time-dependent accretion rate across the inner edge of the simulation domain ($r_{\rm in}=a$), normalized by the initial surface density scale $\Sigma_0$. \[reproduced from MacFadyen & Milosavljevic (2008), ApJ [**672**]{}, 83\]](macfadyen_mdot.eps "fig:"){width="50.00000%"}
MacFadyen and Milosavljevic (MM08) [@macfadyen:08] also developed a sophisticated grid-based code including adaptive mesh refinement to resolve the flows at the inner edge of the circumbinary disk in the SMBH binary context. However, they excise the inner region entirely to avoid excessive demands on their resolution around each black hole so are unable to study the behavior of mini accretion disks. They also use an alpha prescription for viscosity and find qualitatively similar results to the earlier work described above: a gap with $R_{\rm in} \approx 2a$ due to the $m=2$ outer Lindblad resonance, spiral density waves in an eccentric disk, highly variable and periodic accretion, and accretion across the gap of $\sim 20\%$ that expected for a single BH accretion disk with the same mass [@macfadyen:08]. The disk surface density as well as the variable accretion rate are shown in Figure \[fig:macfadyen\]. Recent work by the same group carried out a systematic study of the effect of mass ratio and found significant accretion across the gap for all values of $q=m_2/m_1$ between 0.01 and 1 [@dorazio:12].
The net result of these calculations seems to be that circumbinary gas disks are a viable mechanism for driving the SMBH binary through the final parsec to the GW-driven phase, and supplying sufficient accretion power to be observable throughout. Thus it is particularly perplexing that no such systems have been observed with any degree of certainty. According to simple alpha-disk theory, there should also be a point in the GW evolution where the binary separation is shrinking at such a prodigious rate that the circumbinary disk cannot keep up with it, and effectively decouples from the binary. At that point, gas should flow inwards on the relatively slow timescale corresponding to accretion around a single point mass, and a real gap of evacuated space might form around the SMBHs, which then merge in a near vacuum [@milos:05].
NUMERICAL SIMULATIONS {#simulations}
=====================
Vacuum numerical relativity {#vacuum}
---------------------------
In the context of EM counterparts, the numerical simulation of two equal-mass, non-spinning black holes in a vacuum is just about the simplest problem imaginable. Yet the inherent non-linear behavior of Einstein’s field equations made this a nearly unsolvable Grand Challenge problem, frustrating generations of relativists from the 3+1 formulation of Arnowitt, Deser, and Misner in 1962 [@arnowitt:08], followed shortly by the first attempt at a numerical relativity (NR) simulation on a computer in 1964 [@hahn:64], decades of uneven progress, slowed in large part by the limited computer power of the day (but also by important fundamental instabilities in the formulation of the field equations), to the ultimate solution by Pretorius in 2005 [@pretorius:05] and subsequent deluge of papers in 2006 from multiple groups around the world (for a much more thorough review of this colorful story and the many technical challenges overcome by its participants, see [@centrella:10]).
Here we will review just a few highlights from the recent NR results that are most pertinent to our present subject. For the first 50 years since their original conception, black holes (and general relativity as a whole) were largely relegated to mathematicians as a theoretical curiosity with little possibility of application in astronomy. All this changed in the late 1960s and early 70s when both stellar-mass and super-massive black holes were not only observed, but also understood to be critical energy sources and play a major role in the evolution of galaxies and stars [@thorne:94]. A similar environment was present during the 1990s with regard to binary black holes and gravitational waves. Most believed in their existence, but after decades of false claims and broken promises, the prospect of direct detection of GWs seemed further away than ever. But then in 1999, construction was completed on the two LIGO observatories, and they began taking science data in 2002. At the same time, the space-based LISA concept was formalized with the “Yellow Book,” a report submitted to ESA in 1996, and together with NASA, an international science team was formed in 2001. Astrophysics theory has long been data-driven, but here was a case where large-scale projects were being proposed and even funded based largely on theoretical predictions.
The prospect of real observations and data in turn energized the NR community and provided new motivation to finally solve the binary BH merger problem. Long-duration, accurate waveforms are necessary for both the detection and characterization of gravitational waves. Generic binary sources are fully described by 17 parameters: the BH masses (2), spin vectors (6), binary orbital elements (6), sky position (2), and distance (1). To adequately cover this huge parameter space requires exceedingly clever algorithms and an efficient method for calculating waveforms. Fortunately, most NR studies to date suggest that even the most non-linear phase of the inspiral and merger process produces a relatively smooth waveform, dominated by the leading quadrupole mode [@centrella:10]. Additionally, in the early inspiral and late ringdown phases, relatively simple analytic expressions appear to be quite sufficient in matching the waveforms [@pan:11]. Even more encouraging is the fact that waveforms from different groups using very different methods agree to a high level of accuracy, thus lending confidence to their value as a description of the real world [@baker:07].
In addition to the waveforms, another valuable result from these first merger simulations was the calculation of the mass and spin of the final black hole, demonstrating that the GWs carried away a full $4\%$ of their initial energy in roughly an orbital time, and leave behind a moderately spinning black hole with $a/M=0.7$ [@baker:06a; @campanelli:06].
After the initial breakthrough with equal-mass, non-spinning black holes, the remarkably robust “moving puncture” method was soon applied to a wide variety of systems, including unequal masses [@berti:07], eccentric orbits [@hinder:08], and spinning BHs [@campanelli:06b]. As with test particles around Kerr black holes, when the spins are aligned with the orbital angular momentum, the BHs can survive longer before plunging, ultimately producing more GW power and resulting in a larger final spin. This is another critical result for astrophysics, as the spin evolution of SMBHs via mergers and gas accretion episodes is a potentially powerful diagnostic of galaxy evolution [@berti:08]. Perhaps the most interesting and unexpected result from the NR bonanza was the first accurate calculation of the gravitational recoil, which will be discussed in more detail in the following section.
In addition to the widespread moving puncture method, the NR group at Cornell/Caltech developed a highly accurate spectral method that is particularly well-suited for long evolutions [@boyle:07]. Because it converges exponentially with resolution (as opposed to polynomial convergence for finite-difference methods), the spectral method can generate waveforms with dozens of GW cycles, accurate to a small fraction of phase. These long waveforms are particularly useful for matching the late inspiral to post-Newtonian (PN) equations of motion, the traditional tool of choice for GW data analysis for LIGO and LISA (e.g., [@cutler:93; @apostolatos:94; @kidder:95; @blanchet:06]). The down side of the spectral method has been its relative lack of flexibility, making it very time consuming to set up simulations of new binary configurations, particularly with arbitrary spins. If this problem can be overcome, spectral waveforms will be especially helpful in guiding the development of more robust semi-analytic tools (e.g., the effective-one-body approach of Buonanno [@buonanno:99]) for calculating the inspiral, merger, and ringdown of binary BHs with arbitrary initial conditions.
The natural application for long, high-accuracy waveforms is as templates in the matched-filtering approach to GW data analysis. For LIGO, this is critical to detect most BH mergers, where much of the in-band power will come from the final stages of inspiral and merger. The high signal-to-noise expected from SMBHs with LISA means that most events will probably be detected with high significance even when using a primitive template library [@flanagan:98; @cutler:98]. However, for [*parameter estimation*]{}, high-fidelity waveforms are essential for faithfully reproducing the physical properties of the source. In particular, for spinning BHs, the information contained in the precessing waveform can greatly improve our ability to determine the sky position of the source, and thus improve our prospects for detecting and characterizing any EM counterpart [@lang:08; @thorpe:09; @lang:09].
Gravitational recoil {#recoil}
--------------------
In the general case where there is some asymmetry between the two black holes (e.g., unequal masses or spins), the GW radiation pattern will have a complicated multipole structure. The beating between these different modes leads to a net asymmetry in the momentum flux from the system, ultimately resulting in a recoil or kick imparted on the final merged black hole [@schnittman:08a]. This effect has long been anticipated for any GW source [@bonnor:61; @peres:62; @bekenstein:73], but the specific value of the recoil has been notoriously difficult to calculate using traditional analytic means [@wiseman:92; @favata:04; @blanchet:05; @damour:06]. Because the vast majority of the recoil is generated during the final merger phase, it is a problem uniquely suited for numerical relativity. Indeed, this was one of the first results published in 2006, for the merger of two non-spinning BHs with mass ratio 3:2, giving a kick of $90-100$ km/s [@baker:06b].
Shortly thereafter, a variety of initial configurations were explored, covering a range of mass ratios [@herrmann:07b; @gonzalez:07a], aligned spins [@herrmann:07a; @koppitz:07], and precessing spins [@campanelli:07; @tichy:07]. Arguably the most exciting result came with the discovery of the “superkick” configuration, where two equal-mass black holes have equal and opposite spins aligned in the orbital plane, leading to kicks of $>3000$ km/s [@gonzalez:07b; @campanelli:07; @tichy:07]. If such a situation were realized in nature, the resulting black hole would certainly be ejected from the host galaxy, leaving behind an empty nuclear host [@merritt:04]. Some of the many other possible ramifications include offset AGN, displaced star clusters, or unusual accretion modes. These and other signatures are discussed in detail below in section \[observations\].
Analogous to the PN waveform matching mentioned above, there has been a good deal of analytic modeling of the kicks calculated by the NR simulations [@schnittman:07a; @schnittman:08a; @boyle:08; @racine:09]. Simple empirical fits to the NR data are particularly useful for incorporating the effects of recoil into cosmological N-body simulations that evolve SMBHs along with merging galaxies [@baker:07b; @campanelli:07; @lousto:09; @vanmeter:10b]. While the astrophysical impacts of large kicks are primarily Newtonian in nature (even a kick of $v\sim 3000$ km/s is only $1\%$ of the speed of light), the underlying causes, while only imperfectly understood, clearly point to strong non-linear gravitational forces at work [@pretorius:07; @schnittman:08a; @rezzolla:10; @jaramillo:12; @rezzolla:13].
Pure electromagnetic fields {#EM_fields}
---------------------------
Shortly after the 2006–07 revolution, many groups already began looking for the next big challenge in numerical relativity. One logical direction was the inclusion of electromagnetic fields in the simulations, solving the coupled Einstein-Maxwell equations throughout a black hole merger. The first to do so was Palenzuela et al. [@palenzuela:09], who considered an initial condition with zero electric field and a uniform magnetic field surrounding an equal-mass, non-spinning binary a couple orbits before merger. The subsequent evolution generates E-fields twisted around the two BHs, while the B-field remains roughly vertical, although it does experience some amplification (see Fig. \[fig:palenzuela\]).
![\[fig:palenzuela\] Magnetic and electric field configurations around binary black hole $40M$ ([*left*]{}) and $20M$ ([*right*]{}) before merger. The electric fields get twisted around the black holes, while the magnetic fields remain roughly vertical. \[reproduced from Palenzuela et al. 2009, [*PRL*]{} [ **103**]{}, 081101\]](Palenzuela-1.eps "fig:"){width="45.00000%"} ![\[fig:palenzuela\] Magnetic and electric field configurations around binary black hole $40M$ ([*left*]{}) and $20M$ ([*right*]{}) before merger. The electric fields get twisted around the black holes, while the magnetic fields remain roughly vertical. \[reproduced from Palenzuela et al. 2009, [*PRL*]{} [ **103**]{}, 081101\]](Palenzuela-2.eps "fig:"){width="45.00000%"}
The EM power from this system was estimated by integrating the radial Poynting flux through a spherical shell at large radius. They found only a modest ($30-40\%$) increase in EM energy, but there was a clear transient quadrupolar Poynting burst of power coincident with the GW signal, giving one of the first hints of astrophysical EM counterparts from NR simulations. This work was followed up by a more thorough study in [@moesta:10; @palenzuela:10a], which showed that the EM power $L_{\rm EM}$ scaled like the square of the total BH spin and proportional to $B^2$, as would be expected for a Poynting flux-powered jet [@blandford:77].
Force-free simulations {#force_free}
----------------------
In [@palenzuela:10b; @palenzuela:10c], Palenzuela and collaborators extended their vacuum simulations to include force-free electrodynamics. This is an approximation where a tenuous plasma is present, and can generate currents and magnetic fields, but carries no inertia to push those fields around. They found that any moving, spinning black hole can generate Poynting flux and a Blandford-Znajek-type jet [@blandford:77]. Compared to the vacuum case, force-free simulations of a merging binary predict significant amplification of EM power by a factor of $\sim 10 \times$, coincident with the peak GW power [@palenzuela:10c]. For longer simulations run at higher accuracy, [@moesta:12; @alic:12] found an even greater $L_{\rm
EM}$ amplification of $\sim 30 \times$ that of electro-vacuum.
M/HD simulations {#MHD_simulations}
----------------
As mentioned above in section \[disk\_theory\], if there is an appreciable amount of gas around the binary BH, it is likely in the form of a circumbinary disk. This configuration has thus been the subject of most (magneto)hydrodynamical simulations. SPH simulations of disks that are not aligned with the binary orbit show a warped disk that can precess as a rigid body, and generally suffer more gas leakage across the inner gap, modulated at twice the orbital frequency [@larwood:97; @ivanov:99; @hayasaki:12]. In many cases, accretion disks can form around the individual BHs [@dotti:07; @hayasaki:08].
Massive disks have the ability to drive the binary towards merger on relatively short time scales [@escala:05; @dotti:07; @cuadra:09] and also align the BH spins at the same time [@bogdanovic:07] (although see also [@lodato:09; @lodato:13] for a counter result). Retrograde disks may be even more efficient at shrinking the binary [@nixon:11] and they may also be quite stable [@nixon:12]. Recent simulations by [@roedig:12] show that the binary will evolve due not only to torques from the circumbinary disk, but also from transfer of angular momentum via gas streaming onto the two black holes. They find that the binary does shrink, and eccentricity can still be excited, but not necessarily at the rates predicted by classical theory.
Following merger, the circumbinary disk can also undergo significant disruption due to the gravitational recoil, as well as the sudden change in potential energy due to the mass loss from gravitational waves. These effects lead to caustics forming in the perturbed disk, in turn leading to shock heating and potentially both prompt and long-lived EM afterglows [@oneill:09; @megevand:09; @rossi:10; @corrales:10; @zanotti:10; @ponce:12; @rosotti:12; @zanotti:13]. Any spin alignment would be critically important for both the character of the prompt EM counterpart, as well as the recoil velocity [@lousto:12; @berti:12].
Due to computational limitations, it is generally only possible to include the last few orbits before merger in a full NR simulation. Since there is no time to allow the system to relax into a quasi-steady state, the specific choice of initial conditions is particularly important for these hydrodynamic merger simulations. Some insight can be gained from Newtonian simulations [@shi:12] as well as semi-analytic models [@liu:10; @rafikov:12; @shapiro:13].
If the disk decouples from the binary well before merger, the gas may be quite hot and diffuse around the black holes [@hayasaki:11]. In that case, uniform density diffuse gas may be appropriate. In merger simulations by [@farris:10; @bode:10; @bogdanovic:11], the diffuse gas experiences Bondi-type accretion onto each of the SMBHs, with a bridge of gas connecting the two before merger. Shock heating of the gas could lead to a strong EM counterpart. As a simple estimate for the EM signal, [@bogdanovic:11] use bremsstrahlung radiation to predict roughly Eddington luminosity peaking in the hard X-ray band.
The first hydrodynamic NR simulations with disk-like initial conditions were carried out by [@farris:11] by allowing the disk to relax into a quasi-steady state before turning the GR evolution on. They found disk properties qualitatively similar to classical Newtonian results, with a low-density gap threaded by accretion streams at early times, and largely evacuated at late times when the binary decouples from the disk. Due to the low density and high temperatures in the gap, they estimate the EM power will be dominated by synchrotron (peaking in the IR for $M=10^8 M_\odot$), and reach Eddington luminosity. An analogous calculation was carried out by [@bode:12], yet they find EM luminosity orders of magnitude smaller, perhaps because they do not relax the initial disk for as long.
Most recently, circumbinary disk simulations have moved from purely hydrodynamic to magneto-hydrodynamic (MHD), which allows them to dispense with alpha prescriptions of viscosity and incorporate the true physical mechanism behind angular momentum in accretion disks: magnetic stresses and the magneto-rotational instability [@balbus:98]. Newtonian MHD simulations of circumbinary disks find large-scale $m=1$ modes growing in the outer disk, modulating the accretion flow across the gap [@shi:12]. Similar modes were seen in [@noble:12], who used a similar procedure as [@farris:11] to construct a quasi-stable state before allowing the binary to merge. They find that the MHD disk is able to follow the inspiraling binary to small separations, showing little evidence for the decoupling predicted by classical disk theory. However, the simulations of [@noble:12] use a hybrid space-time based on PN theory [@gallouin:12] that breaks down close to merger. Furthermore, while fully relativistic in its MHD treatment, the individual black holes are excised from the simulation due to computational limitations, making it difficult to estimate EM signatures from the inner flow. Farris et al. [@farris:12] have been able to overcome this issue and put the BHs on the grid with the MHD fluid. They find that the disk decouples at $a \approx 10M$, followed by a decrease in luminosity before merger, and then an increase as the gap fills in and resumes normal accretion, as in [@milos:05].
Giacomazzo et al. [@giacomazzo:12] carried out MHD merger simulations with similar initial conditions to both [@palenzuela:10a] and [@bode:10], with diffuse hot gas threaded by a uniform vertical magnetic field. Unlike in the force-free approximation, the inclusion of significant gas leads to a remarkable amplification of the magnetic field, which is compressed by the accreting fluid. [@giacomazzo:12] found the B-field increased by of a factor of 100 during merger, corresponding to an increase in synchrotron power by a factor of $10^4$, which could easily lead to super-Eddington luminosities from the IR through hard X-ray bands.
The near future promises a self-consistent, integrated picture of binary BH-disk evolution. By combining the various methods described above, we can combine multiple MHD simulations at different scales, using the results from one method as initial conditions for another, and evolve a circumbinary disk from the parsec level through merger and beyond.
Radiation transport {#radiation}
-------------------
Even with high resolution and perfect knowledge of the initial conditions, the value of the GRMHD simulations is limited by the lack of radiation transport and accurate thermodynamics, which have only recently been incorporated into local Newtonian simulations of steady-state accretion disks [@hirose:09a; @hirose:09b]. Significant future work will be required to incorporate the radiation transport into a fully relativistic global framework, required not just for accurate modeling of the dynamics, but also for the prediction of EM signatures that might be compared directly with observations.
![\[fig:pandurata\] A preliminary calculation of the broad-band spectrum produced by the GRMHD merger of [@giacomazzo:12], sampled near the peak of gravitational wave emission. Synchrotron and bremsstrahlung seeds from the magnetized plasma are ray-traced with [Pandurata]{} [@schnittman:13b]. Inverse-Compton scattering off hot electrons in a diffuse corona gives a power-law spectrum with cut-off around $kT_e$. The total mass is $10^7 M_\odot$ and the gas has $T_e = 100$ keV and optical depth of order unity.](pandurata.eps){width="65.00000%"}
Some recent progress has been made by using the relativistic Monte Carlo ray-tracing code [Pandurata]{} as a post-processor for MHD simulations of single accretion disks [@schnittman:13a; @schnittman:13b], reproducing soft and hard X-ray spectral signatures in agreement with observations of stellar-mass black holes. Applying the same ray-tracing approach to the MHD merger simulations of [@giacomazzo:12], we can generate light curves and broad-band spectra, ranging from synchrotron emission in the IR up through inverse-Compton peaking in the X-ray. An example of such a spectrum is shown in Figure \[fig:pandurata\], corresponding to super-Eddington luminosity at the peak of the EM and GW emission. Since the simulation in [@giacomazzo:12] does not include a cooling function, we simply estimate the electron temperature as 100 keV, similar to that seen in typical AGN coronas. Future work will explore the effects of radiative cooling within the NR simulations, as well as incorporating the dynamic metric into the ray-tracing analysis.
Of course, the ultimate goal will be to directly incorporate radiation transport as a dynamical force within the GRMHD simulations. Significant progress has been made recently in developing accurate radiation transport algorithms in a fully covariant framework [@ohsuga:11; @jiang:12; @sadowski:13], and we look forward to seeing them mature to the point where they can be integrated into dynamic GRMHD codes. In addition to [Pandurata]{}, there are a number of other relativistic ray-tracing codes (e.g., [@dolence:09; @shcherbakov:11]), currently based on the Kerr metric, which may also be adopted to the dynamic space times of merging black holes.
OBSERVATIONS: PAST, PRESENT, AND FUTURE {#observations}
=======================================
One way to categorize EM signatures is by the physical mechanism responsible for the emission: stars, hot diffuse gas, or circumbinary/accretion disks. In Figure \[source\_chart\], we show the diversity of these sources, arranged according the spatial and time scales on which they are likely to occur [@schnittman:11]. Over the course of a typical galaxy merger, we should expect the system to evolve from the upper-left to the lower-center to the upper-right regions of the chart. Sampling over the entire observable universe, the number of objects detected in each source class should be proportional to the product of the lifetime and observable flux of that object.
![\[source\_chart\] Selection of potential EM sources, sorted by timescale, typical size of emission region, and physical mechanism (blue/[*italic*]{} = stellar; yellow/Times-Roman = accretion disk; green/[**bold**]{} = diffuse gas/miscellaneous). The evolution of the merger proceeds from the upper-left through the lower-center, to the upper-right.](ss_v2.eps){width="85.00000%"}
Note that most of these effects are fundamentally Newtonian, and many are only indirect evidence of SMBH mergers, as opposed to the prompt EM signatures described above. Yet they are also important in understanding the complete history of binary BHs, as they are crucial for estimating the number of sources one might expect at each stage in a black hole’s evolution. If, for example, we predict a large number of bright binary quasars with separations around $0.1$ pc, and find no evidence for them in any wide-field surveys (as has been the case so far, with limited depth and temporal coverage), we would be forced to revise our theoretical models. But if the same rate calculations accurately predict the number of dual AGN with separations of $\sim
1-10$ kpc, and GW or prompt EM detections are able to confirm the number of actual mergers, then we might infer the lack of binary quasars is due to a lack of observability, as opposed to a lack of existence.
The long-term goal in observing EM signatures will be to eventually fill out a plot like that of Figure \[source\_chart\], determining event rates for each source class, and checking to make sure we can construct a consistent picture of SMBH-galaxy co-evolution. This is indeed an ambitious goal, but one that has met with reasonable success in other fields, such as stellar evolution or even the fossil record of life on Earth.
Stellar Signatures
------------------
On the largest scales, we have strong circumstantial evidence of supermassive BH mergers at the centers of merging galaxies. From large optical surveys of interacting galaxies out to redshifts of $z \sim
1$, we can infer that $5-10\%$ of massive galaxies are merging at any given time, and the majority of galaxies with $M_{\rm gal} \gtrsim 10^{10} M_\odot$ have experienced a major merger in the past 3 Gyr [@bell:06; @mcintosh:08; @deravel:09; @bridge:10], with even higher merger rates at redshifts $z\sim 1-3$ [@conselice:03]. At the same time, high-resolution observations of nearby galactic nuclei find that every large galaxy hosts a SMBH in its center [@kormendy:95]. Yet we see a remarkably small number of dual AGN [@komossa:03; @comerford:09], and only one known source with an actual binary system where the BHs are gravitationally bound to each other [@rodriguez:06; @rodriguez:09]. Taken together, these observations strongly suggest that when galaxies merge, the merger of their central SMBHs inevitably follows, and likely occurs on a relatively short time scale, which would explain the apparent scarcity of binary BHs (although recent estimates by [@hayasaki:10] predict as many as $10\%$ of AGNs with $M\sim 10^7
M_\odot$ might be in close binaries with $a\sim 0.01$ pc). The famous “M-sigma” relationship between the SMBH mass and the velocity dispersion of the surrounding bulge also points to a merger-driven history over a wide range of BH masses and galaxy types [@gultekin:09].
There is additional indirect evidence for SMBH mergers in the stellar distributions of galactic nuclei, with many elliptical galaxies showing light deficits (cores), which correlate strongly with the central BH mass [@kormendy:09]. The cores suggest a history of binary BHs that scour out the nuclear stars via three-body scattering [@milosavljevic:01; @milosavljevic:02; @merritt:07], or even post-merger relaxation of recoiling BHs [@merritt:04; @boylan-kolchin:04; @gualandris:08; @guedes:09].
While essentially all massive nearby galaxies appear to host central SMBHs, it is quite possible that this is not the case at larger redshifts and smaller masses, where major mergers could lead to the complete ejection of the resulting black hole via large recoils. By measuring the occupation fraction of SMBHs in distant galaxies, one could infer merger rates and the distribution of kick velocities [@schnittman:07a; @volonteri:07; @schnittman:07b; @volonteri:08a; @volonteri:10]. The occupation fraction will of course also affect the LISA event rates, especially at high redshift [@sesana:07].
Another indirect signature of BH mergers comes from the population of stars that remain bound to a recoiling black hole that gets ejected from a galactic nucleus [@komossa:08a; @merritt:09; @oleary:09]. These stellar systems will appear similar to globular clusters, yet with smaller spatial extent and much larger velocity dispersions, as the potential is completely dominated by the central SMBH. With multi-object spectrometers on large ground-based telescopes, searching for these stellar clusters in the Milky Way halo or nearby galaxy clusters ($d \lesssim 40$ Mpc) is technically realistic in the immediate future.
Gas Signatures: Accretion Disks
-------------------------------
As discussed above in section \[disk\_theory\], circumbinary disks will likely have a low-density gap within $r\approx 2a$, although may still be able to maintain significant gas accretion across this gap, even forming individual accretion disks around each black hole. The most sophisticated GRMHD simulations suggest that this accretion can be maintained even as the binary is rapidly shrinking due to gravitational radiation [@noble:12]. If the inner disks can survive long enough, the final inspiral may lead to a rapid enhancement of accretion power as the fossil gas is plowed into the central black hole shortly before merger [@armitage:02; @chang:10]. For small values of $q$, a narrow gap could form in the inner disk, changing the AGN spectra in a potentially observable way [@gultekin:12; @mckernan:13].
Regardless of [*how*]{} the gas reaches the central BH region, the simulations described above in section \[simulations\] all seem to agree that even a modest amount of magnetized gas can lead to a strong EM signature. If the primary energy source for heating the gas is gravitational [@vanmeter:10], then typical efficiencies will be on the order of $\sim 1-10$%, comparable to that expected for standard accretion in AGN, although the much shorter timescales could easily lead to super-Eddington transients, depending on the optical depth and cooling mechanisms of the gas[@krolik:10].
However, if the merging BHs are able to generate strong magnetic fields [@palenzuela:09; @moesta:10; @palenzuela:10b; @giacomazzo:12], then hot electrons could easily generate strong synchrotron flux, or highly relativistic jets may be launched along the resulting BH spin axis, converting matter to energy with a Lorentz boost factor of $\Gamma \gg 1$. Even with purely hydrodynamic heating, particularly bright and long-lasting afterglows may be produced in the case of large recoil velocities, which effectively can disrupt the entire disk, leading to strong shocks and dissipation [@lippai:08; @shields:08; @schnittman:08b; @megevand:09; @rossi:10; @anderson:10; @corrales:10; @tanaka:10a; @zanotti:10]. Long-lived afterglows could be discovered in existing multi-wavelength surveys, but successfully identifying them as merger remnants as opposed to obscured AGN or other bright unresolved sources would require improved pipeline analysis of literally millions of point sources, as well as extensive follow-up observations [@schnittman:08b].
For many of these large-kick systems, we may observe quasar activity for millions of years after, with the source displaced from the galactic center, either spatially [@kapoor:76; @loeb:07; @volonteri:08b; @civano:10; @dottori:10; @jonker:10] or spectroscopically [@bonning:07; @komossa:08c; @boroson:09; @robinson:10]. However, large offsets between the redshifts of quasar emission lines and their host galaxies have also been interpreted as evidence of pre-merger binary BHs [@bogdanovic:09; @dotti:09; @tang:09; @dotti:10b] or due to the large relative velocities in merging galaxies [@heckman:09; @shields:09a; @vivek:09; @decarli:10], or “simply” extreme examples of the class of double-peaked emitters, where the line offsets are generally attributed to the disk [@gaskell:88; @eracleous:97; @shields:09b; @chornock:10; @gaskell:10]. An indirect signature for kicked BHs could potentially show up in the statistical properties of active galaxies, in particular in the relative distribution of different classes of AGN in the “unified model” paradigm [@komossa:08b; @blecha:11].
For systems that open up a gap in the circumbinary disk, another EM signature may take the form of a quasar suddenly turning on as the gas refills the gap, months to years after the BH merger [@milos:05; @shapiro:10; @tanaka:10b]. But again, these sources would be difficult to distinguish from normal AGN variability without known GW counterparts. Some limited searches for this type of variability have recently been carried out in the X-ray band [@kanner:13], but for large systematic searches, we will need targeted time-domain wide-field surveys like PTF, Pan-STARRS, and eventually LSST. One of the most valuable scientific products from these time-domain surveys will be a better understanding of what is the range of variability for normal AGN, which will help us distinguish when an EM signal is most likely due to a binary [@tanaka:13].
In addition to the many potential prompt and afterglow signals from merging BHs, there has also been a significant amount of theoretical and observational work focusing on the early precursors of mergers. Following the evolutionary trail in Figure 1, we see that shortly after a galaxy merges, dual AGN may form with typical separations of a few kpc [@komossa:03; @comerford:09], sinking to the center of the merged galaxy on a relatively short timescale ($\lesssim$ 1 Gyr) due to dynamical friction [@begelman:80]. The galaxy merger process is also expected to funnel a great deal of gas to the galactic center, in turn triggering quasar activity [@hernquist:89; @kauffmann:00; @hopkins:08; @green:10]. At separations of $\sim 1$ pc, the BH binary (now “hardened” into a gravitationally bound system) could stall, having depleted its loss cone of stellar scattering and not yet reached the point of gravitational radiation losses [@milosavljevic:03]. Gas dynamical drag from massive disks ($M_{\rm disk} \gg M_{\rm BH}$) leads to a prompt inspiral ($\sim 1-10$ Myr), in most cases able to reach sub-parsec separations, depending on the resolution of the simulation [@escala:04; @kazantzidis:05; @escala:05; @dotti:07; @cuadra:09; @dotti:09b; @dotti:10a].
At this point, a proper binary quasar is formed, with an orbital period of months to decades, which could be identified by periodic accretion [@macfadyen:08; @hayasaki:08; @haiman:09a; @haiman:09b], density waves in the disk [@hayasaki:09], or periodic red-shifted broad emission lines [@bogdanovic:08; @shen:09; @loeb:10; @montuori:11]. If these binary AGN systems do in fact exist, spectroscopic surveys should be able to identify many candidates, which may then be confirmed or ruled out with subsequent observations over relatively short timescales ($\sim 1-10$ yrs), as the line-of-site velocities to the BHs changes by an observable degree. This approach has been attempted with various initial spectroscopic surveys, but as yet, no objects have been confirmed to be binaries by multi-year spectroscopic monitoring [@boroson:09; @lauer:09; @chornock:10; @eracleous:12].
Gas Signatures: Diffuse Gas; “Other”
------------------------------------
In addition to the many disk-related signatures, there are also a number of potential EM counterparts that are caused by the accretion of diffuse gas in the galaxy. For the Poynting flux generated by the simulations of section \[simulations\], transient bursts or modulated jets might be detected in all-sky radio surveys [@kaplan:11; @oshaughnessy:11]. For BHs that get significant kicks at the time of merger, we expect to see occasional episodes of Bondi accretion as the BH oscillates through the gravitational potential of the galaxy over millions of years, as well as off-center AGN activity [@blecha:08; @fujita:09; @guedes:10; @sijacki:10]. On larger spatial scales, the recoiling BH could also produce trails of over-density in the hot interstellar gas of elliptical galaxies [@devecchi:09]. Also on kpc–Mpc scales, X-shaped radio jets have been seen in a number of galaxies, which could possibly be due to the merger and subsequent spin-flip of the central BHs [@merritt:02].
Another potential source of EM counterparts comes not from diffuse gas, or accretion disks, but the occasional capture and tidal disruption of normal stars by the merging BHs. These tidal disruption events (TDEs), which also occurs in “normal” galaxies [@rees:88; @komossa:99; @halpern:04], may be particularly easy to identify in off-center BHs following a large recoil [@komossa:08a]. TDE rates may be strongly increased prior to the merger [@chen:09; @stone:10; @seto:10; @schnittman:10; @chen:11; @wegg:11], but the actual disruption signal may be truncated by the pre-merger binary [@liu:09], and post-merger recoil may also reduce the rates [@li:12]. These TDE events are likely to be seen by the dozen in coming years with Pan-STARRS and LSST [@gezari:09]. In addition to the tidal disruption scenario, in [@schnittman:10] we showed how gas or stars trapped at the stable Lagrange points in a BH binary could evolve during inspiral and eventually lead to enhanced star formation, ejected hyper-velocity stars, highly-shifted narrow emission lines, and short bursts of super-Eddington accretion coincident with the BH merger.
A completely different type of EM counterpart can be seen with pulsar timing arrays (PTAs). In this technique, small time delays ($\lesssim 10$ ns) in the arrival of pulses from millisecond radio pulsars would be direct evidence of extremely low-frequency (nano-Hertz) gravitational waves from massive ($\gtrsim
10^8 M_\odot$) BH binaries [@jenet:06; @sesana:08; @sesana:09; @jenet:09; @seto:09; @pshirkov:10; @vanhaasteren:10; @sesana:10]. By cross-correlating the signals from multiple pulsars around the sky, we can effectively make use of a GW detector the size of the entire galaxy. For now, one of the main impediments to GW astronomy with pulsar timing is the relatively small number of known, stable millisecond radio pulsars. Current surveys are working to increase this number and the uniformity of their distribution on the sky [@lee:13].
Even conservative estimates suggest that PTAs are probably only about ten years away from a positive detection of the GW stochastic background signal from the ensemble of SMBH binaries throughout the universe [@sesana:13]. The probability of resolving an individual source is significantly smaller, but if it were detected, would be close enough ($z \lesssim 1$) to allow for extensive EM follow-up, unlike many of the expected LISA sources at $z \gtrsim 5$. Also, unlike LISA sources, PTA sources would be at an earlier stage in their inspiral and thus be much longer lived, allowing for even more extensive study. A sufficiently large sample of such sources would even allow us to test whether they are evolving due to GW emission or gas-driven migration [@kocsis:11; @tanaka:12; @sesana:12] (a test that might also be done with LISA with only a single source with sufficient signal-to-noise [@yunes:11]).
CONCLUSION
==========
Black holes are fascinating objects. They push our intuition to the limits, and never cease to amaze us with their extreme behavior. For a high-energy theoretical astrophysicist, the only thing more exciting than a real astrophysical black hole is [*two*]{} black holes, destroying everything in their path as they spiral together towards the point of no return. Thus one can easily imagine the frustration that stems from our lack of ability to actually see such an event, despite the fact that it outshines the entire observable universe. And the path forwards does not appear to be a quick one, at least not for gravitational-wave astronomy.
One important step along this path is the engagement of the broader (EM) astronomy community. Direct detection of gravitational waves will not merely be a confirmation of a century-old theory—one more feather in Einstein’s Indian chief head-dress—but the opening of a window through which we can observe the entire universe at once, eagerly listening for the next thing to go bang in the night. And when it does, all our EM eyes can swing over to watch the fireworks go off. With a tool as powerful as coordinated GW/EM observations, we will be able to answer many of the outstanding questions in astrophysics:
How were the first black holes formed? Where did the first quasars come from? What is the galaxy merger rate as a function of galaxy mass, mass ratio, gas fraction, cluster environment, and redshift? What is the mass function and spin distribution of the central BHs in these merging (and non-merging) galaxies? What is the central environment around the BHs, prior to merger: What is the quantity and quality (temperature, density, composition) of gas? What is the stellar distribution (age, mass function, metallicity)? What are the properties of the circumbinary disk? What is the time delay between galaxy merger and BH merger?
These are just a few of the mysteries that will be solved with the routine detection and characterization of SMBH mergers, may we witness them speedily in our days!
We acknowledge helpful conversations with John Baker, Manuela Campanelli, Bruno Giacomazzo, Bernard Kelly, Julian Krolik, Scott Noble, and Cole Miller.
References {#references .unnumbered}
==========
[99]{}
Abell P A
Alic D, Moesta P, Rezzolla L, Zanotti O and Jaramillo J L 2012 36
Anderson M, Lehner L, Megevand M and Neilsen D 2010 044004
Apostolatos T A, Cutler C, Sussman G J and Thorne K S 1994 6274
Armitage P J and Natarajan P 2002 9–12
Armitage P J and Natarajan P 2005 921–927
Arnowitt R, Deser S and Misner C W 2008 [ *Gen. Rel. Grav.*]{} [**40**]{} 1997–2007
Artymowicz P, Clarke C J, Lubow S H, and Pringle J E 1991 L35–L38
Artymowicz P and Lubow S H 1994 651–667
Artymowicz P and Lubow S H 1996 77
Baker J G, Centrella J, Choi D-I, Koppitz M, and van Meter J R 2006 111102
Baker J G, Centrella J, Choi D-I, Koppitz M, van Meter J R and Miller M C 2006 93–96
Baker J G, Campanelli M, Pretorius F and Zlochower Y 2007 S25–S31
Baker J G, Boggs W D, Centrella J, Kelly B J, McWilliams S T, Miller M C and van Meter J R 2007 1140–1144
Balbus S A and Hawley J F 1998 [*Rev. Mod. Phys.*]{} 70 1–53
Buonanno A and Damour T 1999 084006
Baruteau C, Ramirez-Ruiz E and Masset F 2012 L65
Begelman M C, Blandford R D and Rees M J 1980 [*Nature*]{} [**287**]{} 307–309
Bekenstein J D 1973 657
Bell E F 2006 [*Astroph. J.*]{} [**652**]{} 270–276
Berti E, Cardoso V, Gonzalez J, Sperhake U, Hannam M, Husa S and Brugmann B 2007 064034
Berti E and Volonteri M 2008 822
Berti E, Kesden M and Sperhake U 2012 124049
Blanchet L, Quasailah M S S and Will C M 2005 508
Blanchet L 2006 [*LRR*]{} 9 4
Blandford R D and Znajek R L 1977 433
Blecha L and Loeb A 2008 1311–1325
Blecha L, Cox T J, Loeb A and Hernquist L 2011 2154–2182
Bloom J 2009 \[arXiv:0902.1527\]
Bode T, Haas R, Bogdanovic T, Laguna P and Shoemaker D 2010 1117
Bode T, Bogdanovic T, Haas R, Healy J, Laguna P and Shoemaker D 2010 45
Bogdanovic T, Reynolds C S and Miller M C 2007 147
Bogdanovic T, Smith B D, Sigurdsson S and Eracleous M 2008 455–480
Bogdanovic T, Eracleous M and Sigurdsson S 2009 288–292
Bogdanovic T, Bode T, Haas R, Laguna P and Shoemaker D 2011 094020
Bonning E W, Shields G A and Salviander S 2007 13–16
Bonnor W B and Rotenberg M A [*Proc. Royal Soc. A*]{} [**265**]{} 1320
Boroson T A and Lauer T R 2009 [*Nature*]{} [ **458**]{} 53–55
Boyer R H and Lindquist R W 1967 [*J. Mod. Phys.*]{} [**8**]{} 265
Boylan-Kolchin M, Ma C-P and Quataert E 2004 [*Astroph. J. Lett.*]{} [**613**]{} 37–40
Boyle L and Kesden M 2008 024017
Boyle M, Brown D A, Kidder L E, Mroue A H, Pfeiffer H P, Scheel M A, Cook G P and Teukolsky S A 2007 124038
Bridge C R, Carlberg R G and Sullivan M 2010 [ *Astroph. J.*]{} [**709**]{} 1067–1082
Campanelli M, Lousto C, Marronetti P and Zlochower Y 111101
Campanelli M, Lousto C and Zlochower Y 041501
Campanelli M, Lousto C, Zlochower Y and Merritt D 2007 5–8
Centrella J, Baker J G, Kelly B J and van Meter J R 2010 [*Rev. Mod. Phys.*]{} [**82**]{} 3069–3119
Chang P, Strubbe L E, Menou K and Quataert E 2010 2007–2016
Chen X, Madau P, Sesana A and Liu F K 2009 149–152
Chen X, Sesana A, Madau P and Liu F K 2011 13
Chornock R, Bloom J S, Cenko S B, Filippenko A V, Silverman J M, Hicks M D, Lawrence K J, Mendez A J, Rafelski M and Wolfe A M 2010 39–43
Civano F [*et al.*]{} 2010 209–222
Comerford J M 2009 [*Astroph. J.*]{} [**698**]{} 956–965
Conselice C J, Bershady M A, Dickinson M and Papovich C 2003 [*Astron. J.*]{} [**126**]{} 1183–1207
Corrales L R, Haiman Z and MacFadyen A 2010 947–962
Cuadra J, Armitage P J, Alexander R D and Begelman M C 2009 1423–1432
Cutler C, Finn L S, Poisson E and Sussman G J 1993
Cutler C 1998 7089
Damour T and Gopakumar A 2006 124006
Decarli R, Falomo R, Treves A and Barattini M 2010 [*Astron. & Astroph.*]{} [**511**]{} 27
de Ravel L 2009 [*Astron. & Astroph.*]{} [**498**]{} 379–397
Devecchi B, Rasia E, Dotti M, Volonteri M and Colpi M 2009 633–640
Dolence J C, Gammie C F, Moscibrodzka M and Leung P K 387
D’Orazio D J, Haiman Z and MacFadyen A 2012 submitted \[arXiv:1210.0536\]
Dotti M, Colpi M, Haardt F and Mayer L 2007 956–962
Dotti M, Montuori C, Decarli R, Volonteri M, Colpi M and Haardt F 2009 L73–L77
Dotti M, Ruszkowski M, Paredi L, Colpi M, Volonteri M and Haardt F 2009 1640–1646
Dotti M, Volonteri M, Perego A, Colpi M, Ruszkowski M and Haardt F 2010 682–690
Dotti M and Ruszkowski M 2010 37–40
Dottori H, Diaz R J, Albacete-Colombo J F and Mast D 2010 42–46
Elvis M 1978 129
Eracleous M, Halpern J P, Gilbert A M, Newman J A and Filippenko A V 1997 216
Eracleous M, Boroson T A, Halpern J P and Liu J 2012 23
Escala A, Larson R B, Coppi P S and Mardones D 2004 765–777
Escala A, Larson R B, Coppi P S and Mardones D 2005 152–166
Estabrook F B and Wahlquist H D 1975 [ *Gen. Rel. Grav.*]{} [**6**]{}, 439–447
Faller J E, Bender P L, Hall J L, Hils D and Stebbins R T 1989 [*Adv. Space Res.*]{} [**9**]{} 107–111
Farris B D, Liu Y-K and Shapiro S L 2010 084008
Farris B D, Liu Y-K and Shapiro S L 2011 024024
Farris B D, Gold R, Pschalidis V, Etienne Z B and Shapiro S L 2012 221102
Favata M, Hughes S A and Holz D E 2004 5
Flanagan E E and Hughes S A 1998 4535
Fujita Y 2009 1050–1057
Gallouin L, Nakano H, Yunes N and Campanelli M 2012 235013
Gaskell M C 1988 [*LNP*]{} [**307**]{} 61
Gaskell M C 2010 [*Nature*]{} [**463**]{} E1
Gezari S 2009 1367–1379 698, 1367.
Giacomazzo B, Baker J G, Miller M C, Reynolds C S and van Meter J R 2012 15
Goldreich P and Tremaine S 1979 857–871
Gonzalez J A, Hannam M, Sperhake U, Brugmann B and Husa S 2007 091101
Gonzalez J A, Hannam M, Sperhake U, Brugmann B and Husa S 2007 231101
Green P J, Myers A D, Barkhouse W A, Mulchaey J S, Bennert V N, Cox T J and Aldcroft T L 2010 [*Astroph. J.*]{} [ **710**]{} 1578–1588
Gualandris A and Merritt D 2008 [*Astroph. J.*]{} [**678**]{} 780–797
Guedes J, Madau P, Kuhlen M, Diemand J and Zemp M 2009 [*Astroph. J.*]{} [**702**]{} 890–900
Guedes J, Madau P, Mayer L and Callegari S 2011 125
Gultekin K et al. 2009 198–221
Gultekin K and Miller J M 2012 90
Gunther R and Kley W 2002 Å[**387**]{} 550
Gunther R, Schafer C and Kley W 2004 Å[**423**]{} 559
Hahn S G and Lindquist R W 1964 [*Ann. Phys.*]{} [**29**]{} 304
Haiman Z, Kocsis B, Menou K, Lippai Z and Frei Z 2009 [*Class. Quant. Grav.*]{} [**26**]{} 094032
Haiman Z, Kocsis B and Menou K 2009 1952–1969
Halpern J P, Gezari S and Komossa S 2004 572
Hayasaki K, Mineshige S and Ho L C 2008 1134–1140
Hayasaki K and Okazaki A T 2009 5
Hayasaki K, Ueda Y and Isobe N 2010, [*PASJ*]{} [**62**]{} 1351
Hayasaki K 2011 14 5
Hayasaki K, Saito H and Mineshige S 2012 [ *PASJ*]{} submitted \[arXiv:1211.5137\]
Heckman T M, Krolik J H, Moran S M, Schnittman J D and Gezari S 2009 363–367
Hernquist L 1989 [*Nature*]{} [**340**]{} 687
Herrmann F, Hinder I, Shoemaker D, Laguna P and Matzner R A 2007 430–436
Herrmann F, Hinder I, Shoemaker D and Laguna P 2007 S33
Hinder I, Vaishnav B, Herrmann F, Shoemaker D M and Laguna P 2008 081502
Hirose S, Krolik J H and Blaes O 2009 16
Hirose S, Blaes O and Krolik J H 2009 781–788
Hopkins P F, Hernquist L, Cox T J and Keres D 2008 356
Ivanov P B, Papaloizou J C B and Polnarev A G 1999 79
Jaramillo J L, Macedo R P, Moesta P and Rezzolla L 2012 084030
Jenet F A 2006 1571–1576
Jenet F A 2009 \[arXiv:0909.1058\]
Jiang Y-F, Stone J M and Davis S W 2012 14
Jonker P G, Torres M A P, Fabian A C, Heida M, Miniutti G and Pooley D 2010 645–650
Kanner J, Baker J G, Blackburn L, Camp J, Mooley K, Mushotzky R and Ptak A 2013 submitted \[arXiv:1305.5874\]
Kaplan D L, O’Shaugnessy R, Sesana A and Volonteri M 2011 37
Kapoor R C 1976 Pramãna [**7**]{} 334–343
Kauffmann G and Haehnelt M 2000 576
Kazantzidis S, Mayer L, Colpi M, Madau P, Debattista V P, Wadsley J, Stadel J, Quinn T and Moore B 2005 L67–L70
Kidder L E 1995, 821
Kocsis B, Frei Z, Haiman Z and Menou K 2006 27–37
Kocsis B, Haiman Z and Menou K 2008 870–887
Kocsis B and Sesana A 2011 1467
Kocsis B, Haiman Z and Loeb A 2012 2660
Kocsis B, Haiman Z and Loeb A 2012 2680
Komossa S and Bode N 1999 [*Astron. & Astroph.*]{} [**343**]{} 775–787
Komossa S, Burwitz V, Hasinger G, Predehl P, Kaastra J S and Ikebe Y 2003 [*Astroph. J. Lett.*]{} [**582**]{} 15–19
Komossa S and Merritt D 2008a 21–24
Komossa S and Merritt D 2008b 89–92
Komossa S, Zhou H and Lu H 2008 81–84
Koppitz M, Pollney D, Reisswig C, Rezzolla L, Thornburg J, Diener P and Schnetter E 2007 041102
Kormendy J and Richstone D 1995 [*Ann. Rev. Astron. & Astroph.*]{} [**33**]{} 581
Kormendy J, Fisher D B, Cornell M E and Bender R 2009 [*Astroph. J. Suppl.*]{} [**182**]{} 216–309; Kormendy J and Bender R 2009 [*Astroph. J. Lett.*]{} [**691**]{} 142–146
Krolik J H 2010 774–779
Lang R N and Hughes S A 2006 122001
Lang R N and Hughes S A 2008 1184–1200
Lang R N and Hughes S A 2009 [ *Class. Quant. Grav.*]{} [**26**]{} 094035
Larwood J D and Papaloizou J C B 1997 288
Lauer T A and Boroson T R 2009 930–938
Lee K J 2013 688
Li S, Liu F K, Berczik P, Chen X and Sperzem R 2012 65
Lippai Z, Frei Z and Haiman Z 2008 5–8
LISA Assessment Study Report (“Yellow Book”) 2011 ESA/SRE(2011)3
Liu F K, Li S and Chen X 2009 133–137
Liu Y T and Shapiro S L 2010 123011
Lodato G, Nayakshin S, King A R and Pringle J E 2009 1392
Lodato G and Gerosa D 2013 30
Loeb A 2007 041103
Loeb A 2010 047503
Lousto C O and Zlochower Y 2009 064018
Lousto C O, Zlochower Y, Dotti M and Volonteri M 2012 084015
Lynden-Bell D 1969 [*Nature*]{} [**223**]{} 690
MacFadyen A I and Milosavljevic M 2008 83–93
McIntosh D H, Guo Y, Hertzberg J, Katz N, Mo H J, van den Bosch F C, and Yang X 2008 [*Mon. Not. Roy. Astron. Soc.*]{} [**388**]{} 1537–1556
McKernan B, Ford K E S, Kocsis B and Haiman Z 2013 1468
McWilliams S T, Thorpe J I, Baker J G and Kelly B J 2010 064014
Megevand M, Anderson M, Frank J, Hirschmann E W, Lehner L, Liebling S L, Motl P M and Neilsen D 2009 024012
Merrit D and Ekers R D 2002 [*Science*]{} [ **297**]{} 1310–1313
Merritt D, Milosavljevic M, Favata M, Hughes S A and Holz D E 2004 [*Astroph. J. Lett.*]{} [**607**]{} 9–12
Merritt D and Milosavljevic M 2005 [*LRR*]{} [ **8**]{} 8
Merritt D, Mikkola S and Szell A 2007 [ *Astroph. J.*]{} [**671**]{} 53–72
Merritt D, Schnittman J D and Komossa S 2009 1690–1710
Milosavljevic M and Merritt D 2001 [ *Astroph. J.*]{} [**563**]{} 34–62
Milosavljevic M, Merritt D, Rest A and van den Bosch F C 2002 [*Mon. Not. Roy. Astron. Soc.*]{} [**331**]{} 51–55
Milosavljevic M and Merritt D 2003 [ *Astroph. J.*]{} [**596**]{} 860–878
Milosavljevic M and Phinney E S 2005 93–96
Moesta P, Palenzuela C, Rezzolla L, Lehner L, Yoshida S and Pollney D 2010 064017
Moesta P, Alic D, Rezzolla L, Zanotti O and Palenzuela C 2012 32
Montuori C, Dotti M, Colpi M, Decarli R and Haardt F 2011 26
Murray S S 2012 [*SPIE*]{} [**8443**]{} 1L
Nixon C J, King A R and Pringle J E 2011 66
Nixon C J 2012 2597
Noble S C, Mundim B C, Nakano H, Krolik J H, Campanelli M, Zlochower Y and Yunes N 2012 51
Novikov I D and Thorne K D 1973 in [*Black Holes*]{} ed. C DeWitt & B S DeWitt (New York: Gordon and Breach)
O’Leary R M and Loeb A 2009 781–786
O’Neill S M, Miller M C, Bogdanovic T, Reynolds C S and Schnittman J D 2009 859–871
O’Shaughnessy R, Kaplan D L, Sesana A and Kamble A 2011 136
Oda M, Gorenstein P, Gursky H, Kellogg E, Schreier E, Tanenbaum H and Giacconi R 1971 1
Ohsuga K and Mineshige S 2011 2
Ostriker J P and Tremaine S D 1975 113–117
Ostriker J P and Hausman M A 1977 125–129
Palenzuela C, Anderson M, Lehner L, Liebling S L and Neilsen D 2009 081101
Palenzuela C, Lehner L and Yoshida S 2010 084007
Palenzuela C, Garrett T, Lehner L and Liebling S L 2010 044045
Palenzuela C, Lehner L and Liebling S L 2010 [*Science*]{} [**329**]{} 927
Pan Y, Buonanno A, Boyle M, Buchman L T, Kidder L E, Pfeiffer H P and Scheel M A 2011 124052
Peres A 1962 [*Phys. Rev.*]{} [**128**]{} 2471
Peters P C and Mathews J 1963 [*Phys. Rev.*]{} [**131**]{} 435–440
Pnce M, Faber J A and Lombardi J C 2012 71
Pretorius F 2005 121101
Pretorius F 2007 \[arXiv:0710.1338\]
Pringle J E 1991 754–259
Pshirkov M S, Baskaran D and Postnov K A 2010 417–423
Racine E, Buonanno A and Kidder L 2009 044010
Rafikov R R 2012 submitted \[arXiv:1205.5017\]
Rees M J 1984 [*Ann. Rev. Astron. Astroph.*]{} [**22**]{} 471
Rees M J 1988 [*Nature*]{} [**333**]{} 523–528
Rezzolla L, Macedo R P and Jaramillo J L 2010 221101
Rezzolla L 2013 \[arXiv:1303.6464\]
Robinson A, Young S, Axon D J, Kharb P and Smith J E 2010 123–126
Rodriguez C, Taylor G B, Zavala R T, Peck A B, Pollack L K and Romani R W 2006 [*Astroph. J.*]{} [**646**]{} 49–60
Rodriguez C, Taylor G B, Zavala R T, Pihlstrom Y M and Peck A B 2009 37
Roedig C, Dotti M, Sesana A, Cuadra J and Colpi M 2011 3033
Roedig C, Sesana A, Dotti M, Cuadra J, Amaro-Seoane P and Haardt F 2012, Å[**545**]{}, A127
Rosotti G P, Lodato G and Price D J 2012 1958
Rossi E M, Lodato G, Armitage P J, Pringle J E and King A R 2010 2021–2035
Sadowski A, Narayan R, Tchekhovskoy A and Zhu Y 2013 3533
Saltpeter E E 1964 435
Schnittman J D and Buonanno A 2007 [ *Astroph. J. Lett.*]{} [**662**]{} 63–66
Schnittman J D 2007 [ *Astroph. J. Lett.*]{} [**667**]{} 133–136
Schnittman J D, Buonanno A, van Meter J R, Baker J G, Boggs W D, Centrella J, Kelly B J and McWilliams S T 2008 044031
Schnittman J D and Krolik J H 835–844
Schnittman J D 2010 39
Schnittman J D 2011 094021
Schnittman J D, Krolik J H and Noble S C 2013 156
Schnittman J D and Krolik J H 2013 submitted \[arXiv:1302.3214\]
Schwarzschild K 1916 [*Prus. Acad. Sci.*]{} 189
Sesana A 2007 6–10
Sesana A, Vecchio A and Colacino C N 2008 192–209
Sesana A, Vecchio A and Volonteri M 2009 2255-2265
Sesana A and Vecchio A 2010 104008
Sesana A, Roedig C, Reynolds M T and Dotti M 2012 860
Sesana A 2013 1
Seto N 2009 L38–L42
Seto N and Muto T 2010 103004
Shakura N I and Sunyaev R A 1973 [*A& A*]{} [**24**]{} 337
Shapiro S L 2010 024019
Shapiro S L 2013 103009
Shcherbakov R V and Huang L 2011 1052
Shen Y and Loeb A 2009 \[arXiv:0912.0541\]
Shi J-M, Krolik J H, Lubow S H and Hawley J F 2012 118
Shields G A and Bonning E W 2008 758–766
Shields G A, Bonning E W and Salviander S 2009 1367–1373
Shields G A, Rosario D J, Smith K L, Bonning E W, Salviander S, Kalirai J S, Strickler R, Ramirez-Ruiz E, Dutton A A, Treu T and Marshall P J 2009 936–941
Sijacki D, Springel V and Haehnelt M 2011 3656
Spergel D 2013 \[arXiv:1305.5422\]
Stone J M and Norman M L 1992 753–790
Stone N and Loeb A 2010 \[arXiv:1004.4833\]
Takeuchi T, Miyama S M and Lin D N C 1996 832
Tanaka T and Menou K 2010 404–422
Tanaka T, Haiman Z and Menou K 2010 642–651
Tanaka T, Menou K and Haiman Z 2012 705
Tanaka T 2013 in press \[arXiv:1303.6279\] 705
Tang S and Grindlay J 2009 1189–1194
Thorpe J I, McWilliams S T, Kelly B J, Fahey R P, Arnaud K and Baker J G 2009 [*Class. Quant. Grav.*]{} [**26**]{} 094026
Thorne K S and Braginsky V B 1976 1–6
Thorne K S 1994 [*Black Holes and Time Warps: Einstein’s Outrageous Legacy*]{} (New York: W W Norton)
Tichy W and Marronetti P 2007 061502
van Haasteren R and Levin Y 2010 2372–2378
van Meter J R, Wise, J H, Miller M C, Reynolds C S, Centrella J, Baker J G, Boggs W D, Kelly B J and McWilliams S T 2010 89–92
van Meter J R, Miller M C, Baker J G, Boggs W D and Kelly B J 2010 1427–1432
Vivek M, Srianand R, Noterdaeme P, Mohan V and Kuriakosde V C 2009 L6–L9
Volonteri M 2007 5–8
Volonteri M, Lodato G and Natarajan P 2008 1079–1088
Volonteri M and Madau P 2008 57–60
Volonteri M, Gultekin K and Dotti M 2010 2143–2150
Wegg C and Bode N J 2011 8
Wiseman A G 1992 1517
Yunes N, Kocsis B, Loeb A and Haiman Z 2011 171103
Zanotti O, Rezzolla L, Del Zanna L and Palenzuela C 2010 Å[**523**]{} 8
Zanotti O 2013 [*New Astron.*]{} [**17**]{} 331
| ArXiv |
[**Pseudoscalar meson photoproduction on nucleon target**]{}
[**G.H. Arakelyan$^1$, C. Merino$^2$ and Yu.M. Shabelski$^{3}$**]{}\
$^1$A.I.Alikhanyan Scientific Laboratory\
Yerevan Physics Institute\
Yerevan 0036, Armenia\
e-mail: [email protected]\
$^2$Departamento de Física de Partículas, Facultade de Física\
and Instituto Galego de Física de Altas Enerxías (IGFAE)\
Universidade de Santiago de Compostela\
15782 Santiago de Compostela\
Galiza, Spain\
e-mail: [email protected]\
$^{3}$Petersburg Nuclear Physics Institute\
NCR Kurchatov Institute\
Gatchina, St.Petersburg 188350, Russia\
e-mail: [email protected] 0.9 truecm
0.5 truecm
[**Abstract**]{}
We consider the photoproduction of secondary mesons in the framework of the Quark-Gluon String model. At relatively low energies, not only cylindrical, but also planar diagrams have to be accounted for. To estimate the significant contribution of planar diagrams in $\gamma p$ collisions at rather low energies, we have used the expression obtained from the corresponding phenomenological expression for $\pi p$ collisions. The results obtained by the model are compared to the existing SLAC experimental data. The model predictions for light meson production at HERMES energies are also presented.
Introduction
============
The physical photon state is approximately represented by the superposition of a bare photon and of virtual hadronic states having the same quantum numbers as the photon. The bare photon may interact in direct processes. This direct contribution is determined by the lowest order in perturbative QED. The interactions of the hadronic components of the photon (resolved photons) at high enough energies are described within the framework of theoretical models based on Reggeon and Pomeron exchanges, like the Dual-Parton model (DPM), see [@capella] for a review, and the Quark-Gluon String model (QGSM) [@K20; @KTM].
The QGSM is based on the Dual Topological Unitarization (DTU) and on the large 1/N expansion of non-perturbative QCD, and it has been very successful in describing many features of the inclusive hadroproduction of secondaries, both at high [@KTM]-[@Sh], and at comparatively low [@volk; @kaidlow; @amelin] energies.
In the case of photoproduction processes, QGSM was used in [@lugovoi; @aryer] at energies higher then 100 GeV. At these high energies the accounting for only pomeron (cylindrical) diagrams leads to a reasonable agreement. However, at low energies one can nott neglect the contribution of planar diagrams [@volk; @kaidlow; @amelin]. The inclusion of palnar diagrams into the analysis of pp scattering should result in the better description of a wider range of experimental data.
Most interesting it is the comparison of secondary production by photon and hadron beams in their fragmentation regions. The photoproduction ($Q^2$ = 0) points are the boundary ($Q^2\rightarrow$0) for the electroproduction processes [@badelek].
In the present paper we apply QGSM to the low energy photoproduction of pions and kaons by taking into accaount, contrary to [@lugovoi; @aryer], both cylindrical and planar diagrams. We describe the SLAC experimental data [@pi0100; @abe20] on inclusive $x_F$ spectra integrated over $p_T$. The predictions for forthcoming quasireal photoproduction data at HERMES and JLAB energies are also presented.
Inclusive spectra of secondary hadrons in photoproduction processes
===================================================================
Let us now consider the photoproduction processes in the QGSM. The resolved photon can be consider as $$\label{vdm2}
|\gamma> = \frac{1}{6}(4|\overline{u}u> + |\overline{d}d> + |\overline{s}s>) \; ,$$ in agreement with standard Vector Dominance Model (VDM) expansion [@bauer], and so the quarks in Eq. \[vdm2\] can be considered as the valence quarks of a meson. This resolved photon state also has sea quarks and gluons, as usual hadrons.
We can calculate the photoproduction cross section like for a meson-nucleon process[^1] by summing, according to Eq. \[vdm2\], the contribution of $q\overline{q}$ pair collisions with target nucleon, and by taking into account the corresponding coefficients.
At low energies, the inclusive cross section in QGSM consists of two terms, described by two different types of diagrams: cylindrical and planar.
\[onep\]
-12.5cm {width="1.25\hsize"} -1.5cm
$$\label{spectrt}
\frac{dn}{dy}\ = \frac{1}{\sigma_{inel}}\frac{d\sigma}{dy}\ = \frac{dn^{cyl}}{dy} + \frac{dn^{pl}}{dy}$$
In QGSM high energy interactions are considered as takin place via the exchange of one or several Pomerons, and all elastic and inelastic processes result from cutting through or between pomerons [@AGK] (see Fig.1). Each Pomeron corresponds to a cylindrical diagram (see Fig. 1a), and thus, when cutting a Pomeron, two showers of secondaries are produced, as it is shown in Fig. 1b.
For $\gamma p$ interaction, that following Eq. \[vdm2\] is similar to $\pi p$ collisions, the inclusive spectrum of a secondary hadron $h$ produced from a $q \overline{q}$ pair has the form [@KaPi; @Sh]: $$\label{spectr}
\frac{dn^{cyl}}{dy}\ =\frac{x_E}{\sigma_{inel}} \frac{d\sigma^{cyl}}{dx_F}\ =\ \sum_{n=1}^\infty
w_n\phi_n^h (x)\ ,$$ where $x_{F}=2p_{\|}/\sqrt{s}$ is the Feynman variable, $x_{E}=2E/\sqrt{s}$, the functions $\phi_{n}^{h}(x)$ determine the contribution of diagrams with $n$ cut Pomerons, and $w_n$ is the relative weight of these diagrams. Thus, $$\begin{aligned}
\label{spectr1}
\phi_{q\overline{q}p}^h(x) &=& f_{\overline{q}}^{h}(x_+,n)\cdot f_q^h(x_-,n) +
f_q^h(x_+,n)\cdot f_{qq}^h(x_-,n)
\nonumber\\
&+&2(n-1)f_{sea}^h(x_+,n)\cdot f_{sea}^h(x_-,n)\ ,
\\
x_{\pm} &=& \frac12\left[\sqrt{4m_T^2/s+x^2}\ \pm x\right] ,\end{aligned}$$ where $f_{qq}$, $f_q$, and $f_{sea}$ correspond to the contributions of diquarks, valence quarks and sea quarks, respectively. These functions $f_{qq}$, $f_q$, and $f_{sea}$ are determined by the convolution of the diquark and quark distribution functions, $u(x,n)$, with the fragmentation functions, $G^h(z)$, to hadron $h$, e.g. $$f_i^h(x_+,n)\ =\ \int\limits_{x_+}^1u_i(x_1,n)G_i^h(x_+/x_1) dx_1\; ,$$ where $i=q, \overline{q}, qq$-diquarks, and sea quarks.
The fragmentatiion functions $G_i^{\pi}(z)$ have the same value, $a^\pi$ at $z \rightarrow 0$ for all $i$. Similarly, $G_i^K(z)$ have also the same value, $a^K$ at $z \rightarrow 0$ for all $i$. The numerical values of these parameters are [@ampsh1]: $$\centering
a^\pi = 0.68,\ a^K=0.26\; .$$ The diquark and quark distribution functions, which are normalized to unity, as well as the fragmentation functions, are determined from Regge intercepts [@kaidff]. The analytical expressions of these functions for proton are presented in [@KaPi; @Sh]. The distribution functions for quarks and antiquarks in a photon were obtained by using the simplest interpolation of Regge limits at $u_i(x\rightarrow 0)$ and $u_i(x\rightarrow 1)$, following [@KaPi; @Sh]. In the sum of all cylindrical diagrams we have used the weights given in Eq. 1.
At low energies the contribution of planar diagrams becomes significant. In particular, planar diagrams lead to the difference from $\sigma^{\pi^- p}_{tot}$ (Fig. 2a) to $\sigma^{\pi^+ p}_{tot}$(Fig. 2b) total cross sections.
Since the proton contains two $u$ quarks and one $d$ quark, there are two planar diagrams contributing to $\sigma_{tot}^{\pi^- p}$ (Fig. 2a), and only one contributing to $\sigma_{tot}^{\pi^+ p}$ (Fig. 2b). By neglecting the difference in $u$$\overline{u}$ and $d$$\overline{d}$ annihilation, we can consider as equal the contribution by every planar diagram. If we denote this contribution by each planar digram as $\sigma^{\pi p}_{pl}$, the contribution of diagrams in Fig. 2a to the $\sigma_{tot}^{\pi^- p}$ is 2$\sigma^{\pi p}_{pl}$, while the contribution of Fig. 2b to the total $\sigma_{tot}^{\pi^+ p}$ cross section is $\sigma^{\pi p}_{pl}$. Thus, $$\Delta \sigma(\pi^\mp p) = \sigma_{tot}(\pi^- p) - \sigma_{tot}(\pi^+ p) = \sigma_{pl}^{\pi p} \; ,
$$ the cylindrical contributions cancelling each other off into the difference.
\[plsigt\] -17.cm {width="1.5\hsize"} -4.cm
\[planar\]
-17.cm {width="1.5\hsize"}
One can find $\overline{u}$ quark in the photon with probability $\frac{4}{6}$ (see Eq. 1), and so, simply from comparison of the number of diagrams of figs. 2a and 3a at the same energies, there is a contribution to planar photoproduction cross section from the diagram Fig. 3a (strange quarks do not contribute to planar photoproduction) equal to $$\frac{\sigma^{\gamma p(3a)}_{pl}}{\sigma_{inel}^{\gamma p}} = \frac{4}{3}
\frac{\sigma_{pl}^{\pi p}}{\sigma_{inel}^{\pi p}} \; ,$$ and, in a similar way, the contribution to planar photoproduction cross section from the diagram Fig. 3b is $$\frac{\sigma^{\gamma p(3b)}_{pl}}{\sigma_{inel}^{\gamma p}} = \frac{1}{6}
\frac{\sigma_{pl}^{\pi p}}{\sigma_{inel}^{\pi p}}\; ,$$ with $\sigma^{\gamma p}_{tot} \cong \sigma_{inel}^{\gamma p}$ in both cases. Thus, the resulting contribution from planar photoprouction coming from diagrams in figs. 3a and 3b to the inclusive cross section is determined by using similar formula to those in [@K20; @kaidlow]: $$\begin{aligned}
\label{plan}
\centering
\frac{dn^{\gamma p}_{pl}}{dy}\ &=& \frac{\sigma^{\gamma p}_{pl}}{\sigma_{inel}^{\gamma p}}
[\frac{4}{3}G_u^h(x_+)G_{ud}^h(x_-)+ \frac{1}{6}G_d^h(x_+)G_{uu}^h(x_-)] \\
&=& \frac{\Delta \sigma(\pi^\mp p)}{\sigma_{inel}^{\pi p}}[\frac{4}{3}G_u^h(x_+)G_{ud}^h(x_-)+
\frac{1}{6}G_d^h(x_+)G_{uu}^h(x_-)]\end{aligned}$$ The parametrisation of experimental data on $\Delta \sigma({\pi^\mp p})$ at $\sqrt{s} \geq$ 5GeV exists [@pdg]: $$\Delta \sigma({\pi^\mp p}) = 2.161(\frac{s}{s_M})^{-0.544}{\rm mb}\; ,$$ where $s_M = (\mu + m_p + 2.177)^2$, $\mu$ is the pion mass, and $m_p$ is the proton mass. At energies $\sqrt{s}\approx 5 GeV$, where the contribution of planar diagrams is not negligible, $\sigma_{inel}^{\pi p} \approx$ 26 mb. For $K$-mesons photoproduction, the planar diagrams exist only for leading $K^+$ production, since only the u-quark in the photon can create the planar diagram in the collision with the target proton. On the other hand, the $K^-$-meson can be produced in a planar diagram as a slower nonleading particle.
Results of calculations
=======================
In this section, we present the results of the QGSM calculation for pseudoscalar meson photoproduction on a proton target. Thus, in Fig. 4 we compare the QGSM results with the experimental data on $\pi^0$ photoproduction obtained by OMEGA collaboration [@pi0100]. In this experiment the cross sections were measured at photon energies of 110$-$135 GeV, 85$-$110 GeV, and 50$-$85 GeV. The theoretical curves in Fig. 4 have been calculated at photon energies 120 GeV (dashed line), 100 GeV (full line), and 70GeV (dashed-dotted line). As we can see, the theoretical curves obtained at these three energies practically coincide, in correspondence to the scaling shown by the experimental data. At these energies the contribution of planar diagrams is small, and the present calculations are close to the results by [@aryer], where the only contribution from cylindrical diagrams was taken into account.
In Fig. 5 we present the comparison of QGSM calculations for the $x_F$ dependence of the $K^0_S$ inclusive cross section $F(x_F)=(1/\sigma_{tot})d\sigma/dx_F$, integrated over $p_T^2$, to the experimental data by [@abe20] at $E_{\gamma}$= 20 GeV. The dashed line corresponds to the contribution from only cylindrical diagrams, while the full curve represents the sum of contributions from both cylindrical and planar diagrams. As we can see, though the contribution of the planar diagrams is not large, its inclusion leads to a better agreement with the experimental data. However, one has to note that the theoretical result is proportional to $(a^K)^2$ (Eq. 7), and the value of this parameter is mainly known from high energy pp collisions, so its accuracy is estimated not to be better then 10% [@ampsh1].
\[pi0\] -10.cm ![QGSM calculation of the $x_F$ dependence of the $\pi^0$ photoproduction cross section integrated over $p_T^2$, compared to the experimental data [@pi0100]. The full line corresponds to calculations at a photon energy of 100 GeV.](gpi0100.pdf "fig:"){width="1.25\hsize"} -1.cm
\[k0\] -6.5cm ![ QGSM predictions for the $x_F$ dependence of $K^0_S$ photoproduction cross section integrated over $p_T^2$, and compared to the experimental data [@abe20]. The full line corresponds to the calculations for a photon energy of 20 GeV, and the dashed line to the corresponding contribution from only cylindrical diagrams.](k0xfflast.pdf "fig:"){width="1.25\hsize"} -5.5cm
In Fig. 6 we show the model predictions for the inclusive density of $\pi^\pm$ and $K^\pm$ mesons at the energy of HERMES Collaboration $E_{\gamma}$= 17 GeV. The full curves correspond to the inclusive spectra of $\pi^+$ and $K^+$, while dashed lines represent $\pi^-$ and $K^-$ meson photoproduction. We show the summed contribution of both cylindrical and planar diagrams.
\[pi20\] -6.5cm {width="1.25\hsize"} -5.5cm
\[rpi\] -10.5cm {width="1.25\hsize"} -1.5cm
The prediction for the ratio of yields of $\pi^+/\pi^-$-mesons at $E_{\gamma}$= 17 GeV is shown in Fig. 7 by full line. The cylindrical contribution is shown by dashed line. One can see that the planar diagram contribution changes the ratio by 25% in the forward hemisphere at large $x_F$. In Fig. 8 the ratio of yields of $K^+/K^-$-mesons at $E_{\gamma}$= 17 GeV is shown. Only valence $u$-quark contributes for leading $K^+$-meson production in planar diagram.
\[rpi\] -10.5cm {width="1.25\hsize"} -1.cm
Conclusion
==========
We consider a modified QGSM approach for the description of pseudoscalar ($\pi$, $K$) mesons photoproduction on nucleons at relatively low energies. This approach gives reasonable agreement to the experimental data on the $x_F$ dependence for $\pi ^0$ cross sections at $E_{\gamma}$=100 GeV, and for $K^0_S$ at $E_{\gamma}$=20 GeV, by taking into account the planar diagrams contribution that becomes significant at low energies. We also present the model predictions for charged pions and kaons cross sections and for the yields ratios at $E_{\gamma}$=17 Gev (Hermes Collaboration energies).
The comparison of the model results with experimental data allows the estimation of the contribution of the planar diagrams to the particle photoproduction processes. Detailed comparison of the theoretical predictions to experimental data at low energies makes possible to refine the values of the parameters of the model, and, in this way, it improves the reliability of the calculated yields of secondary particles to describe possible future HERMES data of quasireal photoproduction processes.
[**Acknowledgements**]{}
We are grateful to N.Z. Akopov, G.M. Elbakyan, and P.E. Volkovitskii for useful discussions. This paper was supported by the State Committee of Science of Republic of Armenia, Grant-13-1C015, and by Ministerio de Econom[í]{}a y Competitividad of Spain (FPA2011$-$22776), the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042), and Xunta de Galicia, Spain (2011/PC043).
[\*\*]{}
A. Capella, U. Sukhatme, C.I. Tan, and J. Tran Thanh Van, Phys. Rep. [**236**]{}, 225 (1994).
A.B. Kaidalov, Phys. Atom. Nucl., [**66**]{}, (2003), 781.
A.B. Kaidalov and K.A. Ter-Martirosyan, Yad. Fiz. [**39**]{}, 1545 (1984); Yad. Fiz. [**40**]{}, 211 (1984).
G.H. Arakelyan, A. Capella, A.B. Kaidalov, and Yu.M. Shabelski, Eur. Phys. J. [**C26**]{}, 81 (2002).
A.B. Kaidalov, O.I. Piskunova, Yad. Fiz. [**41**]{}, 1278 (1985).
Yu.M. Shabelski, Yad. Fiz. [**44**]{}, 186 (1986).
P.E. Volkovitski, Yad. Fiz. [**44**]{}, 729 (1989).
A.B. Kaidalov [*et al.*]{}, Yad. Fiz. [**49**]{}, 781 (1989); Yad. Fiz. [**40**]{}, 211 (1984).
N.S. Amelin [*et al.*]{}, Yad. Fiz. [**51**]{}, 941 (1990); Yad. Fiz. [**40**]{}, 211 (1984).
V.V. Luigovoi, S.Yu. Sivoklokov, Yu.M. Shabelskii. Phys. Atom. Nucl. [**58**]{}, 67 (1995); Yad. Fiz. [**58**]{} (1995) 72.
G.G. Arakelyan and Sh.S. Eremian, Phys. Atom. Nucl. [**58**]{}, 1241 (1995); Yad. Fiz. [**58**]{}, 132 (1995).
B. Badelek and J. Kwiecinski, Rev. Mod. Phys. [**68**]{} 445 (1996).
R.J. Apsimon [*et al.*]{}, Z. Phys. [**C52**]{}, 397 (1991).
K. Abe [*et al.*]{}, Phys. Rev. [**D32**]{}, 2869 (1985).
T.H. Bauer [*et al.*]{}, Rev. Mod. Phys. [**50**]{}, 261 (1978).
G.H. Arakelyan, C. Merino, C. Pajares, and Yu.M. Shabelski, Eur. Phys. J. [**C54**]{}, 577 (2008); arXiv:0805.2248 \[hep-ph\].
Particle Data Group, Chin. Phys. [**C38**]{}, 090001 (2014).
V.A. Abramovski, V.N. Gribov, and O.V. Kancheli, Yad. Fiz. [**18**]{}, 595 (1973).
A.B. Kaidalov. Yad. Fiz. [**45**]{}, 1432 (1987).
J. Gandsman [*et al.*]{}, Phys. Rev. [**D10**]{}, (1974) 1562.
[^1]: The absolute values of secondary photoproducton cross section are $\alpha_{em}$ times those of meson-nucleon cross section, while the corresponding multiplicities of secondaries $\frac{dn}{dy}$ are of the same order.
| ArXiv |
---
abstract: |
This paper describes TARDIS (Traffic Assignment and Retiming Dynamics with Inherent Stability) which is an algorithmic procedure designed to reallocate traffic within Internet Service Provider (ISP) networks. Recent work has investigated the idea of shifting traffic in time (from peak to off-peak) or in space (by using different links). This work gives a unified scheme for both time and space shifting to reduce costs. Particular attention is given to the commonly used 95th percentile pricing scheme.
The work has three main innovations: firstly, introducing the Shapley Gradient, a way of comparing traffic pricing between different links at different times of day; secondly, a unified way of reallocating traffic in time and/or in space; thirdly, a continuous approximation to this system is proved to be stable. A trace-driven investigation using data from two service providers shows that the algorithm can create large savings in transit costs even when only small proportions of the traffic can be shifted.
author:
- |
Richard G. Clegg,\
Dept of Elec. Eng.\
University College London\
\
Raul Landa\
Dept of Elec. Eng.\
University College London\
\
João Taveira Araújo\
Dept of Elec. Eng.\
University College London\
\
- |
Eleni Mykoniati\
Dept of Elec. Eng.\
University College London\
\
David Griffin\
Dept of Elec. Eng.\
University College London\
\
Miguel Rio\
Dept of Elec. Eng.\
University College London\
\
bibliography:
- 'sigmetrics\_tardis\_2014.bib'
title: 'TARDIS: Stably shifting traffic in space and time'
---
Introduction {#sec:intro}
============
Background {#sec:background}
==========
Definitions {#sec:definitions}
===========
Pricing {#sec:pricing}
=======
Dynamical systems approach {#sec:dynamics}
==========================
Modelling framework {#sec:modelling}
===================
Analysis of user data {#sec:results}
=====================
Conclusions {#sec:conclusions}
===========
*This research has received funding from the Seventh Framework Programme (FP7/2007-2013) of the European Union, through the FUSION project (grant agreement 318205).*
The Shapley gradient price {#sec:shap_indep}
==========================
A stability proof for multiple choice sets {#sec:smith_extension}
==========================================
| ArXiv |
---
abstract: 'Atomistic tight-binding (TB) simulations are performed to calculate the Stark shift of the hyperfine coupling for a single Arsenic (As) donor in Silicon (Si). The role of the central-cell correction is studied by implementing both the static and the non-static dielectric screenings of the donor potential, and by including the effect of the lattice strain close to the donor site. The dielectric screening of the donor potential tunes the value of the quadratic Stark shift parameter ($\eta_2$) from -1.3 $\times$ 10$^{-3} \mu$m$^2$/V$^2$ for the static dielectric screening to -1.72 $\times$ 10$^{-3} \mu$m$^2$/V$^2$ for the non-static dielectric screening. The effect of lattice strain, implemented by a 3.2% change in the As-Si nearest-neighbour bond length, further shifts the value of $\eta_2$ to -1.87 $\times$ 10$^{-3} \mu$m$^2$/V$^2$, resulting in an excellent agreement of theory with the experimentally measured value of -1.9 $\pm$ 0.2 $\times$ 10$^{-3} \mu$m$^2$/V$^2$. Based on our direct comparison of the calculations with the experiment, we conclude that the previously ignored non-static dielectric screening of the donor potential and the lattice strain significantly influence the donor wave function charge density and thereby leads to a better agreement with the available experimental data sets.'
author:
- Muhammad Usman
- Rajib Rahman
- Joe Salfi
- Juanita Bocquel
- Benoit Voisin
- Sven Rogge
- Gerhard Klimeck
- 'Lloyd L. C. Hollenberg'
title: 'Donor hyperfine Stark shift and the role of central-cell corrections in tight-binding theory'
---
0.25cm
Introduction
============
Since the Kane proposal for quantum computing using donor spins in silicon [@Kane_Nature_1998], there has been considerable progress towards the realisation of spin-qubit architectures [@Zwanenburg_RMP_2013; @Lloyd_PRB_2006; @Hill_PRB_2005; @Sousa_PRB_2009]. Notable results include single-atom fabricated devices [@Fuechsle_NN_2012; @Watson_NL_2014; @Weber_Science_2012] and control and measurement of individual donor electron and nuclear spins [@Tyryshkin_Nat_Mat_2012; @Saeedi_Science_2013; @Morello_Nature_2010; @Pla_Nature_2013]. However in building a scalable donor-based quantum computer, an important aspect is understanding and controlling the Stark shift of the donor hyperfine levels.
Accurate theoretical modelling of the donor hyperfine coupling is a challenging problem. First it requires proper incorporation of the valley-orbit (VO) interaction which has been established as a critical parameter to accurately match the experimentally observed energies of the ground state (A$_1$-symmtery) and the excited states (T$_2$ and E symmetries) [@Wellard_Hollenberg_PRB_2005]. Secondly it is essential to perform the calculation of the ground state donor wave function with high precision through proper implementation of the central-cell effects (short-range potential) and the dielectric screening of long-range Coulomb potential.
Earlier studies based on the Kohn and Luttinger’s single-valley effective-mass theory (SV-EMT) [@Kohn_PR_1955] ignored the VO interaction and therefore could not match with the experimental binding energy of the ground state (A$_1$); since then several studies have been performed with incremental improvements in the model. Pantelides and Sah [@Pantelides_Sah_PRB_1974] pointed out that the concept of the central-cell correction is ill defined in SV-EMT and therefore it fails to capture the chemical shift and the splitting of the experimentally observed donor ground state energies which primarily arise from the intervally mixing. Based on this, they presented a multi-valley effective-mass theory (MV-EMT) by explicitly including the central-cell correction along with a non-static dielectric screening of the Coulomb potential representing the donor. Similar EMT based formalisms have been widely applied by various studies later on to investigate the physics of shallow donors [@Saraiva_1; @King_1; @Pica_1].
Overhof and Gerstmann [@Overhof_PRL_2004] applied density-functional theory to successfully calculate the hyperfine ferquency of shallow donors in Si. While their calculations were in excellent agreement with experiment (zero fields), the ab-initio description of the donor wave function was by definition limited to only a few atoms around the donor site. Also they ignored the long-range tail of the Coulomb potential and therefore were unable to match the donor binding energies from their approach. More recently, a much more detailed theoretical calculation was performed by Wellard and Hollenberg [@Wellard_Hollenberg_PRB_2005] based on band-minimum basis (BMB) approach. In their study, by using a core-correcting potential screened by non-static ($k$-dependent) dielectric function, they were able to demostrate excellent agreement with the experimentally measured ground state energy for the P donor in Si (45.5 meV). However the excited state energies remained few meVs off from the experiemental values. Nevertheless, their study clearly highlighted the critical role of the central-cell corrections in theoretical modeling of shallow donors in Si which drastically modifies the charge density of donor wave function and therefore tune the hyperfine coupling and its Stark shift parameters.
Atomistic tight-binding (TB) approach, historically used for the deep level impurities [@Vogl_SSP_1981], has been shown to work remarkably well for the shallow level impurities in Si [@Martins_PRB_2004; @Rahman_PRL_2007; @Weber_Science_2012]. The TB method offers several advantages over EMT and DFT based methodologies, including the capability of inherently incorporating the VO intermixings, calculations over very large supercells (containing several million atoms in the simulation domain) and therefore providing much more detailed description of the donor wave funtion, an easy incorporation of externally applied electric field effects in Hamiltonian, and explicit representation of the short-range and the long-range donor potentials, etc. Martin *et al.* [@Martins_PRB_2004] applied second neares-neighbor sp$^3$s$^*$ TB model to study the effects of an applied electric field on P donor wave functions. Later Rahman *et al.* [@Rahman_PRL_2007] applied a much more sophisticated sp$^3$d$^5$s$^3$ TB Hamiltonian to P donors in Si and bench-marked Stark shift of the donor hyperfine coupling against the BMB calculations. The two theoretical models were found to be in remarkable agreement with each other for P donors, and also exhibited very good agreement with the Stark shift measurement data for the Sb donor in Si [@Bradbury_PRL_2006]; however a direct comparison of the hyperfine Stark shift with experiment was not possible due to the unavailability of any experimental data for the P and As donors. The models were also based on minimal central-cell correction, implemented in terms of short-range correction of donor potential at the donor site and a static dielectric screening of the long-range donor potential tail.
The previously reported experimental data for the hyperfine coupling of the shallow donors (P, As, Sb etc.) in Si [@Feher_PR_1959], and more recent experimental measurements [@Lo_arxiv_2014] of the hyperfine Stark shift for As donor in Si provide excellent opportunities to directly bench-mark TB theory against the experiment data sets. This work for the first time evaluates the role of central-cell corrections in atomistic TB theory through a direct comparison against the experimental data of the hyperfine interaction for a single As donor in Si.
The central-cell corrections in the tight-binding model considered here are implemented by:
- Short-range correction of the donor potential: donor potential is truncated at U$_0$ at the donor site.
- Dielectric screening of the long-range tail of the donor potential: static vs. non-static dielectric screenings.
- Lattice strain around the donor site: changes in the nearest-neighbor bond lengths.
We systematically study the critical significance of the central-cell corrections by including the effect of each central-cell component one-by-one and evaluating its role on the hyperfine coupling and its Stark shift parameter. In each case, we first adjust U$_0$ to match the experimentally measured donor binding energies for the ground and excited states within 1 meV accuracy [@Ahmed_Enc_2009]. We then compute charge density at the nuclear site and its character under the influence of an external electrical field. Our calculations demonstrate that the previously ignored central-cell components, non-static dielectric screening of donor potential and lattice strain, produce significant impact on the donor herperfine Stark shift and therefore lead to match the experimental data with an unprecedent accuracy. Such high precision bench-marking of the theory against the experimental data would be useful in accomplishing high precision control over donor wave functions required in quantum computing.
Methodology
===========
The atomistic simulations are performed using NanoElectronic MOdeling tool NEMO-3D [@Klimeck_1; @Klimeck_2], which has previously shown to quantitatively match the experimental data sets for a variety of nanostructures and nanomaterials, such as shallow donors in Si [@Rahman_PRL_2007; @Weber_Science_2012], III-V alloys [@Usman_1; @Usman_2] and quantum dots [@Usman_3; @Usman_4; @Usman_5], SiGe quantum wells [@Neerav_1], etc. The sp$^3$d$^5$s$^*$ tight-binding parameters for Si material are obtained from Boykin *et al*. [@Boykin_PRB_2004], that have been optimised to accurately reproduce the Si bulk band structure. The As donor is represented by a screened Coulomb potential truncated to U($r_0$)=U$_0$ at the donor site, $r_0$. Here, U$_0$ is an adjustable parameter that represents the central-cell correction at the donor site and has been designed to accurately match the ground state binding energy (A$_1$ = 53.8 meV) of the As donor as measured in the experiment. The size of the simulation domain (Si box around the As donor) is chosen as 32 nm $\times$ 65 nm $\times$ 32 nm, consisting of roughly 3.45 million atoms, with closed boundary conditions in all three dimensions. The surface atoms are passivated by our published method [@Lee_PRB_2004] to avoid any spurious states in the energy range of interest. The multi-million atom real-space Hamiltonian is solved by a prallel Lanczos algorithm to calculate donor single-particle energies and wave functions.
For the study of the effects of the lattic relaxation, the influence of the changed nearest-neighbor bond lengths on the tight-binding Hamiltonian is computed by a generalization of the Harrison’s scalling law [@Boykin_PRB_2004]. In this formulation, the interatomic interaction energies are taken to vary with the bond length $d$ as $(\dfrac{d_0}{d})^\eta$, where d$_0$ is the unrelaxed Si bond length and $\eta$ is a scalling parameter whose magnitude depends on the type of the interaction being considered and is fitted to obtain hydrostatic deformation potentials.
The hyperfine coupling parameter A(0) is directly proportional to the squared magnitude of the ground state wave function at the donor nuclear site, $|\psi(r_0)|^2$ [@Rahman_PRL_2007] and its value is experimentally measured [@Feher_PR_1959] as 1.73 $\times$ 10$^{30}$ m$^{-3}$ for As donor. It is therefore important to theoretically compute the value of $|\psi(r_0)|^2$ at the donor site and compare it with the experimental value. It should be pointed out that in our empirical tight-binding model, the Hamiltonian matrix elements comprising the onsite and nearest-neighbor interactions are optimized numerically to fit the bulk band structure of the host Si material without explicit knowledge of the underlying atomic orbitals. Therefore it is fundamentally not possible to quantitatively determine the value of the hyerperfine coupling A(0) as is possible from the ab-initio type calculations [@Overhof_PRL_2004]. Nevertheless, we apply the methodology published by Lee *et al.* [@Lee_JAP_2005] to estimate the value of $|\psi(r_0)|^2$ from our model, where we have used the value of bulk Si conduction electron at the nuclear site as $\approx$ 9.07 $\times$ 10$^{24}$ cm$^{-3}$ [@Shulman_PR_1956; @Lucy_PRB_2011] and the value of the atomic orbital ratio $\phi_{s^*}(0)/\phi_{s}(0)$ computed to be 0.058 from the assumption of the hydrogen-like atomic orbitals with an effective nuclear charge [@Clementi_JCP_1963]. We believe that this provides a good qualitative comparison of A(0) $ \propto |\psi(r_0)|^2$ from our model with the experimental value, and along with the quantitative match of the donor binding energies (A$_1$, T$_2$, and E) and the Stark shift of hyerfine ($\eta_2$), serve as a benchmark to evaluate the role of the central-cell corrections in the tight-binding theory.
We calculate the Stark shift of the hyperfine interaction as follows [@Rahman_PRL_2007]: the potential due to the electrical field is added in the diagonal of the tight-binding Hamiltonian which distorts the donor wave function and pulls it away from the donor site reducing the field dependent hyperfine coupling, A($\overrightarrow{E}$); the hyperfine coupling A($\overrightarrow{E}$) is directly proportional to $|\psi ( \overrightarrow{E}, r_0 )|^2$, where $r_0$ is the location of donor. The change in A($\overrightarrow{E}$) is parametrized as:
$$\label{eq:hyperfine_coupling}
\Delta A \left( \overrightarrow{E} \right) = A \left( 0 \right) \left( \eta_2 E^2 + \eta_1 E \right)$$
Here $\eta_2$ and $\eta_1$ are the quadratic and linear components of the Stark shift of the hyperfine interaction, respectively. For deeply burried donors (with donor depths typically greater than about 15 nm), the linear component of the Stark shift becomes negligible [@Rahman_PRL_2007]. Therefore we do not provide values of $\eta_1$ in the remainder of this paper which are about two to three orders of magnitude smaller than the values of $\eta_2$.
Screening of donor potential by static dielectric constant
==========================================================
In the first set of simulations, we apply no central-cell correction (U$_0 \rightarrow - \infty$) and the long-range part of the donor potential is Coulomb potential screened by static dielectric constant ($\epsilon(0)$) as given by Eq. \[eq:Static\_donor\_potential\]:
$$\label{eq:Static_donor_potential}
U \left( r \right) = \frac{-e^2}{ \epsilon \left( 0 \right) r}$$
where $\epsilon(0)$ = 11.9 is the static dielectric constant of Si and $e$ is the charge on electron. This case is illustrated by schematic of Fig. \[fig:Fig1\] (a). Such setup leads to a six-fold degenerate set of donor states at a binding energy of $\approx$ 29.6 meV, as would be expected from a simple effective-mass type approximation. The donor wave function density at nuclear site is 3.57$\times$10$^{22}$ m$^{-3}$ which is seven orders of magnitude smaller than the experimental value. This clearly highlights the critical role of the central-cell correction at the donor site to accurately capture the splitting of the donor ground state binding energies and the wave function density at the nuclear site as measured in the experiment.
![Electric field response of hyperfine coupling is plotted for bulk As donor for static dielectric screening of donor potential as given by Eq. \[eq:Static\_donor\_potential\]. The data points are computed directly from the TB simulations and the line plots are fittings of Eq. \[eq:hyperfine\_coupling\]. []{data-label="fig:Fig2"}](fig2.png)
Next, we setup simulations according to the schematic of Fig. \[fig:Fig1\] (b), where we keep the donor potential U($\overrightarrow{r}$) as a Coulomb potential screened by static dielectric constant for Si as given by Eq. \[eq:Static\_donor\_potential\]. Previous tight binding based theoretical studies for the P donors [@Rahman_PRL_2007; @Martins_PRB_2004] and the As donors [@Lansbergen_Nat_Phys_2008] has also used this type of donor potential. We now include central-cell correction at the donor site as a cut-off potential U$_0$ which is tuned to be 2.6342 eV to accurately reproduce the experimental ground state energy, A$_1$=53.1 meV. Further tuning of the onsite TB $d-$orbital energies [@Ahmed_Enc_2009] allowed to match the experimental excited state energies (T$_2$ and E) as listed in table \[tab:table1\]. By applying this model, we compute the value of $|\psi(r_0)|^2$ at the donor nuclear site as 4.05 $\times$ 10$^{30}$, which comes out to be $\approx$ 2.34 times larger than the experimental value. We also compute the electric field response of the hyperfine coupling for the electric field variation from 0 to 0.5 MV/m as shown in the Fig. \[fig:Fig2\]. The quadratic hyperfine Stark shift parameter $\eta_{2}$ is then calculated from the fitting of the TB data by Eq. \[eq:hyperfine\_coupling\] (details of the calculation methodology have been reported in Ref. ) as -1.32 $\times$ 10$^{-3} \mu$m$^2$/V$^2$ compared to the recent experimental value of -1.9 $\pm$ 0.2 $\times$ 10$^{-3} \mu$m$^2$/V$^2$ for the bulk As donor in Si [@Lo_arxiv_2014]. Table \[tab:table2\] provides an overall summary of results for the static dielectric screening of the donor potential. This shows that even with the static dielectric screening of the donor potential, the central-cell correction part provides a reasonably good description of the donor physics.
Since the central-cell effects are implemented through an adjustable parameter U$_0$ at the donor site, we attempt to quantify its effect on the $\eta_2$ by introducing a variation of $\pm$100 meV in its value. Increasing U$_0$ by 100 meV increases the ground state binding energy to 55.6 meV and the value of $\eta_2$ decreases to -1.089 $\times$ 10$^{-3} \mu$m$^2$/V$^2$. On the other hand, decreasing U$_0$ by 100 meV decreases the ground state binding energy to 50.9 meV and the value of $\eta_2$ increases to -1.53 $\times$ 10$^{-3} \mu$m$^2$/V$^2$. This clearly demonstrates that to improve the match with the experimental value of $\eta_2$, the value of U$_0$ should be reduced; however this introduces a large error in the binding energy of the donor ground state which is clearly unacceptable. Therefore we conclude that the current TB model with central-cell parameter U$_0$ and the static dielectric screening of the donor potential provides, at the best, a value of -1.32 $\times$ 10$^{-3} \mu$m$^2$/V$^2$ for the Stark shift of the hyperfine coupling. In the next two sections, we include the effects of non-static dielectric screening of the donor potential and the effect of the nearest-neighbour bond length changes to further evaluate the performance of our TB model.
Screening of donor potential by non-static dielectric function
==============================================================
With the established tight-binding model as our test system, we start further investigation of the central-cell correction, in particular the screening of the donor potential in the vicinity of the As donor. Previous tight-binding calculations [@Rahman_PRL_2007; @Martins_PRB_2004] for the P donor in Si have been based on the static dielectric screening of the donor potential; however Wellard and Hollenberg [@Wellard_Hollenberg_PRB_2005] have already demonstrated the critical importance of the non-static dielectric screening of the donor potential in their band minimum basis (BMB) calculations. By incorporating a non-static screening of the donor potential given in Ref. , they computed an excellent agreement of the donor ground state binding energy with the experimental value. Furthermore, the effect of the non-static dielectric screening was in particular profound on the spatial distribution of the donor wave function around the donor site. Therefore it is critical to investigate the impact of non-static dielectric screening of the donor potential on the values of $|\psi(r_0)|^2$ and $\eta_2$ computed from the tight-binding model. In this section, we investigate this effect by incorporating various non-static dielectric screenings of the donor potentials as reported in the literature.
![Electric field response of hyperfine coupling is plotted for bulk As donor for various screenings of the donor potential. The data points are computed directly from the TB simulations and the line plots are fittings of Eq. \[eq:hyperfine\_coupling\]. []{data-label="fig:Fig3"}](fig3.png)
The screening of the donor potential by a non-static dielectric constant has been a topic of extensive research, and a number of reliable calculations exist for $k$-dependent dielectric function, $\epsilon(k)$, for Si. The most commonly applied dielectric function is obtained by Nara [@Nara_JPSJ_1965]:
$$\label{eq:Nonstatic_dielectric}
\frac{1}{\epsilon(k)} = \frac{A^2 k^2}{k^2 + \alpha^2} + \frac{\left( 1-A \right) k^2}{k^2 + \beta^2} + \frac{1}{\epsilon \left( 0 \right) } \frac{\gamma^2}{k^2 + \gamma^2}$$
where $A$, $\alpha$, $\beta$, and $\gamma$ are fitting constants and have been numerically fitted by various studies for Si (see table 2 for the fitting values reported by various authors). Based on this $k$-dependent dielectric constant, the new screened donor potential in the real space coordinate system is given by:
$$\label{eq:Nonstatic_donor_potential}
U \left( r \right) = \frac{-e^2}{ \epsilon \left( 0 \right) r} \left( 1 + A \epsilon \left( 0 \right) \mathrm{e}^{- \alpha r} + \left( 1-A \right) \epsilon \left( 0 \right) \mathrm{e}^{- \beta r} - \mathrm{e}^{- \gamma r} \right)$$
In our next set of simulations, we apply this donor potential and re-adjust the central-cell correction U$_0$ at the donor site to match the ground and excited state binding energies with the experimental values. The new values of U$_0$ and the corresponding values of the binding energies for A$_1$, E, and T$_2$ states are provided in the table \[tab:table3\] for the four non-static dielectric screenings of the donor potential under consideration in this study. After achieving this excellent agreement of the binding energies with the experimental values, we then compute the values of $|\psi(r_0)|^2$ and the electric field response of the hyperfine coupling as plotted in Fig. \[fig:Fig3\] for the various non-static dielectric screening potentials as listed in table \[tab:table2\]. The calculated values of $|\psi(r_0)|^2$ and $\eta_2$ are listed in table \[tab:table3\]. Overall, the non-static dielectric screening of the donor potential works remarkably well in the tight-binding theory and improves the match with the experimental values of $|\psi(r_0)|^2$ and $\eta_2$. For the non-static dielectric screening provided by Nara & Morita, the agreement of the computed $\eta_2$ with the experimental value is within the range of experimental tolerance. We also find a direct relation of $|\psi(r_0)|^2$ with the central-cell correction parameter U$_0$. The smallest value of U$_0$ is for Richard & Vinsome screening which results in the best match of $|\psi(r_0)|^2$ with the experimental value, different only by a factor of 1.5.
Effect of lattice relaxation
============================
In the calculations performed above so far, we have assumed the crystal lattice as perfect Si lattice where each atom including the As donor is connected to its four nearest neighbor (NN) atoms by unstrained bond lengths of 0.235 nm. In the past tight-binding studies of the donor hyperfine Stark shift [@Rahman_PRL_2007; @Martins_PRB_2004] the effect of lattice strain has been completely ignored based on the assumption that in the presence of the donor, the NN bond length only negligibly changes. However the recent ab-initio study [@Overhof_PRL_2004] suggested a sizeable increase of 3.2% in the NN bond length for the As donors in Si. Our fully atomistic description of the As donor in Si provides an excellent opportunity to investigate the effect of lattice strain. In our next set of simulations, we increase the bond length of the As donor and its four nearest Si neighbors by 3.2%, thereby increasing it from the unstrained value of 0.235 nm to 0.2425 nm. For this study, we choose the non-static dielectric screening of the donor potential as described by Eq. \[eq:Nonstatic\_donor\_potential\] and the fitting parameters provided by Nara $\&$ Morita as given in the table \[tab:table2\]. This setup is schematically shown in Fig. \[fig:Fig1\] (d). Keeping the central-cell correction fixed at 2.2842 eV, we calculate a significant effect of the NN bond length change on the donor binding energies and the donor wave function confinement at the nuclear site. As evident from the third row of the table \[tab:table4\], the donor ground state binding energy A$_1$ decreases by $\approx$ 10 meV and the value of $|\psi(r_0)|^2$ is decreased by a factor of $\approx$ 2.4 as a result of the lattice strain.
Since the binding energies of the donor are adjusted in our model by central-cell correction (by varying U$_0$ and onsite TB energies), we perform further adjustments in U$_0$ by increasing its value to 3.1895 eV to re-establishe the match of the ground state binding energies with the experimental values. Based on this new model, we then recalculate the values of $|\psi(r_0)|^2$ and the Stark shift parameter $\eta_2$, and the corresponding values are provided in the last row of table\[tab:table2\]. The lattice strain only slightly modifies the value of $|\psi(r_0)|^2$ at the donor site, howevere the quadratic Stark shift parameter $\eta_2$ is strongly affected and becomes -1.87 $\times$ 10$^{-3} \mu$m$^2$/V$^2$ which is in remarkable agreement with the exerimental value of -1.9 $\pm$ 0.2 $\times$ 10$^{-3} \mu$m$^2$/V$^2$. Further investigation is needed to establish the connection of the NN bond length change with the value of $|\psi(r_0)|^2$ which would be reported somewhere else.
Conclusions
===========
In conclusion, this work aims to evaluate and benchmark previously established tight-binding model with the recently measured experimental data of the quadratic Stark shift of the As donor hyperfine interaction. The study is systematically performed to investigate the central-cell correction effects in the tight-binding theory. We include central-cell corrections in terms of donor potential cut-off at the nuclear site, static vs. non-static dielectric screenings of the donor potential, and the effect of the lattice strain by changing the As-Si nearest-neighbor bond lengths. Overall our calculations exhibit that tight-binding theory captures the donor physics remarkably well by reproducing the donor binding energy spectra within 1 meV of the expereimentally measured values. When we include the effects of non-static dielectric screening of the donor potential and lattice strain, the computed value of the quadratic Stark shift parameter ($\eta_2$) is calculated to be -1.87 $\times$ 10$^{-3} \mu$m$^2$/V$^2$ which is in excellent agreement with the experimental value of -1.9 $\pm$ 0.2 $\times$ 10$^{-3} \mu$m$^2$/V$^2$. Such detailed bench-marking of theory against the experimental data would allow us to relaibly investigate the single and two donor electron wave functions, especially those relevant for implementing quantum information processing.
***Acknowledgements:*** This work is funded by the ARC Center of Excellence for Quantum Computation and Communication Technology (CE1100001027), and in part by the U.S. Army Research Office (W911NF-08-1-0527). Computational resources are acknowledged from National Science Foundation (NSF) funded Network for Computational Nanotechnology (NCN) through <http://nanohub.org>. NEMO 3D based open source tools are available at: <https://nanohub.org/groups/nemo_3d_distribution>.
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| ArXiv |
---
abstract: 'We have studied the diffusion inside the silica network of sodium atoms initially located outside the surfaces of an amorphous silica film. We have focused our attention on structural and dynamical quantities, and we have found that the local environment of the sodium atoms is close to the local environment of the sodium atoms inside bulk sodo-silicate glasses obtained by quench. This is in agreement with recent experimental results.'
author:
- 'Michaël Rarivomanantsoa$^{a}$, Philippe Jund$^b$ and Rémi Jullien$^c$'
title: 'Sodium diffusion through amorphous silica surfaces: A molecular dynamics study'
---
Introduction
============
Many characteristics of materials such as mechanical resistance, adsorption, corrosion or surface diffusion depend on the physico-chemical properties of the surface. Thus, the interactions between the surfaces with their physico-chemical environment are very important, and in particular for amorphous materials which are of great interest for a wide range of industrial and technological applications (optical fibers coating, catalysis, chromatography or microelectronics). Therefore a great number of studies have for example focused on the interactions between the amorphous silica surfaces with water, experimentally [@exp1] and by molecular dynamics simulations [@md1; @md12; @bakaev; @md10; @md11; @md13].
On the other hand, the sodium silicate glasses entail great interest due to their presence in most of the commercial glasses and geological magmas. They are also often used as simple models for a great number of silicate glasses with more complicated composition. The influence of sodium atoms on the amorphous silica network is the subject of numerous experimental studies: Raman spectroscopy [@brawer; @millan], IR [@brawer; @wong], XPS [@bruckner; @sprenger] and NMR [@sprenger; @silver] from which we have informations about neighboring distances, bond angle distributions or concentration of so-called Q$^{n}$ tetrahedra. In order to improve the insight about the sodium silicate glass structure and to obtain a good understanding of the role of the modifying Na$^+$ cations, Greaves [[*et al.* ]{}]{}have used new promising investigation techniques like EXAFS and MAS NMR [@baker; @greaves]. Despite all these efforts, the structure of sodo-silicate glasses is still a subject of debate. Another means to give informations about this structure is provided by simulations, by either [*ab initio*]{} [@ispas] or classical [@soules; @vessal; @smith; @oviedo; @horbach; @jund] molecular dynamics (MD). In the present work, we are using classical MD simulations, but [*a contrario*]{} to previous simulations, the sodium atoms are not located before hand inside the amorphous silica sample.
Recent experimental studies of the diffusion of Na atoms initially placed at the surface of amorphous silica, using EXAFS spectroscopy [@mazzara], showed that the Na atoms diffuse inside the vitreous silica and once inside the amorphous silica network, the local environment of the Na atoms is characterized by a Na - O distance $d_{{\rm
Na-O}} = 2.3$ Å and by a Na - Si distance $d_{{\rm Na-Si}} = 3.8$ Å. These values are close to the distances characterizing the local environment of Na atoms in sodium silicate glasses obtained by quench.
In this study we have used classical MD simulations in order to reproduce the diffusion of sodium atoms inside a silica matrix and to check that the local environment of the sodium atoms is close to what is found for quenched sodo-silicate glasses. The sodium atoms have been inserted at the surface of thin amorphous silica films under the form of Na$_2$O groups in order to respect the charge neutrality.
Computational method
====================
To simulate the interactions between the different atoms, we use a generalized version [@kramer] of the so-called BKS potential [@bks] where the functional form of the potential remains unchanged: $${\cal{\phi}}\left(\left|\vec{r}_j-\vec{r}_i\right|\right) =
\frac{q_iq_j}{\left|\vec{r}_j-\vec{r}_i\right|}
-A_{ij}\exp\left(-B_{ij}\left|\vec{r}_j-\vec{r}_i\right|\right)
-\frac{C_{ij}}{\left|\vec{r}_j-\vec{r}_i\right|^6}.$$ The potential parameters $A_{ij}$, $B_{ij}$, $C_{ij}$, $q_i$ and $q_j$ involving the silicon and oxygen atoms (describing the interactions inside the amorphous silica network) are extracted from van Beest [[*et al.* ]{}]{}[@bks] and remain unchanged (in particular the partial charges q$_{{\rm Si}} = 2.4\rm e$ and q$_{{\rm O}} = -1.2$e are not modified). The new parameters, devoted to describe the interactions between the sodium atoms and the silica network are given by Kramer [[*et al.* ]{}]{}[@kramer] and are adjusted on [*ab initio*]{} calculations of zeolithes except the partial charge of the sodium atoms whose value q$_{{\rm Na}} = 0.6\rm e$ is chosen in order to respect the system electroneutrality. However, this sodium partial charge does not reproduce the short-range forces and to this purpose, Horbach [[*et al.* ]{}]{}[@horbach] have proposed to vary the charge q$_{{\rm Na}}$ as follows: $$\begin{aligned}
q_{{\rm Na}}(r_{ij})&=&
\left\{
\begin{array}{ll}
0.6\left(1+\ln\left[C\left( r_c-r_{ij}\right)^2+1\right]\right) &
r_{ij} < r_c\\
0.6 & r_{ij} \geqslant r_c
\end{array}
\right.\end{aligned}$$ where $r_{ij}$ is the distance between the particles $i$ and $j$. The parameters $C$ and $r_c$ are adjusted to obtain the experimental structure factor of Na$_2$Si$_2$O$_5$ (NS2) and their values are included in Ref [@horbach]. It is important to note that using this method to model the sodium charge, the system electroneutrality is respected for large distances (in fact for distances $r \geqslant r_c$). Next we assume that the modified BKS potential describes reasonably well the system studied here, for which the sodium atoms are initially located outside the amorphous silica sample. In addition other simulations have shown that this interatomic potential is convenient for various compositions, in particular for NS2, NS3 (Na$_2$Si$_3$O$_7$) [@horbach] and NS4 (Na$_2$Si$_4$O$_9$) [@jund] and we assume it is adapted to model any concentration of modifying Na$^+$ cations inside sodo-silicate glasses.
Our aim here is to obtain a sodo-silicate glass by deposition of sodium atoms at the amorphous silica surface, as it was done experimentally [@mazzara]. Using the [*modus operandi*]{} described in a previous study [@mr] we have generated Amorphous Silica Films (ASF), each containing two free surfaces perpendicular to the $z$-direction. These samples have been made by breaking the periodic boundary conditions along the $z$-direction, normal to the surface, thus creating two free surfaces located at $L/2$ and $-L/2$ with $L=35.8$ Å. In order to evaluate the Coulomb interactions, we used a two-dimensional technique based on a modified Ewald summation to take into account the loss of periodicity in the $z$-direction. For further technical details see Ref [@mr]. Then, instead of initially positioning the sodium atoms inside the silica matrix, like it was done before [@soules; @huang; @smith; @jund; @oviedo], we have deposited 50 Na$_2$O groups inside two layers located at a distance of 4 Å of each free surface as depicted in \[figure1\].
Within the layers, the Na$_2$O groups are assumed to be linear, with $d_{{\rm
Na-O}}=2$ Å [@elliott], and arranged on a pseudoperiodic lattice represented in the zoom of \[figure1\]. Hence the system is made of 100 Na$_2$O groups for 1000 SiO$_2$ molecules, corresponding to a sodo-silicate glass of composition NS10 (Na$_2$Si$_{10}$O$_{21}$) and contains 3300 particles. Since our goal is to study the diffusion of the sodium atoms placed at the amorphous silica surfaces, we fixed the initial temperature of the whole system at 2000 K. Indeed, the simulations of Smith [[*et al.* ]{}]{}[@smith] and Oviedo [[*et al.* ]{}]{}[@oviedo] of sodo-silicate glasses have shown that there is no appreciable sodium diffusion for temperatures below $\approx$ 1500 K. On the other hand, it is worth noticing that Sunyer [[*et al.* ]{}]{}[@sunyer] have found a simulated glass transition temperature $T_g \simeq 2400$ K for a NS4 glass. Therefore we thermalized the sodium layers at 2000 K and placed them at the ASF surfaces, also thermalized at 2000 K.
We have used repulsive ’walls’ at $z=-30$ Å and $z=+30$ Å in order to avoid that some Na$_2$O groups evaporate along the $z$ direction, where the periodic boundary conditions are no more fulfilled. The energy of the repulsive ’walls’ has an exponential shape, $E=E_0\exp\left [-\left ( z_w-z\right )/\sigma \right ]$ where $z_w$ is the wall position, $\sigma=0.1$ Å the distance for which the repulsion energy is diminished by a factor $e$ and $E_0=10$ eV the repulsion energy at the plane $z=z_w$. The value of 30 Å for $z_w$ was chosen in order to place the repulsive walls at a reasonable distance from the Na$_2$O layers and not too far from the ASF surfaces.
We have then performed classical MD simulations, with a timestep $\Delta t=0.7$ fs, using ten statistically independent samples. Since the interactions between the surfaces and the Na$_2$O layers are relatively weak, some Na$_2$O groups may evaporate just before being reflected toward the thin films by the repulsive walls. During this time frame, the system temperature increases up to a temperature of approximately 2800 K. As described by Athanasopoulos [[*et al.* ]{}]{}[@athanasopoulos] this temperature rise is likely due to the approach of the adatoms to the surface of the substrate, dropping in the potential well of the substrate atoms, thus increasing their kinetic energy. This kinetic energy is then transmitted to the substrate with the adsorption. Contrarily to Athanasopoulos [[*et al.* ]{}]{} [@athanasopoulos] and Webb [[*et al.* ]{}]{}[@webb] we have not dissipated this energy excess with a thermal sink region, in fact we have not controlled the temperature at all. Using this device we are able to perform MD simulations of the diffusion of the sodium atoms deposited [*via*]{} Na$_2$O groups at the ASF surfaces. In the following section, we will present the structural and dynamical characteristics of the sodo-silicate film (SSF) obtained in this way.
Results
=======
The observation of the sodium diffusion in the silica network is an important goal of this molecular dynamics simulation. This can be carried out by analyzing the behavior of the density profiles along the normal direction to the surface (the $z$-direction). The density profiles represent the mass densities within slices, of thickness $\Delta z=0.224$ Å, parallel to the surfaces [@mr]. The time evolution of the sodium density profile is represented in \[figure2\](a) and \[figure2\](b) and the time evolution of the total density profile in \[figure2\](c) and \[figure2\](d) .
The sodium atoms enter inside the silica network in the time range 14 - 70 ps ( \[figure2\](a)) and during this time interval the sodium profile exhibits a diffusion front. After 42 ps, the sodium atoms are observed in the entire system illustrating that they have diffused within the whole ASF, as observed experimentally [@mazzara]. This result contrasts with that of MD simulations of the diffusion of platinum (electrically neutral) [@athanasopoulos] and potassium [@garofalini2; @zirl1; @zirl2] but is similar to that of the diffusion of lithium [@garofalini2; @zirl1; @zirl2] at the surface of amorphous silica. Moreover, the sodium profiles ( \[figure2\](a) and (b)) show outward rearrangement at the surface as already pointed out by Zirl [[*et al.* ]{}]{} [@zirl] for glassy sodium aluminosilicate surfaces. This agrees with ion scattering spectroscopy [@kelso] and simulations of sodium silicate glasses [@garofalini] in which high concentrations of sodium ions are found at the surface. For larger times, $t\geqslant 98$ ps represented in \[figure2\](b), the sodium density seems to stabilize around a mean value of 0.3 g.cm$^{-3}$ which corresponds to $\sim 2$ Na atoms per slice ($\Delta z = 0.224$ Å ).
The surface of the system is identified as being the large linear region in which the total density profile decreases. For the short times ( \[figure2\](c)) the surface is located approximately in the range 15 - 20 Å and for larger times ($t\geqslant 98$ ps, \[figure2\](d)), the surface lies in the range 7 - 25 Å. Hence, the introduction of the sodium atoms in the ASF is likely to increase the surface thickness of the system. On the other hand, as observed for the adsorption of platinum atoms on the surface of amorphous silica [@athanasopoulos], the surface position does not seem to vary with time. We have also calculated the silicon and oxygen density profiles, but since they behave like the total density profile, they are not represented here. The non bridging oxygen (NBO) density profile is not represented in \[figure2\] as well since it is close to the Na density profile.
In the time range 98 - 210 ps, the total and sodium density profiles do not evolve with time. In particular, in the region $z \lesssim 5$ Å ( \[figure2\](d)), the total density value remains fluctuating around a mean value of 2.6 g.cm$^{-3}$ and the atom composition is approximately of 370 Si, 760 O and 40 Na which is usually written Na$_{2}$O(SiO$_{2}$)$_{18.5}$ or ’NS18.5’ (it should be noticed that the above mentioned density is significantly larger than the one expected for a “real” NS18.5 glass ($\approx$ 2.3 g.cm$^{-3}$)). Therefore it seems reasonable to consider that the system is in a [*quasi permanent*]{} regime after 210 ps and the forthcoming quantities, structural and dynamical, are calculated for the following 70 ps. It is worth remembering that all the quantities are determined for a system containing 3300 particles and averaged over 10 statistically independent samples.
As usual when studying the structural and dynamical characteristics of free surfaces, the system is divided into several subsystems: here six slices of equal thickness 10 Å. But in order to increase the statistics, the contributions to the physical quantities of the negative and positive slices are averaged. Hence, the system is actually subdivided into three parts, named respectively from the center to the surface, interior, intermediate and external region.
In order to improve the characterization of the local environment of the atoms, we have calculated the radial pair distribution functions for all the pairs $(i,j) \in {\rm [Si,Na,O]}^{2}$ within the three subsystems defined previously. The Na - Na, Na - O and Si - Na pair distribution functions are represented in \[figure3\](a), \[figure3\](b) and \[figure3\](c) respectively and represent the local environment of the sodium atoms for $t\geqslant210$ ps.
At the surfaces of the system the distances are $d_{{\rm Na-O}} \simeq 2.2$ Å, $d_{{\rm Si-Na}} \simeq 3.5$ Å and, due to a lack of statistics, the Na - Na distance is included in the interval $3.3 \lesssim d_{{\rm Na-Na}} \lesssim
3.9$ Å. While slightly smaller, these distances are close to the experimental values found by Mazzara [[*et al.* ]{}]{}($d_{{\rm Na-O}}=2.3$ Å and $d_{{\rm Si-Na}}=3.8$ Å) [@mazzara]. Moreover, the values found in the present work agree with the distances, calculated by MD, corresponding to the sodium environment in sodo-silicate glasses, of several sodium compositions (NS2 [@baker; @smith], NS3 [@horbach] and NS4 [@ispas]), obtained by quench. Therefore, as observed experimentally by Mazzara [[*et al.* ]{}]{}[@mazzara], once within the amorphous silica network, the sodium atoms have the same local environment as in the sodo-silicate glass obtained by quench. This fact is confirmed by the distributions of the $\widehat{{\rm NaONa}}$ and $\widehat{{\rm SiONa}}$ bond angles (not shown) which are close to those determined in quenched sodo-silicate glasses [@oviedo; @sunyer]. In particular, the most probable angles are 90[$^{\rm o}$]{} for $\widehat{{\rm NaONa}}$ and 105[$^{\rm o}$]{} for $\widehat{{\rm SiONa}}$, in agreement with the values found by Oviedo [[*et al.* ]{}]{}[@oviedo] and Sunyer [[*et al.* ]{}]{} [@sunyer].\
The intermolecular distances corresponding to the amorphous silica network structure are $d_{{\rm Si-Si}} \simeq$ 3.1 Å, $d_{{\rm Si-O}} \simeq$ 1.6 Å and $d_{{\rm
O-O}} \simeq$ 2.6 Å. On the other hand, the most probable values exhibited by the distributions of the $\widehat{{\rm OSiO}}$ and $\widehat{{\rm SiOSi}}$ bond angles (not shown) are respectively $\sim 109$[$^{\rm o}$]{} and $\sim 145$[$^{\rm o}$]{}. These distances and bond angle distributions are very similar to those found experimentally or by MD simulations in bulk amorphous silica. The shoulder exhibited at 2.5 Å by the distribution $g_{{\rm Si-Si}}$ at amorphous silica surfaces [@mr] and interpreted as the signature of the twofold rings is not present in the SSF. The absence of the shoulders at 80 and 100[$^{\rm o}$]{} depicted at ASF surfaces [@mr; @md1; @roder] by the $\widehat{{\rm OSiO}}$ and $\widehat{{\rm SiOSi}}$ bond angle distributions confirms that the small sized rings have disappeared as suggested by the radial pair distributions. This result is expected for R$_2$O (with R belonging to the column I) adsorption on glassy silicate surfaces and observed [*via*]{} MD simulations for R = H, Li, Na and K [@md1; @md12; @md11; @garofalini; @garofalini2; @zirl1; @zirl2]. Note that this also occurs for platinum adsorption on sodium aluminosilicate surfaces [@athanasopoulos; @webb] and for H$_2$ adsorption on amorphous silica surfaces [@lopez]. The small rings like two and threefold rings are known to be some of the most reactive sites on the surfaces of silicate glasses since they include strained siloxane bonds that react with water or other adsorbates [@brinker; @bunker]. Since the sodium introduction weakens the amorphous silica network, one important question is to measure the proportion of non bridging oxygens (NBOs). To this purpose, oxygen coordination with silicon was calculated. As expected, there is an important proportion of defective oxygens due to the presence of modifying cations Na$^{+}$ (8.9 % for the SSF to be compared with 1 % for the ASF [@mr]), illustrated by the similarities between the NBO and Na density profiles mentioned previously. Moreover, the NBO concentration is coherent with the concentration of 11 % of NBO calculated by MD simulations in a NS9 system [@huang]. The latter concentration is higher than the one found in this study because of the greater proportion of Na atoms in NS9 compared to 100 Na$_2$O for 1000 SiO$_2$ in the present SSF. At the surface ($z \geqslant 20$ Å), the BOs disappear and correlatively, the defective oxygens become preponderant, as observed for amorphous silica (36.2 % of NBOs at the SSF surfaces, 15 % at the ASF surfaces [@mr] and 10 % at the nanoporous silica surfaces [@beckers]). When analyzing the silicon coordination with oxygen, we can state that the silicon atoms remain coordinated in a tetrahedral way, revealing that the sodium introduction does not modify the silicon environment. The modifications created by the Na$^{+}$ cations are not able to break the SiO$_4$ tetrahedra which are very stable in the amorphous silica network. This agrees with the usual models for the sodo-silicate glasses, [[*ie* ]{}]{}the CRN model of Zachariasen [@zachariasen] and the MRN model of Greaves [@greaves].
One possible way to analyze more precisely the silicon environment consists in calculating the Q$^n$ tetrahedra proportion. A Q$^n$ structure is a SiO$_4$ tetrahedra which contains $n$ BOs. The Q$^n$ proportion is often determined by NMR experiments [@chuang; @maekawa], XPS [@sprenger; @emerson] or by molecular dynamics simulations[@smith], in order to describe the local environment around the silicon atoms. In the SSF, the Q$^2$ and Q$^3$ concentrations (6.8 % and 25.4 % respectively) are weak compared to those determined for NS2, NS3 and NS4 glasses. This result is coherent since the NS2, NS3 and NS4 glasses contain more Na$^+$ cations than the SSF studied in this work. At the surface, the Q$^3$ proportion is 45.9 % which is comparable to the proportions obtained by MD (using the BKS potential) in NS2, NS3 and NS4 glasses and to the experimental proportions in NS3 [@sprenger; @silver] and NS4 [@maekawa; @emerson] glasses. Moreover, it is worth noting that some Q$^1$ appear at the surface. In fact, these structural entities do not allow to create a network but it is conceivable to find those defects forming ’dead ends’ at the surface.
A direct method to confirm the previous assumption concerning the disappearance of small rings consists in analyzing directly the ring size distribution. A ring is a particularly interesting structure because it can be detected using infrared and Raman spectroscopy. In particular the highly strained twofold rings result in infrared-active stretching modes [@bunker; @morrow] at 888 and 908 cm$^{-1}$. In order to determine the probability $P_n$ for a given Si atom, whose coordinate along the normal direction to the surface is in one of the three regions, to be a member of a $n$-fold ring we have used the algorithm described in [@horbach2]. A ring is defined as the shortest path between two oxygen atoms, first neighbors of a given silicon atom and made by Si - O bonds. The ring size is given by the number of silicon atoms contained in the ring.\
The probability $P_n$ is reported in \[figure4\](a) for $n=2,\ldots,9$ and for the three different regions. For comparison, we have also reported $P_n$ determined at the ASF surface and interior [@mr]. In order to improve the medium range order characterization, we have investigated the orientation of the rings computing $<\cos^2\theta>$ for a given ring size, within the three regions of the system, where $\theta$ is the angle between the normal of the surface and the normal of the ring [@mr]. The results are reported in \[figure4\](b) for the three regions and for $n=2,\ldots,9$ together with the results obtained for the ASF surface (dashed line) and interior (dotted line).
For the three regions, the distributions of the sodo-silicate film ( \[figure4\](a)) are closer to the ASF interior than to the ASF surface distributions. Particularly, the probability of a silicon atom to belong to a small sized ring (2, 3 or 4-fold ring) is weak. This confirms the previous conclusion about the disappearance of the small strained sized rings which react with the sodium ions during their adsorption at the amorphous silica surface. As observed for the ring size distributions, the orientation of the rings ( \[figure4\](b)), in the three regions of the sodo-silicate film is similar to that obtained in the interior of the pure silica films [@mr] which means that even at the surface the rings have an isotropic orientation with respect to the surface. This is related to the disappearance of the small sized rings at the surface. Nevertheless, it is worth noting that in the external region the probability of a Si atom to belong to a 5-fold ring is greater than the probability to belong to a 6-fold ring ( \[figure4\](a)) as observed at the ASF [*surfaces*]{} [@mr]. In a sense, the 5-fold rings are not affected by the sodium adsorption in contrast to the small sized rings which disappear with the introduction of the sodium atoms. Also in the external region, the 2 and 3-fold rings (that are still present) are oriented perpendicularly to the surfaces ( \[figure4\](b)) as observed at the ASF surfaces [@mr; @ceresoli].
We have also analyzed the dynamics of the Na$^+$ cations and compared the results obtained in the present ’NS10’ system with those obtained in a NS4 glass. To this purpose, the mean square displacements (MSD) $\left< r^2(t)\right > = \left<\left|r_i(t)-
r_i(0)\right|^2\right>$ have been calculated for each species composing the SSF. \[figure5\] represents the MSD for the BO, NBO, Na and Si atoms within the time frame 0.7 fs - 70 ps after the first 210 ps together with the MSD calculated by Sunyer [[*et al.* ]{}]{}[@sunyer] for NS4 ($t \geqslant 1.5$ ps).
At $\sim$ 2800 K, the MSD of each species exhibits three regimes. For the short times, we observe the so-called ballistic regime where $\left< r^2(t)\right > \sim
t^2$. In this regime, the differences between the species are not really important. For the long times, we recognize the so-called diffusive regime in which $\left< r^2(t)\right>
\sim t$ and in which the Si atoms are the ones that diffuse the less. Similarly to the pure ASF [@mr], the NBOs diffuse more than the BOs. The origin of this feature lies in the fact that the NBOs form only one covalent bond with the Si atoms instead of two for the BOs. The Na atoms diffuse much more (one order of magnitude) than the other species. Between the two former regimes, the MSD exhibits the so-called $\beta$-relaxation, clearly observable for the oxygen and silicon atoms. The phenomenon is not as clear for the sodium atoms, but between $\sim
10^{-1}$ and $\sim 1$ ps, the Na MSD does not behave like $t$ (sub-diffusion) and it is reasonable to consider that the sodium atoms are submitted to the so-called cage effect characterizing the $\beta$-relaxation.\
These results are consistent with the observations made for different sodo-silicate systems [@smith; @oviedo; @horbach] for temperatures close to 2800 K. More precisely the mean square displacements in the SSF are close to those calculated by Sunyer [[*et al.* ]{}]{} [@sunyer] for a quenched NS4 glass at 3000 K. This is somewhat surprising since in a NS10 system, one expects smaller diffusion constants for all the species compared to the diffusion constants obtained in a NS4 system. The modus operandi of the present system (diffusion of the cations through the surface) could explain this observation.
Conclusion
==========
This work was motivated by a recent experimental study [@mazzara] of the local environment of diffusing sodium atoms deposited at the surfaces of thin amorphous silica films. We have reproduced numerically this experiment by classical molecular dynamics simulations after putting Na$_2$O groups at the surfaces of amorphous silica thin films.
We have quantitatively analyzed the temporal evolution of the sodium and total densities and we have checked that the sodium atoms are diffusing inside the amorphous silica network. After a given time, the density profiles are no longer evolving, and we have calculated the structure of the resulting sodo-silicate glass.
Our attention has been focused on the local environment of the sodium atoms. Once inside the thin film, they are preferentially bound to NBOs as predicted by the MRN model of Greaves [@greaves]. The distances and bond angle distributions show that the sodium atoms have a local environment corresponding to the local environment of the sodium atoms in sodo-silicate glasses made by quench, as observed by Mazzara [[*et al.* ]{}]{}[@mazzara]. Moreover, the distances $d_{{\rm Na-O}}=2.2$ Å and $d_{{\rm Si-Na}}\simeq 3.5$ Å are close to the experimental values.
Concerning the amorphous silica network, we have observed [*via*]{} the corresponding distance and bond angle distributions that its short range order is not modified by the introduction of the Na$^+$ cations. We have also calculated the ring size distributions and the orientation of the rings which show that the introduction of the sodium atoms has an influence on the silica network but on larger scales compared to those corresponding to the local environment. The ring size distributions and the orientations are close to the results obtained in the bulk of thin amorphous silica films. This is due to the decrease of the proportion of small rings (particularly two and threefold) which interact with the Na$_2$O groups since they are known to be highly reactive sites for the adsorption of species on amorphous silica surfaces.\
Finally concerning the dynamics of the different atoms we find results similar to those obtained in a NS4 glass at a slightly higher temperature.\
[**Acknowledgments**]{} Calculations have been performed partly at the “Centre Informatique National de l’Enseignement Supérieur” in Montpellier.
[99]{}
D. W. Sindorf and G. E. Maciel, [[J. Am. Chem. Soc. ]{}]{}[**105**]{}, 1487 (1983); D. M. Krol and J. G. van Lierop, [[J. Non-Cryst. Solids ]{}]{}[**63**]{}, 131 (1984); D. M. Krol and J. G. van Lierop, [[J. Non-Cryst. Solids ]{}]{}[**68**]{}, 163 (1984); C. J. Brinker, E. P. Roth, G. W. Scherer and D. R. Tallant, [[J. Non-Cryst. Solids ]{}]{}[**71**]{}, 171 (1985); I.-S. Chuang, D. R. Kinney, C. E. Bronnimann, R. C. Zeigler and G. E. Maciel, [[J. Phys. Chem. ]{}]{}[**96**]{}, 4027 (1992); L. Dubois and B. R. Zegarski, [[J. Phys. Chem. ]{}]{}[**97**]{}, 1665 (1993); D. R. Kinney, I.-S. Chuang and G. E. Maciel, [[J. Am. Chem. Soc. ]{}]{}[**115**]{}, 6786 (1993); I.-S. Chuang, D. R. Kinney and G. E. Maciel, [[J. Am. Chem. Soc. ]{}]{}[**115**]{}, 8696 (1993); A. Grabbe, T. A. Michalske and W. L. Smith, , 4648 (1995)
S. H. Garofalini, , 2069 (1983); S. M. Levine and S. H. Garofalini, 2997 (1987)
B. P. Feuston and S. H. Garofalini, [J. App. Phys. ]{}[**68**]{}, 4830 (1990)
V. A. Bakaev and W. A. Steele, , 9803 (1999)
B. P. Feuston and S. H. Garofalini, , 564 (1989)
S. H. Garofalini, [[J. Non-Cryst. Solids ]{}]{}[**120**]{}, 1 (1990)
M. M. Branda and N. J. Castellani, [Surf. Sci. ]{}[**393**]{}, 171 (1997)
N. Lopez, M. Vitiello, F. Illas and G. Pacchioni, [[J. Non-Cryst. Solids ]{}]{}[**271**]{}, 56 (2000)
S. A. Brawer and W. B. White, , 242 (1975)
P. F. Mc Millan and G. H. Wolf, Rev. Mineral. [**32**]{}, 247 (1995); N. Umekasi, N. Iwamoto, M. Tatsumisago and T. Minami, [[J. Non-Cryst. Solids ]{}]{}[**106**]{}, 77 (1988)
J. Wong and C. A. Angell, Glass Structure by Spectroscopy, M. Dekker, New York (1976)
R. Bruckner, H. U. Chun, H. Goretzki and M. Sammet, [[J. Non-Cryst. Solids ]{}]{}[**42**]{}, 49 (1980)
D. Sprenger, H. Bach, W. Meisel and P. Gütlich, [[J. Non-Cryst. Solids ]{}]{}[**159**]{}, 187 (1993)
A. H. Silver and P. J. Bray, , 984 (1958); J. F. Stebbins, Rev. Mineral. [**18**]{}, 405 (1988)
G. J. Baker, G. N. Greaves, M. Surman and M. Oversluizen, Nucl. Instr. and Meth. in Phys. Res. B [**97**]{}, 375 (1995)
G. N. Greaves, [[J. Non-Cryst. Solids ]{}]{}[**71**]{}, 203 (1985)
S. Ispas, M. Benoit, P. Jund and R. Jullien, , 214206 (2001); T. Uchino and T. Yoko, [[J. Phys. Chem. B ]{}]{}[**102**]{}, 8372 (1998)
T. F. Soules, , 4570 (1979)
B. Vessal, M. Leslie and C. R. A. Catlow, Mol. Phys. [**3**]{}, 123 (1989); B. Vessal, A. Amini, D. Fincham and C. R. A. Catlow, [[Philos. Mag. B ]{}]{}[**60**]{}, 753 (1989); B. Vessal, G. N. Greaves, P. T. Marten, A. V. Chadwick, R. Mole and S. Houde-Walter, Nature [**356**]{}, 504 (1992)
W. Smith, G. N. Greaves and M. J. Gillan, , 3091 (1995)
J. Oviedo and J. F. Sanz, , 9047 (1998)
J. Horbach, W. Kob and K. Binder, [[Philos. Mag. B ]{}]{}[**79**]{}, 1981 (1999)
P. Jund, W. Kob and R. Jullien, , 134303 (2001)
C. Mazzara, J. Jupille, A.-M. Flank, and P. Lagarde, [[J. Phys. Chem. B ]{}]{}[**104**]{}, 3438 (2000)
G. J. Kramer, A. J. M. de Man and R. A. van Santen, [[J. Am. Chem. Soc. ]{}]{}[**64**]{}, 6435 (1991)
B. W. H. van Beest, G. J. Kramer and R. A. van Santen, , 1955 (1990)
C. Huang and A. N. Cormack, , 8180 (1990)
M. Rarivomanantsoa, P. Jund and R. Jullien, J. Phys.: Condens. Matter [**13**]{}, 6707 (2001)
S. D. Elliott and R. Ahlirchs, , 4267 (1998)
E. Sunyer, P. Jund and R. Jullien, [unpublished.]{}
D. C. Athanasopoulos and S. H. Garofalini, , 3775 (1992); D. C. Athanasopoulos and S. H. Garofalini, [Surf. Sci. ]{}[**273**]{}, 129 (1992)
E. B. Webb and S. H. Garofalini, [Surf. Sci. ]{}[**319**]{}, 381 (1994)
D. M. Zirl and S. H. Garofalini, J. Am. Ceram. Soc. [**75**]{}, 2353 (1992)
J. Kelso, C. G. Pantano and S. H. Garofalini, [Surf. Sci. ]{}[**134**]{}, L543 (1983)
S. H. Garofalini and S. M. Levine, [[J. Am. Ceram. Soc. ]{}]{}[**68**]{}, 376 (1985)
S. H. Garofalini and D. M. Zirl, J. Vac. Sci. Technol. A [**6**]{}, 975 (1988)
D. M. Zirl and S. H. Garofalini, Phys. Chem. Glasses [**30**]{}, 155 (1989)
A. Roder, W. Kob and K. Binder, , 7602 (2001)
D. M. Zirl and S. H. Garofalini, [[J. Non-Cryst. Solids ]{}]{}[**122**]{}, 111 (1990)
C. J. Brinker, R. J. Kirkpatrick, D. R. Tallant and B. C. Bunker, [[J. Non-Cryst. Solids ]{}]{}[**99**]{}, 418 (1988); B. C. Bunker, D. M. Haaland, T. A. Michalske and W. L. Smith, [Surf. Sci. ]{}[**222**]{}, 95 (1989); J. Tossell, [[J. Non-Cryst. Solids ]{}]{}[**120**]{}, 13 (1990)
B. C. Bunker, D. M. Haaland, K. J. Ward, T. A. Michalske, W. L. Smith, J. S. Binkley, C. F. Melius and C. A. Balfe, [Surf. Sci. ]{}[**210**]{}, 406 (1989)
J. L. V. Beckers and S. W. de Leeuw, [[J. Non-Cryst. Solids ]{}]{}[**261**]{}, 87 (2000)
W. H. Zachariasen, [[J. Am. Chem. Soc. ]{}]{}[**54**]{}, 3841 (1932)
I.-S. Chuang and G. E. Maciel, [[J. Am. Chem. Soc. ]{}]{}[**118**]{}, 401 (1996)
H. Maekawa, T. Maekawa, K. Kawamura and T. Yokokawa, [[J. Non-Cryst. Solids ]{}]{}[**127**]{}, 53 (1991)
J. F. Emerson, P. E. Stallworth and P. J. Bray, [[J. Non-Cryst. Solids ]{}]{}[**113**]{}, 253 (1989)
B. A. Morrow and A. Devi, Trans. Faraday Soc. [**68**]{}, 403 (1972); B. A. Morrow and I. A. Cody, [[J. Phys. Chem. ]{}]{}[**80**]{}, 1995 (1976)
J. Horbach and W. Kob, , 3169 (1999)
D. Ceresoli, M. Bernasconi, S. Iarlori, M. Parinello and E. Tosatti, , 3887 (2000)
| ArXiv |
---
abstract: |
Given a flat injective ring epimorphism $u\colon R\to U$ between commutative rings, we consider the Gabriel topology ${\mathcal{G}}$ associated to $u$ and the class ${\mathcal{D}}_{\mathcal{G}}$ of ${\mathcal{G}}$-divisible modules. We prove that ${\mathcal{D}}_{\mathcal{G}}$ is an enveloping class if and only if it is the tilting class corresponding to the $1$-tilting module $U\oplus U/R$ and for every ideal $J\in {\mathcal{G}}$ the quotient rings $R/J$ are perfect rings. Equivalently, ${\operatorname{p.dim}}U\leq 1$ and the discrete quotient rings $\mathfrak R/\mathfrak RJ$ of the topological ring $\mathfrak R={\operatorname{End}}(U/R)$ are perfect rings.
Moreover, we show that every enveloping $1$-tilting class over a commutative ring arises from a flat injective ring epimorphism.
address:
- |
Dipartimento di Matematica “Tullio Levi-Civita”\
Università di Padova\
Via Trieste 63, 35121 Padova (Italy)
- |
Dipartimento di Matematica “Tullio Levi-Civita”\
Università di Padova\
Via Trieste 63, 35121 Padova (Italy)
author:
- Silvana Bazzoni
- Giovanna Le Gros
title: Enveloping classes over commutative rings
---
Introduction
============
The classification problem for classes of modules over arbitrary rings is in general very difficult, or even hopeless. Nonetheless, approximation theory was developed as a tool to approximate arbitrary modules by modules in classes where the classification is more manageable. Left and right approximations were first defined in the case of modules over finite dimensional algebras by work of Auslander, Reiten, and Smalø and generalised later by Enochs and Xu for modules over arbitrary rings using the terminology of preenvelopes and precovers.
An important problem in approximation theory is when minimal approximations, that is covers or envelopes, over certain classes exist. In other words, for a certain class ${\mathcal{C}}$, the aim is to characterise the rings over which every module has a minimal approximation in ${\mathcal{C}}$ and furthermore to characterise the class ${\mathcal{C}}$ itself. The most famous positive result of when minimal approximations exist is the construction of an injective envelope for every module. Instead, Bass proved in [@Bass] that projective covers rarely exist. In his paper, Bass introduced and characterised the class of perfect rings which are exactly the rings over which every module admits a projective cover. Among the many characterisations of perfect rings, the most important from the homological point of view is the closure under direct limits of the class of projective modules.
A comparison between the existence of injective envelopes and projective covers shows that the existence of minimal left or right approximations is not a symmetric phenomenon in general.
A class ${\mathcal{C}}$ of modules is called covering, respectively enveloping, if every module admits a ${\mathcal{C}}$-cover, respectively a ${\mathcal{C}}$-envelope.
A cotorsion pair $({\mathcal{A}}, {\mathcal{B}})$ admits (special) ${\mathcal{A}}$-precovers if and only if it admits (special) ${\mathcal{B}}$-preenvelopes. This observation lead to the notion of complete cotorsion pairs, that is cotorsion pairs admitting approximations.
Results by Enochs and Xu ([@Xu Theorem 2.2.6 and 2.2.8]) show that a complete cotorsion pair $({\mathcal{A}}, {\mathcal{B}})$ such that ${\mathcal{A}}$ is closed under direct limits admits both ${\mathcal{A}}$-covers and ${\mathcal{B}}$-envelopes. Note that in the case of the cotorsion pair $({\mathcal{P}}_0, {\mathrm{Mod}\textrm{-}{R}})$, where ${\mathcal{P}}_0$ is the class of projective modules, Bass’s results state that ${\mathcal{P}}_0$ is a covering class if and only if ${\mathcal{P}}_0$ is closed under direct limits.
In this paper we are interested in the conditions under which a class ${\mathcal{C}}$ is enveloping. We will deal with classes of modules over commutative rings and in particular with $1$-tilting classes.
An important starting point is the bijective correspondence between faithful finitely generated Gabriel topologies ${\mathcal{G}}$ and $1$-tilting classes over commutative rings established by Hrbek in [@H]. The tilting class can then be characterised as the class ${\mathcal{D}}_{\mathcal{G}}$ of ${\mathcal{G}}$-divisible modules, that is, the modules $M$ such that $JM=M$ for every $J\in {\mathcal{G}}$.
We prove that if a $1$-tilting class is enveloping, then $R_{\mathcal{G}}$, the ring of quotients with respect to the Gabriel topology ${\mathcal{G}}$, is ${\mathcal{G}}$-divisible, so that $R\to R_{\mathcal{G}}$ is a flat injective ring epimorphism.
It is well known that every flat ring epimorphism $u\colon R\to U$ gives rise to a finitely generated Gabriel topology. We will consider the case of a flat injective ring epimorphism $u\colon R\to U$ between commutative rings and show that if the module $R$ has a ${\mathcal{D}}_{\mathcal{G}}$-envelope, then $U$ has projective dimension at most one. From results by Angeleri Hügel and S[á]{}nchez [@AS], we infer that the module $U\oplus K$, where $K$ is the cokernel of $u$, is a $1$-tilting module with ${\mathcal{D}}_{\mathcal{G}}$ as associated tilting class. In other words, ${\mathcal{D}}_{\mathcal{G}}$ coincides with the class of modules generated by $U$, that is epimorphic images of direct sums of copies of $U$ or also with $K^\perp$, the right Ext-orthogonal of $K$. Assuming furthermore that the class ${\mathcal{D}}_{\mathcal{G}}$ is enveloping, we prove that all the quotient rings $R/J$, for $J\in {\mathcal{G}}$ are perfect rings and so are all the discrete quotient rings of the topological ring $\mathfrak R={\operatorname{End}}(K)$ (Theorems \[T:rjperfect\] and \[T:EndK-properfect\]). In the terminology of [@BP2] this means that $\mathfrak R$ is a pro-perfect topological ring.
Moreover, the converse holds, that is if $\mathfrak R={\operatorname{End}}(K)$ is a pro-perfect topological ring and the projective dimension of $U$ is at most one, then the class of ${\mathcal{G}}$-divisible modules is enveloping (Theorem \[T:characterisation\]).
Consequently, applying results from [@BP2 Section 19], we obtain that ${\mathrm{Add}}(K)$, the class of direct summands of direct sums of copies of $K$, is closed under direct limits. Since ${\mathcal{D}}_{\mathcal{G}}$ coincides with the right Ext-orthogonal of ${\mathrm{Add}}(K)$, we have an instance of the necessity of the closure under direct limits of a class whose right Ext-orthogonal admits envelopes.
Therefore in our situation we prove a converse of the result by Enochs and Xu ([@Xu Theorem 2.2.6]) which states that if a class ${\mathcal{A}}$ of modules is closed under direct limits and extensions and whose right Ext-orthogonal ${\mathcal{A}}^\perp$ admits special preenvelopes with cokernel in ${\mathcal{A}}$, then ${\mathcal{A}}^\perp$ is enveloping.
The case of a non-injective flat ring epimorphism $u\colon R\to U$ is easily reduced to the injective case, since the class of ${\mathcal{G}}$-divisible modules is annihilated by the kernel $I$ of $u$, so all the results proved for $R$ apply to the ring $R/I$ and to the cokernel $K$ of $u$. As a byproduct we obtain that a $1$-tilting torsion class over a commutative ring is enveloping if and only if it arises from a flat injective ring epimorphism with associated Gabriel topology ${\mathcal{G}}$ such that the factor rings $R/J$ are perfect rings for every $J\in {\mathcal{G}}$ (Theorem \[T: tilting-envelope\]).
This provides a partial answer to Problem 1 of [@GT12 Section 13.5] and generalises the result proved in [@B] for the case of commutative domains and divisible modules. The paper is organised as follows. After the necessary preliminaries, in Section \[S:envelope\] we state some general results concerning properties of envelopes with respect to classes of modules.
In Section \[S:gab-top\] we recall the notion of a Gabriel topology and outline the properties of the related ring of quotients.
In Section \[S:tilting-enveloping\], we consider a $1$-tilting class over a commutative ring and its associated Gabriel topology via Hrbek’s results [@H]. We prove that if the $1$-tilting class is enveloping, then the ring of quotients with respect to the Gabriel topology ${\mathcal{G}}$ is ${\mathcal{G}}$-divisible, hence flat.
In Section \[S:compl-endK\] we introduce the completion of a ring with respect to a Gabriel topology and the endomorphism ring of a module as a topological ring. Considering the particular case of a perfect localisation corresponding to a flat injective ring epimorphism $u\colon R\to U$ between commutative rings, we show the isomorphism between the completion of $R$ with respect to the associated Gabriel topology and the topological ring $\mathfrak R={\operatorname{End}}(K)$.
In the main Sections \[S:enveloping\] and \[S:properfect\], we prove a ring theoretic and topological characterisation of commutative rings for which the class of ${\mathcal{G}}$-divisible modules is enveloping where ${\mathcal{G}}$ is the Gabriel topology associated to a flat injective ring epimorphism. Namely, the characterisation in terms of perfectness of the factor rings $R/J$, for every $J\in {\mathcal{G}}$ and the pro-perfectness of the topological ring $\mathfrak R={\operatorname{End}}(K)$.
In Section \[S:notmono\] we extend the results proved in Sections \[S:enveloping\] and \[S:properfect\] to the case of a non-injective flat ring epimorphism
Preliminaries
=============
The ring $R$ will always be associative with a unit and ${\mathrm{Mod}\textrm{-}{R}}$ the category of right $R$-modules.
Let ${\mathcal{C}}$ be a class of right $R$-modules. The right ${\operatorname{Ext}}^1$-orthogonal and right ${\operatorname{Ext}}^\infty$-orthogonal classes of ${\mathcal{C}}$ are defined as follows. $${\mathcal{C}}^{\perp_1} =\{M\in {\mathrm{Mod}\textrm{-}{R}} \ | \ {\operatorname{Ext}}_R^1(C,M)=0 \ {\rm for \
all\ } C\in {\mathcal{C}}\}$$ $${\mathcal{C}}^\perp = \{M\in {\mathrm{Mod}\textrm{-}{R}} \ | \ {\operatorname{Ext}}_R^i(C,M)=0 \ {\rm for \
all\ } C\in {\mathcal{C}}, \ {\rm for \
all\ } i\geq 1 \}$$ The left Ext-orthogonal classes ${}^{\perp_1} {\mathcal{C}}$ and ${}^\perp {\mathcal{C}}$ are defined symmetrically. If the class ${\mathcal{C}}$ has only one element, say ${\mathcal{C}}= \{X\}$, we write $X^{\perp_1}$ instead of $\{X\}^{\perp_1}$, and similarly for the other ${\operatorname{Ext}}$-orthogonal classes.
We will now recall the notions of ${\mathcal{C}}$-preenvelope, special ${\mathcal{C}}$-preenvelope and ${\mathcal{C}}$-envelope for a class ${\mathcal{C}}$ of $R$-modules.
Let ${\mathcal{C}}$ be a class of modules, $N$ a right $R$-module and $C\in {\mathcal{C}}$. A homomorphism $\mu\in {\operatorname{Hom}}_R(N, C)$ is called a ${\mathcal{C}}$-[*preenvelope*]{} (or left approximation) of $N$ if for every homomorphism $f' \in {\operatorname{Hom}}_R(N, C')$ with $C'\in {\mathcal{C}}$ there exists a homomorphism $f\colon C\to C'$ such that $f '=
f \mu$.
A ${\mathcal{C}}$-preenvelope $\mu\in {\operatorname{Hom}}_R(N, C)$ is called a ${\mathcal{C}}$-[*envelope*]{} (or a minimal left approximation) of $N$ if for every endomorphism $f$ of $C$ such that $f \mu=
\mu$, $f$ is an automorphism of $C$.
A ${\mathcal{C}}$-preenvelope $\mu$ of $N$ is said to be [*special*]{} if $\mu$ it is a monomorphism and ${\operatorname{Coker}}\mu\in {}^\perp {\mathcal{C}}$.
The notions of ${\mathcal{C}}$-[*precover*]{} (right approximation), [*special*]{} ${\mathcal{C}}$-[*precover*]{} and of ${\mathcal{C}}$-[*cover*]{} (minimal right approximation) (see [@Xu]) are defined dually.
A class ${\mathcal{C}}$ of $R$-modules is called *enveloping* (*covering*) if every module admits a ${\mathcal{C}}$-envelope (${\mathcal{C}}$-cover).
A pair of classes of modules $({\mathcal{A}}, {\mathcal{B}})$ is a *cotorsion pair* provided that $\mbox{${\mathcal{A}}$} = {}^{\perp_1}
\mbox{${\mathcal{B}}$}
$ and $\mbox{${\mathcal{B}}$} = \mbox{${\mathcal{A}}$} ^{\perp_1}$.
We consider preenvelopes and envelopes for particular classes of modules, that is classes which form the right-hand class of a cotorsion pair.
A cotorsion pair $({\mathcal{A}}, {\mathcal{B}})$ is *complete* provided that every $R$-module $M$ admits a [special ${\mathcal{B}}$-preenvelope]{} or equivalently, every $R$-module $M$ admits a [special ${\mathcal{A}}$-precover]{}.
Results by Enochs and Xu ([@Xu Theorem 2.2.6 and 2.2.8]) show that a complete cotorsion pair $({\mathcal{A}}, {\mathcal{B}})$ such that ${\mathcal{A}}$ is closed under direct limits admits both ${\mathcal{B}}$-envelopes and ${\mathcal{A}}$-covers.
A cotorsion pair $({\mathcal{A}}, {\mathcal{B}})$ is [*hereditary*]{} if for every $A \in {\mathcal{A}}$ and $B \in {\mathcal{B}}$, ${\operatorname{Ext}}^i_R(A, B)=0$ for all $i \geq 1$.\
Given a class $\mathcal{C}$ of modules, the pair $(^{\perp }({\mathcal{C}}^{\perp}),{\mathcal{C}}^{\perp})$ is a (hereditary) cotorsion pair called the cotorsion pair *generated* by ${\mathcal{C}}$, while $(^{\perp }{\mathcal{C}}, (^{\perp }{\mathcal{C}}) ^{\perp})$ is a (hereditary) cotorsion pair called the cotorsion pair *cogenerated* by ${\mathcal{C}}$.
Examples of complete cotorsion pairs are abundant. In fact, by [@ET01 Theorem 10] a cotorsion pair generated by a set of modules is complete.\
For an $R$-module $C$, we let ${\mathrm{Add}}(C)$ denote the class of $R$-modules which are direct summands of direct sums of copies of $C$, and ${\operatorname{Gen}}(C)$ denote the class of $R$-modules which are homomorphic images of direct sums of copies of $C$.\
We now define $1$-tilting and silting modules.\
A right $R$-module $T$ is [*$1$-tilting*]{} if the following conditions hold.
1. ${\operatorname{p.dim}}T \leq1$.
2. ${\operatorname{Ext}}_R^i (T, T^{(\kappa)}) =0$ for every cardinal $\kappa$ and every $i >0$.
3. There exists an exact sequence of the following form where each $T_i$ is in ${\mathrm{Add}}(T)$. $$0 \to R \to T_0 \to T_1 \to 0$$
Equivalently, $T$ is $1$-tilting if and only if $T^\perp = {\operatorname{Gen}}(T)$. The cotorsion pair $({}^\perp(T^{\perp}), T^\perp)$ is called a [*$1$-tilting cotorsion pair*]{} and the torsion class $T^\perp$ is called [*$1$-tilting class*]{}. Two $1$-tilting modules are [*equivalent*]{} if they define the same $1$-tilting class (equivalently, if ${\mathrm{Add}}(T)={\mathrm{Add}}(T')$).
A $1$-tilting class can be generalised in the following way. For a homomorphism $\sigma:P_{-1} \to P_0$ between projective modules in ${\mathrm{Mod}\textrm{-}R}$, consider the following class of modules. $$D_\sigma := \{ X \in {\mathrm{Mod}\textrm{-}R}: {\operatorname{Hom}}_R( \sigma,X) \text{ is surjective}\}$$ An $R$-module $T$ is said to be *silting* if it admits a projective presentation $$P_{-1} \overset{\sigma}\to P_0 \to T \to 0$$ such that ${\operatorname{Gen}}(T) = D_\sigma$. In the case that $\sigma$ is a monomorphism, ${\operatorname{Gen}}(T)$ is a $1$-tilting class.
A ring $R$ is [*left perfect*]{} if every module in ${R\textrm{-}\mathrm{Mod}}$ has a projective cover. By [@Bass], $R$ is left perfect if and only if all flat modules in ${R\textrm{-}\mathrm{Mod}}$ are projective.
An ideal $I$ of $R$ is said to be [*left T-nilpotent*]{} if for every sequence of elements $a_1, a_2, ..., a_i, ...$ in $I$, there exists an $n >0$ such that $a_1 a_2 \cdots a_n =0$. The following proposition for the case of commutative perfect rings is well known.
\[P:perfect\] Suppose $R$ is a commutative ring. The following statements are equivalent for $R$.
- $R$ is perfect
- The Jacobson radical $J(R)$ of $R$, is T-nilpotent and $R/J(R)$ is semi-simple.
- $R$ is a finite product of local rings, each one with a T-nilpotent maximal ideal.
The following fact will be useful. Let $_RF$ be a left $R$-module $_SG_R$ be an $S$-$R$-bimodule such that ${\operatorname{Tor}}_1^R(G, F)=0$. Then, for every left $S$-module $M$ there is an injective map of abelian groups $${\operatorname{Ext}}^1_R(F, {\operatorname{Hom}}_S(G, M))\hookrightarrow{\operatorname{Ext}}^1_S(G\otimes_RF, M)).$$
Envelopes {#S:envelope}
=========
In this section we discuss some useful results in relation to envelopes.
The following result is crucial in connection with the existence of envelopes.
\[P:Xu-env\] [@Xu Proposition 1.2.2] Let ${\mathcal{C}}$ be a class of modules and assume that a module $N$ admits a ${\mathcal{C}}$-envelope. If $\mu\colon N\to C$ is a ${\mathcal{C}}$-preenvelope of $N$, then $C=C'\oplus H$ for some submodules $C'$ and $H$ such that the composition $N\to C\to C'$ is a ${\mathcal{C}}$-envelope of $N$.
We will consider ${\mathcal{C}}$-envelopes where ${\mathcal{C}}$ is a class closed under direct sums and therefore we will make use of the following result which is strongly connected with the notion of T-nilpotency of a ring.
\[T:Xu-sums\][@Xu Theorem 1.4.4 and 1.4.6]
1. Let ${\mathcal{C}}$ be a class closed under countable direct sums. Assume that for every $n\geq 1$, $\mu_n\colon M_n\to C_n$ are ${\mathcal{C}}$-envelopes of $M_n$ and that $\oplus_nM_n$ admits a ${\mathcal{C}}$-envelope. Then $\oplus \mu_n\colon \oplus_nM_n\to \oplus_nC_n$ is a ${\mathcal{C}}$-envelope of $\oplus_nM_n$.
2. Assume that $\oplus \mu_n\colon \oplus_nM_n\to \oplus_nC_n$ is a ${\mathcal{C}}$-envelope of $\oplus_nM_n$ with $M_n\leq C_n$ and let $f_n\colon C_n\to C_{n+1}$ be a family of homomorphisms such that $f_n(M_n)=0$. Then, for every $x\in C_1$ there is an integer $m$ such that $f_m f_{m-1} \dots f_1(x)=0$.
For a complete cotorsion pair $({\mathcal{A}}, {\mathcal{B}})$, we investigate the properties of ${\mathcal{B}}$-envelopes of arbitrary $R$-modules. First of all we state two straightforward lemmas.
\[L:endomorph-env\] Let $0\to N\overset{\mu}\to B\overset{\pi}\to A\to 0$ be an exact sequence. Let $f$ be an endomorphism of $B$ such that $\mu = f \mu$. Then $f(B)\supseteq \mu(N)$ and ${\operatorname{Ker}}f\cap \mu(N)=0$.
\[L:identity-env\] Let $0\to N\overset{\mu}\rightarrow B\overset{\pi}\to A\to 0$ be an exact sequence. For every endomorphism $f$ of $B$, the following are equivalent
1. $\mu = f \mu$.
2. The restriction of $f$ to $\mu(N)$ is the identity of $\mu(N)$.
3. There is a homomorphism $g\in {\operatorname{Hom}}_R(A,B)$ such that $f=id_B-g \pi$.
\(1) $\Leftrightarrow$ (2) This is clear.
\(1) $\Leftrightarrow$ (3) $\mu = f \mu$ if and only if $ (id_B-f) \mu=0$, that is if and only if $\mu(N)$ is contained in ${\operatorname{Ker}}(id_B-f)$. Equivalently, there exists $g\in {\operatorname{Hom}}_R(A,B)$ such that $id_B-f=g \pi $.
\[P:B-envelopes\] Let $({\mathcal{A}}, {\mathcal{B}})$ be a complete cotorsion pair over a ring $R$. Assume that $0\to N\overset{\mu}\to B$ is a ${\mathcal{B}}$-envelope of the $R$-module $N$. Let $\alpha$ be an automorphism of $N$ and let $\beta $ be any endomorphism of $B$ such that $\beta\mu=\mu\alpha$. Then $\beta$ is an automorphism of $B$.
By Wakamatsu’s Lemma (see [@Xu Lemma 2.1.2]), $\mu$ induces an exact sequence $$0\to N\overset{\mu}\to B\overset{\pi}\to A\to 0$$ with $A\in {\mathcal{A}}$. Since $\alpha$ is an automorphism of $N$, it is easy to show that $$0\to N\overset{\mu\alpha}\to B\to A\to 0$$ is a ${\mathcal{B}}$-envelope of $N$. Let $\beta$ be as assumed and consider an endomorphism $g$ of $B$ such that $g\mu\alpha=\mu$. Then $g\beta\mu=\mu$ and thus $g\beta$ is an automorphism of $B$, since $\mu$ is a ${\mathcal{B}}$-envelope. This implies that $\beta$ is a monomorphism so that $\beta(B)\in {\mathcal{B}}$. Since $\mu(N)\subseteq \beta(B)$ there is an epimorphism $\tau\colon B/\mu(N)\to B/\beta(B)$, where $B/\mu(N)$ can be identified with $A$. Consider the diagram: $$\xymatrix{
0\ar[r]&
{\beta(B)}\ar[r]&B\ar[r]^{\rho} & B/\beta(B)\ar[r]&0\\
&& A\ar@{-->}[u]^{h} \ar[ur]_{\tau}}$$ where $\rho$ is the canonical projection and $\tau\pi=\rho$. It can be completed by $h$, since ${\operatorname{Ext}}_R^1(A, \beta(B))=0$. Consider the homomorphism $f=id_B-h\pi$. $f$ is an endomorphism of $B$ satisfying $f\mu=\mu$. By assumption $f$ is an isomorphism, hence, in particular$ f(B)=B$.
Now, $\rho f=\rho-\rho h \pi= \rho-\tau\pi=0$. Hence $ f(B)\subseteq {\operatorname{Ker}}\rho= \beta(B)$; so $\beta(B)=B$ and $\beta$ is an isomorphism.
Gabriel topologies {#S:gab-top}
==================
In this section we briefly introduce Gabriel topologies and discuss some advancements that relate Gabriel topologies to $1$-tilting classes and silting classes over commutative rings as done in [@H] and [@AHHr]. For more detailed discussion on torsion pairs and Gabriel topologies, refer to [@Ste75 Chapters VI and IX].
We will start by giving definitions in the case of a general ring with unit (not necessarily commutative).
Recall that a [*torsion pair*]{} $({\mathcal{E}}, {\mathcal{F}})$ in ${\mathrm{Mod}\textrm{-}R}$ is a pair of classes of modules in ${\mathrm{Mod}\textrm{-}R}$ which are mutually orthogonal with respect to the ${\operatorname{Hom}}$-functor and maximal with respect to this property. The class ${\mathcal{E}}$ is called a [*torsion class*]{} and ${\mathcal{F}}$ a [*torsion-free class*]{}.
A class ${\mathcal{C}}$ of modules is a [ torsion class]{} if and only if it is closed under extensions, direct sums, and epimorphic images. A torsion pair $({\mathcal{E}}, {\mathcal{F}})$ is called [*hereditary*]{} if ${\mathcal{E}}$ is also closed under submodules.
A torsion pair $({\mathcal{E}}, {\mathcal{F}})$ is [*generated*]{} by a class ${\mathcal{C}}$ if ${\mathcal{F}}$ consists of all the modules $F$ such that ${\operatorname{Hom}}_R(C, F)=0$ for every $C\in {\mathcal{C}}$.
A [*(right) Gabriel topology*]{} on $R$ is a filter of right ideals of $R$, denoted ${\mathcal{G}}$, such that the following conditions hold. Recall that for a right ideal $I$ in $R$ and an element $t \in R$, $(I:t) := \{r \in R: tr \in I\}$.
- If $I \in {\mathcal{G}}$ and $r \in R$ then $(I:r) \in {\mathcal{G}}$.
- If $J$ is a right ideal of $R$ and there exists a $I \in {\mathcal{G}}$ such that $(J:t) \in {\mathcal{G}}$ for every $t \in I$, then $J \in {\mathcal{G}}$.
Right Gabriel topologies on $R$ are in bijective correspondence with hereditary torsion pairs in ${\mathrm{Mod}\textrm{-}R}$. Indeed, to each right Gabriel topology ${\mathcal{G}}$, one can associate the following hereditary torsion class. $${\mathcal{E}}_{\mathcal{G}}= \{ M \mid {\mathrm{Ann}}_R (x) \in {\mathcal{G}}\text{ for every } x \in M\}$$ Then, the corresponding torsion pair $({\mathcal{E}}_{\mathcal{G}}, {\mathcal{F}}_{\mathcal{G}})$ is generated by the cyclic modules $R/J$ where $J \in {\mathcal{G}}$. The classes ${\mathcal{E}}_{\mathcal{G}}$ and ${\mathcal{F}}_{\mathcal{G}}$ are referred to as the [*${\mathcal{G}}$-torsion*]{} and [*${\mathcal{G}}$-torsion-free*]{} classes, respectively.
Conversely, if $({\mathcal{E}}, {\mathcal{F}})$ is a hereditary torsion pair in ${\mathrm{Mod}\textrm{-}R}$, the set $$\{J\leq R\mid R/J\in {\mathcal{E}}\}$$ is a right Gabriel topology.
For a right $R$-module $M$ let $t_{\mathcal{G}}(M)$ denote the ${\mathcal{G}}$-torsion submodule of $M$, or sometimes $t(M)$ when the Gabriel topology is clear from context.
The [*module of quotients*]{} of the Gabriel topology ${\mathcal{G}}$ of a right $R$-module $M$ is the module $$M_{\mathcal{G}}:= \varinjlim_{\substack{J \in {\mathcal{G}}}} {\operatorname{Hom}}_R(J, M/t_{\mathcal{G}}(M)).$$ Furthermore, there is a canonical homomorphism $$\psi_M: M\cong {\operatorname{Hom}}_R(R, M) \to M_{\mathcal{G}}.$$ By substituting $M=R$, the assignment gives a ring homomorphism $\psi_R:R \to R_{\mathcal{G}}$ and furthermore, for each $R$-module $M$ the module $M_{\mathcal{G}}$ is both an $R$-module and an $R_{\mathcal{G}}$-module. Both the kernel and cokernel of the map $\psi_M$ are ${\mathcal{G}}$-torsion $R$-modules, and in fact ${\operatorname{Ker}}(\psi_M) = t_{\mathcal{G}}(M)$.
Let $q: {\mathrm{Mod}\textrm{-}R}\to {\mathrm{Mod}\textrm{-}R}_{\mathcal{G}}$ denote the functor that maps each $M$ to its module of quotients. Let $\psi^\ast$ denote the right exact functor ${\mathrm{Mod}\textrm{-}R}\to {\mathrm{Mod}\textrm{-}R}_{\mathcal{G}}$ where $\psi^\ast(M):= M \otimes R_{\mathcal{G}}$. In general, there is a natural transformation $\Theta: \psi^\ast \to q$ with $\Theta_M:M\otimes R_{\mathcal{G}}\to M_{\mathcal{G}}$ which is defined as $m \otimes \eta \mapsto \psi_M(m) \cdot \eta$. That is, for every $M$ the following triangle commutes. $$(\star)\qquad \qquad \xymatrix{
M \ar[rr]^{\psi^\ast(M)} \ar[dr]_{\psi_M}&&M \otimes_R R_{\mathcal{G}}\ar[dl]^{\Theta_M}\\
&M_{\mathcal{G}}&}$$ A right $R$-module is [*${\mathcal{G}}$-closed*]{} if the following natural homomorphisms are all isomorphisms for every $J \in {\mathcal{G}}$. $$M \cong {\operatorname{Hom}}_R(R, M) \to {\operatorname{Hom}}_R (J, M)$$ This amounts to saying that ${\operatorname{Hom}}_R(R/J,M) =0$ for every $J \in {\mathcal{G}}$ (i.e. $M$ is [*${\mathcal{G}}$-torsion-free*]{}) and ${\operatorname{Ext}}^1_R(R/J,M) =0$ for every $J \in {\mathcal{G}}$ (i.e. $M$ is [*${\mathcal{G}}$-injective*]{}). Thus if $M$ is ${\mathcal{G}}$-closed then $M$ is isomorphic to its module of quotients $M_{\mathcal{G}}$. Conversely, every $R$-module of the form $M_{\mathcal{G}}$ is ${\mathcal{G}}$-closed. The ${\mathcal{G}}$-closed modules form a full subcategory of both ${\mathrm{Mod}\textrm{-}R}$ and ${\mathrm{Mod}\textrm{-}R}_{\mathcal{G}}$.
A left $R$-module $N$ is called [*${\mathcal{G}}$-divisible*]{} if $JN = N$ for every $J\in {\mathcal{G}}$. Equivalently, $N$ is ${\mathcal{G}}$-divisible if and only if $R/J \otimes_R N =0$ for each $J \in {\mathcal{G}}$. We denote the class of ${\mathcal{G}}$-divisible modules by ${\mathcal{D}}_{\mathcal{G}}$. It is straightforward to see that ${\mathcal{D}}_{\mathcal{G}}$ is a torsion class in ${R\textrm{-}\mathrm{Mod}}$.
A right Gabriel topology is [*faithful*]{} if ${\operatorname{Hom}}_R(R/J, R) =0$ for every $J \in {\mathcal{G}}$, or equivalently if $R$ is ${\mathcal{G}}$-torsion-free, that is the natural map $\psi_R\colon R \to R_{\mathcal{G}}$ is injective. A right Gabriel topology is [*finitely generated*]{} if it has a basis consisting of finitely generated right ideals, or equivalently if the torsion-free class ${\mathcal{F}}_{\mathcal{G}}$ is closed under direct limits.
In this paper, we will only be concerned with Gabriel topologies over commutative rings. In this setting, much useful research has already done in this direction. Specifically, in [@H], Hrbek showed that over commutative rings the faithful finitely generated Gabriel topologies are in bijective correspondence with $1$-tilting classes, and that the latter are exactly the classes of ${\mathcal{G}}$-divisible modules for some faithful finitely generated Gabriel topology ${\mathcal{G}}$, as stated in the following theorem.
[@H Theorem 3.16] \[T:Hrb-tilting\] Let R be a commutative ring. There are bijections between the following collections.
1. 1-tilting classes ${\mathcal{T}}$.
2. faithful finitely generated Gabriel topologies ${\mathcal{G}}$.
3. faithful hereditary torsion pairs $({\mathcal{E}},{\mathcal{F}})$ of finite type in ${\mathrm{Mod}\textrm{-}R}$.
Moreover, the tilting class ${\mathcal{T}}$ is the class of ${\mathcal{G}}$-divisible modules with respect to the Gabriel topology ${\mathcal{G}}$.
When we refer to the [*Gabriel topology associated to the $1$-tilting class ${\mathcal{T}}$*]{} we will always mean the Gabriel topology in the sense of the above theorem. In addition we will often denote ${\mathcal{A}}$ to be the right ${\operatorname{Ext}}$-orthogonal class to ${\mathcal{D}}_{\mathcal{G}}={\mathcal{T}}$ in the situation just described, so $({\mathcal{A}}, {\mathcal{D}}_{\mathcal{G}})$ will denote the $1$-tilting cotorsion pair.
In [@AHHr] the correspondence between faithfully finitely generated Gabriel topologies and $1$-tilting classes over commutative rings was extended to finitely generated Gabriel topologies which were shown to be in bijective correspondence with silting classes. Thus in this case the class ${\mathcal{D}}_{\mathcal{G}}$ of ${\mathcal{G}}$-divisible modules coincides with the class ${\operatorname{Gen}}T$ for some silting module $T$.
Homological ring epimorphisms {#S:homological}
-----------------------------
There is a special class of Gabriel topologies which behave particularly well and are related to ring epimorphisms. The majority of this paper will be restricted to looking at these Gabriel topologies. The standard examples of these Gabriel topologies over $R$ are localisations of $R$ with respect to a multiplicative subset.
A [*ring epimorphism*]{} is a ring homomorphism $R \overset{u}\to U$ such that $u$ is an epimorphism in the category of unital rings. This is equivalent to the natural map $ U \otimes_R U \to U$ induced by the multiplication in $U$ being an isomorphism, or equivalently that $U \otimes_R (U / u(R)) =0$ (see [@Ste75 Chapter XI.1].
Two ring epimorphisms $R \overset{u}\to U$ and $R \overset{u'}\to U'$ are equivalent if there is a ring isomorphism $\sigma\colon U\to U'$ such that $\sigma u=u'$.
A ring epimorphism is [*homological*]{} if ${\operatorname{Tor}}^R_n(U_R,{}_RU) = 0$ for all $n >0$. A ring epimorphism is called [*(left) flat*]{} if $u$ makes $U$ into a flat left $R$-module. Clearly all flat ring epimorphisms are homological. We will denote the cokernel of $u$ by $K$ and sometimes by $U/R$ or $U/u(R)$.
A left flat ring epimorphism $R \overset{u}\to U$ is called a [*perfect right localisation*]{} of $R$. In this case, by [@Ste75 Chapter XI.2, Theorem 2.1] the family of right ideals $${\mathcal{G}}= \{ J \leq R \mid J U = U \}$$ forms a right Gabriel topology. Moreover, there is a ring isomorphism $\sigma:U \to R_{\mathcal{G}}$ such that $\sigma u: R \to R_{\mathcal{G}}$ is the canonical isomorphism $\psi_R: R \to R_{\mathcal{G}}$, or, in other words, $u$ and $\psi_R$ are equivalent ring epimorphisms. Note also that a right ideal $J$ of $R$ is in ${\mathcal{G}}$ if and only if $R/J \otimes_R U =0$.
We will make use of the characterisations of perfect right localisations from Proposition 3.4 in Chapter XI.3 of Stenström’s book [@Ste75].
In particular, Proposition 3.4 states that the right Gabriel topology ${\mathcal{G}}$ associated to a flat ring epimorphism $R \overset{u}\to U$ is finitely generated and the ${\mathcal{G}}$-torsion submodule $t_{\mathcal{G}}(M)$ of a right $R$-module $M$ is the kernel of the canonical homomorphism $M\to M \otimes_R U $. Thus, $K=U/u(R)$ is ${\mathcal{G}}$-torsion, hence ${\operatorname{Hom}}_R(K, U)=0$. If moreover the flat ring epimorphism $R \overset{u}\to U$ is injective, then $ {\operatorname{Tor}}^R_1(M, K) \cong t_{\mathcal{G}}(M)$ and ${\mathcal{G}}$ is faithful.
\[R:pdU=1\] *From the above observations and results in [@H], when $R$ is commutative and $R \overset{u}\to U$ is a flat injective epimorphism one can associate a $1$-tilting class which is exactly the class of ${\mathcal{G}}$-divisible modules. In the case that additionally ${\operatorname{p.dim}}_R U \leq 1$, one can apply a result from [@AS] which states that $U \oplus K$ is a $1$-tilting module, so there is a $1$-tilting class denoted ${\mathcal{T}}: =(U \oplus K)^\perp = {\operatorname{Gen}}(U)$. In fact, we claim that this is exactly the $1$-tilting class of ${\mathcal{G}}$-divisible modules. Explicitly, the Gabriel topology associated to ${\mathcal{T}}$ in the sense of Theorem \[T:Hrb-tilting\] is exactly the collection of ideals $\{J \mid JM = M \text{ for every } M \in {\mathcal{T}}\}$. The Gabriel topology that arises from the perfect localisation is the collection $\{J \mid JU = U \}$ and since $U \in {\mathcal{T}}= {\operatorname{Gen}}U$, the Gabriel topologies associated to these two $1$-tilting classes are the same. We conclude that the two $1$-tilting classes coincide: ${\operatorname{Gen}}_R(U) = {\mathcal{D}}_{\mathcal{G}}$.\
In [@H Proposition 5.4] the converse is proved: If one starts with a $1$-tilting class ${\mathcal{T}}$ with associated Gabriel topology ${\mathcal{G}}$, so that ${\mathcal{T}}={\mathcal{D}}_{\mathcal{G}}$, then $R_{\mathcal{G}}$ is a perfect localisation and ${\operatorname{p.dim}}R_{\mathcal{G}}\leq 1$ if and only if ${\operatorname{Gen}}R_{\mathcal{G}}= {\mathcal{D}}_{\mathcal{G}}$.*
The following lemma will be useful when working with a Gabriel topology over a commutative ring that arises from a perfect localisation.
\[L:finmanyann\] Let $R$ be a commutative ring, $u:R \to U$ a flat injective ring epimorphism, and ${\mathcal{G}}$ the associated Gabriel topology. Then the annihilators of the elements of $U/R$ form a sub-basis for the Gabriel topology ${\mathcal{G}}$. That is, for every $J\in {\mathcal{G}}$ there exist $z_1, z_2, \dots , z_n \in U$ such that $$\bigcap_{\substack{
0 \leq i \leq n}}
{\mathrm{Ann}}_R(z_i +R) \subseteq J.$$
Every ideal of the form ${\mathrm{Ann}}_R(z+R)$ is an ideal in ${\mathcal{G}}$ since $K=U/R$ is ${\mathcal{G}}$-torsion.
Fix an ideal $J \in {\mathcal{G}}$. Then, $U = JU$, so $1_U = \sum_{0 \leq i \leq n} a_i z_i$ where $a_i \in J$ and $z_i \in U$. We claim that $$\bigcap_{\substack{
0 \leq i \leq n}}
{\mathrm{Ann}}_R(z_i +R) \subseteq J.$$ Take $b \in \bigcap_{\substack{
0 \leq i \leq n}}
{\mathrm{Ann}}_R(z_i +R)$. Then $$b = \sum_{0 \leq i \leq n} b a_i z_i \in J$$ since each $b z_i \in R$, hence $b a_i z_i \in J$, and it follows that $b \in J$.
Enveloping $1$-tilting classes over commutative rings {#S:tilting-enveloping}
=====================================================
For this section, $R$ will always be a commutative ring and ${\mathcal{T}}$ a $1$-tilting class.
By Theorem \[T:Hrb-tilting\] there is a faithful finitely generated Gabriel topology ${\mathcal{G}}$ such that ${\mathcal{T}}$ is the class of ${\mathcal{G}}$-divisible modules. We denote again by $({\mathcal{E}}_{\mathcal{G}}, {\mathcal{F}}_{\mathcal{G}})$ the associated faithful hereditary torsion pair of finite type. We use ${\mathcal{D}}_{\mathcal{G}}$ and ${\mathcal{T}}= {\operatorname{Gen}}T = T^\perp$ interchangeably to denote the $1$-tilting class, and ${\mathcal{A}}$ to denote the right orthogonal class ${}^\perp {\mathcal{D}}_{\mathcal{G}}$.\
The aim of this section is to show that if ${\mathcal{T}}$ is enveloping, then $R_{\mathcal{G}}$, the ring of quotients with respect to ${\mathcal{G}}$, is ${\mathcal{G}}$-divisible and therefore $\psi_R:R \to R_{\mathcal{G}}$ is a perfect localisation of $R$.
In Section \[S:enveloping\], we will moreover, show that $R_{\mathcal{G}}$ has projective dimension at most one, thus the $1$-tilting class arises from the flat injective epimorphism $R\to R_{\mathcal{G}}$ (see Proposition \[P:pd1\], Corollary \[C:U-envelope\]).
Recall that if ${\mathcal{T}}$ is $1$-tilting, ${\mathcal{T}}\cap {}^\perp{\mathcal{T}}= {\mathrm{Add}}(T)$ (see [@GT12 Lemma 13.10]). By (T3) of the definition of a $1$-tilting module we have the following short exact sequence $$\text{(T3) }\quad
0 \to R \overset{\varepsilon}\to T_0 \to T_1 \to 0$$ where $T_0, T_1 \in {\mathrm{Add}}(T)$. In fact, this short exact sequence is a special ${\mathcal{D}}_{\mathcal{G}}$-preenvelope of $R$, and $T_0 \oplus T_1$ is a $1$-tilting module which generates ${\mathcal{T}}$ by [@GT12 Theorem 13.18 and Remark 13.19].
Furthermore, assuming that $R$ has a ${\mathcal{D}}_{\mathcal{G}}$-envelope, we can suppose without loss of generality that the sequence (T3) is the ${\mathcal{D}}_{\mathcal{G}}$-envelope of $R$, since an envelope is extracted from a special preenvelope by passing to direct summands (Proposition \[P:Xu-env\]). For the rest of the section we will denote the ${\mathcal{D}}_{\mathcal{G}}$-envelope of $R$ by $\varepsilon$.
Recall from Section \[S:gab-top\] that for every $M \in {\mathrm{Mod}\textrm{-}R}$ there is the commuting diagram $(\star)$.
Since ${\mathcal{G}}$ is faithful we have the following short exact sequence where $\psi_R$ is a ring homomorphism and $R_{\mathcal{G}}/R$ is ${\mathcal{G}}$-torsion. $$(\dag)\quad
0 \to R \overset{\psi_R} \to R_{\mathcal{G}}\to R_{\mathcal{G}}/R \to 0$$
We begin with some preliminary facts that hold for a general $1$-tilting class ${\mathcal{D}}_{\mathcal{G}}$ and which use only properties of the associated Gabriel topology.
Recall that $D$ is ${\mathcal{G}}$-divisible if and only if $R/J \otimes_R D=0$ for every $J \in {\mathcal{G}}$ if and only if $M\otimes_RD=0$ for every ${\mathcal{G}}$-torsion module $M$.
\[L:tor-Rg\] Let ${\mathcal{D}}_{\mathcal{G}}$ be a $1$-tilting class. Then the following statements hold.
1. If $N$ is a ${\mathcal{G}}$-torsion-free module then the natural map\
$\psi^\ast(N): N \to N \otimes_R R_{\mathcal{G}}$ is a monomorphism.
2. If $D$ is both ${\mathcal{G}}$-divisible and ${\mathcal{G}}$-torsion-free, then $D$ is a $R_{\mathcal{G}}$-module and $D \cong D\otimes_R R_{\mathcal{G}}$ via the natural map\
$\psi^\ast(D)={\operatorname{id}}_D \otimes_R \psi_R:D \otimes_R R \to D \otimes_R R_{\mathcal{G}}$.
3. If ${\operatorname{p.dim}}M \leq 1$, then ${\operatorname{Tor}}^R_1(M, R_{\mathcal{G}})=0$.
4. If ${\operatorname{p.dim}}M \leq 1$ and $M$ is ${\mathcal{G}}$-torsion-free, then\
${\operatorname{Tor}}^R_1(M, R_{\mathcal{G}})=0={\operatorname{Tor}}^R_1(M, R_{\mathcal{G}}/R)$.
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1. Consider the following commuting triangle where $N$ is ${\mathcal{G}}$-torsion-free. $$\xymatrix{
0 \ar[d]& \\
N \ar[r]^{\psi^\ast(N) \hspace{10pt}} \ar[d]_{\psi_N}&N \otimes_R R_{\mathcal{G}}\ar[dl]^{\Theta_N}\\
N_{\mathcal{G}}&}$$ Then $\psi_N$ is a monomorphism and since $\psi_N = \Theta_N \circ \psi^\ast(N)$, also $\psi^\ast(N)$ is a monomorphism.
2. Consider the following commuting diagram where the horizontal sequence is exact by (1) as $D$ is ${\mathcal{G}}$-torsion-free. $$\xymatrix{
&0 \ar[d]& \\
0 \ar[r] &D \ar[r]^{\psi^\ast(D) \hspace{10pt}} \ar[d]_{\psi_D}&D \otimes_R R_{\mathcal{G}}\ar[dl]^{\Theta_D} \ar[r] & D \otimes_R R_{\mathcal{G}}/R \ar[r] &0\\
&D_{\mathcal{G}}&}$$ Additionally, $D\otimes_R R_{\mathcal{G}}/R=0$, since $R_{\mathcal{G}}/R$ is ${\mathcal{G}}$-torsion. Therefore $$\psi^\ast(D):D \to D \otimes_R R_{\mathcal{G}}$$ is an isomorphism.
3. Consider the following exact sequence formed by taking the tensor product of $M$ with the short exact sequence $(\dag)$. $$0 = {\operatorname{Tor}}^R_1(M, R) \to {\operatorname{Tor}}^R_1(M, R_{\mathcal{G}}) \to {\operatorname{Tor}}^R_1(M, R_{\mathcal{G}}/R)$$ By assumption ${\operatorname{p.dim}}M \leq 1$, so there is a projective resolution of $M$,$$0 \to P_1 \overset{\gamma}\to P_0 \to M \to 0$$ where $P_0, P_1$ are projective modules. It follows that ${\operatorname{Tor}}^R_1(M, R_{\mathcal{G}})$ is isomorphic to the kernel of the map $\gamma \otimes_R {\operatorname{id}}_{R_{\mathcal{G}}}$. $$P_1 \otimes_R R_{\mathcal{G}}\xrightarrow{\gamma \otimes_R {\operatorname{id}}_{R_{\mathcal{G}}}} P_0 \otimes_R R_{\mathcal{G}}$$ As $P_1$ is a submodule of $R^{(\alpha)}$ for some cardinal $\alpha$, also $P_1 \otimes_R R_{\mathcal{G}}$ is a submodule of the ${\mathcal{G}}$-torsion-free module $R_{\mathcal{G}}^{(\alpha)}$. Thus ${\operatorname{Tor}}^R_1(M, R_{\mathcal{G}})$ is itself a ${\mathcal{G}}$-torsion-free module.\
However, by applying the tensor product $(- \otimes_R R_{\mathcal{G}}/R)$ to the above projective presentation of $M$, we find that ${\operatorname{Tor}}^R_1(M, R_{\mathcal{G}}/R)$ is a submodule of $P_1 \otimes_R R_{\mathcal{G}}/R$ which is ${\mathcal{G}}$-torsion. Since ${\mathcal{E}}_{\mathcal{G}}$ is a hereditary torsion class also ${\operatorname{Tor}}^R_1(M, R_{\mathcal{G}}/R)$ is ${\mathcal{G}}$-torsion. Therefore, also $ {\operatorname{Tor}}^R_1(M, R_{\mathcal{G}})$ is ${\mathcal{G}}$-torsion since it is a submodule of ${\operatorname{Tor}}^R_1(M, R_{\mathcal{G}}/R)$. We conclude that $ {\operatorname{Tor}}^R_1(M, R_{\mathcal{G}})$ is both ${\mathcal{G}}$-torsion and ${\mathcal{G}}$-torsion-free, hence $ {\operatorname{Tor}}^R_1(M, R_{\mathcal{G}})=0$.
4. Consider the following commuting triangle where $\psi^\ast(M)$ is a monomorphism from (1). $$\xymatrix{
&0 \ar[d]& \\
0 \ar[r]&M \ar[r]^{\psi^\ast(M) \hspace{10pt}} \ar[d]_{\psi_M}&M \otimes_R R_{\mathcal{G}}\ar[dl]^{\Theta_M}\\
&M_{\mathcal{G}}&}$$ By applying the functor $(M \otimes_R -)$ to the short exact sequence $(\dag)$, we have the following exact sequences. $$\xymatrix{
0 \ar[r] & {\operatorname{Tor}}^R_1(M, R_{\mathcal{G}}) \ar[r] & {\operatorname{Tor}}^R_1( M, R_{\mathcal{G}}/R) \ar[r]&0&}$$ $$\xymatrix{
&0 \ar[r] & M \ar[r]^{\psi^\ast(M) \hspace{10pt}} & M \otimes_R R_{\mathcal{G}}\ar[r] & M \otimes_R R_{\mathcal{G}}/R \ar[r] & 0}$$ By (3), ${\operatorname{Tor}}^R_1(M, R_{\mathcal{G}})=0$, thus also ${\operatorname{Tor}}^R_1(M, R_{\mathcal{G}}/R)=0$ as these two modules are isomorphic from the above short exact sequence.
We now show two lemmas about the $1$-tilting module $T_0 \oplus T_1$ and the class ${\mathrm{Add}}(T_0 \oplus T_1)$ assuming that $R$ has a ${\mathcal{D}}_{\mathcal{G}}$-envelope.
\[L:R-env\] Let the following short exact sequence be a ${\mathcal{D}}_{\mathcal{G}}$-envelope of $R$. $$0 \to R \overset{\varepsilon}\to T_0 \to T_1 \to 0$$ Then $T_0$ is ${\mathcal{G}}$-torsion-free and $T_0 \cong T_0 \otimes_R R_{\mathcal{G}}$.
We will show that for every $J \in {\mathcal{G}}$, $T_0 ]$, the annihilator of $J$ in $T_0$ is zero. Set $w:=\varepsilon(1_R)$ and fix a $J \in {\mathcal{G}}$. As $T_0 = JT_0$, $w = \sum_{1 \leq i \leq n} a_i z_i$ where $a_i \in J$ and $z_i \in T_0$. This sum is finite, so we can define the following maps. $$\xymatrix@R=.1cm{
{\mathbf{z}}:R \ar[r] & \bigoplus_{1 \leq i \leq n} T_0 & {\mathbf{a}}:\bigoplus_{1 \leq i \leq n} T_0 \ar[r] & T_0\\
\hspace{15pt}1_R \ar@{|-_{>}}[r] &(z_1, ..., z_n) & \hspace{5pt}(x_1, ..., x_n) \ar@{|-_{>}}[r] & \sum_i a_ix_i}$$ As $\bigoplus_nT_0$ is also ${\mathcal{G}}$-divisible, by the preenvelope property of $\varepsilon$ there exists a map $f:T_0 \to \bigoplus_nT_0$ such that $f \varepsilon = {\mathbf{z}}$. Also, ${\mathbf{a}}{\mathbf{z}}(1_R) = \sum_{1 \leq i \leq n} a_i z_i = w$, so ${\mathbf{a}}{\mathbf{z}} = \varepsilon$ and the following diagram commutes. $$\xymatrix@C=1.8cm@R=1.3cm{
0\ar[r]&
{R}\ar[dr]^{{\mathbf{z}}} \ar[ddr]_\varepsilon \ar[r]^\varepsilon&T_0\ar[r]^{\beta} \ar[d]^{f}& T_1 \ar[r]&0\\
& & \bigoplus_n T_0 \ar^{{\mathbf{a}}}[d]\\
& & T_0}$$ By the envelope property of $\varepsilon$, ${\mathbf{a}}f$ is an automorphism of $T_0$. The restriction of the automorphism ${\mathbf{a}}f$ to $T_0[J]$ is an automorphism of $T_0[J]$, and factors through the module $\bigoplus_nT_0[J]$. However $\mathbf{a}( \bigoplus_nT_0[J]) =0$, so ${\mathbf{a}}f(T_0[J]) =0$, but $\mathbf{a}f$ restricted to $T_0[J]$ is an automorphism, thus $T_0[J] =0$.\
It follows from (3) of Lemma \[L:tor-Rg\] $T_0 \cong T_0 \otimes_R R_{\mathcal{G}}$ since $T_0$ is ${\mathcal{G}}$-divisible.
\[L:addT-tens-tf\] Suppose $R$ has a ${\mathcal{D}}_{\mathcal{G}}$-envelope in ${\mathrm{Mod}\textrm{-}R}$. Then for every $M \in {\mathrm{Add}}(T_0 \oplus T_1)$, $M \otimes_R R_{\mathcal{G}}$ is ${\mathcal{G}}$-torsion-free.
From Lemma \[L:R-env\], $T_0 \cong T_0 \otimes_R R_{\mathcal{G}}$ is ${\mathcal{G}}$-torsion-free. We first show that $T_1 \otimes_R R_{\mathcal{G}}$ is ${\mathcal{G}}$-torsion-free. Consider the following short exact sequence obtained by applying $(- \otimes_R R_{\mathcal{G}})$ to the envelope of $R$, and note that ${\operatorname{Tor}}^R_1(T_1, R_{\mathcal{G}})=0$ by Lemma \[L:tor-Rg\] (3). $$0 \to R_{\mathcal{G}}\to T_0 \otimes_R R_{\mathcal{G}}\to T_1 \otimes_R R_{\mathcal{G}}\to 0$$ As $R_{\mathcal{G}}$ is ${\mathcal{G}}$-closed and $T_0 \otimes_R R_{\mathcal{G}}$ is ${\mathcal{G}}$-torsion-free, by applying the covariant functor ${\operatorname{Hom}}_R(R/J,-)$ to the above sequence for every $J \in {\mathcal{G}}$, we obtain that $T_1 \otimes_R R_{\mathcal{G}}$ must be ${\mathcal{G}}$-torsion-free.\
It is now straightforward to see that the statement holds for any direct summand of $ (T_0 \oplus T_1)^{(\alpha)}$.
We look at ${\mathcal{D}}_{\mathcal{G}}$-envelopes of ${\mathcal{G}}$-torsion modules in ${\mathrm{Mod}\textrm{-}R}$, and find that they are also ${\mathcal{G}}$-torsion.
\[L:torsion-env\] Suppose ${\mathcal{D}}_{\mathcal{G}}$ is enveloping in ${\mathrm{Mod}\textrm{-}R}$ and $M$ is a ${\mathcal{G}}$-torsion $R$-module. Then the ${\mathcal{D}}_{\mathcal{G}}$-envelope of $M$ is ${\mathcal{G}}$-torsion.
To begin with, fix a finitely generated $J \in {\mathcal{G}}$ with a set $\{a_1, \dots, a_t\}$ of generators and consider a ${\mathcal{D}}_{\mathcal{G}}$-envelope $D(J)$ of the cyclic ${\mathcal{G}}$-torsion module $R/J$, denoted as follows. $$0 \to R/J \hookrightarrow D(J) \to A(J) \to 0$$ We will use the T-nilpotency of direct sums of envelopes as in Theorem \[T:Xu-sums\] (2). Consider the following countable direct sum of envelopes of $R/J$ which is itself an envelope, by Theorem \[T:Xu-sums\] (1): $$0 \to \bigoplus_{n}
(R/J)_{n}\hookrightarrow \bigoplus_{n} D(J)_{n} \to \bigoplus_{n}A(J)_{n} \to 0 .$$ Choose an element $a \in J$ and for each $n$ set $f_n\colon D(J)_n\to D(J)_{n+1}$ to be the multiplication by $a$.
Then clearly $(R/J)_n$ vanishes under the action of $f_n $, hence we can apply Theorem \[T:Xu-sums\] (2). For every $d \in D(J)$, there exists an $m$ such that $$f_m \circ \cdots \circ f_2 \circ f_1 (d) = 0 \in D(J)_{(m+1)}.$$ Hence for every $d \in D$ there is an integer $m$ for which $a^m d = 0$.\
Fix $d \in D$ and let $m_i$ be the minimal natural number for which $(a_i)^{m_i}d=0$ and set $m:= \sup\{m_i: 1 \leq i \leq t\}$. Then for a large enough integer $k$ we have that $J^k d =0$ (for example set $k=tm$), and $J^k \in {\mathcal{G}}$. Thus every element of $D(J)$ is annihilated by an ideal contained in ${\mathcal{G}}$, therefore $D(J)$ is ${\mathcal{G}}$-torsion.
Now consider an arbitrary ${\mathcal{G}}$-torsion module $M$. Then $M$ has a presentation $\bigoplus_{\alpha\in \Lambda} R/J_\alpha \overset{p}\to M \to 0$ for a family $\{J_\alpha\}_{\alpha \in \Lambda}$ of ideals of ${\mathcal{G}}$. Since ${\mathcal{G}}$ is of finite type, we may assume that each $J_{\alpha}$ is finitely generated.
Take the push-out of this map with the ${\mathcal{D}}_{\mathcal{G}}$-envelope of $\bigoplus_\alpha R/J_\alpha$. $$\xymatrix{
0 \ar[r]&\bigoplus_{\substack{\alpha \in \Lambda}} R/J_\alpha \ar[r] \ar[d]^p& \bigoplus_{\substack{\alpha \in \Lambda}}D(J_\alpha) \ar[r] \ar[d]& \bigoplus_{\substack{\alpha \in \Lambda}}A(J_\alpha) \ar@{=}[d] \ar[r] & 0\\
0 \ar[r]&M\ar[r] \ar[d] & Z \ar[r] \ar[d]& \bigoplus_{\substack{\alpha \in \Lambda}}A(J_\alpha) \ar[r] & 0\\
&0&0 &&}$$ The bottom short exact sequence forms a preenvelope of $M$. We have shown above that for every $\alpha$ in $A$, $D(J_\alpha)$ is ${\mathcal{G}}$-torsion, so also $Z$ is ${\mathcal{G}}$-torsion. Therefore, as the ${\mathcal{D}}_{\mathcal{G}}$-envelope of $M$ must be a direct summand of $Z$ by Proposition \[P:Xu-env\], also the ${\mathcal{D}}_{\mathcal{G}}$-envelope of $M$ is ${\mathcal{G}}$-torsion.
The following is a corollary to Lemma \[L:addT-tens-tf\] and Lemma \[L:torsion-env\].
\[C:tor-tens-div\] Suppose ${\mathcal{D}}_{\mathcal{G}}$ is enveloping in ${\mathrm{Mod}\textrm{-}R}$ and suppose $M$ is a ${\mathcal{G}}$-torsion $R$-module. Then $M \otimes_R R_{\mathcal{G}}$ is ${\mathcal{G}}$-divisible.
Let the following be a ${\mathcal{D}}_{\mathcal{G}}$-envelope of a ${\mathcal{G}}$-torsion module $M$, where both $D$ and $A$ are ${\mathcal{G}}$-torsion by Lemma \[L:torsion-env\]. $$0 \to M \to D \to A \to 0$$ The module $A$ is ${\mathcal{G}}$-divisible and $R_{\mathcal{G}}/R$ is ${\mathcal{G}}$-torsion so $A \otimes_R R_{\mathcal{G}}/R =0$, hence $A \to A \otimes_R R_{\mathcal{G}}$ is surjective. In particular, $A \otimes_R R_{\mathcal{G}}$ is ${\mathcal{G}}$-torsion. Also as $A \in {\mathrm{Add}}(T_0 \oplus T_1)$, $A \otimes_R R_{\mathcal{G}}$ is ${\mathcal{G}}$-torsion-free by Lemma \[L:addT-tens-tf\] (2). It follows that $A\otimes_R R_{\mathcal{G}}$ is both ${\mathcal{G}}$-torsion and ${\mathcal{G}}$-torsion-free so $A\otimes_R R_{\mathcal{G}}=0$. Additionally as ${\operatorname{p.dim}}A \leq 1$, ${\operatorname{Tor}}^R_1(A, R_{\mathcal{G}})=0$, so the functor $(- \otimes_R R_{\mathcal{G}})$ applied to the envelope of $M$ reduces to the following isomorphism. $$0={\operatorname{Tor}}^R_1(A, R_{\mathcal{G}}) \to M \otimes_R R_{\mathcal{G}}\overset{\cong}\to D \otimes_R R_{\mathcal{G}}\to A \otimes_R R_{\mathcal{G}}=0$$ Hence as $D \otimes_R R_{\mathcal{G}}$ is ${\mathcal{G}}$-divisible, also $M \otimes_R R_{\mathcal{G}}$ is ${\mathcal{G}}$-divisible, as required.
\[P:R\_G-divisible\] Suppose ${\mathcal{D}}_{\mathcal{G}}$ is enveloping in ${\mathrm{Mod}\textrm{-}R}$. Then $R_{\mathcal{G}}$ is ${\mathcal{G}}$-divisible.
We will show that for each $J \in {\mathcal{G}}$, $R/J \otimes_R R_{\mathcal{G}}=0$. Fix a $J \in {\mathcal{G}}$. By Corollary \[C:tor-tens-div\], $R/J \otimes_R R_{\mathcal{G}}$ is ${\mathcal{G}}$-divisible, Thus we have $R/J \otimes_R (R/J \otimes_R R_{\mathcal{G}}) =0$. However $$0=R/J \otimes_R (R/J \otimes_R R_{\mathcal{G}}) \cong (R/J \otimes_R R/J) \otimes_R R_{\mathcal{G}}\cong R/J\otimes_RR_{\mathcal{G}},$$ since $R\to R/J$ is a ring epimorphism, thus $R_{\mathcal{G}}$ is ${\mathcal{G}}$-divisible.
Using the characterisation of a perfect localisation of [@Ste75 Chapter XI.3, Proposition 3.4], we can state the main result of this section.
\[P:tilting-env\] Assume that ${\mathcal{T}}$ is a $1$-tilting class over a commutative ring $R$ such that the class ${\mathcal{T}}$ is enveloping. Then the associated Gabriel topology ${\mathcal{G}}$ of ${\mathcal{T}}$ arises from a perfect localisation.
By Proposition \[P:R\_G-divisible\], $R_{\mathcal{G}}$ is ${\mathcal{G}}$-divisible, hence by [@Ste75 Proposition 3.4 (g)], $\psi\colon R\to R_{\mathcal{G}}$ is flat ring epimorphism and moreover it is injective.
The ${\mathcal{G}}$-completion of $R$ and the endomorphism ring of $K$ {#S:compl-endK}
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The aim of this section is to prove that if $R \overset{u}\to U$ is a commutative flat injective ring epimorphism with associated Gabriel topology ${\mathcal{G}}$, then there is a natural ring isomorphism between the following two rings. $$\Lambda(R) = \varprojlim_{\substack{J \in {\mathcal{G}}}} R/J {\rm \ and\ } {\operatorname{End}}_R(K)=\mathfrak{R}$$ This was mentioned in [@BP2 Remark 19.4], and a much stronger equivalence was shown in [@Pos3]. Also, it follows from this ring isomorphism that $\mathfrak{R}$ is a commutative ring.\
For completeness, we will give an explicit description of the isomorphism between the two rings.\
We will begin by briefly recalling some useful definitions about topological rings specifically referring to Gabriel topologies. Our reference is [@Ste75 Chapter VI.4]. Next we will continue by introducing $u$-contramodules in an analogous way to Positselski in [@Pos]. To finish, we show the ring isomorphism as well as a lemma and a proposition which relate the ${\mathcal{G}}$-torsion $R$-modules $R/J$ to the discrete quotient rings of $\mathfrak{R}$.
Topological rings
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A ring $R$ is a [*topological ring*]{} if it has a topology such that the ring operations are continuous.
A topological ring $R$ is [*right linearly topological*]{} if it has a topology with a basis of neighbourhoods of zero consisting of right ideals of $R$. The ring $R$ with a right Gabriel topology is an example of a right linearly topological ring.
If $R$ is a right linearly topological ring, then the set of right ideals $J$ in a basis $\mathfrak{ B}$ of the topology form a directed set, hence $\{R/J\mid J\in
\mathfrak B\}$ is an inverse system. The [*completion*]{} of $R$ is the module $$\Lambda_{\mathfrak{ B}}(R) := \varprojlim_{\substack{J \in \mathfrak B}} R/J.$$ There is a canonical map $\lambda:R \to \Lambda_\mathfrak{B}(R)$ which sends the element $r\in R$ to $(r +J)_{J\in \mathfrak{ B}}$. If the homomorphism $\lambda_R$ is injective, then $R$ is called [*separated*]{}, which is equivalent to $\bigcap_{J \in \mathfrak{ B}}J =0$. If the map $\lambda$ is surjective, $R$ is called [*complete*]{}.\
The [*projective limit topology*]{} on $\Lambda_{\mathfrak{ B}}(R)$ is the topology where a sub-basis of neighbourhoods of zero is given by the the kernels of the projection maps $\Lambda_{\mathfrak{ B}}(R) \to R/J$. That is, it is the topology induced by the product of the discrete topology on $\prod_{J \in \mathfrak{ B}} R/J$. If the ideals in $\mathfrak{ B}$ are two-sided in $R$, then the module $\Lambda_{\mathfrak{ B}}(R)$ is a ring. Furthermore, it is a linearly topological ring with respect to the projective limit topology. In this case, the ring $\Lambda_{\mathfrak{ B}}(R)$ is both separated and complete with this topology. Each element in $\Lambda_{\mathfrak{ B}}(R)$ is of the form $(r_J +J)_{J\in \mathfrak{ B}}$ with the relation that for $J \subseteq J'$, $r_J - r_{J'} \in J'$. We will simply write $\Lambda(R)$ when the basis $\mathfrak{ B}$ is clear from the context.
\[R:topologies\] *If $W(J)$ is the kernel of the projection $\pi_J\colon\Lambda_{\mathfrak{ B}}(R)\to R/J$, then clearly $W(J)\supseteq \Lambda(R)J$.*
Let $R$ be a linearly topological ring. A right $R$-module $N$ is [*discrete*]{} if for every $x \in N$, the annihilator ideal ${\mathrm{Ann}}_R(x) = \{r \in R: xr =0\}$ is open in the topology of $R$. In case the topology on $R$ is a Gabriel topology ${\mathcal{G}}$ on $R$, then $N$ is discrete if and only if it is ${\mathcal{G}}$-torsion.
A linearly topological ring is [*left pro-perfect*]{} ([@BP2]) if it is separated, complete, and with a base of neighbourhoods of zero formed by two-sided ideals such that all of its discrete quotient rings are perfect.
*For the rest of this subsection, we will be considering a flat injective ring epimorphism of commutative rings denoted $0 \to R \overset{u} \to U$, and we will denote by $K$ the cokernel $U/R$ of $u$.*
Let $\mathfrak{R}$ denote the endomorphism ring ${\operatorname{End}}_R(K)$. Take a finitely generated submodule $F$ of $K$, and consider the ideal formed by the elements of $\mathfrak{R}$ which annihilate $F$. The ideals of this form form a base of neighbourhoods of zero of $\mathfrak{R}$. Note that this is the same as considering ${\operatorname{End}}_R(K)$ with the subspace topology of the product topology on $K^K$ where the topology on $K$ is the discrete topology. We will consider $\mathfrak{R}$ endowed with this topology, which is also called the [*finite topology*]{}.
We will now state the above in terms of a Gabriel topology that arises from a perfect localisation. Let ${\mathcal{G}}$ be the Gabriel topology associated to the flat ring epimorphism $u$. As $K\otimes_RU=0$, $K$ is ${\mathcal{G}}$-torsion, or equivalently a discrete module. Thus there is a natural well-defined action of $\Lambda(R)$ on $K$. In other words, $K$ is a $\Lambda(R)$-module where for every element $(r_J +J)_{J \in {\mathcal{G}}}\in \Lambda (R)$ and every element $z\in U$, the scalar multiplication is defined by $(r_J +J)_{J \in {\mathcal{G}}} \cdot (z+R):= r_{J_z} z +R$ where $J_z:= {\mathrm{Ann}}_R(z+R)$. As well as the natural map $\lambda:R \to \Lambda(R)$, there is also a natural map $\nu:R \to \mathfrak{R}$ where each element of $R$ is mapped to the endomorphism of $K$ which is multiplication by that element.
If $R \overset{u}\to U$ is a flat injective ring epimorphism, then there is a homomorphism $$\alpha: \Lambda(R) = \varprojlim_{\substack{J \in {\mathcal{G}}}} R/J \to \mathfrak{R},$$ where $\alpha$ is induced by the action of $\Lambda(R)$ on $K$. It follows that the following triangle commutes. $$(\ast)\quad \xymatrix{
R \ar[d]_{\nu} \ar[r]^\lambda& \Lambda(R) \ar[dl]^{\alpha} \\
\mathfrak{R} & }$$ The rest of this section is dedicated to showing that $\alpha$ is a ring isomorphism. We will first show that $\alpha$ is injective, but before that we have to recall some terminology.
A module $M$ is [*$U$-h-divisible*]{} if $M$ is an epimorphic image of $U^{(\alpha)}$ for some cardinal $\alpha$. An $R$-module $M$ has a unique $U$-h-divisible submodule denoted $h_U(M)$, and it is the image of the map ${\operatorname{Hom}}_R(U,M) \to {\operatorname{Hom}}_R(R,M) \cong M$. Hence for an $R$-module $M$, by applying the contravariant functor ${\operatorname{Hom}}_R(-,M)$ to the short exact sequence $0 \to R \overset{u}\to U \to K \to 0$ we have the following short exact sequences. $$\label{eq:1contra}
0 \to {\operatorname{Hom}}_R(K, M) \to {\operatorname{Hom}}_R(U,M) \to h_U(M) \to 0$$ $$\label{eq:2contra}
0 \to M / h_U(M) \to {\operatorname{Ext}}^1_R(K,M) \to {\operatorname{Ext}}^1_R(U,M) \to 0$$
By applying the covariant functor ${\operatorname{Hom}}_R(K,-)$ to the same short exact sequence we have the following. $$\label{eq:3contra}
0= {\operatorname{Hom}}_R(K, U) \to {\operatorname{Hom}}_R(K,K) \overset{\delta}\to {\operatorname{Ext}}^1_R(K, R) \to {\operatorname{Ext}}^1_R(K,U) =0, $$ where the last term vanishes since by the flatness of the ring $U$, there is an isomorphism ${\operatorname{Ext}}^1_R(K,U)\cong {\operatorname{Ext}}^1_U(K\otimes_RU,U) =0$. Thus note that ${\operatorname{Hom}}_R(K, K)$ is isomorphic to ${\operatorname{Ext}}^1_R(K, R)$ via $\delta$.
Recall from Lemma \[L:finmanyann\] that the ideals ${\mathrm{Ann}}_R(z +R)$ for $z+R \in K$ form a sub-basis of the topology ${\mathcal{G}}$. Let ${\mathcal{S}}\subset {\mathcal{G}}$ denote denote the ideals of ${\mathcal{G}}$ of the form ${\mathrm{Ann}}_R(z +R)$ for $z+R \in K$. Clearly, the following two intersections of ideals coincide. $$\bigcap_{\substack{J \in {\mathcal{G}}}} J = \bigcap_{\substack{J \in {\mathcal{S}}}} J$$ We begin with some facts about $\Lambda(R)$ and $\mathfrak{R}$.
\[L:mapfacts\] Let $u:R \to U$ be a flat injective ring epimorphism. Then the following hold.
1. The kernel of $\nu: R \to \mathfrak{R}$ is the intersection $\bigcap_{J \in {\mathcal{S}}} J$.
2. The kernel of $\lambda: R \to \Lambda(R)$ is the intersection $\bigcap_{J \in {\mathcal{G}}} J$.
3. The ideal $\bigcap_{J \in {\mathcal{G}}} J$ is the maximal $U$-h-divisible submodule of $R$.
4. The homomorphism $\alpha: \Lambda(R) \to \mathfrak{R}$ is injective.
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1. For $r \in R$, $\nu (r)=0$ if and only if $rK=0$ if and only if $rz \in R$ for every $z \in U$. This amounts to $r \in {\mathrm{Ann}}_R(z +R)$ for every $z\in U$, hence $r \in \bigcap_{J \in {\mathcal{S}}} J$.
2. By the definition of $\lambda$ it is clear that $\lambda (r) =0$ if and only if $r \in J$ for every $J \in {\mathcal{G}}$.
3. First we show that $\bigcap_{J \in {\mathcal{G}}} J \subseteq h_U(R)$. Take $a \in \bigcap_{J \in {\mathcal{G}}} J$. We want to see that multiplication by $a$, $\dot{a}: R \to R$ extends to a map $f:U \to R$ (that is $\dot{a}$ is in the image of the map $u^\ast: {\operatorname{Hom}}_R(U, R) \to {\operatorname{Hom}}_R(R,R)$). By part (1) and its proof, $az \in R$ for every $z \in U$, so we have a well-defined map $\dot{a}: U \to R$, which makes the following triangle commute as desired. $$\xymatrix{
R \ar[d]_{\dot{a}} \ar[r]^u&U \ar[ld]^{\dot{a}} \\
R &}$$ Now take $a \in h_U(R)$. Since $h_U(R)$ is a ${\mathcal{G}}$-divisible submodule of $R$, $a \in J( h_U(R))\leq J$ for each $J \in {\mathcal{G}}$, as required.
4. Take $\eta=(r_J + J)_{J \in {\mathcal{G}}} \in \Lambda(R)$ such that $\alpha(\eta)=0$ or $\eta(z+R)=0$ for each $z \in U$. Then $r_Iz \in R$ where $I = {\mathrm{Ann}}_R(z+R)$. This implies $r_J \in J$ for each $J \in {\mathcal{S}}$, so $\eta=0$.
$u$-contramodules {#S:U-contra}
-----------------
We will begin by discussing a general commutative ring epimorphism $u$ before moving onto a flat injective ring epimorphisms.
Let $u\colon R\to U$ be a ring epimorphism. A [*$u$-contramodule*]{} is an $R$-module $M$ such that $${\operatorname{Hom}}_R(U, M) = 0 = {\operatorname{Ext}}^1_R(U,M).$$
\[L:geiglenz\][@GL91 Proposition 1.1] The category of $u$-contramodules is closed under kernels of morphisms, extensions, infinite products and projective limits in ${R\textrm{-}\mathrm{Mod}}$.
The following two lemmas are proved in [@Pos] for the case of the localisation of $R$ at a multiplicative system. For completeness we include their proofs in our setting.
\[L:pos1.2\][@Pos Lemma 1.2] Let $u\colon R\to U$ be a ring epimorphism and let $M$ be an $R$-module.
1. If ${\operatorname{Hom}}_R(U,M)=0$, then ${\operatorname{Hom}}_R(Z,M)=0$ for any $U$-h-divisible module $Z$.
2. If $M$ is a $u$-contramodule, then ${\operatorname{Ext}}_R^1(Z,M)=0={\operatorname{Hom}}_R(Z,M)$ for any $U$-module $Z$.
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1. By the $U$-h-divisibility of $Z$ there exists a map $U^{(\alpha)} \to Z \to 0$. As ${\operatorname{Hom}}_R(U^{(\alpha)}, M)=0$, it follows that ${\operatorname{Hom}}_R(Z, M)=0$.
2. First note that if $Z \in {\mathrm{Mod}\textrm{-}U}$, there is a short exact sequence $0 \to H \to U^{(\alpha)} \to Z \to 0$ of $U$-modules in ${\mathrm{Mod}\textrm{-}R}$. As ${\operatorname{Ext}}^1_R(U^{(\alpha)}, M)=0$ then ${\operatorname{Hom}}_R(H, M) \cong {\operatorname{Ext}}^1_R(Z,M) $. However, from (1) ${\operatorname{Hom}}_R(H, M)=0$, so also ${\operatorname{Ext}}^1_R(Z,M)=0$.
\[L:oneten\] [@Pos Lemma 1.10] Let $b:A \to B$ and $c:A \to C$ be two $R$-module homomorphisms such that $C$ is a $u$-contramodule while ${\operatorname{Ker}}(b)$ is a $U$-h-divisible $R$-module and ${\operatorname{Coker}}(b)$ is a $U$-module. Then there exists a unique homomorphism $f:B \to C$ such that $c =fb$.
First we show the existence of a homomorphism $f: B \to C$ such that $c =fb$. ${\operatorname{Ker}}b$ is a $U$-h-divisible module, so the composition $c \circ \ker b =0$ by Lemma \[L:pos1.2\] (1), hence the map $c$ factors through $\bar{c}:A/{\operatorname{Ker}}b \to C$ as in the following diagram. $$\xymatrix@C=1.2cm{
{\operatorname{Ker}}b \ar[r]^{\ker b} & A \ar@{->>}[dr] \ar[rdd]_c \ar[rr]^b & & B \ar[r]^{{\operatorname{coker}}b}& {\operatorname{Coker}}b \\
&&A/{\operatorname{Ker}}b \ar[d]^{\bar{c}} \ar@{^{(}->}[ru]\\
& &C}$$ By applying the functor ${\operatorname{Hom}}_R(-,C)$ to the right short exact sequence above we get the following exact sequence. $${\operatorname{Hom}}_R(B, C) \to {\operatorname{Hom}}_R(A/{\operatorname{Ker}}b, C) \to {\operatorname{Ext}}^1_R({\operatorname{Coker}}b, C)$$ By Lemma \[L:pos1.2\] (2), ${\operatorname{Ext}}^1_R({\operatorname{Coker}}b, C)=0$ as ${\operatorname{Coker}}b$ is a $U$-module.
Now we show the uniqueness of such a homomorphism. Suppose $h = f -g$ is such that $hb=0$. Then there exists a homomorphism $\bar{h}: {\operatorname{Coker}}b \to C$ such that $\bar{h} \circ {\operatorname{coker}}b = h$. $$\xymatrix@C=1.2cm{
A \ar[r]^b \ar[rd]& B \ar[d]^h \ar[r]^{{\operatorname{coker}}b} & {\operatorname{Coker}}b \ar[ld]^{\bar{h}}\\
&C&}$$ By assumption, ${\operatorname{Coker}}b$ is a $U$-module, and $C$ is a $u$-contramodule, so ${\operatorname{Hom}}_R({\operatorname{Coker}}b,C) =0$ by Lemma \[L:pos1.2\] (2). Thus $h$ must be the zero homomorphism, so $f =g$.
From now on, $u: R \to U$ will always be a flat injective ring epimorphism.
\[L:endcontra\][@BP2 Lemma 16.2] Let $u:R \to U$ be a flat injective ring epimorphism. Then $\mathfrak{R}$ is a $u$-contramodule and is ${\mathcal{G}}$-torsion-free.
To see that $\mathfrak{R}$ is ${\mathcal{G}}$-torsion-free we note that it is contained in a $U$-module which is always ${\mathcal{G}}$-torsion-free, as follows. $$0 \to {\operatorname{Hom}}_R(K,K) \to {\operatorname{Hom}}_R(U,K)$$ Now we will show that $\mathfrak{R}$ is a $u$-contramodule. By the tensor-hom adjunction, we have the following isomorphism. $${\operatorname{Hom}}_R(U, {\operatorname{Hom}}_R(K,K)) \cong {\operatorname{Hom}}_R(U \otimes_R K, K) =0$$ Similarly, to see that ${\operatorname{Ext}}_R^1(U, \mathfrak{R})=0$, we use the flatness of $U$ so ${\operatorname{Tor}}^R_1(U,K)=0$. Hence there is the following inclusion. $$0 \to {\operatorname{Ext}}_R^1(U, {\operatorname{Hom}}_R(K,K)) \to {\operatorname{Ext}}_{R}^1(U \otimes_R K, K)=0$$
\[L:rjcontra\] Let $u:R \to U$ be a flat injective ring epimorphism with associated Gabriel topology ${\mathcal{G}}$. Then for every $J \in {\mathcal{G}}$, every $R/J$-module $M$ is a $u$-contramodule.
To see that ${\operatorname{Hom}}_R(U,M)=0$, take $f:U \to M$. Then $f(U) = f(JU) = Jf(U) =0$ as $J$ annihilates $M$.
As ${\operatorname{Tor}}^R_i(R/J, U)=0$ and $R \to R/J$ is a ring epimorphism, one has that the following isomorphism. $${\operatorname{Ext}}_R^1(U, M) \cong {\operatorname{Ext}}_{R/J}^1 (R/J \otimes_R U, M) =0$$
\[C:compl-contra\] Let $u:R \to U$ be a flat injective ring epimorphism. Then $\Lambda(R)$ is a $u$-contramodule.
This follows immediately by Lemma \[L:rjcontra\] and by the closure properties of $u$-contramodules in Lemma \[L:geiglenz\].
\[L:coker-nu\] Let $u:R \to U$ be a flat injective ring epimorphism. Then the cokernel of $\nu:R \to \mathfrak{R}$ is a $U$-module.
Recall that $h_U(R)$ is the $U$-h-divisible submodule of $R$ and $\delta$ is as in sequence (\[eq:3contra\]). Consider the following commuting diagram. $$\xymatrix{
0 \ar[r] &R/h_U(R) \ar[r]^\nu \ar@{=}[d] & \mathfrak{R} \ar[r] \ar[d]^\cong_\delta & {\operatorname{Coker}}(\nu) \ar[r] \ar[d] &0\\
0 \ar[r] &R/h_U(R) \ar[r] & {\operatorname{Ext}}^1_R(K,R) \ar[r] & {\operatorname{Ext}}^1_R(U,R) \ar[r] &0}$$ By the five-lemma, the last vertical arrow is an isomorphism, so ${\operatorname{Coker}}(\nu) \cong {\operatorname{Ext}}^1_R(U,R)$ which is a $U$-module, as required.
The isomorphism between the ${\mathcal{G}}$-completion of $R$ and ${\operatorname{End}}(K)$
-------------------------------------------------------------------------------------------
We now prove the main result of this section.
\[P:ringiso\] Let $u:R \to U$ be a flat injective ring epimorphism. Using the notation of subsection 6.1 the morphism $\alpha: \Lambda(R) \to \mathfrak{R}$ is a ring isomorphism.
From $(\ast)$ we have the following commuting triangle: $$\xymatrix{
R \ar[d]_{\nu} \ar[r]^\lambda& \Lambda(R) \ar[dl]^{\alpha} \\
\mathfrak{R} & }$$ From sequences (2) and (3) we have the following exact sequence. $$0 \to h_U(R) \to R \overset{\nu} \to \mathfrak{R} \to {\operatorname{Coker}}(\nu) \to 0$$ where $h_U(R)$ is $U$-h-divisible and ${\operatorname{Coker}}(\nu)$ is a $U$-module by Lemma \[L:coker-nu\]. Both $\Lambda(R)$ and $\mathfrak{R}$ are $u$-contramodules so one can apply Lemma \[L:oneten\] to the two triangles below. That is, firstly, there exists a unique map $\beta$ such that $\beta \nu = \lambda$, and secondly by uniqueness, the identity on $\mathfrak{R}$ is the only homomorphism that makes the triangle on the right below commute. $$\xymatrix{
R \ar[d]_{\lambda} \ar[r]^\nu& \mathfrak{R} \ar[dl]^{\beta} & R \ar[d]_{\nu} \ar[r]^\nu& \mathfrak{R} \ar[dl]^{\text{id}_\mathfrak{R}} \\
\Lambda(R) && \mathfrak{R} & }$$ It follows that since $ \alpha \beta \nu = \alpha \lambda= \nu $, by uniqueness $\alpha \beta = \text{id}_\mathfrak{R}$. Therefore, $\alpha$ is surjective. It was shown in Lemma \[L:mapfacts\] that $\alpha$ is injective, hence $\alpha$ is an isomorphism.
It remains to see that $\alpha$ is a ring homomorphism. First note that if $z\in U$, $s\in R$ and $Jz \subseteq R $, then also $J(sz) \subseteq R$, that is $J \subseteq {\mathrm{Ann}}_R(sz +R)$. Let $\tilde{r}= (r_J+J)_{J \in {\mathcal{G}}}$ and $\tilde{s}= (s_J+J)_{J \in {\mathcal{G}}}$ denote elements of $\Lambda(R)$. Let $L$ denote ${\mathrm{Ann}}_R(z +R)$ and $L_s$ denote ${\mathrm{Ann}}_R(sz +R)$ for a fixed $z+R$ and note that $L \subseteq L_s$. $$\alpha(\tilde{r} \cdot \tilde{s} ):K \to K: z+R \mapsto r_L s_L z +R$$ $$\alpha(\tilde{r}) \alpha(\tilde{s}) = (K \overset{\tilde{r}}\to K) (K \overset{\tilde{s}}\to K) : z +R \mapsto s_L z +R \mapsto r_{L_s}s_Lz+R$$ Then clearly $r_{L_s} - r_L \in L_s$, so the endomorphisms $\alpha(\tilde{r} \cdot \tilde{s} )$ and $\alpha(\tilde{r}) \alpha(\tilde{s})$ are equal.
The following lemma will be useful when passing from the ring $R$ to the complete and separated topological ring $\mathfrak{R}$.
\[L:discreteiso\] Let $u:R \to U$ be a flat injective ring epimorphism with associated Gabriel topology ${\mathcal{G}}$. The $R$-module $R/J$ is isomorphic to $\mathfrak{R}/ J\mathfrak{R}$ and to $\Lambda(R)/J\Lambda(R)$, for every $J\in {\mathcal{G}}$.
$\mathfrak{R}/ J\mathfrak{R}$ and $\Lambda(R)/J\Lambda(R)$ are isomorphic by Proposition \[P:ringiso\]. Both $R/J$ and $\mathfrak{R} / J \mathfrak{R}$ are $R/J$-modules, hence by Lemma \[L:rjcontra\] and Lemma \[L:oneten\], there exists a unique $f$ such that the left triangle below commutes. The map $f$ induces $\bar{f}$ since $J\mathfrak{R}\subseteq {\operatorname{Ker}}f$, so the right triangle below also commutes. $$\xymatrix{
R \ar[d]_{p} \ar[r]^\nu& \mathfrak{R} \ar[dl]^{f} & \mathfrak{R} \ar[d]_{f} \ar[r]^{\pi \hspace{15pt}}& \mathfrak{R} / J\mathfrak{R} \ar[dl]^{\bar{f} }\\
R/J & &R/J & }$$ Let $\bar{\nu}$ be the map induced by $\nu$ as in the following commuting diagram. We will show that $\bar{f}$ and $\bar{\nu}$ are mutually inverse.
$$\xymatrix{
R \ar[r]^\nu \ar[d]_{p} & \mathfrak{R} \ar[d]_{\pi}\\
R/J \ar[r]^{\bar{\nu}} & \mathfrak{R} / J\mathfrak{R}}$$ Then, we have that $\pi \nu = \bar{\nu} p$, and so using the above commuting triangles it follows that $\bar{f} \bar{\nu} p = \bar{f} \pi \nu =f \nu =p $. As $p$ is surjective, $\bar{f} \bar{\nu} = \text{id}_{R/J}$. We now show that $\bar{\nu} \bar{f} = \text{id}_{{\mathfrak{R}}/J\mathfrak{R}}$.
$$\xymatrix{
R \ar[d]_{\pi \nu} \ar[r]^\nu&\mathfrak{R} \ar[dl]^{h} \\
\mathfrak{R} / J\mathfrak{R} & }$$ By uniqueness, $\pi$ is the unique map that fits into the triangle above, that is $\pi \nu = h \nu$ implies that $h = \pi$. So, $$\pi \nu = \bar{\nu} p = \bar{\nu} f \nu = \bar{\nu} \bar{f} \pi \nu$$ Therefore $\pi = \bar{\nu} \bar{f} \pi$, and as $\pi$ is surjective, $\bar{\nu} \bar{f} = \text{id}_{{\mathfrak{R}}/J\mathfrak{R}}$ as required.
\[P:topologies-2\] If $V$ is an open ideal in the topology of $\mathfrak{R}={\operatorname{End}}_R(K)$, then there is $J\in {\mathcal{G}}$ and a surjective ring homomorphism $R/J\to \mathfrak{R}/V$.
By the definition of the topology on $\mathfrak{R}$, if $V$ is an open ideal, then by Proposition \[P:ringiso\], $W=\alpha^{-1}(V)$ is an open ideal in the projective limit topology of $\Lambda(R)$. Hence by Remark \[R:topologies\], there is $J\in {\mathcal{G}}$ such that $W\supseteq \Lambda(R)J$. By Lemma \[L:discreteiso\] there is a surjective ring homomorphism $R/J\to \mathfrak{R}/V.$
When a ${\mathcal{G}}$-divisible class is enveloping {#S:enveloping}
====================================================
For this section, $R$ will always be a commutative ring. Fix a flat injective ring epimorphism $u$ and an exact sequence $$0 \to R \overset{u}\to U \to K \to 0.$$ Denote by ${\mathcal{G}}$ the corresponding Gabriel topology.
The aim of this section is to show that if ${\mathcal{D}}_{\mathcal{G}}$ is enveloping then for each $J \in {\mathcal{G}}$ the ring $R/J$ is perfect. It will follow from Section \[S:properfect\] that also $\mathfrak{R}$ is pro-perfect.
We begin by showing that for a local ring $R$ the rings $R/J$ are perfect, before extending the result to all commutative rings by showing that all ${\mathcal{G}}$-torsion modules (specifically the $R/J$ for $J \in {\mathcal{G}}$) are isomorphic to the direct sum of their localisations.\
In Lemma \[L:R-env\], it was shown that if $\varepsilon:R \to D$ is a ${\mathcal{D}}_{\mathcal{G}}$-envelope of $R$ in ${\mathrm{Mod}\textrm{-}R}$, then $D$ must be ${\mathcal{G}}$-torsion-free. Furthermore, if ${\mathcal{G}}$ arises from a perfect localisation $u:R \to U$ and $R$ has a ${\mathcal{D}}_{\mathcal{G}}$ envelope, then the following proposition allows us to work in the setting that ${\mathcal{D}}_{\mathcal{G}}= {\operatorname{Gen}}U$, thus $({\mathcal{A}}, {\mathcal{D}}_{\mathcal{G}})$ is the $1$-tilting cotorsion pair associated to the $1$-tilting module $U \oplus K$ (see Remark \[R:pdU=1\]).
\[P:pd1\] Let $u:R \to U$ be a (non-trivial) flat injective ring epimorphism and suppose $R$ has a ${\mathcal{D}}_{\mathcal{G}}$-envelope. Then ${\operatorname{p.dim}}_R U \leq 1$.
Let $$0 \to R \overset{\varepsilon}\to D \to D/R \to 0 \eqno(** )$$ denote the ${\mathcal{D}}_{\mathcal{G}}$-envelope of $R$. First we claim that $D$ is a $U$-module by showing that $D$ is ${\mathcal{G}}$-closed, or that $D \cong U \otimes_R D$. Consider the following exact sequence. $$0 \to {\operatorname{Tor}}^R_1( D, K) \to D \to D \otimes_R U \to D \otimes_R K \to 0$$ Therefore we must show that ${\operatorname{Tor}}^R_1(D, K) = 0 = D\otimes_R K$. As $D$ is ${\mathcal{G}}$-divisible and $K$ is ${\mathcal{G}}$-torsion it follows that $D \otimes_R K =0$. By Lemma \[L:R-env\] $D$ is ${\mathcal{G}}$-torsion-free, hence $D\cong D\otimes_R U$ and $D$ is a $U$-module. The cotorsion pair $({\mathcal{A}}, {\mathcal{D}}_{\mathcal{G}})$ is complete, which implies $R$-module $D/R$ is in ${\mathcal{A}}$, so ${\operatorname{p.dim}}_R D/R \leq 1$. From the short exact sequence [$(**)$]{} it follows that also ${\operatorname{p.dim}}_R D \leq 1$. Consider the following short exact sequence of $U$-modules $$0 \to U \to D \otimes_R U \cong D \to D/R \otimes_R U \to 0$$ We now claim that $D/R \otimes_R U$ is $U$-projective. Indeed, take any $Z \in {U\textrm{-}\mathrm{Mod}}$ and note that $Z \in {\mathcal{D}}_{\mathcal{G}}$. Then $0={\operatorname{Ext}}^1_R(D/R, Z)\cong{\operatorname{Ext}}^1_U(D/R\otimes_R U, Z)$. Therefore the short exact sequence above splits in ${\mathrm{Mod}\textrm{-}U}$ and so $U$ is a direct summand of $D$ also as an $R$-module, and the conclusion follows.
\[C:U-envelope\] Let $u:R \to U$ be a (non-trivial) flat injective ring epimorphism and suppose $R$ has a ${\mathcal{D}}_{\mathcal{G}}$-envelope. Then $$0 \to R \overset{u} \to U \to K \to 0$$ is a ${\mathcal{D}}_{\mathcal{G}}$-envelope of $R$.
By Proposition \[P:pd1\] ${\operatorname{p.dim}}U \leq 1$, so from the discussion in Section \[S:gab-top\], $U \oplus K$ is a $1$-tilting module such that $(U \oplus K)^\perp = {\mathcal{D}}_{\mathcal{G}}$. Thus $K \in {\mathcal{A}}$ and so $u$ is a ${\mathcal{D}}_{\mathcal{G}}$-preenvelope. To see that $u$ is an envelope, note that ${\operatorname{Hom}}_R(K,U)=0$, so by Lemma \[L:identity-env\], if $u = f u$, then $f = \text{id}_U$ is an automorphism of $U$, thus $u$ is a ${\mathcal{D}}_{\mathcal{G}}$-envelope as required.
We now begin by showing that when $R$ is a commutative local ring, if ${\mathcal{D}}_{\mathcal{G}}$ is enveloping in ${\mathrm{Mod}\textrm{-}R}$ then for each $J \in {\mathcal{G}}$, $R/J$ is a perfect ring. We will use the ring isomorphism $\alpha: \Lambda(R) \cong \mathfrak{R}$ of Proposition \[P:ringiso\].
\[L:Kindecomp\] Let $R$ be a commutative local ring and $u:R \to U$ a flat injective ring epimorphism and let $K$ denote $U/R$. Then $K$ is indecomposable.
It is enough to show that every idempotent of ${\operatorname{End}}_R(K)$ is either the zero homomorphism or the identity on $K$. Let ${\mathfrak{m}}$ denote the maximal ideal of $R$. Take a non-zero idempotent $e \in {\operatorname{End}}_R(K)$. Then there is an associated element $\alpha^{-1}(e)=\tilde{r}:=(r_J +J)_{J \in {\mathcal{G}}} \in \Lambda(R)$ via the ring isomorphism $\alpha:\Lambda(R) \cong \mathfrak{R}$ of Proposition \[P:ringiso\]. Clearly $\tilde{r}$ is also non-zero and an idempotent in $\Lambda(R)$. We will show this element is the identity in $\Lambda(R)$.
As $\tilde{r}$ is non-zero, there exists a $J_0 \in {\mathcal{G}}$ such that $r_{J_0} \notin J_0$. Also, $\tilde{r} \cdot \tilde{r} - \tilde{r} =0$, hence $$r_{J_0}r_{J_0} - r_{J_0} = r_{J_0}(r_{J_0} -1_R) \in J_0.$$ We claim that $r_{J_0}$ is a unit in $R$. Suppose not, then $r_{J_0} \in {\mathfrak{m}}$, hence $r_{J_0} -1_R$ is a unit, which implies that $r_{J_0} \in J_0$, a contradiction.
Consider some other $J \in {\mathcal{G}}$ such that $J \neq R$. $r_{J \cap J_0} - r_{J_0} \in J_0$, hence $r_{J \cap J_0} \notin J_0$. Therefore, by a similar argument as above, $r_{J \cap J_0}$ is a unit in $R$. As $r_{J \cap J_0} - r_{J} \in J$ and $r_{J \cap J_0}$ is a unit, $r_J \notin J$. Therefore by a similar argument as above $r_J$ is a unit in $R$ for each $J \in {\mathcal{G}}$ and we conclude that $\tilde{r}$ is a unit in $\Lambda(R)$.
Finally, as $r_J(r_J - 1_R) \in J$ for every $J$, and $\tilde{r}:=(r_J +J)_{J \in {\mathcal{G}}} $ is a unit, it follows that $r_J - 1_R \in J$ for each $J$, implying that $\tilde{r}$ is the identity in $\Lambda(R)$.
\[P:localperfect\] Let $R$ be a commutative local ring and consider the $1$-tilting cotorsion pair $({\mathcal{A}}, {\mathcal{D}}_{\mathcal{G}})$ induced by the flat injective ring epimorphism $u:R \to U$. If $ {\mathcal{D}}_{\mathcal{G}}$ is enveloping in ${R\textrm{-}\mathrm{Mod}}$, then $R/J$ is a perfect ring for every $J \in {\mathcal{G}}$.
Let ${\mathfrak{m}}$ denote the maximal ideal of $R$. As $R$ is local, to show that $R/J$ is perfect it is enough to show that for every sequence of elements $\{a_1, a_2, \dots, a_i, \dots \}$ with $a_i \in {\mathfrak{m}}\setminus J$, there exists an $m >0$ such that the product $a_1 a_2 \cdots a_m \in J$ (that is ${\mathfrak{m}}/J$ is T-nilpotent) by Proposition \[P:perfect\].
Fix a $J\in {\mathcal{G}}$ and take $\{a_1, a_2, \dots, a_i, \dots \}$ as above. Consider the following preenvelope of $R/a_iR$. $$0 \to R/a_iR \hookrightarrow U/a_iR \to K \to 0$$ As $R$ is local, by Lemma \[L:Kindecomp\], $K$ is indecomposable, and as $R/a_iR$ is not ${\mathcal{G}}$-divisible this is an envelope of $R/a_iR$.
We will use the T-nilpotency of direct sums of envelopes from Theorem \[T:Xu-sums\]. Consider the following countable direct sum of envelopes of $R/a_iR$ which is itself an envelope by Theorem \[T:Xu-sums\] (1). $$0 \to \bigoplus_{\substack{
i>0
}} R/a_iR \hookrightarrow \bigoplus_{\substack{
i>0
}}U/a_iR \to \bigoplus_{\substack{
i>0
}}K \to 0$$ For each $i>0$, we define a homomorphism $f_i:U/a_iR \to U/a_{i+1}R$ between the direct summands to be the multiplication by the element $a_{i+1}$.
Then clearly $R/a_iR \subseteq U/a_iR$ vanishes under the action of $f_i = \dot{a}_{i+1}$, hence we can apply Theorem \[T:Xu-sums\] (2) to the homomorphisms $\{f_i\}_{i>0}$. So, for every $z + a_1R \in U/a_1R$, there exists an $n>0$ such that $$f_n \cdots f_2 f_1 (z+a_1R) = 0 \in U/a_{n+1}R,$$ which can be rewritten as $$a_{n+1} \cdots a_3 a_2 (z) \in a_{n+1}R.$$ By Lemma \[L:finmanyann\], there exist $z_1, z_2, \dots , z_n \in U$ such that $$\bigcap_{\substack{
0 \leq j \leq n}}
{\mathrm{Ann}}_R(z_j +R) \subseteq J.$$ Let $\Omega = \{z_1, z_2, \dots , z_n\}$. For each $z_j$, there exists an $n_j$ such that $a_{n_j+1} \cdots a_3 a_2$ annihilates $z_j$. That is, $$a_{n_j+1} \cdots a_3 a_2 (z_j) \in a_{n_j+1}R\subseteq R.$$ We now choose an integer $m$ such that $a_{m} \cdots a_3 a_2$ annihilates all the $z_j$ for $a \leq j \leq n$. Set $m = max\{n_j\mid j=1,2\dots, n\}$. Then this $m$ satisfies the following, which finishes the proof. $$a_m a_{m-1} \cdots a_3 a_2 \in \bigcap_{\substack{
0 \leq j \leq n}}
{\mathrm{Ann}}_R(z_j +R) \subseteq J$$
Now we extend the result to general commutative rings. Our assumption is that the Gabriel topology ${\mathcal{G}}$ is arises from a perfect localisation $u:R \to U$ and that the associated $1$-tilting class ${\mathcal{D}}_{\mathcal{G}}$ is enveloping in ${R\textrm{-}\mathrm{Mod}}$.
\[N:simple-env\] There is a preenvelope of the following form induced by the map $u$. $$0 \to R/{\mathfrak{m}}\to U / {\mathfrak{m}}\to K \to 0$$ Let the following sequence denote an envelope of $R/{\mathfrak{m}}$. $$0 \to R/{\mathfrak{m}}\to D({\mathfrak{m}}) \to X({\mathfrak{m}}) \to 0$$ By Proposition \[P:Xu-env\], $D({\mathfrak{m}})$ and $X({\mathfrak{m}})$ are direct summands of $U/{\mathfrak{m}}$ and $K =U/R$ respectively. For convenience we will consider $R/{\mathfrak{m}}$ as a submodule of $D({\mathfrak{m}})$ and $X({\mathfrak{m}})$ as a submodule of $K$.
\[R:tors-facts\]
The following lemma allows us to use Proposition \[P:localperfect\] to say that if $D_{\mathcal{G}}$ is enveloping in $R$, all localisations $R_{\mathfrak{m}}/J_{\mathfrak{m}}$ are perfect rings where ${\mathfrak{m}}$ is a maximal ideal in ${\mathcal{G}}$ and $J \in {\mathcal{G}}$.
\[L:localfacts\] Let $R$ be a commutative ring and consider the $1$-tilting cotorsion pair $({\mathcal{A}}, {\mathcal{D}}_{\mathcal{G}})$ induced from the flat injective ring epimorphism $u:R \to U$. Fix a maximal ideal ${\mathfrak{m}}$ of $R$ and let $u_{\mathfrak{m}}: R_{\mathfrak{m}}\to U_{\mathfrak{m}}$ be the corresponding flat injective ring epimorphism in ${\mathrm{Mod}\textrm{-}R}_{\mathfrak{m}}$. Then the following hold.
1. $K_{\mathfrak{m}}= 0$ if and only if ${\mathfrak{m}}\notin {\mathcal{G}}$.
2. The induced Gabriel topology of $u_{\mathfrak{m}}$ denoted $${\mathcal{G}}({\mathfrak{m}}) = \{L \leq R_{\mathfrak{m}}: L U_{\mathfrak{m}}= U_{\mathfrak{m}}\}$$ contains the localisations ${\mathcal{G}}_{\mathfrak{m}}= \{J_{\mathfrak{m}}: J \in {\mathcal{G}}\}$.
3. Suppose ${\operatorname{p.dim}}U \leq 1$. Then $({\mathcal{A}}_{\mathfrak{m}}, ({\mathcal{D}}_{\mathcal{G}})_{\mathfrak{m}})$ is the $1$-tilting cotorsion pair associated to the flat injective ring epimorphism $u_{\mathfrak{m}}: R_{\mathfrak{m}}\to U_{\mathfrak{m}}$. That is, $({\mathcal{D}}_{{\mathcal{G}}})_{\mathfrak{m}}= {\mathcal{D}}_{{\mathcal{G}}({\mathfrak{m}})}$ and ${\mathcal{A}}_{\mathfrak{m}}= {}^\perp {\mathcal{D}}_{{\mathcal{G}}({\mathfrak{m}})}$.
4. If ${\mathcal{D}}_{\mathcal{G}}$ is enveloping in ${\mathrm{Mod}\textrm{-}R}$, then ${\mathcal{D}}_{{\mathcal{G}}({\mathfrak{m}})}$ is enveloping in ${\mathrm{Mod}\textrm{-}R}_{\mathfrak{m}}$.
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1. Since $K$ is ${\mathcal{G}}$-torsion, this follows by Remark \[R:tors-facts\] (3).
Note that if ${\mathfrak{m}}\notin {\mathcal{G}}$ the rest of the lemma follows trivially.
2. Take $J_{\mathfrak{m}}\in {\mathcal{G}}_{\mathfrak{m}}$. Then $ R_{\mathfrak{m}}/ J_{\mathfrak{m}}\otimes_R U_{\mathfrak{m}}\cong (R/J \otimes_R U) \otimes_R R_{\mathfrak{m}}= 0 $, so $J_{\mathfrak{m}}\in {\mathcal{G}}({\mathfrak{m}})$.
3. That $({\mathcal{A}}_{\mathfrak{m}}, ({\mathcal{D}}_{\mathcal{G}})_{\mathfrak{m}})$ is the $1$-tilting cotorsion pair associated to the $1$-tilting module $(U \oplus K)_{\mathfrak{m}}$ is [@GT12 Proposition 13.50], therefore ${\operatorname{Gen}}(U_{\mathfrak{m}}) = ({\mathcal{D}}_{\mathcal{G}})_{\mathfrak{m}}$ in ${\mathrm{Mod}\textrm{-}R}_{\mathfrak{m}}$. As $u_{\mathfrak{m}}:R_{\mathfrak{m}}\to U_{\mathfrak{m}}$ is a flat injective ring epimorphism and ${\operatorname{p.dim}}_{R_{\mathfrak{m}}}U_{\mathfrak{m}}\leq 1$ the $1$-tilting classes ${\operatorname{Gen}}(U_{\mathfrak{m}})$ and ${\mathcal{D}}_{{\mathcal{G}}({\mathfrak{m}})} $ coincide in ${\mathrm{Mod}\textrm{-}R}_{\mathfrak{m}}$ by [@H Theorem 5.4]. Thus $({\mathcal{D}}_{{\mathcal{G}}})_{\mathfrak{m}}= {\mathcal{D}}_{{\mathcal{G}}({\mathfrak{m}})}$ and it follows that ${\mathcal{A}}_{\mathfrak{m}}= {}^\perp {\mathcal{D}}_{{\mathcal{G}}({\mathfrak{m}})}$.
4. Assume that ${\mathcal{D}}_{\mathcal{G}}$ is enveloping in ${\mathrm{Mod}\textrm{-}R}$ and take some $M \in {\mathrm{Mod}\textrm{-}R}_{\mathfrak{m}}$ with the following ${\mathcal{D}}_{\mathcal{G}}$-envelope. $$0 \to M \to D \to X \to 0$$ We claim that $M$ has a ${\mathcal{D}}_{{\mathcal{G}}({\mathfrak{m}})}$-envelope in ${\mathrm{Mod}\textrm{-}R}_{\mathfrak{m}}$. Since $M \in {\mathrm{Mod}\textrm{-}R}_{\mathfrak{m}}$, $D$ and $X$ are $R_{\mathfrak{m}}$-modules by Proposition \[P:B-envelopes\]. By Proposition \[P:pd1\] ${\operatorname{p.dim}}U \leq 1$. By (3), $({\mathcal{D}}_{{\mathcal{G}}})_{\mathfrak{m}}= {\mathcal{D}}_{{\mathcal{G}}({\mathfrak{m}})}$ so $D \in {\mathcal{D}}_{{\mathcal{G}}({\mathfrak{m}})}$ and $X \in {}^\perp {\mathcal{D}}_{{\mathcal{G}}({\mathfrak{m}})}$. Since $R \to R_{\mathfrak{m}}$ is a ring epimorphism, any direct summand of $D$ which contains $M$ in ${\mathrm{Mod}\textrm{-}R}_{\mathfrak{m}}$ would also be a direct summand in ${\mathrm{Mod}\textrm{-}R}$. Thus we conclude that $0\to M\to D\to X\to 0$ is a ${\mathcal{D}}_{{\mathcal{G}}({\mathfrak{m}})}$-envelope of $M$ in ${\mathrm{Mod}\textrm{-}R}_{\mathfrak{m}}$.
By the above lemma, if ${\mathcal{D}}_{\mathcal{G}}$ is enveloping in ${\mathrm{Mod}\textrm{-}R}$, then ${\mathcal{D}}_{{\mathcal{G}}({\mathfrak{m}})}$ is enveloping in ${\mathrm{Mod}\textrm{-}R}_{\mathfrak{m}}$. Next we show that, under our enveloping assumption, all ${\mathcal{G}}$-torsion modules are isomorphic to the direct sums of their localisations at maximal ideals.\
The proof of the following lemma uses an almost identical argument to the proof of Lemma \[L:torsion-env\].
\[L:Xmnilpotent\] Let $u:R \to U$ be a flat injective ring epimorphism, ${\mathcal{G}}$ the associated Gabriel topology and suppose that ${\mathcal{D}}_{\mathcal{G}}$ is enveloping. Let $D({\mathfrak{m}})$ and $X({\mathfrak{m}})$ be as in Notation \[N:simple-env\] and fix a maximal ideal ${\mathfrak{m}}\in {\mathcal{G}}$. For every element $d \in D({\mathfrak{m}})$ and every element $a \in {\mathfrak{m}}$, there is a natural number $n > 0$ such that $a^n d = 0$. Moreover, for every element $x \in X({\mathfrak{m}})$ and every element $a \in {\mathfrak{m}}$, there is a natural number $n > 0$ such that $a^n x = 0$.
We will use the T-nilpotency of direct sums of envelopes as in Theorem \[T:Xu-sums\] (2). Consider the following countable direct sum of envelopes of $R/{\mathfrak{m}}$ which is itself an envelope by Theorem \[T:Xu-sums\] (1). $$0 \to \bigoplus_{\substack{
0<i
}} (R/{\mathfrak{m}})_{(i)} \to \bigoplus_{\substack{
0<i
}}D({\mathfrak{m}})_{(i)} \to \bigoplus_{\substack{
0<i
}}X({\mathfrak{m}})_{(i)} \to 0$$ For a fixed element $a \in {\mathfrak{m}}$, we choose the homomorphisms $f_i:D({\mathfrak{m}})_{(i)} \to D({\mathfrak{m}})_{(i+1)}$ between the direct summands to all be multiplication by $a$. Then clearly $R/{\mathfrak{m}}\subseteq D({\mathfrak{m}})$ vanishes under the action of $f_i = \dot{a}$, hence we can apply Theorem \[T:Xu-sums\] (2): for every $d \in D({\mathfrak{m}})$, there exists an $n$ such that $$f_n \cdots f_2 f_1 (d) = 0 \in D({\mathfrak{m}})_{(n+1)}.$$ Since each $f_i$ acts as multiplication by $a$, for every $d \in D$ there is an integer $n$ for which $a^n d = 0$, as required.
It is straightforward to see that $X({\mathfrak{m}})$ has the same property as $X({\mathfrak{m}})$ is an epimorphic image of $D({\mathfrak{m}})$.
\[L:suppXm\] Let $u:R \to U$ be a flat injective ring epimorphism and suppose ${\mathcal{D}}_{\mathcal{G}}$ is enveloping. Let ${\mathfrak{m}}\in {\mathcal{G}}$ and let $X({\mathfrak{m}})$ be as in Notation \[N:simple-env\]. The support of $X({\mathfrak{m}})$ is exactly $\{ {\mathfrak{m}}\}$, and each $X({\mathfrak{m}}) \cong X({\mathfrak{m}})_{\mathfrak{m}}$ is $K_{\mathfrak{m}}$.
We claim that $X({\mathfrak{m}})$ is non-zero. Otherwise, $X({\mathfrak{m}})=0$ would imply that $R/{\mathfrak{m}}$ is ${\mathcal{G}}$-divisible, so $R/{\mathfrak{m}}= {\mathfrak{m}}(R/{\mathfrak{m}}) = 0$, a contradiction.
Consider a maximal ideal ${\mathfrak{n}}\neq {\mathfrak{m}}$. Take an element $a \in {\mathfrak{m}}\setminus {\mathfrak{n}}$. Then for any $x \in X({\mathfrak{m}})$, $a^n x = 0$ for some $n > 0$, by Lemma \[L:Xmnilpotent\] and since $a$ is an invertible element in $R_{\mathfrak{n}}$, $x$ is zero in the localisation with respect to ${\mathfrak{n}}$. This holds for any element $x \in X({\mathfrak{m}})$, hence $X({\mathfrak{m}})_{\mathfrak{n}}= 0$.
It follows that since $X({\mathfrak{m}})$ is non-zero, $X({\mathfrak{m}})_{\mathfrak{m}}\neq 0$. As mentioned in Remark \[R:tors-facts\], $X({\mathfrak{m}})$ is an $R_{\mathfrak{m}}$-module and since $X({\mathfrak{m}})$ is a direct summand of $K$, $X({\mathfrak{m}})$ is a direct summand of $K_{\mathfrak{m}}$ which is indecomposable, by Lemma \[L:Kindecomp\]. Therefore $X({\mathfrak{m}})$ is non-zero and is isomorphic to $K_{\mathfrak{m}}$.
\[L:directsumxm\] Let $u:R \to U$ be a flat injective ring epimorphism and suppose ${\mathcal{D}}_{\mathcal{G}}$ is enveloping. Then the sum of the submodules $X({\mathfrak{m}})$ in $K$ is a direct sum.$$\sum_{\substack{
{\mathfrak{m}}\in {\mathcal{G}}}} X({\mathfrak{m}})
=
\bigoplus_{\substack{
{\mathfrak{m}}\in {\mathcal{G}}}} X({\mathfrak{m}})$$
Recall that $X({\mathfrak{m}})$ is non-zero only for ${\mathfrak{m}}\in {\mathcal{G}}$ by Remark \[R:tors-facts\]. Consider an element $$x \in X({\mathfrak{m}}) \cap \sum_{\substack{
{\mathfrak{n}}\neq {\mathfrak{m}}\\ {\mathfrak{n}}\in {\mathcal{G}}}} X({\mathfrak{n}}).$$ We will show that this element must be zero. By Lemma \[L:suppXm\], since $x \in X({\mathfrak{m}})$, $x$ is zero in the localisation with respect to all maximal ideals ${\mathfrak{n}}\neq {\mathfrak{m}}$. But $x$ can also be written as a finite sum of elements $x_i \in X({\mathfrak{n}}_i)$, each of which is zero in the localisation with respect to ${\mathfrak{m}}$, by Lemma \[L:suppXm\]. Therefore, $(x)_{\mathfrak{n}}=0$ for all maximal ideals ${\mathfrak{n}}$, hence $x = 0$ .
\[P:directsumk\] Let $u:R \to U$ be a flat injective ring epimorphism and suppose ${\mathcal{D}}_{\mathcal{G}}$ is enveloping. The module $K$ can be written as a direct sum of its localisations $K_{\mathfrak{m}}$, as follows. $$K \cong \bigoplus_{\substack{
{\mathfrak{m}}\in {\mathcal{G}}}} K_{\mathfrak{m}}= \bigoplus_{\substack{
{\mathfrak{m}}\in {\operatorname{Max}}{R}
}} K_{\mathfrak{m}}$$
From Lemma \[L:directsumxm\], we have the following inclusion. $$\bigoplus_{\substack{
{\mathfrak{m}}\in {\mathcal{G}}}} X({\mathfrak{m}}) \leq K$$ To see that this is an equality we show that these two modules have the same localisation with respect to every ${\mathfrak{m}}$ maximal in $R$. Recall that by Lemma \[L:localfacts\](1) if ${\mathfrak{n}}$ is maximal, then $K_{\mathfrak{n}}= 0$ if and only if ${\mathfrak{n}}\notin {\mathcal{G}}$ and by Lemma \[L:suppXm\], $\text{Supp}(X({\mathfrak{m}})) = \{ {\mathfrak{m}}\}$. Using these lemmas, it follows that for ${\mathfrak{n}}\notin {\mathcal{G}}$, $K_{\mathfrak{n}}= 0 = ( \bigoplus_{\substack{
{\mathfrak{m}}\in {\mathcal{G}}}} X({\mathfrak{m}}))_{\mathfrak{n}}$. Similarly, if ${\mathfrak{m}}\in {\mathcal{G}}$, then $K_{\mathfrak{m}}= X({\mathfrak{m}})_{\mathfrak{m}}$. Hence we have shown the following. $$\bigoplus_{\substack{
{\mathfrak{m}}\in {\mathcal{G}}}} X({\mathfrak{m}}) = K$$ Since $K_{\mathfrak{m}}= X({\mathfrak{m}})_{\mathfrak{m}}$, it only remains to see that $X({\mathfrak{m}}) \cong X({\mathfrak{m}})_{\mathfrak{m}}$, which follows from Remark \[R:tors-facts\].
\[C:torsion-decomposes\] Let $u:R \to U$ be a flat injective ring epimorphism and suppose ${\mathcal{D}}_{\mathcal{G}}$ is enveloping. Then for every ${\mathcal{G}}$-torsion module $M$, the following isomorphism holds. $$M \cong \bigoplus_{\substack{
{\mathfrak{m}}\in {\mathcal{G}}}} M_{\mathfrak{m}}= \bigoplus_{\substack{{\mathfrak{m}}\in {\operatorname{Max}}R}} M_{\mathfrak{m}}$$ Furthermore, it follows that for every $J \in {\mathcal{G}}$, $J$ is contained only in finitely many maximal ideals of $R$.
For the first isomorphism, recall that if an $R$-module $M$ is ${\mathcal{G}}$-torsion, then $M \cong {\operatorname{Tor}}^R_1(M, K)$. Also, note that in this case, $M_{\mathfrak{m}}\cong {\operatorname{Tor}}^R_1(M, K)_{\mathfrak{m}}\cong {\operatorname{Tor}}^{R_{\mathfrak{m}}}_1(M_{\mathfrak{m}}, K_{\mathfrak{m}}) \cong {\operatorname{Tor}}^{R}_1(M, K_{\mathfrak{m}})$. Hence we have the following isomorphisms. $$M \cong {\operatorname{Tor}}^R_1(M, K) \cong {\operatorname{Tor}}^R_1(M, \bigoplus_{\substack{
{\mathfrak{m}}\in {\mathcal{G}}}} K_{\mathfrak{m}}) \cong \bigoplus_{\substack{{\mathfrak{m}}\in {\mathcal{G}}}} {\operatorname{Tor}}^R_1(M, K_{\mathfrak{m}}) \cong \bigoplus_{\substack{{\mathfrak{m}}\in {\mathcal{G}}}} M_{\mathfrak{m}}$$ The fact that $$\bigoplus_{\substack{
{\mathfrak{m}}\in {\mathcal{G}}}} M_{\mathfrak{m}}= \bigoplus_{\substack{{\mathfrak{m}}\in {\operatorname{Max}}R}} M_{\mathfrak{m}}$$ follows from Remark \[R:tors-facts\] (3).\
For the final statement of the proposition, one only has to replace $M$ with the ${\mathcal{G}}$-torsion module $R/J$ where $J \in {\mathcal{G}}$. Hence as $R/J$ is cyclic, it cannot be isomorphic to an infinite direct sum. Therefore, $(R/J)_{\mathfrak{m}}$ is non-zero only for finitely many maximal ideals and the conclusion follows.
We are now in the position to show the main results of this section.
\[T:rjperfect\] Let $u:R \to U$ be a flat injective ring epimorphism and suppose ${\mathcal{D}}_{\mathcal{G}}$ is enveloping. Then $R/J$ is a perfect ring for every $J \in {\mathcal{G}}$.
By Corollary \[C:torsion-decomposes\], every $R/J$ is a finite product of local rings $R_{\mathfrak{m}}/J_{\mathfrak{m}}$. Additionally as $({\mathcal{D}}_{\mathcal{G}})_{\mathfrak{m}}$ is enveloping in ${\mathrm{Mod}\textrm{-}R}_{\mathfrak{m}}$ by Lemma \[L:localfacts\] each $R_{\mathfrak{m}}/J_{\mathfrak{m}}$ is a perfect ring by Proposition \[P:localperfect\]. Therefore, by Proposition \[P:perfect\], $R/J$ itself is perfect.
\[T:EndK-properfect\] Let $u:R \to U$ be a flat injective ring epimorphism and suppose $ {\mathcal{D}}_{\mathcal{G}}$ is enveloping in ${\mathrm{Mod}\textrm{-}R}$. Then the topological ring $\mathfrak{R}={\operatorname{End}}(K)$ is pro-perfect.
Recall that the topology of $\mathfrak{R}$ is given by the annihilators of finitely generated submodules of $K$, so that $\mathfrak{R}={\operatorname{End}}_R(K)$ is separated and complete in its topology. Let $V$ be an open ideal in the topology of $\mathfrak{R}$. By Proposition \[P:topologies-2\] there is $J\in {\mathcal{G}}$ and a surjective ring homomorphism $R/J\to \mathfrak{R}/V$. By Theorem \[T:rjperfect\] $R/J$ is a perfect ring and thus so are the quotient rings $\mathfrak{R}/V$.
${\mathcal{D}}_{\mathcal{G}}$ is enveloping if and only if $\mathfrak{R}$ is pro-perfect {#S:properfect}
========================================================================================
Suppose that $u:R \to U$ is a commutative flat injective ring epimorphism where ${\operatorname{p.dim}}_R U \leq 1$ and denote $K = U/R$. In this section we show that if the endomorphism ring $\mathfrak{R} = {\operatorname{End}}_R(K)$ is pro-perfect, then ${\mathcal{D}}_{\mathcal{G}}$ is enveloping in ${\mathrm{Mod}\textrm{-}R}$. So combining with the results in the Section \[S:enveloping\] we obtain that ${\mathcal{D}}_{\mathcal{G}}$ is enveloping if and only if ${\operatorname{p.dim}}U \leq 1$ and $\mathfrak{R}$ is pro-perfect.
Recall that if ${\operatorname{p.dim}}U \leq 1$, $({\mathcal{A}}, {\mathcal{D}}_{\mathcal{G}})$ denotes the $1$-tilting cotorsion pair associated to the $1$-tilting module $U \oplus K$. The following theorem of Positselski is vital for this section.
\[T:pos-addK\]([@BP2 Theorem 19.6]) Suppose $R$ is a commutative ring and $u:R \to U$ a flat injective ring epimorphism with ${\operatorname{p.dim}}_R U \leq 1$. Then the topological ring $\mathfrak{R}={\operatorname{End}}(K)$ is pro-perfect if and only if $\varinjlim {\mathrm{Add}}(K) = {\mathrm{Add}}(K)$.
A second crucial result that we will use is the following.
\[T:Enochs2\] ([@Xu Theorem 2.2.6]) Assume that ${\mathcal{C}}$ is a class of modules closed under direct limits and extensions. If a module $M$ admits a special ${\mathcal{C}}^{\perp_1}$-preenvelope with cokernel in ${\mathcal{C}}$, then $M$ admits a ${\mathcal{C}}^{\perp_1}$-envelope.
We now show that if $\mathfrak{R}$ is pro-perfect, then ${\mathrm{Add}}(K)$ does in fact satisfy the conditions of Theorem \[T:Enochs2\]. From Theorem \[T:pos-addK\] ${\mathrm{Add}}(K)$ is closed under direct limits. Moreover, ${\mathrm{Add}}(K)$ is closed under extensions as any short exact sequence $0 \to L \to M \to N \to 0$ with $L, N \in {\mathrm{Add}}(K)$ splits.
As the cotorsion pair $({\mathcal{A}}, {\mathcal{D}}_{\mathcal{G}})$ is complete, every $R$-module $M$ has an injective ${\mathcal{D}}_{\mathcal{G}}$-preenvelope, and as ${\mathcal{D}}_{\mathcal{G}}= K^\perp = ({\mathrm{Add}}(K))^\perp$, $M$ has a $({\mathrm{Add}}(K))^\perp$-preenvelope. It remains to be seen that every $M$ has a special preenvelope $\nu$ such that ${\operatorname{Coker}}\nu \in {\mathrm{Add}}(K)$, which we will now show.
\[L:coker-in-AddK\] Suppose $u:R \to U$ is a flat injective ring epimorphism where ${\operatorname{p.dim}}_R U \leq 1$. Let $({\mathcal{A}}, {\mathcal{D}}_{\mathcal{G}})$ be the $1$-tilting cotorsion pair associated to the $1$-tilting module $U \oplus K$. Then every module has a special ${\mathcal{D}}_{\mathcal{G}}$-preenvelope $\nu$ such that ${\operatorname{Coker}}\nu \in {\mathrm{Add}}(K)$.
For every cardinal $\alpha$ the short exact sequence $0 \to R^{(\alpha)} \to U^{(\alpha)} \to K^{(\alpha)} \to 0$ is a ${\mathcal{D}}_{\mathcal{G}}$-preenvelope and is of the desired form. Take an $R$-module $M$ and consider the canonical surjection $R^{(\alpha)} \overset{p}\to M \to 0$. Consider the following pushout $Z$ of $M \gets R^{(\alpha)} \to U^{(\alpha)}$. $$\xymatrix{
&0 \ar[d]& 0 \ar[d]& & \\
&\ker p \ar[d] \ar@{=}[r]& \ker p \ar[d] & & \\
0 \ar[r]&R^{(\alpha)} \ar[r] \ar[d]^p& U^{(\alpha)} \ar[r] \ar[d]& K^{(\alpha)} \ar@{=}[d] \ar[r] & 0\\
0 \ar[r]&M\ar[r] \ar[d] & Z \ar[r] \ar[d]& K^{(\alpha)} \ar[r] & 0\\
&0&0 &&}$$ The module $Z$ is in ${\operatorname{Gen}}U={\mathcal{D}}_{\mathcal{G}}$, and so the bottom row of the above diagram is a ${\mathcal{D}}_{\mathcal{G}}$-preenvelope of $M$ of the desired form.\
The following theorem follows easily from the above discussion.
\[T:divenv\] Suppose $u:R \to U$ is a flat injective ring epimorphism with ${\operatorname{p.dim}}_R U \leq 1$. If the topological ring $\mathfrak{R}$ is pro-perfect, then ${\mathcal{D}}_{\mathcal{G}}$ is enveloping in ${R\textrm{-}\mathrm{Mod}}$.
From Theorem \[T:pos-addK\] and Lemma \[L:coker-in-AddK\], ${\mathrm{Add}}(K)$ does satisfy the conditions of Theorem \[T:Enochs2\]. Thus the conclusion follows, since ${\mathcal{D}}_{\mathcal{G}}={\mathrm{Add}}(K)^\perp$.
Finally combining the above theorem with the results in Section \[S:tilting-enveloping\] and Section \[S:enveloping\] we obtain the two main results of this paper.
\[T:characterisation\] Suppose $u:R \to U$ is a commutative flat injective ring epimorphism, ${\mathcal{G}}$ the associated Gabriel topology and $\mathfrak R$ the topological ring ${\operatorname{End}}_R(K)$. The following are equivalent.
1. ${\mathcal{D}}_{\mathcal{G}}$ is enveloping.
2. ${\operatorname{p.dim}}U \leq 1$ and $R/J$ is a perfect ring for every $J\in {\mathcal{G}}$.
3. ${\operatorname{p.dim}}U \leq 1$ and $\mathfrak{R}$ is pro-perfect.
In particular, if ${\mathcal{D}}_{\mathcal{G}}$ is enveloping then the class ${\mathrm{Add}}(K)$ is closed under direct limits.
(1)$\Rightarrow$(2) Follows by Proposition \[P:pd1\] and Theorem \[T:rjperfect\].
(2)$\Rightarrow$(3) Follows from Lemma \[L:discreteiso\] and Proposition \[P:topologies-2\].
(3)$\Rightarrow$(1) Follows from Theorem \[T:divenv\].
\[T: tilting-envelope\] Assume that $T$ is a $1$-tilting module over a commutative ring $R$ such that the class $T^\perp$ is enveloping. Then there is a flat injective ring epimorphism $u\colon R\to U$ such that $U\oplus U/R$ is equivalent to $T$ and the topological ring $\mathfrak R={\operatorname{End}}(U/R)$ is a pro-perfect ring. Moreover, if ${\mathcal{G}}$ is the associated Gabriel topology, then $R/J$ is perfect ring for every $J\in {\mathcal{G}}$.
By Proposition \[P:tilting-env\], the Gabriel topology ${\mathcal{G}}$ associated to $T^\perp$ arises from a perfect localisation. Moreover, $\psi:R \to R_{\mathcal{G}}$ is injective so by setting $U=R_{\mathcal{G}}$ we can apply Theorem \[T:characterisation\] to conclude.
The case of a non-injective flat ring epimorphism {#S:notmono}
=================================================
Now we extend the results of the previous section to the case of a non-injective flat ring epimorphism $u\colon R \to U$ with $K={\operatorname{Coker}}u$.
As before, the Gabriel topology ${\mathcal{G}}_u=\{J\leq R\mid JU=U\}$ associated to $u$ is finitely generated and the class $${\mathcal{D}}_{{\mathcal{G}}_u}=\{M\in R{\textrm{-}\mathrm{Mod}}\mid JM=M \text{ for every } J\in{\mathcal{G}}_u\}$$ of ${\mathcal{G}}_u$-divisible modules is a torsion class. Moreover, by [@AHHr] it is a silting class, that is there is a silting module $T$ such that ${\operatorname{Gen}}T={\mathcal{D}}_{{\mathcal{G}}_u}$.
The ideal $I$ will denote the kernel of $u$ and $\bar R$ the ring $R/I$ so that there is a flat injective ring epimorphism $\bar u\colon \bar R\to U$.
To the is the associated Gabriel topology ${\mathcal{G}}_{\bar u}=\{L/I\leq \bar R\mid LU=U\}$ in $\bar{R}$ and the following class of $\bar{R}$-modules. $${\mathcal{D}}_{{\mathcal{G}}_{\bar u}}=\{M\in {\bar{R}\textrm{-}\mathrm{Mod}}\mid (L/I)M=M, \text{ for every } L/I\in{\mathcal{G}}_{\bar u}\}.$$ We first note the following
\[L: I-annih-div\] Every module in ${\mathcal{D}}_{{\mathcal{G}}_u}$ is annihilated by $I$, thus ${\mathcal{D}}_{{\mathcal{G}}_u}={\mathcal{D}}_{{\mathcal{G}}_{\bar u}}$.
Note that ${\operatorname{Ker}}u=I$ is the ${\mathcal{G}}_u$-torsion submodule of $R$. Hence for every $b\in I$ there is $J\in {\mathcal{G}}_u$ such that $bJ=0$. Let $M\in {\mathcal{D}}_{{\mathcal{G}}_u}$, then $bM =bJM=0$, thus $IM=0$. We conclude that ${\mathcal{D}}_{{\mathcal{G}}_u}$ can be considered a class in ${\bar{R}\textrm{-}\mathrm{Mod}}$ and coincides with ${\mathcal{D}}_{{\mathcal{G}}_{\bar u}}$.
\[P:same-env\] The class ${\mathcal{D}}_{{\mathcal{G}}_u}$ is enveloping in ${R\textrm{-}\mathrm{Mod}}$ if and only if ${\mathcal{D}}_{{\mathcal{G}}_{\bar u}}$ is enveloping in ${\bar{R}\textrm{-}\mathrm{Mod}}$.
Assume that ${\mathcal{D}}_{{\mathcal{G}}_u}$ is enveloping in ${R\textrm{-}\mathrm{Mod}}$ and let $\bar M\in {\bar{R}\textrm{-}\mathrm{Mod}}$. Consider a ${\mathcal{D}}_{{\mathcal{G}}_u}$-envelope $\bar{\psi}\colon \bar M\to D$ in ${R\textrm{-}\mathrm{Mod}}$. Since $R\to R/I$ is a ring epimorphism and $D$ is annihilated by $I$ by Lemma \[L: I-annih-div\], it is immediate to conclude that $\bar{\psi}$ is also a ${\mathcal{D}}_{{\mathcal{G}}_{\bar u}}$-envelope of $\bar M$.
Conversely, assume that ${\mathcal{D}}_{{\mathcal{G}}_{\bar u}}$ is enveloping in ${\bar{R}\textrm{-}\mathrm{Mod}}$. Take $M\in{R\textrm{-}\mathrm{Mod}}$ and let $\bar{\psi}\colon M/IM\to D$ be a ${\mathcal{D}}_{{\mathcal{G}}_{\bar u}}$-envelope of $ M/IM$ in ${\bar{R}\textrm{-}\mathrm{Mod}}$. Let $\pi\colon M\to M/IM$ be the canonical projection. We claim that $\psi=\bar{\psi} \pi$ is a ${\mathcal{D}}_{{\mathcal{G}}_u}$-envelope of $M$ in ${R\textrm{-}\mathrm{Mod}}$. Indeed, if $f\colon D\to D$ satisfies $f\psi=\psi$, then $f\bar{\psi}\pi =\bar{\psi} \pi$. As $\pi$ is a surjection, $f \bar{\psi} = \bar{\psi}$ and so $f$ is an automorphism of $D$.
Note that ${\operatorname{End}}_R(K)$ is the same as ${\operatorname{End}}_{\bar R}(K)$ both as a ring and as a topological ring. It will be still denoted by $\mathfrak R$. Thus if ${\mathcal{D}}_{{\mathcal{G}}_u}$ is enveloping in ${R\textrm{-}\mathrm{Mod}}$ we can apply the results of the previous sections to the ring $\bar R$, in particular Theorem \[T:characterisation\].
\[T:non-mono\] Let $u\colon R\to U$ be a commutative flat ring epimorphism with kernel $I$. Let ${\mathcal{G}}$ be the associated Gabriel topology and $\mathfrak R$ the topological ring ${\operatorname{End}}_R(K)$. The following are equivalent.
1. ${\mathcal{D}}_{{\mathcal{G}}_u}$ is enveloping.
2. ${\operatorname{p.dim}}_{\bar{R}} U \leq 1$ and $R/L$ is a perfect ring for every $L\in {\mathcal{G}}$ such that $L\supseteq I$.
3. ${\operatorname{p.dim}}_{\bar R} U \leq 1$ and $\mathfrak{R}$ is pro-perfect.
In particular, $U\oplus K$ is a $1$-tilting module over the ring $\bar R$ and since ${\operatorname{Gen}}U$ is contained in ${\bar{R}\textrm{-}\mathrm{Mod}}$, ${\mathcal{D}}_{{\mathcal{G}}_u}={\operatorname{Gen}}U$.
As already noted, results from [@AHHr] imply that ${\operatorname{Gen}}U$ is a silting class. Since we have that $U\oplus K$ is a $1$-tilting module in ${\bar{R}\textrm{-}\mathrm{Mod}}$ inducing the silting class ${\operatorname{Gen}}U$, it is natural to ask the following question.
Is $U\oplus K$ a silting module in ${R\textrm{-}\mathrm{Mod}}$?
[AHH17]{}
Lidia Angeleri Hügel and Michal Hrbek. Silting modules over commutative rings. , (13):4131–4151, 2017.
Hyman Bass. Finitistic dimension and a homological generalization of semi-primary rings. , 95:466–488, 1960.
Silvana Bazzoni. Divisible envelopes, p-1 covers and weak-injective modules. , 9(4):531–542, 2010.
Silvana Bazzoni and Leonid Positselski. Contramodules over pro-perfect topological rings, the covering property in categorical tilting theory, and homological ring epimorphisms. Preprint, arXiv:1808.00937, 2018.
Paul C. Eklof and Jan Trlifaj. How to make [E]{}xt vanish. , 33(1):41–51, 2001.
Werner Geigle and Helmut Lenzing. Perpendicular categories with applications to representations and sheaves. , 144(2):273–343, 1991.
R[ü]{}diger G[ö]{}bel and Jan Trlifaj. , volume 41 of [*de Gruyter Expositions in Mathematics*]{}. Walter de Gruyter GmbH & Co. KG, Berlin, extended edition, 2012. Approximations.
Michal Hrbek. One-tilting classes and modules over commutative rings. , 462:1–22, 2016.
Lidia Angeleri H[ü]{}gel and Javier S[á]{}nchez. Tilting modules arising from ring epimorphisms. , 14(2):217–246, 2011.
Leonid Positselski. Flat ring epimorphisms of countable type. Preprint, arXiv:1808.00937, 2018.
Leonid Positselski. Triangulated [M]{}atlis equivalence. , 17(4):1850067, 44, 2018.
Bo Stenstr[ö]{}m. . Springer-Verlag, New York, 1975. Die Grundlehren der Mathematischen Wissenschaften, Band 217, An introduction to methods of ring theory.
Jinzhong Xu. , volume 1634 of [*Lecture Notes in Mathematics*]{}. Springer-Verlag, Berlin, 1996.
| ArXiv |
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author:
- |
\
HEP Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439, USA\
E-mail:
- |
J. B. Kogut\
Department of Energy, Division of High Energy Physics, Washington, DC 20585, USA\
and\
Dept. of Physics – TQHN, Univ. of Maryland, 82 Regents Dr., College Park, MD 20742, USA\
E-mail:
title: Searching for the elusive critical endpoint at finite temperature and isospin density
---
Introduction
============
Direct simulations of QCD at finite baryon/quark number density are made difficult if not impossible because at finite quark-number chemical potential $\mu$ the fermion determinant is complex. At small chemical potentials, close to the finite temperature transition, various methods have been devised to circumvent this difficulty, series expansions in $\mu$ [@Allton:2002zi; @Allton:2005gk; @Gavai:2003mf], analytic continuation from imaginary $\mu$ [@deForcrand:2002ci; @deForcrand:2006pv; @Azcoiti:2004ri; @Azcoiti:2005tv; @D'Elia:2002gd; @D'Elia:2004at], reweighting methods [@Fodor:2001au; @Fodor:2004nz] and canonical ensemble techniques [@Kratochvila:2005mk; @Alexandru:2005ix].
We adopt a different strategy, and simulate using the magnitude of the fermion determinant and ignoring the phase [@Kogut:2002zg; @Kogut:2005yu]. This can be thought of as considering all quarks to be in isodoublets and introducing a finite chemical potential $\mu_I$ for isospin. In the region of small $\mu/\mu_I$, where the phase is expected to be less important one can argue that the finite $\mu$ and $\mu_I$ transitions might be identical. Since our fermion determinant is positive (or at least non-negative), we can use standard hybrid molecular-dynamics HMD(R) simulations [@Gottlieb:1987mq]. However, for this algorithm, the Binder cumulants used to determine the nature of the finite temperature transition turn out to be strongly dependent on the updating increment $dt$. For this reason we now simulate using the rational hybrid monte-carlo (RHMC) algorithm [@Kennedy:1998cu; @Clark:2005sq], which is exact in the sense of having no $dt$ dependence for observables.
In the low chemical potential domain, the most interesting feature expected in the phase diagram is the critical endpoint, where the finite temperature transition changes from a crossover to a first-order transition as chemical potential is increased. The critical endpoint is expected to lie in the universality class of the 3-dimensional Ising model. For 3 flavours it had been expected that the critical point at zero chemical potentials, where the transition changes from a first order transition to a crossover as mass is increased, would move to higher masses as the chemical potential increases, thereby becoming the critical endpoint. Our preliminary results indicate that this does not happen.
In section 2 we give the fermion action and make a few comments on the RHMC implementation. Section 3 gives our preliminary results. Our conclusions occupy section 4.
QCD at finite isospin density and the RHMC
==========================================
The pseudo-fermion action for QCD at finite $\mu_I$, used for the implementation of the RHMC algorithm is $$S_{pf}=p_\psi^\dag {\cal M}^{-N_f/8} p_\psi
\label{eqn:action}$$ where $p_\psi$ is the momentum conjugate to the pseudo-fermion field $\psi$. $${\cal M} = [D\!\!\!\!/(\frac{1}{2}\mu_I)+m]^\dag
[D\!\!\!\!/(\frac{1}{2}\mu_I)+m] + \lambda^2$$ is the quadratic Dirac operator, and we set $\lambda=0$ for our $\mu_I < m_\pi$ simulations.
To implement the RHMC method we need to know positive upper and lower bounds to the spectrum of ${\cal M}$. $25$ exceeds the upper bound for the $\mu_I$ range of interest. We use a speculative lower bound of $10^{-4}$ since the actual lower bound of the spectrum is unknown. This is justified by varying the choice of lower bounds and comparing the results [@Kogut:2006jg]. For $N_f=3$ we use a $(20,20)$ rational approximation to ${\cal M}^{(\pm 3/16)}$ at the ends of each trajectory, and a $(10,10)$ rational approximation to ${\cal M}^{(-3/8)}$ for the updating.
Simulations and Results
=======================
We are simulating lattice QCD with staggered fermions and $N_f=3$ at quark masses close to $m_c$, the critical mass for $\mu=\mu_I=0$ on $8^3 \times 4$, $12^3 \times 4$ and $16^3 \times 4$ lattices. $m=0.02$, $0.025$, $0.03$, $0.035$, and $\mu_I=0.0,\,0.2,\,0.3$. For our $12^3 \times 4$ simulations we use runs of 300,000 trajectories at each of 4 $\beta$ values close to $\beta_c$, for each $m$ and $\mu_I$. We mostly use $dt=0.05$ for which length-1 trajectories give acceptances of $\sim 70\%$ for the RHMC algorithm.
To determine the nature of the transition, we use 4-th order Binder cumulants [@Binder:1981sa] for the chiral condensate. For any observable $X$ this cumulant is defined by $$B_4(X) = {\langle(X-\langle X \rangle)^4\rangle \over
\langle(X-\langle X \rangle)^2\rangle^2}$$ where the $X$s are lattice averaged quantities. For infinite volumes, $B_4=3$ for a crossover, $B_4=1$ for a first-order transition and $B_4=1.604(1)$ for the 3-dimensional Ising model. Thus, if there is a critical endpoint we would expect $B_4$ to decrease with increasing $\mu_I$, passing through a value close to the Ising value at the critical $\mu_I$.
Figure \[fig:b4mass\] shows our preliminary measurements of the Binder cumulant for the chiral condensate as a function of mass at $\mu_I=\mu=0$ from our $12^3 \times 4$ simulations. Taking the point where the straight-line fit passes through the Ising value as our estimate for the critical mass yields $m_c=0.0264(3)$. Each of the points in this graph were obtained by averaging the Binder cumulants taken from several $\beta$ values close to the transition, and extrapolated to $\beta_c$ which minimizes these cumulants, using Ferrenberg-Swendsen rewieghting [@Ferrenberg:1988yz].
The $\mu_I$ dependence of this Binder cumulant at $\beta_c(\mu_I)$ is shown in figure \[fig:b4m0.03\], for $m=0.03$, a little above $m_c$. It is clear that, rather than decreasing with increasing $\mu_I$, it actually increases slowly. Since $\beta_c$ and hence $T_c$ decrease with increasing $\mu_I$, in physical units $m$ is actually decreasing with increasing $\mu_I$ meaning that at fixed physical $m$ the rise would be even more pronounced. The behaviour at $m=0.035$ is very similar.
Figure \[fig:beta\_c\] shows the dependence of the transition $\beta$, $\beta_c$, on $\mu_I$. As mentioned above, $\beta_c$ and hence the transition temperature $T_c$ fall (slowly) with increasing $\mu_I$ as expected. The fits shown to this preliminary ‘data’ are: $$\begin{aligned}
\beta_c &=& 5.15326(10) - 0.173(2) \mu_I^2 \hspace{0.5in} m=0.035\nonumber \\
\beta_c &=& 5.14386(\:\,8) - 0.172(1) \mu_I^2 \hspace{0.5in} m=0.030\nonumber \\
\beta_c &=& 5.13426(12) - 0.179(4) \mu_I^2 \hspace{0.5in} m=0.025\nonumber \\\end{aligned}$$ which is in reasonable agreement with the results of de Forcrand and Philipsen for the $\mu$ dependence of the transition temperature, obtained from analytic continuation from imaginary $\mu$ if we make the identification $\mu_I=2\mu$.
Figure \[fig:rhmc&hmdr\] shows the $dt$ dependence of the Binder cumulants at the transition for $m=0.035$, $\mu_I=0.2$ in the HMD(R) simulations. The exact RHMC result, which has no $dt$ dependence, is plotted on this graph at $dt=0$. It is clear that the RHMC result is consistent with the $dt \rightarrow 0$ limit of the HMD(R) results. The actual value of $dt$ used in the RHMC simulations was $dt=0.05$, the value of $dt$ for the rightmost point on this graph, showing one advantage of using this new algorithm.
Conclusions
===========
We simulate lattice QCD with 3 flavours of staggered quarks with a small chemical potential $\mu_I < m_\pi$ for isospin, in the neighbourhood of the finite temperature transition from hadronic matter to a quark-gluon plasma. Fourth order Binder cumulants are used to probe the nature of this transition and search for the critical endpoint for masses slightly above the critical mass for zero chemical potentials. Earlier simulations using the HMD(R) algorithm were plagued by large finite $dt$ errors [@Kogut:2005yu]. We now use the RHMC algorithm which is exact in the sense of having no finite $dt$ errors.
We measure the critical mass to be $m_c=0.0264(3)$ for $N_t=4$, in agreement with the recent results of de Forcrand and Philipsen [@deForcrand:2006pv], but considerably below earlier measurements which found values close to $m_c=0.033$ [@Karsch:2001nf; @Christ:2003jk; @deForcrand:2003hx]. These higher values were due to using the HMD(R) algorithm with $dt$ large enough to produce large systematic errors.
For masses greater than $m_c$ we found that the Binder cumulant for the chiral condensate increases with increasing $\mu_I$ and thus shows no evidence for a critical endpoint, contrary to earlier expectations. This also agrees with the observations of de Forcrand and Philipsen [@deForcrand:2006pv] for finite $\mu$, emphasizing the similarities between finite $\mu$ and finite $\mu_I$ for small $\mu$,$\mu_I$, near the finite temperature transition. On these relatively small lattices ($12^3 \times 4$), we really should minimize the Binder cumulant of linear combinations of the chiral condensate, the plaquette and the isospin density to obtain the desired eigenfield of the renormalization group equations, to draw reliable conclusions [@Karsch:2001nf] [^1].
We end with the observation that we have used RHMC simulations where we do not know a positive lower bound for the spectrum of the quadratic Dirac operator. This is done by choosing a speculative lower bound and justifying our choice a postiori. We refer the reader to our recent paper on this subject [@Kogut:2006jg].
Acknowledgements {#acknowledgements .unnumbered}
================
JBK is supported in part by a National Science Foundation grant NSF PHY03-04252. DKS is supported by the U.S. Department of Energy, Division of High Energy Physics, Contract W-31-109-ENG-38. We thank Philippe de Forcrand and Owe Philipsen for helpful discussions. Simulations are performed on Jazz at Argonne, Tungsten and Cobalt at NCSA, Bassi and Jacquard at NERSC, DataStar at SDSC and PC’s in Argonne HEP.
[99]{}
C. R. Allton [*et al.*]{}, Phys. Rev. D [**66**]{}, 074507 (2002) \[arXiv:hep-lat/0204010\]. C. R. Allton [*et al.*]{}, Phys. Rev. D [**71**]{}, 054508 (2005) \[arXiv:hep-lat/0501030\]. R. V. Gavai and S. Gupta, Phys. Rev. D [**68**]{}, 034506 (2003) \[arXiv:hep-lat/0303013\]. P. de Forcrand and O. Philipsen, Nucl. Phys. B [**642**]{}, 290 (2002) \[arXiv:hep-lat/0205016\]. P. de Forcrand and O. Philipsen, arXiv:hep-lat/0607017. V. Azcoiti, G. Di Carlo, A. Galante and V. Laliena, JHEP [**0412**]{}, 010 (2004) \[arXiv:hep-lat/0409157\]. V. Azcoiti, G. Di Carlo, A. Galante and V. Laliena, Nucl. Phys. B [**723**]{}, 77 (2005) \[arXiv:hep-lat/0503010\]. M. D’Elia and M. P. Lombardo, Phys. Rev. D [**67**]{}, 014505 (2003) \[arXiv:hep-lat/0209146\]. M. D’Elia and M. P. Lombardo, Phys. Rev. D [**70**]{}, 074509 (2004) \[arXiv:hep-lat/0406012\]. Z. Fodor and S. D. Katz, Phys. Lett. B [**534**]{}, 87 (2002) \[arXiv:hep-lat/0104001\]. Z. Fodor and S. D. Katz, JHEP [**0404**]{}, 050 (2004) \[arXiv:hep-lat/0402006\]. S. Kratochvila and P. de Forcrand, PoS [**LAT2005**]{}, 167 (2006) \[arXiv:hep-lat/0509143\]. A. Alexandru, M. Faber, I. Horvath and K. F. Liu, Phys. Rev. D [**72**]{}, 114513 (2005) \[arXiv:hep-lat/0507020\]. J. B. Kogut and D. K. Sinclair, Phys. Rev. D [**66**]{}, 034505 (2002) \[arXiv:hep-lat/0202028\]. J. B. Kogut and D. K. Sinclair, arXiv:hep-lat/0509095. S. A. Gottlieb, W. Liu, D. Toussaint, R. L. Renken and R. L. Sugar, Phys. Rev. D [**35**]{}, 2531 (1987). A. D. Kennedy, I. Horvath and S. Sint, Nucl. Phys. Proc. Suppl. [**73**]{}, 834 (1999) \[arXiv:hep-lat/9809092\]. M. A. Clark, P. de Forcrand and A. D. Kennedy, PoS [**LAT2005**]{}, 115 (2006) \[arXiv:hep-lat/0510004\]. J. B. Kogut and D. K. Sinclair, arXiv:hep-lat/0608017. K. Binder, Z. Phys. B [**43**]{}, 119 (1981). A. M. Ferrenberg and R. H. Swendsen, Phys. Rev. Lett. [**61**]{}, 2635 (1988). F. Karsch, E. Laermann and C. Schmidt, Phys. Lett. B [**520**]{}, 41 (2001) \[arXiv:hep-lat/0107020\]. N. H. Christ and X. Liao, Nucl. Phys. Proc. Suppl. [**119**]{}, 514 (2003). P. de Forcrand and O. Philipsen, Nucl. Phys. B [**673**]{}, 170 (2003) \[arXiv:hep-lat/0307020\].
[^1]: We thank Frithjof Karsch for reminding us of this fact
| ArXiv |
---
abstract: 'We report an improved low-energy extrapolation of the cross section for the process $^7\mathrm{Be}(p,\gamma)^8\mathrm{B}$, which determines the $^8$B neutrino flux from the Sun. Our extrapolant is derived from Halo Effective Field Theory (EFT) at next-to-leading order. We apply Bayesian methods to determine the EFT parameters and the low-energy $S$-factor, using measured cross sections and scattering lengths as inputs. Asymptotic normalization coefficients of $^8$B are tightly constrained by existing radiative capture data, and contributions to the cross section beyond external direct capture are detected in the data at $E < 0.5$ MeV. Most importantly, the $S$-factor at zero energy is constrained to be $S(0)= 21.3\pm 0.7$ , which is an uncertainty smaller by a factor of two than previously recommended. That recommendation was based on the full range for $S(0)$ obtained among a discrete set of models judged to be reasonable. In contrast, Halo EFT subsumes all models into a controlled low-energy approximant, where they are characterized by nine parameters at next-to-leading order. These are fit to data, and marginalized over via Monte Carlo integration to produce the improved prediction for $S(E)$.'
author:
- Xilin Zhang
- 'Kenneth M. Nollett'
- 'D. R. Phillips'
bibliography:
- 'nuclear\_reaction.bib'
date: July 2015
title: 'Halo effective field theory constrains the solar ${}^7{\rm Be} + p \rightarrow {}^8{\rm B} + \gamma$ rate'
---
[*Introduction—*]{} A persistent challenge in modeling the Sun and other stars is the need for nuclear cross sections at very low energies [@Adelberger:2010qa; @rolfs-rodney]. Recent years have seen a few measurements at or near the crucial “Gamow peak” energy range for the Sun [@luna; @Adelberger:2010qa], but cross sections at these energies are so small that data almost always lie at higher energies, where experimental count rates are larger. The bulk of the data must be extrapolated to the energies of stellar interiors using nuclear reaction models.
The models available for extrapolation also have limitations. Qualitatively correct models of nonresonant radiative capture reactions, with reacting nuclei treated as interacting particles, have been available since the mid-1960s [@christyduck]. However, these models suffer from weak input constraints and dependence on *ad hoc* assumptions like the shapes of potentials. Developing models with realistically interacting nucleons as their fundamental degrees of freedom is currently a priority for the theoretical community, but progress is slow, and models remain incomplete [@Descouvemont:2004hh; @Navratil:2011sa]. [*Ab initio*]{} calculations employing modern nuclear forces may yield tight constraints in the future.
For the $^7\mathrm{Be}(p,\gamma)^8\mathrm{B}$ reaction – which determines the detected flux of $^8$B decay neutrinos from the Sun – the precision of the astrophysical $S$-factor at solar energies ($\sim
20$ keV) is limited by extrapolation from laboratory energies of typically 0.1–0.5 MeV. A recent evaluation [@Adelberger:2010qa] found the low-energy limit $S(0) = 20.8 \pm 0.7\pm 1.4$ , with the first error reflecting the uncertainties of the measurements. The second accounts for uncertainties in extrapolating those data. It was chosen to cover the full variation among a few extrapolation models thought to be plausible. Since the differences among $S(E)$ shapes for different models were neither well-understood nor represented by continuous parameters, no goodness-of-fit test was used for model selection.
Halo EFT [@vanKolck:1998bw; @Kaplan:1998tg; @Kaplan:1998we; @Bertulani:2002sz; @Bedaque:2003wa; @Hammer:2011ye; @Rupak:2011nk; @Canham:2008jd; @Higa:2008dn; @Ryberg:2013iga], provides a simple, transparent, and systematic way to organize the reaction theory needed for the low-energy extrapolation. The $^7\mathrm{Be}+p$ system is modeled as two interacting particles and described by a Lagrangian expanded in powers of their relative momentum, which is small compared with other momentum scales in the problem. The point-Coulomb part of the interaction can be treated exactly, and the form of the strong interaction is fully determined by the order at which the Lagrangian is truncated [@Ryberg:2013iga; @Zhang:2014zsa; @Zhang:2015; @Ryberg:2014exa]. The coupling constants of the Lagrangian are determined by matching to experiment. This is similar in spirit and in many quantitative details to traditional potential model or $R$-matrix approaches. However, it avoids some arbitrary choices (like Woods-Saxon shapes or matching radii) of these models, is organized explicitly as a low-momentum power series, and allows quantitative estimates of the error arising from model truncation.
The low-energy $S$-factor for $^7\mathrm{Be}(p,\gamma)^8\mathrm{B}$ consists entirely of electric-dipole ($E1$) capture from $s$- and $d$-wave initial states to $p$-wave final states (which dominate $^7\mathrm{Be}+p$ configurations within $^8$B). All models are dominated by “external direct capture,” the part of the $E1$ matrix element arising in the tails of the wave function (out to 100 fm and beyond) [@christyduck; @jennings98]. Models differ in how they combine the tails of the final state with phase shift information and in how they model the non-negligible contribution from short-range, non-asymptotic regions of the wave functions.
Halo EFT includes these mechanisms, and can describe $S(E)$ over the low-energy region (LER) at $E < 0.5$ MeV. Beyond 0.5 MeV, higher-order terms could be important, and resonances unrelated to the $S$-factor in the Gamow peak appear. Compared with a potential model, the EFT has about twice as many adjusted parameters, too many to determine uniquely with existing data. However, calculations of the solar neutrino flux do not require that all parameters be known: it is enough to determine $S(18\pm 6~\mathrm{keV})$. We fit the amplitudes of recently computed next-to-leading-order (NLO) terms [@Zhang:2015] in $E1$ capture to the experimental $S(E)$ data in the LER. We then use Bayesian methods to propagate the (theory and experimental) uncertainties and obtain a rather precise result for $S(20~\mathrm{keV})$.
[*EFT at NLO—*]{} The EFT amplitude for $E1$ capture is organized in an expansion in the ratio of low-momentum and high-momentum scales, $k/\Lambda$. $\Lambda$ is set by the ${{}^{7}\mathrm{Be}}$ binding energy relative to the ${}^3\mathrm{He}+{}^4\mathrm{He}$ threshold, $1.59$ MeV, so $\Lambda \approx 70$ MeV, corresponding to a co-ordinate space cutoff of $\approx 3$ fm. Physics at distances shorter than this is subsumed into contact operators in the Lagrangian. The ${{}^{8}\mathrm{B}}$ ground state, which is 0.1364(10) MeV below the ${{}^{7}\mathrm{Be}}$-p scattering continuum [@AME2012I; @AME2012II], is a shallow p-wave bound state in our EFT: it is bound by contact operators but the wave-function tail should be accurately represented. To ensure this we also include the $J^\pi=\frac{1}{2}^-$ bound excited state of ${{}^{7}\mathrm{Be}}$ in the theory. ${{}^{7}\mathrm{Be}}^*$ is 0.4291 MeV above the ground state; the configuration containing it and the proton is significant in the ${{}^{8}\mathrm{B}}$ ground state [@Zhang:2014zsa]. The large ($\sim 10$ fm) ${{}^{7}\mathrm{Be}}$-p scattering lengths play a key role in the low-energy dynamics; s-wave rescattering in the incoming channels must be accurately described. This also requires that the Coulomb potential be iterated to all orders when computing the scattering and bound state wave functions [@Zhang:2014zsa; @Ryberg:2014exa]. Indeed $Z_{{{}^{7}\mathrm{Be}}} Z_p \alpha_{em} M_{p} \approx k_C = 27$ MeV while the binding momentum of ${{}^{8}\mathrm{B}}$ is 15 MeV, so these low-momentum scales are well separated from $\Lambda$. We generically denote them by $k$, and anticipate that $k/\Lambda \approx 0.2$. Since the EFT incorporates all dynamics at momentum scales $< \Lambda$ its radius of convergence is larger than other efforts at systematic expansions of this $S$-factor [@WilliamsKoonin81; @Baye:2000ig; @Baye:2000gi; @Baye:2005; @Jennings:1998qm; @Jennings:1998ky; @Jennings:1999in; @Cyburt:2004jp; @Mukhamedzhanov:2002du].
The leading-order (LO) amplitude includes only external direct capture. As the ${{}^{7}\mathrm{Be}}$ ground state is $\frac{3}{2}^-$ there are two possible total spin channels, denoted here by $s=1,2$. They correspond, respectively, to ${{}^{3}S_{1}}$ and ${{}^{5}S_{2}}$ components in the incoming scattering state, and ${{}^{3}P_{2}}$ and ${{}^{5}P_{2}}$ configurations in ${{}^{8}\mathrm{B}}$. The parameters that appear at LO are the two asymptotic normalization coefficients (ANCs), $C_{s}$, for the ${{}^{7}\mathrm{Be}}$-p configuration in ${{}^{8}\mathrm{B}}$ in each of the spin channels, together with the corresponding s-wave scattering lengths, $a_{s}$ [@Zhang:2013kja; @Zhang:2014zsa; @Ryberg:2014exa]. The NLO result for $S(E)$, full details of which will be given elsewhere [@Zhang:2015], can be written as:
$$\begin{aligned}
S(E)&=&f(E) \sum_{s} C_{s}^2
\bigg[ \big\vert \mathcal{S}_\mathrm{EC} \left(E;\delta_s(E)\right)
+ \overline{L}_{s} \mathcal{S}_\mathrm{SD} \left(E;\delta_s(E)\right)
+ \epsilon_{s} \mathcal{S}_\mathrm{CX}\left(E;\delta_s(E)\right) \big\vert^2
+|\mathcal{D}_\mathrm{EC}(E)|^2 \bigg] \ . \end{aligned}$$
Here, $f(E)$ is an overall normalization composed of final-state phase space over incoming flux ratio, dipole radiation coupling strength, and a factor related to Coulomb-barrier penetration [@Zhang:2014zsa]. $\mathcal{S}_\mathrm{EC}$ is proportional to the spin-$s$ $E1$ [@Walkecka1995book; @Zhang:2014zsa; @Zhang:2013kja] external direct-capture matrix element between continuum ${{}^{7}\mathrm{Be}}$–p s-wave and ${{}^{8}\mathrm{B}}$ ground-state wave functions. $\mathcal{S}_\mathrm{CX}$ is the contribution from capture with core excitation, i.e. into the ${{}^{7}\mathrm{Be}}^*$-p component of the ground state. Its strength is parameterized by $\epsilon_s$. Since ${{}^{7}\mathrm{Be}}^*$ is spin-half this component only occurs for $s=1$, so $\epsilon_2=0$. Because the inelasticity in ${{}^{7}\mathrm{Be}}$-p s-wave scattering is small [@Navratil:2010jn; @Navratil:2011sa] it is an NLO effect.
Short-distance contributions, $\mathcal{S}_\mathrm{SD}$, are also NLO. They originate from NLO contact terms in the EFT Lagrangian [@Zhang:2015] and account for corrections to the LO result arising from the $E1$ transition at distances $\lsim 3$ fm. The size of these is set by the parameters $\overline{L}_s$, which must be fit to data. $\mathcal{S}_\mathrm{EC}$, $\mathcal{S}_\mathrm{SD}$, and $\mathcal{S}_\mathrm{CX}$ are each functions of energy, $E$, but initial-state interactions mean they also depend on the s-wave phase shifts $\delta_s$. At NLO we parametrize $\delta_s(E)$ by the Coulomb-modified effective-range expansion up to second order in $k^2$, i.e., we include the term proportional to $r_s k^2$, with $r_s$ the effective range (see supplemental material) [@Higa:2008dn; @Koenig:2012bv; @GoldbergerQM]. Finally, $\mathcal{D}_\mathrm{EC}$ is the $E1$ matrix element between the d-wave scattering state and the ${{}^{8}\mathrm{B}}$ bound-state wave function. It is not affected by initial-state interactions up to NLO, and hence is the same for $s=1,\,2$ channels and introduces no new parameters. This leaves us with $9$ parameters in all: $C_{1,2}^2$, $a_{1,2}$ at LO and five more at NLO: $r_{1,2}$, $\overline{L}_{1,2}$, and $\epsilon_1$ [@Zhang:2015].
[*Data—*]{} The 42 data points in our analysis come from all modern experiments with more than one data point for the direct-capture $S$-factor in the LER: Junghans [*et.al.,* ]{} (experiments “BE1” and “BE3”) [@Junghans:2010zz], Filippone [*et.al.,*]{} [@Filippone:1984us], Baby [*et.al.,*]{} [@Baby:2002hj; @Baby:2002ju], and Hammache [*et.al.,*]{} (two measurements published in 1998 and 2001) [@Hammache:1997rz; @Hammache:2001tg]. Ref. [@Adelberger:2010qa] summarizes these experiments, and the common-mode errors (CMEs) we assign are given in the supplemental material. All data are for energies above $0.1$ MeV. We subtracted the $M1$ contribution of the ${{}^{8}\mathrm{B}}$ $1^{+}$ resonance from the data using the resonance parameters of Ref. [@Filippone:1984us]. This has negligible impact for $E \leq 0.5$ MeV, so we retain only points in this region, thus eliminating the resonance’s effects.
[*Bayesian analysis—*]{} To extrapolate $S(E)$ we must use these data to constrain the EFT parameters. We compute the posterior probability distribution function (PDF) of the parameter vector ${\boldsymbol}{g}$ given data, $D$, our theory, $T$, and prior information, $I$. To account for the common-mode errors in the data we introduce data-normalization corrections, $\xi_i$. We then employ Bayes’ theorem to write the desired PDF as: $${\rm pr} \left({\boldsymbol}{g},\{\xi_i\} \vert D;T; I \right)
=
{\rm pr} \left(D \vert {\boldsymbol}{g},\{\xi_i\};T; I \right) {\rm pr} \left({\boldsymbol}{g},\{\xi_i\} \vert I \right) , \label{eqn:bayesian1}$$ with the first factor proportional to the likelihood: $$\ln {\rm pr} \left(D \vert {\boldsymbol}{g},\{\xi_i\};T;I \right) = c - \sum_{j=1}^N \frac{\left[ (1-\xi_j)S({\boldsymbol}{g}; E_j)-D_j\right]^2}{2 \sigma_j^{2}},$$ where $S({\boldsymbol}{g};E_j)$ is the NLO EFT $S$-factor at the energy $E_j$ of the $j$th data point $D_j$, whose statistical uncertainty is $\sigma_j$. The constant $c$ ensures ${\rm pr} \left({\boldsymbol}{g},\{\xi_i\} \vert D;T; I \right)$ is normalized. Since the CME affects all data from a particular experiment in a correlated way there are only five parameters $\xi_i$: one for each experiment.
In Eq. (\[eqn:bayesian1\]) ${\rm pr}\left( {\boldsymbol}{g}, \{ \xi_i\},\vert
I \right)$ is the prior for ${\boldsymbol}{g}$ and $\{\xi_i\}$. We choose independent Gaussian priors for each data set’s $\xi_i$, all centered at $0$ and with width equal to the assigned CMEs. We also choose Gaussian priors for the s-wave scattering lengths $\left(a_{1},\,
a_2\right)$, with centers at the experimental values of Ref. [@Angulo2003], $\left(25,\, -7\right)$ fm, and widths equal to their errors, $\left(9,\, 3\right)$ fm. All the other EFT parameters are assigned flat priors over ranges that correspond to, or exceed, natural values: $0.001 \leq C^2_{1,2} \leq
1\,\mathrm{fm}^{-1}$, $0\leq r_{1,2} \leq 10\, \mathrm{fm} $ [@Phillips:1996ae; @Wigner:1955zz], $-1\leq \epsilon_1\leq 1$, $-10\leq L_{1,2} \leq 10\, \mathrm{fm}$. We do, though, restrict the parameter space by the requirement that there is no s-wave resonance in ${{}^{7}\mathrm{Be}}$-p scattering below $0.6$ MeV.
To determine ${\rm pr}\left({\boldsymbol}{g},\{\xi_i\} \vert D;T;I \right)$, we use a Markov chain Monte Carlo algorithm [@SiviaBayesian96] with Metropolis-Hastings sampling [@Metropolis:1953am], generating $2\times 10^4$ uncorrelated samples in the $14$-dimensional (14d) ${\boldsymbol}{g}$ $\bigoplus$ $\{\xi_i\}$ space. Making histograms, e.g., over two parameters $g_1$ and $g_2$, produces the marginalized distribution, in that case: ${\rm pr} \left(g_{1}, g_{2} \vert D;T;I \right)=$ $\int {\rm pr} \left({\boldsymbol}{g},\{\xi_i\} \vert D;T;I \right)\,$ $d\xi_1 \ldots d\xi_5 dg_3 \ldots dg_9$. Similarly, to compute the PDF of a quantity $F({\boldsymbol}{g})$, e.g., $S(E; {\boldsymbol}{g})$, we construct ${\rm pr}\left(\bar{F}\vert D; T; I\right)$ $\equiv$ $\int {\rm pr} \left({\boldsymbol}{g},\{\xi_i\} \vert D;T;I \right)$ $\delta(\bar{F}-F({\boldsymbol}{g})) d\xi_1 \ldots d \xi_5 d{\boldsymbol}{g}$, and histogramming again suffices.
[*Constraints on parameters and the S-factor—*]{} The tightest parameter constraint we find is on the sum $C_1^2+C_2^2=0.564(23)~\mathrm{fm}^{-1}$, which sets the overall scale of $S(E)$ [^1]. Fig. \[fig:results1\] shows contours of 68% and 95% probability for the 2d joint PDF of the ANCs. Neither ANC is strongly constrained by itself, but they are strongly anticorrelated; the 1d PDF of $C_1^2+ C_2^2$ is shown in the inset. The ellipses in Fig. \[fig:results1\] show ANCs from an *ab initio* variational Monte Carlo calculation (the smaller ellipse) [@Nollett:2011qf] [^2] and inferred from transfer reactions by Tabacaru *et al.* (larger ellipse) [@Tabacaru:2005hv]. These are also shown as error bars in the inset. The *ab initio* ANCs shown compare well with the present results. (The *ab initio* ANCs of Ref. [@Navratil:2011sa] sum to $0.509~\mathrm{fm}^{-1}$ and appear to be in moderate conflict.) Tabacaru *et al.* recognized that their result was $1\sigma$ to $2\sigma$ below existing analyses of $S$-factor data; a $1.8\sigma$ conflict remains in our analysis. We suggest that for $^8$B the combination of simpler reaction mechanism, fewer assumptions, and more precise cross sections makes the capture reaction a better probe of ANCs than transfer reactions.
![(Color online.) 2d distribution for $C_1^2$ (x-axis) and $C_2^2$ (y-axis). Shading represents the 68% and 95% regions. The small circle and ellipse are the $1\sigma$ contours of an [*ab initio*]{} calculation [@Nollett:2011qf] and empirical results [@Tabacaru:2005hv], with their best values marked as red squares. The inset is the histogram and the corresponding smoothed 1d PDF of the quantity $[C_1^2+C_2^2]\times \mathrm{fm}$; the larger and smaller error bars show the empirical and [*ab initio*]{} values.[]{data-label="fig:results1"}](paper_ANC.pdf)
![(Color online.) 2d distribution for $\epsilon_1$ (x-axis) and $\bar{L}_1$ (y-axis). The shaded area is the 68% region. The inset is the histogram and corresponding smoothed 1d PDF of the quantity $0.33\, \bar{L}_1/\mathrm{fm} - \epsilon_1$.[]{data-label="fig:results2"}](paper_LEps.pdf)
Fig. \[fig:results2\] depicts the 2d distribution of $\bar{L}_1$ and $\epsilon_1$. There is a positive correlation: in $S(E)$ below the ${{}^{7}\mathrm{Be}}$-p inelastic threshold, the effect of core excitation, here parameterized by $\epsilon_1$, can be traded against the short-distance contribution to the spin-1 $E1$ matrix element. The inset shows the $1$d distribution of the quantity $0.33\, \bar{L}_1/\mathrm{fm} - \epsilon_1$, for which there is a slight signal of a non-zero value. In contrast, the data prefers a positive $\bar{L}_2$: its 1d pdf yields a 68% interval $-0.58~{\rm fm} < \bar{L}_2 < 7.94~{\rm fm}$.
We now compute the PDF of $S$ at many energies, and extract each median value (the thin solid blue line in Fig. \[fig:results3\]), and 68% interval (shaded region in Fig. \[fig:results3\]). The PDFs for $S$ at $E=0$ and $20~\mathrm{keV}$ are singled out and shown on the left of the figure: the blue line and histogram are for $E=0$ and the red-dashed line is for $E=20$ keV. We found choices of the EFT-parameter vector ${\boldsymbol}{g}$ (given in the supplemental material) that correspond to natural coefficients, produce curves close to the median $S(E)$ curve of Fig. \[fig:results3\], and have large values of the posterior probability.
![(Color online.) The right panel shows the NLO $S$-factor at different energies, including the median values (solid blue curve). Shading indicates the 68% interval. The dashed line is the LO result. The data used for parameter determination are shown, but have not been rescaled in accord with our fitted $\{\xi_i\}$. They are: Junghans [*et.al.*]{}, BE1 and BE3 [@Junghans:2010zz] (filled black circle and filled grey circle), Filippone [*et.al.,*]{} [@Filippone:1984us] (open circle), Baby [*et.al.,*]{} [@Baby:2002hj; @Baby:2002ju] (filled purple diamond), and Hammache [*et.al.,*]{} [@Hammache:1997rz; @Hammache:2001tg] (filled red box). The left panel shows 1d PDFs for $S(0)$ (blue line and histogram) and $S(20~\mathrm{keV})$ (red-dashed line). []{data-label="fig:results3"}](paper_S.pdf)
$S$ (eV b) $S'/S$ ($\mathrm{MeV}^{-1}$) $S''/S$ ($\mathrm{MeV}^{-2} $)
----------- ----------------- ------------------------------ --------------------------------
Median 21.33 \[20.67\] $-1.82$ \[$-1.34$\] 31.96 \[22.30\]
$+\sigma$ 0.66 \[0.60\] 0.12 \[0.12\] 0.33 \[0.34\]
$-\sigma$ 0.69 \[0.63\] 0.12 \[0.12\] 0.37 \[0.38\]
: The median values of $S$, $S'/S$, and $S''/S$ at $E=0$ keV \[$E=20$ keV\], as well as the upper and lower limits of the (asymmetric) 68% interval. The sampling errors are $0.02\%$, $0.07\%$, $0.01\%$ for median values, as estimated from $\left<X^2-\left<X\right>^2\right>^{1/2}/\sqrt{N}$ with $N=2 \times 10^4$.[]{data-label="tab:SdSddS"}
[*$S(20~keV)$ and the thermal reaction rate—*]{}Table \[tab:SdSddS\] compiles median values and 68% intervals for the $S$-factor and its first two derivatives, $S^\prime/S$ and $S^{\prime\prime}/S$, at $E=0$ and $20$ keV. Ref. [@Adelberger:2010qa] recommends $S(0)=20.8\pm 1.6$ (quadrature sum of theory and experimental uncertainties). Our $S(0)$ is consistent with this, but the uncertainty is more than a factor of two smaller. Ref. [@Adelberger:2010qa] also provides effective values of $S^\prime/S=-1.5\pm 0.1~\mathrm{MeV}^{-1}$ and $S^{\prime
\prime}/S=11\pm 4~\mathrm{MeV}^{-2}$. These are not literal derivatives but results of quadratic fits to several plausible models over $0 < E < 50~\mathrm{keV}$, useful for applications. Our values are consistent, considering the large higher derivatives (rapidly changing $S^{\prime\prime}$) left out of quadratic fits.
The important quantity for astrophysics is in fact not $S(E)$ but the thermal reaction rate; derivatives of $S(E)$ are used mainly in a customary approximation to the rate integral [@caughlan62; @rolfs-rodney; @Adelberger:2010qa]. By using our $S^\prime$ and $S^{\prime\prime}$ in a Taylor series for $S(E)$ about $20$ keV, then regrouping terms and applying the approximation formula, we find a rate (given numerically in the supplemental material) that differs from numerical integration of our median $S(E)$ by only 0.01% at temperature $T_9 \equiv T /( 10^9~\mathrm{K}) =
0.016$ (characteristic of the Sun), and 1% at $T_9 = 0.1$ (relevant for novae).
[*How accurate is NLO?—*]{}Our improved precision for $S(0)$ is achieved because, by appropriate choices of its nine parameters, NLO Halo EFT can represent all the models whose disagreement constitutes the 1.4 uncertainty quoted in Ref. [@Adelberger:2010qa]—including the microscopic calculation of Ref. [@Descouvemont:2004hh]. Halo EFT matches their $S(E)$ and phase-shift curves with a precision of 1% or better for $E < 0.5$ MeV, and thus spans the space of models of $E1$ capture in the LER [@Zhang:2015].
The LO curve shown in Fig. \[fig:results3\] employs values of $C_1$, $C_2$, $a_1$, and $a_2$ from the NLO fit. It differs from the NLO curve by $<2$% at $E=0$, and by $< 10$% at $E=0.5$ MeV. This rapid convergence suggests that the naive estimate of N2LO effects in the amplitude, $(k/\Lambda)^2\approx 4 \%$, is conservative. And indeed, we added a term with this $k$-dependence to the model, allowing a natural coefficient that was then marginalized over, and found that it shifted the median and error bars from the NLO result by at most $0.2\%$ in the LER. Finally, we estimate that direct $E2$ and $M1$ contributions to $S$ in the LER are less than $0.01\%$, and radiative corrections are around $0.01\%$.
[*Summary—*]{} We used Halo EFT at next-to-leading order to determine precisely the $^7\mathrm{Be}(p,\gamma)^8\mathrm{B}$ $S$-factor at solar energies. Halo EFT connects all low-energy models by a family of continuous parameters, and marginalization over those parameters represents marginalization over all reasonable models of low-energy physics. Many of the individual EFT parameters are poorly determined by existing $S$-factor data, at $E > 0.1$ MeV, but these data constrain their combinations sufficiently that the extrapolated $S(20~\mathrm{keV})$ is determined to 3%. We estimate that the impact of neglected higher-order terms in the EFT on $S(0)$ is an order of magnitude smaller than this.
Extension of the EFT to higher order and inclusion of couplings between s- and d-wave scattering states is not expected to reduce the uncertainty, although it would provide slightly greater generality in matching possible reaction mechanisms. There is, however, no indication in the literature that coupling to $d$-waves is important for $S(E)$ [@Descouvemont:2004hh] in the LER. Our analysis could perhaps be extended to higher energies, but for $E > 0.5$ MeV, accurate representation of $M1$ resonances is at least as important as reliable calculations of the $E1$ transition.
The most significant source of uncertainty in our extrapolant is, in fact, the $1$ keV uncertainty in the ${{}^{8}\mathrm{B}}$ proton-separation energy, which can shift $S(20~\mathrm{keV})$ by approximately $0.75$%. This could be eliminated by better mass measurements. Further significant improvement in $S(20~\mathrm{keV})$ for ${{}^{7}\mathrm{Be}}(p,\gamma){{}^{8}\mathrm{B}}$ requires stronger constraints on EFT parameters. Better determinations of $s$-wave scattering parameters seem to be of limited utility. The ANCs affect the very-low-energy $S$-factor the most, and so more information on them, from either *ab initio* theory or capture/transfer data, would be useful.
A number of other radiative capture processes whose physics parallels ${{}^{7}\mathrm{Be}}(p,\gamma){{}^{8}\mathrm{B}}$ are important in astrophysics. The formalism developed herein should be applicable to many of them.
[*Acknowledgments—*]{} We thank Carl Brune for several useful discussions, and Barry Davids, Pierre Descouvemont, and Stefan Typel for sharing details of their calculations with us. We are grateful to the Institute for Nuclear Theory for support under Program INT-14-1, “Universality in few-body systems: theoretical challenges and new directions", and Workshop INT-15-58W, “Reactions and structure of exotic nuclei". During both we made significant progress on this project. X.Z. and D.R.P. acknowledge support from the US Department of Energy under grant DE-FG02-93ER-40756. K.M.N. acknowledges support from the Institute of Nuclear and Particle Physics at Ohio University, and from U.S. Department of Energy Awards No. DE-SC 0010 300 and No. DE-FG02-09ER41621 at the University of South Carolina.
Supplemental material
=====================
Common-mode errors for experimental data
----------------------------------------
The quoted common-mode errors for Junghans [*et al.*]{}, sets BE1 and BE3, [@Junghans:2010zz], Filippone [*et.al.,*]{} [@Filippone:1984us], Baby [*et.al.,*]{} [@Baby:2002hj; @Baby:2002ju], and the Hammache [*et.al.,*]{} 1998 data set [@Hammache:1997rz] are $2.7\%$, $2.3\%$, $11.25\%$, $5\%$, and $2.2\%$, respectively. The data of Ref. [@Hammache:2001tg] are a measurement of the absolute $S(186~\mathrm{keV})$ and of the ratios $S(135~\mathrm{keV})/S(186~\mathrm{keV})$ and $S(112~\mathrm{keV})/S(186~\mathrm{keV})$. We treat each of these three quantities as one data point, so they do not need a CME.
EFT details
-----------
The modified-effective-range expansion for s-wave ${{}^{7}\mathrm{Be}}$-p scattering is: $$p\left(\cot \delta_s(E) - i\right)\, \frac{2\pi \eta}{e^{2\pi\eta}-1} =-\frac{1}{a_{s}} + \frac{1}{2} r_{s} p^{2} -2{k_{C}}H(\eta).$$ Here $H(\eta) =\psi(i\eta)+{1}/{(2i\eta)}-\ln(i\eta)$, $\eta \equiv k_C/p$, $k_C \equiv Z_{{{}^{7}\mathrm{Be}}} Z_p \alpha_{em} m_R$ with $m_R$ the ${{}^{7}\mathrm{Be}}$-p reduced mass, $p=\sqrt{2 m_R E}$, and $\psi$ the digamma function [@MathHandBook1].
An EFT parameter set that gives a good fit—as mentioned in the main text—is listed in Table \[tab:aEFTfit\].
$C_{({{}^{3}P_{2}})}^{2}$ ($\mathrm{fm}^{-1}$) $a_{({{}^{3}S_{1}})}$ (fm) $r_{({{}^{3}S_{1}})}$ (fm) $\epsilon_{1} $ $ \overline{L}_{1}$ (fm) $C_{({{}^{5}P_{2}})}^{2}$ ($\mathrm{fm}^{-1}$) $a_{({{}^{5}S_{2}})}$ (fm) $r_{({{}^{5}S_{2}})}$ (fm) $ \overline{L}_{2}$ (fm)
------------------------------------------------ ---------------------------- ---------------------------- ----------------- -------------------------- ------------------------------------------------ ---------------------------- ---------------------------- --------------------------
0.2336 24.44 3.774 -0.04022 1.641 0.3269 -7.680 3.713 0.1612
: A representative EFT parameter set that gives a curve almost on the top of the median value curve (solid blue) in Fig. \[fig:results3\]. The LO curve (dashed black) uses the LO parameters listed here, with the strictly NLO parameters set to zero. Because the parameter space is very degenerate, many such parameter sets could be given that have similar $S(E)$ curves but very different parameter values.[]{data-label="tab:aEFTfit"}
Results for $S$-factor and thermal reaction rate
------------------------------------------------
$E$ (MeV) Median (eV b) $-\sigma$ (eV b) $+\sigma$ (eV b)
----------- --------------- ------------------ ------------------
0. 21.33 0.69 0.66
0.01 20.97 0.65 0.63
0.02 20.67 0.63 0.60
0.03 20.42 0.60 0.58
0.04 20.20 0.57 0.55
0.05 20.02 0.55 0.53
0.1 19.46 0.45 0.44
0.2 19.27 0.34 0.34
0.3 19.65 0.32 0.30
0.4 20.32 0.35 0.31
0.5 21.16 0.42 0.41
: The median values and 68% interval bounds for $S$ in the energy range from 0 to 0.5 MeV. At each energy point, the histogram of $S$ is drawn from the Monte-Carlo simulated ensemble and then is used to compute the median and the bounds. []{data-label="tab:s0to0.5MeV"}
The median values and 68% interval bounds for $S$ in 10 keV intervals to 50 keV and then in 100 keV intervals to 500 keV is listed in Table \[tab:s0to0.5MeV\].
Regrouping the Taylor series for $S(E)$ about $20$ keV into a quadratic and applying the approximations of Refs. [@caughlan62; @rolfs-rodney] yields $$N_A \langle \sigma v \rangle=\frac{2.7648 \times10^5}{T_9^{2/3}} \exp\left(\frac{-10.26256}{T_9^{1/3}}\right)
\times (1 + 0.0406 T_9^{1/3} - 0.5099 T_9^{2/3} - 0.1449 T_9
+0.9397 T_9^{4/3} + 0.6791 T_9^{5/3}),
\label{eq:thermalreactionrate}$$ in units of $\mathrm{cm^3\,s^{-1}\,mol^{-1}}$, where $N_A$ is Avogadro’s number. Up to $T_9=0.6$, the lower and upper limits of the 68% interval for $S(E)$ produce a numerically integrated rate that is $0.969 (1+0.0576 T_9-0.0593 T_9^2)$ and $1.030
(1-0.05 T_9 +0.0511 T_9^2)$ times that of Eq. (\[eq:thermalreactionrate\]). At $T_9 \gtrsim 0.7$ energies beyond the LER, and hence resonances, come into play and so these results no longer hold. We know of no astrophysical environment with such high $T_9$ where $^7\mathrm{Be}(p,\gamma)^8\mathrm{B}$ matters.
[^1]: The second moments of the MCMC sample distribution imply that $C_1^2+ 0.94 C_2^2$ is best constrained, but we consider $C_1^2+
C_2^2$ for simplicity.
[^2]: We recomputed the sampling errors of Ref. [@Nollett:2011qf] in the basis of good $s$, taking more careful account of correlations between ANCs.
| ArXiv |
---
abstract: 'The contemporary society has become more dependent on telecommunication networks. Novel services and technologies supported by such networks, such as cloud computing or e-Health, hold a vital role in modern day living. Large-scale failures are prone to occur, thus being a constant threat to business organizations and individuals. To the best of our knowledge, there are no publicly available reports regarding failure propagation in core transport networks. Furthermore, Software Defined Networking (SDN) is becoming more prevalent in our society and we can envision more SDN-controlled Backbone Transport Networks (BTNs) in the future. For this reason, we investigate the main motivations that could lead to epidemic-like failures in BTNs and SDNTNs. To do so, we enlist the expertise of several research groups with significant background in epidemics, network resiliency, and security. In addition, we consider the experiences of three network providers. Our results illustrate that Dynamic Transport Networks (DTNs) are prone to epidemic-like failures. Moreover, we propose different situations in which a failure can propagate in SDNTNs. We believe that the key findings will aid network engineers and the scientific community to predict this type of disastrous failure scenario and plan adequate survivability strategies.'
author:
- '[^1]'
bibliography:
- 'epidemicsOTN.bib'
title: Unveiling Potential Failure Propagation Scenarios in Core Transport Networks
---
Backbone Transport Networks; SDN-controlled Transport Networks; Failure Propagation; Epidemics.
Introduction\[sec:introduction\]
================================
In the last decade, the Information and Communication Technology sector has substantially increased its dependency on communication networks for both business and pleasure. Additionally, this dependency has increased even more with the inception of the myriad of new emerging technologies and services such as smart-cities, cloud computing, e-Health and the Internet of the Things. Backbone Transport Networks (BTNs) constitute the foundations of the aforementioned network-dependent applications and services. Traditionally, BTNs have been classified in two types:
Static Transport Networks (STNs); and
Dynamic Transport Networks (DTNs).
Typically, BTNs are divided in three differentiated layers: Data Plane (DP), Control Plane (CP), and Network Management Plane (NMP). STNs are centrally controlled by a network management system (NMS) such that the operation is manual and predetermined. On the contrary, DTNs are architectures under Automatically Switched Optical Network (ASON)/Generalized Multi-Protocol Label Switching (GMPLS) CP (or similar) where the management plane is a facilitator, but the actual network control is automated (to some extent) and services are configured and managed via distributed intelligence.
The advent of Software-Defined Networking (SDN) technologies such as OpenFlow might contribute to changing BTNs as we know them today, realizing the concept of SDN-controlled Transport Networks [@McKeown2008Openflow; @mcdysan2013SDTNS]. Currently, the Open Networking Foundation (ONF) has a dedicated working group defining guidelines for applying SDN standards to transport networks. Following the work in this Standards Developing Organization (SDO), major manufacturers of transport network equipment currently offer SDN-based products [@infinera2013ots]. In addition, large-scale SDNTNs such as the B4 Google network have been deployed [@googleB4]. As traditional DTNs, SDNTNs consist of a data and control plane such that the CP can be centralized or distributed [@Gringeri2013SDNTN; @distributed2013sdtns]. The SDN-controller is the main component of the CP, and the applications that run at the controller provide the CP functionalities. Since extensive work regarding the development of standards for SDTNs is ongoing, we strongly believe that next-generation transport networks will be SDN-enabled.
Although BTNs play a pivotal role to ascertain performance and integrity of the aforementioned novel services, their ubiquity is often taken for granted. Recently, network failures of great significance have occurred, re-enforcing the need to take the possibility of such large and potentially catastrophic failures into consideration in the underlying network design [@Habib2013630]. According to the European Network and Information Security Agency (ENISA), in 2011 at least 51 severe outages of communication networks were reported in Europe, where each affected about 400,000 users of fixed and mobile Internet [@CCC2011].
Many different protection and recovery techniques for single failures in communication networks have been extensively analyzed in the last decades. Consequently, in this work we focus on multiple failures, which can be broadly classified as depicted in Fig. \[fig:multfail\]. Multiple failures can be either *static* or *dynamic*. Static multiple failures are essentially one-off failures that affect nodes or links simultaneously at any given point (e.g., an earthquake). Dynamic failures have a temporal dimension. From all possible multiple failures, we focus on the dynamic scenarios and more specifically, in epidemic-like failure scenarios, where failure of one or more nodes might be propagated through the network, possibly resulting in an outbreak.
The aim of this work is to shed light on the following questions: *“Are epidemic-like failures or attacks likely in BTNs?”* and *“Will the upgrade to SDNTNs increase the vulnerability with respect to these type of failures or attacks?”* To do so, we present the state of the art of epidemic-like failure models in Section \[soa\]. Then, we review the main failure propagation model that has been proposed for transport networks in Section \[epidemicsontelecom\]. In Section \[vulnerability\] we discuss whether a failure propagation could occur in BTNs and SDNTNs. Section \[providers\] presents feedback from three network providers based on their experience in addition to the results of our research. Finally, Section \[sec:conclusions\] reviews the main contributions and findings of this work.
![Multiple failures broad classification. An epidemic-like failure is a process where a temporary failure propagates to physical neighbors. A cascading failure that triggers failures in physical neighbors is also considered as an epidemic-like failure.[]{data-label="fig:multfail"}](multfail2.pdf)
What is an epidemic-like failure propagation?\[soa\]
====================================================
Epidemics theory has been used to describe and predict the propagation of diseases, human interactions, natural phenomena, and failures in a wide range of networks. An epidemic-like failure is a dynamic process where a partial/temporary failure propagates to physical neighbors. The spreading of the aforementioned events is formally represented by epidemic models, and can be generally classified in one of the following three families:
- The *Susceptible-Infected* (SI) considers individuals as being either susceptible (S) or infected (I). This family assumes that the infected individuals will remain infected forever, and so can be used for worst-case propagation scenarios ($S\rightarrow I$).
- The *Susceptible-Infected-Susceptible* (SIS) considers that a susceptible individual can become infected on contact with another infected individual, then recovers with some probability of becoming susceptible again. Therefore, individuals might change their state from susceptible to infected, and vice versa, repeatedly ($S\leftrightarrows I$).
- The *Susceptible-Infected-Removed* (SIR), which extends the SI model to take into account a removed state. Here, an individual can be infected just once because when the infected individual recovers, it becomes immune and will no longer pass the infection onto others ($S\rightarrow I \rightarrow R$).
As shown in Fig. \[fig:multfail\], the subset of cascading failures intersects the subset of epidemic failures. Cascading failures are common in most critical infrastructures such as telecommunications, electrical power, rail, and fuel distribution networks [@Strogatz2001]. In telecommunication networks, we consider cascading failures as an epidemic when it occurs due to a malfunctioning in one node of a network which eventually triggers a failure in its neighbors. Real cascading failures in telecommunication networks have been observed in the IP layer of the Internet and in the physical layer of BTNs [@wrap32818]. The propagation of a cascading failure happens gradually in phases: after the initial failure (e.g., a massive broadcast of a routing message with a bug), some of the neighboring nodes get overloaded and fail. This first step leads to further overloading of more nodes and their collapse, constituting the second step and so on. In this way, networks go through multiple stages of cascading failures before they finally stabilize and there are no more failures. It is worth noting that cascading failures in other critical infrastructures, such as power grids, do not necessarily propagate by the physical contact of nodes or links, but by the load balancing in the global network. In such cases, cascading failures are not similar to epidemics, and thus are out of the scope of this work.
A Failure Propagation Model for Telecommunication Networks\[epidemicsontelecom\]
================================================================================
Calle et al. pioneered the research in this field by presenting a new epidemic model called *Susceptible-Infected-Disabled* (SID), which relates each state with a specific functionality of a node in the network [@calle2010multiple]. The state diagram of the SID model (*Susceptible$\leftrightarrows$Infected$\rightarrow$Disabled$\rightarrow$Susceptible*), as seen from a single node, is shown in Fig. \[fig:sid\]. Each node, at each moment of time, can be either susceptible (S), infected (I) or disabled (D). A susceptible node can be infected with probability $\beta$ by receiving the infection from a neighbor (e.g., a bug in the routing or signaling protocol). An infected node can be repaired with probability $\delta_1$ (e.g., the network operator might manually reboot the CP). Finally, the disabled state takes into account the fact that the CP failure eventually affects the DP of the node with probability $\tau$ (e.g., the forwarding tables of the DP become unaccessible). After that, the model states that a repairing time, such as the mean time to repair (MTTR) of the node, determines when it becomes susceptible again ($\gamma$) (e.g., the required time to replace the node).
No operations can be performed during control plane node failures. However, as long as the data plane of the node does not fail, established connections should not fail or be re-routed as a result of control plane node failures. The routing protocol is assumed to be capable of detecting the failure of the control plane and informing all other nodes. Once the routing protocol has converged with this new information, a new connection will not be routed through this node. This same behavior is taken into account by the SID epidemic model.
![State diagram of the SID model and its relationship with the STN and DTN planes.[]{data-label="fig:sid"}](sidstates.pdf)
\
According to the SID model, Fig. \[fig:failurepropagation\] illustrates how a failure can propagate in a BTN. From Fig. \[fig:prop1\], the network operates properly and thus, all nodes are in the susceptible state. At a given time $t$, the network management system updates a module of a controller with software that contains a bug, as shown in Fig. \[fig:prop2\]. As a result, this node becomes infected and propagates the failure (e.g., the bug) to his neighbors, as observed in Fig. \[fig:prop3\]. The epidemic continues to spread while the CP of an infected node eventually affects its DP operation. In this case, the incapability to resolve the problem at the CP might necessitate a complete node replacement (e.g., shutdown), thus impacting the operation of the DP. Consequently, this node becomes disabled as shown in Fig. \[fig:prop4\].
Throughout the last decades, several failures have spread through communication networks. In the early 90s, a rapidly spreading malfunction collapsed the AT&Ts long distance network[^2], causing the loss of more than \$60 million in terms of unconnected calls. In 2002, a failure propagation in the IP layer of the Internet was caused by a vulnerability of the BGP protocol. More recently, a BGP update bug which propagated through Juniper routers caused a major Internet outage in 2011[^3]. In this latter case, routers were reseting and re-establishing its functioning state after five minutes.
Although there are no commercial references nor reports with respect to the occurrence of propagation of failures in BTNs, in the following sections we identify several failure scenarios that could be modelled as epidemic-like spreadings.
Failure Propagation Scenarios in Transport Networks\[vulnerability\]
====================================================================
This section describes BTNs and SDNTNs, showing that both contemporary and future networks might be predisposed to enduring epidemic-like failures.
Backbone Transport Networks
---------------------------
As explained in Section \[sec:introduction\], a BTN is divided into Data Plane (DP), Control Plane (CP) and Network Management Plane (NMP), which have the following functionalities:
- Data plane (or transport plane): responsible for user data transport, usually called data-path.
- Control plane: responsible for connection and resource management, which can be either associated with or separated from the managed DP.
- Management plane: responsible for supervision and management of the whole system (including transport and control planes).
As mentioned, there are two types of BTNs: static and dynamic. In STNs, the control plane does not act autonomously, but takes orders and updates from the NMP, which is eventually driven by a Network Management System (NMS). Thus, the intelligence of STNs is centralized. On the contrary, in DTN architectures such as GMPLS, the control plane dynamically reconfigures the network according to its current state. In this case, the NMP is only used as a facilitator (e.g. initializing or updating software of CP). In DTNs the control plane exists in each physical router and hence, it is distributed. For instance, GMPLS-based DTNs are networks where the control plane runs over an IP/Ethernet network, while the data plane runs over a wavelength routed WDM network.
Failures occurring in BTNs can be classified in two broad groups:
- Faults: They include component failures as a result of natural exhaustion, human errors, and natural disasters.
- Attacks: They are intentional component failures. In fact, components may be selected to maximize the resulting impact of the attack. Furthermore, the identified targets may depend on various criteria such as the number of potentially affected users and additional socio-political and economic considerations.
There are three possible types of failures in the BTNs namely: (a) link; (b) node; and (c) software failures. In 2011 41% of failures were attributed to hardware and software, 12% to human errors, 12% to natural disasters, and 6% to malicious attacks [@CCC2011]. Although the percentage of malicious attacks was the smallest, they resulted in an average of 31 outage hours, as opposed to 17 outage hours for other failures. Therefore, despite the fact that targeted failures are not frequent, they are of paramount importance because they cause major disruptions. Control plane modules may fail due to software or hardware bugs, or protocol logic errors. Recovery from such failures may involve switching to a hot standby, if a redundant software process has been implemented for that module. If the routing and signaling modules are implemented as separate processes, then either one of them may fail independently. If the signaling module fails, then new connections cannot be established through this node, and existing connections cannot be deleted.
In STNs, epidemic-like spreading of failures are rare because CP nodes do not interact between each other (i.e., they only communicate with the NMS). Therefore, if the NMS were compromised, a massive network failure affecting first the CP nodes and then the DP ones could easily occur (e.g., all nodes could be shut down). Nonetheless, this would not be an epidemic-like failure scenario, but a catastrophic static one. On the contrary, in DTNs an error during a software update, or a failure produced by a software itself could induce a CP node failure, which could spread through the network depending on the type of failure (e.g., a message with wrong objects in the fields). For instance, this initial failure, just like an infection, could be caused by the NMS. The spreading itself would be carried out between all nodes of the CP. This scenario is illustrated in Fig. \[fig:failurepropagation\].
SDN-controlled Transport Networks
---------------------------------
As with DTNs, SDNTNs require a DP and a CP. The SDN-controllers constitute the CP of a SDNTN and the network nodes constitute the DP. Applications that run at the controller provide the CP functionalities. Therefore, the SDN-controller can host different applications that support various carrier-class transport technologies such as SDH, DWDM, Ethernet or the suite of protocols of GMPLS.
Fig. \[fig:sdntn\] illustrates a possible scenario with three network providers operating SDNTNs. As observed, the CP can be either centralized (providers A and B) or distributed (provider C) [@Gringeri2013SDNTN; @distributed2013sdtns]. As shown in the bottom left, the SDN-controller might host different applications that run on a Network Operating System (NOS). In addition, there is a component called FlowVisor that acts as a transparent proxy between the DP nodes and the SDN-controller. FlowVisor is a network virtualization layer and its main objective is to make sure that each user of the SDN-controller controls his own virtual network [@sherwood2009flowvisor].
![Possible scenario of SDN transport networks.[]{data-label="fig:sdntn"}](sdntn.pdf)
The emergence of SDNTNs implies an increased reliance on software. On the one hand, this aspect brings many benefits such as network programmability and control logic centralization. On the other hand however, it is also the source of major security concerns. It has been argued that SDN introduces new threat vectors with respect to traditional networks [@vectorsvulnerable2013SDN]. For instance, software faults hold a pivotal role in the reliability of SDNTNs, and significant efforts are made to define tools able to detect SW bugs or errors [@Canini2012NICE]. Additionally, other type of failures could occur due to policy (also called rules) conflicts, given that several users might work on the same physical network, each one on his own virtualized network. In such cases, FlowVisor is in charge to ensure compatibility among all policies.
\
There are several cases in which epidemic-like failures could occur in SDNTNs. We classify these scenarios according to the initial event that triggers the propagation, which can be either a fault or an attack. First, in order to illustrate some potential scenarios in SDNTNs, we assume that a DDoS attack can be launched, for instance, by following the method provided in [@attacking2013SDN]. Consequently, we propose two propagation scenarios:
- *Vertical propagation*, which can be bottom-top or top-bottom: as shown in Fig. \[fig:sdntn\_prop2\], an attacker launches a DoS attack against an OpenFlow switch, which is connected to a primary SDN-controller. In order to increase the resilience of the system, the switch has been assigned a secondary SDN-controller. The switch needs to contact the SDN-controller for every packet. The SDN-controller is not able to handle all the new flow requests and fails. Then, the switch redirects its traffic to the secondary SDN-controller, where the same situation can happen. If the failing SDN-controller is the only controller available for other OpenFlow switches, then the switches lose their connection with the controller and can be considered as failed.
- *Horizontal DP propagation*: as observed in Fig. \[fig:sdntn\_prop1\], an attacker injects huge quantities of traffic to an edge OpenFlow switch (e.g., a DoS attack). The destination of the injected traffic is outside the SDNTN, and to reach it, the traffic has to be routed through core switches and exit via another edge switch. Since SDN allows heterogeneous network scenarios (e.g., hardware from different vendors), we assume that edge OpenFlow switches have high-performance capabilities, while core OpenFlow switches are commodity devices. Therefore, the huge amount of injected traffic can eventually overflow the buffer of core switches, causing successive failures in the DP plane.
Second, it is possible to observe the same scenario shown in Fig. \[fig:sdntn\_prop1\] caused by a fault. In such case, instead of being caused by a DoS attack, the switch failures would occur due to, for instance, a software bug in the routing protocol of the SDN-controller. Lastly, it is worth noting that the proposed horizontal DP propagation, as well as the vertical propagation, are closer to cascading failures. However, in the case of a distributed CP with several SDN-controllers, an epidemic-like spreading could happen if, for instance, the SDN-controllers where infected by masterful worms such as Tuxera[^4].
Transport Network Providers Experience with Large-Scale Failures\[providers\]
=============================================================================
With the purpose of validating some of the failure scenarios mentioned in the previous section, we have approached three network operators asking them about their experience with epidemic and cascading failure occurrence. The three operators were willing to discuss network vulnerabilities and specific spreading patterns within their networks. However, they were only willing to provide this information under a non-disclosure agreement. Consequently, we refer to the three operators as FO, SO and TO, which stand for first, second and third operator, respectively.
According to the experience of the FO with their STN, the most typical errors of hardware (HW) components are closely related to software (SW) bugs. Nonetheless, these failures have no connection to the outside world, i.e., no spreading is possible. The FO stated that their current procedures to operate their STN, which is based on the NMP, could potentially be vulnerable if the management server got compromised. This is highly unlikely, since the NMS has no access to the global Internet and a strong Firewall protection is provided. However, assuming a system breach, the damage to the network could be devastating, i.e., the entire network could fail simultaneously. As to epidemic-like failures, the only situation that the FO could see as problematic would be the SW update process. The SW update happens step-wise, where portions of the network are updated. Furthermore, when an element is updated, it has 2 banks of SW (i.e., an active one and a hot-standby). Typically, the server updates the hot-standby and then switches the operation of the element to that bank of SW, and then updates the primary one. If SW with a bug was loaded to the node, the node could stop operating. Since in STNs the node only communicates with the management server, no direct “infection” to the neighbors can occur. Nevertheless, considering that an entire region is updated (possibly with the same erroneous SW), then several elements can potentially be hit.
The SO uses equipment from a different vendor for their STN, and his experience with that equipment is different than the FO. In general, node HW failures are negligible in comparison with fiber cuts. The SO also suggested that SW updates might cause unresponsive management entity on the node. Furthermore, according to the SO, there is another potential vulnerability: the Digital Communication Network (DCN) interconnecting all nodes in a STN for the purpose of management and configuration information exchange. Typically, this case is related to standard packet-switched network problems, which are well-known from the IP world. Nonetheless, the SO outlined one possible problem: if buggy SW/mis-configurations were installed/occurred in the main routers (the designated router for example), as a result, nodes could start flapping the routes they knew from one port to the other. If one node became confused, he could advertise wrong routes to the rest and if flapping occurred, potentially blocking of the controller entities would be possible. This same scenario could be spread to the rest of nodes, since they also would need to readjust their router interfaces.
The TO has an extensive experience with GMPLS control plane technologies. This expert asserts that in STNs an epidemic spreading of failures is possible only if SW elements are failing, and if some type of distributed network (such as DCN) is deployed between nodes. On the contrary, in DTNs, where GMPLS is assumed to be one of the control plane solutions for an automatic provisioning and operation, this picture might change. The TO stated that there are numerous examples of spreading of failures in the contemporary IP distributed networks (e.g., based on BGP update bugs). Moreover, since the GMPLS CP is mostly a distributed IP-based network, it leads to the expectation that epidemic-like failures become reasonably possible. Three examples were outlined:
1. When a node is affected by a CP failure and it attempts to recover its RSVP-TE state, synchronization becomes very difficult. This process involves synchronization with the neighbors, with the DP and with the NMP. Any of these three steps might fail for different reasons and this can directly impact the knowledge and states of the neighboring nodes. Such a process pertains mainly to the CP and possibly would affect only some of the connections. However, the fact is that an unsuccessful process of synchronization might lead to nodes operating in a state which is erroneous and potentially influencing the node’s service delivery capability.
2. If a node fails, the neighbors of a node must update their state and the state of the network. In time, the neighbors will have moved towards a state they are not “used to be”, i.e., an untypical state (the typical state is when everything is properly functioning). Under this situation, the neighbors might move closer to an unstable state of operation (e.g., if a node switches from a functioning to non-functioning state and vice versa) and this can potentially bring these nodes into a dysfunctional state.
3. Carrier-class components are typically undergoing very stringent tests and the requirements for robustness are extensive. Nevertheless, it is still possible to observe implementations where a failure in one single process might bring other processes in the same node to stop functioning. For example, this can occur if processes (routing and signaling) share CPUs (which might suffer HW/SW problems) or if they share memory (buffer overflows).
Finally, the TO indicated that RSVP-TE is highly vulnerable to problems, and when problems occur, recovering the functional state in the network is much more difficult (due to the synchronization process described earlier). OSPF (Open Shortest Path First) failures are simpler to fix and synchronization between nodes might be established in a shorter period of time. Nevertheless, mis-configured routing is one of the main sources of large-scale failures in networks, as stated in the previous sections of this work. Thus, OSPF is also considered as a potential vulnerability.
Conclusions\[sec:conclusions\]
==============================
In this paper, we have studied the possibility of observing epidemic-like failures in Backbone Transport Networks and in SDN-controlled Transport Networks. We have analyzed both architectures and related each of them to several failure scenarios. Finally, to reinforce our study, three network operators have revealed several vulnerabilities in STNs and DTNs that could eventually lead to epidemic-like failures.
One of the most significant differences between traditional BTNs and SDNTNs is that BTNs are closed systems, so the surface of potential risks that might initiate the propagation of a failure is smaller that the one of SDNTNs, where homogeneous scenarios prevail. In any case, human-induced errors are the major issue, which could be either deliberate or unintentional. For this reason, the robustness of the CP of both DTNs and SDNTNs depends highly on the applied SW engineering principles, and on the way the processes and protocols are configured. Poor network management or mis-configured controllers, or inconsistent protocol implementations might become the root cause of epidemic-like failures in the CP. Additionally, cyber-security plays a pivotal role in DTNs and SDNTNs, given that in such networks the operation of each node relies exclusively on software. Since one of the potential threats is to compromise either the network management plane or the SDN-controller, it will be a matter of how to hack a closed system to gain non-granted access. To conclude, we anticipate that this paper will inspire researchers and professionals to design and implement security mechanisms to enhance the resilience of transport networks under dynamic multiple failure scenarios.
Acknowledgements {#acknowledgements .unnumbered}
================
This work is partially supported by Spanish Ministry of Science and Innovation project TEC 2012-32336, and by the Generalitat de Catalunya research support program SGR-1202. This work is also partially supported by the Secretariat for Universities and Research (SUR) and the Ministry of Economy and Knowledge through AGAUR FI-DGR 2012 and BE-DGR 2012 grants (M. M.)
[^1]: Marc Manzano and Eusebi Calle are with University of Girona, Spain. Anna Manolova Fagertun and Sarah Ruepp are with Technical University of Denmark, Denmark. Caterina Scoglio is with Kansas State University, USA. Ali Sydney is with Raytheon BBN Technologies, USA. Antonio de la Oliva and Alfonso Muñoz are with University Carlos III of Madrid, Spain. Corresponding author: Marc Manzano (email: [email protected] - [email protected]).
[^2]: <http://users.csc.calpoly.edu/~jdalbey/SWE/Papers/att_collapse.html>
[^3]: <http://www.zdnet.com/juniper-fail-seen-as-culprit-in-site-outages-4010024743/>
[^4]: <http://spectrum.ieee.org/telecom/security/the-real-story-of-stuxnet>
| ArXiv |
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PS. \#1[[**\#1**]{}]{} \#1[Nucl. Phys., ]{} \#1[Comm. Math. Phys., ]{} \#1[Phys. Lett., ]{} \#1[Phys. Rev., ]{} \#1[Phys. Rev. Lett., ]{} \#1[Proc. Roc. Soc., ]{} \#1[Prog. Theo. Phys., ]{} \#1[Sov. J. Nucl. Phys., ]{} \#1[Theor. Math. Phys., ]{} \#1[Annals of Phys., ]{} \#1[Proc. Natl. Acad. Sci. USA, ]{}
**$Sp(N_c)$ Gauge Theories and M Theory Fivebrane**
Changhyun Ahn$^{a,}$[^1], Kyungho Oh$^{b,}$[^2] and Radu Tatar$^{c,}$[^3]
[*$^a$ Dept. of Physics, Seoul National University, Seoul 151-742, Korea*]{}
[ *$^b$ Dept. of Mathematics, University of Missouri-St. Louis, St. Louis, Missouri 63121, USA*]{}
[*$^c$ Dept. of Physics, University of Miami, Coral Gables, Florida 33146, USA*]{}
Abstract
0.2in We analyze M theory fivebrane in order to study the moduli space of vacua of $N=1$ supersymmetric $Sp(N_c)$ gauge theories with $N_f$ flavors in four dimensions. We show how the $N=2$ Higgs branch can be encoded in M theory by studying the orientifold which plays a crucial role in our work. When all the quark masses are the same, the surface of the M theory spacetime representing a nontrivial ${\bf S^1}$ bundle over ${\bf R^3}$ develops $A_{N_f-1}$ type singularities at two points where D6 branes are located. Furthermore, by turning off the masses, two singular points on the surface collide and produce $A_{2N_f-1}$ type singularity. The sum of the multiplicities of rational curves on the resolved surface gives the dimension of $N=2$ Higgs branch which agrees with the counting from the brane configuration picture of type IIA string theory. By rotating M theory fivebranes we get the strongly coupled dynamics of $N=1$ theory and describe the vacuum expectation values of the meson field parameterizing Higgs branch which are in complete agreement with the field theory results. Finally, we take the limit where the mass of adjoint chiral multiplet goes to infinity and compare with field theory results. For massive case, we comment on some relations with recent work which deals with $N=1$ duality in the context of M theory.
Introduction
============
One of the most interesting tools used to study nonperturbative dynamics of low energy supersymmetric gauge theories is to understand the D(irichlet) brane dynamics where the gauge theory is realized on the worldvolume of D brane.
This work was pioneered by Hanany and Witten [@hw] where the mirror symmetry of $N=4$ gauge theory in 3 dimensions was interpreted by changing the position of the Neveu-Schwarz(NS)5 brane in spacetime. (see also [@bo1][@bo2]). They took a configuration of type IIB string theory which preserves 1/4 of the supersymmetry and consists of parallel NS5 branes with D3 branes suspended between them and D5 branes located between them. A new aspect of brane dynamics was the creation of D3 brane whenever a D5 brane and NS5 brane are crossing through each other. This was due to the conservation of the linking number(defined as a total magnetic charge for the gauge field coupled with the worldvolume of the both types of NS and D branes).
By T-dualizing the above configuration on one space coordinate, the passage to $N=2$ gauge theory in 4 dimensions can be described as two parallel NS5 branes and D4 branes suspended between them in a flat space in type IIA string theory. When one change the relative orientation of the two NS5 branes [@bar] while keeping their common 4 spacetime dimensions intact, the $N=2$ supersymmetry is broken to $N=1$. The brane configuration[@egk; @egkrs] preserves 1/8 of the supersymmetry and this corresponds to turning on a mass of adjoint field because the distances between D4 branes suspended between the NS5 branes relate to the vacuum expectation values(vevs). The configuration of D4 branes gives the gauge group while the D6 branes give the global flavor group. Using this configuration they described and checked a stringy derivation of Seiberg’s duality for $N=1$ supersymmetric gauge theory with $SU(N_c)$ gauge group with $N_f$ flavors in the fundamental representation which was previously conjectured in [@se1]. This result was generalized to brane configurations with orientifolds which then give $N=1$ supersymmetric theories with gauge group $SO(N_c)$ or $Sp(N_c)$ [@eva; @egkrs]. In this case the NS5 branes have to pass over each other and some strong coupling phenomena have to be considered. Similar results were obtained in [@bh; @bsty; @t] where the moduli space of the supersymmetric gauge theories is geometrically encoded in the brane setup.
Another approach was initiated by Ooguri and Vafa [@ov] where they considered the compactification of IIA string theory on a double elliptically fibered Calabi-Yau threefold. The wrapped D6 branes around three cycles of Calabi-Yau threefold filling also a 4 dimensional spacetime. The transition between electric theory and its magnetic dual appears when a change in the moduli space of Calabi-Yau threefold occurs. Their results were generalized in the papers of [@ao; @a; @ar; @aot] to various other models which reproduce field theory results studied previously.
So far the branes in string theory were considered to be rigid without any bendings. When the branes are intersecting each other, a singularity occurs. In order to avoid that kind of singularities, a very nice simplification was obtained by reinterpreting brane configuration in string theory from the point of view of M theory as was showed by Witten in [@w1]. Then both the D4 branes and NS5 branes come from the fivebranes of M theory (the former is an M theory fivebrane wrapped over $\bf{S^1}$ and the latter is an M theory fivebrane on $\bf{R^{10} \times S^1}$). That is, D4 brane’s worldvolume projects to a five manifold in $\bf{R^{10}}$ and NS5 brane’s worldvolume is placed at a point in $\bf{S^1}$ and fills a six manifold in $\bf{R^{10}}$. To obtain D6 branes one has to use a multiple Taub-NUT space whose metric is complete and smooth. The $N=2$ supersymmetry in four dimensions requires that the worldvolume of M theory fivebrane is $\bf{R^{1,3}}\times \Sigma$ where $\Sigma$ is uniquely identified with the curves [@sw] that appear in the solutions to Coulomb branch of the field theory. The configurations involving orientifolds were considered in[@lll; @bsty1]. The method of brane dynamics was used to study supersymmetric field theories in several dimensions by many authors [@ah; @ba; @k; @cvj1; @hov; @hz; @mmm; @fs; @w2; @biksy; @gomez; @cvj2; @hk; @hsz; @nos; @hy; @noyy; @ss; @bo; @mi].
The original work [@w1] was suited to study the moduli space for $N=2$ supersymmetric theories. By rotating one of the NS5 branes the $N=2$ supersymmetry is broken to $N=1$ [@bar]. In [@w2; @hoo] (see also [@biksy; @ss; @bo]) this was seen from the point of the M theory interpretation, by considering the possible deformation of the curve $\Sigma$. In field theory, the supersymmetry is broken by giving a mass to the adjoint field and if this mass is finite, the $N=1$ field theory can be compared with the previous results obtained in [@ads]. These papers considered the case of unitary groups.
Recently, the exact low energy description of $N=2$ supersymmetric $SU(N_c)$ gauge theories with $N_f$ flavors in 4 dimensions in the framework of M theory fivebrane have been found in [@hoo]. They constructed M fivebrane configuration which encodes the information of Affleck-Dine-Seiberg superpotential [@ads] for $N_f < N_c$. Later, this approach has been used to study the moduli space of vacua of confining phase of $N=1$ supersymmetric gauge theories in four dimensions [@bo]. In terms of brane configuration of IIA string theory, this corresponds to the picture of [@egk] by taking multiples of NS’5 branes rather than a single NS’5 brane.
In the present paper we generalize to the case of symplectic group $Sp(N_c)$ with $N_f$ flavors. The new ingredient that is introduced is the orientifold. We find an interesting picture which differs from the one obtained for unitary group $SU(N_c)$ [@hoo]. This is expected because for $SU(N_c)$ groups we have both baryonic and non-baryonic branches, but in the case of $Sp(N_c)$ we cannot construct any baryon, so we have only non-baryonic branch.
This paper is organized as follows. In section 2 we review the papers of [@ip; @aps; @hms] and study the moduli space of vacua of the $N=1$ theory which is obtained from the $N=2$ theory by adding a mass term to the adjoint chiral multiplet. We discuss for different values of the number of flavors with respect to the number of colors. We also introduce massive matter. In section 3, we start with the setup of M theory fivebrane and discuss the Higgs branches with the resolution of singularities. In section 4, we rotate brane configuration and obtain information about the strong coupling dynamics of $N=1$ theory. In section 5, we take the mass of adjoint field infinite and compare it with field theory results for massless or massive matter. We discuss $N=1$ duality and compare with similar work without D6 branes obtained in [@cs] recently. Finally in section 6, we conclude our results and comment on the outlook in the future directions.
Field Theory Analysis
=====================
Let us review and summarize field theory results already known in the papers of [@ip; @aps; @hms] for future developments. We claim no originality for most of results presented in this section except that we have found the property of meson field $M^{ij}$ having only one kind matrix element which will be discussed in detail later.
$N=2$ Theory
------------
Let us consider $N=2$ supersymmetric $Sp(N_{c})$ gauge theory with matter in the $2N_c$ dimensional representation of $Sp(N_c)$. In terms of $N=1$ superfields, $N=2$ vector multiplet consists of a field strength chiral multiplet $W_{\alpha}^{ab}$ and a scalar chiral multiplet $\Phi_{ab}$, both in the adjoint representation of the gauge group $Sp(N_c)$. The quark hypermultiplets are made of a chiral multiplet $Q^{i}_{a}$ which couples to the Yang-Mills fields where $i = 1,\cdots ,2N_{f}$ are flavor indices( the number of flavors has to be even ) and $a = 1, \cdots , 2N_{c}$ are color indices. The $N=2$ superpotential takes the form: $$\label{super}
W = \sqrt{2} Q^{i}_{a} \Phi^{a}_{b} J^{bc} Q^{i}_{c}
+ \sqrt{2} m_{ij} Q^{i}_{a} J^{ab} Q^{j}_{b},$$ where $J_{ab}$ is the symplectic metric (
[cc]{} 0 & 1\
-1 & 0
) used to raise and lower $Sp(N_{c})$ color indices and $m_{ij}$ is the antisymmetric mass matrix \[mass\] (
[cc]{} 0 & -1\
1 & 0
) ( m\^f\_[1]{}, , m\^f\_[N\_f]{} ). Classically, the global symmetries are the flavor symmetry $O(2N_{f}) = SO(2N_{f})\times \bf{Z_{2}}$ in addition to $U(1)_{R}\times SU(2)_{R}$ chiral R-symmetry. The theory is asymptotically free for $N_{f}$ smaller than $2N_{c}+2$ and generates dynamically a strong coupling scale $\Lambda_{N=2}$ where we denote the $N=2$ theory by writing it in the subscript of $\Lambda$. The instanton factor is proportional to $\Lambda_{N=2}^{2N_{c}+2-N_{f}}$. Then the $U(1)_{R}$ symmetry is anomalous and is broken down to a discrete $\bf{Z_{2N_{c}+2-N_{f}}}$ by instantons.
The moduli space contains the Coulomb and the Higgs branches. The Coulomb branch is parameterized by the gauge invariant order parameters u\_[2k]{}=<(\^[2k]{})>, k=1, , N\_c where $\phi$ is the scalar field in $N=2$ vector multiplet. Up to a gauge transformation $\phi$ can be diagonalized to a complex matrix, $<\phi>=\mbox{diag} ( A_1, \cdots, A_{N_c} )$ where $A_i=
( { a_i \atop 0 }{ 0 \atop -a_i} )$. At a generic point the vevs of $\phi$ breaks the $Sp(N_c)$ gauge symmetry to $U(1)^{N_c}$ and the dynamics of the theory is that of an Abelian Coulomb phase. The Wilsonian effective Lagrangian in the low energy can be made of the multiplets of $A_i$ and $W_i$ where $i=1, 2, \cdots, N_c$. If $k$ $a_i$’s are equal and nonzero then there exists an enhanced $SU(k)$ gauge symmetry. When they are also zero, an enhanced $Sp(k)$ gauge symmetry appears. On the other hand, the Higgs branches are described by gauge invariant quantities which are made from the squarks vevs and which can be written as the meson field $M^{ij} = Q^{i}_{a} J^{ab} Q^{j}_{b}$ because we do not have any baryons.
Breaking $N=2$ to $N=1$
-----------------------
We want now to break $N=2$ supersymmetry down to $N=1$ supersymmetry by turning on a bare mass $\mu$ for the adjoint chiral multiplet $\Phi$. For the moment we consider that all the squarks are massless, so the terms of $m_{ij}$ in (\[super\]) will not enter into our superpotential. The superpotential is expressed as follows: $$\label{e1}
W = \sqrt{2} Q^{i}_{a} \Phi^{a}_{b} J^{bc} Q^{j}_{c} + \mu \;
\mbox{Tr}(\Phi^{2}).$$
When the mass of the adjoint chiral multiplet is much smaller than $\Lambda_{N=2}$, by turning a mass for the adjoint chiral multiplet, the structure of moduli space of vacua for $N=2$ theory is changed. Most of the Coulomb branch is lifted except $2N_{c} + 2 - N_{f}$ points which are related to each other by the action of ${\bf Z_{2N_{c} + 2 - N_{f}}}$ .
When the mass $\mu$ is increased beyond $\Lambda_{N=2}$ we can integrate out the adjoint chiral multiplet in the low-energy theory. Below the scale $\mu$, by a one loop matching between the $N=1$ and $N=2$ theories we obtain the $N=1$ dynamical scale, $\Lambda_{N=1}$ to be: $$\label{scale}
\Lambda_{N=1}^{2(3N_{c}+3-N_{f})} = \mu^{2N_{c} + 2}\Lambda_{N=2}^{2(2N_{c}
+ 2
-N_{f})}.$$
If $\mu$ is much larger than $\Lambda_{N=1}$ but finite, we can integrate out the heavy field $\Phi$ and to obtain a superpotential which is quartic in the squarks and proportional to $1/\mu$. The F-term equation for $\Phi$ from (\[e1\]) gives us to $$Q^{i}_{a}Q^{i}_{c} + \sqrt{2}\mu J_{ab}\Phi^{b}_{c} = 0$$ where we can read off $\Phi^b_c$ $$\Phi^{b}_{c} = \frac{1}{\sqrt{2}\mu} J^{ab} Q^{i}_{a} Q^{i}_{c}$$ or $$\Phi^{2} = - \frac{1}{2\mu^{2}} M^{2}.
\label{fi}$$ We plug this into the superpotential equation and obtain $$\Delta~W = - \frac{1}{2\mu} \mbox{Tr}(M^{2})
\label{pot}$$ which is similar to the equation $(2.5)$ of [@hoo] but without the term involving $(\mbox{Tr}M)^{2}$ because $M$ is traceless antisymmetric in our case and with a minus sign due to (\[fi\]). Therefore, the system below the energy scale $\mu$ can be regarded as the $N=1$ SQCD with the tree level superpotential (\[pot\]) and with the dynamical scale $\Lambda_{N=1}$ given by (\[scale\]). When we take the limit of $\mu\rightarrow \infty$ keeping $\Lambda_{N=1}$ fixed, the superpotential (\[pot\]) vanishes.
Let us start with the discussion for the various values of $N_{f}$ as a function of $N_{c}$.
$\bullet \;\;\; 0\le N_{f}\le N_{c}$
In this range of the number of flavors, as it is well known, a superpotential is dynamically generated [@ads] by strong coupling effects. For a general value of $N_{f}$, the ADS superpotential is given by [@ip]: $$\label{ads}
W_{ADS} = (N_{c} + 1 - N_{f}) \; \omega_{N_{c} + 1 - N_{f}}
(\frac{2^{N_{c} - 1}\Lambda_{N=1}^{3(N_{c}+1)-N_{f}}}{\mbox{Pf}M})^{\frac{1}
{N_{c}+1-N_{f}}}$$ where $\omega_{N_{c} + 1 - N_{f}}$ is an $N_{c}+1-N_{f}$ th root of unity and Pf(Pfaffian) of antisymmetric matrix $M$ has the following relation: $(\mbox{Pf}M)^2=\mbox{det} M$. For $N_{f} = N_{c}$, the gauge group is completely broken for $\mbox{Pf}<M>$ not zero and the ADS superpotential is generated by an instanton in the broken $Sp(N_{c})$.
For large but finite values for $\mu$, the potential obtained after $N=2$ breaking into $N=1$ (\[pot\]) can be described as a perturbation theory to the ordinary $N=1$ theory. Then the total effective superpotential is the sum of $W_{ADS}$ and $\Delta ~W$ W\_[eff]{} = W\_[ADS]{} + W. This form for $W_{eff}$ is exact for any non-zero value of $\mu$. The argument is based on the holomorphic property. The two terms appearing in an analytic function expanded with respect to $1/\mu$ are given by $W_{ADS}$ and $\Delta$W so a term that can be generated are of the form: \[term\] \^[-]{}M\^\_[N=1]{}\^[(3(N\_[c]{}+1)-N\_[f]{})]{} where $\alpha,\gamma$ are non-negative integers. In order to obtain $\alpha,\beta,\gamma$ we use the fact that (\[term\]) is invariant under the axial flavor symmetry $U(1)_{A}$ and has a charge 2 under the R-symmetry $U(1)_{R}$ where $U(1)_R$ is the anomaly free combination of the $U(1)$ R-symmetry group. The charges for $\Lambda_{N=1}, \mu$ and $M$ are given by:
$\Lambda^{3N_{c}+3-N_{f}}_{N=1}$ M $\mu$
------------ ---------------------------------- ---------------------------------- -----------------------------------
$U(1)_{R}$ 0 $\frac{2(N_{f}-N_{c}-1)}{N_{f}}$ $\frac{2(N_{f}-2N_{c}-2)}{N_{f}}$
$U(1)_{A}$ $2N_{f}$ 2 4
Using these values for the charges and applying them in (\[term\]), the condition that the superpotential has $U(1)_{A}$ charge $0$ gives us that $2\alpha -\beta=N_{f}\gamma$ and other condition that the superpotential has charge $2$ under $U(1)_{R}$ becomes $N_{f}(-\alpha+\beta-1) = (N_{c}+1)(-2\alpha+\beta)$. The combination of these two relations leads to: 1 - = (N\_[c]{} + 1 - N\_[f]{}). Because $\gamma\ge 0$ and we are considering the case of $N_{f}<N_{c}$, we have only two solutions for this equation, that is, $\alpha = 0,\gamma = 1/(N_{c}+1-N_{f})$ and $\alpha=1,
\gamma=0$, which exactly correspond to the two terms which appear in (\[term\]). Therefore, it turns out that the superpotential (\[term\]) is exact.
The moduli space of vacua is obtained by extrematizing this superpotential, and we get: $$\label{moduli}
M^{2} = -\mu \; \omega_{N_{c}+1-N_{f}}(
\frac{2^{N_{c}-1}\Lambda_{N=1}^{3(N_{c}+1)-
N_{f}}}{\mbox{Pf}M})^{\frac{1}{N_{c}+1-N_{f}}}.$$
In this moment, by a similarity transformation, $M$ can be brought to different forms. In [@aps; @hms], $M$ has been brought to a form such that $M^{2}=0$ which is the right equation for $M$ whenever we do not consider the ADS potential. But in our case, the equation (\[moduli\]) tells us that $M^{2}$ is not equal with $0$, so we take another form for $M$ after a similarity transformation. We bring $M$ to the simplest form, i.e., with two top-right and bottom-left diagonal blocks, one being minus the other because $M$ is to be antisymmetric. Denote $m_{1}, \cdots, m_{N_{f}}$ by the elements of top-right diagonal block in $M$ and of course $-m_{1}, \cdots, -m_{N_{f}}$ by the elements of the bottom-left diagonal block. In this case we will have an equation like (\[moduli\]) for each $m_{i}$. Since the right hand side is the same for all the $m_{i}$’s, [*they have to be equal*]{}. So all the diagonal entries in the top-right and bottom-left diagonal blocks of [M]{} are equal. This is our [*new*]{} observation which will appear naturally in section 4 due to the symmetry of orientifolding and can be compared with the result of [@hoo] for $SU(N_c)$ case where there were two cases, one with equal diagonal entries and th other with two different entries on the diagonal.
Now since all the top-right diagonal entries are equal with $m \equiv m_1=
\cdots = m_{N_f}$, we find the value for $m$ by solving (\[moduli\]): $$\label{value}
m = 2^{\frac{N_{c}-1}{2(N_{c}+1)-N_{f}}}
\mu\Lambda_{N=2}$$ where we have used the renormalization group( RG ) matching equation (\[scale\]). The values of $m$ in equation (\[value\]) describe the moduli space of the $N=1$ theory in the presence of a perturbation to the ADS superpotential. When $\mu\rightarrow\infty$ and $\Lambda_{N=1}$ are finite, the solution diverges. In this case $\Delta$W is $0$ and divergence of the solution coincides with the fact that there is no supersymmetric vacua in this region of the flavor.
Let us turn on quark mass terms like $\frac{1}{2}m^{ij}M_{ij}$. In this case the effective superpotential is given by W\_[eff]{} = W\_[ADS]{} + W + m M where $W_{ADS}$ and $\Delta ~W$ are given by (\[ads\]) and (\[pot\]) and m is an antisymmetric matrix as in (ref[mass]{}) but where we take $m_{f}=m_{1}^{f} = \cdots = m_{N_{f}}^{f}$ In this case the equation (\[moduli\]) is modified to contain a term $ \mu m M/2$. As we consider the limit $\mu\rightarrow\infty$ by keeping $\Lambda_{N=1}$ finite, the system will be $N=1$ SQCD with massive flavors. Only terms which are proportional to $\mu$ will resist (so the term form the LHS of (\[moduli\]) will be neglected because it does not depend on $\mu$) and thus we obtain the solution for the moduli space to be: \_[N\_[c]{}+1-N\_[f]{}]{} ()\^ = m\_[f]{} m/2 with the solution m\^[N\_[c]{}+1]{}= giving $N_{c}+1$ vacua in accordance with the interpretation of the low energy physics as the pure $N=1$ Yang-Mills theory.
Next we are now increasing the number of flavors.
$\bullet \;\;\; N_{f} = N_{c}+1$.
Now it is obvious that the ADS superpotential vanishes and the classical moduli space of vacua is changed quantum mechanically. It is parameterized by the meson satisfying the constraint[@ip] $$\mbox{Pf}M = 2^{N_{c}-1} \Lambda_{N=1}^{2(N_{c}+1)}.
\label{cons}$$ Again the quartic term $\Delta ~W$ is small for large finite $\mu$ and can be considered as a perturbation to the ordinary $N=1$ theory. By introducing a Lagrange multiplier $X$ in order to impose the constraint(\[cons\]), the effective superpotential will be: $$\label{spt1}
W_{eff} = X (\mbox{Pf}M - 2^{N_{c}-1}\Lambda_{N=1}^{2(N_{c}+1)}) -
\frac{1}{2\mu}
\mbox{Tr}(M^{2}).$$ >From the derivative with respect to $M$, we get: $$M^{2} = \mu \; X \mbox{Pf}M.$$ For the case of $X \neq 0$, again we can bring $M$ by a similarity transformation to the same form as before and this tells us that again all the top-right diagonal entries are the same and we obtain: $$m^{N_{c}+1} = 2^{N_{c} - 1}\Lambda_{N=1}^{2(N_{c}+1)}$$ which leads to, after using the RG equation: $$m = 2^{\frac{N_{c}-1}{N_{c}+1}}\mu\Lambda_{N=2}$$ which gives the moduli space.
$\bullet \;\;\; N_{f}=N_{c}+2$
In this case, the effective potential by adding $\Delta ~W$ is given by: $$W_{eff} = -\frac{\mbox{Pf}M}{2^{N_{c}-1}\Lambda_{N=1}^{2N_{c}+1}}-
\frac{1}{2\mu} \mbox{Tr}(M^{2})$$ which give us after extrematizing: $$M^{2} = -\mu \frac{\mbox{Pf}M}{2^{N_{c}-1}\Lambda_{N=1}^{2N_{c}+1}}.$$ Again $M$ can be brought to a simple form by a similarity transformation and all the diagonal entries are equal. After using the RG equation we get the moduli space given by: $$m = 2^{\frac{N_c-1}{N_c}} \mu\Lambda_{N=2}.$$
$\bullet \;\;\; N_{f} > N_{c}+2$
The theory that we have discussed until now is the electric theory which for this range of the number of flavors has a dual description in terms of a $Sp(N_{f}-N_{c}-2)$ gauge theory with $N_{f}$ flavors $q^{i}$ in the fundamental $(i = 1, \cdots, 2N_{f})$, gauge singlets $M_{ij}$ and a superpotential $$W = \frac{1}{4\lambda}M_{ij}q^{i}_{c}q^{j}_{d} J^{cd}.$$ where the scale $\lambda$ relates the scale $\Lambda_{N=1}$ of the electric theory and the scale $\tilde{\Lambda}_{N=1}$ of the magnetic theory by: $$\Lambda_{N=1}^{3(N_{c}+1)-N_{f}}\tilde{\Lambda}_{N=1}^{3(N_{f}-N_{c}-1)-N_{f}}
= C (-1)^{N_{f}-N_{c}-1} \lambda^{N_{f}}$$ where the constant $C$ was found in [@ip] to be $C=16$. The effective superpotential is given as: W\_[eff]{} = W + W\_[ADS]{} + W. If the vevs for the magnetic quarks are $0$, then the analysis is identical to those for the case $N_{f} < N_{c}$. If the vevs are not zero, then as in [@hoo] we can take a limit to approach $\mbox{Pf}M=0$ and to use the corresponding formula in order to compare with the M theory approach. In the $SU(N_c)$ case, where baryon exist, a specific choice has been taken such that the baryons have a specific interpretation in the M theory picture.
$N=2$ Higgs Branch from M Theory
=================================
In this section we study the moduli space of vacua of $N=2$ supersymmetric QCD by analyzing M theory fivebranes. We will consider the Higgs branch in terms of geometrical picture. Let us first describe the Higgs branch in the type IIA brane configuration.
Following the paper of [@egk], the brane configuration contains three kind of branes: the two parallel NS5 branes extend in the direction $(x^0, x^1, x^2, x^3, x^4, x^5)$, the D4 branes are stretched between two NS5 branes and extend over $(x^0, x^1, x^2, x^3)$ and are finite in the direction of $x^6$, and the D6 branes extend in the direction of $(x^0, x^1, x^2, x^3, x^7, x^8, x^9)$. In order to study symplectic or orthogonal gauge groups, we will consider an O4 orientifold which is parallel to the D4 branes in order to keep the supersymmetry and is not of finite extent in $x^6$ direction. The D4 branes is the only brane which is not intersected by this O4 orientifold. The orientifold gives a spacetime reflection as $(x^4, x^5, x^7, x^8, x^9) \rightarrow
(-x^4, -x^5, -x^7, -x^8, -x^9)$, in addition to the gauging of worldsheet parity $\Omega$. The fixed points of the spacetime symmetry define this O4 planes. Each object which does not lie at the fixed points ( i.e. over the orientifold plane), must have its mirror image. Thus NS5 branes have a mirror in $(x^4,
x^5)$ directions and D6 branes have a mirror in $(x^7, x^8, x^9)$ directions. Another important aspect of the orientifold is its charge, given by the charge of $H^{(6)}=d A^{(5)}$ coming from Ramond Ramond(RR) sector, which is related to the sign of $\Omega^2$. In the natural normalization, where the D4 brane carries one unit of this charge, the charge of the O4 plane is $
\pm 1$, for $\Omega^2= \mp 1$ in the D4 brane sector.
With the above preliminary setup, let us discuss about the two different branches of the theory. The Coulomb branch can be described when all the D4 branes lie between NS5 branes where no squark has vevs. To go to the Higgs branch, the D4 branes are broken on the D6 branes and are suspended between D6 branes being allowed to move on the $(x^{7}, x^{8}, x^{9})$ directions. Together with the gauge field component $A_{6}$ in the $x^{6}$ coordinate this gives two complex parameters to parameterize the location of the D4 branes. In [@hoo], for $SU(N_{c})$ case, the Coulomb branch and the Higgs branch share common directions and this comes from the fact that there are two different eigenvalues for $M$ which correspond to $r$ equal eigenvalues and $N_{c}-r$ equal eigenvalues. By turning on vevs for $r$ squarks, this gives rise to make the $r$ dimensional block of $M$ be nonzero. In brane language, this describes breaking $r$ D4 branes on the D6 branes and suspending the remaining $N_{c}-r$ D4 branes between the two NS5 branes.
In the case of $Sp(N_c)$ gauge theory, there are only two possibilities:
${\bullet}$ All D4 branes are suspended between the two NS5 branes where no squark has vevs.
${\bullet}$ Some of D4 branes are broken on D6 branes [^4].
We, for simplicity, restrict ourselves to the case of all D4 branes being broken on D6 branes. See [@extra; @extra1] for more general cases.
The motion of D4 branes along D6 branes describes the Higgs branch and for each D4 brane suspended between two D6 branes there exist two massless complex scalars parameterizing the fluctuations of the D4 brane. Because of the O4 orientifold we have to take into account D4 branes stretched between two D6 branes. The s-rule [@hw] allows only one D4 brane (and its mirror) between a NS5 brane and a single D6 brane (and its mirror). For $N=2$ theory because we have two NS5 branes, for both of them we have to impose the s-rule. Also, in contrast with $N=1$ theory, there are no complex scalars which correspond to D4 branes stretched between NS’5 branes and D6 branes. However remember that for $N=1$ theory[@egkrs] there are no complex scalar corresponding to a D4 brane stretched between NS5 brane and a D6 brane. The dimension of the Higgs moduli space is obtained by counting all possible breakings of D4 branes on D6 branes as follows: the first D4 brane is broken in $N_{f}-1$ sectors between the D6 branes (therefore the complex dimension is the twice of $N_f-1$), the second D4 branes is broken in $N_{f}-3$ sectors (the complex dimension is twice of $N_f-3$) and so on. But, besides that we have to consider the antisymmetric orientifold projection which eliminates some degrees of freedom, as explained in [@egkrs]. Then the dimension of the Higgs moduli space is given by: \[higgsdim\] 2\[(2N\_[f]{}-2-1) + (2N\_[f]{}-6-1) + + (2N\_[f]{}-4N\_[c]{}+2-1)\]= 4N\_[c]{}(N\_[f]{}-N\_[c]{})-2N\_[c]{} or $4N_{c}N_{f} - 2N_{c}(2N_{c}+1)$ where in the previous equations we have explicitly extracted 1 as a result of the antisymmetric orientifold projection.The overall factor $2$ in the left hand side is due to the mirror D6 branes and the result is very similar to the field theory result except an extra multiplicative factor 2 in the right hand side, because we consider here complex dimensions. In field theory, because of the $N_{f}$ vevs, the gauge symmetry is completely broken and there are $4N_{c}N_{f}-2N_{c}(2N_{c}+1)$ massless neutral hypermultiplets for a $N=2$ supersymmetric theory which thus exactly gives the dimension of the Higgs moduli space. Thus, the field theory results match the brane configuration results.
Let us discuss how the above brane configuration appears in M theory context in terms of a generically smooth single M fivebrane whose worldvolume is ${\bf R^{1,3}} \times \Sigma$ where $\Sigma$ is identified with Seiberg-Witten curves[@as] that determine the solutions to Coulomb branch of the field theory. As usual, we write $v=x^4+i x^5, s=(x^6+i x^{10})/R, t=e^{-s}$ where $x^{10}$ is the eleventh coordinate of M theory which is compactified on a circle of radius $R$. Then the curve $\Sigma$, describing $N=2$ $Sp(N_c)$ gauge theory with $N_f$ flavors, is given[@lll] by an equation in $(v, t)$ space \[ah0\] t\^2-(v\^2 B(v\^2, u\_k)+ \_[N=2]{}\^[2N\_c+2-N\_f ]{} \_[i=1]{}\^[N\_f]{} m\_i )t+ \_[N=2]{}\^[4N\_c+4-2N\_f]{} \_[i=1]{}\^[N\_f]{} (v\^2-[m\_i]{}\^2)=0 where $B(v^2)$ is a polynomial of $v^2$ of degree $N_c$ with the coefficients depending on the moduli $u_k$, $
v^{2N_c} + u_2 v^{2N_c -2} + \cdots + u_{2N_c}
$ and $m_i$ is the mass of quark[^5].
Including D6 Branes
--------------------
In M theory, the type IIA D6 branes are the magnetic dual of the electrically charged D0 branes, which are the Kaluza-Klein monopoles described by a Taub-NUT space. This is derived from a hyper-Kähler solution of the four-dimensional Einstein equation. But we will ignore the hyper-Kähler structure of this Taub-NUT space. Instead, we use one of the complex structures, which can be described by[@lll] \[ah1\] y z=\_[N=2]{}\^[4N\_c+4-2N\_f]{} \_[i=1]{}\^[N\_f]{} (v\^2-[m\_i]{}\^2) in $\bf C^3$. The D6 branes are located at $y=z=0, v=\pm m_i$. This surface, which represents a nontrivial $\bf S^1$ bundle over $\bf R^3$ instead of the flat four dimensional space ${\bf R^3} \times {\bf{S^1}}$ with coordinates $(x^4, x^5, x^6, x^{10})$, is the unfolding of the $A_{2n-1}$ ($n=N_f$) singularity in general. The Riemann surface $\Sigma$ is embedded as a curve in this curved surface and given by \[ah2\] y+z=v\^2 B(v\^2)+ \_[N=2]{}\^[2N\_c+2-N\_f ]{} \_[i=1]{}\^[N\_f]{} m\_i. which reproduces to eq. (\[ah0\]) as we identify $y$ with $t$. >From the symmetries existent in the type II A brane configuration, not all of them are preserved in the M-theory configuration. Our type IIA brane configuration has $U(1)_{4,5}$ and $SU(2)_{7,8,9}$ symmetries interpreted as classical $U(1)$ ans $SU(2)$ R-symmetry groups of the 4 dimensional theory on the brane worldvolume. The classical brane configuration is invariant both under the rotations. One of them, only $SU(2)_{7,8,9}$ is preserved in M theory quantum mechanical configuration but $U(1)_{4,5}$ is broken. This is a the same as saying that the $U(1)_{R}$ symmetry of the $N=2$ supersymmetric field theory is anomalous being broken by instantons. As discussed in section 2, the instanton factor is proportional with $\Lambda_{N=2}^{2N_{c}+2-N_{f}}$. So we have to see what is the charge of this factor under $U(1)_{4,5}$. We see this from equations (\[ah1\]) and (\[ah2\]) by considering $v$ of charge 2. We list below the charges of coordinates and parameter in the table:
$z$ $ y$ $ v$ $\Lambda_{N=2}^{2N_{c}+2-N_{f}}$
------------ ------------ ------ ----------------------------------
$4N_{c}+4$ $4N_{c}+4$ 2 $4N_{c}+4-2N_{f}$
In this case, the full $U(1)_{4,5}$ symmetry is restored, by assigning the instanton charge $(4N_{C}+4-2N_{f})$ to the $\Lambda$ factor.
Note that whenever some $m_i$ are the same, the smooth complex surface (\[ah1\]) develops A-type singularity. But this is misleading since the hyper-Kähler structure becomes singular only if the D6 branes have the same position in $x^6$ and not only in $v$. When D6 branes with the coincident $m_i$’s are separated in the $x^6$ direction, the singular surface (\[ah1\]) must be replaced by a smooth one which is the resolution of $A$-type singularity. We will briefly describe the resolution of the $A$-type singularity. On the resolved surface, we also describe the parity due to orientifolding.
Resolution of the $A$-type Singularity
--------------------------------------
When all bare masses are the same but not zero (say $ m=m_i$), the surface (\[ah1\]) $S$ develops singularities of type $A_{n-1}$ at two points $y=z=0, v=\pm m$. By succession of blowing ups, we obtain a smooth complex surface $\tilde{S}$ which isomorphically maps onto the singular surface $S$ except at the inverse image of the singular points. Over each singular point, there exist $n-1$ rational curves $\bf CP^1$’s on the smooth surface $\tilde{S}$. These rational curves are called the exceptional curves. Let us denote the exceptional curves over the point $y=z=0, v=m$ by $C_1, C_2,
\cdots , C_{n-1}$ and those over the point $y=z=0, v=-m$ by $C'_1, C'_2, \cdots , C'_{n-1}$. Here $C_i$’s (resp. $C'_i$) are arranged so that $C_i$ (resp. $C'_i$) intersects $C_j$ (resp. $C'_j$) only if $i= j\pm 1$. The symmetry due to orientifolding yields the correspondence between $C_i$ and $C'_i$.
When we turn off the bare mass, that is, $m_i =0$ for all $i$, the singularity gets worse. Two singular points on the surface $S$ collides to create the $A_{2n-1}$ singularity. Now there are ${2n-1}$ exceptional curves on the resolved surface, which may be considered as a union of two previous exceptional curves $C_i$ and $C'_i$ and a new rational curve, say $E$ which connects these two exceptional curves. The orientifold provides a reflection between $C_i$ and $C'_i$ while inducing a self-automorphism on $E$. The more precise picture of the resolved surface is as follows: It is covered by $2N_f$ complex planes $U_1, U_2, \cdots , U_{2n}$ with coordinates $(y_1 =y, z_1), (y_2, z_2) ,\cdots , (y_n, x_n =y)$ which are mapped to the singular surface $S$ by U\_i (y\_i, z\_i) {
[l]{} y= y\^i\_i z\^[i-1]{}\_i\
z = y\_i\^[2N\_f-i]{}z\_i\^[2N\_f+1-i]{}\
v= y\_iz\_i
. The planes $U_i$ are glued together by $z_iy_{i+1} =1$ and $y_iz_i =
y_{i+1}z_{i+1}$. The exceptional curve $C_i$ is defined by the locus of $y_i=0$ in $U_i$ and $z_{i+1}=0$ in $U_{i+1}$, the exceptional curve $E$ by $y_n=0$ in $U_n$ and $z_{n+1} =0$ in $U_{n+1}$ and the exceptional curve $C'_i$ by $y_{2n-i}=0$ in $U_{2n-i}$ and $z_{2n-i+1}=0$ in $U_{2n-i+1}$. The separation of the D6 branes in the $x^6$ direction corresponds to the infinitesimal direction on the singular surface $S$. Hence the position of the $D6$-brane may be interpreted as the $2N_f$ intersection points of the exceptional curves.
The Higgs Branch
----------------
In this section, all the bare masses are turned off. In M theory, the transition to the Higgs branch occurs when the fivebrane intersects with the $D6$-branes, which means that the curve $\Sigma$ given by (\[ah2\]) passes through the singular point $y=z=v=0$. As a special case, we will consider when all D4 branes are broken on $D6$ branes in type IIA picture. Write the right hand side of (\[ah2\]) as: \[C\] v\^2B(v\^2) = v\^2(v\^[2N\_c]{} + u\_2 v\^[2N\_c -2]{} + + u\_[2N\_c]{}). Then our case corresponds to $u_k =0$ for all $k$. To describe the Higgs branch, we will study how the curve y+z = v\^[2N\_c]{} looks like in the resolved $A_{2N_f -1}$ surface. Here we ignored the factor $v^2$ in the right hand side of (\[C\]) because it is always contained in the orientifold plane $O4$ and thus does not contribute to the Higgs branch. Away from the singular point $y=z=v=0$, we may regard the curve as embedded in the original $y-z-v$ space because there is no change in the resolved surface in this region. Near the singular point $y=z=v=0$, we have to consider the resolved surface. On the $i$-th patch $U_i$ of the resolved surface, the equation of the curve $\Sigma$ becomes y\^i\_i z\^[i-1]{}\_i + y\_i\^[2N\_f-i]{}z\_i\^[2N\_f+1-i]{} = y\_i\^[2N\_c]{}z\_i\^[2N\_c]{} Now we may factorize this equation according to the range of $i$: For $i=1,\ldots , 2N_c$, we have y\^i\_i z\^[i-1]{}\_i (1 + y\_i\^[2N\_f -2i]{}z\_i\^[2N\_f+ 2-2i]{} - y\_i\^[2N\_c-i]{} z\_i\^[2N\_c+1-i]{} ) =0 , for $ i=2N_c +1, \ldots , 2N_f -2N_c$, y\_i\^[2N\_c]{} z\_i\^[2N\_c]{} (y\_i\^[i-2N\_c ]{} z\_i\^[i-2N\_c-1]{} + y\_i\^[2N\_f -i -2N\_c]{}z\_i\^[2N\_f -i-2N\_c +1]{} - 1) = 0 , and for $ i=2 N_f- 2N_c +1,\ldots ,2N_f$, y\_i\^[2N\_f-i]{}z\_i\^[2N\_f+1-i]{} ( y\_i\^[2i-2N\_f]{} z\_i\^[2i-2 -2N\_f]{} + 1 - y\_i\^[2N\_c-2N\_f +i]{}z\_i\^[2N\_c-2N\_f +i -1]{}) =0. Thus the curve consists of several components. One component, which we call $C$, is the zero of the last factor of the above equations. This extends to the one in the region away from $y=z=v=0$ which we have already considered. The other components are the rational curves $C_1,
\ldots , C_{n-1}, E,
C'_1,\ldots , C'_{n-1}$ with some multiplicities. For convenience, we rename the exceptional curves $E_1, \ldots , E_{2n-1}$ so that $E_i$ is defined by $y_i=0$ on $U_i$ and $z_{i+1} =0$ on $U_{i+1}$. Hence we can see from the above factorization that the component $E_i$ has multiplicity $l_i = i$ for $i=1,\ldots , 2N_c $; $l_i =2N_c $ for $i=2N_c +1, \ldots , 2N_f -2N_c $; and $l_i = {2N_f-i}$ for $i=2 N_f- 2N_c +1,\ldots 2N_f-1$. Note that the component $C$ intersects with $E_{2N_c}$ and $E_{2N_f -2N_c}$.
To count the dimension of the Higgs branch, recall that once the curve degenerates and $\bf CP^1$ components are generated, they can move in the $x^7, x^8, x^9$ directions [@w1]. This motion together with the integration of the chiral two-forms on such $\bf CP^1$’s parameterizes the Higgs branch of the four-dimensional theory. However, we have to omit the component $E_{N_f} =E$ because this corresponds to the D4 brane connecting a D6 brane and its mirror. (Recall that $E$ was created after collision of two singular points which were mirror to each other.) Such a D4 brane is eliminated by the antisymmetric orientifold projection. Hence we have to put $l_{N_f} =0$. Now, after consideration of $\bf Z_2$ symmetry, the quaternionic dimension of the Higgs branch is \_[i=1]{}\^[2N\_f-1]{} l\_i= \_[i=1]{}\^[2N\_c]{} i + (2N\_f-4N\_c )N\_c= 2N\_c(N\_f-N\_c) -N\_c, which is the half of the complex dimension given in (\[higgsdim\]). Perhaps, a more appropriate geometric setting would have been a double covering of a $A_{N_f-1}$ singular surface with the embedded Seiberg-Witten curve. We will leave this for future investigation.
The Rotated Configuration
=========================
As we have seen that $N=2$ supersymmetry can be broken to $N=1$ by inserting a mass term of the adjoint chiral multiplet in field theory approach, we analyze the corresponding configuration in M theory fivebranes. While in the context of IIA picture, this turns out to be the rotation of one of NS5 branes, in order to describe this configuration, let us introduce a complex coordinate w=x\^8+i x\^9. Of course, the fivebranes are positioned at $w=0$ before the rotation. Notice that the D4 brane corresponds classically to an M fivebrane at $v=w=0$ and is extended in the direction of $s$, the NS5 brane is at $s=w=0$ and extended in $v$ and NS’5 brane is at $v=0, s=s_0$ and extended in $w$. Now we rotate only the left NS5 branes and from the behavior of two asymptotic regions which correspond to the two NS5 branes with $v
\rightarrow
\infty$ this rotation leads to the following boundary conditions. \[bdy-cond\] & & w v v , t \~v\^[ 2N\_c+2]{}\
& & w 0 v , t \~\_[N=2]{}\^[2(2N\_c+2-N\_f)]{}v\^[2N\_f-2N\_c-2]{} where the left(right) NS5 brane is related to the first(second) asymptotic boundary condition. Far from the origin of the $(v, w)$ plane which is the location of D4 branes, the location of the NS5 brane in the $s$ plane can be described by $s(NS5)=-2 R(N_f-N_c-1) \ln v$ while the NS’5 brane in the $s$ plane by $ s(NS'5)=-2 R(N_c+1) \ln w$. We discuss about the R-symmetries of the rotated configuration. After rotation, $SU(2)_{7,8,9}$ is broken to $U(1)_{8,9}$. In order this to be true, because of the connection between $v$ and $w$ in (\[bdy-cond\]), $\mu$ has to have charges under $U(1)_{4,5}\times U(1)_{8,9}$. $v$ has charge 2 under $U(1)_{4,5}$ while 0 under $U(1)_{8,9}$. $w$ has charge 0 under $U(1)_{4,5}$ and 2 under $U(1)_{8,9}$. So $\mu$ has $(-2,2)$ charges under $U(1)_{4,5} \times U(1)_{8,9}$. From the equations (\[ah1\]), (\[ah2\]) and (\[bdy-cond\]) we find the following values for the R-symmetry charges:
$v$ $w$ $y=t$ $z$ $\mu $ $\Lambda_{N=2}^{2N_{c}+2-N_{f}} $
-------------- ----- ----- -------------- -------------- -------- ------------------------------------
$U(1)_{4,5}$ 2 0 $4(N_{c}+1)$ $4(N_{c}+1)$ -2 $4(N_{c}+1)-2N_{f}$
$U(1)_{8,9}$ 0 2 0 0 2 0
Since the rotation is only possible at points in moduli space at which all 1-cycles on the curve $\Sigma$ are degenerate [@sw], the curve $\Sigma$ is rational, which means that the functions $v$ and $t$ can be expressed as a rational functions of $w$ after we identify $\Sigma$ with a complex plane $w$ with some deleted points. Because of the symmetry of $v \rightarrow -v, w \rightarrow -w$ due to orientifolding, we can write: v\^2=P(w\^2), t=Q(w\^2). Since $v$ and $t$ become infinity only if $w=0, \infty$, these rational functions are polynomials of $w$ up to a factor of some power of $w$: $P(w^2) = w^{2a}p(w^2), Q(w^2) =
w^{2b}q(w^2)$ where $a$ and $b$ are some integers and $p(w^2)$ and $q(w^2)$ are some polynomials of $w$ with only even degree terms which we may assume non-vanishing at $w=0$. Near one of the points at $w=\infty$, $v$ and $t$ behave as $v\sim \mu^{-1}w$ and $t \sim t^{2N_c +2}$ by (\[bdy-cond\]). Thus the rational functions are of the form P(w\^2) = w\^[2a]{}(w\^[2-2a]{} + )/\^[2]{} Q(w\^2) = \^[-2N\_c -2]{}w\^[2b]{}(w\^[2N\_c +2-2b]{} + ). Around a neighborhood $w=0$, the Riemann surface $\Sigma$ can be parameterized by $1/v$ which goes to zero as $w\to 0$. Since $w$ and $1/v$ are two coordinates around the neighborhood $w=0$ in the compactification of $\Sigma$ and vanish at the same point, they must be linearly related $w \sim 1/v$ in the limit $w \to 0$. The function $P(w^2)$ then takes the form P(w\^2) = . However the equation $v^2 = P(w^2)$ implies that $P(w^2)$ must be a square. Hence we have $w_+ = w_-$ and by letting $w_0^2 =w_{\pm}$ \[rot2\] P(w\^2)= which is a square of $w^2-w_0^2/\mu w$. Since $t\sim v^{2N_f -2N_c -2}$ and $w\sim 1/v$ as $w\to 0$, we get $b= N_c +1 -N_f$ and thus, Q(w\^2) = \^[-2N\_c -2]{}w\^[2(N\_c+1 -N\_f)]{}(w\^[2N\_f]{} + ). For $N_f >0$, by the equation $yz = v^{2N_f}$ defining the space-time, $t=0$ (i.e. $y=0$) implies $v=0$. Therefore the zeros of the polynomial $w^{2n_f} + \cdots$ are $\pm w_0$ of $P(w^2)$. Hence we have Q(w\^2)=\^[-2N\_c-2]{} w\^[2(N\_c+1-N\_f)]{} (w\^2-w\_0\^2)\^[N\_f]{} The value of $w_0$ can be determined by the fact that $v^2$ and $t$ satisfy the relation t+\_[N=2]{}\^[4N\_c+4-2N\_f]{} v\^[2N\_f]{}/ t=v\^2 B(v\^2)+ \_[N=2]{}\^[2N\_c+2-N\_f ]{} \_[i=1]{}\^[N\_f]{} m\_i Then by plugging $v^2$ and $t$ into the above equation we can read off $w_0$ from the lowest order term in power of $w$ && \^[-2N\_c-2]{} w\^[2(N\_c+1-N\_f)]{} (w\^2-w\_0\^2)\^[N\_f]{}+ \_[N=2]{}\^[4N\_c+4-2N\_f]{} v\^[2N\_f]{} \^[2N\_c2]{} w\^[-2(N\_c+1-N\_f)]{} (w\^2-w\_0\^2)\^[-N\_f]{}=\
&& - B(v\^2)- \_[N=2]{}\^[2N\_c+2-N\_f ]{} \_[i=1]{}\^[N\_f]{} m\_i
We want to calculate now $w_{0}$ from the above equation. For this, we will match the lowest order term in powers of $w$ in this equation. Actually we will look for terms with $w^{0}$ i.e., constant terms. In the left hand side the first term will always have a power of $w$, so does not contribute to the lowest order term. In the right hand side the last term will contain at least $w^{2(N_{c}-1)}$ so again does not contribute. After using the expression for $v$, the second term in left hand side will be $$\label{w1}
\Lambda_{N=2}^{4N_c+4-2N_f}(w^{2}-w_{0}^{2})^{N_{f}}
\mu^{2N_{c}+2-2N_{f}}w^{-2N_{c}-2}$$ In the right hand side, from the explicit form of $B(v^{2})$, the only term that can be independent of $w$ is obtained when we take only the highest power of $v$ which then will give $v^{2N_{c}}$. The contribution of this in the right hand side is as follows: $$\label{w2}
\frac{(w^{2}-w_{0}^{2})^{2N_{c}+2}}{\mu^{2N_{c}+2}}w^{-2N_{c}-2}$$ We now extract the lowest order from (\[w1\]) and (\[w2\]) and make them equal to obtain the relation for $w_{0}$ finally: $$\label{w3}
(-1)^{N_{f}} \Lambda_{N=2}^{4N_c+4-2N_f}
w_{0}^{2N_{f}}\mu^{4N_{c}+4-2N_{f}} = w_{0}^{4N_{c}+4}$$ This gives us the value for $w$ to be $$\label{w4}
w_{0} = (-1)^{\frac{N_{f}}{4N_{c}+4-2N_{f}}}(\mu\Lambda_{N=2}).$$ This gives us $w_{0}$ up to a $\bf{Z_{4N_{c}+4-2N_{f}}}$ rotation. We have here $4N_{c}+4-2N_{f}$ instead of $2N_{c}+2-N_{f}$ because of the symmetry $w\leftrightarrow -w$ implied by the orientifold. The rotated curve is now completely determined.
$N=1$ SQCD
==========
We study the $\mu\rightarrow\infty$ limit of our M fivebrane configuration and compare it with the known results in $N=1$ supersymmetric gauge theory. We have considered the rotation of the left NS5-brane, which corresponds to the asymptotic region $t\sim v^{2N_{c}+2}$ before the rotation and to $w\rightarrow\infty,
v\sim\mu^{-1} w$ and $t\sim\mu^{-2N_{c}-2}w^{2N_{c}+2}$ after the rotation. We expect the relation $t\sim v^{2N_{c}+2}$ to hold also in the $\mu\rightarrow \infty$ because the D4 branes still end on the left NS5-brane in this limit.
In order to preserve this relation, we should rescale $t$ by a factor $\mu^{2N_c+2}$ and introduce a new variable \[new\] = \^[2N\_[c]{}+2]{} t which will have the same dependence on $ v$ as $ t$ had before the rotation and this corresponds to the shift of the origin in the direction of $(x^6, x^{10})$. After putting $y =t$ in (\[ah1\]), the space-time is described by z=\^[2N\_[c]{}+2]{}\_[N=2]{}\^[4N\_[c]{}+4-2N\_[f]{}]{}\_[i=1]{}\^[N\_[f]{}]{} (v\^[2]{}-m\_[i]{}\^[2]{}), where $\tilde{y} =\mu^{2N_{c}+2} y$. This equation describes a smooth surface in the limit $\mu \rightarrow\infty$ provided the product \^[2N\_[c]{}+2]{}\^[4N\_[c]{}+4-2N\_[f]{}]{} remains to be finite. We define this product as follows: \_[N=1]{}\^[2(3N\_[c]{}+3-N\_[f]{})]{} = \^[2N\_[c]{}+2]{} \_[N=2]{}\^[4N\_[c]{}+4-2N\_[f]{}]{} which is nothing but the RG matching condition of the four-dimensional field theory. Note that this spacetime and M fivebrane is under the rotation groups $U(1)_{4,5}$ and $U(1)_{8,9}$ in appropriate way discussed before. We want to see what are the deformations of the $N=2$ Coulomb branch after the rotation and the limit $\mu\rightarrow\infty$.
Pure $Sp(N_{c})$ Theory
-----------------------
Without matter, the curve (\[ah0\]) describing the $N=2$ Coulomb branch is given by: t\^[2]{} - C\_[N\_[c]{}+1]{}(v\^[2]{},u\_[k]{})t + \^[4N\_[c]{}+4]{}\_[N=2]{} = 0 where $C_{N_{c}+1}(v^{2},u_{k}) = v^2B(v^2, u_{k})$. This curve is completely degenerate at $(N_{c} +1)$ points on the Coulomb branch. At one these points, the curve has the following form v\^[2]{} = \_[N=2]{}\^[4]{} t\^[-1/(N\_[c]{}+1)]{} + t\^[1/(N\_[c]{}+1)]{}. Thus its rotation is v\^[2]{} &=& \^[2]{} \_[N=2]{}\^[4]{} w\^[-2]{} + \^[-2]{} w\^[2]{}\
t &=& \^[-2N\_[c]{}-2]{}w\^[2N\_[c]{}+2]{}. Now we rescale $t$ as $\tilde{t} = \mu^{2N_{c}+2} t$ and send $\mu$ to $\infty$ by keeping $\Lambda_{N=1}$ finite. It is easy to see that the curve becomes in this limit: v\^[2]{} &=& \_[N=1]{}\^[6]{} \^[-1/(N\_[c]{}+1)]{}\
w\^[2]{} &=& \^[1/(N\_[c]{}+1)]{}. where the RG matching condition is used.
Introducing Massless Matter
---------------------------
For the rotated configuration we use now the expressions for $v^{2}=P(w^{2})$ and $t=Q(w^{2})$ given before in terms of new variables. We again introduce the rescaled $\tilde{t}$ which is then given by \[rot1\] = w\^[2(N\_[c]{} + 1 - N\_[f]{})]{} (w\^[2]{} - w\_[0]{}\^[2]{})\^[N\_[f]{}]{}. For $ v$ and $ w$ we have the relation by remembering that the order parameters $u_k$ are independent of $\mu$, are powers of $\Lambda_{N=2}$ and vanish in the $\mu \rightarrow \infty$ \[rot3\] = \_[N=1]{}\^[6N\_[c]{}+6-2N\_[f]{}]{}v\^[2(N\_[f]{}-N\_[c]{}-1)]{}. When $\mu\rightarrow\infty$, the limit for (\[rot1\]) and (\[rot2\]) is given by the behavior of $w_{0}\sim \mu\Lambda_{N=2}$. By using the relation: \[rel1\] \_[N=2]{} = (\_[N=1]{}\^[3N\_[c]{}+3-N\_[f]{}]{} \^[N\_[c]{}+1-N\_[f]{}]{})\^, we have three regions for $N_f$. Let us see how the curves look like:
$\bullet \;\;\; N_{f} < N_{c} + 1$
(\[rel1\]) tells us that $\mu\Lambda_{N=2}$ diverges and $w_{0}$ also diverges. Therefore the curve becomes infinite in the $x^{6}$ direction. So there is no field theory in four dimensions. This is just the same as saying that there is no supersymmetric vacua in the $N=1$ theory.
$\bullet \;\;\; N_c+1 < N_{f} < 2(N_{c} + 1)$
In this case, from (\[rel1\]) it is easy too see that $\mu\Lambda_{N=2} = 0$ in the limit $\mu\rightarrow\infty$ and (\[rot1\]), (\[rot2\]) and (\[rot3\]) transform into: &=& w\^[2(N\_[c]{}+1)]{}\
v w &=& 0\
v\^[2(N\_[c]{}+1)]{} &=& \^[6(N\_[c]{}+1) - 2N\_[f]{}]{} v\^[2N\_[f]{}]{}.As explained in [@hoo], only the limit $\tilde{t}\ne 0, w=0$ is allowed, so the interpretation of the previous equation is that the curve splits into two components in this limit: $C_{L} (\tilde{t} = w^{2(N_{c} + 1)},
v = 0)$ and $C_{R} (\tilde{t} = \Lambda_{N=1}^{6N_{c}+6-2N_{f}} v^{2N_{f}-2N_{c}-2},
w=0)$ where the component $C_{L}$ corresponds to the NS’5 brane which was rotated and on the other hand, $C_{R}$ refers to the NS5 brane and the attached D4-branes.
${\bullet} \;\;\; N_{f} = N_{c} + 1$
In this case, the RG matching condition tells that $\mu\Lambda_{N=2}$ is equal to $\Lambda_{N=1}^{2}$. The equations (\[rot3\]), (\[rot1\]) and (\[rot2\]) become: &=& \_[N=1]{}\^[4(N\_[c]{}+1)]{}\
&=& (w\^[2]{}-w\_[0]{}\^[2]{})\^[N\_[c]{}+1]{}\
vw &=& 0. The correct interpretation of these equations is that the curve also splits into two components: $C_{L} (\tilde{t} = (w^{2} - w_{0}^{2})^{N_{c} + 1}, v = 0)$ and $C_{R} (\tilde{t} = \Lambda_{N=1}^{4(N_{c}+1)}, w = 0)$.
For $SU(N_c)$ group the cases $N_{f} = N_{c} + 1$ and $N_{f} > N_{c} + 1$ differed from each other because the first different non-baryonic branch roots went to different limits and the second all non-baryonic branch roots have the same limit. That was determined by the fact that $M$, the meson matrix, had 2 different values for the diagonal entries. However in our case, for the $Sp(N_c)$ group, there is only one kind of top-right diagonal entry, so we do not see those difference appeared in $SU(N_c)$ gauge group and also do not have any baryonic branch.
Massive Matter
--------------
Before starting our discussion of introducing matter for our case, let us briefly examine the difference between the results of [@hoo] and [@biksy; @ss] for the case of massive matter when we consider $SU(N_c)$ gauge group. Actually we will just compare the equation (5.28) of [@hoo], and (4.6) and (5.1) of [@ss]. Notice that the second equation in (4.6) of [@ss] and the first one in (5.28) of [@hoo] are the same: v w = (m\_[f]{}\^[N\_[f]{}]{}\_[N=1]{}\^[3N\_[c]{}-N\_[f]{}]{})\^[1/N\_[c]{}]{}. The first equation of (4.6) in [@ss] is not the same as the second one of (5.28) in [@hoo]. Rather the equation of (5.28) looks like the equation (5.1) of [@ss] because there we have the dependence $t-w$. Recall that the vev for $M$ are given for equal squark masses by: m = m\_[f]{}\^\_[N=1]{}\^[ ]{}. The relation between $\tilde{t}$ and $w$ can be rewritten as: \[equi1\] w\^[N\_[f]{} - N\_[c]{}]{} = (w - m)\^[N\_[f]{}]{}. When we write it in terms of $t$ and $v$, this turns out: \[equi2\] v\^[N\_[c]{}]{} = (-1)\^[N\_f]{} \_[N=1]{}\^[3N\_[c]{} - N\_[f]{}]{} (v - m\_[f]{})\^[N\_[f]{}]{} The relations (\[equi1\]) and (\[equi2\]) are just the equivalent of (4.6) and (5.1) in [@ss]. In (\[equi2\]) we have a supplementary power of $\Lambda$ as compared with [@ss]. This is due to the fact that in [@hoo], $t$ has a dimension of mass by its definition, but in [@ss] $t$ is dimensionless. The power of $\Lambda$ is just used to match the dimension of mass. We then find that the same curve can be written in terms of $ t-w$ and $t-v$. The two descriptions correspond to the $N=1$ duality [@se1], between theories with gauge groups $SU(N_{c})$ and $SU(N_{f}-N_{c})$, as we can see from the dependence of $t$ as a function of $v$ and $w$ in (\[equi1\]) and (\[equi2\]). So the electric-magnetic duality can be observed also from the set-up of [@hoo]. The connection between [@hoo] and [@ss] may become clear only after we should introduce D6 branes in the set-up of [@ss] which is a very interesting direction to pursue and investigate.
What happens when we consider our case? If all the quarks have equal mass $m_{f}$, then the curve for $\mu\rightarrow\infty$ becomes: \[s0\] v\^[2]{}w\^[2]{} &=& (m\_[f]{}\^[N\_[f]{}]{}\_[N=1]{}\^[3(N\_[c]{}+1)-N\_[f]{}]{})\^[2/(N\_[c]{}+1)]{}\
&=& w\^[2(N\_[c]{}+1-N\_[f]{})]{}(w\^[2]{}-()\^[2/(N\_[c]{}+1)]{})\^[N\_[f]{}]{}. By the relation which connects the vev of $M$ and the masses of quarks, we obtain: w\^[2(N\_[f]{}-N\_[c]{}-1)]{} = (w\^[2]{}-m\^[2]{})\^[N\_[f]{}]{}. \[s1\] When we write it in terms of $ t$ and $v$, this gives rise to: v\^[2N\_[c]{}+2]{} = (-1)\^[N\_f]{} \_[N=1]{}\^[6(N\_[c]{}+1)-2N\_[f]{}]{} (v\^[2]{}- 2\^ [m\_f]{}\^[2]{})\^[N\_[f]{}]{}. \[s2\] >From the relations (\[s1\]) and (\[s2\]) we again see the duality between the theories with gauge groups $Sp(N_{c})$ and $Sp(N_{f}-N_{c}-2)$. The above relations are similar to the equations (2) and (6) of [@cs]. The equation between $ v$ and $w$ is just the same as those in [@cs]. The electric-magnetic duality appears as an interchange $v-w$, the curve describing the M-theory configuration being unique. Again, in [@cs] $t$ is dimensionless while in our case $\tilde{t}$ has a specific dimension. Therefore we see a power of $\Lambda$ in (\[s2\]). So (\[s1\]) and (\[s2\]) give the electric-magnetic duality in our approach. It would be very interesting to introduce D6 branes in [@cs] and to see the coincidence of the corresponding solution with ours.
Now take the limit $m_{f}\rightarrow\infty$. After we integrate out the massive flavors and use the matching of the running coupling constant \_[N=1]{}\^[3(N\_[c]{}+1)]{}=m\_[f]{}\^[N\_[f]{}]{} \_[N=1]{}\^[3(N\_[c]{}+1)-N\_[f]{}]{} to rewrite the equation (\[s0\]) as v\^[2]{}w\^[2]{} &=& \^[6]{}\_[N=1]{}\
&=& w\^[2(N\_[c]{}+1-N\_[f]{})]{} ( w\^[2]{} - 2\^ )\^[N\_[f]{}]{}. If we keep $\tilde{\Lambda}_{N=1}$ finite while sending $m_{f}$ to $\infty$, this again reduces to the pure Yang-Mills result.
Conclusions
===========
In the present work we considered the M theory description of the supersymmetry breaking from $N=2$ to $N=1$ for the case of symplectic gauge group $Sp(N_c)$ obtaining many aspects of the strong coupling phenomena. In this case no baryon can be constructed so the only Higgs branch is the non-baryonic branch. In field theory approach, starting with a $N=2$ supersymmetric gauge theory and giving mass to the adjoint chiral multiplet, the extremum of the superpotential gave us a unique solution for the expectation value for the meson matrix $M$ in which $\mbox{Tr}M=0$ as opossed to the $SU(N_{c})$ case where $\mbox{M}\ne 0$.
In the M theory fivebrane approach, we discussed first the unrotated configuration which corresponds to an $N=2$ theory, with or without D6 branes. For the case without D6 branes, M theory fivebrane configuration is a single fivebrane with the world volume ${\bf R^{1,3}} \times \Sigma$ where $\Sigma$ is the Seiberg-Witten curve of the gauge group $Sp(N_c)$. By introducing D6 branes, we have considered the complex structure of the corresponding Taub-NUT space which is the same one as those of the ALE space of $A_{2n-1}$-type and resolved the $A_{2n-1}$ singularity.
One of the most important aspects of the $Sp(N_{c})$ gauge theory was the O4 orientifold which is parallel to D4 branes . Its antisymmetric projection which eliminates some degrees of freedom was essential in matching the dimension of the Higgs moduli space in IIA brane approach with the one of the field theory. This observation was used not only in type IIA picture when counting the number of D4 branes suspended between the D6 branes but also in the M theory picture when counting the multiplicities of the rational curves. It is known [@hov; @vafa] that A type singularity by imposing the ${\bf Z_2}$ symmetry, due to the orientifolding, leads to D type singularity. We expect that our M theory fivebrane argument starting from D type singularity can go similarly and will see how the interrelation between two types singularities plays a role.
For rotated branes we used the coordinates $v=x^{4}+ix^{5}$ and $w=x^{8}+ix^{9}$, the position of the D4 branes in the $w$ direction being identified with the eigenvalues of the meson matrix $M$. We found at most one eigenvalue $w_{0}$ for the asymptotic position of the D4 branes which is consistent with the field theory result where only one eigenvalue for $M$ was found. We connected $w_{0}$ with the unique eigenvalue of $M$. In section 5 we have obtained the forms for the rotated curves, for pure gauge group and for massive and massless matter.
In all of our discussions as well as in many exciting works which appeared recently, many results obtained in field theory were rederived in M theory which makes M theory approach an extraordinary laboratory to derive results which were very difficult to obtain only by pure field theory methods. Now we have a clearer view over the strongly coupled phenomena of supersymmetric theories. But we still need to obtain new information like how to introduce flavors in the spinor representation and how to obtain dualities for $N=1$ theories with gauge groups like as $SO\times SO$ or $SO\times SU$, for which M theory approach did not give yet any supplementary information compared with type IIA approach. We hope that this information will be reached in the near future.
[99]{} A. Hanany and E. Witten, Nucl. Phys. [**B492**]{} (1997) 152, hep-th/9611230. J. de Boer, K. Hori, H. Ooguri, Y. Oz, Z. Yin, Nucl. Phys. [**493**]{} (1997) 101, hep-th/9612131. J. de Boer, K. Hori, Y. Oz, Z. Yin, hep-th/9702154. J.L.F. Barbon, Phys. Lett. [**B402**]{} (1997) 59, hep-th/9703051. S. Elitzur, A. Giveon and D. Kutasov, Phys. Lett. [**B400**]{} (1997) 269 hep-th/9702014. S. Elitzur, A. Giveon, D. Kutasov, E. Rabinovici and A. Schwimmer, hep-th/9704104. N. Seiberg, Phys. Lett. [**B435**]{} (1995) 129, hep-th/9411149. N. Evans, C. V. Johnson and A. D. Shapere, hep-th/9703210. J. H. Brodie and A. Hanany, hep-th/9704043. A. Brandhuber, J. Sonnenschein, S. Theisen and S. Yankielowicz, hep-th/9704044. R. Tatar, hep-th/9704198. H. Ooguri and C. Vafa, hep-th/9702180. C. Ahn and K. Oh, hep-th/9704061, to appear in Phys. Lett. [**B**]{}. C. Ahn, hep-th/9705004. C. Ahn and R. Tatar, hep-th/9705106. C. Ahn, K. Oh and R. Tatar, hep-th/9707027. E. Witten, hep-th/9703166. N. Seiberg and E. Witten, Nucl. Phys. [**B426**]{} (1994) 19. K. Landsteiner, E. Lopez and D. Lowe, hep-th/9705199. A. Brandhuber, J. Sonnenschein, S. Theisen and S. Yankielowicz, hep-th/9705232. O. Aharony and A. Hahany, hep-th/9704170. I. Brunner and A. Karch, hep-th/9705022. B. Kol, hep-th/9705031. C. V. Johnson, hep-th/9705148. K. Hori, H. Ooguri and C. Vafa, hep-th/9705220. A. Hanany and A. Zaffaroni, hep-th/9706047. A. Marshakov, M. Martellini and A. Morozov, hep-th/9706050. A. Fayyazuddin and M. Spalinski, hep-th/9706087. E. Witten, hep-th/9706109. A. Brandhuber, N. Itzhaki, V. Kaplunovsky, J. Sonnenschein and S. Yankielowicz, hep-th/9706127. C. Gomez, hep-th/9706131. C. V. Johnson, hep-th/9706155. A. Hanany and K. Hori, hep-th/9707192. A. Hanany, M. J. Strassler and A. Zaffaroni, hep-th/9707244. S. Nam, K. Oh and S.-J. Sin, hep-th/9707247. M. Henningson and P. Yi, hep-th/9707251. T. Nakatsu, K. Ohta, T. Yokono and Y. Yoshida, hep-th/9707258. M. Schmaltz and R. Sundrum, hep-th/9708015. J. de Boer and Y. Oz, hep-th/9708044. A. Mikhailov, hep-th/9708068. K. Hori, H. Ooguri and Y. Oz, hep-th/9706082. I. Affleck, M. Dine and N. Seiberg, Nucl. Phys. [**B241**]{} (1984) 483. K. Intriligator and P. Pouliot, Phys. Lett. [**B353**]{} (1995) 471, hep-th/9505006. P. C. Argyres, M. R. Plesser and A. D. Shapere, Nucl. Phys. [**B483**]{} (1997) 172, hep-th/9608129. T. Hirayama, N. Maekawa and S. Sugimoto, hep-th/9705069. C. Csaki and W. Skiba, hep-th/9708082. J. de Boer, K. Hori, H. Ooguri and Y. Oz, hep-th/9801060. S. Terashima, hep-th/9712172. P.C. Argyres and A.D. Shapere, Nucl. Phys. [**B461**]{} (1996) 437, hep-th/9509175. C. Vafa, hep-th/9707131.
[^1]: [email protected]
[^2]: [email protected]
[^3]: [email protected]
[^4]: There are recent papers on this issue [@extra; @extra1].
[^5]: Note that this $m_i$ is nothing to do with the element of meson field $M$. Unfortunately we used same notation.
| ArXiv |
---
abstract: |
A new kinematic and dynamic study of the halo of the giant elliptical galaxy NGC 5128 is presented. From a spectroscopically confirmed sample of $340$ globular clusters and $780$ planetary nebulae, the rotation amplitude, rotation axis, velocity dispersion, and total dynamical mass are determined for the halo of NGC 5128. The globular cluster kinematics were searched for both radial dependence and metallicity dependence by subdividing the globular cluster sample into 158 metal-rich (\[Fe/H\]$ > -1.0$) and 178 metal-poor (\[Fe/H\]$ <
-1.0$) globular clusters. Our results show that the kinematics of the metal-rich and metal-poor subpopulations are quite similar: over a projected radius of $0-50$ kpc, the mean rotation amplitudes are $47\pm15$ and $31\pm14$ km s$^{-1}$ for the metal-rich and metal-poor populations, respectively. There is a indication within $0-5$ kpc that the metal-poor clusters have a lower rotation signal than in the outer regions of the galaxy. The rotation axis shows an interesting twist at 5 kpc, agreeing with the zero-velocity curve presented by Peng and coworkers. Within 5 kpc, both metal-rich and metal-poor populations have a rotation axis nearly parallel to the north-south direction, which is $0^o$, while beyond 5 kpc the rotation axis twists $\sim180 ^o$. The velocity dispersion displays a steady increase with galactocentric radius for both metallicity populations, with means of $111\pm6$ and $117\pm6$ km s$^{-1}$ within a projected radius of 15 kpc for the metal-rich and metal-poor populations; however, the outermost regions suffer from low number statistics and spatial biases. The planetary nebula kinematics are slightly different. Out to a projected radius of $90$ kpc from the center of NGC 5128, the planetary nebulae have a higher rotation amplitude of $76\pm6$ km s$^{-1}$, and a rotation axis of $170\pm5 ^o$ east of north, with no significant radial deviation in either determined quantity. The velocity dispersion decreases with galactocentric distance. The total mass of NGC 5128 is found using the tracer mass estimator, described by Evans et al., to determine the mass supported by internal random motions and the spherical component of the Jeans equation to determine the mass supported by rotation. We find a total mass of $1.0\pm0.2 \times 10^{12}$ $M_{\odot}$ from the planetary nebula data extending to a projected radius of 90 kpc. The similar kinematics of the metal-rich and metal-poor globular clusters allow us to combine the two subpopulations to determine an independent estimate of the total mass, giving $1.3\pm0.5 \times 10^{12}$ $M_{\odot}$ out to a projected radius of 50 kpc. Lastly, we publish a new and homogeneous catalog of known globular clusters in NGC 5128. This catalog combines all previous definitive cluster identifications from radial velocity studies and $HST$ imaging studies, as well as 80 new globular clusters with radial velocities from a study of M.A. Beasley et al. (in preparation).
author:
- 'Kristin A. Woodley'
- 'William E. Harris'
- 'Michael A. Beasley'
- 'Eric W. Peng'
- 'Terry J. Bridges'
- 'Duncan A. Forbes'
- 'Gretchen L. H. Harris'
title: The Kinematics and Dynamics of the Globular Clusters and Planetary Nebulae of NGC 5128
---
Introduction {#sec:intro}
============
Globular clusters (GCs), as single-age, single-metallicity objects, are excellent tracers of the formation history of their host galaxies, through their dynamics, kinematics, metallicities, and ages. For most galaxies within $\simeq 20$ Mpc, GCs can be identified through photometry and image morphology, from which follow-up radial velocity studies can be carried out with multi-object spectroscopy on 4 and 8 m class telescopes. The ability to target hundreds of objects in a single field has vastly increased the observed samples of GCs confirmed in many galaxies, providing the necessary basis for detailed kinematic and age studies.
Another benefit of using GCs as a kinematic tracer is that they provide a useful, independent basis for comparison with results from planetary nebulae (PNe). This is particularly true given the current debate surrounding the use of PN velocities and the implications for low dark matter halos. For example, [@romanowsky03] reported, based on PN velocities, that three low-luminosity ellipticals revealed declining velocity dispersion profiles and little or no dark matter. However, subsequent simulations of merger-remnant ellipticals suggested that the radial anisotropy of intermediate-age PNe could give rise to the observed profiles within standard halos of dark matter [@dekel05; @mamon05]. Flattening of the galaxy along the line of sight is another possible explanation. One of the ellipticals studied by [@romanowsky03], NGC 3379, has also been investigated using GC kinematics [@pierce06; @bergond06]; both studies found evidence of a dark matter halo. But the two studies of NGC 3379 suffered from small number statistics. Clearly there is a need to directly compare PN and GC kinematics for the same elliptical with sufficiently large numbers of tracer objects.
Previous velocity-based studies of globular cluster systems (GCSs) have shown an intriguing variety of results in their overall kinematics. Prominent recent examples include the following:
1\. [@cote01] performed a kinematic analysis of the GCS in M87 (NGC 4486), the cD galaxy in the Virgo Cluster. With a sample of $280$ GCs, they showed that the entire GCS rotates on an axis matching the photometric [*minor*]{} axis of the galaxy, except for the inner metal-poor sample. Inside the onset of the cD envelope, the metal-poor clusters appear to rotate around the [*major*]{} axis of the galaxy instead. This study also found evidence for an increase in velocity dispersion, $\sigma_v$, with radius, due to the larger scale Virgo Cluster mass distribution. They also showed no strong evidence for a difference in $\sigma_v$ between the metal-poor and metal-rich groups. Using a Virgo mass model, they investigated anisotropy and found that as a whole, the GCS had isotropy, but considered separately, the metal-poor and metal-rich subpopulations had slight anisotropy.
2\. [@cote03] performed a dynamical analysis for M49 (NGC 4472), the other supergiant member of the Virgo Cluster. Using over $260$ GCs, they found that the metal-rich population shows no strong evidence for rotation, while the metal-poor population does rotate about the minor axis of the galaxy. In addition, they found that the metal-poor clusters had an overall higher dispersion than the metal-rich population.
3\. [@richtler04] determined the kinematics for the GCS in NGC 1399, the brightest elliptical in the Fornax Cluster of galaxies. Armed with a sample of over $460$ GCs, they found a marginal rotation signal for the entire GC sample and the outer metal-poor sample, while no rotation was seen for the metal-rich subpopulation. Their projected velocity dispersion showed no radial trend within their determined uncertainty, but the metal-poor clusters had a higher dispersion than the metal-rich clusters. The kinematics of the PNe in NGC 1399 have been most recently studied by [@saglia00] and [@napolitano02] using a small sample of 37 PNe from [@arnaboldi94]. These studies do not indicate any significant deviations in velocity dispersion with radius for the PNe, yet interestingly, [@napolitano02] found stong rotation for the PNe in the inner region of the galaxy.
4\. [@pff04II] studied the GC kinematics of the giant elliptical NGC 5128 from a total of $215$ GCs. Their results showed definite rotation signals in both metallicity groups beyond a 5 kpc distance from the center of the galaxy, as well as similar velocity dispersions in both the metal-poor and the metal-rich populations within a 20 kpc projected radius. A similar PN kinematic study in NGC 5128 [@peng04] showed that the PNe population is rotating around a twisted axis that turns just beyond the 5 kpc distance.
These few detailed kinematic studies of GCSs in elliptical galaxies, show results that appear to differ on a galaxy-by-galaxy basis, without any clear global trends. The age distributions of the GC populations in these same galaxies tend to show a consistent pattern in which the blue, or metal-poor, population is found to be universally old. The red, or metal-rich, population has also been shown to be old in the study of [@strader05]. Their study found that [*both*]{} the metal-poor and metal-rich GCs in a sample of eight galaxies ranging from dwarf to massive ellipticals, all have ages as old as their Galactic GC counterparts. Conversely, in a small sample of studies, the red GC population has also been found to be 2-4 Gyr younger than the metal-poor GC population, and with a wider spread in determined ages [see the studies of @pff04II; @puzia05 among others], although this is not yet a well-established trend. The GCs in NGC 5128 appear to be old, with an intermediate-age population [@pff04II; @beasley06].
NGC 5128 (Centaurus A), the central giant in the Centaurus group of galaxies at a distance of only $\sim4$ Mpc, is a prime candidate for both kinematic and age studies. Its GCS has a specific frequency of $S_N \simeq 2.2\pm0.6$ [@harris06], toward the low end of the giant elliptical range but about twice as high as in typical disk galaxies. Its optical features show faint isophotal shells located in the halo [e.g. @peng02], the prominent dust lane in the inner 5 kpc, the presence of gas, and star formation, all of which suggest that NGC 5128 could be a merger product. [@baade54] first suggested that NGC 5128 could be the result of a merger between two galaxies, a spiral and an elliptical. This idea was followed by the general formation mechanism of disk-disk mergers proposed by [@toomre72]. [@bhh03] found that the metallicity distribution function of the halo field stars could be reproduced by a gas-free (”dry”) disk-disk merger scenario. Recent numerical simulations by [@bekki06] also demonstrate that the PN kinematics observed in NGC 5128 [@peng04] can be reproduced relatively well from a merger of unequal-mass disk galaxies (with one galaxy half the mass of the other) colliding on a highly inclined orbital configuration.
Alternatively, much of NGC 5128 could be a ”red and dead” galaxy, passively evolving since its initial formation as a large seed galaxy [@w06], while undergoing later minor mergers and satellite accretions. Evidence consistent with this scenario is in the halo population of stars in NGC 5128, which have a mean age of $8^{+3.0}_{-3.5}$ Gyr [@rejkuba05]. Its metal-poor GC ages have also been shown to have ages similar to Milky Way GCs, while the metal-rich population appears younger [@pff04II]. Our new spectroscopic study [@beasley06] suggests that NGC 5128 has a trimodal distribution of cluster ages: $\sim50\%$ of metal-rich clusters have ages of $6-8$ Gyr, only a small handful of metal-rich clusters have ages of $1-3$ Gyr, and a large fraction of both metal-rich and metal-poor clusters have ages of $\simeq 12$ Gyr. Lastly, the halo kinematics of NGC 5128 have also been recently shown to match the surrounding satellite galaxies in the low-density Centaurus group, suggesting that NGC 5128 acts like an inner component to its galaxy group [@w06]. Kinematic and age studies with large number of GCs are thus starting to help disentangle the formation of this giant elliptical.
The confirmed GC population in NGC 5128 is now large enough to allow a new kinematic analysis subdivided to explore radial and metallicity dependence, while avoiding small number statistics in almost all regions of the galaxy. The analysis presented here complements the detailed age distribution study provided by [@beasley06]. The results provide a broader picture of the formation scenario of NGC 5128.
The sections of this paper are divided as follows: § \[sec:cat\] contains the full catalog of NGC 5128 GCs with known photometry and radial velocities, § \[sec:kin\] contains the kinematic analysis of GCSs, § \[sec:kin\_PN\] contains the kinematic analysis of the PN population, § \[sec:dyn\] contains the discussion of the dynamical mass of NGC 5128, and § \[sec:concl\] contains our final discussion, as well as concluding remarks.
The Catalog of Globular Clusters in NGC 5128 {#sec:cat}
============================================
Finding GCs in NGC 5128 is challenging. This process begins with photometric surveys of the many thousands of objects projected onto the NGC 5128 field [e.g. @rejkuba01; @hhg04II]; only a few percent of these are the GCs that we seek. This daunting task is made difficult by the galactic latitude of NGC 5128 ($b=19^o$), which means that many foreground stars are present in the field of NGC 5128. Background galaxies are another major contaminant in the field, forcing the use of search criteria such as magnitude, color, and object morphology to help build the necessary candidate list of GCs. Confirmation of such candidate objects can then be done with spectroscopic radial velocity measurements. The GCs in NGC 5128 have radial velocities in the range $v_r = 200-1000$ km s$^{-1}$, while most foreground stars have $v_r < 200$ km s$^{-1}$. Background galaxies have radial velocities of many thousands of km s$^{-1}$ and can easily be eliminated [see the recent studies of @pff04I; @whh05].
Over the past quarter century, there have been seven distinct radial velocity studies to identify GCs in NGC 5128 [@vhh81; @hhvh84; @hhh86; @hghh92; @pff04I; @whh05; @beasley06]. Although [@hghh92] was not a radial velocity study itself, but rather a CCD photometric study of previously confirmed GCs, it does include the GCs determined spectroscopically from [@sharples88]. A recent study with measured GC radial velocities published by [@rejkuba07] has confirmed two new GCs, HCH15 and R122, included in our catalog.
Within this catalog are 80 new GCs with radial velocity measurements from [@beasley06]. The combination of these studies now leads to a confirmed population of 342 GCs.[^1] Also included in our catalog are the new GC candidates from $HST$ STIS imaging from [@hhhm02], labelled C100-C106, and from $HST$ ACS imaging from [@harris06], labelled C111-C179. All these previous studies have their own internal numbering systems, which makes the cluster identifications somewhat confusing at this point. Here we define a new, homogeneous listing combining all this material and with a single numbering system.
Our catalog of the GCs of NGC 5128 is given in Table \[tab:cat\_GC\]. In successive columns, the Table gives the new cluster name in order of increasing right ascension; the previous names of the cluster in the literature; right ascension and declination (J2000); the projected radius from the center of NGC 5128 in arcminutes; the $U$, $B$, $V$, $R$, and $I$ photometric indices and their measured uncertainties; the $C$, $M$, and $T_1$ photometric indices and their uncertainties; the colors $U-B$, $B-V$, $V-R$, $V-I$, $M-T_1$, $C-M$, and $C-T_1$; and, lastly, the weighted mean velocity $v_r$ and its associated uncertainty from all previous studies. All $UBVRI$ photometry is from the imaging survey described in [@pff04I]. The $CMT_1$ data are from [@hhg04II].
The mean velocities are weighted averages with weights on each individual measurement equal to $\varepsilon_{v}^{-2}$ where $\varepsilon_{v}$ is the quoted velocity uncertainty from each study. The uncertainty in the mean velocity is then $<\varepsilon_v> = (\sum \varepsilon_{i}^{-2})^{-1/2}$. There are no individual uncertainties supplied for the velocities for clusters studied by [@hhh86], but their study reports that the mean velocity uncertainty for clusters with $R_{gc} < 11'$ is 25 km s$^{-1}$ and for $R_{gc} > 11'$ is 44 km s$^{-1}$. We have adopted these values accordingly for their clusters.[^2]
The study by [@hghh92] also does not report velocity uncertainties; however, these clusters have all been recently measured by [@pff04I]. The rms scatter of the [@hghh92] values from theirs was 58 km s$^{-1}$. This value has been adopted as the velocity uncertainty of the [@hghh92] clusters in the weighted means.
The weighted mean velocity of cluster C10 does not include the measured value determined by [@pff04I] which is significantly different from other measurements. Also, cluster C27 does not include the measurement of $v_r = 1932\pm203$ km s$^{-1}$ from [@beasley06], indicating that this object is a galaxy. We include C27 as a GC, but with caution.
In the weighted velocity calculations, the velocities and uncertainties of the 27 GCs from [@rejkuba07] have been rounded to the nearest whole number, with any velocity uncertainty below 1 km s$^{-1}$ rounded up to a value of 1.
Lastly, the GC pff\_gc-089 overlaps the previously existing confirmed cluster, C49, within a 0.5” radius; pff\_gc-089 is therefore removed from the catalog of confirmed GCs.
The data in Table \[tab:cat\_GC\] provide the basis for the kinematic study presented in this paper. We use them to derive the rotation amplitude, rotation axis and velocity dispersion in the full catalog of clusters, as well as for the metal-poor (\[Fe/H\] $< -1$) and metal-rich (\[Fe/H\] $> -1$) subpopulations. For this purpose, we define the metallicity of the GCs by transforming the dereddened colors $(C - T_1)_o$ to \[Fe/H\] through the standard conversion [@harris02], calibrated through Milky Way cluster data. A foreground reddening value of $E(B - V) =
0.11$ for NGC 5128, corresponding to $E(C - T_1) = 0.22$, has been adopted. The division of \[Fe/H\] = -1 between metal-rich and metal-poor GCs has been shown as a good split between the two metallicity populations from \[Fe/H\] values converted from $C - T_1$ in [@whh05] and [@hhg04II] for NGC 5128. If no $C$ and/or $T_1$ values are available for the cluster, it is classified as metal-rich or metal-poor through a transformation from $(U-B)_o$ to \[Fe/H\] from [@reed94].
In Figure \[fig:position\] we show the spatial distributions of all the GCs from Table \[tab:cat\_GC\] ([*left*]{}) and the distribution of the known PNe ([*right*]{}). Both systems are spatially biased to the major axis of the galaxy because this is where most of the GC and PN searches have concentrated.
Kinematics of the Globular Cluster System {#sec:kin}
=========================================
Velocity Field {#sec:velfield}
--------------
For the present discussion we adopt a distance of $3.9$ Mpc for NGC 5128. This value is based on four stellar standard candles that each have internal precisions near $\pm 0.2$ mag: the PN luminosity function, the tip of the old-halo red giant branch, the long-period variables, and the Cepheids [@hhp99; @rejkuba04; @ferr06].
HCH15 and R122 have not been included in our kinematic study, as our study was completed before publication of these velocities. The weighted velocities used in this kinematic study do not include the most recent 25 velocities of previously known GCs published in [@rejkuba07]. Note that the velocities published in Table \[tab:cat\_GC\] do, however, include the [@rejkuba07] velocities in the quoted final weighted radial velocities for completeness.
The velocity distribution of the entire sample of 340 is shown in Figure \[fig:gausfit\_all\] ([*top left*]{}), binned in 50 km s$^{-1}$ intervals. A fit with a single Gaussian yields a mean velocity of $546\pm7$ km s$^{-1}$, nicely matching the known systemic velocity of $541\pm7$ km s$^{-1}$ [@hui95]. There is a slight asymmetry at the low-velocity end that is likely due to contamination by a few metal-poor Milky Way halo stars (also seen in the metal-poor subpopulation in the bottom left panel, which has a mean velocity determined by the Gaussian fit as $532\pm13$ km s$^{-1}$).
Selecting the clusters with radial velocity uncertainties less than 50 km s$^{-1}$ leaves 226 clusters, plotted in Fig. \[fig:gausfit\_all\] ([*top right*]{}). The close fit to a single Gaussian is consistent with an isotropic distribution of orbits; the mean velocity is $554\pm5$ km s$^{-1}$. The metal-rich population, with a mean velocity determined by the Gaussian fit of $565\pm11$ km s$^{-1}$, is plotted in the bottom right panel, and also shows no strong asymmetries.
Looking closer at the metal-poor velocity asymmetry, we note that the 15 metal-poor clusters between 250 and 300 km s$^{-1}$ (in the region where contamination by Milky Way field stars could occur) are balanced by only two GCs at the high-velocity end on reflection across the systemic velocity. The same velocity regions in the metal-rich population are nearly equally balanced with four clusters between 250 and 300 km s$^{-1}$ with three clusters at the reflected high-velocity range. Interestingly, the four metal-rich clusters between 250 and 300 km s$^{-1}$ have projected radii $> 17$ kpc even though the metal-rich population is more centrally concentrated than the metal-poor [see @pff04II; @whh05 among others]. The metal-poor clusters between 250 and 300 km s$^{-1}$, conversely, are more evenly distributed, with five clusters between projected radii of 5 and 10 kpc, five clusters between 10 and 20 kpc, and five clusters beyond 20 kpc from the center of NGC 5128. Some of these low-velocity, metal-poor objects could be foreground stars with velocities in the realm of GCs in NGC 5128 ($v_r \gtrsim 250$ km s$^{-1}$). However, with only 340 GCs currently confirmed within $\sim45$’ from the center of NGC 5128, out of an estimated $\simeq
1500$ total clusters within 25’ [@harris06], these metal-poor, low-velocity objects could simply be part of a very incomplete GC sample that is also spatially biased. This potential bias is clearly shown in Figure \[fig:gc\_thetar\], which shows the projected radial distribution as a function of azimuthal angle for our GC sample. Beyond 12 kpc, the two ”voids” coincide with the photometric minor axis of the galaxy, attributed at least partly to incomplete cluster surveys in these regions. These objects should, therefore, not be dropped from the GC catalog without further spectroscopic analysis.
Rotation Amplitude, Rotation Axis, and Velocity Dispersion {#sec:kin_gc}
----------------------------------------------------------
### Mathematic and Analytic Description {#sec:math}
We determine the rotation amplitude and axis of the GCS of NGC 5128 from $$\label{eqn:kin}
v_r(\Theta) = v_{sys} + \Omega R sin(\Theta - \Theta_o)$$ [see @cote01; @richtler04; @w06]. In Equation \[eqn:kin\], $v_r$ is the observed radial velocity of the GCs in the system, $v_{sys}$ is the galaxy’s systemic velocity, $R$ is the projected radial distance of each GC from the center of the system assuming a distance of 3.9 Mpc to NGC 5128, and $\Theta$ is the projected azimuthal angle of the GC measured in degrees east of north. The systemic velocity of NGC 5128 is held constant at $v_{sys}=541$ km s$^{-1}$ [@hui95] for all kinematic calculations. The rotation axis of the GCs, $\Theta_o$, and the product $\Omega R$, the rotation amplitude of the GCs in the system, are the values obtained from the numerical solution. We use a Marquardt-Levenberg non-linear fitting routine [@press92].
Eqn. \[eqn:kin\] assumes spherical symmetry. While this may be a decent assumption for the inner 12 kpc region [it has a low ellipticity of $\sim0.2$; @peng04], true ellipticities for the outer regions of the system are not well known because of the sample bias (see Fig. \[fig:gc\_thetar\]). Future studies to remove these biases are vital to obtaining a sound kinematic solution for the entire system. Eqn. \[eqn:kin\] also assumes that $\Omega$ is only a function of the projected radius and that the rotation axis lies in the plane of the sky. It is not entirely clear how these assumptions, discussed thoroughly in [@cote01], apply to the GC and PN systems of NGC 5128. The $\Omega$ we solve for is, therefore, only a lower limit to the true $\Omega$ if the true rotation axis is not in the plane of the sky.
The projected velocity dispersion is also calculated from the normal condition, $$\label{eqn:veldisp}
\sigma_{v}^{2} = \sum_{i=1}^{N}\frac{(v_{f_i} - v_{sys})^2}{N}$$ where $N$ is the number of clusters in the sample, $v_{f_i}$ is the GC’s radial velocity [*after subtraction of the rotational component determined with Eqn. \[eqn:kin\]*]{}, and $\sigma_v$ is the projected velocity dispersion.
The GCs were assigned individual weights in the sums that combine in quadrature the individual observational uncertainty, $\varepsilon_v$, in $v_r$ and the random velocity component, $\varepsilon_{random}$, of the GCS. The dominance of the latter is evident by the large dispersion in the GC velocities in the kinematic fitting (see Figure \[fig:kin\_plot\]). In other words, the clusters have individual weights, $\omega_i = (\varepsilon_v^2 +
\varepsilon_{random}^2)^{-1}$; the main purpose of this is to assign a bit more importance to the clusters with more securely measured velocities. This random velocity term dominates in nearly every case, leaving the GCs with very similar base weights in the kinematic fitting.
The three kinematic parameters - rotation amplitude, rotation axis, and velocity dispersion - are determined with three different binning methods. The first involves binning the GCs in radially projected circular annuli from the center of NGC 5128. The chosen bins keep a minimum of 15 clusters in each, ranging as high as 124 clusters. The bins are 0-5, 5-10, 10-15, 15-25, and 25-50 kpc. Also, we include 0-50 kpc to determine the overall kinematics of the system.
The second method adopts bins with equal numbers of clusters. The entire population of clusters had nine bins of 38 clusters each, the metal-poor clusters had nine bins of 20 clusters, and the metal-rich clusters had eight bins of 20 clusters. The base weighting is applied to the clusters in both the first and second binning methods.
The third method uses an exponential weighting function, outlined in [@bergond06], to generate a smoothed profile. This method determines each kinematic parameter at the radial position, $R$, of every GC in the entire sample by exponentially weighting all other GCs surrounding that position based on their radial separation, $R - R_i$, following
$$\label{eqn:exp_wei}
w_i(R) = \frac{1}{\sigma_R} exp[\frac{-(R - R_i)^2}{2 \sigma_{R}^{2}}].$$
In Equation \[eqn:exp\_wei\], $w_i$ is the determined weight on each GC in the sample, and $\sigma_R$ is the half-width of the window size. For this study, $\sigma_R$ is incrementally varied in a linear fashion for the total sample from $\sigma_R
= 1.0$ kpc at the radius of the innermost GC in the sample out to $\sigma_R = 4.5$ kpc at the radius of the outermost GC, where the population is lowest. The metal-poor population was given a half-width window of $\sigma_R = 1.0-6.5$ kpc, and the metal-rich population was given a half-width window of $\sigma_R = 2-5.3$ kpc, again from the innermost to outermost cluster. The progressive radial increase in $\sigma_R$ ensured that each point $R$ had roughly equal total weights.
### Rotation Amplitude of the Globular Cluster System {#sec:rotamp}
The kinematic parameters were determined for the entire sample of 340 GCs, as well as the subpopulations of 178 metal-poor and 158 metal-rich GCs (four clusters have unknown metallicity). The kinematic results for the entire population of GCs are shown in Table \[tab:all\_GC\], reproduced almost in full from [@w06], while the results for the metal-poor and metal-rich clusters are shown in Tables \[tab:MP\_GC\] & \[tab:MR\_GC\], respectively. The columns give the radial bin, the mean projected radius in the bin, the radius of the outermost cluster, the number of clusters in the bin, the rotation amplitude, the rotation axis, and the velocity dispersion, with associated uncertainties. These are followed by the mass correction, the pressure-supported mass, the rotationally supported mass, and the total mass in units of solar mass (see § \[sec:dyn\] for the mass discussion). The results for the alternate two methods, using an equal number of GCs per bin and the exponentially weighted GCs, are not shown in tabular form but are included in all of the figures.
Figure \[fig:kin\_plot\] shows the sine fit of Eqn. \[eqn:kin\] for the total population and for the metal-poor and metal-rich subpopulations. All three populations show rotation about a similar axis. As discussed in § \[sec:velfield\], the metal-poor population has more members with low velocities ($V_r \leq 300$ km s$^{-1}$) than the metal-rich population, suggesting possible contamination of Milky Way foreground stars in the sample.
Figures \[fig:rotamp\_final\] & \[fig:rotamp\_metal\] show the rotation amplitude results for the entire population and for the metal-poor and metal-rich subpopulations, respectively. The three kinematic methods, described in Section \[sec:math\], appear to agree relatively well for all three populations of clusters. While there appears to be no extreme difference in rotation amplitude between the cluster populations, the metal-poor subpopulation of clusters has lower rotation in the inner 5 kpc of NGC 5128 than the metal-rich subpopulation. The weighted average of the 0-5 kpc radial bin and the innermost equal-numbered bin, shows that the entire population has a rotation amplitude of $\Omega R = 31\pm17$ km s$^{-1}$, while the metal-poor population has $\Omega R = 17\pm26$ km s$^{-1}$ and the metal-rich population has $\Omega R = 57\pm22$ km s$^{-1}$. [@pff04II] show in their study that the metal-poor population has very little rotation in the central regions, completely consistent with our findings. The rotation amplitude does not appear to differ between the two populations outside of 5 kpc.
### Rotation Axis of the Globular Cluster System {#sec:rotaxis}
The results of the rotation axis solutions are shown in Figures \[fig:rotaxis\_final\] & \[fig:rotaxis\_metal\], again for the entire population and for the metal-poor and metal-rich subpopulations. The solution for $\Theta_0$ agrees well for all three kinematic methods and all subgroups. The inner 5 kpc region has a different rotation axis than the outer regions, demonstrated clearly in all three binning methods. The innermost bin yields weighted averages of $\Theta_o = 369\pm24 ^o$, $\Theta_o
= 25\pm55 ^o$, and $\Theta_o = 352\pm18 ^o$, all of which are equal within their uncertainties. Beyond 5 kpc, the rotation axes for all three populations are in even closer agreement, with averages of $\Theta_o = 189\pm6 ^o$, $\Theta_o = 199\pm7 ^o$, and $\Theta_o = 196\pm7 ^o$ for the entire population, the metal-poor subpopulation, and the metal-rich subpopulation, respectively. The position angle of the photometric major axis of NGC 5128 is $\Theta = 35^o$ and $215^o$ east of north and the photometric minor axis is $\Theta = 119^o$ and $299^o$ east of north [@dufour79]. It appears the GCS is rotating about an axis similar to the photometric major axis for the full extent of the galaxy, with a possible axial twist or counterrotation within 5 kpc.
### Velocity Dispersion of the Globular Cluster System {#sec:veldisp}
Figures \[fig:veldisp\_final\] & \[fig:veldisp\_metal\] show the velocity dispersion for the entire population and for the metal-poor and metal-rich subpopulations. Our results for $\sigma_v$ show no significant differences between the metallicity subpopulations. All three show a relatively flat velocity dispersion ($\sigma_v = 119\pm4$, $\sigma_v = 117\pm6$, and $\sigma_v = 111\pm6$ km s$^{-1}$ within 15 kpc of the center of NGC 5128 for the entire population and for the metal-poor and metal-rich subpopulations, respectively). These results match the previous study of NGC 5128 by [@pff04II], whose determined velocity dispersion for the GCs within 20 kpc ranged between 75 and 150 km s$^{-1}$. At a larger radius, we find that $\sigma_v$ then slowly increases to $\sigma_v > 150$ km s$^{-1}$ towards the outer regions of the halo for all populations. The velocity dispersion of the metal-rich GCs, interestingly, appears higher than that of the metal-poor GCs in the outer regions (although still consistent within the determined uncertainties). In most previous studies, the velocity dispersion of the metal-poor GCs usually appears higher than that of the metal-rich GCs, if there is a notable velocity dispersion difference between the subpopulations [see the studies of @cote03; @richtler04 as examples].
To explore the cause of the distinct rise past 15 kpc a bit further, we have plotted the actual velocity histograms in Figure \[fig:vf\_histo\] for the metal-poor and metal-rich subgroups, subdivided further into inner ($R < 15$ kpc) and outer ($R > 15$ kpc) regions. In the inner 15 kpc, both samples show histograms strongly peaked near $v_f = 0$ and with at least roughly Gaussian falloff to both high and low velocities. By contrast, the histograms for the outer regions ($15-50$ kpc) are noticeably flatter, so that the clusters with larger velocity residuals have relatively more importance to the formal value of $\sigma_v$. Nominally, the flatter shape of the velocity distribution would mean that the outer-halo clusters display anisotropy in the direction of a bias towards more circular orbits. However, such a conclusion would be premature at this point for two reasons. First, the sample size in the outer regions is still too small to lead to high significance, and a direct comparison between the inner and outer histograms (through a Kolmogorov-Smirnov test) does not show a statistically significant difference between them larger than the 70% level. Second, the outer samples may still be spatially biased in favor of objects along the major axis of the halo, as discussed above, and this bias sets in strongly for $R > 12$ kpc (see Fig. \[fig:gc\_thetar\]), very near where we have set the radial divisions in this Figure. This type of velocity distribution can also arise from the accretions of satellite galaxies with their own small numbers of GCs [@bekki03]. We will need to have a larger sample of the outer-halo clusters, and one in which these potential sample biases have been removed, before we can draw any firmer conclusions. However, it needs to be explicitly stated that the outermost point in the kinematic plots for the GCs representing $25-50$ kpc suffers from very high spatial biases and low number statistics ($< 40$ GCs) and covers a large radial interval. The rise in velocity dispersion could be driven purely by systematic effects resulting from the radial gradient of the number density of GCs in this outermost bin [@napolitano01].
Kinematics of the Planetary Nebula System of NGC 5128 {#sec:kin_PN}
=====================================================
NGC 5128 has a large number, 780, of identified PNe with measured radial velocity from the studies of [@peng04] and [@hui95]; these PNe are projected out to 90 kpc assuming a distance to NGC 5128 of 3.9 Mpc. Since these are also old objects, it is of obvious interest to compare them with the GCS. The PNe also have the advantage of giving us the best available look at the kinematics of the halo field stars.
The PN kinematic results are listed in Table \[tab:PN\], with the same columns as Table \[tab:all\_GC\]. The results are shown in Figures \[fig:rotamp\_final\], \[fig:rotaxis\_final\], & \[fig:veldisp\_final\] for the rotation amplitude, rotation axis, and velocity dispersion, respectively. The spatial distribution of the known PNe is, like the GCS, biased toward the major axis at large radii (see Fig. \[fig:position\]). Nevertheless, their kinematics closely resemble the GCs.
The kinematics of the PN system are very consistent among all three binning methods. The rotation amplitude and rotation axis show little radial trend, while the velocity dispersion appears relatively flat within the first 15 kpc at $\sigma_v = 122\pm7$ km s$^{-1}$ and then slowly [*decreases*]{} to $\sigma_v \simeq 85$ km s$^{-1}$ at large galactocentric radius. [@pff04I] show that the velocity dispersion of the PNe drops from a central value of 140 to 75 km s$^{-1}$ in the outer regions of the galaxy, consistent with the findings of this study. Their velocity field analysis led to the discovery of a ”zero-velocity curve” located between the photometric minor axis, $119\pm5 ^o$ east of north [@dufour79], and the north-south direction, for the innermost region of the galaxy. Just beyond 5 kpc, the zero-velocity curve turns and follows a straight line at a $7^o$ angle from the photometric major axis, $35^o$ east of north [@dufour79] [see Figure 7 of @pff04II].
This study does not show a strong change in rotation axis for the PNe in the innermost regions of the galaxy. However, it clearly shows in all three GC populations a significant change in the rotation axis just beyond 5 kpc from the center of the galaxy. A change in axis of 5 kpc outward ($\sim180 ^o$ for the entire population and metal-poor subpopulation and $\sim160 ^o$ for the metal-rich subpopulation) has been found, as discussed in § \[sec:rotaxis\]. Similarly, [@pff04II] show from their sample of 215 clusters that a clear sign of rotation beyond 5 kpc about a misaligned axis appears particularly in their metal-rich subpopulation. The kinematics of the GCs in this study matches the line of zero velocity relatively nicely. Within 5 kpc the rotation axis of the GCS is nearly-parallel to the north-south direction, and beyond 5 kpc the rotation axis is near $200 ^o$ east of north, which is only $\sim10^o$ from the zero-velocity curve.
However, the velocity field of NGC 5128 is complex and not entirely captured by these approximate solutions. The two-dimensional velocity field shown in [@peng04] (see their Figure 7), shows that the photometric major axis (which happens to be very close to our maximum rotation as discussed above) is only $7^o$ from the line of zero velocity. This could lead to a very asymmetric velocity profile that may not be well fit by the sine curve described in Equation \[eqn:kin\]. Biased kinematics, especially the rotation axis, may develop from the sine fit that could lead to a higher estimated velocity dispersion.
[@hui95] similarly studied the kinematics of the PN system in NGC 5128 with a sample of 433 PNe. They obtain a rotation axis of $344\pm10^o$. Our result of $170\pm5^o$ east of north is consistent with their findings on comparing their sine curve fit of their PN data in their Figure 11 to our corresponding fit shown in Fig. \[fig:kin\_plot\] for the GCS, which shares a similar axis to our PN sample (note that in their study, $\phi = 0 ^o$ corresponds to our $\Theta = 305^o$ east of north). Our fits both correspond to a positive rotation amplitude for a rotation axis near $170^o$ east of north and a negative rotation axis near $350^o$ east of north. Therefore, the rotation axis quoted in [@hui95] of $344\pm10^o$ corresponds to a [*negative*]{} rotation amplitude of approximately $70-75$ km s$^{-1}$ (taken from their Figure 11), nicely matching our result of a positive $76\pm6$ km s$^{-1}$ about an axis of $170\pm5^o$ east of north.
Dynamics of NGC 5128 {#sec:dyn}
====================
Both GCs [@cote01; @larsen02; @evans03; @cote03; @beasley04; @pff04II among others] and PNe [@ciardullo93; @hui95; @arnaboldi98; @peng04 among others] can be used to estimate the total dynamical mass of their host galaxies. A variety of tools are in use including derived mass models, the virial mass estimator [@bahcall81], the projected mass estimator [@heisler85], and the tracer mass estimator [@evans03].
NGC 5128 does not have a large X-ray halo [detected by @kraft03; @osullivan01 the latter reporting a measurement of log $L_x = 40.10$ erg s$^{-1}$], such as is evident in other giant ellipticals such as M87 [@cote01] or NGC 4649 [@bridges06]. Thus it is difficult to model the dark matter profile of NGC 5128 with [*a priori*]{} constraints. Without such a mass model, we turn to the tracer mass estimator for the dynamical mass determination. The tracer mass estimator has the distinct advantage over the virial and projected mass estimators that the tracer population does not have to follow the dark matter density in the galaxy - an extremely useful feature for stellar subsystems such as GCs and PNe that might, in principle, have significantly different radial distributions (see [@evans03] for extensive discussion). Below, we determine the mass of NGC 5128 using the tracer populations of GCs and PNe (our mass estimates do not include stellar kinematics in the inner regions).
Mass Determination {#sec:mass}
------------------
The mass contributed by the random internal motion of the galaxy (pressure-supported mass) is determined from the tracer mass estimator as $$\label{eqn:tme}
M_{p} = \frac{C}{GN} \sum_{i}(v_{f_i} - v_{sys})^2R_i$$ where $N$ is the number of objects in the sample and $v_{f_i}$ is the radial velocity of the tracer object [*with the rotation component removed*]{}. For an isotropic population of tracer objects, assumed in this study, the value of $C$ is $$\label{eqn:C}
C = \frac{4(\alpha + \gamma)(4 - \alpha -\gamma)(1-(\frac{r_{in}}{r_{out}})^{(3-\gamma)})}{\pi(3-\gamma)(1-(\frac{r_{in}}{r_{out}})^{(4-\alpha-\gamma)})}$$ where $r_{in}$ and $r_{out}$ are the three-dimensional radii corresponding to the two-dimensional projected radii of the innermost, $R_{in}$, and outermost, $R_{out}$, tracers in the sample. The parameter $\alpha$ is set to zero for an isothermal halo potential in which the system has a flat rotation curve at large distances. Finally, $\gamma$ is the slope of the volume density distribution, which goes as $r^{-\gamma}$, and is found by determining the surface density slope of the sample and deprojecting the slope to three-dimensions. The tracer mass estimator uses a sample of tracer objects defined between $r_{in}$ and $r_{out}$, yet it is important to emphasize that it determines the [*total*]{} enclosed mass within $r_{out}$.
There is also a contribution to the total mass by the rotational component, as determined in § \[sec:rotamp\] for the GCs and § \[sec:kin\_PN\] for the PNe. This mass component is determined from the rotational component of the Jeans equation, $$\label{eqn:rje}
M_{r} = \frac{R_{out}v^{2}_{max}}{G}$$ where $R_{out}$ is the outermost tracer projected radius in the sample and $v_{max}$ is the rotation amplitude. Therefore, the total mass of NGC 5128, $M_t$, is determined by the addition of the mass components supported by rotation, $M_r$, and random internal motion, $M_p$, $$\label{eqn:total_mass}
M_{t} = M_{p} + M_{r}.$$
In the determination of the pressure-supported mass, one must estimate values for $r_{in}$ and $r_{out}$ knowing $R_{in}$ and $R_{out}$. [@evans03] suggest that $r_{in}\simeq
R_{in}$ and $r_{out}\simeq R_{out}$ for distributions taken over a wide angle. However, in this study the inner and outer radii of the chosen bins are at intermediate radial values within the distribution. Their assumption would therefore lead to an underestimate of the determined mass, since the true $r_{out}$ can be quite a bit larger than the projected $R_{out}$. To correct for this contributed uncertainty, distributions of sample tracer populations were generated through Monte Carlo simulations. In the simulations, 340 GCs were randomly placed in a spherically symmetric system extending out to 50 kpc with an $r^{-2}$ projected density, while 780 PNe were placed in the same environment extending out to 90 kpc. From the generated distributions, the value of $C$ in Eqn. \[eqn:C\] was determined for both the real and projected positions of the tracer populations in each designated radial bin. This correction factor, listed in Tables \[tab:MP\_GC\]-\[tab:PN\] as $M_{corr}$, multiplies the pressure-supported mass from Eqn. \[eqn:tme\]. The same correction was applied to the full GC sample and the corresponding $M_{corr}$ values are listed in Table 1 of [@w06]. These values are generally small, but in the worst case they triple $M_p$.
Surface Density Profiles {#sec:surden}
------------------------
In Eqn. \[eqn:C\], the value of $\gamma$ is determined for the tracer populations by deprojecting the slope of the surface density profile to three-dimensions. Figure \[fig:gc\_surden\] shows the surface density profiles for the entire, metal-poor, and metal-rich GC populations, along with the PN profile. The populations were binned, following [@maiz05], into circular annuli of equal numbers of objects, providing the same statistical weight to each bin (although spatial biases may still affect the GC population in the outer regions along the major axis; see Fig. \[fig:gc\_thetar\]). In the inner 5 kpc of all tracer populations, incompleteness due to the obscuration of the dust lane is evident by the flattening of the surface density profile. The innermost objects were, therefore, excluded from the surface density profile fittings. Outside of 5 kpc, the surface densities fit well to power laws, leading to $\gamma = 3.65\pm0.17$, $3.49\pm0.34$, $3.37\pm0.30$, and $3.47\pm0.12$ for the entire GC population, the metal-poor and metal-rich subpopulations of GCs, and the PNe in NGC 5128, respectively. These are all very similar within their uncertainties.
Mass Results {#sec:mass_results}
------------
The similar kinematics we find between the metal-poor and metal-rich subpopulations of GCs in this study strongly justifies the combining of the two populations for the mass determination performed in [@w06]. The GC population provides a [*total*]{} mass estimate of $(1.3\pm0.5) \times 10^{12}$ $M_{\odot}$ from 340 clusters out to a projected radius of 50 kpc. Removing the GCs in our sample with $v_r
\leq 300$ km s$^{-1}$, which will remove all possible contamination from foreground stars, discussed in § \[sec:velfield\], leads to a total mass of $(1.0\pm0.4) \times 10^{12}$ $M_{\odot}$. This mass agrees nicely with our mass determined from our entire GC sample. The PN population provides a total mass of $(1.0\pm0.2) \times 10^{12}$ $M_{\odot}$ from 780 PNe out to 90 kpc in projected radius, agreeing with the GC value within the uncertainty.
We are also able to generate a mass profile of NGC 5128 from the total GC population and the PNe, shown in Figure \[fig:mass\]. The tracer mass estimator determines the [*total*]{} enclosed mass for NGC 5128 within the outermost radius of a given tracer sample. It calculates this total mass using a sample of objects defined within the radial range defined by the sample’s inner and outermost radii. It is therefore possible to use a unique set of tracer objects, denoted by the radial bin range, listed in the first column of Tables \[tab:all\_GC\]-\[tab:PN\], to determine a mass profile from independent mass estimates. The independent binning, leads to sample sizes in the mass determination, in some cases generating higher uncertainties in the total enclosed mass. The most certain mass is the one determined from the full sample of tracers.
In the mass determinations above, we have implicity assumed isotropy for the velocity distributions. But the possibility exists that the PNe (for example) might have radial anisotropy which would produce their gradually falling $\sigma_v(R)$ curve. Replacing Equation \[eqn:C\] in the tracer mass estimator by $$\label{eqn:C_aniso}
C = \frac{16(\alpha + \gamma - 2\beta)(4 - \alpha
-\gamma)(1-(\frac{r_{in}}{r_{out}})^{(3-\gamma)})}{\pi(4 - 3\beta)(3-\gamma)(1-(\frac{r_{in}}{r_{out}})^{(4-\alpha-\gamma)})}$$ which includes the anisotropy parameter, $\beta$, from [@evans03], we find that the mass estimate from the PNe can be forced to agree with the mass estimate from the GCS for a nominal $\beta = 0.8$. For perfect isotropy, $\beta = 0$. This would mean roughly 2:1 radial anisotropy for the PNe in the outer halo. However, we find that any $\beta$ in the wide range of $-10 \leq \beta \leq 1$ would still keep the two methods in agreement within their internal uncertainties, so we are not yet in a position to tightly constrain any anisotropy. It is possible that the GCs may also have anisotropy; it may therefore be too simplistic to find a range of $\beta$ for the PNe for which the masses of the PNe and GCs agree. However, the GCs are likelier to be nearly isotropic than the PNe; the GCs are older, ”hotter” subsystems of the halo. In other studies, the isotropy of the GCS orbits has also been shown to be a good assumption from mass profiles of elliptical galaxies using X-ray observations [@cote01; @cote03; @bridges06 among others].
Both mass estimates can be compared to previous studies. First, we note the total mass determined from the PN data with that of [@peng04]. While the rotationally supported mass was determined here with different values of the mean rotational velocity, they calculated the pressure supported mass using the identical tracer mass estimator technique with exactly the same PN population. The total mass estimate given by [@peng04] is $(5.3\pm0.5) \times 10^{11}$ $M_{\odot}$. Subtracting their rotationally supported mass leaves a pressure supported mass of $\sim 3.4 \times
10^{11}$ $M_{\odot}$, quite different from our $(8.46\pm1.72) \times 10^{11}$ $M_{\odot}$. Recalculating our pressure supported mass estimate with $\gamma = 2.54$, which was used in [@peng04], we are able to reproduce their mass estimate within the uncertainty. The values of $\gamma$ differ between the two studies simply because the $\gamma$ used in [@peng04] was the inverse of the surface density slope instead of the inverse of the volume density slope. Using the correct value of $\gamma = 3.54$, their pressure supported mass estimate would increase to $8.7 \times
10^{11} M_{\odot}$, matching the mass found in this study.
Second, we compare our total mass determined using the GC population with that from [@pff04II]. Using 215 GCs out to 40 kpc, they found a pressuresupported mass of $(3.4\pm0.8) \times 10^{11}$ $M_{\odot}$, again much different from our pressure supported mass of $(1.26\pm0.47) \times 10^{12}$ $M_{\odot}$ using 340 clusters out to 50 kpc. The large difference can again be attributed to their using $\gamma = 2.72$ instead of deprojecting their surface density slope to $\gamma = 3.72$. Using the correct value of $\gamma$, we find a pressure-supported mass of $7.5 \times 10^{11}$ $M_{\odot}$ using the same 215 clusters they used in their study. This corrected estimate is closer to the pressure supported mass determined in our study, but it is not necessarily expected to agree with our result, as our sample contains $130$ more GCs and uses a slightly different $\gamma$ that we have independently redetermined.
Third, the mass determined by the H $_I$ shell study of NGC 5128 by [@schiminovich94] found a mass of $2 \times 10^{11}$ $M_{\odot}$ [*within 15 kpc*]{} assuming a distance of 3.5 Mpc. With the distance of 3.9 Mpc used in this study, the mass determined in their study would increase to $2.2 \times 10^{11}$ $M_{\odot}$, which is $30\%$ smaller than our total mass of $3.89\pm0.94 \times
10^{11}$ $M_{\odot}$ within 15 kpc.
Lastly, we compare our determined mass to a recent study by [@samurovic06]. [@samurovic06] determined a total mass of NGC 5128 using GCs, PNe, and an X-ray data technique. The galaxy mass determined from the GC and PN data was obtained using the tracer mass estimator and the spherical Jeans equation, as performed in our study. However, [@samurovic06] used the volume density slopes determined by [@peng04] and [@pff04II], for the PN and GC data, respectively. They obtained mass estimates for NGC 5128 similar to those of [@peng04] and [@pff04II], discussed above, using an identical PN sample and slightly increased GC sample. They also included an X-ray-modelling mass estimate for NGC 5128 from which they obtained masses of $(7.0\pm0.8) \times 10^{11}$ $M_{\odot}$ out to 50’ and $(11.6\pm1.0) \times 10^{11}$ $M_{\odot}$ out to 80’. This mass estimate is similar to our PNe estimate out to the same radial extent, but the author cautions that it is an overestimate of the true mass of NGC 5128 resulting from a lack of hydrostatic equilibrium in the outer region of the galaxy.
We note here that the mass estimates obtained are higher than those from [@hui95], and [@peng04] derived from the PNe using a two-component mass model, as well as [@samurovic06], using an X-ray modelling technique. This discrepancy is not fully understood and possibly lies in the assumptions that go into the mass estimators with a spatially biased sample. We intend to pursue this issue further with an upcoming larger sample of GCs with less spatial biases.
The mass of NGC 5128 that we find appears to be in the range of other giant elliptical galaxies, such as NGC 1399 [$\sim 2 \times 10^{12}$ $M_{\odot}$ out to 50 kpc; @richtler04], M49 [$\sim 2 \times 10^{12}$ $M_{\odot}$ out to their kinematically studied radius of 35 kpc; @cote03], and M87 [$\sim 9 \times 10^{11}$ $M_{\odot}$ at 20 kpc, the onset of the projected cD envelope; @cote01]. Clearly, it is legitimate to say that NGC 5128 is the largest, most massive galaxy in the neighborhood of the Local Group, and one that can be talked about in the same category as these other giants that reside in larger clusters.
The sample biases mentioned above in our currently available set of both GCs and PNe place limitations on how much we can reasonably interpret the kinematic and dynamic data. We are currently carrying out a set of new spectroscopic programs to increase the tracer sample size and to remove the sample biases, leading to a more complete analysis of the halo velocity field.
Discussion and Conclusions {#sec:concl}
==========================
Angular momentum is an essential quantity for characterizing the sizes, shapes, and formation of galaxies and is often represented as the dimensionless spin parameter,
$$\label{eqn:spin}
\Lambda = \frac{J |E|^{1/2}}{G M^{5/2}}$$
where $J$ is the angular momentum, $E$ is the binding energy, and $M$ is the mass of the galaxy. The spin parameter is representative of a galaxy’s angular momentum compared to the amount of angular momentum needed for pure rotational support: the lower the $\Lambda$-value, the less rotation and rotational support within the galaxy. For an elliptical galaxy in gravitational equilibrium, the spin parameter simplifies to $\Lambda
\sim 0.3 <(\Omega R / \sigma_v)>$ [@fall79], yielding $\Lambda =
0.10$ with $(\Omega R / \sigma_v) = 0.33$ for the entire population of GCs in NGC 5128.
Table \[tab:spin\] shows the spin parameter for four giant galaxies with large GCS kinematic studies, M87, M49, NGC 1399, and NGC 5128. The table columns give the galaxy name, the rotation amplitude, the projected velocity dispersion, and the ratio of the rotation amplitude to the velocity dispersion, followed by the spin parameter. These quantities are shown for the metal-poor and metal-rich populations. What is clearly evident in Table \[tab:spin\] is the strong galaxy to galaxy differences between these four galaxies, already hinted at in § \[sec:intro\]. Though the sample is still quite small, no obvious pattern emerges. There is an indication of metal-poor and metal-rich GCSs having similar spin parameters within the same galaxy. M49 is the only galaxy studied here where this may not be the case. Although the metal-poor and metal-rich cluster spin parameters are consistent within the uncertainties, the metal-rich cluster spin parameter of M49 is also consistent with zero.
In the monolithic collapse scenario, [@peebles69] describes the angular momentum within the galaxy as attributed to the tidal torque transferred from neighbouring proto-galaxies during formation. In this scenario, [@efstathiou79] found that a spin parameter of $\Lambda = 0.06$ for elliptical systems is expected from simulations of the collapse of an isolated protogalactic cloud. But NGC 5128, among many other giant elliptical galaxies, is not in isolation, and therefore not necessarily expected to reproduce such a low spin parameter. Also, the internal rotation axis changes at 5 kpc are not easily explained with only the monolithic collapse scenario. In the monolithic collapse model, the inner regions would be expected to have more pronounced rotations. Yet all four of the galaxies with major kinematic studies presented here do not show a higher rotational signal in the inner regions. In fact, for NGC 1399, the outer region ($R > 6'$) indicates rotation in the metal-poor population that is not evident in the inner regions. Also, a slightly lower rotational signal is present in the inner regions of NGC 5128 for the metal-poor population than in the outer regions.
Hierarchical clustering of cold dark matter also relies on angular momentum in a galaxy being produced by gravitational tidal torques during the growth of initial perturbations. [@sugerman00] have demonstrated that the tidal torque theory predicts an increase in angular momentum during the collapse, and with time, the increase in angular momentum slows. Accretion of satellites and/or merger events is therefore a possible culprit for moving the angular momentum outward, as major mergers of disks and bulges suggest that angular momentum resides largely in the outer regions of the galaxy [@barnes92; @hernquist93].
Alternatively, [@vitvitska02] examine the change in spin parameter in a scenario where the angular momentum in a galaxy is built up by mass accretion. Their results show that the spin parameter changes sharply in major merger events in the galaxy and steadily decreases with small satellite accretion events. They also show that the spin parameter for a galaxy with a major merger after a redshift of $z = 3$ should be notably larger than a galaxy that did not undergo such a major merger. Their study obtains an average of $\Lambda = 0.045$ from $\Lambda$CDM $N$-body simulations for galaxies with halo masses of $(1.1-1.5) \times 10^{12} h^{-1}$ $M_{\odot}$ with $h = 0.7$.
NGC 1399 and M49, with their weak rotation signals, are consistent with the model predictions discussed above, whereby their major formation events could have occurred at early times and with perhaps only minor accretions happening since then. However, NGC 5128 and M87 have spin parameters 2-3 times larger than predicted by the model averages. For NGC 5128, this relatively large rotation (which is nearly independent of both metallicity and radius) may, perhaps, be connected with its history within the Centaurus group environment. The rotation speed and rotation axis for its extended group of satellite galaxies are nearly identical to the NGC 5128 halo [@w06], much as if the accretion events experienced by the central giant have been taking place preferentially along the main axis of the entire group and in the same orientation. The GCS age distribution discussed by [@beasley06] and the mean age for the halo field stars [@rejkuba05] strongly suggest that a high fraction of the stellar population in NGC 5128 formed long ago, with particularly large bursts between 8 and 12 Gyr. Even if the galaxy underwent a significant merger perhaps a few Gyr ago (the traces of which now appear in the halo arcs and shells), the stars in it may already have been old at the time of the merger. Although a very few younger GCs have formed since then, these make up a small minority of what is present, at least for the $R > 5$ kpc halo outside the bulge region that now contains gas and dust.
The situation for M87, with its even larger rotation signature, may require a different sort of individual history. Of the four galaxies compared here, it is at the dynamical center of the richest environment (Virgo), has the most extensive cD-type envelope, and sits within the most massive, extended, and dynamically evolved potential well. A single relatively recent major merger could in principle have caused its present high rotation, but the lack of distinctive tidal features does not necessarily favor such an interpretation and would at least suggest that such a merger should have been with another large elliptical and nondissipative galaxy. [@cote01] discuss an interpretation - at least partially resembling what we suggest for NGC 5128 - that stellar material ”is gradually infalling onto M87 along the so-called principal axis of the Virgo Cluster.”
In conclusion, we have presented a kinematic study of NGC 5128 that makes it now comparable to recent studies of the other giant galaxies, M87, M49, and NGC 1399. Using $340$ GCs ($158$ metal-rich and $178$ metal-poor GCs), we have calculated the rotation amplitude, rotation axis, and velocity dispersion and have searched for radial and metallicity dependences. Our findings show that both metallicity populations rotate with little dependence on projected radius, with $\Omega R = 40\pm10$, 31$\pm$14, and 47$\pm$15 km s$^{-1}$ for the total, metal-poor, and metal-rich populations, respectively. Perhaps the inner 5 kpc shows a slower rotation of the metal-poor population, but more clusters would be needed to confirm this finding. The rotation axis is 189$\pm$12$^o$, 177$\pm$22$^o$, and 202$\pm$15$^o$ east of north for the total, metal-poor, and metal-rich populations out to a 50 kpc projected radius, assuming the velocity field is best fit by a sine curve. The rotation axis does change at 5 kpc, following the zero-velocity curve proposed by [@peng04] or possibly full-on counterrotation. A study with more GCs and lower uncertainties is needed to see what is happening in the innermost 5 kpc of NGC 5128. The velocity dispersion shows a modest increase with galactocentric radius, although the outer regions (especially the metal-rich population) have less reliable statistics; this increase could be driven purely by statistical effects. We find the velocity dispersion we find 123$\pm$5, 117$\pm$7, and 129$\pm$9 km s$^{-1}$ for the total, metal-poor, and metal-rich populations, respectively.
The PN data are also used to determine the kinematics of the halo of NGC 5128. These show results that are encouragingly similar to those of the GC data, except that no rotation axis change is noted with radius, and a [*decrease*]{} in velocity dispersion is found with radius, possibly indicating a difference in orbital anisotropy compared with the GCs. A very similar effect has been noted for the Leo elliptical NGC 3379, although with a much smaller data sample [@romanowsky03; @bergond06; @pierce06]. We also determine the total dynamical mass using both the GCs and the PNe by separately calculating the pressure supported mass with the tracer mass estimator and the rotationally supported mass using the spherical component of the Jeans equation. The total mass is $(1.3\pm0.5) \times 10^{12}$ $M_{\odot}$ from the GC population out to a projected radius of 50 kpc, or $(1.0\pm0.2) \times 10^{12}$ $M_{\odot}$ out to 90 kpc from the PNe.
Overall, we have enough evidence to cautiously conclude that a major episode of star formation occurred about $8-10$ Gyr ago (corresponding to a redshift z = 1.2 - 1.8) and this may have been when the bulk of the visible galaxy was built. We still do not know just why the most metal-poor clusters show up in such relatively large numbers and appear to have ages of $10-12$ Gyr, but this is a common issue in all big galaxies.
This kinematic study and the age study of [@beasley06] on the NGC 5128 cluster system indicate that additional spectroscopic studies to build up both the radial velocity database and age distribution can lead to rich dividends. Large GC samples are clearly needed to remove the current sample biases and to fully understand the complex kinematics and history of this giant elliptical galaxy. It seems clear as well that each galaxy needs to be individually studied to fully understand the different galaxy formation histories. We are continuing these studies particularly for NGC 5128, with the eventual aim of at least doubling the total GC sample size in this unique system.
Acknowledgements: WEH and GLHH acknowledge financial support from NSERC through operating research grants. DAF thanks the ARC for financial support.
Arnaboldi, M., Freeman, K.C., Capaccioli, M., & Ford, H. 1994, ESO Messanger, 76, 40 Arnaboldi, M., Freeman, K.C., Gerhard, O., Matthias, M., Kudritzki, R.P. Méndez, R.H., Capaccioli, M., & Ford, H. 1998, , 507, 759 Baade, W., & Minkowski, R. 1954, , 119, 215 Bahcall, J.N., & Tremaine, S. 1981, , 244, 805 Barnes, J.E. 1992, , 393, 484 Beasley, M.A., Forbes, D.A., Brodie, J.P., & Kissler-Patig, M. 2004, , 347, 1150 Beasley, M.A., Bridges, T., Peng, E., Harris, W.E., Harris, G.L.H., Forbes, D.A., & Mackie, G. 2006, in preparation Bekki, K., Harris, W.E., & Harris, G.L.H. 2003, , 338, 587 Bekki, K., Forbes, D.A., Beasley, M.A., & Couch, W.J. 2003, , 334, 1334 Bekki, K., & Peng, E.W. 2006, , 370, 1737 Bergond, G., Zepf, S.E., Romanowsky, A.J., Sharples, R.M., & Rhode, K.L. 2006, , 448, 155 Bridges, T., Gebhardt, K., Sharples, R., Faifer, F.R., Forte, J.C., Beasley, M.A., Zepf, Z.E., Forbes, D.A., Pierce, M. 2006, , 373, 157 Ciardullo, R., Jacoby, G.H., & Dejonghe, H.B. 1993, , 414, 454 Côté, P., McLaughlin, D.E., Hanes, D.A., Bridges, T.J., Geisler, D., Merritt, D., Hesser, J.E., Harris, G.L.H., & Lee, M.G. 2001, , 559, 828 Côté, P., McLaughlin, D., Cohen, J.G., & Blakesless, J.P. 2003, , 591, 850 Dekel, A., Stoehr, F., Mamon, G.A., Cox, T.J., Novak, G.S., & Primak, J.R., 2005, Nature, 437, 707 Dufour, R.J., Harvel, C.A., Martins, D.M., Schiffer, F.H., Talent, D.L., Wells, D.C., van den Bergh, S., & Talbot, R.J. 1979, , 84, 284 Efstathiou, G., & Jones, B.J.T. 1979, , 186, 133 Evans, N.W., Wilkinson, M.I., Perrett, K.M., & Bridges, T.J. 2003, , 583, 752 Fall, S.M. 1979, Rev. Mod. Phys., 51, 21 Ferrarese, L., Mould, J.R., Stetson, P.B., Tonry, J.L., Blakeslee, J.P., & Ajhar, E.A. 2006, ApJ, in press (astro-ph/0605707) Harris, G.L.H., Geisler, D., Harris, H.C., & Hesser, J.E. 1992, , 104, 613 Harris, G.L.H., Harris, W.E., & Poole, G. 1999, AJ, 117, 855 Harris, W. E., & Harris, G. L. H. 2002, , 123, 3108 Harris, W.E., Harris, G.L.H., Holland, S.T., & McLaughlin, D.E. 2002, , 124, 1435 Harris, G.L.H., Harris, W.E., & Geisler, D. 2004, , 128, 723 Harris, W.E., Harris, G.L.H., Barmby, P., McLaughlin, D.E., & Forbes, D. 2006, , 132, 2187 Heisler, J., Tremaine, S., & Bahcall, J.N. 1985, , 298, 8 Hernquist, I. 1993, , 409, 548 Hesser, J.E., Harris, H.C., van den Bergh, S., & Harris, G.L.H. 1984, , 276,491 Hesser, J.E., Harris, H.C., & Harris, G.L.H. 1986, , 303, L51 Hui, X., Ford, H.C., Freeman, K.C., & Dopita, M.A. 1995, , 449, 592 Kraft, R.P., Váquez, S.E., Forman, W.R., Jones, C., Murray, S.S., Hardcastle, M.J., Worrall, D.M., Churazov, E. 2003, , 592, 129 Larsen, S.S., Brodie, J.P., Beasley, M.A., & Forbes, D.A. 2002, , 124, 828 Lee, H., & Worthey, G. 2005, , 160, 176 Maíz Apellániz, J. & Úbeda, L. 2005, , 629, 7873 Mamon, G.A., & [Ł]{}okas, E.L. 2005, , 362, 95 Napolitano, N.R., Arnaboldi, M., Freeman, K.C., & Capaccioli, M. 2001, , 377, 784 Napolitano, N.R., Arnaboldi, M., & Capaccioli, M. 2002, , 383, 791 O’Sullivan, E., Forbes, D.A., & Ponman, T.J. 2001,, 328, 461 Peebles, P.J.E. 1969, , 155, 393 Peng, E.W., Ford, H.C., Freeman, K.C., & White, R.L. 2002, , 124, 3144 Peng, E.W., Ford, H.C., & Freeman, K.C. 2004, , 150, 367 Peng, E.W., Ford, H.C., & Freeman, K. 2004, , 602, 685 Peng, E.W., Ford, H.C., & Freeman, K. 2004, , 602, 705 Pierce, M., Beasley, M.A., Forbes, D.A., Bridges, T., Gebhardt, K., Faifer, F.R., Forte, J.C., Zepf, S.E., Sharples, R., Hanes, D.A., & Proctor, R. 2006, , 366, 1253 Press, W.H., Teukolsky, S.A., Vetterling, W.T., & Flannery, B.P. 1992, Numerical Recipes in Fortran (2nd Ed., Cambridge: Camdridge University Press) Puzia, T.H., Kissler-Patig, M., Thomas, D., Maraston, C., Saglia, R.P., Bender, R., Goudfrooij, P., & Hempel, M. 2005, , 439, 997 Reed, L.G., Harris, G.L.H., & Harris, W.E. 1994, , 107, 555 Rejkuba, M. 2001, , 269, 812 Rejkuba, M. 2004, , 413, 903 Rejkuba, M., Greggio, L., Harris, W.E., Harris, G.L.H., & Peng, E.W. 2005, , 631, 262 Rejkuba, M., Dubath, P., Minniti, D., & Meylan, G. 2007, , in press (astro-ph/0703385) Richtler, T., Dirsch, B., Gebhardt, K., Geisler, D., Hilker, M., Alonso, M. V., Forte, J. C,. and Grebel, E. K., Infante, L., Larsen, S., Minniti, D., & Rejkuba, M. 2004, , 127, 2094 Romanowsky, A.J., Douglas, N.G., Arnaboldi, M., Kuijken, K., Merrifield, M.R., Napolitano, N.R., Capaccioli, M., & Freeman, K.C., 2003, Science, 301, 1696 Saglia, R.P., Kronawitter, A., Gerhard, O., & Bender, R. 2000, , 119, 153 Samurović, S. 2006, Serb. Astron. J., 173, 35 Schiminovich, D., van Gorkom, J.H., van der Hulst, J.M., & Kasow, S. 1994, , 423, L101 Sharples, R. 1988, IAU Symp. 126: The Harlow-Sharpley Symposium on Globular Cluster Systems in Galaxies, 126, 245 Strader, J., Brodie, J.P., Cenarro, A.J., Beasley, M.A., & Forbes, D.A. 2005, , 130, 1315 Sugerman, B., Summers, F.J., & Kamionkowski, M. 2000, , 311, 762 Thomas, D., Maraston, C., & Korn, A. 2004, , 351, 19 Toomre, A., & Toomre, J. 1972, , 178, 623 van den Bergh, S., Hesser, J.E., & Harris, G.L.H. 1981, , 86, 24 Vitvitska, M., Klypin, A.A., Kravtsov, A.V., Wechsler, R.H., Primack, J.R., & Bullock, J.S. 2002, , 581, 799 Woodley, K.A., Harris, W.E., & Harris, G.L.H. 2005, , 129, 2654 Woodley, K.A. 2006, , 132, 2424
[llllrcccccccccccccccccccccccl]{}
GC0001 & pff\_gc-028 & 13 20 01.16 &-42 56 51.5& 6.46 & 20.11& 19.68& 18.75& 18.20& 17.60& 0.07& 0.02& 0.01& 0.01& 0.03& 19.92& 19.04& 18.22& 0.01& 0.01& 0.01& 0.43& 0.93& 0.55& 1.15& 0.82& 0.88& 1.70& 524$\pm$16\
GC0002 & HH-048 & 13 22 45.36 &-43 07 08.8& 30.26 & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & 856$\pm$50\
GC0003 & HH-080 & 13 23 38.33 &-42 46 22.8& 24.84 & - & - & - & - & - & - & - & - & - & - & 19.45& 18.72& 17.98& 0.02& 0.01& 0.01& - & - & - & - & 0.74& 0.72& 1.46& 497$\pm$33\
GC0004 & HGHH-40/HH-044/C40 & 13 23 42.37 &-43 09 37.8& 21.02 & 19.37& 19.01& 18.12& 17.59& 17.03& 0.04& 0.01& 0.01& 0.01& 0.03& 20.12& 19.28& 18.49& 0.02& 0.01& 0.01& 0.35& 0.89& 0.53& 1.10& 0.79& 0.84& 1.63& 365$\pm$24\
GC0005 & HGHH-01/C1 & 13 23 44.19 &-43 11 11.8& 21.41 & - & - & - & - & - & - & - & - & - & - & 18.73& 18.00& 17.25& 0.06& 0.03& 0.03& - & - & - & - & 0.76& 0.72& 1.48& 642$\pm$1\
GC0006 & pff\_gc-001 & 13 23 49.62 &-43 14 32.0& 22.36 & 20.39& 19.91& 18.91& 18.36& 17.60& 0.09& 0.02& 0.01& 0.01& 0.03& 20.22& 19.25& 18.38& 0.02& 0.01& 0.01& 0.47& 1.01& 0.55& 1.31& 0.87& 0.97& 1.84& 711$\pm$35\
GC0007 & AAT301956 & 13 23 54.52 &-43 20 01.1& 25.41 & 21.22& 21.13& 20.42& 19.98& 19.43& 0.18& 0.06& 0.02& 0.02& 0.04& - & - & - & - & - & - & 0.09& 0.71& 0.44& 0.98& - & - & - & 287$\pm$162\
GC0008 & HH-099 & 13 23 56.70 &-42 59 59.8& 16.66 & - & - & - & - & - & - & - & - & - & - & 21.72& 20.90& 20.01& 0.09& 0.09& 0.06& - & - & - & - & 0.89& 0.82& 1.71& 798$\pm$49\
GC0009 & AAT101931 & 13 23 58.58 &-42 57 17.0& 16.73 & 20.58& 20.57& 19.88& 19.43& 18.96& 0.06& 0.02& 0.01& 0.01& 0.03& 20.60& 20.09& 19.45& 0.01& 0.01& - & 0.02& 0.68& 0.45& 0.92& 0.64& 0.51& 1.15& 590$\pm$144\
GC0010 & AAT101906 & 13 23 58.76 &-43 01 35.2& 16.25 & 20.45& 19.90& 18.89& 18.28& 17.57& 0.06& 0.02& 0.01& 0.01& 0.03& 20.23& 19.11& 18.28& 0.02& 0.03& 0.02& 0.54& 1.01& 0.61& 1.32& 0.83& 1.12& 1.95& 511$\pm$31\
GC0011 & pff\_gc-002 & 13 23 59.51 &-43 17 29.1& 22.94 & 20.70& 20.43& 19.55& 19.02& 18.44& 0.07& 0.02& 0.01& 0.01& 0.03& 20.61& 19.83& 19.07& 0.03& 0.03& 0.01& 0.27& 0.88& 0.53& 1.11& 0.76& 0.78& 1.55& 653$\pm$37\
GC0012 & AAT102120 & 13 23 59.61 &-42 55 19.4& 17.11 & 20.60& 20.59& 19.92& 19.49& 19.03& 0.06& 0.02& 0.01& 0.01& 0.03& 20.61& 20.13& 19.49& 0.01& 0.01& 0.01& 0.02& 0.67& 0.44& 0.89& 0.64& 0.48& 1.12& 293$\pm$83\
GC0013 & HH-001 & 13 24 02.67 &-42 48 32.2& 20.00 & - & - & - & - & - & - & - & - & - & - & 22.52& 21.20& 19.89& 0.08& 0.05& 0.03& - & - & - & - & 1.31& 1.32& 2.63& 775$\pm$74\
GC0014 & pff\_gc-003 & 13 24 03.23 &-43 28 13.9& 31.17 & 20.39& 20.18& 19.31& 18.80& 18.21& 0.15& 0.02& 0.01& 0.01& 0.03& 20.39& 19.64& 18.80& 0.02& 0.02& 0.02& 0.21& 0.87& 0.51& 1.09& 0.84& 0.76& 1.59& 704$\pm$22\
GC0015 & pff\_gc-004 & 13 24 03.74 &-43 35 53.4& 37.98 & 21.12& 20.84& 20.01& 19.49& 18.92& 0.28& 0.03& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.28& 0.83& 0.52& 1.09& - & - & - & 571$\pm$38\
GC0016 & AAT103195 & 13 24 05.98 &-43 03 54.7& 15.18 & 21.57& 21.10& 20.13& 19.58& 18.98& 0.14& 0.03& 0.01& 0.01& 0.03& 21.34& 20.39& 19.50& 0.01& 0.01& 0.01& 0.47& 0.97& 0.56& 1.15& 0.89& 0.95& 1.84& 277$\pm$67\
GC0017 & AAT304867 & 13 24 08.62 & -43 16 27.1& 21.04 & 22.12& 21.15& 19.91& 19.20& 18.54& 0.25& 0.03& 0.01& 0.01& 0.03& 21.62& 20.28& 19.18& 0.01& - & - & 0.97& 1.24& 0.71& 1.36& 1.10& 1.34& 2.44& 305$\pm$56\
GC0018 & AAT305341 & 13 24 10.97 &-43 12 52.8& 18.27 & 22.13& 21.54& 20.51& 19.88& 19.19& 0.23& 0.04& 0.01& 0.01& 0.03& 21.87& 20.80& 19.90& 0.01& 0.01& 0.01& 0.59& 1.03& 0.63& 1.31& 0.89& 1.07& 1.97& 440$\pm$118\
GC0019 & HHH86-28/C28 & 13 24 18.06 &-42 49 01.1& 17.57 & 19.52& 19.33& 18.50& 17.99& 17.45& 0.03& 0.01& 0.01& 0.01& 0.03& 19.46& 18.75& 18.03& 0.02& 0.01& 0.01& 0.19& 0.83& 0.52& 1.05& 0.72& 0.71& 1.43& 461$\pm$22\
GC0020 & pff\_gc-005 & 13 24 18.92 &-43 14 30.1& 18.33 & 21.89& 21.31& 20.29& 19.71& 19.07& 0.18& 0.04& 0.01& 0.01& 0.03& 21.63& 20.64& 19.72& 0.02& 0.02& 0.01& 0.58& 1.02& 0.58& 1.22& 0.91& 1.00& 1.91& 750$\pm$33\
GC0021 & WHH-1/HH-096 & 13 24 21.40 &-43 02 36.8& 12.19 & 19.14& 18.85& 18.01& 17.48& 16.98& 0.03& 0.01& 0.01& 0.01& 0.03& 19.11& 18.33& 17.63& 0.02& 0.02& 0.03& 0.29& 0.84& 0.53& 1.03& 0.70& 0.78& 1.48& 583$\pm$29\
GC0022 & pff\_gc-006 & 13 24 23.72 &-43 07 52.1& 13.48 & 20.22& 20.00& 19.22& 18.70& 18.20& 0.05& 0.02& 0.01& 0.01& 0.03& 20.10& 19.46& 18.73& 0.01& 0.01& 0.01& 0.21& 0.78& 0.52& 1.02& 0.73& 0.65& 1.37& 644$\pm$28\
GC0023 & WHH-2 & 13 24 23.98 &-42 54 10.7& 13.56 & 20.57& 20.45& 19.67& 19.17& 18.65& 0.06& 0.02& 0.01& 0.01& 0.03& 20.54& 19.94& 19.27& 0.03& 0.03& 0.02& 0.12& 0.78& 0.50& 1.02& 0.68& 0.59& 1.27& 582$\pm$81\
GC0024 & pff\_gc-007 & 13 24 24.15 &-42 54 20.6& 13.45 & 21.85& 21.16& 20.12& 19.53& 18.82& 0.17& 0.03& 0.01& 0.01& 0.03& 21.54& 20.50& 19.60& 0.02& 0.02& 0.01& 0.69& 1.04& 0.59& 1.30& 0.90& 1.04& 1.94& 617$\pm$25\
GC0025 & AAT308432 & 13 24 25.55 &-43 21 35.6& 23.38 & 20.98& 20.43& 19.40& 18.80& 18.11& 0.27& 0.02& 0.01& 0.01& 0.03& 20.83& 19.78& 18.84& 0.03& 0.01& 0.02& 0.54& 1.04& 0.60& 1.29& 0.94& 1.05& 1.99& 835$\pm$83\
GC0026 & C111 & 13 24 26.97 &-43 17 20.0& 19.62 & - & - & - & - & - & - & - & - & - & - & 22.54& 22.25& 21.36& 0.04& 0.04& 0.04& - & - & - & - & 0.89& 0.29& 1.17& -\
GC0027 & AAT106695 & 13 24 28.18 &-42 53 04.3& 13.54 & 20.95& 20.75& 19.63& 18.96& 18.30& 0.09& 0.03& 0.01& 0.01& 0.03& 21.16& 20.17& 19.19& 0.06& 0.06& 0.04& 0.20& 1.12& 0.67& 1.33& 0.98& 0.99& 1.96& 835$\pm$83\
GC0028 & AAT106880 & 13 24 28.44 &-42 57 52.9& 11.30 & 21.11& 20.94& 20.14& 19.61& 19.11& 0.10& 0.03& 0.01& 0.01& 0.03& 21.06& 20.43& 19.65& 0.01& 0.01& 0.01& 0.18& 0.80& 0.53& 1.03& 0.79& 0.63& 1.41& 558$\pm$98\
GC0029 & pff\_gc-008 & 13 24 29.20 &-43 21 56.5& 23.38 & 21.40& 20.93& 19.94& 19.41& 18.75& 0.35& 0.03& 0.01& 0.01& 0.03& 21.24& 20.31& 19.43& 0.02& 0.01& 0.01& 0.47& 0.99& 0.54& 1.19& 0.87& 0.93& 1.81& 466$\pm$38\
GC0030 & AAT107060 & 13 24 29.23 &-43 08 36.6& 13.02 & 21.81& 21.26& 20.26& 19.63& 19.00& 0.19& 0.04& 0.01& 0.01& 0.03& 21.60& 20.62& 19.69& 0.01& 0.01& 0.01& 0.55& 1.01& 0.63& 1.25& 0.93& 0.99& 1.92& 600$\pm$58\
GC0031 & AAT107145 & 13 24 29.73 &-43 02 06.5& 10.62 & 21.20& 20.93& 20.08& 19.50& 18.97& 0.11& 0.03& 0.01& 0.01& 0.03& 21.16& 20.33& 19.55& 0.03& 0.02& 0.04& 0.27& 0.85& 0.58& 1.11& 0.78& 0.82& 1.60& 595$\pm$202\
GC0032 & pff\_gc-009 & 13 24 31.35 &-43 11 26.7& 14.55 & 20.82& 20.58& 19.77& 19.25& 18.70& 0.07& 0.02& 0.01& 0.01& 0.03& 20.73& 20.05& 19.26& 0.03& 0.02& 0.01& 0.23& 0.81& 0.52& 1.07& 0.80& 0.68& 1.47& 683$\pm$38\
GC0033 & WHH-3 & 13 24 32.17 &-43 10 56.9& 14.10 & 20.56& 20.32& 19.49& 18.97& 18.43& 0.06& 0.02& 0.01& 0.01& 0.03& 20.46& 19.76& 18.99& 0.02& 0.02& 0.01& 0.25& 0.83& 0.52& 1.06& 0.76& 0.71& 1.47& 709$\pm$54\
GC0034 & C112 & 13 24 32.66 &-43 18 48.8& 20.32 & - & - & - & - & - & - & - & - & - & - & 22.98& 22.22& 21.31& 0.03& 0.02& 0.02& - & - & - & - & 0.91& 0.75& 1.67& -\
GC0035 & pff\_gc-010 & 13 24 33.09 &-43 18 44.8& 20.23 & 20.45& 20.36& 19.68& 19.26& 18.84& 0.06& 0.02& 0.01& 0.01& 0.03& 20.45& 19.91& 19.24& 0.01& 0.01& 0.01& 0.09& 0.68& 0.42& 0.85& 0.67& 0.54& 1.21& 344$\pm$58\
GC0036 & AAT107977 & 13 24 34.63 &-43 12 50.5& 15.18 & 21.73& 21.18& 20.18& 19.57& 18.89& 0.17& 0.04& 0.01& 0.01& 0.03& 21.52& 20.54& 19.59& 0.02& 0.02& 0.01& 0.54& 1.01& 0.60& 1.28& 0.95& 0.98& 1.93& 517$\pm$123\
GC0037 & pff\_gc-011 & 13 24 36.87 &-43 19 16.2& 20.36 & 20.01& 19.82& 19.03& 18.53& 18.00& 0.13& 0.02& 0.01& 0.01& 0.03& 19.95& 19.33& 18.55& 0.01& 0.01& 0.01& 0.18& 0.79& 0.50& 1.04& 0.78& 0.62& 1.40& 616$\pm$41\
GC0038 & C113 & 13 24 37.75 &-43 16 26.5& 17.80 & - & - & - & - & - & - & - & - & - & - & 20.50& 19.87& 19.12& 0.02& 0.02& 0.01& - & - & - & - & 0.75& 0.63& 1.38& -\
GC0039 & pff\_gc-012 & 13 24 38.77 &-43 06 26.6& 10.38 & 21.24& 20.51& 19.45& 18.83& 18.15& 0.13& 0.02& 0.01& 0.01& 0.03& 20.92& 19.78& 18.85& 0.01& 0.01& 0.01& 0.73& 1.06& 0.62& 1.30& 0.93& 1.14& 2.07& 573$\pm$21\
GC0040 & HGHH-41/C41 & 13 24 38.98 &-43 20 06.4& 20.94 & 20.20& 19.59& 18.59& 17.94& 17.32& 0.06& 0.02& 0.01& 0.01& 0.03& 19.95& 18.92& 17.97& 0.02& 0.01& 0.01& 0.61& 1.00& 0.65& 1.27& 0.95& 1.03& 1.98& 363$\pm$1\
GC0041 &HGHH-29/C29 & 13 24 40.39 &-43 18 05.3& 19.01 & 19.77& 19.15& 18.15& 17.54& 16.89& 0.04& 0.01& 0.01& 0.01& 0.03& 19.46& 18.37& 17.53& 0.04& 0.03& 0.02& 0.62& 1.00& 0.61& 1.26& 0.84& 1.09& 1.92& 726$\pm$1\
GC0042 &pff\_gc-013 & 13 24 40.42 &-43 35 04.9& 35.01 & 20.06& 19.83& 19.00& 18.49& 17.98& 0.15& 0.02& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.24& 0.83& 0.51& 1.03& - & - & - & 727$\pm$31\
GC0043 &C114 & 13 24 40.48 &-42 53 35.3& 11.46 & - & - & - & - & - & - & - & - & - & - & 23.46& 22.39& 21.42& 0.05& 0.04& 0.03& - & - & - & - & 0.97& 1.07& 2.04& -\
GC0044 & WHH-4/HH-024 & 13 24 40.60 &-43 13 18.1& 14.89 & 20.71& 20.12& 19.12& 18.50& 17.85& 0.07& 0.02& 0.01& 0.01& 0.03& 20.46& 19.40& 18.53& 0.03& 0.02& 0.01& 0.60& 1.00& 0.62& 1.27& 0.87& 1.06& 1.93& 688$\pm$25\
GC0045 &pff\_gc-014 & 13 24 41.05 &-42 59 48.4& 8.62 & 21.60& 21.17& 20.23& 19.67& 18.94& 0.26& 0.06& 0.02& 0.02& 0.03& 21.47& 20.56& 19.76& 0.04& 0.03& 0.04& 0.43& 0.94& 0.57& 1.30& 0.80& 0.91& 1.71& 690$\pm$34\
GC0046 &pff\_gc-015 & 13 24 41.20 &-43 01 45.6& 8.51 & 20.27& 20.01& 19.22& 18.74& 18.15& 0.06& 0.02& 0.01& 0.01& 0.03& 20.12& 19.40& 18.70& 0.05& 0.04& 0.05& 0.26& 0.79& 0.47& 1.06& 0.70& 0.72& 1.42& 533$\pm$25\
GC0047 &AAT109380 & 13 24 43.58 &-43 08 43.2& 11.05 & 19.64& 19.52& 18.80& 18.32& 17.87& 0.04& 0.01& 0.01& 0.01& 0.03& 19.60& 19.02& 18.30& 0.01& - & - & 0.12& 0.73& 0.48& 0.93& 0.72& 0.59& 1.31& 465$\pm$38\
GC0048 &pff\_gc-016 & 13 24 43.60 &-42 53 07.3& 11.36 & 21.38& 20.87& 19.85& 19.26& 18.58& 0.12& 0.03& 0.01& 0.01& 0.03& 21.19& 20.21& 19.33& 0.02& 0.02& 0.01& 0.51& 1.02& 0.60& 1.27& 0.88& 0.98& 1.85& 505$\pm$17\
GC0049 &pff\_gc-017 & 13 24 43.63 &-42 58 16.4& 8.54 & 20.54& 20.32& 19.51& 18.99& 18.47& 0.07& 0.02& 0.01& 0.01& 0.03& 20.44& 19.78& 19.01& 0.01& 0.01& 0.01& 0.22& 0.81& 0.52& 1.04& 0.77& 0.66& 1.43& 963$\pm$50\
GC0050 & WHH-5 & 13 24 44.58 &-43 02 47.3& 8.04 & 20.47& 20.03& 19.06& 18.49& 17.87& 0.08& 0.02& 0.01& 0.01& 0.03& 20.29& 19.19& 18.90& 0.05& 0.04& 0.04& 0.44& 0.97& 0.57& 1.19& 0.29& 1.09& 1.39& 671$\pm$41\
GC0051 & HH-054 & 13 24 45.22 &-43 23 19.2& 23.49 & - & - & - & - & - & - & - & - & - & - & 20.92& 20.08& 19.25& 0.05& 0.03& 0.04& - & - & - & - & 0.82& 0.84& 1.66& 998$\pm$250\
GC0052 &AAT109711 & 13 24 45.35 &-42 59 33.5& 7.89 & 19.88& 19.87& 19.30& 18.90& 18.33& 0.06& 0.02& 0.01& 0.01& 0.03& 19.84& 19.48& 18.91& 0.02& 0.02& 0.04& 0.02& 0.56& 0.40& 0.98& 0.58& 0.36& 0.94& 276$\pm$59\
GC0053 &AAT109788 & 13 24 45.78 &-43 02 24.5& 7.75 & 21.82& 21.02& 19.89& 19.23& 18.56& 0.25& 0.04& 0.01& 0.01& 0.03& 21.39& 20.17& 19.43& 0.02& 0.02& 0.03& 0.80& 1.14& 0.65& 1.33& 0.74& 1.22& 1.96& 527$\pm$30\
GC0054 &HGHH-G019/G19 & 13 24 46.46 &-43 04 11.6& 8.12 & 20.16& 19.95& 19.16& 18.64& 18.13& 0.06& 0.02& 0.01& 0.01& 0.03& 20.06& 19.38& 18.64& 0.04& 0.03& 0.03& 0.21& 0.79& 0.52& 1.03& 0.75& 0.67& 1.42& 710$\pm$22\
GC0055 &HGHH-09/C9 & 13 24 46.72 &-43 01 18.4& 7.48 & 20.00& 19.76& 18.95& 18.40& 17.90& 0.05& 0.02& 0.01& 0.01& 0.03& 19.87& 19.13& 18.42& 0.04& 0.04& 0.04& 0.24& 0.81& 0.56& 1.05& 0.70& 0.74& 1.45& 501$\pm$21\
GC0056 &pff\_gc-018 & 13 24 47.10 &-43 06 01.7& 8.87 & 20.90& 20.03& 18.91& 18.25& 17.54& 0.10& 0.02& 0.01& 0.01& 0.03& 20.45& 19.23& 18.28& 0.02& 0.01& 0.01& 0.87& 1.12& 0.66& 1.37& 0.95& 1.22& 2.17& 534$\pm$15\
GC0057 &HGHH-G277/G277 & 13 24 47.37 &-42 58 29.8& 7.82 & 20.26& 19.96& 19.10& 18.56& 18.02& 0.06& 0.02& 0.01& 0.01& 0.03& 20.14& 19.37& 18.61& 0.01& 0.01& 0.01& 0.30& 0.87& 0.53& 1.08& 0.76& 0.77& 1.53& 714$\pm$33\
GC0058 & WHH-6 & 13 24 47.37 &-42 57 51.2& 8.06 & 21.33& 20.75& 19.73& 19.12& 18.44& 0.15& 0.03& 0.01& 0.01& 0.03& 21.08& 20.07& 19.15& 0.01& 0.01& 0.01& 0.58& 1.02& 0.62& 1.29& 0.93& 1.00& 1.93& 685$\pm$43\
GC0059 &AAT110138 & 13 24 47.61 &-43 10 48.5& 12.12 & 20.87& 20.53& 19.67& 19.11& 18.57& 0.10& 0.03& 0.01& 0.01& 0.03& 20.75& 19.98& 19.16& 0.03& 0.02& 0.01& 0.34& 0.87& 0.56& 1.10& 0.82& 0.77& 1.59& 358$\pm$137\
GC0060 &HHH86-10/C10 & 13 24 48.06 &-43 08 14.2& 10.13 & 19.87& 19.44& 18.49& 17.91& 17.33& 0.05& 0.01& 0.01& 0.01& 0.03& 19.66& 18.75& 17.93& 0.03& 0.03& 0.01& 0.44& 0.95& 0.58& 1.16& 0.81& 0.91& 1.73& 829$\pm$22\
GC0061 &C115 & 13 24 48.71 &-42 52 35.5& 11.12 & - & - & - & - & - & - & - & - & - & - & 20.88& 20.23& 19.51& 0.02& 0.02& 0.01& - & - & - & - & 0.73& 0.65& 1.37& -\
GC0062 &pff\_gc-019 & 13 24 48.97 &-42 57 48.4& 7.81 & 20.10& 19.87& 19.05& 18.53& 18.00& 0.05& 0.02& 0.01& 0.01& 0.03& 20.00& 19.31& 18.57& 0.03& 0.03& 0.01& 0.23& 0.81& 0.53& 1.05& 0.74& 0.69& 1.43& 607$\pm$27\
GC0063 &AAT110410 & 13 24 49.38 &-43 08 17.7& 10.00 & 21.47& 21.18& 20.34& 19.79& 19.23& 0.16& 0.04& 0.02& 0.01& 0.03& 21.37& 20.66& 19.81& 0.03& 0.02& 0.01& 0.29& 0.84& 0.55& 1.11& 0.85& 0.71& 1.56& 580$\pm$81\
GC0064 &AAT110551 & 13 24 50.09 &-43 07 36.2& 9.42 & 21.43& 21.08& 20.44& 20.04& 19.49& 0.16& 0.04& 0.02& 0.02& 0.03& 21.19& 20.70& 20.03& 0.02& 0.02& 0.01& 0.35& 0.64& 0.41& 0.96& 0.67& 0.49& 1.16& 615$\pm$76\
GC0065 &pff\_gc-020 & 13 24 50.48 &-42 59 49.0& 6.92 & 20.57& 20.33& 19.53& 19.00& 18.47& 0.09& 0.03& 0.01& 0.01& 0.03& 20.42& 19.58& 19.68& 0.05& 0.04& 0.04& 0.24& 0.80& 0.53& 1.06&-0.11& 0.85& 0.74& 331$\pm$47\
GC0066 &HGHH-42/C42 & 13 24 50.87 &-43 01 22.9& 6.72 & 20.53& 19.91& 18.86& 18.24& 17.53& 0.09& 0.02& 0.01& 0.01& 0.03& 20.21& 19.11& 18.21& 0.04& 0.03& 0.05& 0.62& 1.05& 0.63& 1.33& 0.90& 1.10& 2.00& 552$\pm$12\
GC0067 &VHH81-02/C2 & 13 24 51.49 &-43 12 11.1& 12.86 & 19.64& 19.33& 18.50& 18.01& 17.42& 0.04& 0.01& 0.01& 0.01& 0.03& 19.48& 18.72& 17.94& 0.04& 0.03& 0.01& 0.31& 0.83& 0.49& 1.07& 0.78& 0.77& 1.55& 628$\pm$22\
GC0068 &C100 & 13 24 51.80 &-43 04 33.7& 7.33 & - & - & 20.08& - & - & - & - & - & - & - & 20.44& 19.72& 19.03& 0.08& 0.08& 0.10& - & - & - & 1.28& 0.69& 0.72& 1.41& -\
GC0069 &AAT111033 & 13 24 52.98 &-43 11 55.8& 12.50 & 21.17& 20.93& 20.12& 19.66& 19.13& 0.11& 0.03& 0.01& 0.01& 0.03& 21.10& 20.40& 19.64& 0.01& 0.01& 0.01& 0.24& 0.81& 0.46& 0.99& 0.76& 0.70& 1.47& 302$\pm$166\
GC0070 &HGHH-G302/G302 & 13 24 53.29 &-43 04 34.8& 7.15 & 20.30& 20.01& 19.20& 18.69& 18.16& 0.07& 0.02& 0.01& 0.01& 0.03& 20.18& 19.50& 18.73& 0.02& 0.01& 0.01& 0.28& 0.81& 0.52& 1.04& 0.77& 0.68& 1.45& 558$\pm$43\
GC0071 &AAT111185 & 13 24 54.00 &-43 04 24.4& 6.96 & 20.92& 20.76& 19.92& 19.41& 18.92& 0.11& 0.03& 0.01& 0.01& 0.03& 20.90& 20.28& 19.37& 0.02& 0.02& 0.02& 0.16& 0.84& 0.51& 1.00& 0.91& 0.62& 1.53& 466$\pm$87\
GC0072 &pff\_gc-021 & 13 24 54.18 &-42 54 50.4& 8.78 & 20.21& 20.03& 19.25& 18.75& 18.24& 0.06& 0.02& 0.01& 0.01& 0.03& 20.13& 19.52& 18.81& 0.03& 0.02& 0.01& 0.18& 0.78& 0.50& 1.01& 0.70& 0.62& 1.32& 594$\pm$50\
GC0073 &pff\_gc-022 & 13 24 54.33 &-43 03 15.5& 6.44 & 20.72& 20.58& 19.84& 19.37& 18.90& 0.11& 0.03& 0.01& 0.01& 0.03& 20.69& 20.09& 19.37& 0.03& 0.02& 0.02& 0.14& 0.74& 0.48& 0.95& 0.72& 0.60& 1.32& 619$\pm$44\
GC0074 &HHH86-30/C30 & 13 24 54.35 &-42 53 24.8& 9.84 & 18.76& 18.24& 17.25& 16.67& 16.02& 0.03& 0.01& 0.01& 0.01& 0.03& 18.47& 17.49& 16.68& 0.03& 0.04& 0.02& 0.52& 0.98& 0.58& 1.23& 0.81& 0.98& 1.79& 778$\pm$13\
GC0075 &AAT111296 & 13 24 54.49 &-43 05 34.7& 7.50 & 21.70& 20.99& 19.96& 19.31& 18.61& 0.23& 0.04& 0.01& 0.01& 0.03& 21.40& 20.28& 19.33& 0.02& 0.01& 0.01& 0.71& 1.03& 0.65& 1.35& 0.96& 1.11& 2.07& 695$\pm$45\
GC0076 &pff\_gc-023 & 13 24 54.55 &-42 48 58.7& 13.59 & 20.55& 20.30& 19.44& 18.90& 18.37& 0.06& 0.02& 0.01& 0.01& 0.03& 20.48& 19.74& 18.97& 0.01& 0.01& 0.01& 0.25& 0.86& 0.54& 1.08& 0.77& 0.74& 1.51& 457$\pm$31\
GC0077 &HGHH-11/C11 & 13 24 54.73 &-43 01 21.7& 6.01 & 19.55& 18.96& 17.91& 17.26& 16.61& 0.05& 0.01& 0.01& 0.01& 0.03& 19.21& 17.98& 17.20& 0.06& 0.05& 0.06& 0.59& 1.05& 0.66& 1.30& 0.79& 1.23& 2.01& 753$\pm$1\
GC0078 &AAT111406 & 13 24 55.29 &-43 03 15.6& 6.28 & 21.84& 21.15& 20.13& 19.50& 18.80& 0.30& 0.05& 0.02& 0.01& 0.03& 21.66& 20.70& 18.80& 0.03& 0.02& 0.01& 0.69& 1.03& 0.63& 1.33& 1.90& 0.96& 2.86& 669$\pm$77\
GC0079 &C116 & 13 24 55.46 &-43 09 58.5& 10.61 & - & - & - & - & - & - & - & - & - & - & 23.80& 22.74& 21.84& 0.06& 0.02& 0.03& - & - & - & - & 0.90& 1.07& 1.97& -\
GC0080 & HH-052 & 13 24 55.71 &-43 22 48.4& 22.43 & - & - & - & - & - & - & - & - & - & - & 22.58& 21.61& 20.42& 0.05& 0.03& 0.03& - & - & - & - & 1.19& 0.97& 2.16& 921$\pm$146\
GC0081 &pff\_gc-024 & 13 24 55.71 &-43 20 39.1& 20.36 & 21.07& 20.73& 19.84& 19.31& 18.73& 0.28& 0.03& 0.01& 0.01& 0.03& 21.00& 20.17& 19.34& 0.02& 0.01& 0.01& 0.34& 0.89& 0.53& 1.11& 0.83& 0.83& 1.66& 279$\pm$38\
GC0082 &C117 & 13 24 56.06 &-42 54 29.6& 8.80 & - & - & - & - & - & - & - & - & - & - & 21.27& 20.18& 19.26& 0.01& 0.02& 0.01& - & - & - & - & 0.92& 1.09& 2.01& -\
GC0083 &AAT111563 & 13 24 56.08 &-43 10 16.4& 10.79 & 21.34& 21.03& 20.45& 20.05& 19.62& 0.15& 0.04& 0.02& 0.02& 0.03& 21.14& 20.73& 20.05& 0.02& 0.01& 0.02& 0.31& 0.58& 0.41& 0.83& 0.68& 0.41& 1.09& 649$\pm$102\
GC0084 &HGHH-G279 & 13 24 56.27 &-43 03 23.4& 6.15 & 19.97& 19.91& 19.45& 19.17& 18.88& 0.06& 0.02& 0.01& 0.01& 0.03& 19.76& 19.38& 19.83& 0.01& 0.01& 0.01& 0.06& 0.45& 0.29& 0.57&-0.45& 0.38&-0.07& 366$\pm$34\
GC0085 &C118 & 13 24 57.17 &-43 08 42.6& 9.39 & - & - & - & - & - & - & - & - & - & - & 22.60& 21.76& 20.67& 0.04& 0.03& 0.02& - & - & - & - & 1.08& 0.84& 1.92& -\
GC0086 &HGHH-31/C31 & 13 24 57.44 &-43 01 08.1& 5.52 & 20.09& 19.42& 18.38& 17.75& 17.06& 0.08& 0.02& 0.01& 0.01& 0.03& 19.73& 18.59& 17.71& 0.04& 0.03& 0.04& 0.67& 1.04& 0.62& 1.31& 0.88& 1.14& 2.02& 690$\pm$18\
GC0087 &HGHH-G369 & 13 24 57.52 &-42 59 23.3& 5.78 & 19.82& 19.58& 18.74& 18.22& 17.69& 0.06& 0.02& 0.01& 0.01& 0.03& 19.67& 18.96& 18.24& 0.04& 0.03& 0.05& 0.24& 0.83& 0.53& 1.06& 0.72& 0.71& 1.44& 512$\pm$17\
GC0088 &pff\_gc-025 & 13 24 57.56 &-43 05 32.8& 7.04 & 21.50& 21.02& 20.02& 19.39& 18.67& 0.20& 0.04& 0.02& 0.01& 0.03& 21.41& 20.41& 19.45& 0.04& 0.03& 0.02& 0.47& 1.00& 0.63& 1.35& 0.96& 1.00& 1.97& 923$\pm$30\
GC0089 &C119 & 13 24 57.69 &-42 55 48.4& 7.64 & - & - & - & - & - & - & - & - & - & - & 21.16& 21.02& 20.67& 0.03& 0.03& 0.02& - & - & - & - & 0.35& 0.14& 0.49& -\
GC0090 &C120 & 13 24 57.95 &-42 52 04.9& 10.56 & - & - & - & - & - & - & - & - & - & - & 23.26& 22.34& 21.43& 0.04& 0.04& 0.03& - & - & - & - & 0.90& 0.92& 1.83& -\
GC0091 &VHH81-03/C3 & 13 24 58.21 &-42 56 10.0& 7.33 & 19.34& 18.73& 17.71& 17.10& 16.44& 0.04& 0.01& 0.01& 0.01& 0.03& 19.02& 17.88& 17.08& 0.04& 0.03& 0.02& 0.61& 1.02& 0.61& 1.26& 0.80& 1.14& 1.94& 562$\pm$2\
GC0092 &C121 & 13 24 58.42 &-43 08 21.2& 8.97 & - & - & - & - & - & - & - & - & - & - & 23.53& 23.01& 22.45& 0.05& 0.04& 0.09& - & - & - & - & 0.56& 0.52& 1.08& -\
GC0093 &pff\_gc-026 & 13 24 58.45 &-42 42 53.3& 19.02 & 20.44& 20.33& 19.52& 19.07& 18.50& 0.08& 0.02& 0.01& 0.01& 0.03& 20.47& 19.82& 19.09& 0.02& 0.01& 0.01& 0.12& 0.81& 0.45& 1.02& 0.73& 0.65& 1.37& 490$\pm$67\
GC0094 &C122 & 13 24 59.01 &-43 08 21.4& 8.91 & - & - & - & - & - & - & - & - & - & - & - & - & 22.33& - & - & - & - & - & - & - & - & - & - & -\
GC0095 &C123 & 13 24 59.92 &-43 09 08.6& 9.46 & - & - & - & - & - & - & - & - & - & - & 21.88& 21.10& 20.24& 0.02& 0.01& 0.01& - & - & - & - & 0.86& 0.78& 1.64& -\
GC0096 &AAT112158 & 13 25 00.15 &-42 54 09.0& 8.61 & 21.99& 21.43& 20.43& 19.86& 19.23& 0.25& 0.05& 0.02& 0.01& 0.03& 21.75& 20.78& 19.88& 0.01& 0.01& 0.01& 0.55& 1.00& 0.57& 1.20& 0.90& 0.96& 1.87& 699$\pm$43\
GC0097 &C124 & 13 25 00.37 &-43 10 46.9& 10.85 & - & - & - & - & - & - & - & - & - & - & 23.07& 22.36& 21.57& 0.03& 0.02& 0.03& - & - & - & - & 0.78& 0.71& 1.50& -\
GC0098 &pff\_gc-027 & 13 25 00.64 &-43 05 30.3& 6.58 & 21.22& 20.75& 19.74& 19.13& 18.45& 0.17& 0.04& 0.01& 0.01& 0.03& 21.05& 20.06& 19.11& 0.01& 0.01& 0.01& 0.47& 1.01& 0.61& 1.29& 0.95& 0.99& 1.94& 524$\pm$34\
GC0099 &C125 & 13 25 00.83 &-43 11 10.6& 11.16 & - & - & - & - & - & - & - & - & - & - & 22.34& 21.55& 20.66& 0.03& 0.02& 0.02& - & - & - & - & 0.89& 0.78& 1.67& -\
GC0100 &C126 & 13 25 00.91 &-43 09 14.5& 9.45 & - & - & - & - & - & - & - & - & - & - & 24.02& 23.54& 22.66& 0.04& 0.05& 0.08& - & - & - & - & 0.88& 0.48& 1.35& -\
GC0101 &C127 & 13 25 01.32 &-43 08 43.4& 8.97 & - & - & - & - & - & - & - & - & - & - & 23.21& 22.42& 21.62& 0.04& 0.02& 0.03& - & - & - & - & 0.80& 0.79& 1.59& -\
GC0102 &C128 & 13 25 01.46 &-43 08 33.0& 8.81 & - & - & - & - & - & - & - & - & - & - & 23.08& 22.04& 20.95& 0.05& 0.03& 0.02& - & - & - & - & 1.09& 1.04& 2.13& -\
GC0103 &pff\_gc-029 & 13 25 01.60 &-42 54 40.9& 8.03 & 19.97& 19.75& 19.37& 19.06& 18.71& 0.05& 0.02& 0.01& 0.01& 0.03& 19.72& 19.50& 19.10& 0.03& 0.03& 0.01& 0.22& 0.38& 0.31& 0.65& 0.40& 0.22& 0.62& 570$\pm$38\
GC0104 &C129 & 13 25 01.63 &-42 50 51.3& 11.34 & - & - & - & - & - & - & - & - & - & - & 22.59& 21.71& 20.83& 0.04& 0.02& 0.02& - & - & - & - & 0.88& 0.88& 1.76& -\
GC0105 &pff\_gc-030 & 13 25 01.73 &-43 00 09.9& 4.83 & 21.68& 20.94& 19.89& 19.26& 18.54& 0.36& 0.06& 0.02& 0.02& 0.03& 21.21& 19.93& 20.43& 0.05& 0.04& 0.02& 0.73& 1.05& 0.64& 1.35&-0.50& 1.29& 0.78& 357$\pm$29\
GC0106 &HGHH-04/C4 & 13 25 01.83 &-43 09 25.4& 9.52 & 19.10& 18.86& 18.04& 17.50& 16.98& 0.03& 0.01& 0.01& 0.01& 0.03& 18.95& 18.24& 17.50& 0.04& 0.03& 0.01& 0.23& 0.82& 0.54& 1.06& 0.74& 0.71& 1.45& 689$\pm$16\
GC0107 &C130 & 13 25 01.86 &-42 52 27.8& 9.88 & - & - & - & - & - & - & - & - & - & - & 21.22& 20.58& 19.85& 0.01& 0.01& 0.01& - & - & - & - & 0.73& 0.64& 1.37& -\
GC0108 &pff\_gc-031 & 13 25 02.76 &-43 11 21.2& 11.17 & 20.51& 20.29& 19.48& 18.99& 18.42& 0.07& 0.02& 0.01& 0.01& 0.03& 20.43& 19.75& 18.95& 0.03& 0.02& 0.01& 0.23& 0.81& 0.49& 1.06& 0.80& 0.68& 1.48& 444$\pm$33\
GC0109 &HGHH-G176 & 13 25 03.13 &-42 56 25.1& 6.51 & 20.47& 19.94& 18.93& 18.32& 17.67& 0.09& 0.02& 0.01& 0.01& 0.03& 20.24& 19.23& 18.36& 0.01& 0.02& 0.01& 0.54& 1.01& 0.61& 1.26& 0.87& 1.01& 1.88& 551$\pm$13\
GC0110 &HGHH-G066 & 13 25 03.18 &-43 03 02.5& 4.85 & 20.26& 19.67& 18.68& 18.15& 17.44& 0.09& 0.02& 0.01& 0.01& 0.03& 19.94& 18.96& 18.05& 0.01& 0.01& 0.02& 0.59& 0.99& 0.52& 1.24& 0.91& 0.98& 1.89& 576$\pm$8\
GC0111 &pff\_gc-032 & 13 25 03.24 &-42 57 40.5& 5.65 & 20.92& 20.59& 19.73& 19.18& 18.61& 0.14& 0.04& 0.01& 0.01& 0.03& 20.81& 20.05& 19.22& 0.01& 0.02& 0.01& 0.33& 0.86& 0.55& 1.12& 0.83& 0.76& 1.59& 648$\pm$29\
GC0112 &pff\_gc-033 & 13 25 03.34 &-43 15 27.4& 14.98 & 21.10& 20.49& 19.46& 18.89& 18.21& 0.33& 0.03& 0.01& 0.01& 0.03& 20.83& 19.79& 18.90& 0.02& 0.02& 0.01& 0.61& 1.02& 0.57& 1.25& 0.89& 1.04& 1.93& 531$\pm$18\
GC0113 &HHH86-32/C32 & 13 25 03.37 &-42 50 46.2& 11.28 & 20.22& 19.50& 18.44& 17.81& 17.15& 0.05& 0.01& 0.01& 0.01& 0.03& 19.86& 18.71& 17.85& 0.03& 0.03& 0.01& 0.72& 1.06& 0.63& 1.29& 0.85& 1.15& 2.01& 718$\pm$11\
GC0114 &pff\_gc-034 & 13 25 03.37 &-43 11 39.6& 11.41 & 20.98& 20.70& 19.91& 19.35& 18.84& 0.09& 0.03& 0.01& 0.01& 0.03& 20.89& 20.19& 19.38& 0.03& 0.02& 0.01& 0.28& 0.80& 0.56& 1.06& 0.81& 0.69& 1.50& 605$\pm$46\
GC0115 &C131 & 13 25 03.67 &-42 51 21.7& 10.72 & - & - & - & - & - & - & - & - & - & - & 21.57& 20.98& 20.20& 0.02& 0.02& 0.01& - & - & - & - & 0.78& 0.59& 1.36& -\
GC0116 &AAT112752 & 13 25 04.12 &-43 00 19.6& 4.37 & 20.51& 20.16& 19.34& 18.88& 18.44& 0.15& 0.04& 0.02& 0.01& 0.03& 20.35& 19.58& 18.87& 0.02& 0.02& 0.03& 0.35& 0.82& 0.46& 0.90& 0.72& 0.76& 1.48& 679$\pm$82\
GC0117 &pff\_gc-035 & 13 25 04.48 &-43 10 48.4& 10.54 & 21.32& 20.73& 19.71& 19.07& 18.44& 0.14& 0.03& 0.01& 0.01& 0.03& 21.13& 20.07& 19.13& 0.03& 0.02& 0.01& 0.59& 1.02& 0.64& 1.27& 0.94& 1.06& 2.00& 627$\pm$22\
GC0118 &AAT112964 & 13 25 04.61 &-43 07 21.7& 7.50 & 20.45& 20.39& 19.61& 19.13& 18.68& 0.08& 0.03& 0.01& 0.01& 0.03& 20.47& 19.90& 19.11& 0.01& 0.01& 0.01& 0.06& 0.78& 0.49& 0.94& 0.78& 0.57& 1.35& 456$\pm$118\
GC0119 &HGHH-43/C43 & 13 25 04.81 &-43 09 38.8& 9.47 & 19.74& 19.45& 18.60& 18.07& 17.53& 0.04& 0.01& 0.01& 0.01& 0.03& 19.59& 18.85& 18.07& 0.03& 0.02& 0.01& 0.29& 0.85& 0.53& 1.07& 0.78& 0.74& 1.52& 518$\pm$25\
GC0120 & WHH-7 & 13 25 05.02 &-42 57 15.0& 5.68 & 18.96& 18.39& 17.41& 16.81& 16.19& 0.04& 0.01& 0.01& 0.01& 0.03& 18.69& 17.60& 16.83& 0.02& 0.03& 0.01& 0.57& 0.98& 0.60& 1.22& 0.77& 1.09& 1.86& 722$\pm$20\
GC0121 &HGHH-G035 & 13 25 05.29 &-42 58 05.8& 5.09 & 21.02& 20.59& 19.62& 19.01& 18.41& 0.17& 0.04& 0.01& 0.01& 0.03& 20.87& 19.95& 19.08& 0.01& 0.02& 0.01& 0.43& 0.97& 0.61& 1.21& 0.87& 0.92& 1.80& 776$\pm$26\
GC0122 &pff\_gc-036 & 13 25 05.46 &-43 14 02.6& 13.52 & 20.09& 19.83& 19.04& 18.52& 17.91& 0.05& 0.02& 0.01& 0.01& 0.03& 19.96& 19.27& 18.49& 0.02& 0.02& 0.01& 0.26& 0.79& 0.52& 1.13& 0.78& 0.69& 1.47& 666$\pm$30\
GC0123 &HGHH-12/C12/R281 & 13 25 05.72 &-43 10 30.7& 10.18 & - & - & - & - & - & - & - & - & - & - & 19.34& 18.23& 17.36& 0.04& 0.03& 0.02& - & - & - & - & 0.87& 1.11& 1.98& 440$\pm$1\
GC0124 &HGHH-G342 & 13 25 05.83 &-42 59 00.6& 4.52 & 19.65& 19.14& 18.18& 17.57& 16.96& 0.08& 0.02& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.51& 0.96& 0.61& 1.22& - & - & - & 553$\pm$21\
GC0125 &HHH86-13/C13 & 13 25 06.25 &-43 15 11.6& 14.58 & 20.02& 19.60& 18.68& 18.13& 17.51& 0.04& 0.01& 0.01& 0.01& 0.03& 19.82& 18.90& 18.12& 0.03& 0.03& 0.02& 0.41& 0.93& 0.55& 1.16& 0.78& 0.92& 1.70& 601$\pm$12\
GC0126 & R276 & 13 25 07.33 &-43 08 29.6& 8.23 & 21.93& 21.62& 20.78& 20.26& 19.73& 0.26& 0.07& 0.02& 0.02& 0.04& 21.88& 21.16& 20.32& 0.03& 0.03& 0.02& 0.31& 0.83& 0.52& 1.05& 0.84& 0.71& 1.55& 550$\pm$28\
GC0127 &AAT113428 & 13 25 07.33 &-43 06 20.6& 6.38 & 21.25& 20.92& 20.04& 19.49& 18.93& 0.16& 0.04& 0.01& 0.01& 0.03& 21.14& 20.36& 19.51& 0.02& 0.01& 0.01& 0.33& 0.88& 0.54& 1.11& 0.85& 0.78& 1.63& 657$\pm$67\
GC0128 &pff\_gc-037 & 13 25 07.48 &-43 12 29.4& 11.93 & 21.60& 21.29& 20.42& 19.91& 19.33& 0.55& 0.05& 0.02& 0.01& 0.03& 21.46& 20.73& 19.90& 0.02& 0.01& 0.01& 0.31& 0.87& 0.50& 1.08& 0.83& 0.73& 1.56& 554$\pm$23\
GC0129 & WHH-8 & 13 25 07.62 &-43 01 15.2& 3.66 & 19.83& 19.16& 18.12& 17.46& 16.82& 0.13& 0.03& 0.01& 0.01& 0.03& 19.48& 18.32& 17.45& 0.03& 0.03& 0.04& 0.67& 1.04& 0.66& 1.30& 0.87& 1.16& 2.03& 690$\pm$32\
GC0130 & WHH-9 & 13 25 08.51 &-43 02 57.4& 3.93 & 20.59& 19.95& 18.91& 18.37& 17.64& 0.17& 0.03& 0.01& 0.01& 0.03& 20.28& 19.17& 18.30& 0.05& 0.04& 0.05& 0.64& 1.04& 0.54& 1.27& 0.87& 1.11& 1.98& 315$\pm$100\
GC0131 &C132 & 13 25 08.79 &-43 09 09.6& 8.72 & - & - & - & - & - & - & - & - & - & - & 20.38& 19.69& 18.90& 0.02& 0.02& 0.01& - & - & - & - & 0.79& 0.69& 1.48& -\
GC0132 & R271 & 13 25 08.81 &-43 09 09.5& 8.72 & 20.49& 20.23& 19.42& 18.91& 18.39& 0.08& 0.02& 0.01& 0.01& 0.03& 20.38& 19.69& 18.90& 0.02& 0.02& 0.01& 0.26& 0.80& 0.52& 1.03& 0.79& 0.69& 1.48& 436$\pm$45\
GC0133 &pff\_gc-038 & 13 25 08.82 &-43 04 14.9& 4.63 & 20.83& 20.29& 19.29& 18.69& 18.03& 0.17& 0.04& 0.01& 0.01& 0.03& 20.58& 19.57& 18.71& 0.03& 0.03& 0.03& 0.54& 1.00& 0.61& 1.26& 0.87& 1.01& 1.88& 431$\pm$19\
GC0134 &pff\_gc-039 & 13 25 09.10 &-42 24 00.9& 37.29 & 20.01& 19.77& 18.95& 18.44& 17.91& 0.17& 0.02& 0.01& 0.01& 0.03& 19.92& 19.22& 18.50& 0.01& 0.01& 0.01& 0.23& 0.82& 0.51& 1.04& 0.72& 0.70& 1.42& 387$\pm$27\
GC0135 & K-029 & 13 25 09.19 &-42 58 59.2& 4.00 & 19.13& 18.65& 17.71& 17.13& 16.55& 0.08& 0.02& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.48& 0.94& 0.58& 1.16& - & - & - & 677$\pm$44\
GC0136 &pff\_gc-040 & 13 25 09.54 &-42 55 18.5& 6.71 & 20.09& 19.89& 19.12& 18.60& 18.11& 0.06& 0.02& 0.01& 0.01& 0.03& 20.00& 19.37& 18.66& 0.02& 0.02& 0.01& 0.20& 0.76& 0.52& 1.01& 0.71& 0.62& 1.34& 504$\pm$35\
GC0137 &HGHH-G329 & 13 25 10.20 &-43 02 06.7& 3.33 & 21.73& 20.89& 19.71& 19.07& 18.38& 0.69& 0.11& 0.03& 0.02& 0.03& 21.34& 20.04& 19.07& 0.02& 0.01& 0.03& 0.84& 1.18& 0.64& 1.33& 0.97& 1.29& 2.26& 502$\pm$16\
GC0138 & K-033 & 13 25 10.25 &-42 55 09.5& 6.78 & 21.07& 20.47& 19.46& 18.90& 18.21& 0.13& 0.03& 0.01& 0.01& 0.03& 20.78& 19.77& 18.86& 0.01& 0.01& 0.01& 0.60& 1.01& 0.57& 1.25& 0.91& 1.02& 1.93& 582$\pm$33\
GC0139 & K-034 & 13 25 10.27 &-42 53 33.1& 8.23 & 19.33& 18.79& 17.80& 17.26& 16.55& 0.04& 0.01& 0.01& 0.01& 0.03& 19.14& 18.03& 17.25& 0.03& 0.03& 0.02& 0.55& 0.99& 0.54& 1.25& 0.79& 1.11& 1.89& 464$\pm$42\
GC0140 &HHH86-14/C14 & 13 25 10.49 &-42 44 52.6& 16.57 & 19.23& 18.84& 17.94& 17.40& 16.79& 0.03& 0.01& 0.01& 0.01& 0.03& 19.06& 18.16& 17.41& 0.01& 0.02& 0.01& 0.39& 0.90& 0.54& 1.15& 0.75& 0.90& 1.65& 705$\pm$10\
GC0141 &AAT113992 & 13 25 10.51 &-43 03 24.0& 3.85 & 21.95& 21.39& 20.40& 19.79& 19.19& 0.58& 0.11& 0.03& 0.03& 0.04& 21.77& 20.78& 19.82& 0.03& 0.03& 0.01& 0.57& 0.99& 0.61& 1.20& 0.97& 0.99& 1.95& 648$\pm$58\
GC0142 &C133 & 13 25 11.05 &-43 01 32.3& 3.05 & - & - & - & - & - & - & - & - & - & - & - & - & 19.30& - & - & - & - & - & - & - & - & - & - & -\
GC0143 &HGHH-G348 & 13 25 11.10 &-42 58 03.0& 4.32 & 19.90& 19.66& 18.92& 18.46& 18.03& 0.10& 0.03& 0.01& 0.01& 0.03& 19.80& 19.15& 18.45& 0.01& 0.01& - & 0.24& 0.74& 0.46& 0.89& 0.70& 0.65& 1.35& 416$\pm$30\
GC0144 &pff\_gc-041 & 13 25 11.17 &-43 03 09.6& 3.62 & 20.57& 20.32& 19.51& 18.99& 18.48& 0.19& 0.05& 0.02& 0.02& 0.03& 20.48& 19.77& 19.01& 0.02& 0.02& 0.02& 0.24& 0.81& 0.52& 1.03& 0.76& 0.71& 1.47& 456$\pm$29\
GC0145 &HGHH-G327 & 13 25 11.98 &-43 04 19.3& 4.27 & 21.46& 20.83& 19.78& 19.16& 18.56& 0.34& 0.06& 0.02& 0.02& 0.03& 21.16& 20.11& 19.19& 0.01& 0.01& 0.01& 0.63& 1.05& 0.61& 1.22& 0.92& 1.05& 1.97& 608$\pm$19\
GC0146 &HGHH-G081 & 13 25 12.11 &-42 57 25.2& 4.68 & 19.90& 19.52& 18.62& 18.05& 17.49& 0.08& 0.02& 0.01& 0.01& 0.03& 19.72& 18.88& 18.10& 0.03& 0.03& 0.01& 0.38& 0.90& 0.57& 1.13& 0.77& 0.85& 1.62& 536$\pm$43\
GC0147 &pff\_gc-042 & 13 25 12.21 &-43 16 33.9& 15.67 & 19.91& 19.74& 18.97& 18.47& 17.91& 0.12& 0.02& 0.01& 0.01& 0.03& 19.82& 19.19& 18.47& 0.02& 0.02& 0.01& 0.17& 0.78& 0.50& 1.05& 0.72& 0.63& 1.35& 627$\pm$23\
GC0148 &AAT114302 & 13 25 12.34 &-42 58 07.7& 4.11 & 21.87& 21.39& 20.44& 19.90& 19.35& 0.53& 0.11& 0.03& 0.03& 0.04& 21.64& 20.76& 19.97& 0.02& 0.01& 0.01& 0.47& 0.95& 0.55& 1.09& 0.79& 0.88& 1.67& 754$\pm$143\
GC0149 &pff\_gc-043 & 13 25 12.45 &-43 14 07.4& 13.27 & 20.51& 20.22& 19.40& 18.89& 18.33& 0.20& 0.02& 0.01& 0.01& 0.03& 20.38& 19.67& 18.90& 0.03& 0.02& 0.01& 0.29& 0.82& 0.52& 1.07& 0.77& 0.70& 1.48& 527$\pm$33\
GC0150 & WHH-10 & 13 25 12.84 &-42 56 59.8& 4.95 & 20.62& 20.36& 19.57& 19.07& 18.48& 0.12& 0.03& 0.01& 0.01& 0.03& 20.43& 19.90& 19.10& 0.03& 0.02& 0.01& 0.26& 0.79& 0.50& 1.09& 0.80& 0.54& 1.33& 664$\pm$141\
GC0151 & R261 & 13 25 12.90 &-43 07 59.1& 7.35 & 19.65& 19.15& 18.20& 17.60& 17.03& 0.05& 0.01& 0.01& 0.01& 0.03& 19.42& 18.49& 17.62& 0.02& 0.01& 0.01& 0.51& 0.94& 0.60& 1.17& 0.88& 0.93& 1.81& 615$\pm$4\
GC0152 &pff\_gc-044 & 13 25 13.19 &-43 16 35.6& 15.67 & 20.29& 20.11& 19.37& 18.87& 18.33& 0.16& 0.02& 0.01& 0.01& 0.03& 20.19& 19.59& 18.87& 0.02& 0.02& 0.01& 0.17& 0.74& 0.50& 1.04& 0.72& 0.60& 1.32& 568$\pm$53\
GC0153 &C134 & 13 25 13.20 &-43 02 31.3& 2.97 & - & - & - & - & - & - & - & - & - & - & 22.45& 21.64& 20.71& 0.07& 0.05& 0.07& - & - & - & - & 0.93& 0.80& 1.74& -\
GC0154 &pff\_gc-045 & 13 25 13.31 &-42 52 12.4& 9.31 & 21.57& 20.86& 19.82& 19.21& 18.53& 0.19& 0.04& 0.01& 0.01& 0.03& 21.38& 20.32& 19.40& 0.07& 0.06& 0.04& 0.71& 1.04& 0.62& 1.29& 0.93& 1.06& 1.99& 474$\pm$23\
GC0155 & R259 & 13 25 13.88 &-43 07 32.5& 6.87 & 22.13& 21.70& 20.70& 20.11& 19.48& 0.37& 0.08& 0.02& 0.02& 0.03& 21.96& 21.07& 20.10& 0.02& 0.01& 0.01& 0.43& 1.00& 0.60& 1.23& 0.96& 0.89& 1.85& 628$\pm$48\
GC0156 &HGHH-G271 & 13 25 13.95 &-42 57 42.6& 4.25 & 19.45& 19.36& 18.72& 18.32& 17.90& 0.07& 0.02& 0.01& 0.01& 0.03& 19.40& 18.92& 18.30& 0.01& 0.01& - & 0.09& 0.63& 0.40& 0.82& 0.62& 0.47& 1.09& 353$\pm$27\
GC0157 &C135 & 13 25 14.07 &-43 00 51.8& 2.49 & - & - & - & - & - & - & - & - & - & - & - & - & 19.90& - & - & - & - & - & - & - & - & - & - & -\
GC0158 &C136 & 13 25 14.07 &-43 03 35.0& 3.47 & - & - & - & - & - & - & - & - & - & - & 22.48& 21.50& 21.19& 0.04& 0.02& 0.02& - & - & - & - & 0.30& 0.99& 1.29& -\
GC0159 & WHH-11/K-051 & 13 25 14.24 &-43 07 23.5& 6.71 & 21.26& 20.63& 19.55& 18.90& 18.21& 0.18& 0.04& 0.01& 0.01& 0.03& 21.02& 19.95& 18.95& 0.03& 0.02& 0.01& 0.64& 1.07& 0.65& 1.34& 1.00& 1.06& 2.07& 582$\pm$30\
GC0160 &pff\_gc-046 & 13 25 14.83 &-43 41 10.6& 40.10 & 19.91& 19.72& 18.93& 18.43& 17.95& 0.11& 0.02& 0.01& 0.01& 0.03& 19.87& 19.14& 18.37& 0.01& - & 0.01& 0.20& 0.79& 0.50& 0.98& 0.77& 0.73& 1.50& 532$\pm$21\
GC0161 &pff\_gc-047 & 13 25 15.12 &-42 50 30.4& 10.88 & 20.92& 20.72& 19.87& 19.37& 18.87& 0.09& 0.03& 0.01& 0.01& 0.03& 20.87& 20.22& 19.41& 0.02& 0.02& 0.01& 0.20& 0.85& 0.50& 0.99& 0.81& 0.65& 1.46& 717$\pm$55\
GC0162 &AAT114769 & 13 25 15.12 &-42 57 45.7& 4.08 & 21.20& 21.00& 20.25& 19.76& 19.25& 0.30& 0.09& 0.03& 0.03& 0.04& 21.11& 20.52& 19.80& 0.02& 0.02& 0.01& 0.20& 0.75& 0.49& 1.00& 0.71& 0.59& 1.30& 650$\pm$169\
GC0163 & R257 & 13 25 15.24 &-43 08 39.2& 7.84 & 22.29& 21.77& 20.78& 20.18& 19.58& 0.35& 0.07& 0.02& 0.02& 0.03& 22.02& 21.09& 20.16& 0.02& 0.02& 0.02& 0.51& 1.00& 0.60& 1.20& 0.92& 0.93& 1.85& 339$\pm$51\
GC0164 &pff\_gc-048 & 13 25 15.79 &-42 49 15.1& 12.09 & 20.92& 20.67& 19.85& 19.30& 18.78& 0.09& 0.03& 0.01& 0.01& 0.03& 20.85& 20.13& 19.41& 0.02& 0.02& 0.02& 0.25& 0.82& 0.55& 1.07& 0.71& 0.72& 1.44& 535$\pm$53\
GC0165 &AAT114913 & 13 25 15.93 &-43 06 03.3& 5.35 & 21.69& 21.33& 20.39& 19.88& 19.29& 0.34& 0.08& 0.02& 0.02& 0.04& 21.60& 20.72& 19.86& 0.02& 0.02& 0.02& 0.36& 0.94& 0.52& 1.10& 0.86& 0.88& 1.74& 563$\pm$63\
GC0166 &pff\_gc-049 & 13 25 16.06 &-43 05 06.5& 4.49 & 20.74& 20.45& 19.59& 19.05& 18.50& 0.15& 0.04& 0.01& 0.01& 0.03& 20.64& 19.85& 19.07& 0.03& 0.03& 0.03& 0.29& 0.86& 0.54& 1.09& 0.78& 0.79& 1.57& 674$\pm$32\
GC0167 &C137 & 13 25 16.06 &-43 02 19.3& 2.42 & - & - & - & - & - & - & - & - & - & - & - & - & 19.04& - & - & - & - & - & - & - & - & - & - & -\
GC0168 &VHH81-05/C5 & 13 25 16.12 &-42 52 58.2& 8.44 & 18.70& 18.49& 17.68& 17.13& 16.68& 0.03& 0.01& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.21& 0.81& 0.55& 1.00& - & - & - & 555$\pm$3\
GC0169 & HCH01 & 13 25 16.22 &-42 59 43.4& 2.52 & 18.69& 18.36& 17.47& 16.94& 16.40& 0.52& 0.15& 0.04& 0.03& 0.04& - & - & - & - & - & - & 0.33& 0.89& 0.53& 1.07& - & - & - & 649$\pm$45\
GC0170 &HHH86-33/C33 & 13 25 16.26 &-42 50 53.3& 10.47 & 19.58& 19.34& 18.50& 18.02& 17.50& 0.04& 0.01& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.24& 0.84& 0.48& 1.00& - & - & - & 596$\pm$18\
GC0171 &AAT114993 & 13 25 16.44 &-43 03 33.1& 3.16 & 22.49& 21.50& 20.44& 19.82& 19.32& 1.41& 0.19&.382& 0.308& 0.224& 21.95& 20.81& 19.83& 0.02& 0.02& 0.03& 0.99& 1.06& 0.62& 1.12& 0.98& 1.14& 2.12& 352$\pm$136\
GC0172 & HCH02 & 13 25 16.69 &-43 02 08.7& 2.23 & 19.27& 18.87& 17.93& 17.37& 16.76& 0.98& 0.26& 0.07& 0.05& 0.05& - & - & - & - & - & - & 0.40& 0.94& 0.56& 1.17& - & - & - & 300$\pm$2\
GC0173 &pff\_gc-050 & 13 25 16.73 &-42 50 18.4& 11.02 & 21.63& 21.28& 20.39& 19.86& 19.27& 0.16& 0.04& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.35& 0.89& 0.54& 1.12& - & - & - & 688$\pm$70\
GC0174 &C138 & 13 25 16.91 &-43 03 08.0& 2.79 & - & - & - & - & - & - & - & - & - & - & 21.53& 20.79& 19.97& 0.02& 0.02& 0.03& - & - & - & - & 0.81& 0.74& 1.55& -\
GC0175 & R254 & 13 25 16.96 &-43 09 28.0& 8.54 & 20.52& 20.26& 19.43& 18.91& 18.33& 0.09& 0.03& 0.01& 0.01& 0.03& 20.42& 19.65& 18.91& 0.04& 0.03& 0.03& 0.27& 0.82& 0.52& 1.10& 0.73& 0.77& 1.51& 561$\pm$27\
GC0176 &C139 & 13 25 17.06 &-43 02 44.6& 2.50 & - & - & - & - & - & - & - & - & - & - & 20.73& 19.62& 18.86& 0.03& 0.02& 0.04& - & - & - & - & 0.76& 1.11& 1.87& -\
GC0177 &HGHH-G219 & 13 25 17.31 &-42 58 46.6& 3.03 & 20.69& 19.92& 18.84& 18.20& 17.55& 0.43& 0.08& 0.02& 0.02& 0.03& - & - & - & - & - & - & 0.77& 1.09& 0.63& 1.29& - & - & - & 535$\pm$24\
GC0178 & R253 & 13 25 17.33 &-43 08 39.0& 7.74 & 20.75& 20.49& 19.67& 19.14& 18.64& 0.09& 0.03& 0.01& 0.01& 0.03& 20.68& 20.00 & 19.47& 0.03& 0.03& 0.05& 0.26& 0.81& 0.53& 1.04& 0.53& 0.69& 1.22& 486$\pm$52\
GC0179 &C140 & 13 25 17.42 &-43 03 25.2& 2.94 & - & - & - & - & - & - & - & - & - & - & 21.40& 20.65& 19.83& 0.03& 0.02& 0.03& - & - & - & - & 0.82& 0.75& 1.57& -\
GC0180 &C141 & 13 25 18.14 &-43 02 50.9& 2.43 & - & - & - & - & - & - & - & - & - & - & 22.23& 21.55& 20.95& 0.06& 0.05& 0.06& - & - & - & - & 0.60& 0.68& 1.29& -\
GC0181 & WHH-12 & 13 25 18.27 &-42 53 04.8& 8.25 & 20.36& 20.00& 19.07& 18.47& 17.97& 0.07& 0.02& 0.01& 0.01& 0.03& 20.23& 19.35& 18.71& 0.04& 0.04& 0.01& 0.36& 0.93& 0.60& 1.10& 0.64& 0.89& 1.52& 558$\pm$99\
GC0182 &AAT115339 & 13 25 18.44 &-43 04 09.8& 3.45 & 21.19& 20.91& 20.11& 19.53& 19.01& 0.39& 0.10& 0.03& 0.03& 0.04& 21.11& 20.38& 19.56& 0.02& 0.01& 0.02& 0.27& 0.81& 0.58& 1.09& 0.81& 0.73& 1.55& 618$\pm$108\
GC0183 &C142 & 13 25 18.50 &-43 01 16.4& 1.67 & - & - & - & - & - & - & - & - & - & - & - & - & 17.64& - & - & - & - & - & - & - & - & - & - & -\
GC0184 &AAT320656 & 13 25 19.13 &-43 12 03.8& 11.03 & 20.61& 20.50& 19.81& 19.35& 18.89& 0.08& 0.03& 0.01& 0.01& 0.03& 20.57& 20.01& 19.35& 0.02& 0.01& 0.02& 0.10& 0.70& 0.46& 0.92& 0.66& 0.56& 1.22& 453$\pm$233\
GC0185 &HGHH-G331 & 13 25 19.50 &-43 02 28.4& 1.99 & 20.00& 19.60& 18.92& 18.47& 17.93& 0.55& 0.14& 0.05& 0.04& 0.05& - & - & - & - & - & - & 0.40& 0.68& 0.44& 0.99& - & - & - & 371$\pm$22\
GC0186 &AAT115561 & 13 25 19.83 &-42 58 27.2& 3.05 & 22.13& 21.47& 20.50& 19.92& 19.29& 1.44& 0.26& 0.07& 0.06& 0.06& - & 20.98& 20.15& 1.00& 0.04& 0.02& 0.66& 0.98& 0.57& 1.20& 0.83& - & - & 514$\pm$46\
GC0187 & R247 & 13 25 19.99 &-43 07 44.1& 6.73 & 21.86& 21.20& 20.11& 19.40& 18.62& 0.28& 0.05& 0.02& 0.01& 0.03& 21.73& 20.65& 19.75& 0.05& 0.04& 0.04& 0.66& 1.10& 0.70& 1.49& 0.90& 1.08& 1.98& 662$\pm$48\
GC0188 &AAT115605 & 13 25 20.44 &-42 54 08.5& 7.13 & 21.44& 21.22& 20.47& 20.00& 19.50& 0.18& 0.05& 0.02& 0.02& 0.03& 21.36& 20.76& 20.11& 0.01& 0.03& 0.01& 0.22& 0.75& 0.47& 0.97& 0.66& 0.60& 1.25& 713$\pm$131\
GC0189 &AAT115679 & 13 25 20.72 &-43 06 35.9& 5.60 & 21.93& 21.49& 20.44& 19.80& 19.15& 0.40& 0.09& 0.02& 0.02& 0.03& 21.71& 20.79& 19.84& 0.02& 0.02& 0.03& 0.44& 1.05& 0.64& 1.30& 0.95& 0.92& 1.87& 511$\pm$165\
GC0190 & WHH-13/HH-090 & 13 25 21.29 &-42 49 17.7& 11.91 & 20.75& 20.22& 19.20& 18.61& 17.95& 0.08& 0.02& 0.01& 0.01& 0.03& 20.59& 19.52& 18.74& 0.03& 0.03& 0.02&0.53& 1.02& 0.59& 1.25& 0.78& 1.08& 1.85& 444$\pm$33\
GC0191 &AAT321194 & 13 25 21.32 &-43 23 59.3& 22.87 & 21.46& 21.03& 20.06& 19.48& 18.82& 0.38& 0.03& 0.01& 0.01& 0.03& 21.33& 20.42& 19.53& 0.03& 0.02& 0.01& 0.43& 0.97& 0.58& 1.24& 0.90& 0.90& 1.80& 254$\pm$169\
GC0192 &HGHH-06/C6 & 13 25 22.19 &-43 02 45.6& 1.89 & 18.65& 18.17& 17.21& 16.61& 16.03& 0.24& 0.06& 0.02& 0.01& 0.03& - & - & - & - & - & - & 0.48& 0.96& 0.60& 1.18& - & - & - & 855$\pm$2\
GC0193 &pff\_gc-051 & 13 25 22.35 &-43 15 00.1& 13.89 & 21.15& 20.73& 19.79& 19.20& 18.61& 0.10& 0.02& 0.01& 0.01& 0.03& 21.00& 20.10& 19.22& 0.03& 0.03& 0.02& 0.42& 0.94& 0.59& 1.18& 0.88& 0.90& 1.78& 468$\pm$39\
GC0194 &C143 & 13 25 23.20 &-43 03 12.9& 2.22 & - & - & - & - & - & - & - & - & - & - & 21.87& 21.35& 20.52& 0.05& 0.04& 0.06& - & - & - & - & 0.82& 0.52& 1.34& -\
GC0195 &AAT116025 & 13 25 23.46 &-42 53 26.2& 7.75 & 21.23& 20.88& 20.00& 19.48& 18.85& 0.14& 0.04& 0.01& 0.01& 0.03& 21.16& 20.32& 19.62& 0.04& 0.05& 0.01& 0.35& 0.88& 0.51& 1.15& 0.70& 0.84& 1.54& 545$\pm$64\
GC0196 &AAT116220 & 13 25 24.40 &-43 07 58.9& 6.86 & 20.76& 20.39& 19.47& 18.91& 18.36& 0.11& 0.03& 0.01& 0.01& 0.03& 20.62& 19.72& 18.92& 0.03& 0.03& 0.03& 0.37& 0.92& 0.56& 1.11& 0.80& 0.90& 1.70& 524$\pm$41\
GC0197 &AAT116385 & 13 25 25.39 &-42 58 21.5& 2.82 & 21.71& 21.23& 20.26& 19.59& 18.99& 1.22& 0.26& 0.07& 0.05& 0.06& 21.58& 20.68& 19.94& 0.02& 0.02& 0.01& 0.48& 0.97& 0.66& 1.27& 0.74& 0.90& 1.64& 550$\pm$40\
GC0198 & WHH-14 & 13 25 25.49 &-42 56 31.2& 4.64 & 20.99& 20.83& 20.06& 19.56& 19.05& 0.18& 0.05& 0.02& 0.02& 0.03& 20.98& 20.36& 19.68& 0.02& 0.03& 0.01& 0.15& 0.77& 0.50& 1.01& 0.68& 0.62& 1.30& 461$\pm$75\
GC0199 &AAT204119 & 13 25 25.70 &-42 37 40.9& 23.47 & 20.59& 20.49& 19.81& 19.35& 18.93& 0.18& 0.02& 0.01& 0.01& 0.03& 20.60& 20.04& 19.42& 0.01& - & - & 0.10& 0.67& 0.47& 0.88& 0.62& 0.56& 1.18& 404$\pm$74\
GC0200 &pff\_gc-052 & 13 25 25.75 &-43 05 16.5& 4.14 & 20.88& 20.70& 19.88& 19.39& 18.86& 0.19& 0.06& 0.02& 0.02& 0.03& 20.86& 20.17& 19.42& 0.03& 0.03& 0.03& 0.18& 0.82& 0.48& 1.01& 0.75& 0.69& 1.44& 462$\pm$52\
GC0201 &HGHH-46/C46 & 13 25 25.97 &-43 03 25.7& 2.30 & 19.74& 19.71& 19.10& 18.74& 18.30& 0.37& 0.12& 0.04& 0.04& 0.05& 19.77& 19.30& 18.64& 0.01& 0.02& 0.22& 0.03& 0.61& 0.36& 0.80& 0.66& 0.47& 1.13& 508$\pm$19\
GC0202 &C144 & 13 25 26.28 &-43 04 38.5& 3.50 & - & - & - & - & - & - & - & - & - & - & 23.22& 22.36& 21.96& 0.06& 0.05& 0.05& - & - & - & - & 0.41& 0.85& 1.26& -\
GC0203 &AAT116531 & 13 25 26.75 &-43 08 53.4& 7.74 & 20.92& 20.65& 19.83& 19.34& 18.83& 0.10& 0.03& 0.01& 0.01& 0.03& 20.77& 19.95& 19.94& 0.07& 0.06& 0.02& 0.27& 0.82& 0.49& 1.00& 0.01& 0.82& 0.83& 267$\pm$71\
GC0204 & WHH-15 & 13 25 26.78 &-42 52 39.9& 8.48 & 20.73& 20.56& 19.81& 19.36& 18.87& 0.09& 0.03& 0.01& 0.01& 0.03& 20.68& 20.80& 19.53& 0.03& 0.02& 0.01& 0.16& 0.75& 0.46& 0.94& 1.27&-0.12& 1.15& 513$\pm$53\
GC0205 & R235 & 13 25 26.82 &-43 09 40.5& 8.53 & 20.55& 20.30& 19.53& 19.02& 18.51& 0.08& 0.02& 0.01& 0.01& 0.03& 20.42& 19.73& 19.01& 0.04& 0.03& 0.04& 0.25& 0.77& 0.51& 1.03& 0.72& 0.69& 1.41& 498$\pm$28\
GC0206 & WHH-16/K-102 & 13 25 27.97 &-43 04 02.2& 2.89 & 20.90& 20.23& 19.18& 18.56& 17.90& 0.56& 0.11& 0.03& 0.02& 0.04& 20.64& 19.52& 18.60& 0.04& 0.03& 0.03& 0.66& 1.06& 0.61& 1.28& 0.93& 1.11& 2.04& 661$\pm$47\
GC0207 & HCH13 & 13 25 28.69 &-43 02 55.0& 1.78 & 21.32& 20.82& 19.89& 19.30& 18.63& 2.53& 0.55& 0.15& 0.12& 0.12& - & - & - & - & - & - & 0.50& 0.93& 0.59& 1.26& - & - & - & 641$\pm$24\
GC0208 &C145 & 13 25 28.81 &-43 04 21.6& 3.22 & - & - & - & - & - & - & - & - & - & - & 19.32& 18.54& 17.81& 0.01& 0.01& 0.02& - & - & - & - & 0.73& 0.78& 1.51& -\
GC0209 & WHH-17 & 13 25 29.25 &-42 57 47.1& 3.38 & 19.68& 19.39& 18.57& 18.06& 17.48& 0.15& 0.04& 0.02& 0.01& 0.03& 19.54& 18.77& 18.13& 0.02& 0.03& - & 0.29& 0.82& 0.51& 1.09& 0.65& 0.76& 1.41& 619$\pm$45\
GC0210 &AAT116969 & 13 25 29.41 &-42 53 25.6& 7.73 & 21.55& 21.28& 20.46& 20.10& 19.48& 0.21& 0.06& 0.02& 0.02& 0.03& 21.45& 20.77& 20.10& 0.03& 0.02& 0.01& 0.27& 0.82& 0.35& 0.98& 0.68& 0.68& 1.36& 446$\pm$61\
GC0211 &HGHH-G169 & 13 25 29.43 &-42 58 09.9& 3.00 & 21.39& 20.58& 19.49& 18.92& 18.12& 0.81& 0.13& 0.03& 0.03& 0.04& 21.00& 19.80& 18.94& 0.01& 0.02& 0.01& 0.81& 1.10& 0.57& 1.37& 0.86& 1.21& 2.07& 643$\pm$24\
GC0212 &pff\_gc-053 & 13 25 29.62 &-42 54 44.5& 6.42 & 20.91& 20.29& 19.26& 18.73& 17.98& 0.14& 0.03& 0.01& 0.01& 0.03& 20.65& 19.54& 18.78& 0.03& 0.04& 0.01& 0.62& 1.03& 0.53& 1.29& 0.76& 1.11& 1.86& 439$\pm$20\
GC0213 & R229 & 13 25 29.74 &-43 11 42.8& 10.57 & 20.70& 20.49& 19.72& 19.20& 18.69& 0.08& 0.02& 0.01& 0.01& 0.03& 20.62& 19.94& 19.22& 0.04& 0.03& 0.04& 0.21& 0.77& 0.52& 1.03& 0.73& 0.68& 1.41& 517$\pm$66\
GC0214 &HCH15 & 13 25 29.80 &-43 00 07.0& 1.10 & 19.27& 18.87& 17.93& 17.37& 16.76& 0.98& 0.26& 0.07& 0.05& 0.05& - & - & - & - & - & - & 0.04& 0.94& 0.56& 1.17& - & - & - & 519$\pm$1\
GC0215 &C146 & 13 25 29.87 &-43 05 09.2& 4.03 & - & - & - & - & - & - & - & - & - & - & 21.71& 20.78& 19.94& 0.03& 0.02& 0.02& - & - & - & - & 0.84& 0.93& 1.77& -\
GC0216 & WHH-18 & 13 25 30.07 &-42 56 46.9& 4.39 & 19.85& 19.36& 18.36& 17.85& 17.16& 0.09& 0.02& 0.01& 0.01& 0.03& 19.63& 18.60& 17.88& 0.01& 0.03& - & 0.50& 0.99& 0.51& 1.20& 0.72& 1.03& 1.75& 752$\pm$25\
GC0217 &pff\_gc-054 & 13 25 30.28 &-43 41 53.6& 40.75 & 20.65& 20.60& 19.99& 19.57& 19.09& 0.19& 0.03& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.05& 0.62& 0.41& 0.89& - & - & - & 297$\pm$40\
GC0218 & HCH16 & 13 25 30.29 &-42 59 34.8& 1.64 & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & 458$\pm$2\
GC0219 &HHH86-15/C15/R226 & 13 25 30.41 &-43 11 49.6& 10.69 & 20.11& 19.56& 18.56& 17.94& 17.35& 0.06& 0.02& 0.01& 0.01& 0.03& 19.80& 18.68& 17.92& 0.05& 0.05& 0.05& 0.55& 1.00& 0.62& 1.21& 0.76& 1.12& 1.88& 644$\pm$1\
GC0220 &C147 & 13 25 30.65 &-43 03 47.1& 2.70 & - & - & - & - & - & - & - & - & - & - & 21.58& 20.88& 20.05& 0.05& 0.05& 0.03& - & - & - & - & 0.83& 0.69& 1.52& -\
GC0221 &pff\_gc-055 & 13 25 30.72 &-42 48 13.4& 12.94 & 20.87& 20.68& 19.92& 19.44& 18.97& 0.08& 0.03& 0.01& 0.01& 0.03& 20.83& 20.20& 19.54& 0.02& 0.02& 0.02& 0.19& 0.76& 0.48& 0.96& 0.66& 0.62& 1.28& 485$\pm$47\
GC0222 &AAT205071 & 13 25 30.74 &-42 30 16.0& 30.89 & 19.65& 19.61& 18.86& 18.37& 17.96& 0.09& 0.01& 0.01& 0.01& 0.03& 19.68& 19.10& 18.50& 0.02& 0.02& - & 0.04& 0.74& 0.49& 0.90& 0.60& 0.58& 1.18& 288$\pm$61\
GC0223 & WHH-19 & 13 25 31.03 &-42 50 14.9& 10.92 & 18.47& 18.16& 17.29& 16.78& 16.24& 0.02& 0.01& 0.01& 0.01& 0.03& 18.08& 17.67& 16.91& 0.06& 0.06& 0.02& 0.31& 0.87& 0.51& 1.05& 0.76& 0.42& 1.18& 451$\pm$40\
GC0224 &AAT117287 & 13 25 31.08 &-43 04 17.0& 3.20 & 22.20& 21.44& 20.36& 19.82& 19.13& 1.22& 0.20& 0.05& 0.04& 0.05& 21.82& 20.56& 20.45& 0.04& 0.03& 0.03& 0.76& 1.08& 0.54& 1.23& 0.11& 1.27& 1.37& 554$\pm$60\
GC0225 &HGHH-G292 & 13 25 31.48 &-42 58 08.3& 3.09 & 21.77& 21.05& 19.97& 19.33& 18.64& 1.04& 0.18& 0.05& 0.03& 0.04& 21.48& 20.85& 19.53& 0.01& 0.02& 0.01& 0.72& 1.08& 0.64& 1.34& 1.33& 0.63& 1.95& 655$\pm$43\
GC0226 & HCH18 & 13 25 31.60 &-43 00 02.8& 1.32 & 21.06& 20.99& 19.93& 19.20& 18.43& 2.42& 0.76& 0.19& 0.13& 0.12& - & - & - & - & - & - & 0.07& 1.06& 0.73& 1.50& - & - & - & 455$\pm$1\
GC0227 &AAT117322 & 13 25 31.73 &-42 55 15.7& 5.93 & 21.14& 20.91& 20.08& 19.55& 19.07& 0.19& 0.05& 0.02& 0.02& 0.03& 21.07& 20.38& 19.75& 0.04& 0.05& 0.01& 0.22& 0.84& 0.52& 1.01& 0.63& 0.69& 1.32& 636$\pm$133\
GC0228 &HGHH-44/C44 & 13 25 31.73 &-43 19 22.6& 18.25 & 19.76& 19.50& 18.69& 18.14& 17.66& 0.04& 0.01& 0.01& 0.01& 0.03& 19.59& 18.85& 18.15& 0.03& 0.02& 0.01& 0.26& 0.80& 0.55& 1.03& 0.70& 0.74& 1.44& 505$\pm$1\
GC0229 &C148 & 13 25 31.75 &-43 05 46.0& 4.68 & - & - & - & - & - & - & - & - & - & - & 21.25& 20.79& 20.21& 0.07& 0.06& 0.05& - & - & - & - & 0.58& 0.46& 1.04& -\
GC0230 & R224/C149 & 13 25 32.33 &-43 07 17.1& 6.20 & 21.07& 20.90& 20.15& 19.64& 19.19& 0.15& 0.04& 0.02& 0.02& 0.03& 21.02& 20.42& 19.68& 0.03& 0.02& 0.02& 0.16& 0.76& 0.50& 0.96& 0.74& 0.60& 1.34& 389$\pm$45\
GC0231 &HGHH-G359 & 13 25 32.42 &-42 58 50.2& 2.47 & 20.44& 19.86& 18.86& 18.24& 17.64& 0.45& 0.10& 0.03& 0.02& 0.04& 20.13& 19.09& 18.33& 0.01& 0.01& - & 0.58& 1.00& 0.62& 1.22& 0.76& 1.04& 1.80& 489$\pm$34\
GC0232 &pff\_gc-056 & 13 25 32.80 &-42 56 24.4& 4.83 & 19.62& 19.43& 18.64& 18.15& 17.64& 0.06& 0.02& 0.01& 0.01& 0.03& 19.51& 18.82& 18.22& 0.03& 0.04& - & 0.19& 0.79& 0.48& 1.00& 0.60& 0.69& 1.29& 306$\pm$27\
GC0233 & R223 & 13 25 32.80 &-43 07 02.2& 5.97 & 20.08& 19.67& 18.77& 18.23& 17.64& 0.07& 0.02& 0.01& 0.01& 0.03& 19.89& 18.98& 18.19& 0.03& 0.03& 0.03& 0.41& 0.91& 0.54& 1.13& 0.79& 0.92& 1.71& 776$\pm$1\
GC0234 & K-131 & 13 25 32.88 &-43 04 29.2& 3.48 & 21.23& 20.44& 19.37& 18.77& 18.12& 0.38& 0.07& 0.02& 0.02& 0.03& 20.84& 19.65& 18.74& 0.03& 0.02& 0.03& 0.79& 1.08& 0.59& 1.24& 0.91& 1.20& 2.10& 639$\pm$44\
GC0235 &pff\_gc-057 & 13 25 33.17 &-42 59 03.2& 2.33 & 22.21& 21.38& 20.35& 19.75& 18.95& 2.11& 0.37& 0.09& 0.07& 0.07& 21.67& 20.75& 19.86& 0.03& 0.02& 0.01& 0.82& 1.04& 0.60& 1.40& 0.90& 0.91& 1.81& 515$\pm$12\
GC0236 &C150 & 13 25 33.82 &-43 02 49.6& 2.03 & - & - & - & - & - & - & - & - & - & - & - & - & 19.78& - & - & - & - & - & - & - & - & - & - & -\
GC0237 &C151 & 13 25 33.93 &-43 03 51.4& 2.94 & - & - & - & - & - & - & - & - & - & - & 22.27& 20.90& 19.95& 0.03& 0.01& 0.03& - & - & - & - & 0.95& 1.37& 2.32& -\
GC0238 & MAGJM-11 & 13 25 33.94 &-42 59 39.4& 1.89 & 19.80& 19.64& 18.87& 18.44& 17.83& 0.80& 0.25& 0.07& 0.06& 0.07& - & - & - & - & - & - & 0.16& 0.77& 0.43& 1.04& - & - & - & 444$\pm$17\
GC0239 &HGHH-G206 & 13 25 34.10 &-42 59 00.7& 2.44 & 20.68& 20.09& 19.10& 18.58& 17.91& 0.56& 0.12& 0.03& 0.03& 0.04& 20.37& 19.36& 18.61& 0.02& 0.03& 0.01& 0.58& 0.99& 0.52& 1.19& 0.75& 1.00& 1.75& 600$\pm$24\
GC0240 &HGHH-45/C45 & 13 25 34.25 &-42 56 59.1& 4.33 & 19.99& 19.81& 19.03& 18.60& 18.05& 0.10& 0.03& 0.01& 0.01& 0.03& 19.91& 19.17& 18.67& 0.08& 0.07& 0.02& 0.18& 0.78& 0.43& 0.98& 0.50& 0.75& 1.25& 612$\pm$34\
GC0241 & WHH-20 & 13 25 34.36 &-42 51 05.9& 10.12 & 20.30& 19.93& 19.01& 18.52& 17.90& 0.06& 0.02& 0.01& 0.01& 0.03& 20.11& 19.25& 18.59& 0.04& 0.05& 0.01& 0.37& 0.92& 0.49& 1.10& 0.66& 0.86& 1.52& 259$\pm$33\
GC0242 &C152 & 13 25 34.64 &-43 03 16.4& 2.48 & - & - & - & - & - & - & - & - & - & - & - & - & 17.82& - & - & - & - & - & - & - & - & - & - & -\
GC0243 &C153 & 13 25 34.64 &-43 03 27.8& 2.65 & - & - & - & - & - & - & - & - & - & - & - & - & 18.23& - & - & - & - & - & - & - & - & - & - & -\
GC0244 & HCH21 & 13 25 34.65 &-43 03 27.7& 2.65 & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & 663$\pm$2\
GC0245 &C154 & 13 25 34.71 &-43 03 30.2& 2.69 & - & - & - & - & - & - & - & - & - & - & - & - & 19.58& - & - & - & - & - & - & - & - & - & - & -\
GC0246 &HHH86-16/C16 & 13 25 35.00 &-42 36 05.0& 25.10 & 20.01& 19.57& 18.59& 18.00& 17.41& 0.12& 0.01& 0.01& 0.01& 0.03& 19.85& 18.83& 18.12& 0.03& 0.03& 0.01& 0.44& 0.98& 0.59& 1.19& 0.70& 1.02& 1.73& 538$\pm$16\
GC0247 &pff\_gc-058 & 13 25 35.12 &-42 56 45.3& 4.60 & 19.67& 19.47& 18.68& 18.19& 17.70& 0.07& 0.02& 0.01& 0.01& 0.03& 19.55& 18.85& 18.29& 0.03& 0.03& 0.01& 0.20& 0.78& 0.49& 0.99& 0.56& 0.70& 1.26& 365$\pm$22\
GC0248 & K-144 & 13 25 35.16 &-42 53 01.0& 8.25 & 22.10& 21.31& 20.25& 19.63& 18.94& 0.33& 0.05& 0.02& 0.01& 0.03& 21.72& 20.44& 19.75& 0.01& 0.01& 0.01& 0.79& 1.06& 0.62& 1.31& 0.69& 1.28& 1.97& 593$\pm$43\
GC0249 & WHH-21 & 13 25 35.22 &-43 12 01.5& 10.97 & 19.64& 19.60& 19.00& 18.63& 18.25& 0.04& 0.01& 0.01& 0.01& 0.03& 19.61& 19.15& 18.62& 0.02& 0.01& 0.02& 0.04& 0.60& 0.37& 0.75& 0.53& 0.46& 0.99& 243$\pm$61\
GC0250 & WHH-22 & 13 25 35.31 &-43 05 29.0& 4.56 & - & - & - & - & - & - & - & - & - & - & 19.66& 18.80& 18.03& 0.04& 0.04& 0.04& - & - & - & - & 0.77& 0.86& 1.63& 492$\pm$126\
GC0251 & MAGJM-08 & 13 25 35.50 &-42 59 35.2& 2.12 & 20.76& 20.18& 19.17& 18.68& 17.94& 1.15& 0.24& 0.06& 0.05& 0.05& - & - & - & - & - & - & 0.58& 1.01& 0.49& 1.22& - & - & - & 701$\pm$28\
GC0252 & R215 & 13 25 35.64 &-43 08 36.8& 7.61 & 22.06& 21.68& 20.80& 20.27& 19.70& 0.27& 0.06& 0.02& 0.02& 0.03& 21.98& 21.18& 20.35& 0.03& 0.03& 0.04& 0.38& 0.87& 0.53& 1.11& 0.83& 0.79& 1.63& 543$\pm$57\
GC0253 & R213 & 13 25 35.93 &-43 07 27.9& 6.50 & 22.70& 22.03& 20.92& 20.35& 19.62& 0.56& 0.10& 0.03& 0.02& 0.03& 22.34& 21.26& 20.34& 0.03& 0.01& 0.02& 0.67& 1.11& 0.58& 1.30& 0.92& 1.08& 2.00& 532$\pm$52\
GC0254 &AAT117899 & 13 25 36.05 &-42 53 40.3& 7.63 & 21.70& 21.17& 20.17& 19.62& 18.97& 0.23& 0.05& 0.02& 0.01& 0.03& 21.51& 20.51& 19.70& 0.03& 0.03& 0.01& 0.53& 1.01& 0.54& 1.20& 0.81& 1.01& 1.81& 609$\pm$116\
GC0255 &C155 & 13 25 36.47 &-43 08 03.5& 7.10 & - & - & - & - & - & - & - & - & - & - & 22.98& 22.21& 21.36& 0.06& 0.05& 0.05& - & - & - & - & 0.85& 0.77& 1.63& -\
GC0256 & MAGJM-06 & 13 25 36.69 &-42 59 59.2& 2.02 & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & 443$\pm$15\
GC0257 &AAT118198 & 13 25 37.47 &-43 05 44.9& 4.94 & 21.49& 20.68& 19.64& 18.98& 18.29& 0.31& 0.05& 0.02& 0.01& 0.03& 21.14& 19.94& 19.03& 0.04& 0.02& 0.03& 0.81& 1.04& 0.66& 1.35& 0.91& 1.20& 2.11& 575$\pm$36\
GC0258 &AAT118314 & 13 25 38.03 &-43 16 59.2& 15.95 & 20.69& 20.62& 19.89& 19.45& 18.98& 0.08& 0.03& 0.01& 0.01& 0.03& 20.67& 20.13& 19.42& 0.01& 0.01& 0.01& 0.08& 0.72& 0.44& 0.92& 0.71& 0.53& 1.24& 257$\pm$59\
GC0259 & R209 & 13 25 38.13 &-43 13 02.2& 12.04 & 21.92& 21.51& 20.62& 20.05& 19.43& 0.20& 0.05& 0.02& 0.01& 0.03& 21.91& 20.95& 20.13& 0.04& 0.04& 0.04& 0.41& 0.90& 0.57& 1.19& 0.83& 0.96& 1.79& 558$\pm$40\
GC0260 &C156 & 13 25 38.43 &-43 05 02.6& 4.37 & - & - & - & - & - & - & - & - & - & - & 19.76& 18.55& 17.65& 0.02& 0.02& 0.03& - & - & - & - & 0.90& 1.21& 2.11& -\
GC0261 &C157 & 13 25 38.45 &-43 03 28.9& 3.06 & - & - & - & - & - & - & - & - & - & - & - & - & 19.21& - & - & - & - & - & - & - & - & - & - & -\
GC0262 &HGHH-G268 & 13 25 38.61 &-42 59 19.6& 2.71 & 20.23& 19.83& 18.93& 18.36& 17.79& 0.34& 0.08& 0.03& 0.02& 0.04& 20.05& 19.15& 18.44& 0.02& 0.02& - & 0.40& 0.91& 0.56& 1.13& 0.71& 0.91& 1.62& 436$\pm$43\
GC0263 &C158 & 13 25 38.76 &-43 05 34.5& 4.88 & - & - & - & - & - & - & - & - & - & - & 21.67& 20.86& 20.06& 0.05& 0.04& 0.05& - & - & - & - & 0.80& 0.81& 1.61& -\
GC0264 &C159 & 13 25 39.17 &-43 04 33.8& 4.02 & - & - & - & - & - & - & - & - & - & - & 21.91& 20.82& 19.93& 0.02& 0.02& 0.02& - & - & - & - & 0.90& 1.09& 1.99& -\
GC0265 & MAGJM-01 & 13 25 39.33 &-43 00 48.8& 2.17 & 19.90& 19.52& 18.61& 18.06& 17.50& 0.32& 0.08& 0.02& 0.02& 0.03& - & - & - & - & - & - & 0.37& 0.91& 0.55& 1.11& - & - & - & 645$\pm$36\
GC0266 &pff\_gc-059 & 13 25 39.65 &-43 04 01.4& 3.62 & 21.53& 20.97& 19.96& 19.35& 18.73& 0.55& 0.11& 0.03& 0.02& 0.04& 21.42& 20.33& 19.48& 0.05& 0.04& 0.04& 0.56& 1.01& 0.61& 1.23& 0.86& 1.09& 1.94& 525$\pm$27\
GC0267 &HGHH-17/C17 & 13 25 39.73 &-42 55 59.2& 5.62 & 18.82& 18.51& 17.63& 17.10& 16.57& 0.04& 0.01& 0.01& 0.01& 0.03& 18.61& 17.72& 17.19& 0.04& 0.05& 0.01& 0.32& 0.88& 0.53& 1.06& 0.53& 0.89& 1.42& 782$\pm$2\
GC0268 &HHH86-18/C18/K-163 & 13 25 39.88 &-43 05 01.9& 4.49 & 18.79& 18.42& 17.53& 16.97& 16.43& 0.04& 0.01& 0.01& 0.01& 0.03& 18.49& 17.51& 16.89& 0.05& 0.04& 0.05& 0.38& 0.89& 0.56& 1.10& 0.62& 0.99& 1.60& 480$\pm$2\
GC0269 &C160 & 13 25 40.09 &-43 03 07.1& 3.01 & - & - & - & - & - & - & - & - & - & - & - & - & 19.99& - & - & - & - & - & - & - & - & - & - & -\
GC0270 &C101 & 13 25 40.47 &-42 56 02.7& 5.65 & - & - & 20.34& - & - & - & - & - & - & - & 21.84& 20.90& 20.08& 0.03& 0.04& 0.01& - & - & - & - & 0.82& 0.94& 1.76& -\
GC0271 & R204/C161 & 13 25 40.52 &-43 07 17.9& 6.59 & 21.58& 20.84& 19.80& 19.16& 18.41& 0.21& 0.04& 0.01& 0.01& 0.03& 21.21& 20.15& 19.26& 0.04& 0.03& 0.03& 0.74& 1.05& 0.63& 1.39& 0.89& 1.06& 1.95& 425$\pm$32\
GC0272 &HGHH-34/C34 & 13 25 40.61 &-43 21 13.6& 20.22 & 19.37& 19.01& 18.12& 17.59& 17.03& 0.04& 0.01& 0.01& 0.01& 0.03& 19.64& 18.54& 17.76& 0.04& 0.03& 0.02& 0.35& 0.89& 0.53& 1.10& 0.77& 1.10& 1.87& 669$\pm$18\
GC0273 &C162 & 13 25 40.87 &-43 05 00.4& 4.56 & - & - & - & - & - & - & - & - & - & - & 21.95& 21.41& 20.91& 0.06& 0.05& 0.01& - & - & - & - & 0.51& 0.54& 1.05& -\
GC0274 & R203 & 13 25 40.90 &-43 08 16.0& 7.52 & 20.66& 20.56& 19.84& 19.36& 18.88& 0.09& 0.03& 0.01& 0.01& 0.03& 20.62& 20.08& 19.38& 0.04& 0.03& 0.03& 0.09& 0.73& 0.48& 0.96& 0.70& 0.54& 1.24& 455$\pm$35\
GC0275 &AAT118874 & 13 25 41.36 &-42 58 08.9& 3.91 & 21.55& 21.02& 19.99& 19.43& 18.81& 0.44& 0.09& 0.03& 0.02& 0.04& 21.49& 20.30& 19.57& 0.05& 0.02& - & 0.53& 1.03& 0.56& 1.18& 0.73& 1.19& 1.92& 570$\pm$82\
GC0276 &C163 & 13 25 41.63 &-43 03 45.8& 3.66 & - & - & - & - & - & - & - & - & - & - & 21.98& 21.25& 20.43& 0.05& 0.04& 0.04& - & - & - & - & 0.81& 0.73& 1.55& -\
GC0277 & R202 & 13 25 42.00 &-43 10 42.2& 9.91 & 20.34& 20.08& 19.26& 18.72& 18.22& 0.06& 0.02& 0.01& 0.01& 0.03& 20.35& 19.74& 18.94& 0.07& 0.03& 0.06& 0.27& 0.81& 0.54& 1.05& 0.80& 0.61& 1.41& 286$\pm$45\
GC0278 &C164 & 13 25 42.09 &-43 03 19.5& 3.43 & - & - & - & - & - & - & - & - & - & - & - & - & 19.60& - & - & - & - & - & - & - & - & - & - & -\
GC0279 &HGHH-G370 & 13 25 42.25 &-42 59 17.0& 3.26 & 19.61& 19.29& 18.39& 17.86& 17.32& 0.17& 0.05& 0.02& 0.01& 0.03& 19.45& 18.59& 17.93& 0.01& 0.03& 0.01& 0.32& 0.90& 0.54& 1.08& 0.66& 0.86& 1.52& 507$\pm$34\
GC0280 &pff\_gc-060 & 13 25 42.43 &-42 59 02.6& 3.43 & 20.11& 19.74& 18.81& 18.24& 17.66& 0.20& 0.05& 0.02& 0.01& 0.03& 19.96& 19.02& 18.33& 0.02& 0.03& 0.01& 0.37& 0.93& 0.57& 1.15& 0.69& 0.95& 1.63& 890$\pm$19\
GC0281 &AAT119058 & 13 25 42.53 &-43 03 41.5& 3.73 & 21.85& 21.42& 20.39& 19.77& 19.18& 0.60& 0.13& 0.04& 0.03& 0.04& 21.69& 20.74& 19.86& 0.04& 0.03& 0.04& 0.43& 1.03& 0.62& 1.21& 0.88& 0.95& 1.83& 385$\pm$87\
GC0282 &pff\_gc-061 & 13 25 42.62 &-42 45 10.9& 16.20 & 21.59& 21.15& 20.25& 19.69& 19.12& 0.14& 0.03& 0.01& 0.01& 0.03& 21.42& 20.55& 19.75& 0.01& 0.01& 0.01& 0.44& 0.90& 0.55& 1.13& 0.80& 0.87& 1.67& 500$\pm$48\
GC0283 &pff\_gc-062 & 13 25 43.23 &-42 58 37.4& 3.81 & 21.13& 20.46& 19.42& 18.83& 18.18& 0.34& 0.06& 0.02& 0.02& 0.03& 20.79& 19.71& 18.92& 0.01& 0.01& 0.01& 0.67& 1.04& 0.59& 1.24& 0.80& 1.08& 1.88& 697$\pm$37\
GC0284 &HGHH-19/C19 & 13 25 43.40 &-43 07 22.8& 6.87 & 19.37& 19.01& 18.12& 17.59& 17.03& 0.04& 0.01& 0.01& 0.01& 0.03& 18.96& 17.83& 18.64& 0.05& 0.04& 0.01& 0.35& 0.89& 0.53& 1.10&-0.80& 1.13& 0.32& 632$\pm$10\
GC0285 &C165 & 13 25 43.43 &-43 04 56.5& 4.77 & - & - & - & - & - & - & - & - & - & - & 20.10& 18.98& 18.17& 0.05& 0.04& 0.05& - & - & - & - & 0.81& 1.12& 1.93& -\
GC0286 &pff\_gc-063 & 13 25 43.80 &-43 07 54.9& 7.39 & 20.66& 20.55& 19.83& 19.37& 18.89& 0.08& 0.03& 0.01& 0.01& 0.03& 20.63& 20.13& 19.45& 0.04& 0.03& 0.03& 0.11& 0.72& 0.46& 0.94& 0.67& 0.50& 1.18& 554$\pm$75\
GC0287 &pff\_gc-064 & 13 25 43.90 &-42 50 42.7& 10.85 & 21.16& 20.67& 19.66& 19.07& 18.41& 0.11& 0.03& 0.01& 0.01& 0.03& 20.96& 19.97& 19.20& 0.02& 0.03& 0.01& 0.50& 1.01& 0.59& 1.25& 0.76& 0.99& 1.76& 560$\pm$24\
GC0288 &HGHH-35/C35 & 13 25 44.21 &-42 58 59.4& 3.72 & 19.92& 19.51& 18.58& 18.01& 17.43& 0.13& 0.03& 0.01& 0.01& 0.03& 19.74& 18.79& 18.11& 0.02& 0.02& 0.01& 0.41& 0.93& 0.57& 1.14& 0.68& 0.94& 1.63& 544$\pm$13\
GC0289 &C166 & 13 25 44.90 &-43 04 21.1& 4.50 & - & - & - & - & - & - & - & - & - & - & - & - & 20.72& - & - & - & - & - & - & - & - & - & - & -\
GC0290 &AAT208065 & 13 25 45.77 &-42 34 18.0& 27.05 & 21.79& 21.45& 20.09& 19.40& 18.71& 0.49& 0.05& 0.01& 0.01& 0.03& 22.06& 20.70& 19.66& 0.04& 0.03& 0.02& 0.33& 1.36& 0.69& 1.39& 1.04& 1.36& 2.40& 836$\pm$41\
GC0291 & WHH-23 & 13 25 45.90 &-42 57 20.2& 5.07 & 19.26& 19.12& 18.37& 17.89& 17.41& 0.05& 0.02& 0.01& 0.01& 0.03& 19.13& 18.50& 18.00& 0.03& 0.05& 0.01& 0.14& 0.75& 0.48& 0.96& 0.50& 0.64& 1.14& 286$\pm$63\
GC0292 &C167 & 13 25 45.97 &-43 06 45.4& 6.54 & - & - & - & - & - & - & - & - & - & - & - & - & 20.41& - & - & - & - & - & - & - & - & - & - & -\
GC0293 & WHH-24 & 13 25 46.00 &-42 56 53.0& 5.43 & 20.64& 20.43& 19.63& 19.12& 18.62& 0.12& 0.04& 0.01& 0.01& 0.03& 20.54& 19.91& 19.25& 0.02& 0.03& 0.01& 0.21& 0.80& 0.51& 1.01& 0.66& 0.63& 1.29& 566$\pm$48\
GC0294 &AAT119596 & 13 25 46.06 &-43 08 24.5& 8.01 & 21.19& 20.69& 19.70& 19.10& 18.51& 0.12& 0.03& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.50& 1.00& 0.60& 1.19& - & - & - & 573$\pm$53\
GC0295 &HGHH-G378 & 13 25 46.13 &-43 01 22.9& 3.39 & 21.30& 20.59& 19.53& 18.86& 18.22& 0.55& 0.10& 0.03& 0.02& 0.03& - & - & - & - & - & - & 0.71& 1.06& 0.67& 1.31& - & - & - & 487$\pm$31\
GC0296 &AAT208206 & 13 25 46.49 &-42 34 54.1& 26.47 & 20.08& 20.10& 19.47& 19.07& 18.69& 0.12& 0.02& 0.01& 0.01& 0.03& 20.13& 19.73& 19.19& 0.01& 0.01& 0.01&-0.02& 0.63& 0.40& 0.78& 0.55& 0.40& 0.94& 293$\pm$115\
GC0297 &HGHH-G284 & 13 25 46.59 &-42 57 03.0& 5.37 & 21.47& 20.89& 19.87& 19.28& 18.63& 0.26& 0.05& 0.02& 0.01& 0.03& 21.26& 20.20& 19.41& 0.01& 0.01& 0.01& 0.57& 1.03& 0.58& 1.24& 0.79& 1.06& 1.85& 479$\pm$12\
GC0298 &AAT119697 & 13 25 46.68 &-42 53 48.6& 8.12 & 21.61& 21.06& 20.05& 19.49& 18.87& 0.23& 0.05& 0.02& 0.01& 0.03& 21.42& 20.42& 19.65& 0.02& 0.02& 0.01& 0.54& 1.01& 0.57& 1.18& 0.77& 1.01& 1.78& 633$\pm$108\
GC0299 &pff\_gc-065 & 13 25 46.92 &-43 08 06.6& 7.81 & 20.60& 20.17& 19.23& 18.66& 18.08& 0.08& 0.02& 0.01& 0.01& 0.03& 20.43& 19.55& 18.71& 0.01& 0.01& 0.01& 0.42& 0.94& 0.57& 1.15& 0.83& 0.88& 1.72& 450$\pm$44\
GC0300 &HGHH-G204 & 13 25 46.99 &-43 02 05.4& 3.66 & 19.35& 19.12& 18.33& 17.83& 17.31& 0.12& 0.04& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.23& 0.79& 0.51& 1.02& - & - & - & 690$\pm$18\
GC0301 &pff\_gc-066 & 13 25 47.14 &-43 06 08.8& 6.14 & 21.21& 20.82& 19.92& 19.37& 18.80& 0.15& 0.04& 0.01& 0.01& 0.03& 21.07& 20.21& 19.44& 0.01& 0.01& 0.01& 0.39& 0.91& 0.55& 1.11& 0.77& 0.86& 1.63& 530$\pm$44\
GC0302 &HGHH-G357 & 13 25 47.78 &-43 00 43.4& 3.71 & 20.87& 20.55& 19.71& 19.24& 18.71& 0.26& 0.07& 0.02& 0.02& 0.04& 20.71& 19.96& 19.31& 0.01& 0.02& 0.01& 0.33& 0.83& 0.47& 1.00& 0.66& 0.75& 1.41& 664$\pm$31\
GC0303 &AAT119894 & 13 25 47.92 &-42 55 52.4& 6.45 & 21.45& 21.30& 20.25& 19.59& 18.92& 0.23& 0.07& 0.02& 0.02& 0.03& 21.68& 20.75& 19.85& 0.05& 0.04& 0.03& 0.15& 1.05& 0.66& 1.33& 0.90& 0.93& 1.84& 448$\pm$90\
GC0304 &C168 & 13 25 48.46 &-43 07 12.5& 7.16 & - & - & - & - & - & - & - & - & - & - & - & - & 19.71& - & - & - & - & - & - & - & - & - & - & -\
GC0305 &HGHH-G251 & 13 25 48.54 &-42 57 41.2& 5.16 & 20.50& 19.93& 18.93& 18.33& 17.69& 0.13& 0.03& 0.01& 0.01& 0.03& 20.23& 19.22& 18.46& 0.02& 0.01& 0.01& 0.57& 1.00& 0.60& 1.24& 0.76& 1.01& 1.77& 574$\pm$27\
GC0306 &pff\_gc-067 & 13 25 48.77 &-43 11 38.7& 11.19 & 20.51& 20.35& 19.60& 19.08& 18.60& 0.07& 0.02& 0.01& 0.01& 0.03& 20.47& 19.87& 19.14& 0.01& 0.01& 0.01& 0.16& 0.75& 0.52& 1.00& 0.74& 0.60& 1.33& 632$\pm$43\
GC0307 &pff\_gc-068 & 13 25 49.27 &-43 02 20.4& 4.13 & 20.73& 20.41& 19.55& 18.99& 18.48& 0.20& 0.05& 0.02& 0.02& 0.03& 20.65& 20.17& 19.13& 0.03& 0.04& 0.01& 0.32& 0.85& 0.56& 1.08& 1.04& 0.48& 1.52& 393$\pm$26\
GC0308 &HGHH-20/C20 & 13 25 49.69 &-42 54 49.3& 7.50 & 19.18& 18.83& 17.95& 17.43& 16.88& 0.04& 0.01& 0.01& 0.01& 0.03& 19.01& 18.19& 17.54& 0.03& 0.02& 0.01& 0.35& 0.88& 0.53& 1.07& 0.65& 0.82& 1.47& 744$\pm$9\
GC0309 &AAT120259 & 13 25 49.73 &-43 05 04.7& 5.64 & 21.95& 21.16& 20.12& 19.51& 18.80& 0.35& 0.06& 0.02& 0.01& 0.03& 21.57& 20.48& 19.56& 0.01& 0.01& 0.01& 0.79& 1.04& 0.62& 1.32& 0.92& 1.09& 2.01& 477$\pm$54\
GC0310 &HGHH-48/C48 & 13 25 49.82 &-42 50 15.3& 11.62 & 19.63& 19.46& 18.67& 18.16& 17.67& 0.04& 0.01& 0.01& 0.01& 0.03& 19.56& 18.92& 18.28& 0.02& 0.02& 0.01& 0.17& 0.79& 0.51& 1.01& 0.64& 0.64& 1.28& 547$\pm$19\
GC0311 &pff\_gc-069 & 13 25 49.93 &-42 40 08.2& 21.40 & 21.55& 20.90& 19.87& 19.26& 18.62& 0.41& 0.03& 0.01& 0.01& 0.03& 21.29& 20.24& 19.39& 0.01& 0.01& 0.01& 0.64& 1.03& 0.61& 1.25& 0.85& 1.05& 1.90& 518$\pm$31\
GC0312 &HGHH-47/C47 & 13 25 49.95 &-42 52 09.4& 9.87 & 19.80& 19.55& 18.69& 18.18& 17.65& 0.05& 0.02& 0.01& 0.01& 0.03& 19.70& 18.97& 18.30& 0.03& 0.02& 0.01& 0.26& 0.86& 0.51& 1.04& 0.67& 0.73& 1.40& 589$\pm$22\
GC0313 &AAT120336 & 13 25 50.22 &-43 06 08.5& 6.48 & 21.66& 21.14& 20.17& 19.57& 18.96& 0.23& 0.05& 0.02& 0.01& 0.03& 21.48& 20.50& 19.67& 0.02& 0.01& 0.01& 0.52& 0.97& 0.60& 1.21& 0.83& 0.97& 1.81& 452$\pm$67\
GC0314 & WHH-25 & 13 25 50.34 &-43 04 08.2& 5.12 & 21.33& 20.93& 19.97& 19.42& 18.83& 0.22& 0.05& 0.02& 0.01& 0.03& 21.19& 20.27& 19.57& 0.02& 0.02& 0.01& 0.40& 0.95& 0.55& 1.14& 0.71& 0.92& 1.62& 525$\pm$50\
GC0315 &AAT120355 & 13 25 50.37 &-43 00 32.6& 4.20 & 21.26& 21.14& 20.42& 20.02& 19.58& 0.28& 0.09& 0.03& 0.03& 0.04& 21.32& 20.75& 20.17& 0.03& 0.03& 0.01& 0.12& 0.72& 0.39& 0.84& 0.59& 0.56& 1.15& 548$\pm$77\
GC0316 &pff\_gc-070 & 13 25 50.40 &-42 58 02.3& 5.20 & 20.39& 20.23& 19.42& 18.94& 18.45& 0.12& 0.04& 0.01& 0.01& 0.03& 20.35& 19.67& 19.06& 0.01& 0.01& 0.01& 0.16& 0.81& 0.48& 0.97& 0.61& 0.68& 1.29& 556$\pm$49\
GC0317 &C169 & 13 25 51.01 &-42 55 36.3& 7.00 & - & - & - & - & - & - & - & - & - & - & 21.60& 21.00& 20.39& 0.03& 0.03& 0.02& - & - & - & - & 0.62& 0.60& 1.21& -\
GC0318 &AAT120515 & 13 25 51.31 &-42 59 29.4& 4.64 & 21.11& 20.81& 19.92& 19.38& 18.81& 0.21& 0.06& 0.02& 0.02& 0.03& 21.04& 20.24& 19.46& 0.04& 0.04& 0.01& 0.30& 0.90& 0.53& 1.11& 0.77& 0.81& 1.58& 461$\pm$141\
GC0319 &pff\_gc-071 & 13 25 51.54 &-42 59 46.8& 4.58 & 20.79& 20.59& 19.82& 19.34& 18.83& 0.16& 0.05& 0.02& 0.02& 0.03& 20.71& 20.07& 19.45& 0.01& 0.01& 0.01& 0.20& 0.77& 0.48& 0.99& 0.62& 0.64& 1.27& 475$\pm$41\
GC0320 &C102 & 13 25 52.07 &-42 59 14.4& 4.87 & - & - & 21.43& - & - & - & - & - & - & - & 22.71& 21.87& 21.04& 0.02& 0.02& 0.02& - & - & - & - & 0.83& 0.84& 1.67& -\
GC0321 &pff\_gc-072 & 13 25 52.14 &-42 58 30.2& 5.20 & 20.68& 20.45& 19.62& 19.12& 18.54& 0.15& 0.04& 0.02& 0.01& 0.03& 20.64& 19.92& 19.25& 0.02& 0.02& 0.01& 0.23& 0.83& 0.51& 1.08& 0.67& 0.72& 1.39& 504$\pm$22\
GC0322 &HGHH-21/C21 & 13 25 52.74 &-43 05 46.4& 6.52 & 19.16& 18.76& 17.87& 17.32& 16.77& 0.04& 0.01& 0.01& 0.01& 0.03& 18.97& 18.17& 17.40& 0.03& 0.02& 0.01& 0.40& 0.89& 0.55& 1.11& 0.77& 0.80& 1.58& 462$\pm$2\
GC0323 &pff\_gc-073 & 13 25 52.78 &-42 58 41.7& 5.21 & 21.36& 20.94& 19.97& 19.38& 18.74& 0.25& 0.06& 0.02& 0.02& 0.03& 21.25& 20.29& 19.50& 0.02& 0.01& 0.01& 0.42& 0.98& 0.58& 1.22& 0.79& 0.97& 1.76& 401$\pm$29\
GC0324 &HGHH-G256 & 13 25 52.88 &-43 02 00.0& 4.70 & 20.69& 20.18& 19.00& 18.35& 17.65& 0.18& 0.04& 0.01& 0.01& 0.03& 20.50& 19.31& 18.44& 0.03& 0.04& 0.01& 0.51& 1.17& 0.65& 1.36& 0.87& 1.20& 2.06& 495$\pm$18\
GC0325 &AAT209412 & 13 25 53.30 &-42 30 52.7& 30.63 & 21.94& 21.23& 20.22& 19.60& 18.97& 0.52& 0.04& 0.01& 0.01& 0.03& 21.69& 20.64& 19.76& 0.01& 0.01& 0.01& 0.71& 1.01& 0.62& 1.25& 0.88& 1.05& 1.92& 998$\pm$135\
GC0326 &pff\_gc-074 & 13 25 53.37 &-42 51 12.4& 11.00 & 21.16& 20.83& 20.00& 19.48& 18.90& 0.11& 0.03& 0.01& 0.01& 0.03& 21.04& 20.27& 19.56& 0.01& 0.01& 0.01& 0.33& 0.83& 0.52& 1.10& 0.71& 0.77& 1.48& 471$\pm$24\
GC0327 &pff\_gc-075 & 13 25 53.50 &-43 03 56.6& 5.50 & 20.96& 20.45& 19.48& 18.90& 18.28& 0.14& 0.03& 0.01& 0.01& 0.03& 20.78& 19.76& 19.01& 0.02& 0.03& 0.01& 0.51& 0.97& 0.58& 1.20& 0.75& 1.02& 1.77& 748$\pm$18\
GC0328 &HGHH-22/C22 & 13 25 53.57 &-42 59 07.6& 5.16 & 19.39& 19.06& 18.15& 17.62& 17.07& 0.05& 0.02& 0.01& 0.01& 0.03& 19.21& 18.34& 17.70& 0.02& 0.03& - & 0.34& 0.90& 0.54& 1.09& 0.64& 0.87& 1.52& 578$\pm$1\
GC0329 &pff\_gc-076 & 13 25 53.75 &-43 19 48.6& 19.26 & 20.67& 20.06& 19.07& 18.41& 17.78& 0.08& 0.02& 0.01& 0.01& 0.03& 20.44& 19.44& 18.52& 0.02& 0.01& 0.01& 0.62& 0.99& 0.66& 1.29& 0.92& 1.00& 1.92& 368$\pm$15\
GC0330 &AAT120976 & 13 25 54.28 &-42 56 20.6& 6.84 & 21.51& 21.23& 20.38& 19.84& 19.27& 0.23& 0.06& 0.02& 0.02& 0.03& 21.50& 20.73& 19.99& 0.03& 0.02& 0.02& 0.28& 0.85& 0.54& 1.11& 0.74& 0.77& 1.51& 595$\pm$69\
GC0331 &AAT328533 & 13 25 54.39 &-43 18 40.1& 18.19 & 21.79& 21.26& 20.31& 19.74& 19.09& 0.16& 0.04& 0.01& 0.01& 0.03& 21.62& 20.64& 19.77& 0.01& 0.01& 0.01& 0.53& 0.95& 0.58& 1.23& 0.87& 0.98& 1.85& 577$\pm$89\
GC0332 &HGHH-23/C23 & 13 25 54.58 &-42 59 25.4& 5.22 & 18.92& 18.29& 17.22& 16.62& 15.95& 0.04& 0.01& 0.01& 0.01& 0.03& 18.59& 17.44& 16.69& 0.01& 0.03& - & 0.63& 1.07& 0.60& 1.28& 0.75& 1.15& 1.90& 674$\pm$1\
GC0333 &C103 & 13 25 54.98 &-42 59 15.4& 5.36 & - & - & 18.88& - & - & - & - & - & - & - & 20.43& 20.44& 18.44& 0.01& 0.02& 0.02& - & - & - & - & 2.00&-0.01& 1.99& -\
GC0334 &C170 & 13 25 56.11 &-42 56 12.9& 7.17 & - & - & - & - & - & - & - & - & - & - & 23.31& 22.83& 22.16& 0.05& 0.04& 0.06& - & - & - & - & 0.67& 0.48& 1.15& -\
GC0335 &AAT121367 & 13 25 56.26 &-43 01 32.9& 5.25 & 21.06& 20.92& 20.24& 19.82& 19.39& 0.25& 0.08& 0.03& 0.03& 0.04& 21.01& 20.45& 19.85& 0.01& 0.01& 0.01& 0.13& 0.68& 0.42& 0.85& 0.60& 0.56& 1.15& 438$\pm$80\
GC0336 & WHH-26 & 13 25 56.59 &-42 51 46.6& 10.76 & 20.50& 20.02& 19.05& 18.43& 17.81& 0.07& 0.02& 0.01& 0.01& 0.03& 20.33& 19.36& 18.56& 0.02& 0.01& 0.01& 0.48& 0.97& 0.62& 1.24& 0.79& 0.97& 1.76& 412$\pm$36\
GC0337 &AAT329209 & 13 25 57.28 &-43 41 09.0& 40.37 & 20.87& 20.70& 19.87& 19.40& 18.90& 0.21& 0.02& 0.01& 0.01& 0.03& 20.78& 20.12& 19.35& 0.01& - & 0.01& 0.17& 0.82& 0.47& 0.97& 0.77& 0.66& 1.43& 601$\pm$65\
GC0338 &C171 & 13 25 57.78 &-42 55 36.1& 7.82 & - & - & - & - & - & - & - & - & - & - & 22.25& 21.45& 20.67& 0.04& 0.03& 0.02& - & - & - & - & 0.78& 0.79& 1.58& -\
GC0339 &C172 & 13 25 57.95 &-42 53 04.3& 9.79 & - & - & - & - & - & - & - & - & - & - & 22.15& 21.45& 20.91& 0.02& 0.01& 0.01& - & - & - & - & 0.55& 0.70& 1.24& -\
GC0340 &pff\_gc-077 & 13 25 58.15 &-42 31 38.2& 30.03 & 20.57& 20.32& 19.50& 18.96& 18.47& 0.16& 0.02& 0.01& 0.01& 0.03& 20.55& 19.83& 19.09& 0.02& 0.01& 0.01& 0.24& 0.83& 0.53& 1.03& 0.74& 0.71& 1.45& 675$\pm$41\
GC0341 &pff\_gc-078 & 13 25 58.47 &-43 08 06.3& 8.96 & 20.93& 20.34& 19.30& 18.69& 18.02& 0.10& 0.02& 0.01& 0.01& 0.03& 20.70& 19.68& 18.75& 0.02& 0.01& 0.01& 0.59& 1.03& 0.62& 1.28& 0.93& 1.02& 1.95& 545$\pm$18\
GC0342 &HGHH-G143 & 13 25 58.69 &-43 07 11.0& 8.29 & 20.31& 20.02& 19.21& 18.68& 18.18& 0.06& 0.02& 0.01& 0.01& 0.03& 20.22& 19.52& 18.77& 0.02& 0.01& 0.01& 0.29& 0.81& 0.53& 1.03& 0.75& 0.70& 1.45& 503$\pm$38\
GC0343 &pff\_gc-079 & 13 25 58.91 &-42 53 18.9& 9.70 & 20.74& 20.43& 19.58& 19.06& 18.52& 0.10& 0.03& 0.01& 0.01& 0.03& 20.62& 19.86& 19.14& 0.02& 0.01& 0.01& 0.30& 0.86& 0.51& 1.06& 0.72& 0.77& 1.48& 410$\pm$21\
GC0344 &AAT121826/C104 & 13 25 59.49 &-42 55 30.8& 8.10 & 20.91& 20.65& 19.81& 19.28& 18.73& 0.13& 0.03& 0.01& 0.01& 0.03& 20.81& 20.15& 19.40& 0.02& 0.02& 0.01& 0.26& 0.83& 0.53& 1.08& 0.74& 0.66& 1.40& 448$\pm$98\
GC0345 &pff\_gc-080 & 13 25 59.55 &-42 32 39.4& 29.08 & 21.10& 20.80& 19.96& 19.40& 18.84& 0.25& 0.03& 0.01& 0.01& 0.03& 21.07& 20.32& 19.56& 0.02& 0.01& 0.01& 0.29& 0.85& 0.56& 1.11& 0.75& 0.75& 1.51& 598$\pm$56\
GC0346 &C173 & 13 25 59.57 &-42 55 01.5& 8.46 & - & - & - & - & - & - & - & - & - & - & 23.15& 22.00& 21.06& 0.04& 0.03& 0.02& - & - & - & - & 0.94& 1.15& 2.09& -\
GC0347 &C174 & 13 25 59.63 &-42 55 15.7& 8.30 & - & - & - & - & - & - & - & - & - & - & 23.24& 22.42& 21.42& 0.04& 0.04& 0.03& - & - & - & - & 0.99& 0.82& 1.81& -\
GC0348 &pff\_gc-081 & 13 26 00.15 &-42 49 00.7& 13.51 & 20.43& 20.17& 19.34& 18.80& 18.28& 0.07& 0.02& 0.01& 0.01& 0.03& 20.36& 19.61& 18.93& 0.02& 0.01& 0.01& 0.26& 0.83& 0.53& 1.05& 0.69& 0.75& 1.43& 304$\pm$31\
GC0349 & K-217 & 13 26 00.81 &-43 09 40.1& 10.46 & 21.63& 21.09& 20.09& 19.54& 18.91& 0.16& 0.03& 0.01& 0.01& 0.03& 21.42& 20.48& 19.57& 0.01& 0.01& 0.01& 0.54& 0.99& 0.56& 1.19& 0.91& 0.94& 1.85& 315$\pm$157\
GC0350 &C175 & 13 26 00.93 &-42 58 28.9& 6.65 & - & - & - & - & - & - & - & - & - & - & 23.51& 22.53& 21.84& 0.06& 0.04& 0.03& - & - & - & - & 0.69& 0.97& 1.67& -\
GC0351 &pff\_gc-082 & 13 26 00.98 &-42 22 03.4& 39.56 & 20.83& 20.17& 19.16& 18.58& 17.89& 0.21& 0.02& 0.01& 0.01& 0.03& 20.51& 19.53& 18.69& 0.02& 0.01& 0.01& 0.66& 1.01& 0.58& 1.27& 0.84& 0.98& 1.82& 573$\pm$19\
GC0352 &AAT122146 & 13 26 01.00 &-43 06 55.3& 8.40 & 21.14& 20.90& 20.05& 19.53& 19.04& 0.11& 0.03& 0.01& 0.01& 0.03& 21.09& 20.38& 19.58& 0.01& 0.01& 0.01& 0.24& 0.85& 0.52& 1.01& 0.80& 0.71& 1.51& 517$\pm$99\
GC0353 &HGHH-G221 & 13 26 01.11 &-42 55 13.5& 8.52 & 20.87& 20.24& 19.30& 18.75& 18.13& 0.10& 0.02& 0.01& 0.01& 0.03& 20.54& 19.60& 18.83& 0.01& 0.01& 0.01& 0.62& 0.94& 0.55& 1.16& 0.77& 0.94& 1.72& 390$\pm$15\
GC0354 &AAT329848 & 13 26 01.29 &-43 34 15.5& 33.68 & 19.44& 19.19& 18.39& 17.89& 17.37& 0.08& 0.01& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.25& 0.79& 0.50& 1.03& - & - & - & 558$\pm$46\
GC0355 &pff\_gc-083 & 13 26 01.83 &-42 58 15.0& 6.89 & 21.23& 20.80& 19.86& 19.30& 18.72& 0.16& 0.04& 0.01& 0.01& 0.03& 21.12& 20.21& 19.44& 0.02& 0.02& 0.01& 0.42& 0.95& 0.55& 1.13& 0.77& 0.91& 1.68& 458$\pm$31\
GC0356 &pff\_gc-084 & 13 26 02.25 &-43 08 55.6& 10.03 & 21.31& 20.80& 19.84& 19.24& 18.63& 0.12& 0.03& 0.01& 0.01& 0.03& 21.13& 20.19& 19.32& 0.02& 0.01& 0.01& 0.51& 0.97& 0.59& 1.21& 0.87& 0.93& 1.80& 458$\pm$38\
GC0357 &AAT122445 & 13 26 02.43 &-43 00 11.7& 6.43 & 21.13& 20.99& 20.28& 19.81& 19.42& 0.15& 0.05& 0.02& 0.02& 0.03& 21.08& 20.50& 19.92& 0.01& 0.02& 0.01& 0.14& 0.71& 0.47& 0.86& 0.58& 0.58& 1.16& 342$\pm$98\
GC0358 &C176 & 13 26 02.79 &-42 57 05.0& 7.61 & - & - & - & - & - & - & - & - & - & - & 22.35& 21.92& 21.30& 0.02& 0.02& 0.03& - & - & - & - & 0.62& 0.43& 1.05& -\
GC0359 &HGHH-25/C25 & 13 26 02.85 &-42 56 57.0& 7.69 & 20.17& 19.56& 18.49& 17.85& 17.17& 0.07& 0.02& 0.01& 0.01& 0.03& 19.92& 18.83& 17.97& 0.03& 0.02& 0.01& 0.61& 1.07& 0.64& 1.32& 0.86& 1.09& 1.95& 703$\pm$9\
GC0360 &AAT122526 & 13 26 02.90 &-43 05 43.0& 7.90 & 21.89& 21.35& 20.36& 19.73& 19.05& 0.24& 0.05& 0.02& 0.01& 0.03& 21.72& 20.72& 19.81& 0.02& 0.01& 0.01& 0.54& 1.00& 0.62& 1.31& 0.90& 1.00& 1.90& 506$\pm$50\
GC0361 &C177 & 13 26 03.20 &-42 54 30.1& 9.30 & - & - & - & - & - & - & - & - & - & - & 22.95& 22.11& 21.09& 0.11& 0.07& 0.05& - & - & - & - & 1.02& 0.84& 1.87& -\
GC0362 &C178 & 13 26 03.85 &-42 56 45.3& 7.95 & - & - & - & - & - & - & - & - & - & - & 23.02& 22.32& 21.53& 0.04& 0.04& 0.03& - & - & - & - & 0.79& 0.69& 1.48& -\
GC0363 &HGHH-G293/G293 & 13 26 04.20 &-42 55 44.7& 8.60 & 20.11& 19.90& 19.10& 18.61& 18.11& 0.06& 0.02& 0.01& 0.01& 0.03& 20.04& 19.36& 18.69& 0.01& 0.01& 0.01& 0.21& 0.80& 0.49& 0.99& 0.67& 0.69& 1.35& 581$\pm$15\
GC0364 &AAT122808 & 13 26 04.61 &-43 09 10.2& 10.49 & 21.05& 20.87& 20.12& 19.66& 19.18& 0.10& 0.03& 0.01& 0.01& 0.03& 21.00& 20.45& 19.69& 0.01& 0.01& 0.01& 0.18& 0.75& 0.46& 0.94& 0.76& 0.55& 1.31& 264$\pm$131\
GC0365 &AAT122794 & 13 26 04.69 &-42 47 35.1& 15.16 & 21.89& 21.15& 20.00& 19.29& 18.69& 0.21& 0.04& 0.01& 0.01& 0.03& 21.60& 20.37& 19.38& 0.01& 0.01& 0.01& 0.74& 1.15& 0.71& 1.31& 0.98& 1.23& 2.22& 336$\pm$161\
GC0366 &C105 & 13 26 05.12 &-42 55 37.0& 8.80 & - & - & 22.01& - & - & - & - & - & - & - & 23.68& 22.58& 21.76& 0.07& 0.04& 0.04& - & - & - & - & 0.82& 1.10& 1.92& -\
GC0367 &HGHH-07/C7 & 13 26 05.41 &-42 56 32.4& 8.30 & 18.38& 18.03& 17.17& 16.65& 16.08& 0.03& 0.01& 0.01& 0.01& 0.03& 18.18& 17.33& 16.64& 0.03& 0.03& 0.01& 0.35& 0.86& 0.53& 1.09& 0.68& 0.85& 1.53& 595$\pm$1\
GC0368 &C106 & 13 26 06.15 &-42 56 45.4& 8.32 & - & - & 21.28& - & - & - & - & - & - & - & 22.54& 21.46& 20.66& 0.03& 0.01& 0.01& - & - & - & - & 0.80& 1.08& 1.88& -\
GC0369 &pff\_gc-085 & 13 26 06.42 &-43 00 38.1& 7.11 & 20.39& 20.20& 19.43& 18.99& 18.47& 0.07& 0.02& 0.01& 0.01& 0.03& 20.29& 19.64& 19.03& 0.02& 0.02& 0.01& 0.20& 0.76& 0.44& 0.96& 0.61& 0.65& 1.26& 548$\pm$31\
GC0370 &pff\_gc-086 & 13 26 06.55 &-43 06 14.5& 8.75 & 21.48& 20.93& 19.95& 19.37& 18.76& 0.16& 0.03& 0.01& 0.01& 0.03& 21.29& 20.32& 19.45& 0.02& 0.01& 0.01& 0.56& 0.97& 0.59& 1.20& 0.87& 0.97& 1.84& 440$\pm$26\
GC0371 &pff\_gc-087 & 13 26 06.87 &-42 33 17.3& 28.77 & 20.26& 19.96& 19.11& 18.56& 18.05& 0.14& 0.02& 0.01& 0.01& 0.03& 20.17& 19.42& 18.70& 0.02& 0.01& 0.01& 0.30& 0.86& 0.55& 1.05& 0.73& 0.75& 1.47& 830$\pm$29\
GC0372 &HGHH-G170 & 13 26 06.93 &-42 57 35.1& 8.02 & 20.78& 20.21& 19.22& 18.65& 17.97& 0.11& 0.02& 0.01& 0.01& 0.03& 20.57& 19.56& 18.73& 0.02& 0.02& 0.01& 0.57& 0.99& 0.57& 1.25& 0.83& 1.01& 1.84& 636$\pm$27\
GC0373 &AAT123188 & 13 26 06.94 &-43 07 52.6& 9.85 & 21.70& 21.30& 20.40& 19.85& 19.27& 0.17& 0.04& 0.01& 0.01& 0.03& 21.57& 20.71& 19.94& 0.02& 0.01& 0.01& 0.40& 0.90& 0.56& 1.13& 0.77& 0.86& 1.63& 364$\pm$56\
GC0374 &HGHH-36/C36/R113 & 13 26 07.73 &-42 52 00.3& 11.72 & 19.42& 19.19& 18.35& 17.81& 17.33& 0.04& 0.01& 0.01& 0.01& 0.03& 19.32& 18.61& 17.94& 0.02& 0.01& 0.01& 0.23& 0.84& 0.54& 1.03& 0.67& 0.71& 1.38& 703$\pm$1\
GC0375 &AAT123453 & 13 26 08.38 &-42 59 18.9& 7.67 & 21.29& 20.95& 20.09& 19.55& 19.03& 0.17& 0.04& 0.02& 0.01& 0.03& 21.20& 20.40& 19.67& 0.03& 0.02& 0.01& 0.34& 0.86& 0.54& 1.06& 0.73& 0.80& 1.53& 257$\pm$154\
GC0376 &pff\_gc-088 & 13 26 08.86 &-43 01 21.4& 7.54 & 19.99& 19.79& 18.99& 18.46& 18.00& 0.05& 0.02& 0.01& 0.01& 0.03& 19.86& 19.14& 18.59& 0.03& 0.03& 0.01& 0.20& 0.80& 0.53& 0.99& 0.55& 0.72& 1.27& 554$\pm$29\
GC0377 &AAT123656 & 13 26 09.61 &-43 07 05.9& 9.71 & 21.77& 21.13& 20.08& 19.46& 18.79& 0.19& 0.04& 0.01& 0.01& 0.03& 21.57& 20.51& 19.59& 0.03& 0.02& 0.02& 0.64& 1.05& 0.63& 1.29& 0.92& 1.07& 1.99& 380$\pm$93\
GC0378 & R111 & 13 26 09.71 &-42 50 29.5& 13.14 & 22.07& 21.57& 20.59& 19.91& 19.09& 0.28& 0.06& 0.02& 0.02& 0.03& 21.96& 21.03& 20.20& 0.03& 0.03& 0.02& 0.51& 0.97& 0.68& 1.50& 0.82& 0.94& 1.76& 717$\pm$48\
GC0379 &C179 & 13 26 09.87 &-42 56 36.0& 8.96 & - & - & - & - & - & - & - & - & - & - & 22.13& 21.71& 21.04& 0.03& 0.03& 0.01& - & - & - & - & 0.67& 0.42& 1.09& -\
GC0380 &HGHH-37/C37/R116 & 13 26 10.58 &-42 53 42.7& 10.81 & 19.86& 19.38& 18.43& 17.87& 17.26& 0.04& 0.01& 0.01& 0.01& 0.03& 19.65& 18.72& 17.96& 0.01& 0.01& 0.01& 0.48& 0.95& 0.56& 1.17& 0.76& 0.93& 1.69& 612$\pm$1\
GC0381 & WHH-27 & 13 26 12.82 &-43 09 09.2& 11.51 & 20.27& 19.68& 18.66& 18.06& 17.42& 0.05& 0.01& 0.01& 0.01& 0.03& 20.02& 19.03& 18.13& 0.02& 0.01& 0.01& 0.59& 1.02& 0.60& 1.24& 0.90& 0.99& 1.89& 545$\pm$60\
GC0382 & WHH-28 & 13 26 14.18 &-43 08 30.4& 11.25 & 20.21& 19.98& 19.17& 18.65& 18.14& 0.05& 0.02& 0.01& 0.01& 0.03& 20.13& 19.48& 18.71& 0.02& 0.01& 0.01& 0.23& 0.81& 0.52& 1.03& 0.77& 0.65& 1.41& 506$\pm$123\
GC0383 &HHH86-26/C26 & 13 26 15.27 &-42 48 29.4& 15.36 & 19.99& 19.26& 18.13& 17.47& 16.77& 0.05& 0.01& 0.01& 0.01& 0.03& 19.66& 18.45& 17.59& 0.04& 0.03& 0.02& 0.73& 1.13& 0.66& 1.35& 0.86& 1.21& 2.07& 377$\pm$14\
GC0384 &R122 & 13 26 15.95 &-42 55 00.5& 10.76 & 19.02& 18.78& 18.02& 17.58& 17.14& 0.03& 0.01& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.24& 0.76& 0.44& 0.88& - & - & - & 588$\pm$2\
GC0385 &AAT125079 & 13 26 17.29 &-43 06 39.3& 10.62 & 21.13& 20.90& 20.09& 19.59& 19.06& 0.10& 0.03& 0.01& 0.01& 0.03& 21.07& 20.41& 19.64& 0.02& 0.01& 0.01& 0.23& 0.81& 0.51& 1.03& 0.77& 0.66& 1.43& 513$\pm$183\
GC0386 &C50/K-233 & 13 26 19.66 &-43 03 18.6& 9.76 & 20.09& 19.68& 18.74& 18.17& 17.57& 0.05& 0.02& 0.01& 0.01& 0.03& 19.91& 18.92& 18.28& 0.04& 0.04& 0.01& 0.41& 0.93& 0.58& 1.17& 0.64& 1.00& 1.63& 615$\pm$58\
GC0387 &C49/pff\_gc-089 & 13 26 20.20 &-43 10 35.7& 13.48 & 19.90& 19.73& 18.95& 18.45& 17.95& 0.12& 0.02& 0.01& 0.01& 0.03& 19.85& 19.30& 18.53& 0.03& 0.02& 0.01& 0.17& 0.78& 0.50& 1.00& 0.77& 0.56& 1.32& 538$\pm$30\
GC0388 &pff\_gc-090 & 13 26 20.53 &-43 03 18.5& 9.91 & 20.90& 20.34& 19.33& 18.74& 18.10& 0.10& 0.02& 0.01& 0.01& 0.03& 20.70& 19.61& 18.85& 0.03& 0.03& 0.01& 0.56& 1.00& 0.60& 1.24& 0.75& 1.09& 1.85& 486$\pm$26\
GC0389 &AAT215171 & 13 26 20.66 &-42 38 32.0& 24.60 & 21.03& 20.63& 19.69& 19.11& 18.53& 0.27& 0.03& 0.01& 0.01& 0.03& 20.97& 20.06& 19.26& 0.02& 0.02& 0.01& 0.40& 0.94& 0.58& 1.15& 0.80& 0.90& 1.70& 527$\pm$48\
GC0390 & R107 & 13 26 21.11 &-42 48 41.1& 15.84 & 21.53& 20.82& 19.75& 19.13& 18.48& 0.16& 0.03& 0.01& 0.01& 0.03& 21.23& 20.16& 19.30& 0.05& 0.03& 0.02& 0.71& 1.07& 0.62& 1.27& 0.86& 1.06& 1.92& 405$\pm$28\
GC0391 &pff\_gc-091 & 13 26 21.14 &-43 42 24.6& 42.41 & 20.42& 20.13& 19.29& 18.77& 18.21& 0.15& 0.02& 0.01& 0.01& 0.03& 20.30& 19.56& 18.72& 0.01& 0.01& 0.01& 0.29& 0.84& 0.52& 1.08& 0.84& 0.74& 1.58& 623$\pm$37\
GC0392 &pff\_gc-092 & 13 26 21.31 &-42 57 19.1& 10.53 & 21.36& 20.86& 19.92& 19.35& 18.76& 0.14& 0.03& 0.01& 0.01& 0.03& 21.18& 20.27& 19.46& 0.02& 0.02& 0.01& 0.50& 0.93& 0.57& 1.16& 0.81& 0.92& 1.73& 462$\pm$27\
GC0393 & R117 & 13 26 21.99 &-42 53 45.5& 12.38 & 20.52& 20.41& 19.68& 19.23& 18.75& 0.07& 0.02& 0.01& 0.01& 0.03& 20.50& 19.98& 19.32& 0.02& 0.02& 0.01& 0.10& 0.73& 0.45& 0.93& 0.66& 0.53& 1.18& 484$\pm$26\
GC0394 & R118 & 13 26 22.01 &-42 54 26.5& 11.99 & 20.73& 20.60& 19.87& 19.38& 18.93& 0.08& 0.03& 0.01& 0.01& 0.03& 20.72& 20.19& 19.50& 0.03& 0.02& 0.01& 0.13& 0.73& 0.49& 0.95& 0.70& 0.52& 1.22& 440$\pm$73\
GC0395 & WHH-29 & 13 26 22.08 &-43 09 10.7& 12.79 & 20.97& 20.70& 19.82& 19.28& 18.73& 0.08& 0.02& 0.01& 0.01& 0.03& 20.92& 20.17& 19.38& 0.03& 0.02& 0.01& 0.27& 0.88& 0.54& 1.09& 0.79& 0.75& 1.54& 505$\pm$78\
GC0396 &pff\_gc-093 & 13 26 22.65 &-42 46 49.8& 17.50 & 20.58& 20.39& 19.62& 19.13& 18.66& 0.08& 0.03& 0.01& 0.01& 0.03& 20.52& 19.91& 19.23& 0.02& 0.01& 0.01& 0.20& 0.77& 0.48& 0.96& 0.69& 0.61& 1.30& 576$\pm$42\
GC0397 & WHH-30 & 13 26 23.60 &-43 03 43.9& 10.55 & 20.56& 20.25& 19.38& 18.83& 18.30& 0.10& 0.03& 0.01& 0.01& 0.03& 20.45& 19.62& 18.96& 0.03& 0.03& 0.01& 0.31& 0.87& 0.55& 1.08& 0.65& 0.83& 1.49& 470$\pm$66\
GC0398 &pff\_gc-094 & 13 26 23.66 &-43 00 45.6& 10.25 & 20.99& 20.77& 19.97& 19.49& 18.93& 0.09& 0.03& 0.01& 0.01& 0.03& 20.92& 20.24& 19.57& 0.03& 0.02& 0.01& 0.22& 0.80& 0.47& 1.04& 0.67& 0.68& 1.35& 334$\pm$64\
GC0399 &HHH86-38/C38/R123 & 13 26 23.78 &-42 54 01.1& 12.50 & 19.67& 19.30& 18.41& 17.87& 17.32& 0.04& 0.01& 0.01& 0.01& 0.03& 19.51& 18.68& 17.96& 0.02& 0.02& 0.01& 0.37& 0.89& 0.54& 1.09& 0.72& 0.83& 1.54& 405$\pm$1\
GC0400 &HGHH-51 & 13 26 23.86 &-42 47 17.1& 17.26 & 19.63& 19.34& 18.47& 17.93& 17.38& 0.05& 0.02& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.29& 0.87& 0.54& 1.09& - & - & - & 343$\pm$43\
GC0401 &AAT335187 & 13 26 23.95 &-43 17 44.4& 19.53 & 21.19& 21.11& 20.46& 20.04& 19.60& 0.10& 0.03& 0.01& 0.01& 0.03& 21.20& 20.72& 20.08& 0.01& 0.01& 0.01& 0.08& 0.65& 0.42& 0.86& 0.64& 0.49& 1.12& 277$\pm$158\
GC0402 &pff\_gc-095 & 13 26 25.50 &-42 57 06.2& 11.33 & 21.13& 20.60& 19.60& 19.02& 18.40& 0.11& 0.02& 0.01& 0.01& 0.03& 20.97& 19.97& 19.17& 0.04& 0.03& 0.02& 0.53& 1.00& 0.58& 1.20& 0.80& 0.99& 1.79& 405$\pm$18\
GC0403 & R124 & 13 26 28.87 &-42 52 36.4& 14.08 & 20.32& 20.17& 19.42& 18.96& 18.43& 0.05& 0.02& 0.01& 0.01& 0.03& 20.28& 19.69& 19.02& 0.02& 0.02& 0.01& 0.15& 0.75& 0.46& 0.99& 0.68& 0.58& 1.26& 541$\pm$35\
GC0404 &pff\_gc-096 & 13 26 30.29 &-42 34 41.7& 28.83 & 21.24& 20.73& 19.73& 19.10& 18.42& 0.31& 0.03& 0.01& 0.01& 0.03& 21.08& 20.09& 19.24& 0.01& 0.01& 0.01& 0.52& 1.00& 0.63& 1.30& 0.85& 0.99& 1.83& 532$\pm$25\
GC0405 & R105 & 13 26 33.55 &-42 51 00.9& 15.74 & 21.28& 20.78& 19.86& 19.28& 18.70& 0.11& 0.03& 0.01& 0.01& 0.03& 21.18& 20.25& 19.46& 0.02& 0.01& 0.01& 0.50& 0.92& 0.58& 1.15& 0.79& 0.93& 1.72& 534$\pm$34\
GC0406 &HGHH-27/C27 & 13 26 37.99 &-42 45 49.9& 20.00 & 19.60& 19.39& 18.60& 18.11& 17.62& 0.03& 0.01& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.21& 0.78& 0.49& 0.98& - & - & - & 492$\pm$37\
GC0407 & WHH-31 & 13 26 41.43 &-43 11 25.0& 16.96 & 20.76& 20.44& 19.50& 18.94& 18.36& 0.07& 0.02& 0.01& 0.01& 0.03& 20.67& 19.89& 19.05& 0.03& 0.02& 0.02& 0.32& 0.94& 0.56& 1.14& 0.85& 0.77& 1.62& 573$\pm$67\
GC0408 &HHH86-39/C39 & 13 26 42.03 &-43 07 44.8& 15.12 & 18.72& 18.33& 17.43& 16.91& 16.42& 0.02& 0.01& 0.01& 0.01& 0.03& 18.57& 17.73& 16.92& 0.01& 0.01& - & 0.39& 0.89& 0.53& 1.01& 0.81& 0.83& 1.65& 271$\pm$20\
GC0409 &pff\_gc-097 & 13 26 45.40 &-43 26 34.1& 29.13 & 19.75& 19.51& 18.72& 18.23& 17.76& 0.10& 0.01& 0.01& 0.01& 0.03& 19.66& 19.04& 18.28& 0.02& 0.01& 0.01& 0.24& 0.79& 0.49& 0.96& 0.76& 0.62& 1.38& 599$\pm$30\
GC0410 & HH-017 & 13 26 49.31 &-43 04 57.8& 15.41 & 23.21& 22.54& 20.80& 19.87& 19.06& 0.67& 0.11& 0.02& 0.01& 0.03& 23.32& 21.63& 20.13& 0.05& 0.03& 0.02& 0.67& 1.75& 0.93& 1.73& 1.50& 1.68& 3.19& 839$\pm$63\
GC0411 &pff\_gc-098 & 13 26 53.94 &-43 19 17.7& 24.05 & 19.54& 19.17& 18.28& 17.73& 17.16& 0.09& 0.01& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.37& 0.89& 0.55& 1.12& - & - & - & 631$\pm$18\
GC0412 &AAT222977 & 13 26 58.91 &-42 38 53.4& 27.82 & 21.18& 21.02& 20.34& 19.84& 19.38& 0.28& 0.03& 0.01& 0.01& 0.03& 21.16& 20.64& 20.02& 0.02& 0.01& 0.01& 0.17& 0.68& 0.50& 0.96& 0.62& 0.52& 1.14& 655$\pm$112\
GC0413 & HH-060 & 13 26 59.78 &-42 55 26.5& 17.79 & - & - & - & - & - & - & - & - & - & - & 23.41& 21.69& 20.14& 0.05& 0.04& 0.02& - & - & - & - & 1.55& 1.72& 3.26& 811$\pm$32\
GC0414 &pff\_gc-099 & 13 26 59.82 &-42 32 40.5& 33.09 & 20.96& 20.72& 19.94& 19.40& 18.92& 0.23& 0.03& 0.01& 0.01& 0.03& 20.92& 20.29& 19.59& 0.02& 0.01& 0.02& 0.25& 0.78& 0.53& 1.02& 0.70& 0.63& 1.33& 425$\pm$34\
GC0415 &pff\_gc-100 & 13 27 03.41 &-42 27 17.2& 38.12 & 19.92& 19.37& 18.41& 17.81& 17.17& 0.11& 0.01& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.56& 0.96& 0.60& 1.24& - & - & - & 513$\pm$10\
GC0416 &pff\_gc-101 & 13 27 21.56 &-42 38 41.4& 30.63 & 19.89& 19.51& 18.72& 18.20& 17.74& 0.10& 0.01& 0.01& 0.01& 0.03& 19.74& 19.02& 18.39& 0.01& 0.01& - & 0.38& 0.79& 0.51& 0.97& 0.63& 0.72& 1.35& 263$\pm$16\
GC0417 &pff\_gc-102 & 13 28 18.45 &-42 33 12.5& 41.90 & 20.38& 20.11& 19.29& 18.76& 18.24& 0.14& 0.02& 0.01& 0.01& 0.03& 20.28& 19.59& 18.94& 0.04& 0.04& 0.01& 0.27& 0.83& 0.52& 1.04& 0.65& 0.70& 1.34& 428$\pm$40\
[rrrrrrrrrrr]{}
0-50 & 12.9 & 48.8 & 340 & 40$\pm$10 & 189$\pm$12 & 123$\pm$5 & 1.0 & 125.8$\pm$46.5 & 3.1$\pm$1.3 & 128.9$\pm$46.5\
0-5 & 3.64 & 4.96 & 54 & 24$\pm$21 & 334$\pm$59 & 120$\pm$12 & - & - & - & -\
5-10 & 7.65 & 9.96 & 124 & 43$\pm$15 & 195$\pm$20 & 112$\pm$8 & 2.3 & 37.4$\pm$6.4 & 0.4$\pm$0.3 & 37.8$\pm$6.4\
10-15 & 12.4 & 14.9 & 68 & 83$\pm$25 & 195$\pm$12 & 105$\pm$10 & 1.6 & 36.5$\pm$9.3 & 2.4$\pm$1.5 & 38.9$\pm$9.4\
15-25 & 19.0 & 24.3 & 56 & 35$\pm$26 & 184$\pm$34 & 147$\pm$16 & 1.3 & 89.5$\pm$27.4 & 0.7$\pm$1.0 & 90.2$\pm$27.4\
25-50 & 34.7 & 48.8 & 39 & 96$\pm$45 & 169$\pm$17 & 148$\pm$21 & 1.0 & 184.7$\pm$84.5 & 10.4$\pm$9.8 & 195.1$\pm$85.1\
[rrrrrrrrrrr]{}
0-50 & 13.8 & 48.1 & 178 & 31$\pm$14 & 177$\pm$22 & 117$\pm$7 & 1.0 & 94.5$\pm$58.5 & 1.1$\pm$1.0 & 95.6 $\pm$58.6\
0-5 & 3.84 & 4.96 & 22 & 16$\pm$38 & 373$\pm$120& 99$\pm$19 & - & - & - & -\
5-10 & 7.76 & 9.96 & 66 & 30$\pm$21 & 223$\pm$39 & 116$\pm$12 & 2.1 & 38.9$\pm$12.4 & 0.2$\pm$0.3 & 39.1$\pm$12.4\
10-15 & 12.7 & 14.8 & 41 & 97$\pm$38 & 199$\pm$13 & 102$\pm$15 & 1.5 & 31.4$\pm$15.1 & 3.3$\pm$2.5 & 34.6$\pm$15.3\
15-25 & 18.8 & 24.3 & 26 & 45$\pm$31 & 139$\pm$49 & 112$\pm$20 & 1.3 & 59.7$\pm$32.7 & 1.2$\pm$1.6 & 60.8$\pm$32.7\
25-50 & 36.7 & 48.1 & 23 & 61$\pm$43 & 137$\pm$50 & 141$\pm$24 & 1.0 & 122.4$\pm$82.9 & 4.2$\pm$5.9 & 126.7$\pm$83.1\
[rrrrrrrrrrr]{}
0-50 & 12.0 & 48.8 & 158 & 47$\pm$15 & 202$\pm$15 & 129$\pm$9 & 1.0 & 116.5$\pm$73.8 & 4.3$\pm$2.3 & 120.8$\pm$73.9\
0-5 & 3.61 & 4.98 & 29 & 44$\pm$27 & 308$\pm$47 & 134$\pm$23 & - & - & - & -\
5-10 & 7.52 & 9.93 & 58 & 65$\pm$22 & 180$\pm$19 & 105$\pm$11 & 1.5 & 17.4$\pm$7.7 & 1.0$\pm$0.7 & 18.3$\pm$7.7\
10-15 & 12.0 & 14.9 & 27 & 74$\pm$34 & 186$\pm$22 & 108$\pm$17 & 1.2 & 23.8$\pm$11.3 & 1.9$\pm$1.8 & 25.7$\pm$13.3\
15-25 & 19.3 & 24.3 & 30 & 50$\pm$36 & 223$\pm$32 & 168$\pm$24 & 1.1 & 72.1$\pm$46.2 & 1.2$\pm$1.7 & 73.3$\pm$46.3\
25-50 & 31.7 & 48.8 & 15 & 102$\pm$96 & 191$\pm$21 & 146$\pm$65 & 1.0 & 219.1$\pm$174.6 & 11.8$\pm$22.2 & 230.9$\pm$176.0\
[rrrrrrrrrrr]{}
0-90 &14.1 & 88.2 & 780 & 76$\pm$6 & 170$\pm$5 & 118$\pm$13 & 1.0 & 84.6$\pm$17.2 &11.9$\pm$1.9 & 96.5$\pm$17.3\
0-5 & 3.3 & 4.99 & 184 & 75$\pm$16 & 179$\pm$11 & 131$\pm$7 & - & - & - & -\
5-10 & 7.6 & 9.98 & 211 & 82$\pm$12 & 177$\pm$8 & 120$\pm$6 & 3.0 & 43.9$\pm$3.6 & 1.6$\pm$0.5 & 45.4$\pm$3.6\
10-15 &11.8 & 14.8 & 138 & 76$\pm$15 & 171$\pm$10 & 118$\pm$7 & 2.0 & 42.3$\pm$5.0 & 2.0$\pm$0.8 & 44.2$\pm$5.1\
15-20 &17.4 & 20.0 & 71 & 96$\pm$24 & 169$\pm$12 & 116$\pm$10 & 1.7 & 47.7$\pm$5.6 & 4.3$\pm$2.1 & 52.0$\pm$6.9\
20-30 &24.6 & 30.0 & 87 & 76$\pm$17 & 157$\pm$17 & 108$\pm$8 & 1.5 & 53.2$\pm$8.4 & 4.0$\pm$1.8 & 57.2$\pm$8.6\
30-40 &34.6 & 39.8 & 50 & 44$\pm$15 & 132$\pm$38 & 87$\pm$9 & 1.3 & 40.7$\pm$7.4 & 1.8$\pm$1.2 & 42.3$\pm$7.5\
40-80 &48.7 & 71.2 & 36 & 61$\pm$45 & 183$\pm$22 & 85$\pm$11 & 1.0 & 48.6$\pm$11.5 & 6.2$\pm$9.1 & 54.8$\pm$14.7\
[lrrrrrrrr]{}
NGC 5128 & 31$\pm$14 & 117$\pm$7 & 0.26$\pm$0.12 & 0.08$\pm$0.04 & 47$\pm$15 & 129$\pm$9 & 0.36$\pm$0.11 & 0.11$\pm$0.03\
M87 &186$^{+58}_{-41}$ &397$^{+36}_{-14}$ & 0.47$^{+0.13}_{-0.11}$& 0.14$^{+0.04}_{-0.03}$ &155$^{+53}_{-37}$ &365$^{+38}_{-18}$ & 0.43$^{+0.14}_{-0.12}$ & 0.13$^{+0.04}_{-0.04}$\
M49 & 93$^{+69}_{-37}$ &342$^{+33}_{-18}$ & 0.27$^{+0.19}_{-0.11}$& 0.08$^{+0.06}_{-0.03}$ &-26$^{+64}_{-79}$ &265$^{+34}_{-13}$ & 0.10$^{+0.27}_{-0.25}$ & 0.03$^{+0.08}_{-0.08}$\
NGC 1399 & 15$\pm$26 &291$\pm$14 & 0.05$\pm$0.09 & 0.02$\pm$0.03 & 7$\pm$24 & 255$\pm$13 & 0.03$\pm$0.09 & 0.01$\pm$0.03\
![Sine curve fit for the GCs in NGC 5128 ([*circles*]{}) with a fixed systemic velocity of $v_{sys} = 541$ km s$^{-1}$, for 0-50 kpc from the center of NGC 5128. The top panel shows all 340 GCs with rotation amplitude $\Omega R = 40\pm10 $ km s$^{-1}$ and rotation axis $\Theta_o = 189\pm12^{o}$ east of north. The middle panel shows the 178 metal-poor clusters with $\Omega R = 31\pm14$ km s$^{-1}$ and $\Theta_o = 177\pm22^{o}$ east of north, and the bottom panel shows the 158 metal-rich clusters with $\Omega R = 47\pm15$ km s$^{-1}$ and $\Theta_o = 202\pm15^{o}$ east of north. The squares represent the weighted velocities in $72^{o}$ bins.[]{data-label="fig:kin_plot"}](f4.eps)
[^1]: The confirmed GC list in [@w06] containing 343 GCs has been reduced to 340 based on recent spectroscopic and imaging studies. Object 304867 with a high radial velocity of $305\pm56$ km s$^{-1}$ appears to be an M-type star based on the strong molecular bands in its spectrum, see [@beasley06] for further discussion. Objects pff\_gc-010 and 114993 are also rejected as GCs because of their starlike appearance under $HST$ ACS imaging; see [@harris06]. However, newly confirmed GCs HCH15 and R122 have now been added [@rejkuba07].
[^2]: [@hhh86] report C32 at a distance of $R_{gc} = 10.8'$, but more recently [@pff04I] claim a distance of $R_{gc} =
11.25'$, so it has an adopted uncertainty of 44 km s$^{-1}$ in the weighted mean.
| ArXiv |
---
abstract: 'In this paper, we develop a fully discrete Galerkin method for solving initial value fractional integro-differential equations(FIDEs). We consider Generalized Jacobi polynomials(GJPs) with indexes corresponding to the number of homogeneous initial conditions as natural basis functions for the approximate solution. The fractional derivatives are used in the Caputo sense. The numerical solvability of algebraic system obtained from implementation of proposed method for a special case of FIDEs is investigated. We also provide a suitable convergence analysis to approximate solutions under a more general regularity assumption on the exact solution.'
author:
- |
P. Mokhtary\
\
\
title: 'Discrete Galerkin Method for Fractional Integro-Differential Equations'
---
[**Subject Classification:**]{}[34A08; 65L60]{}
[**Keywords:**]{} Fractional integro-differential equation(FIDE), Galerkin Method, Generalized Jacobi Polynomials(GJPs), Caputo derivative.
Introduction
============
In this paper, we provide a convergent numerical scheme for solving FIDE $$\label{1}
\left\{\begin{array}{l}
\mathcal D^q u(x)=p(x) u(x)+f(x)+\lambda \int\limits_0^x{K(x,t) u(t) dt},~~~ x \in \Omega=[0,1],\\
\\
u(0)=0,
\end{array}\right.$$ where $q\in \mathbb R^+ \bigcap (0,1)$. The symbol $\mathbb R^+$ is the collection of all positive real numbers. $p(x)$ and $f(x)$ are given continuous functions and $K(x,t)$ is a given sufficiently smooth kernel function, $u(x)$ is the unknown function.
Note that the condition $u(0)=0$ is not restrictive, due to the fact that (\[1\]) with nonhomogeneous initial condition $u(0)=d,~~d
\neq 0$ can be converted to the following homogeneous FIDE $$\left\{\begin{array}{l}
\mathcal D^q \tilde u(x)=p(x) \tilde u(x)+\tilde f(x)+\lambda \int\limits_0^x{K(x,t) \tilde u(t) dt},~~~ x \in \Omega=[0,1],\\
\\
\tilde u(0)=0,
\end{array}\right.$$ by the simple transformation $\tilde u(x)=u(x)-d$, where $\tilde
f(x)=f(x)+d\bigg(p(x)+\lambda \int_{0}^{x}{K(x,t)dt}\bigg)$.
Such kind of equations arising in the mathematical modeling of various physical phenomena, such as heat conduction, materials with memory, combined conduction, convection and radiation problems([@r2], [@r5], [@r20], [@r21]).
$\mathcal D^q u(x)$ denotes the fractional Caputo differential operator of order $q$ and defines as([@r8], [@r13], [@r22]) $$\label{2} \mathcal D^q u(x) = \mathcal I^{1-q} u'(x),$$ where $$\label{3}
\mathcal I^\mu u(x)=\frac{1}{\Gamma{(\mu)}}
\int\limits_0^x{(x-s)^{\mu-1} u(s) ds},$$ is the fractional integral operator from order $\mu$. $\Gamma{(\mu)}$ is the well known Gamma function. The following relation holds[@r8] $$\label{20}\mathcal I^q(\mathcal D^q
u(x))=u(x)-u(0).$$
From the relation above, it is easy to check that (\[1\]) is equivalent with the following weakly singular Volterra integral equation $$\label{5}
u(x)=g(x)+\lambda \int\limits_0^x{{\bar K}(x,t) u(t) dt}.$$
Here $g(x)=\mathcal I^q f(x)$ and ${\bar
K}(x,t)=\frac{(x-t)^{q-1}}{\Gamma{(q)}}p(t)+\int\limits_t^x{\frac{(x-s)^{q-1}}{\Gamma{(q)}}
K(s,t)ds}.$ From the well known existence and uniqueness Theorems([@r3], [@r7]), it can be concluded that if the following conditions are fulfilled
- $f(x) \in C^l(\Omega),~~l \ge 1$
- $p(x) \in C^l(\Omega),~~l \ge 1$
- $K(x,t) \in C^l(D),~~ D=\{(x,t);0 \le t \le x\le
1\},~~l \ge1$
- $K(x,x)\neq 0$,
the regularity of the unique solution $u(x)$ of (\[5\]) and also (\[1\]) is described by $$\label{6}
u(x)=\sum\limits_{(j,k)}{\gamma_{j,k} x^{j+kq}}+U_l(x;q) \in
C^l(0,1]\bigcap C(\Omega),\hspace{.5 cm} \text{with} \hspace{.5 cm}
|u'(x)| \le C_q x^{q-1},$$ where the coefficients $\gamma_{j,k}$ are some constants, $U_l(.;q)
\in C^l(\Omega)$ and $(j,k):=\{(j,k):~~j,k \in \mathbb
N_0,~j+kq<l\}$. Here $\mathbb N_0=\mathbb N \bigcup \{0\}$, where the symbol $\mathbb N$ denotes the collection of all natural numbers. Thus, we must expect the first derivative of the solution to has a discontinuity at the origin. More precisely, if the given functions $g(x), p(x)$ and $K(x,t)$ are real analytic in their domains then it can be concluded that there is a function $U=U(z_1,z_2)$ real and analytic at $(0,0)$, so that solutions of (\[5\])and also (\[1\]) can be written as $u(x)=U(x,x^q)$([@r3], [@r7]).
Recently, several numerical methods for the numerical solution of FIDE’s have been proposed. In [@r19], fractional differential transform method was developed to solve FIDE’s with nonlocal boundary conditions. In [@r23], Rawashdeh studied the numerical solution of FIDE’s by polynomial spline functions. In [@r1], an analytical solution for a class of FIDE’s was proposed. Adomian decomposition method to solve nonlinear FIDE’s was proposed in [@r17]. In [@r25], authors solved fractional nonlinear Volterra integro differential equations using the second kind Chebyshev wavelets. In [@r11], Taylor expansion approach was presented for solving a class of linear FIDE’s including those of Fredholm and Volterra types. In [@r16], authors were solved FIDE’s by adopting Hybrid Collocation method to an equivalent integral equation of convolution type. In [@r12], Chebyshev Pseudospectral method was implemented to solve linear and nonlinear system of FIDE’s. In [@r15], authors proposed an analyzed spectral Jacobi Collocation method for the numerical solution of general linear FIDE’s. In [@r9], authors applied Collocation method to solve the nonlinear FIDE’s. In [@r18], Mokhtary and Ghoreishi, proved the $L^2$ convergence of Legendre Tau method for the numerical solution of nonlinear FIDE’s.
Many of the techniques mentioned above or have not proper convergence analysis or if any, very restrictive conditions including smoothness of the exact solution are assumed. In this paper we will consider non smooth solutions of (\[1\]). In this case although the discrete Galerkin method can be implemented directly but this method leads to very poor numerical results. Thus it is necessary to introduce a regularization procedure that allows us to improve the smoothness of the given functions and then to approximate the solution with a satisfactory order of convergence. To this end, we propose a regularization process which the original equation (\[1\]) will be changed into a new equation which possesses a more regularity properties by taking a suitable coordinate transformation. Our logic in choosing proper transformation is based upon the formal asymptotic expansion of the exact solution in (\[6\]). Consider (\[1\]), using the variable transformation $$\label{6xx}
x=v^{\frac{1}{q}},\;\; v=x^{q},\;\; t=w^{\frac{1}{q}},\;\; w=t^q,$$ we can change (\[1\]) to the following equation $$\label{6x}
\mathcal M^q \bar u(v)=\bar p(v) \bar u(v)+\bar
f(v)+\lambda\int\limits_0^v{\tilde{K}(v,w) \bar{u}(w)dw},$$ where $$\begin{aligned}
\label{rv4}
\nonumber\bar p(v)&=&p(v^{\frac{1}{q}}),\;\; \bar
f(v)=f(v^{\frac{1}{q}}),~~{\tilde
K}(v,w)=\frac{w^{{\frac{1}{q}}-1}}{q}
K(v^{\frac{1}{q}},w^{\frac{1}{q}}).\\
\mathcal M^q \bar
u(v)&=&\frac{1}{\Gamma{(1-q)}}\int\limits_0^v{(v^{\frac{1}{q}}-w^{\frac{1}{q}})^{-q}
{\bar u}'(w)dw}.\end{aligned}$$
From (\[6\]), the exact solution $\bar u(v)$ can be written as $\bar{u}(v)=u(v^{\frac{1}{q}})=\sum\limits_{(j,k)}{\gamma_{j,k}
v^{\frac{j}{q}+k}}+U_l(v^{\frac{1}{q}};q)$. It can be easily seen that $\bar u'(v) \in C(\Omega)$. It is trivial that for $q=\frac{1}{n},~n \in \mathbb N$, the unknown function $\bar u(v)$ will be in the form $$\bar{u}(v)=u(v^n)=\sum\limits_{(j,k)}{\gamma_{j,k}
v^{nj+k}}+U_l(v^n;q), \quad n \in \mathbb N,$$ which is infinitely smooth. Then we can deduce that the solution $\bar u(v)$ of the new equation (\[6x\]) possesses better regularity and discrete Galerkin theory can be applied conveniently to obtain high order accuracy.
In the sequel, we introduce the discrete Galerkin solution $\bar
u_N(v)$ based upon GJPs to (\[6x\]). Since the exact solutions of (\[1\]) can be written as $u(x)=\bar u(v)$ then we define $u_N(x)=\bar u_N(v),\; x, v \in \Omega$ as the approximate solution of (\[1\]).
Spectral Galerkin method is one of the weighted residual methods(WRM), in which approximations are defined in terms of truncated series expansions, such that residual which should be exactly equal to zero, is forced to be zero only in an approximate sense. It is well known that, in this method, the expansion functions must satisfy in the supplementary conditions. The two main characteristics behind the approach are that, first it reduces the given problems to those of solving a system of algebraic equations, and in general converges exponentially and almost always supplies the most terse representation of a smooth solution([@a13], [@a14], [@aa26]).
In this article, we use shifted GJPs on $\Omega$, which are mutually orthogonal with respect to the shifted weight function $\delta^{\alpha,\beta}(v)=(2-2v)^\alpha(2v)^\beta$ on $\Omega$ where $\alpha, \beta$ belong to one of the following index sets $$\begin{aligned}
{\mathcal N_1}&=&\{(\alpha,\beta); \alpha, \beta \le -1, ~\alpha,
\beta \in \mathbb Z\},\quad~~~~~~~~~~~~~~ {\mathcal
N_2}=\{(\alpha,\beta); \alpha \le -1, \beta > -1,~~\alpha \in
\mathbb Z, \beta \in \mathbb R\},
\\
{\mathcal N_3}&=&\{(\alpha,\beta); \alpha>-1, \beta \le -1,~\alpha
\in \mathbb R, \beta \in \mathbb Z\},\quad {\mathcal
N_4}=\{(\alpha,\beta); \alpha, \beta
> -1,~\alpha, \beta \in \mathbb R\},\end{aligned}$$ where the symbol $\mathbb Z$ is the collection of all integer numbers. The main advantage of GJPs is that these polynomials, with indexes corresponding to the number of homogeneous initial conditions in a given FIDE, are the natural basis functions to the Galerkin approximation of this problem([@a15], [@a16]).
The organization of this paper is as follows: we begin by reviewing some preliminaries which are required for establishing our results in Section 2. In Section 3, we introduce the discrete Galerkin method based on the GJPs and its application to (\[6x\]). Numerical solvability of the algebraic system obtained from discrete Galerkin discretization of a special case of (\[6x\]) with $0<q<\frac{1}{2}$ and $\bar p(v)=1$ based on GJPs is given in Section 4. Convergence analysis of the proposed scheme is provided in Section 5. Numerical experiments are carried out in Section 6.
Preliminaries and Notations
===========================
In this section, we review the basic definitions and properties that are required in the sequel.
Defining weighted inner product $$\Big(u_1,u_2\Big)_{\alpha, \beta}=\int_{\Omega}{u_1(v) u_2(v) \delta^{\alpha, \beta}(v)
dv},$$ and discrete Jacobi-Gauss inner product $$\bigg(u_1,u_2\bigg)_{N,\alpha,\beta}=\sum\limits_{k=0}^N{u_1(v_k^{\alpha,\beta}) u_2(v_k^{\alpha,\beta}) \delta_k^{\alpha,\beta}},$$ we recall the following norms over $\Omega$ $$\|u\|_{\alpha,\beta}^2=\Big(u,u\Big)_{\alpha,\beta}, \quad
\|u\|_{N,\alpha,\beta}^2=\Big(u,u\Big)_{N,\alpha,\beta},\quad
\|u\|_{\infty}=\sup_{v \in \Omega} |u(v)|.$$
Here, $v_k^{\alpha,\beta}$ and $\delta_k^{\alpha,\beta}$ are the shifted Jacobi Gauss quadrature nodal points on $\Omega$ and corresponding weights respectively.
The non-uniformly Jacobi-weighted Sobolev space denotes by $B_{\alpha,\beta}^{k}(\Omega)$ and defines as follows $$B_{\alpha , \beta}^{k}(\Omega)=\{ u: \|u^{(s)}\|_{\alpha+s,\beta+s}
< \infty;~~ 0 \le s\le k\},$$ equipped with the norm and semi-norm $$||u||_{\alpha,\beta,k}^2=\sum\limits_{s = 0}^k
||u^{(s)}||_{\alpha+s,\beta+s}^2, \quad
|u|_{\alpha,\beta,k}=||u^{(k)}||_{\alpha+k,\beta+k}.$$
The space $B_{\alpha,\beta}^{k}(\Omega)$ distinguishes itself from the usual weighted Sobolev space $H_{\alpha,\beta}^{k}(\Omega)$ by involving different weight functions for derivatives of different orders. The usual weighted Sobolev space $H_{\alpha,\beta}^{k}(\Omega)$ is defined as $$H_{\alpha , \beta}^{k}(\Omega)=\{ u: \|u^{(s)}\|_{\alpha,\beta} <
\infty;~~ 0 \le s\le k\},$$ equipped with the norm $$||u||_{H_{\alpha,\beta}^{k}(\Omega)}^2=\sum\limits_{s = 0}^k
||u^{(s)}||_{\alpha,\beta}^2.$$
We denote the shifted GJPs on $\Omega$ by $G_n^{\alpha,\beta}(v)$ and define as $$\label{7} G_n^{\alpha,\beta}(v) = \left\{
\begin{array}{l}
{(2 -2v)^{-\alpha}}{(2v)^{-\beta}}J_{n-n_0}^{-\alpha,-\beta}(v),~ (\alpha,\beta) \in {\mathcal N_1}, ~~n_0 = -(\alpha+\beta),\\
\\
{(2 -2v)^{-\alpha}}J_{n-n_0}^{-\alpha,\beta}(v), \quad \quad \quad ~~~\;\;(\alpha,\beta) \in {\mathcal N_2},~~ n_0 = -\alpha, \\
\\
{(2v)^{-\beta}}J_{n-n_0}^{\alpha,-\beta}(v), \quad \quad \quad ~~~\;\;(\alpha,\beta) \in {\mathcal N_3},~~ n_0 = -\beta,\\
\\
J_{n-n_0}^{\alpha,\beta}(v),\quad \quad \quad \;\;\quad \quad \quad~~~~~~~ \;(\alpha,\beta) \in {\mathcal N_4},~~ n_0=0,\\
\end{array} \right.$$ where $J_n^{\alpha,\beta}(v)$ is the classical shifted Jacobi polynomials on $\Omega$; see [@aa26]. An important fact is that the shifted GJPs $\{G_n^{\alpha,\beta}(v); n \ge 1 \}$ form a complete orthogonal system in $L_{\alpha,\beta}^2(\Omega)$; see([@a15], [@a16]). To present a Galerkin solution for (\[6x\]) it is fundamental that the basis functions in the approximate solution satisfy in the homogeneous initial condition. To this end, since $G_n^{0,-1}(0) = 0,\;\; n \ge 1$, then we can consider $\{G_n^{0,-1}(v),~~n\geq 1\}$ as suitable basis functions to the Galerkin solution of (\[6x\]).
From (\[7\]) and the following formula [@aa16] $$J_i^{\alpha,\beta}(v)=\sum\limits_{k=0}^{i}{(-1)^{i-k}\frac{\Gamma{(i+\beta+1)}\Gamma{(i+k+\alpha+\beta+1)}}
{\Gamma{(k+\beta+1)}\Gamma{(i+\alpha+\beta+1)}(i-k)!k!}v^k},\quad
\alpha,\beta\in \mathcal N_4,$$ we can obtain the following explicit formula for $G_i^{0,-1}(v)$ $$\label{8} G_i^{0,-1}(v)= (2v)J_{i-1}^{0,1}(v)=2
\sum_{k=0}^{i-1}(-1)^{i-1-k} \frac{(i+k)!}{(k+1)! (i-1-k)!
k!}v^{k+1},~~~~~i \ge 1.$$
For any continuous function $Z(v)$ on $\Omega$, we define the Legendre Gauss interpolation operator $\mathcal I_N$, as $$\label{9}
\mathcal I_N Z(v) =\sum\limits_{s=0}^N
{\frac{\bigg(Z,J_s^{0,0}\bigg)_{N,0,0}}{\|J_s^{0,0}\|_{N,0,0}^2}
J_s^{0,0}(v)}.$$
Let $\mathcal P_N$ be the space of all algebraic polynomials of degree up to $N$. We introduce Legendre projection $\Pi_N:
L^2(\Omega) \to \mathcal P_N$ which is a mapping such that for any $Z(v) \in L^2(\Omega)$, $$\label{cc5}
\bigg(Z-\Pi_N Z, \phi\bigg)_{0,0}=0,\quad \forall \phi \in \mathcal
P_N.$$
Discrete Galerkin Approach
==========================
In this section, we present the numerical solution of (\[6x\]) by using the discrete Galerkin method based on GJPs.
Let $$\label{10c}
\tilde u_N(v)=\sum\limits_{i=1}^N{b_i G_{i}^{0,-1}(v)},$$ be the Galerkin solution of (\[6x\]). It is trivial that $\tilde
u_N(0)=0.$
Galerkin formulation of (\[6x\]) is to find $\tilde u_N(v)$, such that $$\label{rv17}
\bigg(\mathcal M^q \tilde u_N,G_i^{0,-1}\bigg)_{0,-1}=\bigg(\bar
p(v) \tilde u_N(v),G_i^{0,-1}\bigg)_{0,-1}+\bigg(\bar
f(v),G_i^{0,-1}\bigg)_{0,-1}+\lambda\bigg(\mathcal K(\tilde
u_N),G_i^{0,-1}\bigg)_{0,-1}, ~~i=1,2,...,N$$ where $\mathcal K(\tilde u_N)=\int\limits_0^v{\tilde K(v,w)\tilde
u_N(w)dw}.$
Applying transformation $w(\theta)=v \theta,~~\theta \in \Omega$ we get $$\label{rv5}
\mathcal K(\tilde u_N)=\mathcal K_\theta(\tilde u_N)=v
\int\limits_0^1{\tilde K(v,w(\theta)) \tilde u_N(w(\theta))d\theta}.$$
Substituting (\[rv5\]) in (\[rv17\]) yields $$\begin{gathered}
\label{12}
\bigg(\mathcal M^q \tilde u_N,G_i^{0,-1}\bigg)_{0,-1}=\bigg(\bar
p(v) \tilde u_N(v),G_i^{0,-1}\bigg)_{0,-1}+\bigg(\bar
f(v),G_i^{0,-1}\bigg)_{0,-1}+\lambda\bigg(\mathcal K_\theta(\tilde
u_N),G_i^{0,-1}\bigg)_{0,-1},\\ i=1,2,...,N.\end{gathered}$$
Inserting (\[10c\]) in (\[12\]) we get $$\begin{gathered}
\label{13}
\sum\limits_{j=1}^{N}{b_j \bigg\{\bigg(\mathcal M^q
G_j^{0,-1}(v),G_i^{0,-1}\bigg)_{0,-1}-\bigg(\bar p(v)
G_j^{0,-1}(v),G_i^{0,-1}\bigg)_{0,-1}-\lambda\bigg(\mathcal
K_{\theta}(G_j^{0,-1}),G_i^{0,-1}\bigg)_{0,-1}\bigg\}}
\\ =\bigg(\bar f(v),G_i^{0,-1}\bigg)_{0,-1},
~~i=1,2,...,N.\hspace{.01 cm}\end{gathered}$$
Following the relation $G_i^{0,-1}(v)
\delta^{0,-1}(v)=G_{i-1}^{0,1}(v)$, we can rewrite (\[13\]) as $$\begin{gathered}
\label{14}
\sum\limits_{j=1}^{N}{b_j \bigg\{\bigg(\mathcal M^q
G_j^{0,-1}(v),G_{i-1}^{0,1}\bigg)_{0,0}-\bigg(\bar p(v)
G_j^{0,-1}(v),G_{i-1}^{0,1}\bigg)_{0,0}-\lambda\bigg(\mathcal
K_{\theta}(G_j^{0,-1}),G_{i-1}^{0,1}\bigg)_{0,0}\bigg\}}
\\=\bigg(\bar
f(v),G_{i-1}^{0,1}\bigg)_{0,0},~~ i=1,2,...,N.\end{gathered}$$
Now, we try to find an explicit form for $\mathcal M^q G_j^{0,-1}$. To this end, using (\[8\]) we have $$\begin{aligned}
\label{15}
\mathcal M^q G_j^{0,-1}(v)&=&
\frac{2}{\Gamma{(1-q)}}\sum\limits_{k=0}^{j-1}{(-1)^{j-1-k}\frac{(j+k)!}{(k+1)!k!
(j-1-k)!}
\int\limits_0^v{\big(v^{\frac{1}{q}}-w^{\frac{1}{q}}\big)^{-q}(w^{k+1})'dw}}\\
\nonumber
&=&\frac{2}{\Gamma{(1-q)}}\sum\limits_{k=0}^{j-1}{(-1)^{j-1-k}\frac{(j+k)!}{(k!)^2
(j-1-k)!}
\int\limits_0^v{\big(v^{\frac{1}{q}}-w^{\frac{1}{q}}\big)^{-q} w^{k}
dw}}.\end{aligned}$$
Applying the relation[@ax15] $$\int\limits_0^v{\bigg(v^{\frac{1}{q}}-w^{\frac{1}{q}}\bigg)^{-q} w^k
dw}=\bigg(\frac{q \pi \csc{(\pi q)}\Gamma{(q+qk)}}{\Gamma{(q)}
\Gamma{(1+k q)}}\bigg)v^k,~~ k \ge 0,$$ in (\[15\]) we can obtain the following explicit formula for $\mathcal M^q G_j^{0,-1}$: $$\label{15x}
\mathcal M^q G_j^{0,-1}(v)
=\frac{2}{\Gamma{(1-q)}}\sum\limits_{k=0}^{j-1}{(-1)^{j-1-k}\frac{(j+k)!}{(k!)^2
(j-1-k)!}\bigg(\frac{q \pi \csc{(\pi q)}\Gamma{(q+qk)}}{\Gamma{(q)}
\Gamma{(1+k q)}}\bigg)v^k}=:\Psi_{j,q}(v),$$
Substituting (\[15x\]) in (\[14\]) we obtain $$\begin{gathered}
\label{16}
\sum\limits_{j=1}^{N}{b_j
\bigg\{\bigg(\Psi_{j,q}(v),G_{i-1}^{0,1}\bigg)_{0,0}-\bigg(\bar p(v)
G_j^{0,-1}(v),G_{i-1}^{0,1}\bigg)_{0,0}-\lambda\bigg(\mathcal
K_{\theta}(G_j^{0,-1}),G_{i-1}^{0,1}\bigg)_{0,0}\bigg\}}\\=\bigg(\bar
f(v),G_{i-1}^{0,1}\bigg)_{0,0},~~i=1,2,...,N.\end{gathered}$$
In this position, we approximate the integral terms of (\[16\]) using $(N+1)-$point Legendre Gauss quadrature formula. Our discrete Galerkin method is to seek $$\label{10}\bar
u_N(v)=\sum\limits_{i=1}^N{a_i G_{i}^{0,-1}(v)},$$ such that coefficients $\{a_j\}_{j=1}^N$ satisfies in the following algebraic system of linear equations $$\begin{gathered}
\label{17}
\sum\limits_{j=1}^{N}{a_j
\bigg\{\bigg(\Psi_{j,q}(v),G_{i-1}^{0,1}\bigg)_{0,0}-\bigg(\bar p(v)
G_j^{0,-1}(v),G_{i-1}^{0,1}\bigg)_{N,0,0}-\lambda\bigg(\mathcal
K_{N,\theta}(G_j^{0,-1}),G_{i-1}^{0,1}\bigg)_{N,0,0}\bigg\}}
\\=\bigg(\bar f(v),G_{i-1}^{0,1}\bigg)_{N,0,0}, i=1,2,...,N,\end{gathered}$$ where $$\label{rv6}
\mathcal K_{N,\theta}(G_j^{0,-1})=v \sum\limits_{k=0}^{N}{\tilde
K(v,w(\theta_k)) G_j^{0,-1}(w(\theta_k))\delta_k}.$$
Here $\{\theta_k\}_{k=0}^N$ and $\{\delta_k\}_{k=0}^N$ are the shifted Legendre Gauss quadrature points on $\Omega$ and corresponding weights respectively. Note that, from (\[15x\]) we can see that $\Psi_{j,q}(v)$ is a polynomial from degree at most $N$, then we have $\bigg(\Psi_{j,q}(v),G_{i-1}^{0,1}\bigg)_{N,0,0}=\bigg(\Psi_{j,q}(v),G_{i-1}^{0,1}\bigg)_{0,0}.$ It is trivial that the solution of (\[17\]) gives us unknown coefficients $\{a_i\}_{i=1}^N$ in (\[10\]).
Existence and Uniqueness Theorem for Discrete Galerkin Algebraic System
=======================================================================
The main object of this section is providing an existence and uniqueness Theorem for a special case of the discrete Galerkin algebraic system of equations (\[17\]) with $\bar p(v)=1$ and $0<q<\frac{1}{2}$. Throughout the paper, $C_i$ will denote a generic positive constant that is independent on $N$.
First, we give some preliminaries which will be used in the sequel.
Let $\mathcal X, \mathcal Y$ be normed spaces. A linear operator $\mathcal A: \mathcal X \to \mathcal Y $ is compact if the set $\{\mathcal A {x}|~ ||x||_{\mathcal X}\leq 1\}$ has compact closure in $\mathcal Y$.
\[rt1\]\[\[rvv1\], \[rvv2\], \[rvv3\]\] Assume $\mathcal X,~\mathcal Y$ be two banach spaces. Let $$\label{rv7}v=\mathcal A v+f,$$ be a linear operator equation where $\mathcal A:\mathcal X \to
\mathcal Y$ is a linear continuous operator, and the operator $I-\mathcal A$ is continuously invertible. As an approximation solution of (\[rv7\]) we consider the equation $$\label{rv8c}
v_N=\mathcal A_N v_N+\mathcal B_N f,$$ which can be rewritten as $$\label{rv8}
v_N=\tilde{\mathcal B}_N \mathcal A v_N+\mathcal S_N v_N+\mathcal
B_N f,$$ where $\mathcal A_N$ is a linear continuous operator in a closed subspace $\tilde{\mathcal Y}$ of $\mathcal Y$. $\mathcal B_N,
\tilde{\mathcal B}_N:\mathcal Y\to \tilde{\mathcal Y}$ are linear continuous projection operators and $\mathcal S_N=\mathcal
A_N-\mathcal{\tilde B}_N \mathcal A$ is a linear operator in $\tilde{\mathcal Y}$. If the following conditions are fulfilled
- for any $Z \in \tilde{\mathcal Y}$ we have $\|\mathcal S_N Z\|\to 0$ as $N \to \infty$
- $\|\mathcal A-\tilde{\mathcal B}_N \mathcal A\| \to 0$ as $N \to \infty$
- $\|f-\mathcal B_N f\| \to 0$ as $N \to \infty$
then (\[rv8\]) possesses a uniquely solution $v_N \in
\tilde{\mathcal Y}$, for a sufficiently large $N$.
\[rl1\] [@rvv4] Let $\mathcal X, \mathcal Y$ be banach spaces and $\tilde{\mathcal Y}$ be a subspace of $\mathcal Y$. Let $\tilde{\mathcal B}_N:\mathcal Y \to \tilde{\mathcal Y}$ be a family of linear continuous projection operators with $$\tilde{\mathcal B}_N y \to y~~ \text{as}~~ N \to \infty, ~~ y \in
\mathcal Y.$$
Assume that linear operator $\mathcal A:\mathcal X \to \mathcal Y$ be compact. Then $$\|\mathcal A-\tilde{\mathcal B}_N \mathcal A\| \to 0~~\text{as}~~ N
\to \infty.$$
\[l1\] (Interpolation error bound[@aa26]) Let $\mathcal I_N Z$ be the interpolation polynomial approximation of the function $Z(v)$ defined in (\[9\]). For any $Z(v) \in
B_{0,0}^k(\Omega)$ with $k \ge 1$, we have $$\|Z-\mathcal I_N Z\|_{0,0} \le C N^{-k} |Z|_{0,0,k}.$$
\[l3\] [@r6] For every bounded function $Z(v)$, there exists a constant $C$ independent of $Z$ such that $$\sup\limits_{N}\|\mathcal I_N Z\|_{0,0} \le C \sup\limits_{v}|Z(v)|.$$
\[l4\] (Legendre Gauss quadrature error bound[@aa26]) If $Z(v) \in B_{0,0}^k(\Omega)$ for some $k \ge 1$ and $\Phi \in
\mathcal P_N$, then for the Legendre Gauss integration we have
$$\bigg|(Z,\Phi)_{0,0}-(Z,\Phi)_{N,0,0}\bigg| \le C N^{-k}
\|Z\|_{0,0,k} \|\Phi\|_{0,0}.$$
Now we intend to prove existence and uniqueness Theorem for a special case of the discrete Galerkin system (\[17\]) with $\bar
p(v)=1$ and $0<q<\frac{1}{2}$.
(Existence and Uniqueness)Let $0<q<\frac{1}{2}$ and $\bar p(v)=1$. If (\[6x\]) has a uniquely solution $\bar u(v)$ then the linear discrete Galerkin system (\[17\]) has a uniquely solution $\bar
u_N(v) \in \mathcal P_N$ for sufficiently large $N$.
Our strategy in proof is based on two steps. First, we try to represent (\[17\]) in the operator form (\[rv8\]). Then by applying Theorem \[rt1\] to operator form obtained in the first step the desired result have been concluded.
[**Step 1:**]{} In this step, we show that the discrete Galerkin system (\[17\]) can be written in the operator form (\[rv8\]). To this end, consider (\[6x\]) and define $$\mathcal{\bar R}_N(v)=\mathcal M^q \bar u_N(v)-\bar u_N(v)-\lambda
\mathcal K_{N,\theta}(\bar u_N)-\bar f(v).$$
According to the proposed method, we have $$\bigg(\mathcal{\bar R}_N(v),G_{i-1}^{0,1}(v)\bigg)_{N,0,0}=0,\quad
i=1, 2,...,N.$$
From interpolation and Legendre Gauss quadrature properties, we can write $$\label{18}
\bigg(\mathcal{\bar
R}_N(v),G_{i-1}^{0,1}(v)\bigg)_{N,0,0}=\bigg(\mathcal
I_N(\mathcal{\bar
R}_N),G_{i-1}^{0,1}(v)\bigg)_{N,0,0}=\bigg(\mathcal
I_N(\mathcal{\bar R}_N),G_{i-1}^{0,1}(v)\bigg)_{0,0}=0,\quad i=1,
2,...,N.$$
Since $\mathcal I_N(\mathcal{\bar R}_N(v))$ is a polynomial, it can be represented by a linear orthogonal polynomial expansion based on $\big\{G_i^{0,-1}(v)\big\}_{i=0}^N$, as $$\label{cc6}
\mathcal I_N(\mathcal{\bar
R}_N)=\sum\limits_{i=1}^{N}{\frac{\bigg(\mathcal I_N(\mathcal{\bar
R}_N),G_{i}^{0,-1}\bigg)_{0,-1}}{\|G_i^{0,-1}\|_{0,-1}^2}
G_i^{0,-1}(v)}=\sum\limits_{i=1}^{N}{\frac{\bigg(\mathcal
I_N(\mathcal{\bar
R}_N),G_{i-1}^{0,1}\bigg)_{0,0}}{\|G_i^{0,-1}\|_{0,-1}^2}
G_i^{0,-1}(v)}.$$
Using the relations (\[18\]) and (\[cc6\]) yields $\mathcal
I_N(\mathcal{\bar R}_N)=0$. Thus $$\mathcal I_N\bigg(\mathcal M^q \bar u_N-\bar u_N(v)-\lambda \mathcal
\mathcal K_{N,\theta}(\bar u_N)-\bar f\bigg)=0,$$ which can be rewritten as $$\label{rv9c}
\bar u_N(v)=\mathcal I_N\bigg(\mathcal M^q -\lambda \mathcal
\mathcal K_{N,\theta}\bigg)\bar u_N-\mathcal I_N \bar f=\mathcal I_N
\mathcal T_N \bar u_N -\mathcal I_N \bar f,$$ and thereby $$\label{rv9}
\bar u_N(v)=\Pi_N \mathcal T \bar u_N+\mathcal S_N \bar u_N
-\mathcal I_N \bar f,$$ where $\mathcal I_N$ and $\Pi_N$ are defined in (\[9\]) and (\[cc5\]) respectively and $$\begin{aligned}
\mathcal T&=&\mathcal M^q-\lambda \mathcal K_\theta,\\
\mathcal T_N&=&\mathcal M^q -\lambda \mathcal \mathcal K_{N,\theta},\\
\mathcal S_N&=&\mathcal I_N \mathcal T_N-\Pi_N \mathcal T.\end{aligned}$$
Since (\[rv9\]) is the form in which the discrete Galerkin method is implemented, as it leads directly to the equivalent linear system (\[17\]). It can be easily check that (\[rv9\]) can be considered in the operator form (\[rv8\]) by assuming $$\begin{aligned}
\label{cc1}
\nonumber\mathcal X&=&H_{0,0}^1(\Omega),~~\mathcal
Y=L^2(\Omega),~~\tilde{\mathcal
Y}=\mathcal P_N,\\
v_N&=&\bar u_N,~~\quad ~~\tilde{\mathcal B_N}=\Pi_N,~~\quad \mathcal
B_N=\mathcal
I_N,\\
\nonumber\mathcal A&=&\mathcal T,~~\quad ~~ \mathcal A_N=\mathcal
I_N \mathcal T_N,\end{aligned}$$ which can be completed the desired result of step 1.
[**Step 2:**]{} In this step we intend to apply Theorem \[rt1\] with the assumptions (\[cc1\]) to prove the Theorem. To this end, following Theorem \[rt1\] we must show that $$\begin{aligned}
\label{cc3}
\nonumber&&1)~\text{for any}~ Z \in \mathcal P_N ~\text{we have}~
\|\mathcal S_N Z\|_{0,0}=\|\mathcal I_N \mathcal T_N Z-\Pi_N
\mathcal T Z\|_{0,0}
\to 0 ~\text{as}~ N \to \infty,\\
&&2) \|\mathcal T-\Pi_N \mathcal T\|_{0,0} \to 0~ \text{as}~ N \to
\infty,\\
\nonumber&&3) \|\bar f-\mathcal I_N \bar f\|_{0,0} \to 0 ~\text{as}~
N \to \infty.\end{aligned}$$
First, we prove the first condition in (\[cc3\]). For this, we can write $$\label{rv13}
\|\mathcal S_N Z\|_{0,0}=\|\bigg(\mathcal I_N \mathcal T_N-\Pi_N
\mathcal T\bigg)Z\|_{0,0}\le \|\mathcal I_N\bigg(\mathcal
T_N-\mathcal T\bigg)Z\|_{0,0}+\|\bigg(\mathcal
I_N-\Pi_N\bigg)\mathcal T Z\|_{0,0}.$$
Since $\mathcal M^q Z \in \mathcal P_N$ for any $Z \in \mathcal P_N$ then $$\label{rv14}
\|\mathcal S_N Z\|_{0,0}\le \lambda\Bigg(\|\mathcal
I_N\bigg(\mathcal K_{N,\theta}-\mathcal
K_{\theta}\bigg)Z\|_{0,0}+\|\bigg(\mathcal I_N-\Pi_N\bigg)\mathcal
K_\theta Z\|_{0,0}\Bigg).$$
Using Lemmas \[l3\] and \[l4\] and the relations (\[rv5\]) and (\[rv6\]) we can obtain $$\label{25rv}
\|\mathcal I_N\bigg(\mathcal K_{N,\theta}-\mathcal
K_{\theta}\bigg)z\|_{0,0} \le \sup\limits_{v \in \Omega}{|\mathcal
K_\theta(z(v))-\mathcal K_{N,\theta}(z(v))|} \le C_1 N^{-k_1}
\sup\limits_{v \in \Omega}{\bigg(\|\tilde
K(v,w(\theta))\|_{0,0,k_1}\|v Z(w(\theta))\|_{0,0}\bigg)},$$ where norms $\|\tilde K(v,w(\theta))\|_{0,0,k_1}$ and $\|v
Z(w(\theta))\|_{0,0}$ are applied with respect to the variable $\theta$.
According to Lemma \[l4\] we have $$\label{cc7}
\Bigg|\bigg(\mathcal K_\theta,J_s^{0,0}\bigg)_{0,0}-\bigg(\mathcal
K_\theta,J_s^{0,0}\bigg)_{N,0,0}\Bigg| \le CN^{-k_2} \|\mathcal
K_\theta\|_{0,0,k_2} \|J_s^{0,0}\|_{0,0},~~s=0,1,...,N,$$ where norm $\|\mathcal K_\theta\|_{0,0,k_2}$ is applied with respect to the variable $v$. Using (\[cc7\]) and the relations (\[9\]), (\[cc5\]) we can yields $$\label{rv15}
\|\bigg(\mathcal I_N-\Pi_N\bigg)\mathcal K_\theta Z\|_{0,0} \to
0~~\text{as}~~N \to \infty.$$
Substituting (\[25rv\]) and (\[rv15\]) in (\[rv14\]) we can conclude the first condition in (\[cc3\]). Applying Lemma \[l4\] gives us the third condition in (\[cc3\]). To complete the proof, it is sufficient that we prove the second condition of (\[cc3\]). To this end, we apply Lemma \[rl1\] with the assumptions (\[cc1\]). Since $\|y-\Pi_N y\|_{0,0} \to 0~\text{as}~N \to
\infty$ for $y \in L^2(\Omega)$(see\[\]), the second condition in (\[cc3\]) can be achieved by proving compactness of the operator $\mathcal T$. Since $\tilde k(v,w)$ is a continuous kernel, then the operator $\mathcal T$ will be compact if $\mathcal M^q(\bar
u):H_{0,0}^1(\Omega) \to L^2(\Omega)$ be a compact operator. For this, define $M=\big[\int_{0}^{1}{\int_{0}^{1}{\bigg(v^{\frac{1}{q}}-w^{\frac{1}{q}}\bigg)^{-2q}
dw}dv}\big]^{\frac{1}{2}}< \infty$. For $\bar u \in
H_{0,0}^1(\Omega)$, using Cauchy-Schwartz inequality we have $$\begin{aligned}
\|\mathcal M^q \bar u\|_{0,0}^2&=&
\int_{0}^{1}{|\int_{0}^{v}{\bigg(v^{\frac{1}{q}}-w^{\frac{1}{q}}\bigg)^{-q}
\bar u'(w) dw}|^2 dv} \le
\int_{0}^{1}\bigg({\int_{0}^{v}{|\bigg(v^{\frac{1}{q}}-w^{\frac{1}{q}}\bigg)^{-q} \bar u'(w)| dw}\bigg)^2 dv} \\
& \le &
\int_{0}^{1}\bigg({\int_{0}^{1}{|\bigg(v^{\frac{1}{q}}-w^{\frac{1}{q}}\bigg)^{-q}
\bar u'(w)| dw}\bigg)^2 dv} \le
\int_{0}^{1}{\bigg(\int_{0}^{1}{\bigg(v^{\frac{1}{q}}-w^{\frac{1}{q}}\bigg)^{-2q}dw}
\int_{0}^{1}{|\bar u'(w)|^2dw}\bigg)dv} \\
& \le & M^2 \|\bar u'\|_{0,0}^2 \le M^2 \|\bar
u\|_{H_{0,0}^1(\Omega)}^2.\end{aligned}$$
So $\mathcal M^q \bar u \in L^2(\Omega)$ and $\|\mathcal M^q\|=\sup
\bigg\{\frac{\|\mathcal M^q \bar u\|_{0,0}}{\|\bar
u\|_{H_{0,0}^1(\Omega)}}~~\Big|~~ \bar u \ne 0, \bar u \in
H_{0,0}^1(\Omega)\bigg\} \le M < \infty$. Therefore $\mathcal
M^q:H_{0,0}^{1}(\Omega) \to L^2(\Omega)$ defined by (\[rv4\]), is a bounded operator. If we proceed same as Theorem 3.4 in \[\[arr1\]\], we can conclude compactness of the operator $\mathcal
M^q(\bar u)$. Then the operator $\mathcal T$ is a compact operator and from Lemma \[rl1\] we have $$\|\mathcal T-\Pi_N \mathcal T\|_{0,0} \to 0~\text{as}~ N\to \infty$$
In this position, all conditions in (\[cc3\]) that are required to deduce existence and uniqueness of solutions of the discrete Galerkin system (\[17\]) have been proved and then the proof is completed.
Convergence analysis
====================
In this section, we will try to provide a reliable convergence analysis which theoretically justify convergence of the proposed discrete Galerkin method for the numerical solution of a special case of (\[6x\]) with $\bar p(v)=1$.
In the sequel, our discussion is based on these Lemmas:
[@aa26]\[l2\]For any $Z \in B_{0,0}^1(\Omega)$, we have $$\|\mathcal I_{N}Z\|_{0,0} \le
C\bigg(\|Z\|_{0,0}+N^{-1}\|Z'\|_{1,1}\bigg).$$
\[l5\] The fractional integral operator $\mathcal
I^\mu$ is bounded in $L^2(\Omega)$ and $$\|\mathcal I^\mu Z\|_{0,0} \le C \|Z\|_{0,0},~~ Z \in L^2(\Omega).$$
\[l6\] [@r6](Gronwall inequality)Assume that $Z(v)$ is a non-negative, locally integrable function defined on $\Omega$ which satisfies $$Z(v) \le b(v)+B \int_{0}^{v}{(v-w)^\alpha w^\beta Z(w) dw}, \quad w
\in \Omega,$$ where $\alpha, \beta>-1$, $b(v) \ge 0$ and $B \ge 0$. Then, there exist a constant $C$ such that $$Z(v) \le b(v)+ C \int_{0}^{v}{(v-w)^\alpha w^\beta b(w) dw}, \quad w
\in \Omega.$$
Now, we state and prove the main result of this section regarding the error analysis of the proposed method for the numerical solution of a special case of (\[6x\]) with $\bar p(v)=1$.
\[t1\] (Convergence)Let $u(x)$ and $\bar u(v)$ are the exact solutions of the equations (\[1\]) and (\[6x\]) respectively that is related by $u(x)=\bar u(v)$. Assume that $u_N(x)=\bar u_N(v)$ be the approximate solution of (\[1\]), where $\bar u_N(v)$ is the discrete Galerkin solution of the transformed equation (\[6x\]). If the following conditions are fulfilled
1. $\bar f(v) \in B_{0,0}^{k_1}(\Omega)$ for $k_1 \ge 1$,
2. $\mathcal K(\bar u) \in B_{0,0}^{k_2}(\Omega)$ for $k_2 \ge 1$,
then for sufficiently large $N$ we have $$\|e_N(u)\|_{0,0} \le C N^{-\xi}\bigg(|\bar f|_{0,0,k_1}+|\mathcal
K(\bar u)|_{0,0,k_2}\bigg)$$ where $\xi=\min\{k_1,k_2\}$ and $e_N(u)= u(x)-u_N(x)$ denotes the error function.
Since $\mathcal M^q \bar u_N$ is a polynomial from degree of at most $N$ then we can rewrite (\[rv9c\]) as $$\label{7x}
\mathcal M^q \bar u_N-\bar u_N(v)-\mathcal I_N\bigg(\lambda \mathcal
\mathcal K_{N,\theta}(\bar u_N)\bigg)=\mathcal I_N \bar f.$$
Subtracting (\[6x\]) from (\[7x\]) and some simple manipulations we can obtain $$\label{19}
\mathcal M^q \bar e_N=\bar e_N(v)+e_{\mathcal I_N}(\bar f)+\lambda
\mathcal K(\bar e_N)+\lambda e_{\mathcal I_N}(\mathcal K(\bar
u_N))+\lambda \mathcal I_N\bigg(\mathcal K_{\theta}(\bar
u_N)-\mathcal K_{N,\theta}(\bar u_N)\bigg),$$ where $e_{\mathcal I_N}(g)=g-\mathcal I_N(g)$ denotes the interpolating error function and $\bar e_N(v)=\bar u(v)-\bar u_N(v)$ is the error function of approximating (\[6x\]) using discrete Galerkin solution $\bar u_N(v)$.
Applying the transformation (\[6xx\]) to the operator $\mathcal
I^q u$ we get $$\mathcal M_1^q \bar
u=\frac{1}{\Gamma{(q)}}\int\limits_0^v{\bigg(v^{\frac{1}{q}}-w^{\frac{1}{q}}\bigg)^{q-1}
\bar u(w) \frac{w^{\frac{1}{q}-1}}{q}dw}.$$
Following (\[20\]) it is easy to check that $$\label{7xx}
\mathcal M_1^q \big(\mathcal M^q \bar u(v)\big)=\bar u(v)-\bar u(0).$$
Applying the operator $\mathcal M_1^q$ on the both sides of (\[19\]) and using (\[7xx\]) we can yield $$\label{rv1}
|\bar e_N(v)| \le \mathcal M_1^q(|\bar e_N|+|\lambda \mathcal
K(\bar e_N)|)+\bigg|\mathcal M_1^q\bigg(e_{\mathcal I_N}(\bar
f)+\lambda e_{\mathcal I_N}(\mathcal K(\bar u_N))+\lambda \mathcal
I_N\bigg(\mathcal K_{\theta}(\bar u_N)-\mathcal K_{N,\theta}(\bar
u_N)\bigg)\bigg)\bigg|.$$
Since $$\begin{aligned}
\label{rv2}
\nonumber\mathcal M_1^q \int\limits_{0}^{v}{|\tilde K(v,w) \bar
e_N(w)|
dw}&=&\frac{1}{\Gamma{(q)}}\int\limits_{0}^{v}{\int\limits_{0}^{w}{\big(v^{\frac{1}{q}}-w^{\frac{1}{q}}\big)^{q-1}
\frac{w^{\frac{1}{q}-1}}{q} |\tilde K(w,s)||\bar e_N(s)|ds}dw}\\
\nonumber&=&\frac{1}{\Gamma{(q)}}\int\limits_{0}^{v}{\int\limits_{s}^{v}{\big(v^{\frac{1}{q}}-w^{\frac{1}{q}}\big)^{q-1}
\frac{w^{\frac{1}{q}-1}}{q} |\tilde K(w,s)||\bar e_N(s)|dw}ds}\\
&=&\frac{1}{\Gamma{(q)}}\int\limits_{0}^{v}{\tilde K_1(v,s) |\bar
e_N(s)|ds}\le \frac{\|\tilde
K_1(v,s)\|_\infty}{\Gamma{(q)}}\int\limits_{0}^{v}{|\bar e_N(s)|ds},\end{aligned}$$ then (\[rv1\]) can be rewritten as $$\begin{aligned}
\label{rv3}
\nonumber|\bar e_N(v)| &\le &
\frac{1}{\Gamma{(q)}}\bigg(\int\limits_{0}^{v}{\big(v^{\frac{1}{q}}-w^{\frac{1}{q}}\big)^{q-1}
\frac{w^{\frac{1}{q}-1}}{q}|\bar e_N(w)|dw}+|\lambda|\|\tilde
K_1(v,s)\|_\infty \int\limits_{0}^{v}{|\bar e_N(w)|
dw}\bigg)\\
\nonumber &+&\bigg|\mathcal M_1^q\bigg(e_{\mathcal I_N}(\bar
f)+\lambda e_{\mathcal I_N}(\mathcal K(\bar u_N))+\lambda \mathcal
I_N\bigg(\mathcal K_{\theta}(\bar u_N)-\mathcal K_{N,\theta}(\bar
u_N)\bigg)\bigg)\bigg|\\ &\le& \tilde C
\bigg(\int\limits_{0}^{v}{\big(v^{\frac{1}{q}}-w^{\frac{1}{q}}\big)^{q-1}
|\bar e_N(w)|dw}\bigg)\\ \nonumber &+&\bigg|\mathcal
M_1^q\bigg(e_{\mathcal I_N}(\bar f) +\lambda e_{\mathcal
I_N}(\mathcal K(\bar u_N))+\lambda \mathcal I_N\bigg(\mathcal
K_{\theta}(\bar u_N)-\mathcal K_{N,\theta}(\bar
u_N)\bigg)\bigg)\bigg|,\end{aligned}$$ where $$\tilde K_1(v,s)=\int\limits_{s}^{v}{\big(v^{\frac{1}{q}}-w^{\frac{1}{q}}\big)^{q-1}
\frac{w^{\frac{1}{q}-1}}{q} |\tilde K(w,s)|dw},\quad \tilde
C=\max\bigg\{\frac{1}{q \Gamma{(q)}},\frac{|\lambda|\|\tilde
K_1(v,s)\|_\infty }{\Gamma{(q)}}\bigg\}
\|1+\big(v^{\frac{1}{q}}-w^{\frac{1}{q}}\big)^{1-q}\|_\infty.$$
Using Gronwall inequality(Lemma \[l6\]) in (\[rv3\]) yields $$\label{21c}
\|\bar e_N\|_{0,\frac{1}{q}-1} \le C_1 \bigg(\|\mathcal
M_1^q\bigg(e_{\mathcal I_N}(\bar f)+\lambda e_{\mathcal
I_N}(\mathcal K(\bar u_N))+\lambda \mathcal I_N\bigg(\mathcal
K_{\theta}(\bar u_N)-\mathcal K_{N,\theta}(\bar
u_N)\bigg)\bigg)\|_{0,\frac{1}{q}-1}\bigg).$$
It can be checked that by applying the transformation (\[6\]) we obtain $$\label{cc8}\|\bar
e_N(v)\|_{0,\frac{1}{q}-1}^2=q
2^{\frac{1}{2}-1}\|e_N(x)\|_{0,0}^2.$$
On the other hand, by applying (\[6\]) and Lemma \[l5\] we can get $$\label{cc4}
\|\mathcal M_1^q \bar Z(v)\|_{0,\frac{1}{q}-1}^2=q
2^{\frac{1}{q}-1}\|\mathcal I^q Z(x)\|_{0,0}^2 \le C q
2^{\frac{1}{q}-1}\|Z(x)\|_{0,0}^2=C \|\bar
Z(v)\|_{0,\frac{1}{q}-1}^2,$$ where $Z(x)$ is a given function and $\bar Z(v)=z(v^{\frac{1}{q}})$. Using the relations (\[cc8\]) and (\[cc4\]) in (\[21c\]) we can obtain $$\begin{aligned}
\label{21cc}
\sqrt{q 2^{\frac{1}{q}-1}}\|e_N\|_{0,0} &\le & C_2 \| e_{\mathcal
I_N}(\bar f)+\lambda e_{\mathcal I_N}(\mathcal K(\bar u_N))+\lambda
\mathcal I_N\bigg(\mathcal K_{\theta}(\bar u_N)-\mathcal
K_{N,\theta}(\bar u_N)\bigg)\|_{0,\frac{1}{q}-1} \\
\nonumber&\le & C_2 \bigg(\| e_{\mathcal I_N}(\bar
f)\|_{0,\frac{1}{q}-1}+|\lambda|\|e_{\mathcal I_N}(\mathcal K(\bar
u_N))\|_{0,\frac{1}{q}-1}+|\lambda|\|\mathcal I_N\bigg(\mathcal
K_{\theta}(\bar u_N)-\mathcal K_{N,\theta}(\bar
u_N)\bigg)\|_{0,\frac{1}{q}-1}\bigg).\end{aligned}$$
Since $\delta^{0,\frac{1}{q}-1} \le \delta^{0,0}$ then we have $\|.\|_{0,\frac{1}{q}-1} \le \|.\|_{0,0}$ and thereby (\[21cc\]) can be written as $$\label{21}
\|e_N\|_{0,0} \le C_3 \bigg(\mathcal L_1+\mathcal L_2+\mathcal
L_3\bigg),$$ where $$\mathcal L_1=\|e_{\mathcal I_N}(\bar f)\|_{0,0}, \quad \mathcal
L_2=\|e_{\mathcal I_N}(\mathcal K(\bar u_N))\|_{0,0}, \quad \mathcal
L_3=\|\mathcal I_N\big(\mathcal K_{\theta}(\bar u_N)-\mathcal
K_{N,\theta}(\bar u_N)\big)\|_{0,0}.$$
Now, it is sufficient to find suitable upper bounds to the above norms. According to Lemma \[l1\] we have $$\begin{aligned}
\label{22}
\mathcal L_1&=&\|e_{\mathcal I_N}(\bar f)\|_{0,0} \le C_3 N^{-k_1}
|\bar f|_{0,0,k_1},\\
\nonumber \\
\nonumber\mathcal L_2&=&\|e_{\mathcal I_N}(\mathcal K(\bar
u_N))\|_{0,0} \le C_4 N^{-k_2}|\mathcal K(\bar u-\bar
e_N(u))|_{0,0,k_2}.\end{aligned}$$
Using Lemmas \[l3\] and \[l4\] we can obtain $$\begin{aligned}
\label{25}
\nonumber \mathcal L_3=\|\mathcal I_N\big(\mathcal K_\theta(\bar
u_N)-\mathcal K_{N,\theta}(\bar u_N)\big)\|_{0,0} &\le &
\sup\limits_{v \in
\Omega}{|\mathcal K_\theta(\bar u_N)-\mathcal K_{N,\theta}(\bar u_N)|}\\
\nonumber &\le & C_7 N^{-k_3} \sup\limits_{v \in
\Omega}{\bigg(\|\tilde
K(v,w(\theta))\|_{0,0,k_3}\|v \bar u_N(w(\theta))\|_{0,0}\bigg)}\\
\nonumber &\le & C_7 N^{-k_3}\bigg(\sup\limits_{v \in
\Omega}{\|\tilde
K(v,w(\theta))\|_{0,0,k_3}\bigg)}\|\bar u_N(w)\|_{0,0}\\
&\le & C_7 N^{-k_3} \bigg(\sup\limits_{v \in \Omega}{\|\tilde
K(v,w(\theta))\|_{0,0,k_3}\bigg)}\bigg(\|\bar u\|_{0,0}+\|\bar
e_N\|_{0,0}\bigg),\end{aligned}$$ where norms $\|\tilde K(v,w(\theta))\|_{0,0,k_3}$ and $\|v \bar
u_N(w(\theta))\|_{0,0}$ are applied with respect to the variable $\theta$.
For sufficiently large $N$ the desired result can be concluded by inserting (\[22\]) and (\[25\]) in (\[21\]).
Numerical Results
=================
In this section we apply a program written in Mathematica to a numerical example to demonstrate the accuracy of the proposed method and effectiveness of applying GJPs. the “Numerical Error” always refers to the $L^2$-norm of the obtained error function.
\[e1\]Consider the following FIDE $$\mathcal D^{\frac{1}{2}}u(x)=u(x)+f(x)+\frac{1}{2}\int\limits_0^x{\sqrt{x t} u(t)
dt},\quad u(0)=0,$$ with the exact solution $u(x)=\frac{\sin{x}}{\sqrt{x}}$ and $$f(x) =\frac{-x+\sqrt{x}\bigg(\sqrt{x}
\cos{x}+\sqrt{\pi}\bigg(J_0(\frac{x}{2})\cos{\frac{x}{2}}-J_1(\frac{x}{2})\sin{\frac{x}{2}}\bigg)\bigg)-2
\sin{x}}{2 \sqrt{x}}.$$
Here $J_n(z)$ gives the Bessel function of the first kind.
This example has a singularity at the origin, i.e., $$u'(x) \sim \frac{1}{\sqrt{x}}.$$
In the theory presented in the previous section, our main concern is the regularity of the transformed solution. For the present problem by applying coordinate transformation $x=v^2$, the infinitely smooth solution $$\bar u(v)=u(v^2)=\frac{\sin{v^2}}{v},$$ is obtained. The main purpose is to check the convergence behavior of the numerical solutions with respect to the approximation degree $N$. Numerical results obtained are given in the Table 1 and Figure 1. As expected, the errors show an exponential decay, since in this semi-log representation(Figure 1) one observes that the error variations are essentially linear versus the degrees of approximation.
Table $1$: The numerical errors of example \[e1\].
N
---- ------------------------ -- --
2 $7.86 \times 10^{-3}$
4 $4.71 \times 10^{-5}$
6 $2.15 \times 10^{-6}$
8 $1.05 \times 10^{-8}$
10 $4.42 \times 10^{-10}$
12 $2.05 \times 10^{-12}$
14 $2.64 \times 10^{-14}$
16 $1.91 \times 10^{-16}$
=2.8 in =2.8 in
[9]{} \[rvv4\] K. E. Atkinson, [*The Numerical Solution of Integral Equations of the Second Kind,*]{} Cambridge, (1997).
\[r1\]F. Awawdeh, E. A. Rawashdeh, H. M. Jaradat, Analytic solution of fractional integro-differential equations, [*Ann. Univ. Craiova math. Comput. Sci. Ser.*]{} [**38**]{} (2011), 1-10.
R. L. Bagley, P. J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, [*J. Rheol.,*]{} [**27**]{}(1983), 201-210.
\[r3\]H. Brunner, [*Collocation Methods for Volterra and Related Functional Equations,*]{} Cambridge University Press, Cambridge, (2004).
\[r5\] M. Caputo, Linear models of dissipation whose Q is almost frequency independent II, [*Geophys. J. Roy. Astron. Soc.,*]{} [**13**]{}(1967), 529-539.
\[a13\] C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, [*Spectral Methods, Fundamentals in Single Domains,*]{} Springer-Verlag, Berlin, (2006).
\[r6\]Y. Chen Y., T. Tang, Convergence analysis of the Jacobi spectral collocation methods for volterra integral equations with a weakly singular kernel, [*Math. Comput.*]{} [**79**]{}(2010), 147-167.
\[r8\]K. Diethelm, [*The Analysis of Fractional Differential Equations,*]{} Springer-Verlag Berlin, (2010).
\[aa16\] E. H. Doha, A. H. Bhrawy, S. S. Ezz-Eldien, A new Jacobi operational matrix: an application for solving fractional differential equations, [*Appl. Math. Model.,*]{} [**36**]{} (2012), 4931-4943.
\[r9\]M. R. Eslahchi, M. Dehghan, M. Parvizi, Application of the Collocation method for solving nonlinear fractional integro-differential equations, [*J. Comput. Appl. Math.,*]{} [**257**]{} (2014), 105-128.
\[ax15\] F. Ghoreishi, P. Mokhtary, Spectreal Collocation method for multi-order fractional differential equations, [*Int. J. Comput. Math.,*]{} [**11**]{}(2014), no. 5, 23pp.
\[a15\] Ben-Yu Guo , Jie Shen, Li-Lian Wang, Optimal Spectral-Galerkin methods using Generalized Jacobi polynomials, [*J. Sci. Comput.,*]{} [**27**]{}(2006), 305-322.
\[a16\] Ben-Yu Guo, Jie Shen, Li-Lian Wang, Generalized Jacobi polynomials/functions and their applications, [*Appl. Numer. Math.,*]{} [**59**]{}(2009), 1011-1028.
\[r11\] L. Huang, X. F. Li, Y. L. Zhao, X. Y. Duan, Approximate solution of fractional integro-differential eqations by Taylor expansion method, [*Comput. Math. Appl.,*]{} [**62**]{} (2011), 1127-1134.
\[a14\] J. S. Hesthaven, S. Gottlieb, D. Gottlieb, [*Spectral Methods for Time-Dependent Problems,*]{} Cambridge University Press, (2007).
\[rvv1\] L. V. Kantrovich, Functional analysis and applied mathematics,[*Usp. Mat. Nauk,*]{} [**3**]{} (1984), 89-185.
\[rvv2\] L. V. Kantrovich, G. P. Akilov, [*Functional Analysis in Normed Spaces(Funktsional’nyi analiz v normirovannykh prostranstvakh),*]{} Fizmatgiz, Moscow, (1959).
\[r11\] L. Huang, X. F. Li, Y. L. Zhao, X. Y. Duan, Approximate solution of fractional integro-differential eqations by Taylor expansion method, [*Comput. Math. Appl.,*]{} [**62**]{} (2011), 1127-1134.
\[r13\]A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, [*Theory and Applications of Fractional Differential Equations,*]{} Elsevier, Amesterdam, (2006).
\[r12\]M. M. Khader, N. H. Sweilam, On the approximate solutions for system of fractional integro-differential equations using Chebyshev pseudo-spectral method, [*Appl. Math. Model.,*]{} [**27**]{}(2013), no. 24, 819-9828.
\[r17\]R. C. Mittal, R. Nigam, Solution of fractional calculus and fractional integro-differential equations by Adomian decomposition method, [*Int. J. Appli. Math. Mech.,*]{} [**4**]{}(2008), no. 4, 87-94.
\[r16\]Xiaohua Ma, C. Huang, Numerical solution of fractional integro-differential equations by Hybrid Collocation method, [*Appl. Math. Comput.,*]{} [**219**]{} (2013), 6750-6760.
\[r15\]Xiaohua Ma, C. Huang, Spectral Collocation method for linear fractional integro-differential equations, [*Appl. Math. Model.*]{} [**38**]{} (2014), no. 4, 1434-1448.
\[r18\]P. Mokhtary, F. Ghoreishi, The $L^2$-convergence of the Legendre spectral Tau matrix formulation for nonlinear fractional integro differential equations, [*Numer. Algorithms*]{}, [**58**]{} (2011), 475-496.
\[r19\]D. Nazari, S. Shahmorad, Application of the fractionl differential transform method to fractional-order integro-differential equations with nonlocal boundary conditions, [*J. Comput. Appl. Math.,*]{} [**234**]{} (2010), 883-891.
\[r20\]K. B. Oldham, J. Spanier, [*The Fractional Calculus,*]{} Academic Press, New York, (1974).
\[r21\]W. E. Olmstead, R. A. Handelsman, Diffusion in a semi-infinite region with nonlinear surface dissipation, [*SIAM Rev.*]{} [**18**]{} (1976), 275-291.
\[r22\]I. Podlubny, [*Fractional Differential Equations,*]{} Academic Press, (1999).
\[arr1\] D. Porter, David S. G. Stirling, [*Integral Equations, A Practical Treatment, from Spectral Theory to Applications*]{}, Cambridge University Press, New York, (1990).
\[r23\]E. A. Rawashdeh, Nemerical solution of fractional integro-differential equations by Collocation method, [*Appl. Math. Comput,*]{} [**176**]{}(2006), 1-6.
\[aa26\] Jie Shen, Tao Tang, Li-Lian Wang,[*Spectral Methods, Algorithms, Analysis and Applications,*]{} Springer, (2011).
\[r7\] Li Xianjuan, T. Tang, Convergence analysis of Jacobi spectral Collocation methods for Abel-Volterra integral equations of second kind, [*Front. Math. China,*]{} [**7**]{}(2012), no. 1, 69-84.
\[r25\]Li Zhu, Qibin Fan, Numerical solution of nonlinear fractional order Volterra integro differential equations by SCW, [*Commun. Nonlinear Sci. Numer. Simul.,*]{} [**18**]{} (2013), no. 15, 1203-1213.
\[rvv3\]G. M. Vainikko, Galerkin’s perturbation method and the general theory of approximate methods for non-linear equations,[*USSR Computational Mathematics and Mathematical Physics ,*]{} [**7**]{}(1967), no. 4, 1-41.
| ArXiv |
---
author:
- Wensheng Cheng
- Yan Zhang
- Xu Lei
- Wen Yang
- Guisong Xia
bibliography:
- 'segmentation.bib'
- 'change\_detection.bib'
title: Semantic Change Pattern Analysis
---
| ArXiv |
---
abstract: 'The escape fraction, [$f_{\rm esc}$]{}, of ionizing photons from early galaxies is a crucial parameter for determining whether the observed galaxies at $z \geq 6$ are able to reionize the high-redshift intergalactic medium. Previous attempts to measure [$f_{\rm esc}$]{} have found a wide range of values, varying from less than 0.01 to nearly 1. Rather than finding a single value of $f_{esc}$, we clarify through modeling how internal properties of galaxies affect [$f_{\rm esc}$]{} through the density and distribution of neutral hydrogen within the galaxy, along with the rate of ionizing photons production. We find that the escape fraction depends sensitively on the covering factor of clumps, along with the density of the clumped and interclump medium. One must therefore be cautious when dealing with an inhomogeneous medium. Fewer, high-density clumps lead to a greater escape fraction than more numerous low-density clumps. When more ionizing photons are produced in a starburst, [$f_{\rm esc}$]{} increases, as photons escape more readily from the gas layers. Large variations in the predicted escape fraction, caused by differences in the hydrogen distribution, may explain the large observed differences in [$f_{\rm esc}$]{} among galaxies. Values of [$f_{\rm esc}$]{} must also be consistent with the reionization history. High-mass galaxies alone are unable to reionize the universe, because [$f_{\rm esc}$]{} $> 1$ would be required. Small galaxies are needed to achieve reionization, with greater mean escape fraction in the past.'
author:
- 'Elizabeth R. Fernandez, and J. Michael Shull'
title: The Effect of Galactic Properties on the Escape Fraction of Ionizing Photons
---
INTRODUCTION {#sec:introduction}
============
Observations of the cosmic microwave background optical depth made with the [*[Wilkinson Microwave Anisotropy Probe (WMAP)]{}*]{} [@kogut/etal:2003; @spergel/etal:2003; @page/etal:2007; @spergel/etal:2007; @dunkley/etal:2008; @komatsu/etal:2008; @wmap7] suggest that the universe was reionized sometime between $6<z<12$. Because massive stars are efficient producers of ultraviolet photons, they are the most likely candidates for the majority of reionization. However, in order for early star-forming galaxies to reionize the universe, their ionizing radiation must be able to escape from the halos, in which neutral hydrogen () is the dominant source of Lyman continuum (LyC) opacity. The escape fraction, [$f_{\rm esc}$]{}, of ionizing photons is a key parameter for starburst galaxies at $z > 6$, which are believed to produce the bulk of the photons that reionize the universe [@robertson/etal:2010; @trenti/etal:2010; @Bouwens/etal:2010].
The predicted values of escape fraction span a large range from $0.01 \lesssim f_{\rm esc} < 1$, derived from a variety of theoretical and observational studies of varying complexity. Various properties of the host galaxy, its stars, or its environment are thought to affect the number of ionizing photons that escape into the intergalactic medium (IGM). For example, @ricotti/shull:2000 studied [$f_{\rm esc}$]{} in spherical halos using a Strömgren approach. @wood/loeb:2000 assumed an isothermal, exponential disk galaxy and followed an ionization front through the galaxy using three-dimensional Monte Carlo radiative transfer. Both @wood/loeb:2000 and @ricotti/shull:2000 state that [$f_{\rm esc}$]{} varies greatly, from $<0.01$ to $1$, depending on galaxy mass, with larger galaxies giving smaller values of [$f_{\rm esc}$]{}. A similar dependence with galaxy mass is also seen by the simulations of @yajima/etal:2010, because larger galaxies tend to have star formation buried within dense hydrogen clouds, while smaller galaxies often had clearer paths for escaping ionizing radiation. @gnedin/etal:2008, on the other hand, ran a high resolution N-body simulation with adaptive-mesh refinement in a cosmological context. Contrary to @ricotti/shull:2000, @wood/loeb:2000 and @yajima/etal:2010, they state that lower-mass galaxies have significantly smaller [$f_{\rm esc}$]{}, as the result of a declining star formation rate. In addition, above a critical halo mass, [$f_{\rm esc}$]{} does not change by much. The model of @gnedin/etal:2008 allowed for the star formation rate to increase with the mass of the galaxy at a higher rate than a linear proportionality would allow. The larger galaxies also tended to have star formation occurring in the outskirts of the galaxy, which made it easier for ionizing photons to escape. Their model included a distribution of gas within the galaxy, which created free sight-lines out of the galaxy. @wise/cen:2008 used adaptive mesh hydrodynamical simulations on dwarf galaxies. Even though their simulations covered a different mass range than the larger galaxies studied by @gnedin/etal:2008, they found much higher value of [$f_{\rm esc}$]{} than would be expected from extrapolating results from @gnedin/etal:2008 to lower masses. @wise/cen:2008 attribute this difference to the irregular morphology of their dwarf galaxies with a turbulent and clumpy interstellar medium (ISM), allowing for large values of [$f_{\rm esc}$]{}.
Others have also looked at how the shape and morphology of the galaxy can affect [$f_{\rm esc}$]{}. @dove/shull:1994, using a Strömgren model, studied how [$f_{\rm esc}$]{} varies with various disk density distributions. In addition, many authors have found that superbubbles and shells can trap radiation until blowout, seen in analytical models of @dove/shull/ferrara:2000 as well as in hydrodynamical simulations of @fujita/etal:2003. The analytical model by @clark/oey:2002 showed that high star formation rates can raise the porosity of the ISM and thereby increase [$f_{\rm esc}$]{}. In addition to bubbles and structure caused from supernovae, galaxies can have a clumpy ISM whose inhomogeneities affect [$f_{\rm esc}$]{}. For example, dense clumps could reduce [$f_{\rm esc}$]{} [@dove/shull/ferrara:2000]. On the other hand, @boisse:1990, @hobson/scheuer:1993, @witt/gordon:1996, and @wood/loeb:2000 all found that clumps in a randomly distributed medium cause [$f_{\rm esc}$]{} to rise, while @ciardi/etal:2002 found that the effects of clumps depend on the ionization rate.
A host of other galaxy parameters have been tested analytically and with simulations. Increasing the baryon mass fraction lowers [$f_{\rm esc}$]{} for smaller halos, but increases it at masses greater than $10^8 \: M_{\sun}$ [@wise/cen:2008]. Star formation history changes the amount of ionizing photons and neutral hydrogen, causing [$f_{\rm esc}$]{} to vary from $0.12$ to $0.20$ for coeval star formation and from $0.04$ to $0.10$ for a time-distributed starburst [@dove/shull/ferrara:2000]. Other galactic quantities, such as spin [@wise/cen:2008] or dust content [@gnedin/etal:2008], do not seem to affect the escape fraction.
Observations have also been used to constrain [$f_{\rm esc}$]{}, especially at $z\lesssim3$. Searches for escaping Lyman continuum radiation at redshifts $z \lesssim 1-2$ have found escape fractions of at most a few percent
[@bland/maloney:2002; @bridge:2010; @cowie/etal:2009; @tumlinson/etal:1999; @deharveng:2001; @grimes/etal:2007; @grimes:2009; @heckman/etal:2001; @Leitherer/etal:1995; @malkan:2003; @Siana/etal:2007]. @hurwitz/etal:1997 saw large variations in the escape fraction, and @hoopes:2007 and @bergvall/etal:2006 saw a relatively high escape fraction of $10\%$. @ferguson/etal:2001 observed [$f_{\rm esc}$]{} $\approx 0.2$ at $z\approx1$. @hanish/etal:2010 do not see a difference in [$f_{\rm esc}$]{} between starbursts and normal galaxies. @siana/etal:2010 also found low escape fractions at $z \approx 1.3$ and showed that no more than $8\%$ of galaxies at this redshift can have $f_{\rm esc,rel} > 0.5$. Note that $f_{\rm esc,rel}$, which the authors use to compare their results to other surveys, is defined as the ratio of escaping LyC photons to escaping 1500 $\AA$ photons. In our own Galaxy, @bland/maloney:1999 and @putman/etal:2003 found an escape fraction of only a few percent. Observations using $\gamma$-ray bursts [@chen:2007] show [$f_{\rm esc}$]{} $\approx 0.02$ at $z \approx 2$. At higher redshift ($z \approx 3$), [$f_{\rm esc}$]{} seems to vary drastically from galaxy to galaxy [@shapley/etal:2006; @Iwata/etal:2009; @vanzella:2010], with a few galaxies having very large escape fractions. Some studies have found low values of the escape fraction at $z \approx 3$ [@fernandez-soto/etal:2003; @Giallongo/etal:2002; @heckman/etal:2001; @inoue/etal:2005; @wyithe:2010], while others have found significant LyC leakage [@Steidel:2002; @shapley/etal:2006]. This large variation from galaxy to galaxy suggests a dependence on viewing angle and could indicate the patchiness and structure of neutral hydrogen within the galaxy [@bergvall/etal:2006; @deharveng:2001; @grimes/etal:2007; @heckman/etal:2001]. From these observations, one infers that the fundamental properties of the galaxies change with time or that [$f_{\rm esc}$]{} increases with increasing redshift [@bridge:2010; @cowie/etal:2009; @inoue/etal:2006; @Iwata/etal:2009; @Siana/etal:2007]. @Bouwens/etal:2010 looked at the blue color of high redshift ($z \approx 7$) galaxies and argued that the nebular component must be reduced. This would suggest a much larger escape fraction in the past.
The minimum mass of galaxy formation can also put limitations on [$f_{\rm esc}$]{}. Observations of Ly$\alpha$ absorption toward high-redshift quasars, combined with the UV luminosity function of galaxies, can limit [$f_{\rm esc}$]{} from a redshift of $5.5$ to $6$, with [$f_{\rm esc}$]{} $ \sim$ 0.20–0.45 if the halos producing these photons are larger than $10^{10} \: M_{\sun}$. This can decrease to [$f_{\rm esc}$]{} $\sim$ 0.05–0.1 if halos down to $10^8 \: M_{\sun}$ are included as sources of escaping ionizing photons [@srbinovsky/wyithe:2008].
It is clear that many factors can affect [$f_{\rm esc}$]{}, and the problem is quite complicated. Cosmological simulations that predict the escape fraction provide a more accurate estimate for [$f_{\rm esc}$]{}. However, many parameters of the galaxy change at once, and it becomes difficult to understand how a single parameter can affect the escape fraction. In addition, trends may be difficult to understand because of the manner in which some physical processes are included or neglected. Analytic models can show clearer trends, even though they may be over-simplified and miss important physics. Therefore, rather than predicting a quantitative value for [$f_{\rm esc}$]{}
, we seek to understand how properties of galaxies and their internal structure affect the escape fraction. Because our model is simplified, the values of [$f_{\rm esc}$]{}are not exact, but rather illustrate trends caused by various galactic properties. In section \[sec:Method\], we explain our method of tracing photons that escape the galaxy. In section \[sec:Results\], we explain our results and compare our results to previous literature in section \[sec:Lit\]. In section \[sec:reion\], we consider constraints from reionization and we conclude in section \[sec:Conclusions\]. Throughout, we use the cosmological parameters from WMAP-7 [@wmap7].
METHODOLOGY {#sec:Method}
===========
Properties of the Galaxy
------------------------
We use an exponential hyperbolic secant profile [@spitzer:1942] to describe the density of an isothermal disk in a halo of mass $M_{\rm halo}$: $$n_H(Z) = n_0 \exp[-r/r_{h}]\: \rm{sech}^2\left(\frac{Z}{z_0}\right),
\label{eq:densitysech}$$ [@spitzer:1942] where $n_0$ is the number density of hydrogen at the center of the galaxy, $Z$ is the height above the galaxy mid-plane, and $r_{h}$ is the scale radius: $$r_{h} = \frac{j_d \lambda }{\sqrt{2}m_d} r_{\rm vir}$$ [@mo/etal:1998]. The parameter $j_d$ is the fraction of the halo’s angular momentum in the disk, $\lambda$ is the spin parameter, $m_d$ is the fraction of the halo in the disk ($m_d=\Omega_b/\Omega_m$), and $r_{vir}$ is the virial radius. As in @wood/loeb:2000, we assume $j_d/m_d = 1$ and $\lambda = 0.05$. The virial radius is $$r_{vir}=(0.76 {\rm kpc}) \left(\frac{M_{halo}}{10^8 M_{\sun}
h^{-1}}\right)^{1/3}\left(\frac{\Omega_m}{\Omega(z_f)}\frac{\Delta_c}{200}\right)^{-1/3}
\left(\frac{1+z_f}{10}\right)^{-1} h^{-1} \: ,
\label{eq:rvir}$$ [@navarro/etal:1997] where $\Delta_c = 18 \pi^2 +82d-39d^2$ and $d=\Omega_{zf}-1$. $\Omega_{zf}$ is the local value of $\Omega_m$ at the redshift of galaxy formation, $z_f$. The dependence of the virial radius on $z_f$ will affect the density of the disk, with smaller disks of higher density forming earlier. The disk scale height, $z_0$, is given by $$z_0 = \left(\frac{\langle v^2 \rangle}{2\pi G \rho_0}\right)^{1/2} = \left(\frac{M_{halo}}{2\pi \rho_0 r_{vir}}\right)^{1/2},$$ where $\langle v^2 \rangle$ is the mean square of the velocity and $\rho_0$ is the central density. In a real galaxy, there will be non-thermal motions of the gas. However, to simplify the calculations, we assume that the gas is virialized, feels the gravity of the disk, and therefore follows the relation $\langle v^2 \rangle = G M_{halo} / r_{vir}$. However, radiative cooling can cause the disk to be thinner than this. The central density is solved for in a self-consistent way after the halo mass and the redshift of formation are specified. We use $15 r_{h}$ and $2z_0$ as the limits of the radius and height of the disk, respectively. The structure of real galaxies is more complicated. The addition of stars in the halo will change the gas distribution through feedback, heating, and gravitational effects. For purposes of simplicity, we ignore these effects.
The mass of the disk (stars and gas) is taken as $M_{\rm disk} = m_d \, M_{\rm halo}$, where $m_d$ is the fraction of matter that is incorporated into the disk. The upper limit of $m_d$ is $\Omega_b / \Omega_m$. The mass of the stars within the disk is $M_* = M_{\rm disk} f_* $, where $f_*$ is the star formation efficiency, which describes the fraction of baryons that form into stars.[^1] The remainder of the mass of the disk is in gas, distributed according to equation \[eq:densitysech\] with gas temperature of $10^4$ K.
The number of ionizing photons is related to $f_*$, considering either Population III (metal-free) or Population II (metal-poor, $Z = 0.02 Z_\sun$) stars. The total number of ionizing photons per second from the entire stellar population, $Q_{\rm pop}$ is $$Q_{pop} = \frac{\int_{m_1}^{m_2}\overline{Q}_H(m) f(m) dm}{\int_{m_1}^{m_2} m f(m) dm}
\times M_* \; ,$$ where $m$ is the mass of the star, and $m_1$ and $m_2$ are the upper and lower mass limits of the mass spectrum, given by $f(m)$. For a less massive distribution of stars, we use the Salpeter initial mass spectrum [@salpeter:1955]: $$f(m) \propto m^{-2.35},
\label{eq:salpeter}$$ with $m_1=0.4 \; M_\sun$ and $m_2 = 150 \; M_\sun$. The Larson initial mass spectrum illustrates a case with heavier stars [@larson:1998]: $$f(m)\propto m^{-1}\left(1+\frac{m}{m_c}\right)^{-1.35},$$ with $m_1 = 1 \; M_\sun$, $m_2=500 \; M_\sun$, and $m_c = 250 \; M_\sun$ for Population III stars and $m_1 = 1 \; M_\sun$, $m_2=150 \; M_\sun$, and $m_c = 50 \; M_\sun$ for Population II stars. We define $\overline{Q}_H$ as the number of ionizing photons emitted per second per star, averaged over the star’s lifetime. For Population III stars of mass parameter $x \equiv \log_{10}(m/M_\sun)$, this is $$\begin{aligned}
\log_{10}\left[\overline{Q}_H/{\rm s^{-1}}\right]
&=&\left\{
\begin{array}{ll}
43.61 + 4.90x - 0.83x^2 & 9-500~M_\sun \; ,\\
39.29 + 8.55x & 5-9~M_\sun \; ,\\
0 & \rm {otherwise} \; ,
\end{array}\right.\end{aligned}$$ and for Population II stars, $$\begin{aligned}
\log_{10}\left[\overline{Q}_H/{\rm s^{-1}}\right]
&=&\left\{
\begin{array}{ll}
27.80 + 30.68x - 14.80x^2
+ 2.50x^3 & \geq 5 M_\sun\\
0 & \rm {otherwise} \; ,
\end{array}\right.\end{aligned}$$ as given in Table 6 of @schaerer:2002.
Calculating the Escape Fraction
-------------------------------
We place the stars at the center of the galaxy. An ionized H region develops around the stars, where the number of ionizing photons emitted per second by the stellar population, $Q_{\rm pop}$, is balanced by recombinations, such that $$Q_{pop} = \frac{4}{3} \pi r_s^3 n_H^2 \alpha_B(T).$$ Here, $\alpha_B$ is the case-B recombination rate coefficient of hydrogen and $T$ is the temperature of the gas (we assume $T=10^4K$). The radius of this region, called the Strömgren radius, is $$r_s= \left(\frac{3 Q_{\rm pop}}{4 \pi n_H^2 \alpha_B}\right)^{1/3}.$$ This radius is simple to evaluate in the case of a uniform medium, but if we are concerned with clumps and a disk with a density profile, the density will be changing with location. Although we are calculating [$f_{\rm esc}$]{} at a moment in time, in reality the region is not static, and the ionization front will propagate at a flux-limited speed.
We assume that all stars are placed at the center of the galaxy. In reality, star formation will be distributed throughout the galaxy. If stars are closer to the edge of the galaxy, their photons will have less hydrogen to traverse, and hence will escape more easily. Therefore, the results we present will be lower limits of the escape fraction. We integrate along the path length that a photon takes in order to escape the galaxy, following the formalism in @dove/shull:1994. We can then calculate the escape fraction of ionizing photons along each ray emanating from the center of the galaxy by equating the number of ionizing photons to the number of hydrogen atoms across its path. If there are more photons than hydrogen atoms, the ray can break out of the disk; otherwise, no photons escape and the escape fraction is zero. The escape fraction along a path, $\eta$, thus depends on the amount of hydrogen the ray transverses, which depends on its angle $\theta$, measured from the axis perpendicular to the disk: $$\eta(\theta) = 1 -\frac{4 \pi \alpha_B}{Q_{pop}}\int^{\infty}_0 n_H^2(Z)r^2 dr \; .$$ Photons are more likely to escape out of the top and bottom of the disk, rather than the sides, because there is less path length to traverse. This creates a critical angle, beyond which photons no longer will escape the galaxy. The total escape fraction, [$f_{\rm esc}$]{}, is then found by integrating over all angles $\theta$ and the solid angle $\Omega$: $$\begin{aligned}
f_{\rm esc}(Q_{\rm pop}) &=& \int \int \frac{\eta(\theta) }{4\pi} d\theta d\Omega\\
&=& \int \frac{1}{2}\eta(\theta)\: \sin(\theta)\: d\theta \; .\end{aligned}$$ We take into account the whole disk (top and bottom) so that an [$f_{\rm esc}$]{} of $1$ means that all photons produced are escaping into the IGM.
Adding Clumps
-------------
A medium with clumps can be described with the density contrast $C = n_{c}/n_{ic}$ between the clumps (density $n_{c}$), and the interclump medium (density $n_{ic}$). The percentage of volume taken up by the clumps is described by the volume filling factor $f_{V}$. We randomly distribute clumps throughout the galaxy. We define $n_{\rm mean}$ as the density the medium would have if it was not clumpy, given by equation \[eq:densitysech\]. The density at each point is given by $$n_{c}=\frac{n_{\rm mean}}{f_{V}+(1-f_{V})/C}
\label{eq:clump}$$ if the point is in a clump and $$n_{ic}=\frac{n_{\rm mean}}{f_{V}(C-1)+1}
\label{eq:ic}$$ if the point is not in a clump, similar to [@wood/loeb:2000]. In this way, the galaxy retains the same interstellar gas mass, independent of $f_{V}$ and $C$. As $C$ increases, the density of the clumps increases as the density of the non-clumped medium falls. Similarly, if $f_V$ is larger, more of the medium is contained in less dense clumps. We trace photons on their path through the galaxy and track whether or not they encounter a clump. At each step on the path out of the galaxy, a random number is generated. A clump exists if this random number is less than the volume filling factor. As the filling factor increases, clumps can merge, forming larger, arbitrarily shaped clumps. Counting the number of photons exiting the galaxy then leads to [$f_{\rm esc}$]{}. The covering factor is computed by counting how many clumps intersect a ray as it travels out of the galaxy.
RESULTS {#sec:Results}
=======
Properties of the Clumps
------------------------
In the first calculations, we placed Population III stars with a Larson mass spectrum and a star formation efficiency $f_* = 0.5$ in a halo of $M_{\rm halo} = 10^9 M_\sun$, with a redshift of formation of $z_f = 10$. The clumps have diameter $10^{17}$ cm, unless otherwise stated. The top panel of Figure \[fig:varyff\] shows [$f_{\rm esc}$]{} as a function of $f_V$ for various values of $C$. The case with no clumps is equivalent to $C=1$. As clumps are introduced, [$f_{\rm esc}$]{}quickly falls, but rises again as $f_V$ rises. This is because the clumps become less dense (since more of the medium is in clumps and the mass of the galaxy must be kept constant). In addition, [$f_{\rm esc}$]{} drops as $C$ increases, showing that denser clumps with a less dense interclump medium stops more ionizing radiation than a more evenly distributed medium. The clumps are small enough that essentially every ray traversing the galaxy encounters one of these very dense clumps and is diminished.
In the bottom panel of Figure \[fig:varyff\], the same population of stars is shown for various values of $f_V$ as a function of $\log(C)$. As $C$ increases, [$f_{\rm esc}$]{} becomes low for small values of $f_V$. Again, this is because of a few very dense clumps that stop essentially all radiation. As $f_V$ increases, more of the medium is in clumps, and therefore the density of the clumps decreases. The combined effect is an increase of [$f_{\rm esc}$]{}. The solid black line shows the case with no clumps, or when $f_V=0$. For $f_V=0$, [$f_{\rm esc}$]{} equals the case with no clumps ($C=1$), as it should. Above $C \sim 10-100$, increasing $C$ no longer affects [$f_{\rm esc}$]{}.
![ Escape fraction of ionizing photons out of the disk as a function of the clump volume filling factor $f_V$ for various values of the clumping factor $C$ ([*[top panel]{}*]{}) and $\log(C)$ for various values of $f_V$ ([*[bottom panel]{}*]{}). Shown for a $10^9 M_\sun$ halo at $z_f=10$, with $f_* = 0.5$ and Population III stars with a Larson mass spectrum. The fraction of matter incorporated into the disk, $m_d$, scales with the escape fraction. []{data-label="fig:varyff"}](figure1a.eps "fig:"){width="120mm"} ![ Escape fraction of ionizing photons out of the disk as a function of the clump volume filling factor $f_V$ for various values of the clumping factor $C$ ([*[top panel]{}*]{}) and $\log(C)$ for various values of $f_V$ ([*[bottom panel]{}*]{}). Shown for a $10^9 M_\sun$ halo at $z_f=10$, with $f_* = 0.5$ and Population III stars with a Larson mass spectrum. The fraction of matter incorporated into the disk, $m_d$, scales with the escape fraction. []{data-label="fig:varyff"}](figure1b.eps "fig:"){width="120mm"}
So far, we have only been exploring the results of small clumps ($10^{17}\: \rm{cm}$, or $\sim 0.3\: \rm{pc}$) in diameter. What would happen if we were to increase the size of these clumps? In this case, a ray would traverse fewer clumps as it travels out of the galaxy (the covering factor will fall), but any given clump would be larger. As shown in Figure \[fig:clumpsize\], $f_{esc}$ rises as the clumps increase in size. For very low values of $f_V$, only a few clumps exist and not every ray comes in contact with a clump, increasing the escape fraction above the case with no clumps.
To illustrate this further, Figure \[fig:clumplog\] shows [$f_{\rm esc}$]{} against $f_V$ for a large clump size ($10^{19}$ cm in diameter). In the top panel, $C$ is varied for $m_d = 0.01$ and $m_d = 0.17$. The left-most vertical line represents the value of $f_V$ needed for a photon traversing the longest path length to pass through an average of one clump for both $m_d=0.01$ and $m_d=0.17$. The right-most vertical solid line represents the value of $f_V$ for a photon traversing the shortest path length to pass through an average of one clump for $m_d=0.17$, while the dashed line is the shortest path for $m_d=0.01$. To the right of these lines, all path lengths intersect a clump. In other words, here the covering factor is greater than one. To the left of these lines, there are clump-free path lengths out of the galaxy. For $m_d=0.17$, we see that if $f_V$ is low enough some rays will pass through fewer than one clump, on average, [$f_{\rm esc}$]{} is much greater than the no-clump case. For very low values of $f_V$, there are so few clumps that the interclump medium approaches the case with no clumps. Therefore, the plot of [$f_{\rm esc}$]{} is peaked in the region where there are some paths that do not intersect a clump. These results are averaged over ten runs, while Figure \[fig:clumplog2\] shows the distribution for each run.
The bottom panel of Figure \[fig:clumplog\] shows [$f_{\rm esc}$]{} for various values of $m_d$. As $m_d$ increases, the disk is less massive, and the escape fraction for a non-clumped galaxy rises. For $m_d = 0.01$, the escape fraction of a non-clumped galaxy is $\sim 1$. In this case, once clumps are added, the escape fraction decreases because regions dense enough to stop ionizing radiation are finally introduced. When the covering factor is greater than one, the clumps grow less dense, causing the escape fraction to rise again. For all other cases, [$f_{\rm esc}$]{} peaks when the covering factor is less than one, and falls below the escape fraction for the non-clumped case when the covering factor is greater than one.
![ Effect of large clumps on escape fraction: [$f_{\rm esc}$]{} out of the disk is shown for a galaxy in a $10^9 M_\odot$ halo, with $z_f = 10$, $f_* = 0.1$, and a Pop III Larson initial mass spectrum, with $m_d=0.17$. If the clumps are very large and $f_V$ is very low, there are cases where [$f_{\rm esc}$]{} is larger than the no-clump case. In the top left corner, the results are averaged over ten runs to reduce noise. In the remaining three panels, the distributions are shown for each run. In each case, the black points represent a case with no clumps. Galaxy properties are the same as in Figure \[fig:varyff\]. []{data-label="fig:clumpsize"}](figure2a.eps "fig:"){width="80mm"} ![ Effect of large clumps on escape fraction: [$f_{\rm esc}$]{} out of the disk is shown for a galaxy in a $10^9 M_\odot$ halo, with $z_f = 10$, $f_* = 0.1$, and a Pop III Larson initial mass spectrum, with $m_d=0.17$. If the clumps are very large and $f_V$ is very low, there are cases where [$f_{\rm esc}$]{} is larger than the no-clump case. In the top left corner, the results are averaged over ten runs to reduce noise. In the remaining three panels, the distributions are shown for each run. In each case, the black points represent a case with no clumps. Galaxy properties are the same as in Figure \[fig:varyff\]. []{data-label="fig:clumpsize"}](figure2b.eps "fig:"){width="80mm"} ![ Effect of large clumps on escape fraction: [$f_{\rm esc}$]{} out of the disk is shown for a galaxy in a $10^9 M_\odot$ halo, with $z_f = 10$, $f_* = 0.1$, and a Pop III Larson initial mass spectrum, with $m_d=0.17$. If the clumps are very large and $f_V$ is very low, there are cases where [$f_{\rm esc}$]{} is larger than the no-clump case. In the top left corner, the results are averaged over ten runs to reduce noise. In the remaining three panels, the distributions are shown for each run. In each case, the black points represent a case with no clumps. Galaxy properties are the same as in Figure \[fig:varyff\]. []{data-label="fig:clumpsize"}](figure2c.eps "fig:"){width="80mm"} ![ Effect of large clumps on escape fraction: [$f_{\rm esc}$]{} out of the disk is shown for a galaxy in a $10^9 M_\odot$ halo, with $z_f = 10$, $f_* = 0.1$, and a Pop III Larson initial mass spectrum, with $m_d=0.17$. If the clumps are very large and $f_V$ is very low, there are cases where [$f_{\rm esc}$]{} is larger than the no-clump case. In the top left corner, the results are averaged over ten runs to reduce noise. In the remaining three panels, the distributions are shown for each run. In each case, the black points represent a case with no clumps. Galaxy properties are the same as in Figure \[fig:varyff\]. []{data-label="fig:clumpsize"}](figure2d.eps "fig:"){width="80mm"}
![[*[Top:]{}*]{} The [$f_{\rm esc}$]{} out of the disk is shown for a galaxy in a $10^9 M_\odot$ halo, with $z_f=10$, $f_* = 0.1$, and a Population III Larson initial mass spectrum. The left-most solid vertical line represents the value of the volume filling factor $f_V$ needed for a photon transversing the longest path length to pass through an average of one clump for a galaxy with $m_d = 0.17$, while the right-most solid vertical line represents the value of $f_V$ needed for a photon transversing the shortest path length to pass through an average of one clump for a galaxy with $m_d = 0.17$. Therefore, to the right of this line, the covering factor is greater than one, and all path lengths intersect a clump, while to the left, there are clump-free path lengths out of the galaxy. For a galaxy with $m_d=0.01$ the dashed vertical line represents the value of $f_V$ needed for a photon transversing the shortest path length to pass through an average of one clump for a galaxy. The value of $f_V$ needed for a photon traversing the longest path overlaps with the case where $m_d = 0.17$, because as $m_d$ changes, the scale height of the galaxy changes, but the radius remains the same. [*[Bottom]{}*]{}: The dependence on the escape fraction as $m_d$ is varied. The same population is shown, with $C = 1$ for the solid lines and $C = 1000$ for the dashed lines. The results are averaged over ten runs to reduce noise. []{data-label="fig:clumplog"}](figure3a.eps "fig:"){width="95mm"} ![[*[Top:]{}*]{} The [$f_{\rm esc}$]{} out of the disk is shown for a galaxy in a $10^9 M_\odot$ halo, with $z_f=10$, $f_* = 0.1$, and a Population III Larson initial mass spectrum. The left-most solid vertical line represents the value of the volume filling factor $f_V$ needed for a photon transversing the longest path length to pass through an average of one clump for a galaxy with $m_d = 0.17$, while the right-most solid vertical line represents the value of $f_V$ needed for a photon transversing the shortest path length to pass through an average of one clump for a galaxy with $m_d = 0.17$. Therefore, to the right of this line, the covering factor is greater than one, and all path lengths intersect a clump, while to the left, there are clump-free path lengths out of the galaxy. For a galaxy with $m_d=0.01$ the dashed vertical line represents the value of $f_V$ needed for a photon transversing the shortest path length to pass through an average of one clump for a galaxy. The value of $f_V$ needed for a photon traversing the longest path overlaps with the case where $m_d = 0.17$, because as $m_d$ changes, the scale height of the galaxy changes, but the radius remains the same. [*[Bottom]{}*]{}: The dependence on the escape fraction as $m_d$ is varied. The same population is shown, with $C = 1$ for the solid lines and $C = 1000$ for the dashed lines. The results are averaged over ten runs to reduce noise. []{data-label="fig:clumplog"}](figure3b.eps "fig:"){width="95mm"}
![Distribution of [$f_{\rm esc}$]{} for each run shown in the top panel of Figure \[fig:clumplog\], with $m_d = 0.17$ in the top panel and $m_d = 0.01$ in the bottom panel. []{data-label="fig:clumplog2"}](figure4a.eps "fig:"){width="120mm"} ![Distribution of [$f_{\rm esc}$]{} for each run shown in the top panel of Figure \[fig:clumplog\], with $m_d = 0.17$ in the top panel and $m_d = 0.01$ in the bottom panel. []{data-label="fig:clumplog2"}](figure4b.eps "fig:"){width="120mm"}
Star formation is more likely to take place in clumps. Star formation will also have an effect on gas clumping produced by stellar winds and supernova shells. Because of this, clumps are likely to be distributed around locations of star formation. To analyze this effect, we have defined a region 100 pc from the center of star formation in the galaxy. Inside this region, the volume filling factor is $f_{\rm V,near}$, and outside of this region, the volume filling factor is $f_{\rm V,far}$. For a case where clumps are only located near to the star formation center, $f_{\rm V,far} = 0$. If $f_{\rm V,far} < f_{\rm V,near}$, there are fewer clumps in the outer portions of the galaxy than near the star formation center.
Results are shown in Figure \[fig:clumpdist\]. The black solid line shows the case with no clumps, where $f_{V}=0$ throughout, and the red triple-dot dashed line represents the case where $f_V=0.3$ throughout. For the case where $f_{\rm V,near}=0.3$ and $f_{\rm V,far}=0$, clumps are only near stars. This results in a higher escape fraction than if the entire galaxy had $f_{V}=0.3$. When there are some clumps far from star formation, in the case where $f_{\rm V,near}=0.3$ and $f_{\rm V,far}=0.1$, the escape fraction falls below the case where if the entire galaxy had $f_{V}=0.3$. This is a result of clumps becoming more dense as $f_V$ increases.
We have chosen a 100 pc radius of the region near to the star formation. If this region is much larger, $\sim 1$ kpc, the escape fraction will be equal to the case $f_V = f_{\rm V,near}$, where $f_V$ is the volume filling factor if the galaxy has the same value throughout. If the region is much smaller, $\sim10$ pc, the escape fraction will be equal to the case where $f_V = f_{\rm V,far}$. Overall, the distribution of clumps does not change the escape fraction significantly.
Properties of Stars and the Galaxy
----------------------------------
In Figure \[fig:Q1\], we analyze how the stellar population affects the escape fraction. In the top panel, $f_*$ is held constant as $f_V$ increases. In the bottom panel, $f_V$ is held constant as $f_*$ increases. Both plots show metal-free (Population III) stars and metal-poor (Population II) stars, as well as stars with a heavy Larson initial mass spectrum and a light Salpeter initial mass spectrum. In both cases, [$f_{\rm esc}$]{} is proportional to the number of ionizing photons that are emitted by the stars, with heavier stars or stars with fewer metals more likely to produce photons that can escape the nebula. This is because when more ionizing photons are produced, the critical angle where photons can break free from the halo increases, and hence more photons escape. When the galaxy forms more stars (higher $f_*$), there is the added effect of less hydrogen remaining in the galaxy to absorb ionizing photons. Therefore, [$f_{\rm esc}$]{} increases greatly as $f_*$ increases.
![The $f_{esc}$ for the disk is shown for stars of varied masses and metallicities and various values of $f_V$ with $f_* = 0.5$ ([*[top panel]{}*]{}) and various values of $f_*$ for $f_V=0.1$ ([*[bottom panel]{}*]{}). Very high values of $f_*$ ($0.9$) approach a case in which all ionizing photons are escaping. Both are shown for a $10^9 M_\sun$ halo at $z_f=10$. In each case, the highest line is for Population III Larson, followed by Population II Larson, Population III Salpeter, and Population II Salpeter. For $f_* =0.3$, [$f_{\rm esc}$]{} $\sim 0$ for both Salpeter cases. Please see the electronic edition for a color version of this plot. []{data-label="fig:Q1"}](figure6a.eps "fig:"){width="120mm"} ![The $f_{esc}$ for the disk is shown for stars of varied masses and metallicities and various values of $f_V$ with $f_* = 0.5$ ([*[top panel]{}*]{}) and various values of $f_*$ for $f_V=0.1$ ([*[bottom panel]{}*]{}). Very high values of $f_*$ ($0.9$) approach a case in which all ionizing photons are escaping. Both are shown for a $10^9 M_\sun$ halo at $z_f=10$. In each case, the highest line is for Population III Larson, followed by Population II Larson, Population III Salpeter, and Population II Salpeter. For $f_* =0.3$, [$f_{\rm esc}$]{} $\sim 0$ for both Salpeter cases. Please see the electronic edition for a color version of this plot. []{data-label="fig:Q1"}](figure6b.eps "fig:"){width="120mm"}
The star formation redshift, $z_f$, is varied in Figure \[fig:galaxy1\]. As $z_f$ increases, the galaxy is smaller and more concentrated. Therefore, it is easiest for photons to escape from less dense disks at low redshifts. At redshifts where we expect reionization to take place, it is harder for photons to escape the galaxy. This problem may be remedied by the high-redshift expectation of more massive or metal-free stars with higher values of $f_*$.
COMPARISON TO PREVIOUS LITERATURE {#sec:Lit}
=================================
As noted in the introduction, there have been many previous studies that calculated the number of ionizing photons emitted from high redshift halos, resulting in a wide range of values for the escape fraction. Various factors are proposed that affect the number of ionizing photons that escape into the IGM, in particular the effects of a clumpy ISM. @boisse:1990 and @witt/gordon:1996 found that clumps increase transmission, and @hobson/scheuer:1993 found that a three-phase medium (clumps grouped together, rather than randomly distributed) further increases transmission. Very dense clumps (with $C=10^6$) were studied by @wood/loeb:2000, who found that clumps increase [$f_{\rm esc}$]{} over the case with no clumps. For very small values of $f_V$, their [$f_{\rm esc}$]{} was very high, because most of the density is in a few very dense clumps, and most lines of sight do not encounter a clump. Their clump size is 13.2 pc, which is similar to our largest clump size, $5 \times 10^{19}$ cm. Their results are consistent with our findings for clumps with large radius, low $f_V$, and high $C$, where most rays do not encounter a clump.
@ciardi/etal:2002 included the effect of clumps using a fractal distribution of the ISM with $C = 4-8$. They noted that this distribution of clumps increases [$f_{\rm esc}$]{} in cases with lower ionization rate because there are clearer sight lines. They found that [$f_{\rm esc}$]{} is more sensitive to the gas distribution than to the stellar distribution.
@dove/shull/ferrara:2000 reported that [$f_{\rm esc}$]{} decreases as clumps are added. This results from the fact that adding clumps does not change the density of the interclump medium. In their model, as clumps are added, the mass of hydrogen in the galaxy increases. On the other hand, our current method decreases the density of the interclump medium as clumps are added or become denser to keep the overall mass of the galaxy constant. Photons are more likely to escape along paths with lower density, as in irregular galaxies and along certain lines of sight [@gnedin/etal:2008; @wise/cen:2008]. Shells, such as those created by supernova remnants (SNRs), can trap ionizing photons until the bubble blows out of the disk, allowing photons a clear path to escape and causing [$f_{\rm esc}$]{}to rise [@dove/shull:1994; @dove/shull/ferrara:2000; @fujita/etal:2003]. These SNRs or superbubbles create porosity in the ISM, and above a critical star formation rate, [$f_{\rm esc}$]{} rises [@clark/oey:2002]. This is similar to what is seen in our results. As with a dense clump, a shell will essentially stop all radiation, while a clear path, similar to the case with a low $f_V$, allows many free paths along which radiation can escape. Our model extends the previous work by enlarging the parameter space. We see how low values of $f_V$, changing the clump size, and location of the clumps affect the results. This heavy dependence on how the clumping can affect the escape fraction can explain why such a variation is seen in observations in the escape fraction from galaxy to galaxy.
Previous work differs as to whether or not [$f_{\rm esc}$]{} increases or decreases with redshift. @ricotti/shull:2000 state that [$f_{\rm esc}$]{} decreases with increasing redshift for a fixed halo mass, but is consistent with higher escape fraction from dwarf galaxies. This assumes that the star formation efficiency is proportional to the baryonic content in a galaxy. However, other studies seem to indicate that [$f_{\rm esc}$]{} increases with redshift. High resolution simulations of @razoumov/sl:2006 state that [$f_{\rm esc}$]{}increases with redshift from $z = 2.39$ (where [$f_{\rm esc}$]{} = 0.01–0.02) to $z = 3.8$ (where [$f_{\rm esc}$]{} = 0.06–0.1). This is a result of higher gas clumping and lower star formation rates at lower redshifts, causing the escape fraction to fall. At higher redshifts, the simulations of @razoumov/sl:2009 see this trend continue, with an [$f_{\rm esc}$]{} $\approx 0.8$ at $z \approx 10.4$ that declines with time. This trend has also been seen observationally from $0<z<7$ [@bridge:2010; @cowie/etal:2009; @inoue/etal:2006; @Iwata/etal:2009; @Siana/etal:2007; @Bouwens/etal:2010]. (However, @vanzella/etal:2010 point out that observational measurements of the escape fraction can be contaminated by lower redshift interlopers.) Other simulations have given different results. @gnedin/etal:2008 say that [$f_{\rm esc}$]{} changes little from $3 < z < 9$, always being about 0.01–0.03. This difference could possibly be attributed to how the models deal with star formation efficiencies within galaxies. @wood/loeb:2000 state that since disk density increases with redshift, [$f_{\rm esc}$]{} will fall as the formation redshift increases, ranging from [$f_{\rm esc}$]{} = 0.01–1. We found that [$f_{\rm esc}$]{}decreases with increasing redshift of formation, since disks are more dense. However, this assumes that the types of stars and $f_*$ remain constant. If $f_*$ is larger at higher redshifts and if stars are more massive and have a lower metallicity (likely), the number of ionizing photons should increase, which could cause [$f_{\rm esc}$]{} to increase, despite a denser disk. It is also possible that the disk morphology does not exist at high redshifts. Therefore, in order to understand the evolution of [$f_{\rm esc}$]{} with redshift, one must understand the evolution of other properties, namely, $f_*$, $Q_{\rm pop}$, and the density distribution of gas within a galaxy.
CONSTRAINTS FROM REIONIZATION {#sec:reion}
=============================
If galaxies are responsible for keeping the universe reionized, there must be a minimum number of photons that can escape these galaxies to be consistent with reionization. The star formation rate ($\dot{\rho}$) that corresponds to a star formation efficiency $f_*$ is given by: $$\dot{\rho}(z) =
0.536~M_\sun~{\rm yr^{-1}}~{\rm Mpc}^{-3}
\left(\frac{f_*}{0.1}\frac{\Omega_bh^2}{0.02}\right)
\left(\frac{\Omega_mh^2}{0.14}\right)^{1/2} \left(\frac{1+z}{10}\right)^{3/2}
y_{\rm min}(z) e^{-y_{\rm min}^2(z)/2}
\label{eq:ferKom}$$ [@fernandez/komatsu:2006], assuming a Press-Schechter mass function [@press/schechter:1974], with $$y_{\rm min}(z) \equiv \frac{\delta_c}{\sigma[M_{\rm min}(z)]D(z)} \; .$$ Here, $\delta_c$ is the overdensity, $\sigma(M)$ is the present-day rms amplitude of mass fluctuations, and $M_{\rm min}$ is the minimum mass of halos that create stars.
Similarly, the [$f_{\rm esc}$]{} needed to reionize the universe can be related to the critical star formation rate, $\dot{\rho}_{\rm crit}$, needed to keep the universe ionized: $$\dot{\rho}_{\rm crit}(z) = (0.012 \: M_\sun \: {\rm{yr}}^{-1} \: {\rm{Mpc}}^{-3} \: )
\left[\frac{1+z}{8}\right]^3\left[\frac{C_H/5}{f_{esc}/0.5}\right]\left[\frac{0.004}{Q_{\rm LyC}}\right]T_4^{-0.845}
\label{eq:trenti}$$ [@madau/etal:1999; @trenti/shull:inprep], which results from the number of photoionizations needed to balance recombinations to keep the universe ionized at $z=7$. Here, $C_H$ is the clumping of the IGM (which we scale to a typical value of $C_H=5$), $T_4$ is the temperature of the IGM in units of $10^4$ K, and $Q_{\rm LyC}$ is the conversion factor from $\dot{\rho}(z)$ to the total number of Lyman continuum photons produced per $M_{\odot}$ of star formation, $$Q_{\rm LyC} \equiv \frac{N_{\rm LyC}/10^{63}}{\dot{\rho}_{\rm crit}t_{\rm rec}},$$ where $N_{\rm LyC}$ is the number of Lyman continuum (LyC) photons produced by a star and $t_{\rm rec}$ is the hydrogen recombination timescale. We assume that $Q_{\rm LyC}=0.004$, which is reasonable for a low-metallicity stellar population.
By requiring the star formation rate to be at least as large as the critical value, we can solve for the value of [$f_{\rm esc}$]{} needed to reionize the universe. Results are shown in the top panel of Figure \[fig:reion1\] plotted at two redshifts, $z=7$ and $z=10$. We also consider stars forming in galaxies down to a minimum mass $$M_{min} = (10^8~M_{\odot}) \left[ \frac{(1+z)}{10}\right] ^{-1.5} \; ,$$ [@barkana/loeb], or if smaller halos are suppressed and only those above $10^9M_\sun$ are forming stars. If the required [$f_{\rm esc}$]{} exceeds $1$ (shown by the dashed line), the given population cannot reionize the universe. As redshift decreases, it becomes much easier to keep the universe reionized, and a smaller [$f_{\rm esc}$]{} is needed, as expected.
At higher redshifts, it is harder for stars to keep the IGM ionized because the gas density is higher and there are fewer massive halos forming stars. If small halos are suppressed, the remaining high-mass halos have a much harder time keeping the universe ionized. It is interesting to note that the universe cannot be reionized at $z = 8-10$ if only larger halos ($M_h > 10^9 M_\sun$) are producing ionizing photons that escape into the IGM. At $z = 9$, $f_*$ must always be above $0.1$ for all cases shown in order for the universe to be reionized. At $z=7$, $f_*$ can be very low. Alternately, one can set equations \[eq:ferKom\] and \[eq:trenti\] equal and constrain directly the value of $f_{esc}f_*Q_{LyC}/C_{IGM}$. Any values greater than this would be able to reionize the universe. This is shown in the bottom panel of Figure \[fig:reion1\].
Observations of high redshift galaxies at $z = 7-10$ [@robertson/etal:2010; @Bouwens/etal:2010] show that currently observed galaxies, along with an escape fraction that is compatible with lower redshift observations ([$f_{\rm esc}$]{} $\approx 0.2$), are not able to reionize the universe. Therefore, the faint end slope of the luminosity function must be very steep to create a greater contribution to reionization from high-redshift small galaxies, the escape fraction at high redshifts was probably much higher, the IMF was top heavy and made of low metallicity stars, and/or the clumping factor of the IGM was low [@bunker/etal:2004; @bunker/etal:2010; @gnedin:2008; @gonzalez/etal:2010; @lehnert/bremer:2003; @mclure/etal:2010; @oesch/etal:2010a; @oesch/etal:2010b; @ouchi/etal:2009; @richard/etal:2008; @stiavelli/etal:2004; @yan/windhorst:2004]. In addition, observations of Ly$\alpha$ absorption toward quasars and the UV luminosity function suggest that [$f_{\rm esc}$]{} is allowed to be much lower if low-mass galaxies are allowed to be sources of ionizing photons [@srbinovsky/wyithe:2008]. This is consistent with our findings; small galaxies need to be included and their star formation not suppressed to permit [$f_{\rm esc}$]{} to be sufficiently high to ionize the universe. Low values of [$f_{\rm esc}$]{}, consistent with the low redshift universe, would require high, and in some cases unreasonably high, values of $f_*$. Therefore, the escape fraction was almost certainly higher in the past than it is today.
![[*[Top:]{}*]{} The [$f_{\rm esc}$]{} needed from a population of galaxies with various values of $f_*$ to reionize the universe at $z = 7$ or $10$. If the required [$f_{\rm esc}$]{} lies above $1$ (dashed line), the population cannot reionize the universe. [*Bottom:*]{} The product $f_{\rm esc} f_*Q_{LyC}/C_{IGM}$, which is directly constrained by reionization. The region above each line has a value of $f_{\rm esc}f_*Q_{LyC}/C_{IGM}$ large enough to reionize the universe. Please see the electronic edition for a color version of this plot. []{data-label="fig:reion1"}](figure8a.eps "fig:"){width="120mm"} ![[*[Top:]{}*]{} The [$f_{\rm esc}$]{} needed from a population of galaxies with various values of $f_*$ to reionize the universe at $z = 7$ or $10$. If the required [$f_{\rm esc}$]{} lies above $1$ (dashed line), the population cannot reionize the universe. [*Bottom:*]{} The product $f_{\rm esc} f_*Q_{LyC}/C_{IGM}$, which is directly constrained by reionization. The region above each line has a value of $f_{\rm esc}f_*Q_{LyC}/C_{IGM}$ large enough to reionize the universe. Please see the electronic edition for a color version of this plot. []{data-label="fig:reion1"}](figure8b.eps "fig:"){width="120mm"}
CONCLUSIONS {#sec:Conclusions}
===========
We have explored how the internal properties of galaxies can affect the amount of escaping ionizing radiation. The properties of clumps within the galaxy have the strongest effect on [$f_{\rm esc}$]{}. When the covering factor is much less than one, only a few, dense clumps exist. These clumps only stop a small fraction of the ionizing radiation, and therefore, the escape fraction will be large. As the covering factor increases to one, the escape fraction will fall, eventually to become smaller than the case with a non-clumped galaxy. (The exception to this is if the non-clumped galaxy has an escape fraction of about one. In this case the addition of clumps will only cause the escape fraction to fall around a covering factor of one.) This indicates that the escape fraction is very sensitive to the way the ISM is distributed, causing the escape fraction to vary from 1 to 0 in some cases.
The escape fraction depends on the star formation efficiency and the population of stars formed at the center of the galaxy. As the number of ionizing photons increases, the critical angle at which photons can escape the galaxy will also increase, which will have a direct effect on the escape fraction. Since disks were more likely more dense at higher redshifts, the escape fractions will be lower, unless a change in the star formation efficiency or stellar population would increase the number of ionizing photons. Therefore, the escape fraction depends directly on the number of ionizing photons and the distribution of hydrogen. These variations can be extreme and can help to explain why some observations see large differences in the escape fraction from galaxy to galaxy. Other factors, such as the mass of the galaxy and the redshift, affect the escape fraction indirectly (as the number of ionizing photons and distribution of hydrogen will change with mass or redshift).
At high redshifts, small galaxies are probably needed to help complete reionization, consistent with recent observations. In addition, low values of the escape fraction, $f_{\rm esc} \approx 0.02$, similar to what is seen at low redshifts, are probably insufficient to allow reionization to be completed. Otherwise, they would require very high values of the star formation efficiency.
We acknowledge support from the University of Colorado Astrophysical Theory Program through grants from NASA (NNX07AG77G) and NSF (AST07-07474).
[99]{} Barkana, R., & Loeb, A. 2001, Phys. Rep., 349, 125 Bergvall, N., Zackrisson, E., Andersson, B.-G., Arnberg, D., Masegosa, J., & …stlin, G. 2006, A&A, 448, 513 Bland-Hawthorn, J., & Maloney, P. R. 2002, ASPC, 254, 267 Bland-Hawthorn, J., & Maloney, P. R., 1999, ApJ, 510, L33 Boissé, P. 1990, A&A, 228, 483 Bouwens, R. J., et al. 2010, , 708, L69 Bridge, C. R., et al. 2010, , 720, 465 Bunker, A. J., Stanway, E. R., Ellis, R. S., & McMahon, R. G. 2004 MNRAS, 355, 374 Bunker, A. J., et al. 2010, MNRAS, 409, 855 Chen, H. W., Prochaska, J. X., & Gnedin, N. Y. 2007, ApJ, 667, L125
Ciardi, B., Bianchi, S., & Ferrara, A. 2002, MNRAS, 331, 463
Clarke, C., & Oey, M. S. 2002, MNRAS, 337, 1299 Cowie, L. L., Barger, A. J., & Trouille, L. 2009, ,692,1476 Deharveng, J-M., Buat, V., Le Brun, V., Milliard, B., Kunth, D., Shull, J. M., & Gry, C. 2001, A&A, 375, 805 Dove, J. B., & Shull, J. M. 1994, , 430, 222
Dove, J. B., Shull, J. M., & Ferrara, A. 2000, , 531, 846 Dunkley, J., et al. 2009, ApJS, 180, 306 Ferguson, H. C. 2001, in Deep Fields, Proc. ESO Workshop, ed. S. Cristiani, A. Renzini, R. E. Williams (Garching: Springer-Verlag), 54
Fernandez, E. R., & Komatsu, E. 2006, ApJ, 646, 703 Fernandez-Soto, A., Lanzetta, K. M., & Chen, H.-W. 2003, MNRAS, 342, 1215 Fujita, A., Martin, C. L., Mac Low, M. M., & Abel, T. 2003, , 599, 50 Giallongo, E., Cristiani, S., D’Odorico, S., & Fontana, A. 2002, , 568, L9 Gnedin, N. Y. 2008, , 673, L1
Gnedin, N. Y., Kravtsov, A. V., & Chen, H. -W. 2008, , 672, 765 Gonz‡lez, V., LabbŽ, I., Bouwens, R. J., Illingworth, G., Franx, M., Kriek, M., & Brammer, G. B. 2010, , 713, 115 Grimes, J. P., et al. 2007, , 668, 891 Grimes, J. P., et al. 2009, ,181, 272 Hanish, D. J., Oey, M. S., Rigby, J. R., de Mello, D. F., & Lee, J. C. 2010, , 725, 2029
Heckman, T. M., Sembach, K. R., Meurer, G. R., Leitherer, C., Calzetti, D., & Martin, C. L. 2001, , 558, 56 Hobson, M. P., & Scheuer, P. A. 1993, , 264, 145 Hoopes, C. G., et al. 2007, ApJS, 173, 441 Hurwitz, M., Jelinsky, P., Van Dyke Dixon, W. 1997, , 481, L31 Inoue, A. K., Iwata, I., Deharveng, J.-M., Buat, V., & Burgarella, D. 2005 A&A, 435, 471 Inoue, A. K., Iwata, I., & Deharveng, J.-M. 2006, , 371, L1 Iwata, I., et al. 2009, , 692, 1287 Kogut, A., et al. 2003, , 148, 161 Komatsu, E., et al. 2009, ApJS, 180, 330 Komatsu, E., et al. 2011, ApJS, 192, 18 Larson, R. B. 1998, , 301, 569
Lehnert, M. D., & Bremer, M. 2003, , 593, 630
Leitherer, C., Ferguson, H. C., Heckman, T. M., Lowenthan, J. D. 1995, , 454, L19
Madau, P., Haardt, F., & Rees, M. J. 1999, , 514, 648
Malkan, M., Webb, W., Konopacky, Q. 2002, , 598, 878 McLure, R. J., Dunlop, J. S., Cirasuolo, M., Koekemoer, A. M., Sabbi, E., Stark, D. P., Targett, T. A., & Ellis, R. S.2010, MNRAS, 403, 960 Mo, H. J., Mao, S., & White, S. D. 1998, MNRAS, 295, 319 Navarro, J. F., Frenk, C. S., & White, S. D. M. 1997, , 490, 493 Oesch, P. A., et al. 2010, , 709, L16 Oesch, P. A., et al. 2010, , 690, 1350 Ouchi, M., et al. 2009, , 706, 1136
Page, L., et al. 2007, ApJS, 170, 335 Press, W. H., & Schechter, P. 1974, , 187, 425 Putman, M. E., Bland-Hawthorn, J., Veilleux, S., Gibson, B. K., Freeman, K. C., & Maloney, P. R. 2003 , ApJ, 597, 948
Razoumov, A., & Sommer-Larsen, J. 2006, , 651, L89
Razoumov, A., & Sommer-Larsen, J. 2010, , 710, 1239 Richard, J., Stark, Daniel P., Ellis, Richard S., George, Matthew R., Egami, Eiichi, Kneib, Jean-Paul, Smith, & Graham P. 2008, , 685, 705 Ricotti, M., & Shull, J. M. 2000, , 542, 548
Robertson, B. E., Ellis, R. S., Dunlop, J. S., McLure, R. J., & Stark, D. P. 2010, , 468, 49
Salpeter, E. E. 1955, , 121, 161 Schaerer, D. 2002, , 382, 28
Shapley, A. E., Steidel, C. C., Pettini, M., Adelberger, K. L., & Erb, D. K. 2006, , 651, 688
Siana, B., et al. 2007, , 668, 62 Siana, B., et al. 2010, , 723, 241 Spergel, D. N., et al. 2003, ApJS, 148, 175 Spergel, D. N., et al. 2007, ApJS, 170, 377 Spitzer, L. 1942, , 95, 329 Srbinovsky, J. A., & Wyithe, J. S. B. 2008, PASA, 27, 110 Steidel, C. C., Pettini, M, & Adelberger, K. L. 2001, , 546, 665 Stiavelli, M., Fall, S. M., & Panagia, N. 2004, , 610, L1 Shull, J. M., & Trenti, M. 2010, in prep Trenti, M., Stiavelli, M., Bouwens, R. J., Oesch, P., Shull, J. M., Illingworth, G. D., Bradley, L. D., & Carollo, C. M. 2010, , 714, 202
Tumlinson, J., Giroux, M. L., Shull, J. M., & Stocke, J. T. 1999, , 118, 2148
Vanzella, E., Siana, B., Cristiani, S, & Nonino, M. 2010, MNRAS, 404, 1672 Vanzella, E., et al. 2010, , 725, 1011
Wise, J. H., & Cen, R. 2009, , 693, 984
Witt, A. N., & Gordon, K. D. 1996, , 463, 681
Wood, L., & Loeb, A. 2000, ApJ, 545, 86
Wyithe, J. S. B., Hopkins, A. M., Kistler, M. D., Yuksel, H., & Beacom, J. F. 2010, MNRAS 401, 2561
Yajima, H., Jun-Hwan, C., & Nagamine, K. 2010, MNRAS , in press, arXiv:1002.3346
Yan, H., & Windhorst, R. A. 2004, , 612, L93
[^1]: The star formation efficiency is defined by the fraction of baryons in stars at any given time. Therefore, the escape fraction calculated is the escape fraction at that point in the galaxy’s lifetime. The total escape fraction of the galaxy will depend on the lifetime of the burst and the star formation history, which can be obtained by integrating $f_*$ over the duration of the burst, taking into account the propagation of the ionizing front within the galaxy.
| ArXiv |
---
author:
- 'Michael Beck, Sebastian Henningsen'
bibliography:
- 'references.bib'
date: 'February, 2017'
title: |
Technical Report\
The Stochastic Network Calculator
---
Introduction and Overview
=========================
In this technical report, we provide an in-depth description of the Stochastic Network Calculator tool, henceforth just called “Calculator”. This tool is designed to compute and automatically optimize performance bounds in queueing networks using the methodology of stochastic network calculus (SNC). For a detailed introduction to SNC, see the publicly available thesis of Beck [@Beck:thesis], which shares the notations and definitions used in this document. Other introductory texts are the books [@Chang:book; @Jiang:book] and the surveys [@Fidler:survey; @Fidler:guide].
The Stochastic Network Calculator is an open source project and can be found on github at <https://github.com/scriptkitty/SNC>; it was also presented in [@Beck:SNCalc; @Beck:SNCalc2]. We structure this report as follows:
- We give the essential notations, definitions, and results of SNC in Sections \[sec:Essentials\] and \[sec:Theoretical Results\]. Section \[sec:Modeling with SNC\] focuses on modeling queueing systems, service elements, and arrivals within the language of SNC.
- Section \[sec:Code Structure\] gives an overview on the calculator’s code structure and its workflow.
- In Section \[sec:Code Representation\] we give more detail on how the concepts of SNC are represented in the code.
- How to access and extend the Calculator is topic of Section \[sec:APIs and Extensions\].
- We wrap things up with a full example in Section \[sec:full\_example\]. It describes the modeling steps we have undertaken to produce the results for our presentation at the IEEE Infocom conference 2017 [@Beck:SNCalc2].
We refrain from displaying larger chunks of code in this report for two reasons: (1) The Calculator is under development and the code-base might change at any time. (2) The code is extensively commented; hence, instead of enlargening this report to unbearable lengths, it will be more useful for developers to read about the code’s details in their own context.
Essentials of Stochastic Network Calculus {#sec:Essentials}
=========================================
This Section orients itself on the notations and definitions made in [@Beck:thesis].
We partition time into time-slots of unit length and consider a fluid data model. In this scenario we define a *flow* as a cumulative function of data:
A *flow* is a cumulative function $$\begin{aligned}
A\,:\,\mathbb{N}&\rightarrow \mathbb{R}_0^+ \\
t&\mapsto A(t)\end{aligned}$$
The interpretation of $A(t)$ is the (cumulative) amount of data arriving up to time $t$. Correspondingly the doubly-indexed function $A(s,t)$ describes the amount of data arriving in the interval $(s,t]$.
In SNC a stochastic bound on the amount of arrivals is needed. Without such a bound the total number of arrivals in some interval could be arbitrarily large, thus, making an analysis of the system impossible. The Calculator is based on the MGF approach of SNC. Flows and other stochastic processes are represented by their respective MGF which are upper bounded.
\[def:Arrival-Bound\] The MGF of some quantity $A(t)$ at $\theta$ is defined by $$\phi_{A(t)}(\theta) = \mathbb{E}(e^{\theta A(t)}),$$ where $\mathbb{E}$ denotes the expectation of a random variable.
We have an MGF-bound for a flow $A$ and the value $\theta>0$, if $$\phi_{A(s,t)}(\theta) \leq e^{\theta \rho(\theta)(t.s) + \theta \sigma(\theta)}$$ holds for all time pairs $s\leq t$.
(0,0) node(start) ++(1,0) node\[circle,draw\](U) [$U$]{} ++(1,0) node(end) ; (start) – node\[above\] [$A$]{}(U); (U) – node\[above\] [$B$]{}(end);
The second basic quantity in a queueing system is the amount of service per time-slot. The relationship between these two and the resulting output (seeFigure \[fig:single-queue\]) is given by $$B(t) \geq A\otimes U(0,t) = \min_{0\leq s\leq t}\{A(0,s)+U(s,t)\}.$$ Here $U$ is a bivariate function (or stochastic process) that describes the service process’s behavior. For example a constant rate server takes the form$U(s,t) = r_U(t-s)$, where $r_U$ is the service element’s rate.
\[def:Service-Bound\] A service element is a dynamic $U$-server, if it fulfills for any input-output pair $A$ and $B$: $$B(t)\geq A\otimes U(0,t).$$
Such a server is MGF-bounded for some $\theta>0$, if $U$ fulfills $$\phi_{U(s,t)}(-\theta)\leq e^{\theta\rho(\theta)(t-s) + \theta\sigma(\theta)}$$ for all pairs $s\leq t$.
Note the minus-sign in the above definition to indicate that the service is bounded from below, whereas the arrivals are bounded from above.
We are particularly interested in two performance measures that puts a system’s arrivals and departures into context.
The *backlog* at time $t$ for an input-output pair $A$ and $B$ is defined by $$\mathfrak{b}(t) = A(t) - B(t).$$ The *virtual delay* at time $t$ is defined by $$\mathfrak{d}(t) = \min\{s\geq 0 \mid A(t) \leq B(t+s)\}.$$
Note that in these definitions we make two assumptions about the queueing system: (1) As the backlog is defined by the difference of $A$ and $B$, we assume the system to be loss-free – all data that has not yet departed from the system must still be queued in it. (2) We only consider the *virtual* delay. This is the time until we see an amount of departures from the system, which is *equivalent* to the accumulated arrivals up to a time $t$. For FIFO-systems its virtual delay coincides with its delay; in non-FIFO systems, however, this does not need to be the case.
Modeling with Stochastic Network Calculus {#sec:Modeling with SNC}
=========================================
Modeling the Network
--------------------
(-1,1) node(source\_1)\[align=center\] [Source]{} (0,-1) node(source\_2)\[align=center\] [Source]{};
(source\_1) – (0,1); (source\_2) – (1,-1);
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(1.65,-1.2) – ++(0,0.4); (1.6,-1.2) – ++(0,0.4); (1.55,-1.2) – ++(0,0.4);
(1,1) node\[circle, draw, minimum size = 0.5cm\](Server\_1) (2,-1) node\[circle, draw, minimum size = 0.5cm\](Server\_2);
(Server\_1) – (3.5,0.3); (Server\_2) – (3.5,-0.3);
(3.5,0.05) – ++(0.75,0) – ++(0,0.5) – ++(-0.75,0); (3.5,-0.55) – ++(0.75,0) – ++(0,0.5) – ++(-0.75,0);
(4.15, 0.1) – ++(0,0.4); (4.1, 0.1) – ++(0,0.4); (4.05, 0.1) – ++(0,0.4);
(4.15, -0.5) – ++(0,0.4); (4.1, -0.5) – ++(0,0.4);
(4.85,0) node\[circle, draw, minimum size = 1.2cm\](Server\_3) ++(0,0.3) node\[circle, minimum size = 1.2cm\](dummy\_top) ++(0,-0.6) node\[circle, minimum size = 1.2cm\](dummy\_bottom);
(7,0.3) node\[align = center\](Destination\_1)[Departures]{} (7,-0.3) node\[align = center\](Destination\_2)[Departures]{};
(dummy\_top) – (Destination\_1); (dummy\_bottom) – (Destination\_2);
(a)
(-0.75,0) node(source)\[draw, circle\] [$e$]{} (1,1) node(U)\[draw, circle\] [$U$]{} (1,-1) node(V)\[draw, circle\] [$V$]{} (3,0) node(W)\[draw, circle\] [$W$]{} ++(0.05,0.25) node(top\_dummy)\[circle, text = white\] [$\cdot$]{} ++(0,-0.5) node(bottom\_dummy)\[circle, text = white\] [$\cdot$]{} ++(2,0.25) node(destination)\[draw, circle\] [$e^\prime$]{} ++(-0.2,0.05) node(dest\_top\_dummy)\[circle, text = white\] ++(0,-0.1) node(dest\_bottom\_dummy)\[circle, text = white\] ;
(source) – (U); (U) – (top\_dummy); (top\_dummy) – (dest\_top\_dummy); (source) – (V); (V) – (bottom\_dummy); (bottom\_dummy) – (dest\_bottom\_dummy);
(b)
In SNC we consider a queueing system such as a communication network as a collection of flows and service elements. These can be represented as nodes and edges as shown in Figure \[fig:graph-representation\]. In this transformation we replace each flow’s route by a sequence of directed edges, such that each hop of the flow through the network is mapped to one edge; furthermore, we introduce two extra nodes. They represent the “outside” of the network. Each flow originates from $e$ and leaves the network via $e^\prime$. Note that in such a scenario there does not need to be a one to one correspondence between nodes and physical entities. In the graph representation one node with several inputs just represents one resource that is expended by several flows of arrivals.
For example: A router can have several interfaces each leading to another router. In this scenario data packets do not queue up in the router, but rather in each of its interfaces; hence, the nodes in the corresponding graph represent a single interface only and not the entire router. This leads to different topologies between the physical network (in terms of routers and their connections) and the graph of service elements and flows.
Modeling the Arrival Processes
------------------------------
Now, we give more details on the modeling of data flows and present some of the arrival bounds currently implemented in the tool. As introduced in the previous chapter we use a fluid model with discrete time-slots in the Calculator. This means we are interested in the arrival’s distribution per time-slot; or more precisely: in their moment generating function (MGF). The easiest (and bland) example is a stream of data with constant rate:
Assume a source sends $r$ data per time-slot (for example100 Mb/sec.). It immediately follows: $A(s,t)=r(t-s)$ and as there is no randomness involved it also holds $\phi_{A(s,t)}(\theta)=e^{\theta r(t-s)}$ for the MGF. To achieve a bound conform to Definition \[def:Arrival-Bound\] we define $\rho(\theta)=r$ and $\sigma(\theta)=0$.
We can construct a simple random model by assuming that the arrivals per time-slot are stochastically independent and follow the same distribution (an i.i.d. assumption).
\[ex:Exponential-Increments\] Assume a source sends in time-slot $t$ an amount of data equal to $a_t$. Here $a_t$ are stochastically independent and exponentially distributed with a common parameter $\lambda$. The exponential distribution has density $$f(x) = \lambda e^{-\lambda x} \qquad \text{ for all } x\geq 0$$ and $f(x)=0$ for all x < 0. From this we can derive the MGF for a single increment as $\phi_{a(t)}(\theta)=\frac{\lambda}{\lambda-\theta}$. Due to the stochastic independence of the increments we have: $$\phi_{A(s,t)}(\theta)=\prod_{r=s+1}^t \phi_{a(r)}(\theta)=\left(\frac{\lambda}{\lambda - \theta}\right);$$ hence, an MGF-bound for this type of arrivals is given by $\rho(\theta) = 1/\theta \log(\frac{\lambda}{\lambda - \theta})$ and $\sigma(\theta) = 0 $.
The above example can be easily generalized to increments with other distributions, as long as their MGF can be computed.
As of today we can roughly divide the methodology of SNC by the way of how to bound the involved stochastic processes (for details refer to [@Fidler:survey]). The Calculator uses MGF-bounds as in Definitions \[def:Arrival-Bound\] and \[def:Service-Bound\]. In the next example we show how to convert bounds from the “other” branch of SNC to MGF-bounds.
We say a flow has exponentially bounded burstiness (follows the EBB-model), if $$\mathbb{P}(A(s,t)>\rho(t-s)+\varepsilon)\leq Me^{d \varepsilon}$$ holds for all pairs $s\leq t$ and $\varepsilon>0$. In this model we call $M$ the prefactor and $d$ the bound’s decay; the parameter $\rho$ represents the arrival’s long-term rate. We can convert such a bound to an MGF-bound (see for example [@Li_eff_bandwidth_2007] or Lemma 10.1 in [@Beck:thesis]) via $$\phi_{A(s,t)}(\theta) \leq \int_0^1 e^{\theta (\rho(t-s)+\varepsilon)}\mathrm{d}\varepsilon^\prime.$$ Here $\varepsilon = -1/d \log(\tfrac{\varepsilon^\prime}{M})$. Solving the above integral leads to $$\phi_{A(s,t)}(\theta) \leq e^{\theta \rho(t-s)} \frac{1}{M^{\nicefrac{\theta}{d}}(1 - \nicefrac{\theta}{d})}$$ and we can define an MGF-bound for flows following the EBB-model by $\rho(\theta) = \rho$ and $\sigma(\theta) = -1/\theta \log(M^{\nicefrac{\theta}{d}}(1 - \nicefrac{\theta}{d}))$.
The above example is important as it allows us to use results from the tailbounded branch of SNC inside the Calculator. The EBB-model contains important traffic classes such as Markov-modulated On-Off processes (see [@Li_eff_bandwidth_2007]).
The next example is influenced by classical queueing theory and is a way to handle the underlying flow being defined in a continuous time setting.
Assume a Poisson jump process on $\mathbb{R}$, meaning the interarrival times between any two jumps are independent and exponentially distributed for some intensity parameter $\mu$. At each jump a packet arrives and the sequence of packets forms the increment process $a_i$ with $i \in \mathbb{N}$. The total number of arrivals in a (discrete-timed) interval $(s,t]$ is given by $$A(s,t) = \sum_{i \in N(s,t)}a_i,$$ where $N(s,t)$ is the set of jumps which occur in the interval $(s,t]$; now, assume that the increment process $a_i$ is i.i.d. (for stochastically independent exponential distributions we have the traditional M/M/1-model of queueing theory); then, we can calculate the MGF as $$\begin{aligned}
\phi_{A(s,t)}(\theta) & = \sum_{k=0}^\infty \mathbb{E}\left(e^{\theta A(s,t)} \mid N(s,t) = k\right)\mathbb{P}(N(s,t) = k) \\
& = e^{\mu(t-s)(\phi_{a_i}(\theta) - 1)};\end{aligned}$$ hence, the flow is MGF-bounded with $\rho(\theta) = \mu/\theta (\phi_{a_i}(\theta) - 1)$ and $\sigma(\theta)=0$.
In our last example for modeling arrivals we only assume that their distribution are stationary. Instead of having detailed information on their distribution, we model them as the aggregate of sub-flows that (each by their own) have passed through a token bucket shaper.
Assume a subflow $A_i$. We say that $A_i$ has passed through a token bucket shaper, if for all pairs $s\leq t$ it holds $A(s,t)\leq \rho_i(t-s) + \sigma_i$. The rate $\rho_i$ is the shaper’s token refreshing rate and $\sigma_i$ is its bucket size; now, assume that the stochastic processes $A_i$ are stationary, meaning $A_i(s,t)$ is equal in distribution to any shift performed to the interval $(s,t)$. For the aggregate $\sum_i A_i =: A$ the following bound ([@Massoulie:tokenbuckets]) holds: $$\phi_{A(s,t)}(\theta) \leq e^{\theta \sum_i \rho_i(t-s)}\left(1/2 e^{\theta \sum_i \sigma_i} + 1/2 e^{-\theta \sum_i \sigma_i}\right).$$ By defining $\rho(\theta) = \sum_i \rho_i$ and $\sigma(\theta) = 1/\theta \log(1/2 e^{\theta \sum_i \sigma_i} + 1/2 e^{-\theta \sum_i \sigma_i})$ we have an MGF-bound in the sense of Definition \[def:Arrival-Bound\].
With the above examples we see how to derive MGF-bounds from several models: We covered stochastically independent increments, a conversion from tailbounds, the traditional M/M/1-model, and the result of aggregated shaped traffics. All these bounds are implemented and available in the Calculator.
Modeling the Service Process
----------------------------
The modeling of service elements is usually much easier, as the randomness of service times usually comes from flows interfering with the service element; in fact, the Calculator has currently only one kind of service element implemented, which is the constant rate service.
We can model a service process $U$ by a constant rate server, i.e., $U(s,t) = r(t-s)$ for some rate $r$. Similarly to the constant rate arrivals the MGF simply is $$\phi_{U(s,t)}(-\theta) = e^{-r(t-s)}$$ and we achieve an MGF as in Definition \[def:Service-Bound\] by defining $\rho(\theta)=-r$ and $\sigma(\theta)=0$.
We want to briefly point out that, when we deal with a wired system, service elements can usually be modeled as constant rate servers. One should bear in mind, however, that the situation becomes fundamentally different in wireless scenarios. In wireless scenarios, the channel characteristics and properties of wireless nodes have to be taken into account. More details on the latter can be found in [@jiang:servermodel] and [@Jiang:IWQoS2010], where a router’s service is parametrized via statistical methods and measurements. This method can likely be applied as a general approach to get a more detailed service description of real-world systems. The modeling of fading channels is addressed in [@Fidler:fading-channels].
In the next section we reason why this simple service model still allows to analyze a wide variety of networks.
Theoretical Results {#sec:Theoretical Results}
===================
Performance Bounds
------------------
For a system with a single node and a single arrival as in Figure \[fig:single-queue\], we have the following performance bounds:
\[thm:Fundamental-Theorem\] Consider the system in Figure \[fig:single-queue\] and assume that the MGF-bounds $$\begin{aligned}
\phi_{A(s,t)}(\theta) &\leq e^{\theta\rho_A(\theta)(t-s) + \theta\sigma_A(\theta)}\\
\phi_{U(s,t)}(-\theta) &\leq e^{\theta\rho_U(\theta)(t-s) + \theta\sigma_U(\theta)}\end{aligned}$$ hold for $A$ and $U$ and some $\theta > 0$. If $A$ and $U$ are stochastically independent, then for all $t>0$ the following bounds hold: $$\begin{aligned}
\mathbb{P}(\mathfrak{b}(t)>N) &\leq e^{\theta N}e^{\theta \sigma_A(\theta) + \theta \sigma_U(\theta)} \cdot \frac{1}{1 - e^{\theta (\rho_A(\theta)+\rho_U(\theta))}} \\
\mathbb{P}(\mathfrak{d}(t)>T) &\leq e^{\theta \rho_U(\theta)T}e^{\theta \sigma_A(\theta) + \theta \sigma_U(\theta)} \cdot \frac{1}{1 - e^{\theta (\rho_A(\theta)+\rho_U(\theta))}},\end{aligned}$$ if $\rho_A(\theta)+\rho_U(\theta) < 0$.
Proofs for the above theorem can for example be found in [@Beck:thesis; @fidler-iwqos06].
This theorem gives us a method to achieve stochastic bounds on the virtual delay and backlog of a single server with a single input. This raises the question on how to achieve performance bounds on more complex networks. The idea here is to reduce a network to the single-flow-single-node case. To illustrate this we give an example of a slightly more complex network.
Assume the same network as above, but instead of a single flow entering the service element we have two flows $A_1$ and $A_2$ as input, each with their own bounding functions $\rho_i(\theta)$ and $\sigma_i(\theta)$ ($i\in\{1,2\}$). In this scenario we might be interested in the total backlog which can accumulate at the service element. For using the above result, we make an important observation: If $A_1$ and $A_2$ are stochastically independent, we can derive from their MGF-bounds a new bound for the aggregated arrivals: $$\begin{aligned}
\phi_{A_1(s,t) + A_2(s,t)}(\theta) & = \mathbb{E}(e^{\theta(A_1(s,t) + A_2(s,t))})=\phi_{A_1(s,t)}(\theta)\phi_{A_2(s,t)}(\theta) \\
& \leq e^{\theta \rho_1(\theta)(t-s) + \theta\sigma_1(\theta)}e^{\theta \rho_2(\theta)(t-s) + \theta\sigma_2(\theta)} \\
& = e^{\theta (\rho_1(\theta) + \rho_2(\theta))(t-s) + \theta(\sigma_1(\theta) + \sigma_2(\theta))}. \\\end{aligned}$$ By defining $\rho_A(\theta) = \rho_1(\theta) + \rho_2(\theta)$ and $\sigma_A(\theta) = \sigma_1(\theta) + \sigma_2(\theta)$ we can use Theorem \[thm:Fundamental-Theorem\] again and calculate the aggregated flow’s backlog.
It is important to describe exactly what happened in the above example: We have reduced a network consisting of two flows and a service element to a network with only a single flow and a service element. For this we calculated a new MGF-bound consisting of MGF-bounds we have known before. This idea of reducing networks makes SNC a powerful theory.
Reduction of Networks
---------------------
In this subsection we generalize the above result and show four methods for reducing a network’s complexity. The first of these network operations is a repetition of the above example. Proofs and many more details for the following results can for example be found in [@Beck:thesis].
\[lem:Multiplexing\] Assume a service element has two stochastically independent input flows $A_1$ and $A_2$ with MGF-bounds $\rho_i(\theta)$ and $\sigma_i(\theta)$; then, the aggregate has an MGF-bound with bounding functions $\rho(\theta) = \rho_1(\theta)+\rho_2(\theta)$ and $\sigma(\theta) = \sigma_1(\theta) + \sigma_2(\theta)$.
The next lemma simplifies a tandem of two service elements into a single service element which describes the end-to-end service.
(0,0) node(start) ++(1,0) node\[circle,draw\](U\_1) [$U_1$]{} ++(1.5,0) node\[circle,draw\](U\_2) [$U_2$]{} ++(1,0) node(end) ; (start) – node\[above\] [$A$]{}(U\_1); (U\_1) – (U\_2); (U\_2) – (end);
\[lem:Convolution\] Assume a tandem network as in Figure \[fig:Tandem-Network\], where the processes $U_1$ and $U_2$ are stochastically independent and MGF-bounded by the functions $\rho_i(\theta)$ and $\sigma_i(\theta)$; then, we can merge the two service elements into a single service element with input $A$ and output $C$, representing the end-to-end behavior. It has an MGF-bound with bounding functions $\rho(\theta) = \max(\rho_1(\theta),\rho_2(\theta))$ and $$\sigma(\theta) = \sigma_1(\theta) + \sigma_2(\theta) - 1/\theta \log(1 - e^{-\theta |\rho_1(\theta) - \rho_2(\theta)|}).$$
The next lemma ties in with a service element’s scheduling discipline. Consider again the situation in Figure \[fig:single-queue\], but with two input flows $A_1$ and $A_2$; now, instead of the system’s performance with respect to the aggregated flow we might be interested in the performance for a particular flow. Considering a subflow raises the question of how the flows’ arrivals are scheduled inside the service element. From the perspective of SNC the easiest scheduling policy is the strict priority policy (or arbitrary multiplexing): In this policy the flow with lower priority only receives service, if there are no arrivals of the higher priority flow enqueued.
\[lem:Demultiplexing\] Assume the above described scenario and that $A_1$ and $U$ are stochastically independent with bounding functions $\rho_A(\theta),\sigma_A(\theta)$ and $\rho_U(\theta),\sigma_U(\theta)$, respectively. This system can be reduced to a single-flow-single-node system for flow $A_2$ and a service element $U_l$ with MGF-bound $$\phi_{U_l(s,t)}(-\theta) \leq e^{\theta(\rho_A(\theta)+\rho_U(\theta))(t-s) + \theta(\sigma_A(\theta)+\sigma_U(\theta))}.$$
More elaborate scheduling policies have been analyzed in SNC. At this stage, however, the Calculator has only implemented the above method for calculating leftover service. Note that this is a worst case view with respect to the scheduling algorithm. By this we mean that any other scheduling, like FIFO or WFQ, gives more service to $A_2$ than arbitrary multiplexing does; therefore, the result of the above lemma can always be used as a lower bound for the service $A_2$ receives.
The next result is needed to produce results for intermediate nodes or flows. It gives an MGF-bound for a service element’s output.
\[lem:Deconvolution\] Assume the scenario as in Figure \[fig:single-queue\] again. If $A$ and $U$ are stochastically independent and MGF-bounded by bounding functions $\rho_A(\theta),\sigma_A(\theta)$ and $\rho_U(\theta),\sigma_U(\theta)$, respectively, we have for the output flow $B$: $$\phi_{B(s,t)}(\theta) \leq e^{\theta\rho_A(\theta)(t-s) + \theta(\sigma_A(\theta)+\sigma_U(\theta))}\cdot \frac{1}{1 - e^{\theta(\rho_A(\theta)+\rho_U(\theta))}},$$ if $\rho_A(\theta)+\rho_U(\theta) < 0$. By this $B$ is MGF-bounded with $\rho_B(\theta) = \rho_A(\theta)$ and $$\sigma_B(\theta) =\sigma_A(\theta) + \sigma_U(\theta) - 1/\theta \log(1 - e^{\theta(\rho_A(\theta) + \rho_U(\theta))}).$$
All of the above results required some independence assumption between the analyzed objects. For the analysis of stochastically dependent objects, we use Hölder’s inequality:
Let $X$ and $Y$ be two stochastic processes. It holds $$\mathbb{E}(XY) \leq (\mathbb{E}(X^p))^{\nicefrac{1}{p}}(\mathbb{E}(Y^q))^{\nicefrac{1}{q}}$$ for all pairs $p,q$ such that $\tfrac{1}{p} + \tfrac{1}{q} = 1$. In particular we have $$\phi_{XY}(\theta) \leq (\phi_{X}(p\theta))^{\tfrac{1}{p}}(\phi_{Y}(q\theta))^{\tfrac{1}{q}}.$$
When we apply this inequality to the above results we get a modified set of network operations. For more details, we refer again to [@Beck:thesis].
In the case of stochastic dependence the bounding functions in Lemma \[lem:Multiplexing\] change to $\rho(\theta) = \rho_1(p\theta) + \rho_2(q\theta)$ and $\sigma(\theta) = \sigma_1(p\theta) + \sigma_2(q\theta)$.
\[lem:Dependent-Convolution\] In the case of stochastic dependence the bounding functions in Lemma \[lem:Convolution\] change to $\rho(\theta) = \max(\rho_1(p\theta) + \rho_2(q\theta))$ and $$\sigma(\theta) = \sigma_1(p\theta) + \sigma_2(q\theta) - 1/\theta \log(1 - e^{-\theta |\rho_1(p\theta) - \rho_2(q\theta)|}).$$
In the case of stochastic dependence the bounding functions in Lemma \[lem:Demultiplexing\] change to $\rho(\theta) = \rho_A(p\theta) + \rho_U(q\theta)$ and $\sigma(\theta) = \sigma_A(p\theta) + \sigma_U(q\theta)$.
In the case of stochastic dependence the bounding functions in Lemma \[lem:Deconvolution\] change to $\rho_B(\theta) = \rho_A(p\theta) $ and $$\sigma_B(\theta) =\sigma_A(p\theta) + \sigma_U(q\theta) - 1/\theta \log(1 - e^{\theta(\rho_A(p\theta) + \rho_U(q\theta))}).$$
Now, we show how these network operations work together to reduce a complex network to the single-node-single-flow case. These examples are taken directly from Chapter 1 of [@Beck:thesis] lifted to MGF-bounded calculus.
(-1,0) node(label) [$\mathcal{G}:$]{} (0,0.25) node(Origin\_1) ++(1,0) node(dummy\_U\_top)\[text = white\][$\cdots$]{} ++(0,-0.25) node(U)\[circle, draw\][$U$]{} ++(1,0.25) node(dummy\_V\_top)\[text = white\][$\cdots$]{} ++(0,-0.25) node(V)\[circle, draw\][$V$]{} ++(1,0.25) node(Destination\_1) (0,-0.25) node(Origin\_2) ++(1,0) node(dummy\_U\_bottom)\[text = white\][$\cdots$]{} ++(1,0) node(dummy\_V\_bottom)\[text = white\][$\cdots$]{} ++(1,0) node(Destination\_2) ;
(Origin\_1) – node\[above\] [${A}_1$]{}(dummy\_U\_top); (dummy\_U\_top) – (dummy\_V\_top); (dummy\_V\_top) – (Destination\_1);
(Origin\_2) – node\[below\] [${A}_2$]{}(dummy\_U\_bottom); (dummy\_U\_bottom) – (dummy\_V\_bottom); (dummy\_V\_bottom) – (Destination\_2);
We consider the network of Figure \[fig:2-Nodes-2-Flows\] and assume that the following MGF-bounds on the involved elements hold: $$\begin{aligned}
\phi_{A_1(s,t)}(\theta) & \leq e^{\theta\rho_{A_1}(\theta)(t-s) + \theta\sigma_{A_1}(\theta)} \\
\phi_{A_2(s,t)}(\theta) & \leq e^{\theta\rho_{A_2}(\theta)(t-s) + \theta\sigma_{A_2}(\theta)} \\
\phi_{U(s,t)}(\theta) & \leq e^{\theta\rho_{U}(\theta)(t-s) + \theta\sigma_{U}(\theta)} \\
\phi_{V(s,t)}(\theta) & \leq e^{\theta\rho_{V}(\theta)(t-s) + \theta\sigma_{V}(\theta)}.\end{aligned}$$
We present three examples for reducing the network using the operations defined in the lemmata above.
Consider the graph $\mathcal{G}$ given in Figure \[fig:2-Nodes-2-Flows\]. After merging both arrivals the graph can be simplified in two ways: Either apply Lemma \[lem:Convolution\] to the two service elements (resulting in graph $\mathcal{G}_{1}$ in Figure \[fig:Reduction-Example-Aggregate-First\]) or calculate an output bound for the first node’s departures (resulting in graph $\mathcal{G}_{1}^{\prime}$ in Figure \[fig:Reduction-Example-Aggregate-First\]). The graphs $\mathcal{G}_{1}$ and $\mathcal{G}_{1}^{\prime}$ describe the system for both arrivals aggregated and as such, can also be used to calculate performance bounds for only one of the flows. The difference between these two methods is that the graph $\mathcal{G}_{1}$ describes the system’s end-to-end behavior, whereas $\mathcal{G}_{1}^{\prime}$ describes the behavior at the service element $V$. The MGF-bounds of the quantities appearing in Figure \[fig:Reduction-Example-Aggregate-First\] can be calculated using Lemmas \[lem:Multiplexing\]-\[lem:Deconvolution\]. To show how these work together we derive here the bounding functions for the MGF-bound on $A^\prime := (A_1\oplus A_2)\oslash U$: First we combine the MGF-bounds of $A_1$ and $A_2$ into the MGF-bound $$\phi_{A_1(s,t)+A_2(s,t)}(\theta) \leq e^{\theta(\rho_{A_1}(\theta)+\rho_{A_2}(\theta))(t-s) + \theta(\sigma_{A_1}(\theta)+\sigma_{A_2}(\theta))}.$$ Next, we apply Lemma \[lem:Deconvolution\] to the aggregate and the service process $U$, resulting in $$\phi_{A^\prime(s,t)}(\theta) \leq e^{\theta(\rho_{A_1}(\theta)+\rho_{A_2}(\theta))(t-s) + \theta(\sigma_{A_1}(\theta)+\sigma_{A_2}(\theta))} \frac{1}{1 - e^{\theta(\rho_{A_1}(\theta)+\rho_{A_2}(\theta)+\rho_U(\theta))}},$$ if $\rho_{A_1}(\theta)+\rho_{A_2}(\theta)+\rho_U(\theta)<0$.
(-1,0) node(label) [$\mathcal{G}_1:$]{} ++(1.5,0) node(Origin) [${A}_1 \oplus {A}_2$]{} ++(2.5,0) node(Service)\[ellipse, draw\][$U\otimes V$]{};
(Origin) – (Service);
\(a) Convolution after multiplexing
(-1,0) node(label) [$\mathcal{G}_1^\prime:$]{} ++(2,0) node(Origin) [$({A}_1 \oplus {A}_2)\oslash U$]{} ++(2,0) node(Service)\[circle, draw\][$V$]{};
(Origin) – (Service);
\(b) Deconvolution after multiplexing.
\[ex:Subtract-First\] Another method to reduce $\mathcal{G}$ is to subtract one of the flows – say $A_{2}$ – first. Afterwards either Lemma \[lem:Convolution\] or Lemma \[lem:Deconvolution\] can be applied, leading to the graphs $\mathcal{G}_{2}$ and $\mathcal{G}_{2}^{\prime}$ in Figure \[fig:Reduction-Example-Subtract-First\]. The graph $\mathcal{G}_{2}$ describes an end-to-end behavior, whereas $\mathcal{G}_{2}^{\prime}$ is the local analysis at the second node. In contrast to the previous example, the flows are considered separately throughout the whole analysis. This approach proves to be better in general topologies in which flows interfere only locally. Note that by following this approach there occurs a stochastic dependency in graph $\mathcal{G}_2$ when we use \[lem:Convolution\]: The process $A_2$ appears in both service descriptions. As a consequence we need to use its variation formulated in Lemma \[lem:Dependent-Convolution\], which introduces a set of Hölder parameters; similarly, in graph $\mathcal{G}^\prime_2$ we have to employ a variation of Theorem \[thm:Fundamental-Theorem\] when we want to calculate performance bounds (the process $A_2$ appears in the arrivals and in the service description).
(-1,0) node(label) [$\mathcal{G}_2:$]{} ++(1,0) node(Origin) [${A}_1$]{} ++(4.5,0) node(Service)\[ellipse, draw\][$[U\ominus{A}_2]^+\otimes[V\ominus({A}_2\oslash U)]^+$]{};
(Origin) – (Service);
\(a) Convolution after subtraction.
(-1,0) node(label) [$\mathcal{G}_2^\prime:$]{} ++(2,0) node(Origin) [${A}_1 \oslash [U\ominus {A}_2]^+$]{} ++(4,0) node(Service)\[ellipse, draw\][$[V\ominus({A}_2\oslash U)]^+$]{};
(Origin) – (Service);
Instead of merging one of the edges first, one can also use Lemma \[lem:Convolution\] to merge the two service elements first. The resulting node is labeled by $U\otimes V$. The graph $\mathcal{G}_{3}$ in Figure \[fig:Reduction-Example-Convolve-First\](a) equals $\mathcal{G}_{1}$; indeed, just the order of aggregation and convolution was switched. Subtracting a crossflow from the convoluted service element, instead, would lead to Figure \[fig:Reduction-Example-Convolve-First\](b). This last graph $\mathcal{G}^\prime_3$ is generally assumed to yield the best end-to-end bounds for flow the flow $A_1$; however, this strategy of convoluting before calculating leftover services cannot be applied in general feedforward networks.
(-1,0) node(label) [$\mathcal{G}_3:$]{} ++(1.5,0) node(Origin) [${A}_1 \oplus {A}_2$]{} ++(2.5,0) node(Service)\[ellipse, draw\][$U\otimes V$]{};
(Origin) – (Service);
\(a) Multiplexing after convolution.
(-1,0) node(label) [$\mathcal{G}_3^\prime:$]{} ++(1,0) node(Origin) [${A}_1$]{} ++(3,0) node(Service)\[ellipse, draw\][$[(U\otimes V)\ominus {A}_2]^+$]{};
(Origin) – (Service);
\(b) Subtraction afer convolution.
We see there are several ways of reducing even this simple example of a network. The results differ in quality and also in what exactly we want to analyze (the performance with respect to a single flow vs. the aggregate and the end-to-end performance vs. the performance at the network’s second node). Note also that the choice of network operations applied may or may not result in Hölder parameters appearing in the resulting performance bounds; therefore any automatic process that performs these actions must keep track whether stochastic dependencies occur and where exactly Hölder parameters must be introduced.
End-to-end Results
------------------
Now, we discuss SNC’s capabilities to perform an end-to-end analysis of a queueing system. Often one is interested in the end-to-end delay of a tandem of servers as in Figure \[fig:Tandem-Network\], but with $n$ service elements instead of two. A typical scenario would be the end-to-end delay between a client and a server with several routers or switches in between.
Given such a network we could theoretically calculate an end-to-end delay bound in two ways: (1) We could start by reducing the network to the first service element and calculate a delay bound for this element in isolation; next, we reduce the original network to the second service element and calculate another local delay bound and so on. All these single-node delay bounds can be combined into an end-to-end delay bound by “adding them up”. While this approach works in theory, we know the resulting bounds to be very loose in general. (2) The other approach is to use Lemma \[lem:Convolution\] to get an end-to-end description for the system and use it to derive delay bound directly. From the theory of network calculus we know that this approach is beneficial.
Still, in this second course of action there exists a problem: Inspecting Lemma \[lem:Convolution\] reveals that with each application of it a term of the form $\frac{1}{1 - e^{\theta |\rho_i(\theta) - \rho_{i+1}(\theta)|}}$ enters the equations. These terms worsen the delay bounds, especially when the quantitites $\rho_i(\theta)$ and $\rho_{i+1}(\theta)$ are similarly sized (in fact, if they should be equal the lemma cannot deliver this result at all). The next theorem shows a method for avoiding these terms completely (see Theorem 3.1 in [@Beck:thesis] and also [@fidler-iwqos06]). This can be seen as an end-to-end convolution, whereas the successively applying Lemma \[lem:Convolution\] would compare to a node-by-node convolution.
\[thm:End-to-End\] Fix some $\theta>0$ and consider a sequence of two service elements as in Lemma \[lem:Convolution\]; further, let $A$ be MGF-bounded with functions $\rho_A$ and $\sigma_A$. Let $A$, $U$, and $V$ be stochastically independent. Under the stability condition $\rho_{A}(\theta)<-\rho_{U}(\theta)\wedge-\rho_{V}(\theta)$, the end-to-end performance bounds $$\begin{aligned}
\mathbb{P}(\mathfrak{d}(t)>T)\leq e^{-\theta \rho_A(\theta)T} \frac{e^{\theta(\sigma_{A}(\theta)+\sigma_{U}(\theta)+\sigma_{V}(\theta))}} {(1-e^{\theta(\rho_{U}(\theta)+\rho_{A}(\theta))})(1-e^{\theta(\rho_{V}(\theta)+\rho_{A}(\theta))})}\end{aligned}$$ holds for all $t$ and $T$.
Above theorem easily generalizes to $N$ hops. Denoting the bounding functions of the $i$-th server by $\rho_{i}$ and $\sigma_i$ we have $$\mathbb{P}(\mathfrak{d}(t)>T)\leq e^{-\theta \rho_A(\theta)T} \frac{e^{\theta \sigma_A(\theta) + \sum_i \theta \sigma_i(\theta)}} {\prod_i 1 - e^{\theta(\rho_i(\theta) +\rho_A(\theta))}}$$ under the stability condition $\rho_A(\theta) < \bigwedge_i - \rho_i(\theta)$.
For stochastically dependent services or arrivals, the introduction of Hölder parameters is needed similarly to the previous subsection.
Note that by using lemmata \[lem:Multiplexing\] - \[lem:Deconvolution\] (or their respective variants for stochastically dependent cases) we can reduce any feedforward network to a tandem of $N$ service elements for any flow of interest with $N$ hops. In doing so, however, the exact sequence of performed network operations will determine the number of Hölder parameters. The optimal way of reducing the network to the tandem is not known and subject to current research.
Code Structure: An Overview on the Calculator {#sec:Code Structure}
=============================================
Now, we give an overview on the Calculator and its code structure. The work-flow with the program is the following.
1. The network must be modeled and given to the Calculator by the user. This requires deriving MGF-bounds for all input-flows and service-elements. If pre-existing stochastic dependencies are known, they must be given to the program. Otheriwse the program will assume stochastic independence. This does only include the initial stochastic processes given to the program, for any intermediate results the tool will keep track of stochastic dependencies by itself. For example the stochastic dependencies occurring in Example \[ex:Subtract-First\] will be recognized by the tool. Networks can either be entered through the GUI or by loading a file holding the description.
2. After giving the network to the tool it can perform an *analysis* of it for any given flow and node of interest (or flow and path of interest). This translates into using Lemmata \[lem:Multiplexing\] - \[lem:Deconvolution\] (and their variants) until the network has been reduced to one on which Theorem \[thm:Fundamental-Theorem\] can be applied. This step is performed entirely on a symbolic level, meaning: The Calculator works on the level of functions and composes these as defined by Lemmata \[lem:Multiplexing\] - \[lem:Deconvolution\]. As a last action of this analysis step the tool applies Theorem \[thm:Fundamental-Theorem\] (again on the level of functions) to compute the function that describes the delay- or backlog bound.
3. In the *optimization*-step the tool takes the function from the analysis-step and optimizes it by all the parameters it includes. The parameters to be optimized include at least $\theta$ and might include any number of additional Hölder parameters; consequently, it is important that any optimization-method implemented to the tool is flexible to the actual number of optimization parameters occurring. This step is usually the one that takes the most computational time.
The tool reflects the above roadmap by consisting of different exchangeable parts. Figure \[fig:Calculator-Modules\] presents these modules.
- The GUI is the interface between the program’s core and the user. We have implemented a simple GUI for the tool, which allows to construct and manipulate the network by hand. It also gives access to the program’s analysis-part and optimization-part. Note that the GUI is not necessary to use the calculator, instead, the provided packages can be mostly used like a library.
- The Calculator is the core of the program. It is the interface between the other models and relays commands and information as needed.
- The Network stores all the needed topology. This includes the flows and nodes with their parameters, but also MGF-bounds on service processes and flows and Hölder parameters that are created during the analysis-step.
- The Analysis is responsible to performing the algebraic part. It is coded entirely on a symbolic level.
- The Optimizer has the task to “fill” the functions given by the Analysis with numerical values. Following an optimization strategy (or heuristic) it will find a near optimal set of parameters and calculate the corresponding performance bound.
(0,0) node\[ellipse, draw\](User) [User]{} ++(0,-1) node\[rectangle, draw\](GUI) [GUI]{} ++(0,-1) node\[rectangle, draw\](Calculator) [Calculator]{} ++(-4,-1) node(Network) [Network]{} ++(0,-1) node\[rectangle, draw\](Flow) [Flows]{} ++(0,-1) node\[rectangle, draw\](Hoelder) [Hoelder]{} ++(0,-1) node\[rectangle, draw\](Vertex) [Vertices]{}
++(4,3) node\[rectangle, draw\](Analysis) [Analysis]{} ++(4,0) node\[rectangle, draw\](Optimizer) [Optimizer]{};
(User) – (GUI); (GUI) – (Calculator); (Calculator) – (-3,-2.85); (-5,-2.75) rectangle (-3,-6.3); (Calculator) – (Analysis); (Calculator) – (Optimizer);
The calculator’s main class is the [SNC]{} class, which is a singleton, bridging the communication between GUI and backend. Alternatively, the [SNC]{} class can be used directly. It provides a command-pattern-style interface; accordingly, the underlying network is altered through sending commands to the [SNC]{} class; moreover, these commands are stored in an [undoRedoStack]{}, which is be used to track, undo and redo changes. Note that loosening connections and facilitating the use of the backend without a GUI is still work in progress.
When exceptions occur in the backend, the control flow will try to repair things as good as possible. If this is not possible, a generic runtime exception is thrown. These exceptions are specified in [misc]{} at the moment. The design choice for generic exceptions extending the java build-in [RuntimeException]{} stems from the fact that we wanted to avoid methods with a variety of checked exceptions. As a result, the code is less cluttered and more readable. We are aware that this topic is under debate, especially in the java community and that this practice needs thorough documentation.
Currently the code is organized into the following packages all packages start with [unikl.disco.]{}, which we omit in this list:
- [calculator]{} This package contains only the main class, called [SNC]{}. It is the core of the program.
- [calculator.commands]{} This package contains the various commands by which the network is manipulated (adding a flow, removing a vertex, etc.)
- [calculator.gui]{} This package includes all classes needed to generate, display, and interact with the tool’s GUI. Except for the [FlowEditor]{} class, the GUI is modular, easy to change and extend since actions are separated from markup.
- [calculator.network]{} This package includes all classes related to topological information like [Flow]{} and [Vertex]{}. It further contains the classes needed to perform an analysis.
- [calculator.optimization]{} This class contains all classes needed to evaluate and optimize performance bounds. Note that the result of an analysis is given as an object of type [Flow]{} and belongs to the [calculator.network]{} package. The bound’s information must be “translated” into backlog- and delay bounds first, before numerical values can be provided. This is why we find in the [optimization]{}-package classes like [BacklogBound]{} or [BoundType]{}.
- [calculator.symbolic\_math]{} This is a collection of algebraic manipulations. We find for example an [AdditiveComposition]{} here, which just combines two symbolic functions into a new one. When evaluated with a set of parameters this returns the sum of the two atom-functions evaluated with the same set of parameters; furthermore, we find in this package symbolic representations of MGF-bounds for flows ([Arrival]{}) and service elements ([Service]{}).
- [calculator.symbolic\_math.functions]{} Some arrival classes or specific manipulations (such as Lemma \[lem:Deconvolution\]) require the repeated usage of very specific algebraic manipulations. This package collects these operations.
- [misc]{} A package containing miscellaneous classes not fitting well anywhere else and generic runtime exceptions.
Input & Output {#sec:input_output}
==============
There are several different methods to input and output data when using the calculator:
- Using the functions provided by the GUI
- Writing/reading networks from/to text files
- Using custom defined functions when the calculator is used as a library
The GUI provides methods for adding, removing, subtracting and convoluting flows and vertices. This is useful for quick tests and experimentation but cumbersome for larger networks. To that end, we implemented a simple text-file based interface for saving and loading networks. Note that at the moment only networks without dependencies can be loaded/saved! Extending these methods to the general case is subject to future work.
The file format is specified as follows:
- A line starting with “\#” is a comment and ignored
- First, the interfaces (vertices) are specified, each in its own line
- After the last interface definition, an “EOI” ends the interface block
- Then, the flows are specified, again, each in its own line, until “EOF” ends the document
An interface line has the following syntax (without the linebreak):
I <vertexname>, <scheduling policy>, <type of service>,
<parameters> ...
At the moment only FIFO scheduling and constant rate (“CR”) service are supported. A flow line has the following syntax (again, without the linebreak):
F <flowname>, <number of vertices on route>,
<name of first hop>:<priority at this hop>, ...,
<type of arrival at first hop>, <parameters> ...
The priority is a natural number with 0 being the highest. At the moment, the following arrival types are possible (with parameters):
- CONSTANT – service rate ($\geq 0$!)
- EXPONENTIAL – mean
- EBB (exponentially bounded burstiness) – rate, decay, prefactor
- STATIONARYTB – rate, bucket, \[maxTheta\]
where [maxTheta]{} is optional. In the following we see a sample network with three vertices and one exponentially distributed flow.
# Configuration of a simple network
# Interface configuration. Unit: Mbps
I v1, FIFO, CR, 1
I v2, FIFO, CR, 3
I v3, FIFO, CR, 4
EOI
# Traffic configuration. Unit Mbps or Mb
# One flow with the route v1->v2->v3
F F1, 3, v1:1, v2:1, v3:2, EXPONENTIAL, 2
EOF
Code Representation of SNC Results and Concepts {#sec:Code Representation}
===============================================
Now we elaborate on some of the most important classes of the Calculator and how they represent core-concepts of SNC.
The [Network]{}-class
---------------------
This class is responsible for storing a network’s topology and manipulating its elements. Its key members are three [Map]{}s:
- [flows]{} is of type [Map<Integer,Flow>]{} and is a collection of flows, each with a unique ID. Each flow represents one flow’s entire path through the network. See also the subsection about [flows]{} below.
- [vertices]{} is of type [Map<Integer,Vertex>]{} and is a collection of vertices, each with a unique ID. Each vertex represents one service element of the network. It does not matter how many flows this service element has to process, it will always be modeled by a single [Vertex]{}. See also the below subsection about [vertices]{}.
- [hoelders]{} is of type [Map<Integer,Hoelder>]{}. Each newly introduced Hölder parameter (actually the pair of parameters $p$ and $q$ are defined by $1/p +1/q = 1$ and can be represented by a single variable) is collected in this object and receives a unique ID. This data-structure is needed to keep track of and distinguish the introduced parameters.
These [Map]{}s are created and manipulated by various methods of the [Network]{}-class. Some of these methods are straightforward such as [addVertex]{}, [addFlow]{}, and [setServiceAt]{}. Others methods are more involved and directly reflect core concepts of SNC: The method [computeLeftover]{} for example manipulates the network like follows: For a specific node it identifies the flow that has priority at this service element. It then calculates the leftover service description after serving this flow (Lemma \[lem:Demultiplexing\]) and gives this description to this [Vertex]{}; furthermore, the method gives as output an object of type [Arrival]{}, which encodes the MGF bound on this node’s output for the just served flow (Lemma \[lem:Deconvolution\]).
The [Flow]{}-class and the [Arrival]{}-class
--------------------------------------------
These two classes are closely related: The [Flow]{} class can be thought of as the topological information of a flow through the network. It contains a [List]{} of integer-IDs that describes the flow’s path through the network and a [List]{} of corresponding priorities. It further has a [List]{} of [Arrival]{}-objects. These objects describe the flow’s MGF-bounds at a given node. Usually a flow added to the network only has a single [Arrival]{}-object in this list, which is the MGF-bound at that flow’s the ingress node. Every [Flow]{}-object keeps track of for how many hops arrival-bounds are known in the integer variable [established\_arrivals]{}.
An [Arrival]{}-object most important members are the two [SymbolicFunction]{}s [rho]{} and [sigma]{}. These directly represent the bounding-functions $\rho$ and $\sigma$ of an MGF-bound (see Definition \[def:Arrival-Bound\]); further, important members are two[Set<Integer>]{}s, which keep track of the flows’ and services’ IDs this arrival is stochastically dependent to, respectively.
The [Vertex]{}-class and the [Service]{}-class
----------------------------------------------
Similarly to [Flow]{} and [Arrival]{} these two classes are closely connected to each other. Each [Vertex]{}-object has a member of type [Service]{}, which describes its service via an MGF-bound (Definition \[def:Service-Bound\]); furthermore, a [Vertex]{}-object has members [priorities]{} (of type[Map<Integer,Integer>]{}) and [incoming]{} (of type [Map<Integer, Arrival>]{}) to identify which flow would receive the full service and what set of flows are incoming to this node.
An [Service]{}-object is the equivalent of an [Arrival]{}-object on the service side. It also contains two [SymbolicFunction]{}s called [rho]{} and [sigma]{} and two [Set<Integer>]{} to keep track of stochastic dependencies.
The [SymbolicFunction]{}-interface
----------------------------------
This interface lies at the core of the symbolic computations made to analyze a network. Each MGF-bound is represented by two functions $\rho$ and $\sigma$, which find their representation as [SymbolicFunction]{} in the code. This interface’s most important method is the [getValue]{}-method. It takes a [double]{} (the $\theta$) and a [Map<Integer,Hoelder>]{} (the – possibly empty – set of Hölder parameters) as input and evaluates the function at this point. A simple example is the [ConstantFunction]{}-class, which implements this interface. When the method [getValue]{} is called, an object of this kind just returns a constant value, given that the [Map<Integer,Hoelder>]{} was empty. Mathematically written such an object just represents $f(\theta) = c$, which can for example be found in the MGF-bound of constant rate arrivals or service elements.
The modeling power here lies in the composition of [SymbolicFunctions]{}; for example, when we want to merge two constant rate arrivals, their MGF-bound would contain $\rho_{agg}(\theta) = r_1 + r_2$ with $\rho_1(\theta) = r_1$ and $\rho_2(\theta)= r_2$ being the subflows’ rates, respectively. The class [AdditiveComposition]{} implements the [SymbolicFunction]{}-interface itself and has two members of type [SymbolicFunction]{}. These are called atom-functions; in this scenario the atom-functions would be two[ConstantFunction]{}s with rates $r_1$ and $r_2$. When the [getValue]{}-method of[AdditiveComposition]{} is called it will relay the given parameters to its atom-functions and get their values ($r_1$ and $r_2$) and return their sum to the caller; indeed, [AdditiveComposition]{} is just a representation of the plus-sign in$r_1 + r_2 = \rho_1(\theta) + \rho_2(\theta)$.
The [AbstractAnalysis]{}-class
------------------------------
The abstract class [AbstractAnalysis]{} defines the methods and members an analysis of a network needs. It serves as a starting point for concrete analysis classes. Its members include a network’s topological information together with the indices of the flow of interest and the service of interest, so the analysis knows what performance the caller is interested in. The important method here is the [analyze]{}-method of the [Analyzer]{} interface, which every Analysis has to implement. This method (to be defined by any realization of this abstract class) gives as output an object of type [Arrival]{}, which represents a performance bound; in fact, remembering the bounds from Theorem \[thm:Fundamental-Theorem\] $$\begin{aligned}
\mathbb{P}(\mathfrak{b}(t)>N) &\leq e^{\theta N}e^{\theta \sigma_A(\theta) + \theta \sigma_U(\theta)} \cdot \frac{1}{1 - e^{\theta (\rho_A(\theta)+\rho_U(\theta))}} \\
\mathbb{P}(\mathfrak{d}(t)>T) &\leq e^{\theta \rho_U(\theta)T}e^{\theta \sigma_A(\theta) + \theta \sigma_U(\theta)} \cdot \frac{1}{1 - e^{\theta (\rho_A(\theta)+\rho_U(\theta))}}.\end{aligned}$$ we see that we can split these bounds into a part that depends on the bound’s value ($N$ or $T$, respectively) and a factor that does not depend on the bound’s value. So, we can also write: $$\mathbb{P}(\mathfrak{b}(t)>N) \leq e^{\theta \rho_\mathfrak{b}(\theta) N + \theta \sigma_\mathfrak{b}(\theta)}$$ with $\rho_\mathfrak{b}(\theta) := 1$ and $\sigma_\mathfrak{b} := \sigma_A(\theta) + \theta_U(\theta) - 1/\theta \log(1 - e^{\theta (\rho_A(\theta) + \rho_U(\theta))})$. And: $$\mathbb{P}(\mathfrak{d}(t)>T) \leq e^{\theta \rho_\mathfrak{d}(\theta) T + \theta \sigma_\mathfrak{d}(\theta)}$$ with $\rho_\mathfrak{d}(\theta) = \theta \rho_U(\theta)$ and $\sigma_\mathfrak{d} = \sigma_\mathfrak{b}$. This representation has the advantage that we can use the already implemented operations for MGF-bounds for our performance bounds; for this reason, the output of the [analyze]{}-method is an object of type [Arrival]{}, which is how the code represents an MGF-bound.
The [AbstractOptimizer]{}-class
-------------------------------
Similar to the [AbstractAnalysis]{} class, this abstract class serves as a starting point for implementing optimizers. It implements the [minimize]{}-method of the [Optimizer]{} interface, which every optimizer has to implement. This method takes as input the granularity for which the continuous space of optimization parameters should be discretized too and returns the minimal value found by the optimization algorithm. Its most important member is [bound]{}, which is basically the MGF-bound presented in the previous subsection. Together with the class [BoundType]{} and the interface [Optimizable]{} the function to be optimized is defined. This can either be a backlog- or delay bound as defined in the previous subsection or it can be their inverse functions, i.e., the smallest bound $N$ or $T$ that can be found for a given violation probability $\varepsilon$. for this the bounds from Theorem \[thm:Fundamental-Theorem\] must be solved for $N$ and $T$.
APIs and Extending the Calculator {#sec:APIs and Extensions}
=================================
The interfaces and abstract classes provide a good starting point when extending the calculator. The backend makes heavy use of the factory pattern. As long as new classes implement the necessary interfaces, extending the behavior is easy. The only exception being the [FlowEditor]{} of the GUI, which we are planning to rewrite as soon as possible.
In this section we describe in more detail how a user can implement his own models into it. For this we cover four cases; we describe how users can implement their own
- arrival model to the Calculator and its GUI, given some known MGF-bounds.
- service model to the Calculator and its GUI, given some known MGF-bounds.
- method of analysis to the Calculator and its GUI.
- method of parameter optimization to the Calculator and its GUI.
These descriptions are subject to changes of the code and we strongly recommend to pay attention to the code’s documentation before implementing any of the above.
Adding Arrival Models
---------------------
To add a new arrival model to the calculator we need to be able to write the arrivals in an MGF-bounded form as in Definition \[def:Arrival-Bound\]. We consider for this documentation an arrival that has an exponential amount of data arriving in each time step with rate parameter $\lambda$ as example (seeExample \[ex:Exponential-Increments\]): $$\mathbb{E}(e^{\theta(A(t)-A(s))})\leq\left(\frac{\lambda}{\lambda-\theta}\right)^{t-s}\qquad\text{for all }\theta<\lambda.$$ In this case $\rho(\theta)=\tfrac{1}{\theta}\log(\tfrac{\lambda}{\lambda-\theta})$ and $\sigma(\theta)=0$.
When we have appropriate $\sigma$ and $\rho$ we can implement the arrival model, by performing changes in the following classes:
1. In [ArrivalFactory]{} we write a new method[buildMyModel(parameter1,...)]{} with any input parameters needed for your model (like a rate-parameter, etc.). In this function we construct the $\sigma$ and $\rho$ as symbolic functions. For this we might have to write our own new symbolic functions. These go into the package[uni.disco.calculator.symbolic\_math.functions]{}. See the otheralready implemented arrival models for examples.
2. We add our new model in the list of arrival types in the class [ArrivalType]{}.
3. To make our new model available in the GUI we need to change the class [FlowEditor]{}
1. First we need to prepare the dialog so it can collect the parameters from users’ input. Under the comment-line “Adds the cards for the arrival” we can find one card for each already implemented arrival model. We add our arrival model here appropriately.
2. We add our newly created card to [topCardContainer]{} in the directly subsequent lines.
3. A bit further down the code we can find the action the dialog should perform after the [APPROVE\_OPTION]{}. We add our own [if]{}-clause and follow the examples of the already implemented flows in how to generate the [Arrival]{}-object from the parameters put in by the user. Make sure to use [return;]{} to jump out of the [if]{}-clause, whenever a parameter could not have been read from the input-fields or was initialized incorrectly (e.g., a negative rate was given).
Adding Service Models
---------------------
Adding new service models is completely parallel to how to add new arrival models. Again MGF-bounds must be available to implement a new model (see Definition \[def:Service-Bound\]). Changes to the code must be made in the classes: [ServiceFactory]{}, [ServiceType]{}, and [VertexEditor]{}. For exact details, compare to the changes being performed for adding new arrival models.
Adding a new Analysis
---------------------
To add a new method for analyzing a network we follow these steps:
- We construct a new class extending the [AbstractAnalysis]{}-class. We must make sure that the output for the [analyze]{}-method produces the required performance measure in an MGF-bound format and is an [Arrival]{}-object.
- Next we add the new analysis in the class [AnalysisType]{}.
- We add a new case in the class [AnalysisFactory]{}. Should our analysis require more parameters than the one offered by the [getAnalyzer]{}-method of [AnalysisFactory]{} we must make corresponding changes to the factory. When doing so these changes must be propagated to the classes [AnalysisDialog]{} and the [analyzeNetwork]{}-method of the [SNC]{}-class. In this case, however, we would recommend to switch to a builder pattern instead.
Adding a new Optimization
-------------------------
To add a new method for optimization a performance bound we follow these steps:
- We construct a new class extending the [AbstractOptimizer]{}-class. The new optimizer must define the [minimize]{}-method: The code for how to find a near optimal value goes in here.
- Afterwards we add the new optimizer to the [OptimizationType]{}-class and as a new case in the [OptimizationFactory]{}. As with adding new analyses there might be more parameters needed than the optimization factory can currently offer. In this case changes must be propagated to the [OptimizationDialog]{} and to the method called[optimizeSymbolicFunction]{} in the [SNC]{}-class. Again, when existing methods have to be changed anyway, it would be advisable to use a more general approach, such as a builder pattern.
A Full Example {#sec:full_example}
==============
(-0.5,0.25) node(Origin\_top) [$A_x$]{} ++(0,-0.5) node(Origin\_bottom) [$A$]{} ++(1.5,0.25) node(U\_1)\[circle, draw\][$U_1$]{} ++(1.5,0) node(U\_2)\[circle, draw\][$U_2$]{} ++(1.5,0) node(dots)\[circle, draw\] [$U_3$]{} ++(1.5,0) node(U\_n)\[circle, draw\][$U_4$]{} ++(1.5,0.25) node(dummy\_Destination\_top) ++(0,-0.5) node(dummy\_Destination\_bottom);
(1,0.25) node(dummy\_U\_1\_top)\[text = white\][$U_1$]{} ++(0,-0.5) node(dummy\_U\_1\_bottom)\[text = white\][$U_1$]{} ++(1.5,0.5) node(dummy\_U\_2\_top)\[text = white\][$U_2$]{} ++(0,-0.5) node(dummy\_U\_2\_bottom)\[text = white\][$U_2$]{} ++(1.5,0.5) node(dummy\_dots\_top)\[text = white\][$\ldots$]{} ++(0,-0.5) node(dummy\_dots\_bottom)\[text = white\][$\ldots$]{} ++(1.5,0.5) node(dummy\_U\_n\_top)\[text = white\][$U_n$]{} ++(0,-0.5) node(dummy\_U\_n\_bottom)\[text = white\][$U_n$]{};
(Origin\_top) – (dummy\_U\_1\_top); (Origin\_bottom) – (dummy\_U\_1\_bottom);
(dummy\_U\_1\_top) – (dummy\_U\_2\_top); (dummy\_U\_1\_bottom) – (dummy\_U\_2\_bottom);
(dummy\_U\_2\_top) – (dummy\_dots\_top); (dummy\_U\_2\_bottom) – (dummy\_dots\_bottom);
(dummy\_dots\_top) – (dummy\_U\_n\_top); (dummy\_dots\_bottom) – (dummy\_U\_n\_bottom);
(dummy\_U\_n\_top) – (dummy\_Destination\_top); (dummy\_U\_n\_bottom) – (dummy\_Destination\_bottom);
(1,1.25) node(Rung1) [$A^1$]{} ++(1.5,0) node(Rung2) [$A^2$]{} ++(1.5,0) node(Rung3) [$A^3$]{} ++(1.5,0) node(Rung4) [$A^4$]{}; (1, 1) – (U\_1); (2.5, 1) – (U\_2); (4, 1) – (dots); (5.5, 1) – (U\_n);
(U\_1) – (1,-1); (U\_2) – (2.5,-1); (dots) – (4,-1); (U\_n) – (5.5,-1);
In this section we will give a full walk-through on our modeling steps for the results presented in [@Beck:SNCalc2]. In this scenario we consider the topology in Figure \[fig:ladder-topology\] with 2, 3, or 4 service elements in tandem. In this network we consider the flow of interest as having a low-priority under the crossing flow $A_x$. This can be interpret as our flow of interest lying in the “backgronud” traffic that flows from end-to-end. The rung-flows $A^1,\ldots, A^4$ are interfering with the service elements in a FIFO- or WFQ-fashion. We also conducted NS3 simulations for the same scenario to make the analytical results comparable.
Arrival Model
-------------
We used the following approach for producing arrivals in NS3-simulations: Each arrival consists of a constant stream of data with $x$ MB/s, where $x$ is a value that changes each 0.1 seconds and is exponentially distributed. The subsequent values of $x$ are stochastically independent from each other for all flows and each time-slot. Here the parameters of exponential distributions is chosen, such that the expected datarate for the flow of interest $A$ is equal to 20 MB/s, the crossflow’s expected datarate is 40 MB/s, and each rung-flow’s expected rate is20 MB/s.
To model these arrivals in the Calculator we use the MGF-bounds as derived in Example \[ex:Exponential-Increments\]. Notice that this model is slightly different from the simulations, since the model assumes that the complete bulk of arrivals of one time-slot (with length of 0.1 seconds) arrive in an instant, whereas our simulation streams these arrivals with a constant rate over each single time-slot. We will make up for this difference when modeling service elements.
Service Model
-------------
In the NS3-simulation we use a 100 MB/s link-speed between each node. The natural method to model these is to define a constant rate server with rate $r = 100$ MB/s; however, we want to account for the differences in the model and the simulation when it comes to the flow’s burstiness. Note that in the simulations the service elements start working on the data “as it comes in”, meaning the processing starts from simulation time zero onwards; instead, in our SNC model we would wait one full time-slot and consider all the arrivals of the first 0.1 seconds to arrive in one batch at time $t= 0.1$ s. As the service rate is constant there is basically a shift of service by one time-slot between the simulation and the model. For this reason we define the service’s MGF-bound by the functions $$\begin{aligned}
\rho_{S^\prime}(\theta) & = - 10 \\
\sigma_{S^\prime}(\theta) & = - 10 \end{aligned}$$ The unit chosen here is MB per time-slot (100 MB/sec = 10 MB/0.1 sec). The above MGF-bound differs from a constant rate MGF-bound by having one additional time-slot of service in $\sigma_S$, which is available right at the start of the model. We have implemented a corresponding service model into the Calculator as described in the previous Section.
So far we have not discussed how the service elements schedule the flows. Our simulations work either by a FIFO- or WFQ-scheduling, when it comes to decide whether a packet from the flow of interest or another flow will be processed. So far the Calculator does not have leftover service descriptions implemented for these scheduling disciplines; however, using a leftover scheduling will lead to overly pessimistic results; instead, we have decided to neglect the crossflows’ burstiness entirely and subtract the expected number of the rung-flows’ arrivals from the constant rate server. This means, we have to subtract the value 2 from our service rates, leading us to the bounding functions $\rho_S(\theta) = -8$ and $\sigma_S(\theta) = -8$.
End-to-End Analysis
-------------------
As a last step we need to account for the crossflow $A_x$, which joins our flow of interest for the entire path. We need to take this crossflow into account when we want to use Theorem \[thm:End-to-End\]; in fact, this slightly modifies the proof of this theorem in a straightforward manner. Having this result at hand we implemented a new [Analysis]{} to the Calculator, which uses this end-to-end result. There was no need to implement a new optimization method, as the ones implemented can already cope with this scenario.
With the analysis and the service model implemented the results about end-to-end delay can be produced using the Calculator. Note that calling the analysis repeatedly is needed to produce the graphs in [@Beck:SNCalc2]. Since the GUI does not support such a repeated calling we accessed the Calculators methods and classes directly instead. This allowed to automatically loop through an increasing given violation probability.
| ArXiv |
---
abstract: 'Let $X$ be a smooth irreducible complex algebraic variety of dimension $n$ and $L$ a very ample line bundle on $X$. Given a toric degeneration of $(X,L)$ satisfying some natural technical hypotheses, we construct a deformation $\{J_s\}$ of the complex structure on $X$ and bases $\mathcal{B}_s$ of $H^0(X,L, J_s)$ so that $J_0$ is the standard complex structure and, in the limit as $s \to \infty$, the basis elements approach dirac-delta distributions centered at Bohr-Sommerfeld fibers of a moment map associated to $X$ and its toric degeneration. The theory of Newton-Okounkov bodies and its associated toric degenerations shows that the technical hypotheses mentioned above hold in some generality. Our results significantly generalize previous results in geometric quantization which prove “independence of polarization” between Kähler quantizations and real polarizations. As an example, in the case of general flag varieties $X=G/B$ and for certain choices of $\lambda$, our result geometrically constructs a continuous degeneration of the (dual) canonical basis of $V_{\lambda}^*$ to a collection of dirac delta functions supported at the Bohr-Sommerfeld fibres corresponding exactly to the lattice points of a Littelmann-Berenstein-Zelevinsky string polytope $\Delta_{\underline{w}_0}(\lambda) \cap {{\mathbb{Z}}}^{\dim(G/B)}$.'
address:
- |
Department of Math and Computer Science\
Mount Allison University\
67 York St.\
Sackville, NB, E4L 1E6\
Canada
- |
Department of Mathematics and Statistics\
McMaster University\
1280 Main Street West\
Hamilton, Ontario L8S4K1\
Canada
- |
Department of Mathematics\
University of Pittsburgh\
301 Thackeray Hall\
Pittsburgh, PA, 15260\
USA
author:
- Mark Hamilton
- Megumi Harada
- Kiumars Kaveh
title: 'Convergence of polarizations, toric degenerations, and Newton-Okounkov bodies'
---
[^1]
[^2]
[^3]
Introduction
============
The motivation for the present manuscript arose from two rather different research areas: the theory of geometric quantization in symplectic geometry on the one hand, and the algebraic-geometric theory of Newton-Okounkov bodies - particularly in its relation to representation theory - on the other. Since we do not expect all readers of this paper to be familiar with both theories, we begin with a brief description of each.
We begin with a sketch of geometric quantization. As is well-known, symplectic geometry (Hamiltonian flows on symplectic manifolds) is the mathematical language for formulating classical physics, whereas it is the language of linear algebra and representation theory (unitary flows on Hilbert spaces) which forms the basis for formulating quantum physics. It has been a long-standing question within symplectic geometry to understand, from a purely mathematical and geometric perspective, the relation between the classical picture and the quantum picture, in terms of both the phase spaces and the defining equations of the dynamics. In one direction, to go from “quantum” to “classical”, one can “take a classical limit”. The reverse direction, i.e. that of systematically associating to a symplectic manifold $(M,\omega)$ a Hilbert space $Q(M,\omega)$ and to similarly relate, for instance, Hamilton’s equations on $(M,\omega)$ to Schrödinger-type equations on $Q(M,\omega)$, is generally referred to as the theory of *quantization*. In this manuscript, we deal specifically with *geometric quantization*, a theory which associates to a symplectic manifold $(M,\omega)$ a Hilbert space $Q(M,\omega)$.
For a fixed $(M,\omega)$, it turns out that there are many possible ways of constructing a suitable Hilbert space $Q(M,\omega)$. To describe the choices we first set some notation. First suppose that $[\omega]$ is an integral cohomology class. Next, let $(L,\nabla, h)$ be a Hermitian line bundle with connection satisfying $\mathrm{curv}(\nabla)=\omega$. Such a triple is called a *pre-quantum line bundle*, or sometimes a *pre-quantization*. Also required is a *polarization,* of which the two main types are as follows. A *Kähler polarization* is a choice of compatible complex structure $J$ on $M$. Given such a $J$, one can define the quantization $Q(M, \omega)$ to be $H^0(M, L, J)$, the space of holomorphic sections of $L$ with respect to this complex structure $J$. On the other hand, one may also consider a (possibly singular) *real polarization* of $M$, which is a foliation of $M$ into Lagrangian submanifolds. Among the Lagrangian leaves one can define a special (usually finite, if $M$ is compact) subset called the *Bohr-Sommerfeld leaves*. There is not yet an agreed-upon “correct” definition of the corresponding Hilbert space for a real polarization, but one approach which has been investigated, and which will be used in this manuscript, is to consider distributional sections supported on the set of Bohr-Sommerfeld leaves. Based on the above discussion, the following natural question arises: *Is the quantization $Q(M,\omega)$ “independent of polarization,” i.e., independent of the choices made?* More specifically, we can ask: does the quantization coming from a Kähler polarization agree with the quantization coming from a real polarization? The results of this manuscript confirms independence of polarization in a rather large class of examples, significantly extending previously known results which were restricted to special cases such as toric varieties and flag varieties.
We next briefly motivate the theory of Newton-Okounkov bodies. The famous Atiyah-Guillemin-Sternberg and Kirwan convexity theorems link equivariant symplectic and algebraic geometry to the combinatorics of polytopes. In the case of a toric variety $X$, the combinatorics of its moment map polytope $\Delta$ fully encodes the geometry of $X$, but this fails in the general case. In his influential work, Okounkov constructed (circa roughly 1996), for an (irreducible) projective variety $X \subseteq {\mathbb{P}}(V)$ equipped with an action of a reductive algebraic group $G$, a convex body $\tilde{\Delta}$ and a natural projection from $\tilde{\Delta}$ to the moment polytope $\Delta$ of $X$. Moreover, the volumes of the fibers of this projection encode the asymptotics of the multiplicities of the irreducible representations appearing in the homogeneous coordinate ring of $X$, or in other words, the Duistermaat-Heckman measure [@Okounkov-BM; @Okounkov-log-concave]. Recently, Askold Khovanskii and the third author (also independently Lazarsfeld and Mustata [@LazMus]) vastly generalized Okounkov’s ideas [@KavKho], and in particular constructed such $\tilde{\Delta}$ (called *Newton-Okounkov bodies* or sometimes simply *Okounkov bodies*) even without presence of any group action. In the setting studied by Okounkov, the maximum possible (real) dimension of the Newton-Okounkov body $\tilde{\Delta}$ is the transcendence degree of ${{\mathbb{C}}}(X)^U$ where $U$ is a maximal unipotent subgroup of $G$; when there is no group action (as in the setting studied in [@LazMus; @KavKho]) we have $\dim_{{{\mathbb{R}}}}(\tilde{\Delta}) = \dim_{{{\mathbb{C}}}}(X).$ Hence one interpretation of the results of Okounkov, Lazarsfeld-Mustata and Kaveh-Khovanskii is that there *is* a convex geometric/combinatorial object of ‘maximal’ dimension associated to $X$, even when $X$ is not a toric variety. This represents a vast expansion of the possible settings in which combinatorial methods may be used to analyze the geometry of algebraic varieties. There is promise of a rich theory which interacts with a wide range of inter-related areas: for instance, the third author showed [@Kav-cry] that the *Littelmann-Berenstein-Zelevinsky string polytopes* from representation theory, which generalize the well-known *Gel’fand-Cetlin polytopes*, are examples of $\tilde{\Delta}$. In the long-term, one can expect further applications to Schubert calculus and to geometric representation theory (e.g. see [@KST]).
We now turn attention to the present manuscript. Firstly we should explain that the two seemingly disparate research areas mentioned above are related due to the results in [@HarKav], which uses a certain toric degeneration that arises from (the semigroup associated to) a Newton-Okounkov body [@Anderson] to construct *integrable systems*[^4] on a wide class of projective varieties. Integrable systems are highly special Hamiltonian systems on symplectic (or, in our setting, Kähler) manifolds, and naturally give rise to (singular) real polarizations. Therefore, the theory of Newton-Okounkov bodies and their associated toric degenerations provide a natural setting in which to examine the theory of geometric quantization.
Before describing the statement of our main result (Theorem \[theorem:main\]) in more detail we first recall the content of two manuscripts of Baier, Florentino, Mourao, and Nunes [@BFMN] and the first author and Konno [@HamKon], on which much of the current manuscript is based.
As already mentioned, a natural question that arises in the theory of geometric quantization is that of independence of polarization, i.e., the isomorphism class of a geometric quantization should be independent of the choices made. In the above context, this means we wish to show $\dim H^0(X, L, J)$ is equal to the number of Bohr-Sommerfeld fibres. Nunes and his collaborators initiated a “convergence of polarizations” approach to this question. Specifically, they deform the complex structure on $X$ in such a way that the [Kähler]{} polarization it defines converges, in a suitable sense to be further explained in Section \[sec-main-result\], to the real polarization on the same manifold. (See [@nunes] for an overview of this program.) Although there are more general versions of this theory, in this paper we focus particularly on the case of symplectic toric manifolds as described in [@BFMN], where the [Kähler]{}polarization converges to the (singular) real polarization given by the fibres of the moment map (i.e., the integrable system) for a torus action. Indeed, for the case of a symplectic toric manifold $X$ associated to a Delzant polytope $\Delta$, it is well-known (see for example [@Ham-toric]) that there is a natural basis $\{ \sigma^m {\mid}m\in \Delta\cap {{\mathbb{Z}}}^n\}$ of the space $H^0(X,L, J)$ of holomorphic sections of $L$ that is indexed by the integer lattice points in $\Delta$. It is also well-known that the Bohr-Sommerfeld fibres in $X$ are exactly the moment map fibres over precisely the same set of integer lattice points $\Delta \cap {{\mathbb{Z}}}^n$. In particular, the dimensions of the two quantizations agree. This is often seen as one of the most basic and motivational examples of the phenomenon of “independence of polarization”.
The first author and Konno extend the results of Baier, Florentino, Mourao, and Nunes [@BFMN], which only apply to toric manifolds, to the case of the complete flag variety ${{\mathcal F}\ell ags}({{\mathbb{C}}}^n)$ by making use of a toric degeneration of ${{\mathcal F}\ell ags}({{\mathbb{C}}}^n)$ as constructed in [@Kogan-Miller]. The precise definition of a toric degeneration is given in Section \[sec-main-result\]; roughly, it is a (flat) family of algebraic varieties over ${{\mathbb{C}}}$ with generic fiber isomorphic to a given variety (in this case ${{\mathcal F}\ell ags}({{\mathbb{C}}}^n)$) and special fiber a toric variety. The gradient-Hamiltonian-flow technique pioneered by Ruan [@Ruan] allows one to “pull back” the integrable system on the special fiber to one on the original variety ${{\mathcal F}\ell ags}({{\mathbb{C}}}^n)$ and also enables the authors to apply the techniques of [@BFMN] to ${{\mathcal F}\ell ags}({{\mathbb{C}}}^n)$.
With the above as motivation, we now describe the main result of this manuscript, although we do not give the full and precise statement due to its rather technical nature. Let $X$ be a smooth, irreducible complex algebraic variety with $\dim_{{{\mathbb{C}}}}(X) = n$, equipped with prequantum data $(L, \nabla, h)$. Suppose $X$ admits a toric degeneration ${{\mathcal X}}$ as above (and made precise in Section \[sec-main-result\]). Under these assumptions, we can construct from the toric degeneration ${{\mathcal X}}$ an integrable system $\mu: X \to {{\mathbb{R}}}^n$ on $X$ as in [@HarKav]. Very roughly, the main result of this paper then states the following (see Theorem \[theorem:main\] for the precise statement).
**Theorem A.** Under some natural technical hypotheses on $X$ and its toric degeneration ${{\mathcal X}}$, there exists a continuous deformation $\{J_s\}_{s
\in [0,\infty)}$ of the complex structure on (the underlying smooth manifold of) $X$ such that $J_0$ is the original complex structure on $X$, and in the limit as $s$ goes to $\infty$, the Kähler polarization defined by $J_s$ converges to the (singular) real polarization associated to the integrable system $\mu$ on $X$.
A notable feature of the above theorem is that, following the work of [@BFMN; @HamKon], Theorem A gives an explicit correspondence between specific elements of the Kähler and the real quantization (rather than just an equality of dimensions). The theorem above places additional hypotheses on $X$ and its toric degeneration, so one immediately then asks: when do these hypotheses hold? The second result of this manuscript, made precise in Theorem \[theorem:toric deg from NOBY\], is that the construction given in [@HarKav] gives a large class of examples on which Theorem A applies, with the caveat that it is necessary to replace the original line bundle $L$ with a suitable tensor power (or, equiv galently, the original symplectic form with a positive integer multiple thereof). Roughly, our result (Theorem \[theorem:toric deg from NOBY\]) states the following.
**Theorem B.** Let $X$ be as above, equipped with the line bundle $L$. Then the construction of the toric degeneration given by valuations (as in [@HarKav]) can be made to satisfy the additional technical hypotheses in Theorem A for the pair $(X, L^{\otimes d})$ for sufficiently large $d$ and thus gives ‘convergence of polarization’ in these cases.
It is also worth mentioning that, in the precise statement of our main Theorem A, as given in Theorem \[theorem:main\] below, we assert an *existence* of a certain basis of holomorphic sections with appropriate convergence properties, which then implies Theorem A. In the general case considered in Theorem A, this basis is not very explicit for some of the values of the deformation parameter. However, in the situation of Theorem B where the toric degeneration arises from a valuation as above, we additionally show in Theorem \[thm-sections-lin-ind\] that this basis can be chosen to be both natural and explicit throughout the deformation.
As already mentioned, there are several indications of interesting connections between the theory of Newton-Okounkov bodies and representation theory. Indeed, putting the results of [@Kav-cry] and [@HarKav] together, we obtain an integrable system on a flag variety $G/B$ whose moment map image is precisely the Littelmann-Berenstein-Zelevinsky string polytope $\Delta_{\underline{w}_0}(\lambda)$. This construction uses the so-called *(dual) canonical basis* of $V_{\lambda}^*$, the dual space of the $G$-module $V_\lambda$ with highest weight $\lambda$. The elements of this basis are parametrized by the lattice points $\Delta_{\underline{w}_0}(\lambda) \cap {{\mathbb{Z}}}^{\dim(G/B)}$. The integrable system gives rise to a real polarization of $L_\lambda \to G/B$ whose Bohr-Sommerfeld fibers are in one-to-one correspondence with $\Delta_{\underline{w}_0}(\lambda) \cap {{\mathbb{Z}}}^{\dim(G/B)}$, where $L_\lambda$ is the usual pullback line bundle from the Plücker embedding associated to weight $\lambda$. Moreover, in this context, the Borel-Weil theorem implies that the Kähler quantization $H^0(G/B, L_\lambda)$ can be identified with $V_{\lambda}^*$. Hence, in this special case and for sufficiently large multiples of $\lambda$, our Theorem A geometrically constructs a continuous degeneration of a basis of $V_{\lambda}^*$ to a collection of dirac delta functions supported at the Bohr-Sommerfeld fibres corresponding exactly to the lattice points $\Delta_{\underline{w}_0}(\lambda) \cap {{\mathbb{Z}}}^{\dim(G/B)}$.
The paper is organized as follows. In Section \[sec-main-result\] we recall the necessary definitions and give a full and precise statement of our main Theorem \[theorem:main\]. We set up the necessary family of complex structures, based largely on the work of [@BFMN] and [@HamKon], in Section \[sec:variation\]. The proof of Theorem \[theorem:main\] occupies Section \[sec:proof\]. We then show in Section \[sec:NOBY\] that the construction in [@HarKav] gives many examples of toric degenerations satisyfing the hypotheses of Theorem \[theorem:main\].
We close with some brief comments on open questions. Firstly, we believe that the proof of our main results can be modified to work for an embedding of the toric degeneration into $Y
\times {{\mathbb{C}}}$ where $Y$ is any smooth projective toric variety (instead of just a projective space). Secondly, we also believe that the constructions in this paper should descend to a GIT quotient by a torus action. We leave these open for future work.
**Acknowledgements.** We thank Yael Karshon for providing the opportunity for us to learn about each other’s past work, thus helping to initiate this collaboration.
Statement of the main theorem {#sec-main-result}
=============================
This section is devoted to the full and precise formulation of both the hypotheses for, and the statement of, our main theorem. We first provide a quick overview of geometric quantization and then dive straight into the technicalities of our theorem. Some key motivational remarks, which may aid a reader unfamiliar with this material, are contained in Remark \[remark:motivation\].
We begin with definitions in the theory of geometric quantization. For details see e.g. [@Woodhouse]. Let $(X,\omega)$ be a symplectic manifold, i.e. $X$ is a smooth manifold and $\omega$ is a closed non-degenerate differential $2$-form on $X$. Suppose $({L}, h, \nabla)$ is a complex line bundle with Hermitian structure $h$ and a connection $\nabla$ satisfying $\text{curv}(\nabla) = \omega$ and with parallel transport preserving $h$. Such a triple is called a [*prequantum line bundle*]{} (or sometimes *prequantum data*, or *prequantization*) of $(X,\omega)$. Note that for a prequantum line bundle to exist, $[\omega]$ must be an integral cohomology class.
To pass from a prequantization to a quantization, we must choose a *polarization,* which is an integrable complex Lagrangian distribution on $X$. We only deal with two types of polarizations in this manuscript, as follows. Firstly, a [*Kähler polarization*]{} on $(X,\omega)$ is a compatible complex structure $J$. Then the corresponding [*Kähler quantization*]{} of $X$ is defined to be the space of holomorphic sections $H^0(X,L)$ of $L$, where $L$ is equipped with the holomorphic structure specified by $J$. Secondly, a [*(singular) real polarization*]{} on $X$ is a (singular) foliation of $X$ into Lagrangian submanifolds. Let $P$ denote the distribution in $TX$ corresponding to a real polarization. By abuse of language, we frequently refer to both the foliation and the distribution as a real polarization. A special case, of much recent interest in this area, is the (singular) foliation given by the fibres of a completely integrable system $F\colon X \to {{\mathbb{R}}}^n$. In this setting, a section $\sigma$ of ${L}{\bigr\rvert_{U}}$ over some open set $U\subset X$ is said to be [*flat along the leaves*]{} or [*leafwise flat*]{} if it is covariantly constant with respect to $\nabla$ in directions tangent to $P$. Leafwise flat sections always exist locally, but not usually globally. A leaf $\ell$ of the real polarization $P$ is a [*Bohr-Sommerfeld leaf*]{} if there exists a (nonzero) section $\sigma$ that is flat along the leaves and defined on all of $\ell$. The set of Bohr-Sommerfeld leaves is typically discrete in the space of leaves.
There is not at present a single agreed-upon definition of quantization using a real polarization. The basic philosophy is that the quantization corresponding to a real polarization “should” be given by leafwise flat sections over Bohr-Sommerfeld fibres, but since there are no globally defined leafwise flat sections (the set of Bohr-Sommerfeld fibers being usually discrete), this is not straightforward. One possible approach is to relax the requirement that the sections be smooth and look at distributional sections supported on the set of Bohr-Sommerfeld leaves. Several examples have been investigated using this approach; see [@nunes] and references therein. This is also the approach we take in this manuscript.
We now set up the terminology and notation required for the statement of our main theorem. Let $X$ be a smooth, irreducible complex algebraic variety with $\dim_{{\mathbb{C}}}(X)=n$. We suppose in addition that $X$ is equipped with prequantum data $(\omega, J, L, h, \nabla)$ as above, where $(\omega, J)$ is a Kähler structure on $X$ and $L$ is a very ample line bundle over $X$ with Chern class equal to the Kähler class (i.e. $c_1(L)=[\omega] \in
H^2(X,{{\mathbb{Z}}})$). In [@HarKav], using ideas from [@NNU], a toric degeneration is used to construct an integrable system on $X$ which is a Hamiltonian $T^n$-action on an open dense subset of $X$. Recall that a *toric degeneration* of $X$ in the sense of [@HarKav] is a flat family $\pi: {{\mathcal X}}\to {{\mathbb{C}}}$ of irreducible varieties such that the family is trivial over ${{\mathbb{C}}}^*= {{\mathbb{C}}}\setminus \{0\}$ with each fiber isomorphic to $X$, and the (possibly singular) central fiber $X_0 := \pi^{-1}(0)$ is a toric variety with respect to a complex torus ${{\mathbb{T}}}_0$. In particular there exists a fiber-preserving isomorphism $\varrho: X \times {{\mathbb{C}}}^* \to \pi^{-1}({{\mathbb{C}}}^*)$ from the trivial fiber bundle $X \times {{\mathbb{C}}}^* \to {{\mathbb{C}}}^*$ to $\pi^{-1}({{\mathbb{C}}}^*)$ and it follows by assumption on $X$ that ${{\mathcal X}}$ is smooth away from $X_0$. For a fixed $t \in {{\mathbb{C}}}^*$, let $X_t := \pi^{-1}(t)$ denote the fiber of the family ${{\mathcal X}}$ and let $\varrho_t$ denote the restriction of $\varrho$ to $X \times \{t\}$. By assumption, $\varrho_1$ is an isomorphism from $X \cong X\times \{t\}$ to $X_1$. We will frequently identify $X$ with $X_1$ using this isomorphism $\varrho_1$.
In this manuscript, following [@HarKav] we assume that $X$ admits a toric degeneration with the additional property that the family ${{\mathcal X}}$ can be embedded in ${{\mathcal{P}}}\times {{\mathbb{C}}}$ (where ${{\mathcal{P}}}\cong {\mathbb{P}}^N$ is a projective space for an appropriate choice of $N$), as an algebraic subvariety such that
- the map $\pi: {{\mathcal X}}\to {{\mathbb{C}}}$ is the restriction to ${{\mathcal X}}$ of the usual projection ${{\mathcal{P}}}\times {{\mathbb{C}}}\to {{\mathbb{C}}}$ to the second factor, and
- the action of ${{\mathbb{T}}}_0$ on $X_0$ extends to a linear action of ${{\mathbb{T}}}_0$ on ${{\mathcal{P}}}\cong {{\mathcal{P}}}\times\{0\}$.
Sometimes by abuse of notation we think of $X_t \subseteq {{\mathcal{P}}}\times \{t\}$ as a subvariety of ${{\mathcal{P}}}$ via the natural identification ${{\mathcal{P}}}\cong {{\mathcal{P}}}\times \{t\}$. Next we equip the ambient projective space ${{\mathcal{P}}}$ with prequantum data $(\omega_{{\mathcal{P}}},
J_{{\mathcal{P}}}, L_{{\mathcal{P}}}\cong \mathcal{O}(1), \nabla_{{\mathcal{P}}}, h_{{\mathcal{P}}})$. In addition to the prequantum data on ${{\mathcal{P}}}$, we need data on ${{\mathcal{P}}}\times {{\mathbb{C}}}$. We let $\Omega = (\omega_{{\mathcal{P}}}, \omega_{std})$ denote the product Kähler structure on ${{\mathcal{P}}}\times {{\mathbb{C}}}$ where $\omega_{std}$ is the standard symplectic structure $\frac{i}{2} dz \wedge d\bar{z}$ on ${{\mathbb{C}}}$ with respect to the usual complex coordinate $z$. Moreover, by pulling back via the projection $\pi_1: {{\mathcal{P}}}\times {{\mathbb{C}}}\to {{\mathcal{P}}}$ to the first factor, we also have a line bundle $\pi_1^*L_{{\mathcal{P}}}$ on ${{\mathcal{P}}}\times {{\mathbb{C}}}$; this restricts to a line bundle $L_{{\mathcal X}}$ on the family ${{\mathcal X}}$. Let $\omega_t := \Omega \vert_{X_t}$ (respectively $L_t := L_{{\mathcal X}}\vert_{X_t}$) denote the restriction of $\Omega$ (respectively $L_{{\mathcal X}}$) to the fiber $X_t = \pi^{-1}(t)$. Moreover, pulling back the prequantum data on ${{\mathcal{P}}}$ to ${{\mathcal{P}}}\times {{\mathbb{C}}}$ via $\pi_1$ and restricting to ${{\mathcal X}}$ yields prequantum data on ${{\mathcal X}}$. Let $\nabla_t$ and $h_t$ denote the restrictions of $\nabla_{{\mathcal X}}$ and $h_{{\mathcal X}}$, respectively, to the fiber $X_t$. With this notation in place we can state further assumptions on our toric degeneration (also see [@HarKav]):
- Under the isomorphism $\varrho_1: X \to X_1$, the prequantum data $(\omega_1, L_1, \nabla_1, h_1)$ on $X_1$ pulls back to the prequantum data $(\omega, L, \nabla, h)$ on $X$.
- The Kähler form $\Omega$ on ${{\mathcal{P}}}\times {{\mathbb{C}}}$ is $T_0$-invariant, where $T_0 \cong (S^1)^n$ is the compact torus subgroup of the complex torus ${{\mathbb{T}}}_0 \cong ({{\mathbb{C}}}^*)^n$ acting on the toric variety $X_0$.
In this context, it was shown in [@HarKav] that $X$ admits an integrable system which is a Hamiltonian torus action on an open dense subset of $X$. We quote the following.
[@HarKav Theorem (A) in Introduction] \[theorem:HarKav\] Let $X$ be a smooth, irreducible complex algebraic variety with $\dim_{{\mathbb{C}}}(X)=n$ equipped with a Kähler structure $\omega$. Let $\pi: {{\mathcal X}}\to {{\mathbb{C}}}$ be a toric degeneration of $X$ in the sense described above. Suppose that $\pi: {{\mathcal X}}\to {{\mathbb{C}}}$ additionally satisfies assumptions (a)-(d). Then:
1. there exists a surjective continuous map $\phi: X \to X_0$ which is a symplectomorphism on a dense open subset $U \subset X$ (in the classical topology),
2. there exists a completely integrable system $\mu = (F_1, \ldots, F_n)$ on $(X, \omega)$ such that its moment map image $\Delta$ coincides with the moment map image of $(X_0, \omega_0)$ (which is a polytope).
3. Let $U \subset X$ be the open dense subset of $X$ from (1). Then the integrable system $\mu=(F_1, \ldots, F_n)$ generates a Hamiltonian torus action on $U$, and the inverse image [$\mu^{-1}(\Delta^\circ)$ of the interior of $\Delta$ under the moment map $\mu: X \to {{\mathbb{R}}}^n$ of the integrable system]{} lies in the open subset $U$.
The main result of the present manuscript extends the above result by additionally working with the prequantum data. We first state one additional assumption on the family ${{\mathcal X}}$. Since ${{\mathcal{P}}}\cong {\mathbb{P}}^N$ is a standard projective space, there is a complex torus ${{\mathbb{T}}}_{{\mathcal{P}}}\cong ({{\mathbb{C}}}^*)^N$ acting in the standard fashion on ${{\mathcal{P}}}$. By assumption (b) above, the torus ${{\mathbb{T}}}_0$ acting on $X_0$ extends to a linear action on ${{\mathcal{P}}}$, i.e. there is an inclusion homomorphism $\iota: {{\mathbb{T}}}_0 {\hookrightarrow}{{\mathbb{T}}}_{{\mathcal{P}}}$ inducing the action of ${{\mathbb{T}}}_0$ on ${{\mathcal{P}}}$, and this action preserves $X_0 \subset {{\mathcal{P}}}$. Similarly there is an inclusion (by abuse of notation also denoted $\iota$) of compact subgroups $\iota:
T_0 \cong (S^1)^n {\hookrightarrow}T_{{\mathcal{P}}}\cong (S^1)^N$. Let $\iota^*: {\mathfrak{t}}_{{\mathcal{P}}}^* \to
{\mathfrak{t}}_0^*$ denote the corresponding dual projection. Let $\Delta_{{\mathcal{P}}}\subseteq {\mathfrak{t}}_{{\mathcal{P}}}^*$ denote the moment polytope (i.e. the moment map image) of ${{\mathcal{P}}}$ associated to the Hamiltonian action on ${{\mathcal{P}}}$ of $T_{{\mathcal{P}}}$, and let $\Delta_0 \subseteq {\mathfrak{t}}_0^*$ denote the moment polytope of $X_0$ with respect to ${{\mathbb{T}}}_0$. We will make the following genericity assumption on $X_0$:
- the special fiber $X_0 \subseteq {{\mathcal{P}}}$ of our toric degeneration is the closure of the ${{\mathbb{T}}}_0$-orbit through $[1:1:1:\cdots:1] \in {{\mathcal{P}}}$.
From this it follows by standard Hamiltonian-geometry arguments that $\iota^*(\Delta_{{\mathcal{P}}}) = \Delta_0$. We now define $$\label{eq:def W_0}
W_0 := \iota^*(\Delta_{{\mathcal{P}}}\cap {{\mathbb{Z}}}^N) \subseteq \Delta_0 \cap {{\mathbb{Z}}}^n.$$
It is not necessarily the case that $\iota^*(\Delta_{{\mathcal{P}}}\cap {{\mathbb{Z}}}^N) =
\Delta_0 \cap {{\mathbb{Z}}}^n$ even if $\iota^*(\Delta_{{\mathcal{P}}}) = \Delta_0$, as can be seen from the case when $X_0$ is the closure of the image of the embedding ${{\mathbb{C}}}^* \to {\mathbb{P}}^2$ given by $t \mapsto [t^2:t^3:1]$.
We call an element of $W_0$ an *interior point* if it is in the interior if $\Delta_0$, and a *boundary point* if it is on the boundary of $\Delta_0$. We will show in Proposition \[prop:w0-bohr-s\] that $W_0$ is the Bohr-Sommerfeld set of the integrable system defined in Theorem \[theorem:HarKav\].
We can now state the main result of this paper.
\[theorem:main\] Let $X$ be a smooth irreducible complex algebraic variety with $\dim_{{\mathbb{C}}}(X)=n$. Suppose $X$ is equipped with prequantum data $(\omega, J, L, h, \nabla)$ as above. Let $\pi: {{\mathcal X}}\to {{\mathbb{C}}}$ be a toric degeneration of $X$ satisfying assumptions (a)-(e). Let $\mu: X \to {{\mathbb{R}}}^n$ denote the integrable system associated to the toric degeneration ${{\mathcal X}}$ as in Theorem \[theorem:HarKav\]. We additionally assume the following properties hold.
1. The restriction map $H^0({{\mathcal{P}}}, L_{{\mathcal{P}}}) \to H^0(X, L)$ (where we identify $X \cong X_1, L \cong L_1$) is surjective.
2. The restriction to the respective lattices $\iota^*: ({\mathfrak{t}}_{{\mathcal{P}}}^*)_{{{\mathbb{Z}}}} \to
({\mathfrak{t}}_0^*)_{{{\mathbb{Z}}}}$ of the dual projection is surjective.
3. The dimension of the space of holomorphic sections of $L \to X$ is the cardinality of $W_0$, i.e. $\dim_{{\mathbb{C}}}(H^0(X,L)) = \lvert W_0 \rvert$.
Then there exists a continuous one-parameter family $\{J_s\}_{s \in [0, \infty)}$ of complex structures on the underlying $C^\infty$-manifold of $X$ such that the following holds.
- For $s=0$, the complex structure $J_0$ agrees with the original complex structure on $X$.
- For each $s \in [0, \infty)$ the triple $(X, \omega, J_s)$ is Kähler and the Hermitian line bundle $({L}, h, \nabla)$ induces a holomorphic structure $\overline{\partial}^s$ on ${L}$.
- For each $s \in [0, \infty)$ there exists a basis $\{ \sigma_s^m \mid m \in W_0\}$ of $H^0(X, {L}, \overline{\partial}^s)$ such that for all interior points $m \in W_0$ the section $\frac{\sigma_s^m}{{\lVert\sigma_s^m\rVert}_{L^1(X)}}$ converges to a delta function supported on the Bohr-Sommerfeld fiber $\mu^{-1}(m)$ in the following sense: there exist a covariantly constant section $\delta_m$ of $(L, X, \nabla)_{\mu^{-1}(m)}$ and a measure $d\theta_m$ on $\mu^{-1}(m)$ such that, for any smooth section $\tau$ of the dual line bundle $L^*$ over $X$, we have: $$\label{eq:equation in main theorem}
\lim_{s \to \infty} \int_X \left\langle \tau, \frac{\sigma^m_s}{\|
\sigma^m_s \|_{L^1(X)}} \right \rangle d(vol) =
\int_{\mu^{-1}(m)} \langle \tau, \delta_m \rangle
d\theta_m$$ where $\| \cdot \|_{L^1(X)}$ denotes the $L^1$-norm with respect to the symplectic volume.
\[remark:motivation\] The essential idea behind Theorem \[theorem:main\] is a construction due to Baier, Florentino, Mourao, and Nunes [@BFMN] of a varying set $\{\chi_s\}_{s \in [0,\infty)}$ (to be defined in Section \[sec:variation\]) of diffeomorphisms of the underlying smooth manifold of the ambient projective space ${{\mathcal{P}}}$ which is designed to have certain convergence properties. Specifically, let $\{\sigma^m\}_{m \in \Delta_{{{\mathcal{P}}}} \cap {{\mathbb{Z}}}^n}$ denote the natural basis of $H^0({{\mathcal{P}}}, L_{{\mathcal{P}}})$ already mentioned above (see e.g. [@Ham-toric]) with respect to the original complex structure. In [@BFMN] the authors construct the diffeomorphisms $\chi_s$ precisely so that a pullback $\sigma^m_s$ of $\sigma^m$, defined using the $\chi_s$ at time $s$, has the form (for large enough $s$) of a “bell curve” centred at the Bohr-Sommerfeld fiber $\mu_{{{\mathcal{P}}}}^{-1}(m)$ and, as $s \to \infty$ and with appropriate normalizations, the bell curve gets narrower and narrower, thus converging to a dirac-delta distribution supported on the Bohr-Sommerfeld fiber. The bulk of the arguments in the present paper are devoted to taking this fundamental construction of [@BFMN] for the projective space ${{\mathcal{P}}}$ and making the necessary adjustments to apply them to our more general situation. We rely heavily on [@HamKon], which already worked out some of the necessary steps for the case of the full flag variety.
- For $m\in W_0$ a boundary point, we are confident that similar arguments will show that the support of the section $\sigma^m_s$ localizes around the Bohr-Sommerfeld fiber $\mu^{-1}(m)$; doing so in this paper, however, would require including many more details of the constructions in [@BFMN] and [@HamKon] than we felt was desirable. On the other hand, for the full statement of the convergence of sections, we do not have a sufficiently concrete topological description of the fiber $\mu^{-1}(m)$ to construct an analogue of the measure $d\theta_m$ for fibres over boundary points of $W_0$.
- The normalization factor $\lVert \sigma^m_s \rVert_{L^1(X)}$ in guarantees that the “area under the bell curve” mentioned in Remark \[remark:motivation\] is always equal to $1$ as $s$ varies.
- There are different versions of convergence in functional analysis, and the notion used in Theorem \[theorem:main\] is called “weak convergence”. In particular, note that our convergence assertion is not uniform in the space of test sections $\tau$.
As mentioned in the introduction, the purpose of Sections \[sec:variation\] and \[sec:proof\] is to prove Theorem \[theorem:main\]. We show that the theory of Newton-Okounkov bodies and their associated toric degenerations provides a large class of examples satisfying the hypotheses of Theorem \[theorem:main\] in Section \[sec:NOBY\].
Variation of complex structures and bases of holomorphic sections {#sec:variation}
=================================================================
In order to prove Theorem \[theorem:main\] we rely on work of Guillemin and Abreu [@Abreu; @Guill-book; @Guill-K] and, more recently, of Baier, Florentino, Mourao, and Nunes [@BFMN]. Moreover, the first author and Konno [@HamKon] have results similar to our Theorem \[theorem:main\] for the special case of flag manifolds and its Gel’fand-Tsetlin integrable system. In this section we recall the relevant background and establish the preliminary results required to prove the results in our (more general) case.
The gradient-Hamiltonian flow
-----------------------------
Let $X$ be a smooth, irreducible complex algebraic variety and $\pi: {{\mathcal X}}\to {{\mathbb{C}}}$ be a toric degeneration of $X$ satisfying assumptions (a)-(d) as above. We equip (the smooth locus of) ${{\mathcal X}}$ with the Kähler form $\omega_{{\mathcal X}}:= \Omega \vert_{{{\mathcal X}}}$ as in Section \[sec-main-result\]. The proof of our main result will use the gradient-Hamiltonian techniques of [@HarKav] which we now briefly recall.
Following Ruan [@Ruan], we define the gradient-Hamiltonian vector field corresponding to $\pi$ on the smooth locus ${{\mathcal X}}_{smooth}$ of ${{\mathcal X}}$ as follows. Let $\nabla({\operatorname{Re}}(\pi))$ denote the gradient vector field on ${{\mathcal X}}_{smooth}$ associated to the real part ${\operatorname{Re}}(\pi)$, with respect to the Kähler metric $\omega_{{\mathcal X}}$. Since $\omega_{{\mathcal X}}$ is Kähler and $\pi$ is holomorphic, the Cauchy-Riemann equations imply that $\nabla({\operatorname{Re}}(\pi))$ is related to the Hamiltonian vector field $\xi_{{\operatorname{Im}}(\pi)}$ of the imaginary part ${\operatorname{Im}}(\pi)$ with respect to $\omega_{{\mathcal X}}$ by $$\label{gradient Re pi and Hamiltonian Im pi}
\nabla({\operatorname{Re}}(\pi)) = - \xi_{{\operatorname{Im}}(\pi)}.$$ Let $Z$ denote the closed subset of ${{\mathcal X}}$ which is the union of the singular locus of ${{\mathcal X}}$ and the critical set of ${\operatorname{Re}}(\pi)$, i.e. the set on which $\nabla({\operatorname{Re}}(\pi)) = 0$. The *gradient-Hamiltonian vector field* $V_\pi$, which is defined only on the open set ${{\mathcal X}}\setminus Z$, is by definition $$\label{def-grad-Hamiltonian}
V_\pi := - \frac{\nabla({\operatorname{Re}}(\pi))}{\|\nabla({\operatorname{Re}}(\pi))\|^2}.$$ Where defined, $V_\pi$ is smooth. For $t \in {{\mathbb{R}}}_{\geq 0}$ let $\phi_t$ denote the time-$t$ flow corresponding to $V_\pi$. Note that since $V_\pi$ may not be complete, $\phi_t$ for a given $t$ is not necessarily defined on all of ${{\mathcal X}}\setminus Z$; this issue is dealt with in the next proposition.
The gradient-Hamiltonian flow is the tool which allows us to relate the geometry of different fibres of the toric degeneration. Recall that $X_t$ denotes the fiber $\pi^{-1}(t)$ and that we often identify $X_1$ with the original variety $X$ using the isomorphism $\varrho_1$ above. We now record some facts, which hold under our assumptions, assembled from [@HarKav Sections 2-4] and which are also used in the proof of [@HarKav Theorem (A)].
\[prop-grad-Hamiltonian\] In the setting above, we have the following.
- Let $s,t \in {{\mathbb{R}}}$ with $s \geq t > 0$. Where defined, the flow $\phi_t$ takes $X_s \cap ({{\mathcal X}}\setminus Z)$ to $X_{s-t}$. In particular, where defined, $\phi_t$ takes a point $x \in
X_t$ to a point in the fiber $X_0$. Moreover, for $s>t>0$, the map $\phi_t$ is defined on all of $X_s$ and it is a diffeomorphism from $X_s$ to $X_{s-t}$.
- Where defined, the flow $\phi_t$ preserves the symplectic structures, i.e., if $x \in X_z \cap ({{\mathcal X}}\setminus Z)$ is a point where $\phi_t(x)$ is defined, then $\phi_t^*(\omega_{z-t})_{\phi_t(x)} = (\omega_z)_x$. In particular, for $s>t>0$, the map $\phi_t$ is a symplectomorphism between $X_s$ and $X_{s-t}$.
- For $s=t$, there exists an open dense subset $U_t=U_s$ of $X_t$ and an open dense subset $U_0 \subset X_0$ in the smooth locus of $X_0$ such that $\phi_t$ is a symplectomorphism from $U_t$ to $U_0$. Moreover, $\phi_t$ extends continuously to a map $\phi_t: X_t
\to X_0$.
Using the gradient-Hamiltonian flows, for each $0<t\leq 1$ we construct an integrable $\mu_t \colon X_t \to {{\mathbb{R}}}^n$ by pulling back the standard integrable system $\mu_0: X_0 \to {{\mathbb{R}}}^n$ (arising from the structure of $X_0$ as a toric variety) on $X_0$ through the maps $\phi_t$ [@HarKav Theorem 5.2]. More precisely, we define $$\label{eq:definition mu_t}
\mu_t := \phi_t^* \mu_0: X_t \to {{\mathbb{R}}}^n.$$ As a result, the moment map image $\mu_t(X_t)$ for each $0 < t \leq 1$ is equal to the moment map image $\Delta_0 := \mu_0(X_0)$ of the toric variety $X_0$. In particular, we have an integrable system $\mu: X \to {{\mathbb{R}}}^n$ on $X\cong X_1$ whose image is $\Delta_0$. For further details we refer to [@HarKav].
In what follows it will sometimes be useful to refer to the families of $X_t$ and $U_t$ with $t\in [0,1]$, and so we define $$\label{eq:X-and-U-fam}
\begin{split}
{{\mathcal X}}_{[0,1]} &:= \pi{^{-1}}\bigl([0,1]\bigr)\subset {{\mathcal X}}\\
\mathcal{U}_{[0,1]} & := \{ x \in {{\mathcal X}}_{[0,1]} {\mid}x \in U_{\pi(x)}\}.
\end{split}$$
The varying complex structure {#ss:varying-cplx-str}
-----------------------------
We now define a family $\{J_{s,t}\}$ of complex structures on $X \cong
X_1$ where $s, t$ are real parameters with $0 \leq s < \infty$ and $0
< t \leq 1$. In Section \[sec:proof\] we will choose an appropriate continuous function $t = t(s)$ of $s$ and thus define a $1$-parameter family $\{J_s = J_{s,t(s)}\}$ of complex structures which will satisfy the properties asserted in our Theorem \[theorem:main\].
For details we refer to [@BFMN; @HamKon] but we briefly set some notation. Recall that ${{\mathcal{P}}}\cong {\mathbb{P}}^N$ is a standard projective space. By slight abuse of notation we denote also by ${{\mathcal{P}}}$ the underlying smooth manifold. Since the usual projective space ${\mathbb{P}}^N$ is naturally a Kähler manifold with Kähler structure $({{\mathcal{P}}}, \omega_{{\mathcal{P}}}, J_{{\mathcal{P}}})$, we may consider ${{\mathcal{P}}}$ as a symplectic manifold $({{\mathcal{P}}}, \omega_{{\mathcal{P}}})$ or as a complex manifold $({{\mathcal{P}}}, J_{{\mathcal{P}}})$. In [@BFMN] the authors construct a family of diffeomorphisms $$\label{eq:chi_s}
\chi_s: {{\mathcal{P}}}\to {{\mathcal{P}}}$$ for $s \in {{\mathbb{R}}}$ with $0 \leq s < \infty$ which satisfy the following:
- $\chi_0: {{\mathcal{P}}}\to {{\mathcal{P}}}$ is the identity function, and
- for any $s$ with $0 \leq s < \infty$, the triple $({{\mathcal{P}}}, \omega_{{\mathcal{P}}}, \chi_s^*(J_{{\mathcal{P}}}))$ is a Kähler structure on ${{\mathcal{P}}}$.
The family $\{\chi_s\}$ in satisying ($\chi$-1) and ($\chi$-2) is not uniquely determined; the general construction given in [@BFMN] could yield many such choices of $\{\chi_s\}$. We interpret the diffeomorphisms $\chi_s$ as giving rise to a one-parameter family of Kähler structures on ${{\mathcal{P}}}$ with respect to the same symplectic structure but with varying complex structure.
In this paper, we wish to use the varying complex structures $\chi_s^* J_{{\mathcal{P}}}$ on ${{\mathcal{P}}}$ to define a family of complex structures on $X$. However, this is not completely straightforward because a smooth submanifold $X$ of ${{\mathcal{P}}}$ may be a complex submanifold of ${{\mathcal{P}}}$ for the original complex structure $J_{{\mathcal{P}}}$ but may *not* be a complex submanifold of ${{\mathcal{P}}}$ equipped with the altered complex structure $\chi_s^*J_{{\mathcal{P}}}$. To address this issue, the first author and Konno prove the following [@HamKon Proposition 6.1].
\[prop:embedding of submanifold\] ([@HamKon Proposition 6.1]) Let $V$ be a smooth submanifold of ${{\mathcal{P}}}$ with associated embedding $\rho: V
{\hookrightarrow}{{\mathcal{P}}}$. Assume that $V$ is a complex submanifold of ${{\mathcal{P}}}$ with respect to the complex structure $J_{{\mathcal{P}}}$ and let $\omega_V := \rho^*(\omega_{{\mathcal{P}}})$ denote the corresponding Kähler form on $V$. Let $\chi_s: {{\mathcal{P}}}\to {{\mathcal{P}}}$ for $s$ a real parameter, $0 \leq s <
\infty$, be a family of diffeomorphisms as in satisfying ($\chi$-1) and ($\chi$-2). Then there exists a family $\{\rho_s\}_{0 \leq s < \infty}$ of embeddings $\rho_s: V {\hookrightarrow}{{\mathcal{P}}}$ such that
1. for all $s$ with $0 \leq s < \infty$ we have $\rho_s^*\omega_{{\mathcal{P}}}\vert_V = \omega_V$,
2. for all $s$ with $0 \leq s < \infty$ the image $\rho_s(V)
\subset {{\mathcal{P}}}$ is a complex submanifold of $({{\mathcal{P}}}, \chi_s^*(J_{{\mathcal{P}}}))$, and
3. $\rho_0 = \rho$.
In particular, for any $s$ with $0 \leq s < \infty$ the pair $\left(\omega_V, \rho_s^*\left(\chi_s^*(J_{{\mathcal{P}}}) \vert_{\rho_s(V)}\right)\right)$ is a Kähler structure on $V$. Furthermore, for each choice of family $\{\chi_s\}$ as in , the family of embeddings $\{\rho_s\}$ satisfying the above conditions is unique.
Each fiber $X_t$ (for $0 < t \leq 1$) of our family ${{\mathcal X}}$ is a complex submanifold of ${{\mathcal{P}}}$ with respect to the original complex structure $J_{{\mathcal{P}}}$. Hence, applying Proposition \[prop:embedding of submanifold\] to each $X_t$, we obtain embeddings $$\label{def:rho s t}
\rho_{s,t}: X_t \to {{\mathcal{P}}}$$ where $s, t$ are real parameters with $0 \leq s < \infty$ and $0 <
t \leq 1$. We can now define a family $\{J_{s,t}\}$ of complex structures on $X=X_1$. We have the following key diagram (note that it is not commutative): $$\label{eq:key diagram}
\xymatrix{
& \rho_{s,t}(X_t) \ar[r]^{\subset} & ({{\mathcal{P}}},\omega_{{\mathcal{P}}}) \ar[r]^{\chi_s} & ({{\mathcal{P}}},J_{{\mathcal{P}}}) \\
X=X_1 \ar[r]^{\phi_{1-t}} & X_t \ar[u]^{\rho_{s,t}} \ar[r]^{\phi_t} & X_0 \ar[u] & \\
}$$ and we have the following.
Let $s,t \in {{\mathbb{R}}}$ with $0 \leq s < \infty$ and $0 < t \leq
1$. Let $\phi_{1-t}: X \cong X_1 \to X_t$ be the gradient-Hamiltonian flow and let $\{\chi_s\}$ be a choice of diffeomorphisms as in and $\rho_{s,t}: X_t {\hookrightarrow}{{\mathcal{P}}}$ be the corresponding embeddings in . The complex structure $J_{s,t}$ on $X$ is then defined by $$\label{eq:definition Jst}
J_{s,t} ;= (\rho_{s,t} \circ \phi_{1-t})^*(\chi_s^* J_{{\mathcal{P}}}\vert_{\rho_{s,t}(X_t)})$$ Equivalently, $J_{s,t}$ is the pullback $(\chi_s \circ \rho_{s,t} \circ \phi_{1-t})^* J_{{\mathcal{P}}}$.
Since both $\rho_{s,t}$ and $\phi_{1-t}$ behave well with respect to the symplectic structures (Propositions \[prop-grad-Hamiltonian\](b) and \[prop:embedding of submanifold\](a)), the following is immediate.
\[lemma:pairs Kahler\] Let $s, t \in {{\mathbb{R}}}$ with $0 \leq s < \infty$ and $0 < t \leq 1$. Then the triple $(X\cong X_1, \omega = \omega_1, J_{s,t})$ is Kähler.
In what follows, we will need two further properties of these embeddings $\rho_{s,t}$ and complex structures $J_{s,t}$, the first of which requires some additional hypotheses on the family $\{\chi_s\}$, as we now explain. As mentioned above, the family $\{\chi_s\}$ given in is not unique. However, when ${{\mathcal{P}}}$ is equipped with a complex torus action and $V$ happens to be the closure in ${{\mathcal{P}}}$ of a torus orbit, then it turns out that the family $\{\chi_s\}$ can be chosen in such a way that $V$ remains a complex submanifold for *all* of the complex structures $\chi_s^* J_{{\mathcal{P}}}$, not just the original complex structure $J_{{\mathcal{P}}}$. Before proceeding it should be noted that although the statement of Proposition \[prop:embedding of submanifold\] (equivalently [@HamKon Proposition 6.5]) contains the hypothesis that $V$ is smooth, it is shown in the proof of [@HamKon Proposition 6.5] that the argument for [@HamKon Proposition 6.1] can be extended in this special case to give a well-defined embedding $\rho$ of $V$, with analogous properties.
\[prop:toric submanifolds don’t move\] Let ${{\mathbb{H}}}\subset {{\mathbb{T}}}_{{\mathcal{P}}}$ be a complex subtorus, acting on ${{\mathcal{P}}}$ by restriction of the standard ${{\mathbb{T}}}_{{\mathcal{P}}}$-action on ${{\mathcal{P}}}$. Let $V$ denote the (possibly singular) closure in ${{\mathcal{P}}}$ of the ${{\mathbb{H}}}$-orbit of $[1:1:\cdots:1] \in {{\mathcal{P}}}$. Then there exists a choice of a family $\{\chi_s\}$ as in , satisfying the assumptions ($\chi$-1) and ($\chi$-2), such that for all $s$ with $0 \leq s < \infty$ we have $\rho_s = \rho_0$, where $\rho_s$ is the (unique) embedding associated to $\chi_s$ constructed in Proposition \[prop:embedding of submanifold\] above. In particular, on the smooth locus of $V$, the complex structures $\chi_s^* J_{{\mathcal{P}}}$ and $J_{{\mathcal{P}}}$ agree, for all $s$.
Since the technical aspects of the construction of the above family $\{\chi_s\}$ are not used in this manuscript, we do not discuss it further here; for details see [@HamKon; @BFMN]. In the setting of this manuscript, the fiber $X_0 :=
\pi^{-1}(0)$ over $0$ of our family ${{\mathcal X}}$ is by assumption the closure of the ${{\mathbb{T}}}_0$-orbit of $[1:1:\cdots:1]$ for ${{\mathbb{T}}}_0 \subset {{\mathbb{T}}}_{{\mathcal{P}}}$ a subtorus of ${{\mathbb{T}}}_{{\mathcal{P}}}$. Hence Proposition \[prop:toric submanifolds don’t move\] applies, and we therefore obtain a family $\{\chi_s\}$ of diffeomorphisms as in (satisfying the assumptions ($\chi$-1) and ($\chi$-2)) such that the associated $\rho_s$’s leave $X_0$ invariant, i.e., $\rho_s=\rho_0$ on $X_0$ for all $0 \leq s < \infty$. This will be crucial in what follows so we now record, by way of emphasis, that
> **henceforth, we assume that the family $\{\chi_s\}$ is chosen in such a way that the conclusion of Proposition \[prop:toric submanifolds don’t move\] holds.**
Given this choice of $\{\chi_s\}$, in our later arguments we need to know that the maps $\rho_{s,t}$ defined in satisfy some continuity conditions with respect to the parameter $t$. We record the following.
\[prop:rho-smooth-in-t\] Let $s \in {{\mathbb{R}}}$ with $0 \leq s < \infty$. Let ${{\mathcal X}}\subseteq {{\mathcal{P}}}\times {{\mathbb{C}}}$ be the toric degeneration as above and $\{\chi_s\}$ be a family of diffeomorphisms as above, chosen so that the conclusion of Proposition \[prop:toric submanifolds don’t move\] holds. Then
1. the map ${{\mathcal X}}_{[0,1]} \to {{\mathcal{P}}}$ given by $x \mapsto
\rho_{s,\pi(x)}(x)$ is continuous, and
2. the map ${\mathcal{U}}_{[0,1]} \to {{\mathcal{P}}}\times {{\mathbb{C}}}$ given by $x \mapsto
(\rho_{s,\pi(x)}(x), \pi(x))$ is a diffeomorphism onto its image.
We first review the construction of the map $\rho_s$ from [@HamKon]. Let $\chi_s\colon {{\mathcal{P}}}\to {{\mathcal{P}}}$ be the diffeomorphisms from . Let $\psi_s = \chi_0 \circ \chi_s{^{-1}}\colon {{\mathcal{P}}}\to {{\mathcal{P}}}$ and let $\omega_s = (\chi_s{^{-1}})^* \omega_{{\mathcal{P}}}$. By property ($\chi$-2) of the family $\{\chi_s\}$ we know that $(\omega_s, J_{{\mathcal{P}}})$ is a Kähler structure on ${{\mathcal{P}}}$. Thus, any submanifold $V$ of ${{\mathcal{P}}}$ which is complex with respect to $J_{{\mathcal{P}}}$ is also a symplectic submanifold with respect to $\omega_s$. Hence we can define a time-dependent vector field $\mathbb{V}_s$ on ${{\mathcal{P}}}$ by $$(\mathbb{V}_s)_{\psi_s(p)} = \frac{d}{d\tau} \psi_{s+\tau} (p) \Bigr\rvert_{\tau=0}$$ for $p\in {{\mathcal{P}}}$. Following [@HamKon] we further define a vector field $Y_s$ on $V$ by $$\label{eq:Y_s}
\iota_{Y_s}\bigl(\omega_s\vert_V\bigr)
= -\psi_s^* \bigl(\iota_{\mathbb{V}_s}\omega_{{\mathcal{P}}}\bigr)\bigr\rvert_V.$$ Letting $\varphi_s$ denote the corresponding flow of the vector field $Y_s$, again following [@HamKon] we finally define $\rho_s$ by $$\rho_s := \chi_s^{-1} \circ \rho_0 \circ \varphi_s \circ \chi_0
\bigr\rvert_{V}.$$ The construction just recounted deals with a single submanifold $V
\subseteq {{\mathcal{P}}}$. To prove the proposition we must show that this construction can be extended to one on a family ${{\mathcal X}}\subseteq {{\mathcal{P}}}\times {{\mathbb{C}}}$ in a way which guarantees the claimed smoothness and continuity properties with respect to the extra parameter. To do this, we first define $\widehat{\chi}_s: {{\mathcal{P}}}\times {{\mathbb{C}}}\to {{\mathcal{P}}}\times {{\mathbb{C}}}$ by $\widehat{\chi}_s(x,t) = (\chi_s(x), t)$ for $(x,t) \in {{\mathcal{P}}}\times {{\mathbb{C}}}$ and $\widehat{\psi}_s := \widehat{\chi}_0 \circ \widehat{\chi}_s^{-1}$. We then define a time-dependent vector field $\widehat{\mathbb{V}}_s$ on ${{\mathcal{P}}}\times {{\mathbb{C}}}$ by $$(\widehat{\mathbb{V}}_s)_{(\psi_s(p),t)} := \frac{d}{d\tau}
\widehat{\psi}_{s+\tau}(p,t) \Bigr\rvert_{\tau=0}$$ and a vector field $\widehat{\mathbb{Y}}_s$ on ${{\mathcal X}}_{smooth}$, the smooth locus of ${{\mathcal X}}$, by $$\label{eq:def hat Y_s}
\iota_{\widehat{\mathbb{Y}}_s}(\widehat{\omega}_s) =
\iota_{\widehat{\mathbb{Y}}_s}((\widehat{\chi}_s^{-1})^* \Omega) = -
\widehat{\psi}_s^*(\iota_{\widehat{\mathbb{V}}_s} \Omega) \bigr\vert_{{{\mathcal X}}_{smooth}}$$ where $\Omega$ is the product Kähler structure on $\mathcal{P} \times {{\mathbb{C}}}$. Note that $\widehat{\mathbb{V}}_s = (\mathbb{V}_s,0)$ by definition of $\widehat{\psi}_s$ and by construction $\widehat{\mathbb{Y}}_s$ is a smooth vector field on ${{\mathcal X}}_{smooth}$. We wish to analyze the relation between $Y_s$, defined via the above construction from [@HamKon] on each $X_t$ separately, and $\widehat{\mathbb{Y}}_s$, for which we need some preliminaries. Let $pr_1:
{{\mathcal{P}}}\times {{\mathbb{C}}}\to {{\mathcal{P}}}$ be the projection to the first factor and $\mathcal{V} \subseteq T{{\mathcal X}}_{smooth}$ denote the vertical subbundle of $T{{\mathcal X}}_{smooth}$ with respect to $pr_1$, i.e., $\mathcal{V}_x :=
\ker(d(pr_1)_x)$ for each $x \in {{\mathcal X}}_{smooth}$. Then $\mathcal{V}$ is a smooth symplectic subbundle of $T{{\mathcal X}}_{smooth}$ with respect to $\Omega
\vert_{{{\mathcal X}}_{smooth}}$, so there is a canonical decomposition $T{{\mathcal X}}_{smooth} \cong \mathcal{V} \oplus \mathcal{V}^{\Omega}$ and the projection $T{{\mathcal X}}_{smooth} \to \mathcal{V}$ is smooth.
We now claim that $$\label{eq:form is a pullback from base}
- \widehat{\psi}_s^*(\iota_{\widehat{\mathbb{V}}_s} \Omega) = pr_1^* \big( -
\Psi_s^*(\iota_{\mathbb{V}_s} \omega_{{{\mathcal{P}}}}) \big)$$ as $1$-forms on ${{\mathcal{P}}}\times {{\mathbb{C}}}$. Indeed, for any $w \in T({{\mathcal{P}}}\times {{\mathbb{C}}})
\cong T{{\mathcal{P}}}\oplus T{{\mathbb{C}}}$, we may decompose $w=(w_{{\mathcal{P}}}, w_{{\mathbb{C}}})$ into its two factors and compute $$\begin{split}
\widehat{\psi}_s^*(\iota_{\widehat{\mathbb{V}}_s} \Omega)(w) & =
\Omega(\widehat{\mathbb{V}}_s, (\widehat{\psi}_s)_*(w)) \\
& = \omega_{{{\mathcal{P}}}}(\mathbb{V}_s, ((\widehat{\psi}_s)_*w)_{{{\mathcal{P}}}}) \textup{
since } \widehat{\mathbb{V}}_s = (\mathbb{V}_s, 0) \\
& = \omega_{{{\mathcal{P}}}}(\mathbb{V}_s, (\Psi_s)_*(\omega_{{{\mathcal{P}}}})) \textup{ since
$\widehat{\psi}_s$ acts as the identity on the ${{\mathbb{C}}}$ factor} \\
& = (\Psi_s)_*(\iota_{\mathbb{V}_s} \omega_{{{\mathcal{P}}}})(w_{{{\mathcal{P}}}}) \\
& = pr_1^*(\Psi_s^*(\iota_{\mathbb{V}_s} \omega_{{{\mathcal{P}}}}))(w).
\end{split}$$ Now suppose $Z \in \mathcal{V} \subseteq T{{\mathcal X}}_{smooth}$, so $Z = (Z_{{\mathcal{P}}},
0)$ where $Z_{{{\mathcal{P}}}} \in T{{\mathcal{P}}}\cap T{{\mathcal X}}_{smooth}$. We have $$\begin{split}
\iota_{\widehat{\mathbb{Y}}_s}(\widehat{\omega}_s)(Z) & = -
\widehat{\psi}_s^*(\iota_{\widehat{\mathbb{V}}_s} \Omega)(Z) \\
& = pr_1^*(- \Psi_s^*(\iota_{\mathbb{V}_s} \omega_{{{\mathcal{P}}}}))(Z) \\
& = - \Psi_s^*(\iota_{\mathbb{V}_s} \omega_{{{\mathcal{P}}}})(Z_{{\mathcal{P}}}) \\
& = \iota_{Y_s}(\omega_s)(Z_{{\mathcal{P}}})
\end{split}$$ where the first equality is by , the second by , and the last is the definition of $Y_s$ on each fiber. From this it follows that the symplectic-orthogonal projection $(\widehat{Y}_s)_{vert}$ of $\widehat{Y}_s$ to the vertical subbundle $\mathcal{V} \subseteq T{{\mathcal X}}_{smooth}$ agrees, fiberwise, with the vector field $Y_s$ defined using the original construction from [@HamKon]. Since the symplectic-orthogonal projection is smooth, as argued above, it follows that the vector field $Y_s$, considered together on all of ${{\mathcal X}}_{smooth}$, is smooth on ${{\mathcal X}}_{smooth}$. Let $\widehat{\varphi}_s$ denote the flow corresponding to $(\widehat{Y}_s)_{vert}$, which exists since it exists fiberwise for each $X_t$ with $t\neq 0$, and for $X_0$, the argument in the proof of [@HamKon Proposition 6.1] shows that a flow exists and extends continuously to all of $X_0$. Hence the statement (1) of the proposition now follows.
To prove (2), we first observe that the above argument shows that the map $\mathcal{U}_{[0,1]} \to {{\mathcal{P}}}\times {{\mathbb{C}}}$ given by $x \mapsto
(\rho_{s, \pi(x)}(x), \pi(x))$ is smooth. So it suffices to show that this map is smoothly invertible. We know that for each fixed $t \in
{{\mathbb{C}}}$ with $t \neq 0$, the map $\rho_{s,t}: X_t \to {{\mathcal{P}}}$ is an embedding. Moreover, Proposition \[prop:toric submanifolds don’t move\] implies that on $U_0 \subseteq X_0$ we have $\rho_{s,0}=\rho_0$, hence $\rho_{s,0}$ is also an embedding. It follows that $x \mapsto (\rho_{s,\pi(x)}(x), \pi(x))$ is injective, so it is bijective on its image. It remains to show that the inverse map is also smooth. Since $\mathcal{U}_{[0,1]}$ lies in ${{\mathcal X}}_{smooth}$ we may decompose $T\mathcal{U}_{[0,1]}$ into the vertical subbundle $\mathcal{V}$ and its complement $\mathcal{V}^{\Omega}$. With respect to this decomposition and the standard decomposition $T{{\mathcal{P}}}\oplus T{{\mathbb{C}}}$ of ${{\mathcal{P}}}\times {{\mathbb{C}}}$, the derivative of the above map at a point in $\pi^{-1}(t)$ is of the form $$\begin{bmatrix} (\rho_{s,t})_* & \star \\ 0 & I \end{bmatrix}$$ where $I$ is an isomorphism and $(\rho_{s,t})_*$ is injective. Thus the whole derivative is also injective, and it follows that the inverse mapping is smooth.
Pullbacks of prequantum data {#subsec:lifts}
----------------------------
The main result of this manuscript deals with quantizations, and in particular with sections of certain prequantum line bundles. In this section, we show that the gradient-Hamiltonian flows $\phi_{1-t}$ and the embeddings $\rho_{s,t}$ from Section \[ss:varying-cplx-str\] lift to the total spaces of the relevant line bundles. This will be crucial for our constructions below. Recall that we have the prequantum data $(L_{{\mathcal{P}}}, \nabla_{{\mathcal{P}}}, h_{{\mathcal{P}}})$ and $(L_{{\mathcal X}}, \nabla_{{\mathcal X}}, h_{{\mathcal X}})$ respectively on ${{\mathcal{P}}}$ and ${{\mathcal X}}$ and that the latter restricts to give prequantum data on the fibers $X_t$, which we denote by $(L_t, \nabla_t, h_t)$.
We first recall that the horizontal lift of the gradient-Hamiltonian flow with respect to the connection $\nabla_{{\mathcal X}}$ preserves the connections and Hermitian metrics on each fiber [@HamKon Proposition 4.3].
\[lemma:lift-grH\]([@HamKon Proposition 4.3])
Let $s, t \in {{\mathbb{R}}}$ with $s\geq t > 0$.
1. If $s>t$, there exists a unique horizontal lift $\tilde{\phi}_t:
L_s \to L_{s-t}$ of the gradient-Hamiltonian flow $\phi_t: X_s \to
X_{s-t}$ to the total spaces of the prequantum line bundles. The lift $\tilde{\phi}_t$ is an isomorphism of line bundles and also preserves the fiberwise connections and Hermitian structures, i.e., $\tilde{\phi}_t^* \nabla_{s-t} = \nabla_s$ and $\tilde{\phi}_t^* h_{s-t} = h_s$.
2. If $s=t$, there there exists a unique horizontal lift $\tilde{\phi}_{s=t}:
L_t \vert_{U_t} \to L_0 \vert_{U_0}$ of the gradient-Hamiltonian flow $\phi_t: U_t \to
U_0$ to the total spaces of the (restricted) prequantum line bundles. The lift $\tilde{\phi}_t$ is an isomorphism of line bundles and also preserves the fiberwise connections and Hermitian structures, i.e., $\tilde{\phi}_t^* \nabla_{0} = \nabla_t$ and $\tilde{\phi}_t^* h_{0} = h_t$ (restricted to $U_t$ and $U_0$).
3. The map $L_0 \vert_{U_0} \times [0,1] \to L_{{\mathcal X}}$ given by $(x,t) \mapsto
\tilde{\phi}_t^{-1}(x)$ is smooth, where the domain $L_0 \vert_{U_0}
\times [0,1]$ is given the standard smooth structure induced from the product structure.
The argument is essentially the same as in [@HamKon]. For the statement in (3) we note that the gradient-Hamiltonian flow on ${{\mathcal X}}\setminus Z$ is smooth and hence its horizontal lift is also smooth. Now an argument similar to that in the proof of Proposition \[prop:rho-smooth-in-t\] yields the result.
We remark that it follows from the above lemma that the following diagram commutes for $t$ with $0 < t < 1$: $$\xymatrix{
L_1 \ar[d] \ar[r]^{\tilde{\phi}_{1-t}} & L_{t} \ar[d] \\
X\cong X_1 \ar[r]^{\phi_{1-t}} & X_{t} \\
}$$
We will also need a similar statement for the embeddings $\rho_{s,t}$ [@HamKon Proposition6.3(1)].
\[lemma:lift-rhos\] ([@HamKon Proposition 6.3(1)]) There exists a lift $\tilde{\rho}_{s,t}: L_t \to L_{{\mathcal{P}}}\vert_{\rho_{s,t}(X_t)}$ of $\rho_{s,t}$ to the total spaces of the line bundles $L_t$ and $L_{{\mathcal{P}}}\vert_{\rho_{s,t}(X_t)}$ which identifies the prequantum data. In particular, the following diagram commutes: $$\xymatrix{
L_t \ar[d] \ar[r]^{\tilde{\rho}_{s,t}} & L_{{\mathcal{P}}}\vert_{\rho_{s,t}(X_t)} \ar[d] \\
X_t \ar[r]^{\rho_{s,t}} & \rho_{s,t}(X_t) \\
}$$
We need a smoothness property for the map $\widetilde{\rho}_{s,t}$, analogous to Proposition \[prop:rho-smooth-in-t\] for the $\rho_{s,t}$.
\[prop:rho-tilde-smooth-in-t\] Let $s \in {{\mathbb{R}}}$, $0 \leq s < \infty$ be fixed. Let ${{\mathcal X}}\subseteq {{\mathcal{P}}}\times {{\mathbb{C}}}$ be the toric degeneration as above and $\{\chi_s\}$ be the family of diffeomorphisms as above (in particular chosen so that the conclusion of Proposition \[prop:toric submanifolds don’t move\] holds). For any $0 \leq t \leq 1$, let $\rho_{s,t}:
X_t {\hookrightarrow}{{\mathcal{P}}}$ be the embedding defined in Proposition \[prop:embedding of submanifold\], and let $\tilde{\rho}_{s,t}$ be the lifting of $\rho_{s,t}$ as defined in Lemma \[lemma:lift-rhos\]. Let $L_{\mathcal{U}}$ denote the restriction of the line bundle $L_{{\mathcal X}}$ to the open subset ${\mathcal{U}}_{[0,1]}$ defined in . Then the map $$\label{eq:formula for tilderhos}
\tilde{\rho}_s: L_{\mathcal{U}}\to L_{{\mathcal{P}}}\times {{\mathbb{C}}}, \quad
(x,\xi) \mapsto (\tilde{\rho}_{s,\pi(x)}(\xi), \pi(x))$$ where the pair $(x,\xi)$ consists of a point $x \in {\mathcal{U}}$ and $\xi \in
L_x$, is a diffeomorphism onto its image.
We recall the construction of $\tilde{\rho}_{s,t}$ from [@HamKon] and globalize it to the family ${{\mathcal X}}_{smooth}$. Let $\Theta\colon {{\mathcal X}}_{smooth}{\times}[0,\infty) \to {{\mathcal{P}}}{\times}{{\mathbb{C}}}$ be the map $(x,s)\mapsto \bigl( \rho_{s,\pi(x)}(x), \pi(x)\bigr)$. Let $(L',\nabla',h')=\Theta^*(L_{{{\mathcal{P}}}{\times}{{\mathbb{C}}}}, \nabla_{{{\mathcal{P}}}{\times}{{\mathbb{C}}}}, h_{{{\mathcal{P}}}{\times}{{\mathbb{C}}}})$. Note that $L'\vert_{{{\mathcal X}}_{smooth}{\times}\{s_0\}} = \rho_{s_0}^* L_{{{\mathcal X}}}$.
Let $\mathcal{Z}\in \operatorname{Vect}(L')$ be the horizontal lift to $L'$ with respect to $\nabla'$ of the vector field ${\frac{{\partial}}{{\partial}s}}$ on ${{\mathcal X}}_{smooth}{\times}[0,\infty)$. Then the flow of $\mathcal{Z}$ through time $s$ induces a diffomorphism between the bundles $\rho_0^* L_{{{\mathcal X}}}$ and $\rho_{s}^* L_{{{\mathcal X}}}$ over ${{\mathcal X}}_{smooth}{\times}\{0\}$ and ${{\mathcal X}}_{smooth}{\times}\{s\}$. This is the same as a diffeomorphism $L_{{{\mathcal X}}} \to L_{\rho_{s_0}({{\mathcal X}})}$ lifting the map $\rho_{s_0}$, and we denote it by $\tilde{\rho}_s$. Since $\mathcal{U}_{[0,1]}$ is a subset of ${{\mathcal X}}_{smooth}$ by construction, the map $\tilde{\rho}_s$ is defined on $L_{\mathcal{U}}$, and is a diffeomorphism onto its image.
If we restrict the map $\Theta$ to the (smooth locus in the) fibre $X_t{\times}[0,\infty)$ we obtain a map $\Theta_t \colon (x,s) \mapsto \rho_{s,t}(x)$, which agrees with the map used in [@HamKon Claim 6.4] to construct the lift of $\rho_{s,t}$ over the submanifold $X_t$. Furthermore, it is clear that $\Theta_t^*(L_{{{\mathcal{P}}}{\times}{{\mathbb{C}}}}, \nabla_{{{\mathcal{P}}}{\times}{{\mathbb{C}}}}, h_{{{\mathcal{P}}}{\times}{{\mathbb{C}}}})$ restricts to $(L',\nabla',h')$ on $X_t{\times}[0,\infty)$; this agrees with the data used in the construction in [@HamKon]. Therefore the lifting $\tilde{\rho_s}$ constructed above agrees on $U_t$ with the map $\tilde{\rho}_{s,t}$ as constructed in [@HamKon], and the formula given in agrees with the map $\tilde{\rho}_s$ constructed in the previous paragraph.
Finally, we analyze the behavior of the diffeomorphisms $\chi_s: {{\mathcal{P}}}\to {{\mathcal{P}}}$ of with respect to the prequantum data. Recall that $L_{{\mathcal{P}}}$ is a holomorphic line bundle with respect to the canonical complex structure $J_{{\mathcal{P}}}$ on ${{\mathcal{P}}}$ (i.e. its transition functions are holomorphic), and hence there exists a differential operator $\overline{\partial}$ defining the space of holomorphic sections $H^0({{\mathcal{P}}}, L_{{\mathcal{P}}}, \overline{\partial})$ of $L_{{\mathcal{P}}}$ over $({{\mathcal{P}}}, J_{{\mathcal{P}}})$. We recall the following [@HamKon Theorem 5.3(A)].
([@HamKon Theorem 5.3(A)]) \[lemma:chi-tilde\] There exists a lift $\tilde{\chi}_s$ of $\chi_s$ to an isomorphism of the line bundle $L_{{\mathcal{P}}}$ such that the following diagram commutes $$\xymatrix{
L_{{\mathcal{P}}}\ar[d] \ar[r]^{\tilde{\chi}_{s}} & L_{{\mathcal{P}}}\ar[d] \\
{{\mathcal{P}}}\ar[r]^{\chi_{s}} & {{\mathcal{P}}}\\
}$$ and such that the connection $\nabla_{{\mathcal{P}}}$ is the canonical Chern connection for the Hermitian holomorphic line bundle $(L_{{\mathcal{P}}}, h_{{\mathcal{P}}},
\tilde{\chi}_s^*\overline{\partial})$.
From now on we notate $$\overline{\partial}_s := \tilde{\chi}_s^*(\overline{\partial})$$ and denote by $H^0({{\mathcal{P}}}, L_{{\mathcal{P}}}, \overline{\partial}_s)$ the corresponding space of sections. From the definitions of the respective holomorphic structures, it is immediate that the pullback by $\tilde{\chi}_s^*$ of a section which is holomorphic with respect to $\overline{\partial}$ is holomorphic with respect to $\overline{\partial}_s$.
The pullback $\tilde{\chi}_s^*(\sigma)$ of a section $\sigma \in H^0({{\mathcal{P}}}, L_{{\mathcal{P}}}, \overline{\partial})$ is an element of $H^0({{\mathcal{P}}}, L_{{\mathcal{P}}},
\overline{\partial}_s)$.
In fact, in our arguments below we will need to pull back sections to the original variety $X$ via the diagram $$\label{eq:big diagram for pulling back sections}
\xymatrix{
L \ar[r]^{\tilde{\phi}_{1-t}} \ar[d] & L_t \ar[r]^{\tilde{\rho}_{s,t}}
\ar[d] & L_{{\mathcal{P}}}\ar[d] \ar[r]^{\tilde{\chi}_{s}} & L_{{\mathcal{P}}}\ar[d] \\
X \ar[r]^{\phi_{1-t}} & X_t \ar[r]^{\rho_{s,t}} & {{\mathcal{P}}}\ar[r]^{\chi_{s}} & {{\mathcal{P}}}\\
}$$ obtained by composing the three diagrams above. We record the following.
Let $s, t \in {{\mathbb{R}}}$ with $0 \leq s< \infty$ and $0
\leq t < 1$. Following notation as above, the pullbacks $\tilde{\chi}^*_s$, $\tilde{\rho}_{s,t}^*$ and $\tilde{\phi}^*_{1-t}$ preserve holomorphic sections, so in particular there is a well-defined map $$\tilde{\phi}^*_{1-t} \circ \tilde{\rho}_{s,t}^* \circ
\tilde{\chi}^*_s: H^0({{\mathcal{P}}}, L_{{\mathcal{P}}}, \overline{\partial}_0) \to H^0(X, L,
\overline{\partial}_{s,t})$$ where $\overline{\partial}_0$ denotes the standard holomorphic structure on $({{\mathcal{P}}}={\mathbb{P}}^N, \omega_{{\mathcal{P}}}, J_{{\mathcal{P}}})$.
The differential operator $\overline{\partial}_{s,t}$ is associated to the complex structure $\tilde{\phi}_{1-t}^* \tilde{\rho}_{s,t}^* \tilde{\chi}_s^*(J_{{\mathcal{P}}})$ obtained by pullback, so the result is immediate from the definitions.
We also record the important fact that the Bohr-Sommerfeld fibres of the integrable system on $X$ correspond to those on $X_0$. In particular, the Bohr-Sommerfeld set is the set $W_0$ defined in .
\[prop:w0-bohr-s\] Let $W_0 := \iota^*(\Delta_{{\mathcal{P}}}\cap {{\mathbb{Z}}}^N) \subseteq \Delta_0 \cap {{\mathbb{Z}}}^n$ as defined in . Then the Bohr-Sommerfeld fibres in $X$ are precisely the preimages of points in $W_0$ under the integrable system $\mu$ constructed in .
Since the gradient-Hamiltonian flow lifts to the line bundle preserving the connection, it follows that the gradient-Hamiltonian flow maps Bohr-Sommerfeld fibres in $X$ to Bohr-Sommerfeld fibres in $X_0$. Since the Bohr-Sommerfeld fibres of the torus moment map are the preimages of $W_0$, and the integrable system $\mu\colon X \to {{\mathbb{R}}}^n$ was constructed by pulling back the moment map for the torus action on $X_0$, we obtain the result.
Varying bases of sections $\{\sigma^m_{s,t}\}$ of $H^0(X, L, \overline{\partial}_{s,t})$ {#subsec:varying bases}
----------------------------------------------------------------------------------------
Our main result, Theorem \[theorem:main\], asserts the existence of a basis of sections $\{\sigma^m_s\}$, indexed by $m \in W_0$ and dependent on a real parameter $s$, where each $\sigma^m_s$ is holomorphic with respect to the complex structure $J_s$. In this section, we take a step in this direction by using the results of Section \[subsec:lifts\] to define sections $\sigma^m_{s,t}$ in $H^0(X,L,\overline{\partial}_{s,t})$.
The sections $\sigma^m_{s,t}$ are constructed using certain standard sections of $({{\mathcal{P}}}, L_{{\mathcal{P}}})$ which we now recall. It is well-known that ${{\mathcal{P}}}$ is a toric variety with respect to the standard action of its torus $T_{{\mathcal{P}}}$. For any integer lattice point in $\Delta_{{\mathcal{P}}}\cap {{\mathbb{Z}}}^N$, there is a well-known method (see e.g. [@Ham-toric]) to associate to it a holomorphic section in $H^0({{\mathcal{P}}}, L_{{\mathcal{P}}}, \overline{\partial}_0)$. In fact, this association yields a bijective correspondence between $\Delta_{{\mathcal{P}}}\cap {{\mathbb{Z}}}^N$ and a basis for $H^0({{\mathcal{P}}}, L_{{\mathcal{P}}}, \overline{\partial}_0)$ which we denote as $$\tilde{m} \in \Delta_{{\mathcal{P}}}\cap {{\mathbb{Z}}}^N \mapsto \sigma^{\tilde{m}} \in
H^0({{\mathcal{P}}}, L_{{\mathcal{P}}}, \overline{\partial}_0).$$ Now recall that $W_0 := \iota^*(\Delta_{{\mathcal{P}}}\cap {{\mathbb{Z}}}^N)$ is defined to be precisely the lattice points in $\Delta_0$ which lie in the image of $\iota^*$ of $\Delta_{{\mathcal{P}}}\cap {{\mathbb{Z}}}^N$. Thus for any $m \in W_0$, by assumption there exists a preimage $\tilde{m}$ of $m$ under $\iota^*$.
We can now define our set of sections $\sigma^m_{s,t}$. The sections depend on a choice of preimage $\tilde{m}$ for each $m \in W_0$, but their essential properties - such as those asserted in Theorem \[theorem:main\] - are independent of these choices. For this reason and for simplicity we suppress this choice from the notation. Specifically, we have the following. (It may be helpful for the reader to refer to the diagram .)
\[definition:sigma s t\] For each $m \in W_0$, let $\tilde{m} \in \Delta_{{\mathcal{P}}}\cap {{\mathbb{Z}}}^N$ denote a fixed choice of preimage of $m$ under $\iota^*$. Let $s, t \in {{\mathbb{R}}}$ with $0 \leq s < \infty$ and $0 < t \leq 1$. We define $$\label{eq:def sections}
\sigma^{m}_{s,t} := \tilde{\phi}^*_{1-t} \tilde{\rho}^*_{s,t}
\tilde{\chi}^*_s \sigma^{\tilde{m}} \in H^0(X, L,
\overline{\partial}_{s,t}).$$
In the next section we will find an appropriate function $t=t(s)$ so that the bases $\{\sigma^m_s :=
\sigma^m_{s,t=t(s)}\}$ will satisfy the convergence conditions asserted in Theorem \[theorem:main\] with respect to the complex structures $J_s
:= J_{s,t=t(s)}$ and $\overline{\partial}_s :=
\overline{\partial}_{s,t=t(s)}$.
Proof of the main theorem {#sec:proof}
=========================
We now proceed to a proof of the main result of this manuscript, Theorem \[theorem:main\]. Much of this section is devoted to proving results about the sections $\sigma^m_{s,t}$ defined in Section \[subsec:varying bases\], under certain hypotheses on the parameters $s$ and $t$. At the end of this section we choose an appropriate function $t=t(s)$ so that the sections $\sigma^m_s := \sigma^m_{s,t(s)}$ depend only on the single parameter $s$ and have the correct convergence properties.
We begin our discussion with a statement about supports. Specifically, part of the assertion of Theorem \[theorem:main\] is that a certain (normalized) section weakly converges to a dirac-delta function on the corresponding Bohr-Sommerfeld fiber. In particular, the support of $\sigma^m_{s,t=t(s)}$ must concentrate on a neighborhood of the Bohr-Sommerfeld fiber as $s$ gets large. We make this precise in the proposition below, for which we need some preliminaries. Let $m \in W_0 \subset {\mathfrak{t}}_0^*$. For a real number $\eta>0$, let $B_\eta(m)$ denote the open ball of radius $\eta$ around $m$ with respect to the usual metric on ${\mathfrak{t}}_0^*$. We introduce the following notation: $$\label{eq:definition B_eta}
B_\eta(m) := \mu^{-1}(B_\eta(m)), \quad
B_{\eta,t}(m) := \mu_t^{-1}(B_\eta(m)), \quad
B_{\eta,0}(m) := \mu_0^{-1}(B_\eta(m))$$ where $\mu, \mu_t$ and $\mu_0$ are the moment maps for the integrable systems on $X, X_t$ and $X_0$ respectively. When the point $m$ is clear from context, we sometimes write $B_{\eta,t} =
B_{\eta,t}(m)$, etc. We also let $d(vol)$ denote the symplectic volume form on $X_t$ and $\rho_{s,t}(X_t)$ for all $s$ and $t$; since the relevant maps between these spaces preserve symplectic structures, the ambiguity in this notation does not pose problems. We let ${\lvert\cdot\rvert}$ denote the norm with respect to the hermitian metric on all the line bundles; again, all relevant maps preserve the hermitian metric so there is no ambiguity. We let ${\lVert\cdot\rVert}_{L^1(\cdot)}$ denote the $L^1$-norm of a section over some space; for the sake of space and readability we will occasionally omit the explicit mention of the space in the notation and write simply ${\lVert\cdot\rVert}$.
We can now state and prove the following.
\[prop:support on fibers\] Let $m \in W_0$ be an interior point and let $\sigma^m_{s,t} \in H^0(X, L,
\overline{\partial}_{s,t})$ be the section defined in . Then there exists a continuous function $t' = t'(s): [0,\infty) \to [0,1]$ such that for every $\epsilon>0$ and $\eta>0$, there exists $s_0 > 0$ such that $$\label{eq:support on fibers estimate}
\int_{X \setminus B_\eta} \bigg\lvert \frac{\sigma^m_{s, t}}
{{\lVert\sigma^m_{s, t}\rVert}_{L^1(X)}} \bigg\rvert d(vol) < \epsilon$$ for all $s > s_0$ and $0 \leq t \leq t'(s)$, and moreover, $\lim_{s
\to \infty} t'(s) = 0$.
We believe the result of Proposition \[prop:support on fibers\] holds for boundary points as well as interior points but we restrict ourselves to interior points for the purposes of this paper.
The proof of Proposition \[prop:support on fibers\] requires several steps, the first of which states that the analogous result is true for the special fiber $X_0$. We quote the following.
\[lemma:support-section-converges-X0\] ([@HamKon Proposition 6.6, (3) and (4)]) Let $m \in W_0$ be an interior point and let $\tilde{m} \in \Delta_{{\mathcal{P}}}\cap {{\mathbb{Z}}}^N$ denote the preimage of $m$ fixed in Definition \[definition:sigma s t\]. For any $\epsilon>0$ and $\eta>0$, there exists $s_0>0$ such that $$\int_{X_0 \setminus B_{\eta,0}} \bigg\lvert
\frac{\tilde{\chi}^*_s(\sigma^{\tilde{m}})}
{{\lVert\tilde{\chi}^*_s(\sigma^{\tilde{m}})\rVert}_{L^1(X_0)}}
\bigg\rvert d(vol) < \epsilon$$ for all $s>s_0$.
The following estimate will also be useful. Roughly, it says that for any fixed $s>0$ and any $\epsilon>0$, there are fibers $X_t$ of the family which are sufficiently close to $X_0$ such that the two maps $\rho_{s,t}$ and $\phi_t$ do not differ on $X_t$ by more than distance $\epsilon$.
\[lemma:rho-st-close\] Let $s, t \in {{\mathbb{R}}}$ with $0 < s < \infty$ and $0 \leq t
\leq 1$, let $\rho_{s,t}: X_t \to {{\mathcal{P}}}$ be the embedding in Proposition \[prop:embedding of submanifold\], let $\phi_t \colon X_t \to X_0$ denote the gradient-Hamiltonian flow, and let $\epsilon>0$. Then there exists $t_0>0$ such that for any $x\in X_t$ with $0<t<t_0$, $$d_{{\mathcal{P}}}(\rho_{s,t}(x), \phi_{t}(x))<\epsilon$$ where $d_{{\mathcal{P}}}$ denotes the distance function on ${{\mathcal{P}}}$ induced from the Kähler metric on $({{\mathcal{P}}}, \omega_{{\mathcal{P}}}, J_{{\mathcal{P}}})$.
First, we show that we can choose $t$ small enough that $\phi_t(x)$ is close to $x$, uniformly in $X_t$. Note that this part of the argument is independent of the parameter $s$. Let $$B := \Bigl\{ (x,t) \in {{\mathcal X}}{\times}[0,1] {\mid}\pi(x) \in [0,1],\
t\leq \pi(x) \Bigr\}
\subseteq {{\mathcal{P}}}\times [0,1] \times [0,1] \subseteq {{\mathcal{P}}}\times {{\mathbb{C}}}\times [0,1].$$ Then $B$ is a closed subset of the compact space ${{\mathcal{P}}}{\times}[0,1]
\times [0,1]$, and is therefore compact. Consider the map $\Psi: B \to B$ given by $\Psi(x,t) = (\phi_t(x),
\pi(x)-t)$. Note that $\Psi$ is well-defined since $t \leq \pi(x)$ by assumption so $\pi(x)-t \geq 0$. It follows from [@HarKav Part 1, Theorem 4.1] that $\Psi$ is continuous as a function from $B$ to itself, and hence uniformly continous. In particular, this implies that for any $\delta>0$, there exists a $t_0>0$ such that for any $t<t_0$ we have $d_{{\mathcal{P}}}(\phi_t(x), x) <\delta$ for any $x \in X_t$.
We next analyze the embeddings $\rho_{s,t} \colon X_t {\hookrightarrow}\mathcal{P}$. Recall that ${{\mathcal X}}_{[0,1]}$ denotes $\pi{^{-1}}([0,1])\subset {{\mathcal X}}$. For a fixed $s$, let $f_s$ denote the map ${{\mathcal X}}_{[0,1]} \to {{\mathcal{P}}}$ given by $x \mapsto \rho_{s,\pi(x)}(x)$. Then $f_s$ is continuous by Proposition \[prop:rho-smooth-in-t\], and therefore also uniformly continuous since ${{\mathcal X}}_{[0,1]}$ is compact. Recall from Proposition \[prop:toric submanifolds don’t move\] that $\rho_{s,0}=\operatorname{id}$ for all $s$. Thus for any $x \in X_t$ we have $$f_s(\phi_t(x)) = \rho_{s,0}(\phi_t(x)) = \phi_t(x)$$ since $\phi_t(x) \in X_0$. Now let $\epsilon>0$ be given. Choose $\delta>0$ such that $d_{{\mathcal{P}}}(x,y)<\delta$ implies $d_{{\mathcal{P}}}(f_s(x),f_s(y))<\epsilon$ for any $x$, $y \in {{\mathcal X}}_{[0,1]}$. Then choose $t_0$ so that $0<t<t_0$ implies $d_{{\mathcal{P}}}(\phi_t(x),x)<\delta$ for all $x \in X_t$, as above. Then for all $x\in X_t$ with $0<t<t_0$, $$d_{{\mathcal{P}}}(\rho_{s,t}(x) , \phi_t(x)) =
d_{{\mathcal{P}}}(f_s(x) , f_s(\phi_t(x))) < \epsilon$$ as required.
We are now ready to prove Proposition \[prop:support on fibers\]. The idea of the proof is to combine two separate estimates, as we now sketch. On the one hand, we will use Lemma \[lemma:rho-st-close\] to argue that we can make the integral over $X$ close to the analogous integral over $X_0$. On the other hand, we know from Lemma \[lemma:support-section-converges-X0\] that the integral over $X_0$ can be made arbitrarily small. Putting these estimates together gives the result. We make this precise in the proof below.
We first note that by Lemma \[lemma:support-section-converges-X0\] there exists $s_0>0$ such that for any $s>s_0$ we have $$\label{eq:integral on X_0}
\int_{X_0 \setminus B_{\eta,0}} \bigg\lvert
\frac{\tilde{\chi}^*_s(\sigma^{\tilde{m}})}
{\|\tilde{\chi}^*_s(\sigma^{\tilde{m}})\|_{L^1(X_0)}}
\bigg\rvert {{\hspace{1mm}}}d(vol) <
\frac{\epsilon}{2}.$$
Next we notice that since $\tilde{\chi}_s^*(\sigma^{\tilde{m}})$ is a holomorphic section of a hermitian line bundle, its norm ${\lvert\tilde{\chi}_s^*(\sigma^{\tilde{m}})\rvert}$ is a continuous function on ${{\mathcal{P}}}$. Since $\rho_{s,t}$ is continuous in $t$, as noted in Proposition \[prop:rho-smooth-in-t\], the function $t\mapsto {\lVert\tilde{\chi}_s^*\sigma^{\tilde{m}}\circ \rho_{s,t}\rVert}_{L^1(X_t)}$ is a continous function on $t$. Define $C_0$ and $C_1$, respectively, to be the minumum and maximum values of ${\lVert\tilde{\chi}_s^* \sigma^{\tilde{m}} \circ \rho_{s,t}\rVert}_{L^1(X_t)}$ for $t\in [0,1]$, and note that $C_0\neq 0$ for sufficiently small $t$ because $\tilde{\chi}_s^*(\sigma^{\tilde{m}})$ is not identically zero [@HamKon Theorem 5.3 (a5)], and thus $\tilde{\chi}_s^* \sigma^{\tilde{m}} \circ \rho_{s,t}$ is not the zero section for sufficiently small $t$.
Similarly, since $\tilde{\chi}_s^*(\sigma^{\tilde{m}})$ is a holomorphic section of a hermitian line bundle on the compact set ${{\mathcal{P}}}$, its norm ${\lvert\tilde{\chi}_s^*(\sigma^{\tilde{m}})\rvert}$ is a continuous, and hence uniformly continuous, function on ${{\mathcal{P}}}$. Thus for any $\epsilon>0$, there exists a $\delta>0$ such that if $d_{{\mathcal{P}}}(x,x') < \delta$ for $x, x' \in {{\mathcal{P}}}$, then $$\bigg\lvert {\lvert\tilde{\chi}_s^*(\sigma^{\tilde{m}})\rvert}(x)
- {\lvert\tilde{\chi}^*_s(\sigma^{\tilde{m}})\rvert} (x') \bigg\rvert <
\frac{\epsilon C_0 \|\tilde{\chi}_s^* \sigma^{\tilde{m}}\|_{L^1(X_0)}}
{4 C_1 \operatorname{vol}(X_t)}.$$ Moreover, by Lemma \[lemma:rho-st-close\], for a fixed $s$ with $s>s_0$ as above, we know there exists $t_0=t_0(s)>0$ such that for any $t$ with $0<t<t_0=t_0(s)$ and any $x \in X_t$ we have $$d_{{\mathcal{P}}}(\rho_{s,t}(x), \phi_t(x)) < \delta.$$ For what follows we will also choose $t_0(s)$ sufficiently small so that $\tilde{\chi}_s^*(\sigma^{\tilde{m}}) \circ \rho_{s,t}$ is not identically zero for $0 < t < t_0$, so in particular $C_0 > 0$. Now, we have that for all $x \in X_t$ with $0 < t < t_0=t_0(s)$, $$\label{eq:sigma-m-s-close}
\bigg\lvert
{\bigl\lvert\tilde{\chi}_s^*(\sigma^{\tilde{m}})\bigr\rvert}\bigl(\rho_{s,t}(x)\bigr)
- {\bigl\lvert\tilde{\chi}^*_s(\sigma^{\tilde{m}})\bigr\rvert}\bigl(\phi_{t}(x)\bigr) \bigg\rvert
< \frac{\epsilon C_0 \|\tilde{\chi}_s^* \sigma^{\tilde{m}}\|_{L^1(X_0)}}
{4 C_1 \operatorname{vol}(X_t)}
\leq \frac{\epsilon \|\tilde{\chi}_s^* \sigma^{\tilde{m}}\|_{L^1(X_0)}}{4 \operatorname{vol}(X_t)}$$ where the last inequality is because $C_0/C_1\leq 1$.
Next we recall that the sections $\sigma^m_{s,t}$ on $X$ in Definition \[definition:sigma s t\] are given by a sequence of pullbacks. In particular, since both $\tilde{\phi}_{1-t}$ and $\tilde{\rho}_{s,t}$ preserve the hermitian metric and $\phi_{1-t}$ preserves symplectic structures, we have that $$\label{eq:pullback integrals}
\int_{X \setminus B_\eta} {\lvert\sigma^m_{s,t}\rvert} {{\hspace{1mm}}}d(vol) =
\int_{X_t\setminus B_{\eta,t}}
{\bigl\lvert\tilde{\rho}^*_{s,t} \tilde{\chi}^*_s(\sigma^{\tilde{m}})\bigr\rvert} {{\hspace{1mm}}}d(vol) =
\int_{X_t \setminus B_{\eta,t}} {\bigl\lvert\tilde{\chi}_s^*(\sigma^{\tilde{m}})\bigr\rvert}
\circ \rho_{s,t} {{\hspace{1mm}}}d(vol),$$ where we also use that $\phi_{1-t}^{-1}(B_{\eta, t}) = B_\eta$ by construction of the moment maps $\mu_t$; for the same reason, the $L^1$-norms satisfy $$\label{eq:sigma^m norm equality}
\big\| \sigma^m_{s,t} \big\|_{L^1(X)} =
\int_{X} {\lvert\sigma^m_{s,t}\rvert} {{\hspace{1mm}}}d(vol) =
\int_{X_t} {\bigl\lvert\tilde{\chi}_s^*(\sigma^{\tilde{m}})\bigr\rvert} \circ \rho_{s,t}
{{\hspace{1mm}}}d(vol) =
\big\|\tilde{\chi}_s^*(\sigma^{\tilde{m}}) \circ \rho_{s,t}\big\|_{L^1(X_t)}.$$ Similarly, since $\phi_t$ and $\tilde{\phi}_t$ preserve the relevant structures we have $$\label{eq:pullbacks to X_0 and X_t}
\int_{X_0 \setminus B_{\eta,0}}
{\bigl\lvert\tilde{\chi}_s^*(\sigma^{\tilde{m}})\bigr\rvert} {{\hspace{1mm}}}d(vol) =
\int_{X_t\setminus B_{\eta,t}}
{\bigl\lvert\tilde{\phi}_t^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})\bigr\rvert}{{\hspace{1mm}}}d(vol) =
\int_{X_t\setminus B_{\eta,t}} {\bigl\lvert\tilde{\chi}_s^*(\sigma^{\tilde{m}})\bigr\rvert}
\circ \phi_t {{\hspace{1mm}}}d(vol).$$ In this case, the $L^1$-norms satisfy $$\label{eq:norms-phi-t}
\big\|\tilde{\chi}_s^*(\sigma^{\tilde{m}}) \big\|_{L^1(X_0)} =
\int_{X_0} {\bigl\lvert\tilde{\chi}_s^*(\sigma^{\tilde{m}})\bigr\rvert} {{\hspace{1mm}}}d(vol)
= \int_{X_t} {\bigl\lvert\tilde{\chi}_s^*(\sigma^{\tilde{m}})\bigr\rvert}
\circ \phi_t {{\hspace{1mm}}}d(vol)
= \big \| \tilde{\chi}_s^*(\sigma^{\tilde{m}}) \circ \phi_t \big\|_{L^1(X_t)};$$ however, because $\phi_t$ preserves the symplectic structure, the above norms are equal for all $t\in [0,1]$.
Because of the normalizing factors in the denominators, we will need to show that ${\lVert\tilde{\chi}_s^* \sigma^{\tilde{m}}\circ \rho_{s,t}\rVert}_{L^1(X_t)}$ is close to ${\lVert\tilde{\chi}_s^* \sigma^{\tilde{m}}\circ \phi_t\rVert}_{L^1(X_t)}$, which we do as follows: For $s>s_0$ and for all $x$ in $X_t$, $0<t\leq t_0(s)$, $$\label{eq:estimate-on-norms}
\begin{split}
\Big\lvert {\lVert\tilde{\chi}_s^* \sigma^{\tilde{m}}\circ \rho_{s,t}\rVert}
- {\lVert\tilde{\chi}_s^* \sigma^{\tilde{m}}\circ \phi_t\rVert}
\Big\rvert
& \leq
\int_{X_t}
\Big\lvert {\bigl\lvert\tilde{\chi}_s^* \sigma^{\tilde{m}}\circ \rho_{s,t}\bigr\rvert}
- {\bigl\lvert\tilde{\chi}_s^* \sigma^{\tilde{m}}\circ \phi_t\bigr\rvert}
\Big\rvert {{\hspace{1mm}}}d(vol) \\
& \leq
\int_{X_t} \frac{\epsilon C_0 \|\tilde{\chi}_s^* \sigma^{\tilde{m}}\|_{L^1(X_0)}}
{4 C_1 \operatorname{vol}(X_t)} {{\hspace{1mm}}}d(vol)
= \frac{\epsilon C_0 \|\tilde{\chi}_s^* \sigma^{\tilde{m}}\|_{L^1(X_0)}}{4 C_1}
\end{split}$$ where the last inequality comes from .
Now, using the fact that $${\left\lvert\frac{a}{h} - \frac{b}{k}\right\rvert} = {\left\lvert\frac{a(k-h) + h(a-b)}{hk}\right\rvert}
\leq \frac{{\lverta\rvert}{\lvertk-h\rvert}}{{\lverthk\rvert}} + \frac{{\lverta-b\rvert}}{{\lvertk\rvert}}$$ we obtain that, for our fixed value of $s>s_0$ and for all $x \in X_t$ with $0<t\leq t_0(s)$, the expression $$\left\lvert
\frac{{\lvert\tilde{\chi}_s^*\sigma^{\tilde{m}}\rvert}\bigl(\rho_{s,t}(x)\bigr)}
{{\lVert\tilde{\chi}_s^* \sigma^{\tilde{m}}\circ \rho_{s,t}\rVert}}
- \frac{{\lvert\tilde{\chi}_s^* \sigma^{\tilde{m}}\rvert}\bigl(\phi_t(x)\bigr)}
{{{\lVert\tilde{\chi}_s^* \sigma^{\tilde{m}}\circ \phi_t\rVert}}}
\right\rvert$$ is less than or equal to $$\label{eq:huge-ugly-mess-with-normalizations}
\begin{split}
\frac{{\bigl\lvert\tilde{\chi}_s^* \sigma^{\tilde{m}}\bigl(\rho_{s,t}(x)\bigr)\bigr\rvert}}
{{\lVert\tilde{\chi}_s^* \sigma^{\tilde{m}}\circ \rho_{s,t}\rVert}
{\lVert\tilde{\chi}_s^* \sigma^{\tilde{m}}\circ \phi_t\rVert}} \;
\Big\lvert
{\lVert\tilde{\chi}_s^* \sigma^{\tilde{m}}\circ \rho_{s,t}\rVert}
- {\lVert\tilde{\chi}_s^* \sigma^{\tilde{m}}\circ \phi_t\rVert}
\Big\rvert
+ \frac{\Bigl\lvert
{\bigl\lvert\tilde{\chi}_s^*(\sigma^{\tilde{m}})\bigr\rvert}\bigl(\rho_{s,t}(x)\bigr)
- {\bigl\lvert\tilde{\chi}^*_s(\sigma^{\tilde{m}})\bigr\rvert}\bigl(\phi_{t}(x)\bigr)
\Bigr\rvert}
{{{\lVert\tilde{\chi}_s^* \sigma^{\tilde{m}}\circ \phi_t\rVert}}}.
\end{split}$$ Using on the first term, the second estimate in on the second term, the fact that ${\bigl\lvert\tilde{\chi}_s^*(\sigma^{\tilde{m}})\bigl(\rho_{s,t}(x)\bigr)\bigr\rvert} \leq
\frac{C_1}{\operatorname{vol}(X_t)}$ by the definition of $C_1$, the fact that $C_0$ is the minimum value of ${\lVert\tilde{\chi}_s^*\sigma^{\tilde{m}} \circ \rho_{s,t}\rVert}$ (and hence $\frac{1}{{\lVert\tilde{\chi}_s^*\sigma^{\tilde{m}} \circ \rho_{s,t}\rVert}} \leq \frac{1}{C_0}$), and the equality , this becomes $$\label{eq:normalized-estimate-rho-phi-sigma}
\begin{split}
\left\lvert
\frac{{\bigl\lvert\tilde{\chi}_s^*\sigma^{\tilde{m}}\bigr\rvert}\bigl(\rho_{s,t}(x)\bigr)}
{{\lVert\tilde{\chi}_s^* \sigma^{\tilde{m}}\circ \rho_{s,t}\rVert}}
- \frac{{\bigl\lvert\tilde{\chi}_s^* \sigma^{\tilde{m}}\bigr\rvert}\bigl(\phi_t(x)\bigr)}
{{{\lVert\tilde{\chi}_s^* \sigma^{\tilde{m}}\circ \phi_t\rVert}}}
\right\rvert
& \leq
\frac{C_1}{C_0 \operatorname{vol}(X_t)
{\lVert\tilde{\chi}_s^* \sigma^{\tilde{m}}\circ \phi_{t}\rVert}}
\frac{\epsilon C_0 \|\tilde{\chi}_s^* \sigma^{\tilde{m}}\|}{4 C_1}
+
\frac{1}{{{\lVert\tilde{\chi}_s^* \sigma^{\tilde{m}}\circ \phi_t\rVert}}}
\frac{\epsilon \|\tilde{\chi}_s^* \sigma^{\tilde{m}}\|}{4 \operatorname{vol}(X_t)} \\
& = \frac{\epsilon}{2\operatorname{vol}(X_t)}.
\end{split}$$
Putting everything together, for $s>s_0$ and $t<t_0$ for the chosen $s_0, t_0(s)$ as above, we have $$\label{eq:final estimate for supports}
\begin{split}
\int_{X \setminus B_\eta}
\frac{{\lvert\sigma^m_{s,t}\rvert}}{{\lVert\sigma^m_{s,t}\rVert}} {{\hspace{1mm}}}d(vol) & =
\int_{X \setminus B_\eta}
\frac{{\lvert\sigma^m_{s,t}\rvert}}{{\lVert\sigma^m_{s,t}\rVert}} {{\hspace{1mm}}}d(vol) -
\int_{X_0 \setminus B_{\eta,0}}
\frac{{\lvert\tilde{\chi}^*_s(\sigma^{\tilde{m}})\rvert}}
{{\lVert\tilde{\chi}^*_s(\sigma^{\tilde{m}})\rVert}}
{{\hspace{1mm}}}d(vol) +
\int_{X_0 \setminus B_{\eta,0}}
\frac{{\lvert\tilde{\chi}^*_s(\sigma^{\tilde{m}})\rvert}}
{{\lVert\tilde{\chi}^*_s(\sigma^{\tilde{m}})\rVert}}
{{\hspace{1mm}}}d(vol) \\
& = \int_{X_t \setminus B_{\eta,t}}
\frac{{\lvert\tilde{\chi}^*_s(\sigma^{\tilde{m}})\rvert} \circ \rho_{s,t}}
{{\bigl\lVert\tilde{\chi}_s^*(\sigma^{\tilde{m}}) \circ \rho_{s,t}\bigr\rVert}}
{{\hspace{1mm}}}d(vol) -
\int_{X_t \setminus B_{\eta,t}}
\frac{ {\lvert\tilde{\chi}^*_s(\sigma^{\tilde{m}})\rvert} \circ \phi_t}
{{\bigl\lVert\tilde{\chi}^*_s(\sigma^{\tilde{m}})\circ \phi_t\bigr\rVert}}
{{\hspace{1mm}}}d(vol)
+ \int_{X_0 \setminus B_{\eta,0}}
\frac{{\lvert \tilde{\chi}^*_s(\sigma^{\tilde{m}}) \rvert}}
{{\bigl\lVert\tilde{\chi}^*_s(\sigma^{\tilde{m}})\bigr\rVert}}
{{\hspace{1mm}}}d(vol) \\
& \leq \bigg\lvert
\int_{X_t \setminus B_{\eta,t}}
\frac{{\lvert\tilde{\chi}^*_s(\sigma^{\tilde{m}})\rvert} \circ \rho_{s,t}}
{{\bigl\lVert\tilde{\chi}_s^*(\sigma^{\tilde{m}}) \circ \rho_{s,t}\bigr\rVert}}
{{\hspace{1mm}}}d(vol) -
\int_{X_t \setminus B_{\eta,t}}
\frac{{\lvert\tilde{\chi}^*_s(\sigma^{\tilde{m}})\rvert} \circ \phi_t}
{{\lVert\tilde{\chi}^*_s(\sigma^{\tilde{m}})\circ \phi_t\rVert}}
{{\hspace{1mm}}}d(vol) \bigg \rvert +
\int_{X_0 \setminus B_{\eta,0}}
\frac{{\lvert\tilde{\chi}^*_s(\sigma^{\tilde{m}})\rvert}}
{{\lVert\tilde{\chi}^*_s(\sigma^{\tilde{m}})\rVert}}
{{\hspace{1mm}}}d(vol) \\
& \leq
\int_{X_t \setminus B_{\eta,t}} \bigg \lvert
\frac{{\lvert\tilde{\chi}^*_s(\sigma^{\tilde{m}})\rvert} \circ \rho_{s,t}}
{{\bigl\lVert\tilde{\chi}_s^*(\sigma^{\tilde{m}}) \circ \rho_{s,t}\bigr\rVert}} -
\frac{{\lvert\tilde{\chi}^*_s(\sigma^{\tilde{m}})\rvert} \circ \phi_t}
{{\lVert\tilde{\chi}^*_s(\sigma^{\tilde{m}})\circ \phi_t\rVert}}
\bigg\rvert {{\hspace{1mm}}}d(vol) +
\int_{X_0 \setminus B_{\eta,0}}
\frac{{\lvert\tilde{\chi}^*_s(\sigma^{\tilde{m}})\rvert}}
{{\lVert\tilde{\chi}^*_s(\sigma^{\tilde{m}})\rVert}}
{{\hspace{1mm}}}d(vol) \\
& \leq
\int_{X_t \setminus B_{\eta,t}} \frac{\epsilon}{2 \operatorname{vol}(X_t)} {{\hspace{1mm}}}d(vol) +
\int_{X_0 \setminus B_{\eta,0}}
\frac{{\lvert\tilde{\chi}^*_s(\sigma^{\tilde{m}})\rvert}}
{{\lVert\tilde{\chi}^*_s(\sigma^{\tilde{m}})\rVert}} {{\hspace{1mm}}}d(vol)\\
& \leq \frac{\epsilon}{2 \operatorname{vol}(X_t)} \cdot \operatorname{vol}(X_t) +
\frac{\epsilon}{2} = \epsilon
\end{split}$$ as required, where the second equality uses , , , and , the second-to-last inequality uses , and the last inequality uses .
Finally, we wish to prove that there exists $t'=t'(s): [0, \infty) \to
[0,1]$ a continuous function of $s$ such that holds for $\sigma^m_{s,t}$ for all $s$ and all $t$ with $0 \leq t \leq t'(s)$, and also such that $t'(s) \to 0$ as $s \to
\infty$. To see this, notice first that immediately before we made a choice of $t_0(s)$ which depended on $s$. In the subsequent argument we proved statements that hold for all $t$ with $0<t<t_0$. In particular, these statements are still true if we replace $t_0$ by a smaller choice of $t_0$. From this it follows that we may assume without loss of generality that $t_0 = t_0(s)$ is a monotone non-increasing function of $s$.
A standard result from real analysis (see e.g. [@Tao Lemma 1.6.31(iii)]) says that a bounded, non-decreasing monotone function $F\colon {{\mathbb{R}}}\to {{\mathbb{R}}}$ can be written as $F=F_c + F_{pp}$, where $F_c$ is a continuous monotone non-decreasing function and $F_{pp}$ is a jump function. Looking at the definition of $F_{pp}$ in the proof in [@Tao], it is obvious that $F_{pp}$ is nonnegative; a little more thought shows that if $F(x)$ is nonnegative then $F_{pp}(x) \leq F(x)$ for all $x$, so if $F(x)$ is nonnegative then so is $F_c$. Taking $F(s) = t_0(-s)$ for $s\leq 0$ (and constant for $s\geq 0$), we can apply this decomposition to obtain that $t_0(s)$ for $s>0$ is the sum of a nonnegative jump function and a continuous positive non-increasing function $t'(s)$ which therefore satisfies $0\leq t'(s) \leq t_0(s)$. By shrinking $t'(s)$ if necessary, we can arrange that $t'(s)\to 0$ as $s\to\infty$. Since the inequalities at each stage are true for the chosen $s$ and for all $t$ with $0<t\leq t_0(s)$, they are true for $0<t<t'(s)$, and we are finished.
Next, the fiber $\mu^{-1}_0(m)$ is diffeomorphic to a torus and lies entirely within the open dense torus orbit of $X_0$, and thus it is possible to obtain much more refined information about the behavior of the family $\{\sigma^m_{s,t}\}$ in the limit. Specifically, let $\Gamma(X, L^*)$ denote the space of *smooth* (not necessarily holomorphic) sections of the dual complex line bundle and let $\langle \cdot, \cdot \rangle$ denote the usual pairing between $L^*$ and $L$. For $\sigma \in \Gamma(X,L^*)$ we let $\| \sigma \|_{L^1(X)}$ denote its $L^1$-norm with respect to the Hermitian metric on $L$, i.e. $\| \sigma \|_{L^1(X)} = \int_X \lvert \sigma \rvert {{\hspace{1mm}}}d(vol)$. We have the following.
\[proposition:interior\] Let $m \in \Delta_0 \cap {{\mathbb{Z}}}^n$ be an interior lattice point. Let $\tau \in \Gamma(X_1, {L}_1^*)$. Then there exist a covariantly constant section $\delta_m$ of $(L
\vert_{\mu^{-1}(m)}, \nabla \vert_{\mu^{-1}(m)})$, a measure $d\theta_m$ on $\mu^{-1}(m)$, and a continuous function $t=t(s)$ satisfying $\lim_{s \to
\infty} t(s)=0$ such that $$\label{eq:main}
\lim_{s \to \infty} \int_{X} \bigg\langle \tau, \frac{\sigma^m_{s,
t(s)}}{\| \sigma^m_{s,t(s)}\|_{L^1(X)}} \bigg \rangle {{\hspace{1mm}}}d(vol) =
\int_{\mu^{-1}(m)} \langle \tau, \delta_m \rangle {{\hspace{1mm}}}d\theta_m.$$
The idea of the proof is similar to that of Proposition \[prop:support on fibers\] above and requires a number of steps. Namely, we will relate the LHS of to a limit of integrals on $X_{t(s)}$ and then approximate the integral on $X_{t(s)}$ by one on $X_0$. We then use the fact that the analogous statement to Proposition \[proposition:interior\] is already known on $X_0$; this is the content of the following lemma.
\[lemma:estimate on X\_0\] Let $U_0$ denote the open dense ${{\mathbb{T}}}_0$-orbit in $X_0$ and let $m \in
\Delta_0 \cap {{\mathbb{Z}}}^n$ be an interior lattice point. Let $\tilde{m}$ be the fixed choice of preimage of $m$ under $\iota^*$ as in Definition \[definition:sigma s t\]. Then there exists a covariantly constant section $\delta_{m,0}$ of $(L_{{\mathcal{P}}}\vert_{\mu_0^{-1}(m)}, \nabla \vert_{\mu_0^{-1}(m)})$ over $\mu_0^{-1}(m)$ and a measure $d\theta_{m,0}$ on $\mu_0^{-1}(m)$ such that, for any smooth section $\tau \in \Gamma(U_0,
L^*_{{\mathcal{P}}}\vert_{V_0})$ of the dual line bundle, we have $$\lim_{s \to \infty} \int_{X_0}
\bigg\langle \tau, \frac{\tilde{\chi}^*_s(\sigma^{\tilde{m}})}
{\|\tilde{\chi}^*_s(\sigma^{\tilde{m}}) \|_{L^1(X_0)}} \bigg \rangle
{{\hspace{1mm}}}d(vol) =
\int_{\mu_0^{-1}(m)} \langle \tau, \delta_{m,0} \rangle {{\hspace{1mm}}}d\theta_{m,0}.$$
This is essentially the content of an argument given in [@HamKon]. Specifically, the $V_{symp}$ and $V_{comp}$ in the proof of [@HamKon Proposition 6.6(4)] can be identified with our $X_0$. Similarly, their $T^\ell_{{\mathbb{C}}}$ (respectively $T^n_{{\mathbb{C}}}$) is our ${{\mathbb{T}}}_0$ (respectively ${{\mathbb{T}}}_{{\mathcal{P}}}$). Finally, to apply the argument in [@HamKon] it is necessary that $X_0$ is the closure of the ${{\mathbb{T}}}_0$-orbit of $[1:1:\cdots:1]$ and that $\iota^*$ is surjective on lattices, which hold by our assumptions (e) and (g), respectively, as stated in Section \[sec-main-result\]. Thus, the argument of [@HamKon] applies.
In order to translate the previous lemma to a statement concerning other fibers, we need some additional information. The next lemma recalls some results from [@HarKav] and also constructs compact subsets $K_t$ which will be useful for obtaining estimates. Let $\Delta^{\circ}_0$ denote the interior of the moment polytope.
\[lemma:ut-kt\] Let $t \in [0,1]$. Then there exists an open subset $U_t \subseteq
X_t$ and a compact subset $K_t \subseteq U_t$ such that:
1. \[l:u0-smooth\] For $t=0$, $U_0$ equals $\mu_0^{-1}(\Delta^{\circ}_0) \subseteq X_0$, the open dense ${{\mathbb{T}}}_0$-orbit in $X_0$. In particular, $U_0$ lies in the smooth locus of $X_0$.
2. \[l:gr-H-ut\] The gradient-Hamiltonian flow $\phi_s\colon U_t \to U_{t-s}$ is a diffeomorphism for all $0\leq s\leq t$. In particular, $\phi_t$ is a diffeomorphism from $U_t$ to $U_0$.
3. \[l:gr-H-kt\]
The flow $\phi_s: U_t \to U_{t-s}$ in (2) identifies $K_t$ with $K_{t-s}$, i.e., $\phi_s(K_t) = K_{t-s}$ for all $0\leq s \leq t$.
4. \[l:k-contains-bs\] The subset $K_t$ contains a neighborhood of every interior Bohr-Sommerfeld fiber. More precisely, there exists some $\eta>0$ such that, for any $m \in \Delta^{\circ}_0\cap {{\mathbb{Z}}}^n$, the neighborhood $B_{\eta,t}(m)$ as in is contained in $K_t$.
In order to satisfy (1), we first define $U_0 :=
\mu_0^{-1}(\Delta^{\circ}_0)$. From [@HarKav Corollary 3.3] it follows that $U_0$ is contained in the locus of points in ${{\mathcal X}}$ where the gradient-Hamiltonian vector field is defined. Moreover, as in the proof of [@HarKav Theorem 5.2], we know the gradient-Hamiltonian flow is well-defined on all of $X_t$ for any $t \neq 0$. By [@HarKav Lemma 2.5] we may now define $U_t :=
\phi_{-t}(U_0)=\phi_t^{-1}(U_0)$ from which it is immediate that $\phi_s: U_t \to U_{t-s}$ is a diffeomorphism from $U_t$ to $U_{t-s}$ (with inverse $\phi_{-s}$) for any $0 \leq s \leq t$. This proves (2).
It remains to define the compact subsets $K_t$ and to prove the claims (3) and (4). Let $C \subset
\Delta^{\circ}_0$ be a connected closed subset containing within its interior every interior lattice point, i.e., if $m \in \Delta_0^\circ \cap {{\mathbb{Z}}}^n$ then $m \in
C^\circ$; such a $C$ clearly exists. We define $K_t := \mu_t^{-1}(C)$. Then $K_t$ is closed since $\mu_t$ is continuous. Since $C \subseteq \Delta_0^\circ$ and we saw $\mu_0^{-1}(\Delta_0^\circ) \subseteq U_0$ above, it also follows from the definition of the $\mu_t$ that $K_t \subseteq
U_t$. Moreover, since $\mu_{t-s} \circ \phi_s = \mu_t$ by construction of the integrable systems , and the $\phi_t$ are diffeomorphisms on the $U_t$, it follows that $\phi_s(K_t)=K_{t-s}$ for all $0 \leq s \leq t$. This proves (3). Finally, since $C$ contains only a finite number of lattice points, there exists some $\eta>0$ such that for all interior lattice points $m$, the ball $B_{\eta}(m)$ is contained in $C$. This proves (4) and completes the proof.
Roughly, the idea in what follows is to replace the integrals in previous proofs by integrals over $K_t$. By using Proposition \[prop:support on fibers\] we will be able to control the error terms. Then, since $K_t$ is compact by assumption, we will be able to use a uniform continuity argument. We begin with an estimate which is uniform on $K_t$ for all sufficiently small $t$. This will be a key component of the proof of Proposition \[proposition:interior\].
\[last-bloody-technical-argument-left\]
Let $\tilde{m}$ be a preimage of an interior lattice point $m \in
\Delta_0 \cap {{\mathbb{Z}}}^n$. There exists a continuous function $t''=t''(s):
{{\mathbb{R}}}_{>0} \to [0,1]$ such that for any $\varepsilon > 0$ and any $s \in
{{\mathbb{R}}}_{>0}$, the following holds: if $t \in [0,1]$ satisfies $0 \leq t
\leq t''(s)$ then $$\label{eq:estimate on Kt}
\biggl\lvert
\frac{\tilde{\rho}_{s,t}^* \tilde{\chi}_s^*(\sigma^{\tilde{m}})(x)}
{\|\tilde{\rho}_{s,t}^* \tilde{\chi}_s^*(\sigma^{\tilde{m}})\|_{L^1(X_t)}}
- \frac{\tilde{\phi}_t^* \tilde{\chi}_s^*(\sigma^{\tilde{m}})(x)}
{\|\tilde{\phi}_t^* \tilde{\chi}_s^*(\sigma^{\tilde{m}})\|_{L^1(X_t)}}
\biggr\rvert < \varepsilon.$$ for all $x \in K_t$.
The idea of the proof is to show that the LHS of is (the norm of) a continuous section of a line bundle over a suitable family that is equal to zero for all $x \in K_0$ (i.e. when $t=0$) for any value of $s$; then we use uniform continuity.
Recall that ${{\mathcal X}}_{[0,1]}$ denotes the restriction of our family ${{\mathcal X}}$ to the subset $[0,1] \subseteq {{\mathbb{C}}}$, i.e. ${{\mathcal X}}_{[0,1]} := \pi^{-1}([0,1])
\subset {{\mathcal X}}$, and $\mathcal{U}_{[0,1]} = \{ x \in {{\mathcal X}}_{[0,1]} {\mid}x \in U_{\pi(x)}\}$ is the family of open dense subsets $U_t$ in each fibre, as in . We also define $$\begin{aligned}
{\mathcal{K}}_{[0,1]} := \{ x \in {{\mathcal X}}_{[0,1]} {\mid}x \in K_{\pi(x)}\}\end{aligned}$$ to be the family of the compact sets $K_t$ from Lemma \[lemma:ut-kt\] over ${[0,1]}$. Let $L_{\mathcal{K}}$ (respectively $L_{\mathcal{U}}$) denote $L_{{\mathcal X}}\vert_{{\mathcal{K}}_{[0,1]}}$ (respectively $L_{{\mathcal X}}\vert_{\mathcal{U}_{[0,1]}}$).
Now fix $s \in {{\mathbb{R}}}$ with $s>0$ and let $x \in U_t \subset X_t$ for any $t$. Recall that $\phi_t$ is the gradient-Hamiltonian flow, so in particular $\phi_t: X_t \to X_0$ takes $X_t$ to $X_0$ and $U_t$ to $U_0$ (cf. Lemma \[lemma:ut-kt\]). In the LHS of , the expression $\tilde{\phi}_t^*
\tilde{\chi}_s^*(\sigma^{\tilde{m}})(x)$ is by definition equal to $\big( \tilde{\phi}_t^{-1} \circ \tilde{\chi}_s^*(\sigma^{\tilde{m}})
\vert_{X_0} \circ \phi_t \big)(x)$. In particular, $\tilde{\phi}_t^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})$ sends each $U_t$ to $L_t$ by definition, so it is a section of the bundle $L_t \to U_t$. Putting these together for all $U_t$ for $t \in [0,1]$, we obtain a section of $L_{\mathcal{U}} \to \mathcal{U}_{[0,1]}$. The $L^1$-norm in the denominator is independent of $t$, since it can be written as $$\|\tilde{\phi}_t^* \tilde{\chi}_s^*(\sigma^{\tilde{m}})\|_{L^1(X_t)}
= \int_{X_t} {\bigl\lvert \tilde{\phi}_t^* \tilde{\chi}_s^*(\sigma^{\tilde{m}})\bigr\rvert} {{\hspace{1mm}}}d(vol)
= \int_{X_0} {\bigl\lvert\tilde{\chi}_s^*(\sigma^{\tilde{m}})\bigr\rvert} {{\hspace{1mm}}}d(vol)$$ since $\phi_t$ preserves the Hermitian and symplectic structures, and so the denominator of the second term is constant on $\mathcal{U}_{[0,1]}$.
Recall that $\sigma^{\tilde{m}}$ is holomorphic (hence smooth), $\chi_s$ is smooth, and the map $\phi: \mathcal{U}_{[0,1]} \to U_0$ given by $x \mapsto \phi_{\pi(x)}(x)$ is smooth (since it is the flow of a smooth vector field). Together with the last statement in Lemma \[lemma:lift-grH\] we may conclude that the section of $L_{\mathcal{U}} \to \mathcal{U}_{[0,1]}$ obtained above (sending $x
\in U_t$ to $\tilde{\phi}_{\pi(x)}^*
\tilde{\chi}_s^*(\sigma^{\tilde{m}})(x)$) is smooth. Finally, recalling that when $t=0$ the maps $\phi_0: U_0 \to U_0$ and $\tilde{\phi}_0: L_0 \vert_{U_0} \to L_0 \vert_{U_0}$ are both equal to the identity, we conclude that the above section restricts on $U_0$ to be equal to $\frac{\tilde{\chi}_s^*(\sigma^{\tilde{m}})}
{{\lVert\tilde{\chi}_s^*(\sigma^{\tilde{m}})\rVert}}\Big\vert_{U_0}$.
Next, we consider the expression $\tilde{\rho}_{s,t}^*
\tilde{\chi}_s^*(\sigma^{\tilde{m}})$ contained in the LHS of . Recall that $\rho_{s,t}$ is the embedding $X_t \to {{\mathcal{P}}}$ given in and specified in Proposition \[prop:embedding of submanifold\], and $\tilde{\rho}_{s,t}$ is the lift of $\rho_{s,t}$ to the line bundles (cf. Lemma \[lemma:lift-rhos\]). In particular, for $x \in U_t$ the expression $\tilde{\rho}_{s,t}^*
\tilde{\chi}_s^*(\sigma^{\tilde{m}})(x)$ is by definition equal to $\big( \tilde{\rho}_{s,t}^{-1} \circ
\tilde{\chi}_s(\sigma^{\tilde{m}}) \vert_{\rho_{s,t}(U_t)} \circ
\rho_{s,t} \big)(x)$. As in the above case, by construction $\tilde{\rho}_{s,t}^* \circ \tilde{\chi}_s^*(\sigma^{\tilde{m}})$ sends each $U_t$ to $L_t$ by definition, so it is a section of $L_t
\to U_t$ and by putting these together we obtain a section of $L_{\mathcal{U}} \to \mathcal{U}_{[0,1]}$. Similarly to the above case, the last statement of Proposition \[prop:rho-tilde-smooth-in-t\] allows us to conclude that this section is continuous. In this case the $L^1$-norm in the denominator is no longer independent of $t$, but it will be continuous in $t$, and never zero since the section is not the zero section. Moreover, by our assumption on $\{\chi_s\}$ and Proposition \[prop:toric submanifolds don’t move\] as well as Proposition \[prop:rho-tilde-smooth-in-t\] we also know that $\rho_{s,0}=\operatorname{id}$ for all $s$ and its lift $\tilde{\rho}_{s,0}$ acts as the identity on $L_0
\vert_{U_0}$, so $\tilde{\rho}_{s,0}^*\tilde{\chi}_s^*(\sigma^{\tilde{m}}) = \tilde{\chi}_s^*(\sigma^{\tilde{m}})$.
Now for $x\in\mathcal{K}_{[0,1]}$, let $$h_s(x) =
\frac{\tilde{\rho}_{s,t}^* \tilde{\chi}_s^*(\sigma^{\tilde{m}})(x)}
{{\lVert\tilde{\rho}_{s,t}^* \tilde{\chi}_s^*(\sigma^{\tilde{m}})\rVert}}
- \frac{\tilde{\phi}_t^* \tilde{\chi}_s^*(\sigma^{\tilde{m}})(x)}
{{\lVert\tilde{\phi}_t^* \tilde{\chi}_s^*(\sigma^{\tilde{m}})\rVert}}.$$ By the preceding discussion, $h_s$ is a continuous section of $L_{{\mathcal{K}}}$ over ${\mathcal{K}}_{[0,1]}$ and in particular its absolute value $|h_s|$ is a continuous function from ${\mathcal{K}}_{[0,1]}$ to ${{\mathbb{R}}}$. Moreover, by the above, $h_s(x) = 0$ for $x\in X_0$.
By the continuity of $h_s$, and since we know $K_0 \subseteq
h_s^{-1}(0)$, we may cover $K_0$ with open sets $U_\alpha$ with the property that $|h_s(x)| < \varepsilon$ for all $U_\alpha$. Extend this to an open cover of ${\mathcal{K}}_{[0,1]}$. The compactness of ${\mathcal{K}}_{[0,1]}$ implies that there exists a finite subcover, a subset of which is a finite cover of the subset $K_0$ of ${\mathcal{K}}_{[0,1]}$. In particular, from the construction of the cover, we may conclude that there exists some $t''(s)>0$ such that for all $0 \leq t < t''(s)$ and $x \in K_t$ we have ${\lverth_s(x)\rvert}<\varepsilon$. Now an argument similar to that in the proof of Proposition \[prop:support on fibers\] shows that in fact we may choose $t''(s)$ to be a continuous function of $s$, as claimed.
We are finally ready for a proof of Proposition \[proposition:interior\].
We first define the covariantly constant section $\delta_m$ and measure $d\theta_m$. Recall that the integrable system on the variety $X$ is defined by pulling back that on $X_0$ via $\phi_1$. In particular $\phi_1$ induces a pullback of the action-angle coordinates on $U_0$ to $U=U_1=\phi_1^{-1}(U_0)$ and a diffeomorphism of tori $\mu_0^{-1}(m) \cong \mu^{-1}(m)
\cong (S^1)^n$. Hence by using $\phi_1$ we may pull back the covariantly constant $\delta_{m,0}$ and measure $d\theta_{m,0}$ on $U_0$ of Lemma \[lemma:estimate on X\_0\] to a covariantly constant $\delta_m$ and $d\theta_m$ respectively. Let $\tau \in \Gamma(U, {L}^*_{{{\mathcal{P}}}} \vert_{U})$ be a test section. For simplicity of notation we let $(\phi_1)_*\tau$ denote the section of $L^*_{{\mathcal{P}}}\vert_{U_0}$ obtained by using the identifications $\phi_1: U=U_1 \to
U_0$ and $\tilde{\phi}_1: L_{{\mathcal{P}}}\vert_{U} \to L_{{\mathcal{P}}}\vert_{U_0}$, so $(\phi_1)_*\tau (x):= \tilde{\phi}_1 \circ \tau \circ \phi_1^{-1}(x)$. Then by definition we have $$\int_{\mu_0^{-1}(m)} \langle (\phi_1)_* \tau, \delta_{m,0} \rangle
{{\hspace{1mm}}}d\theta_{m,0} = \int_{\mu^{-1}(m)} \langle \tau, \delta_m \rangle
{{\hspace{1mm}}}d\theta_m$$ for any test section. Now let $\epsilon>0$. From the above it suffices to prove that we can find a continuous function $t=t(s)$ and an $s_0>0$ such that for all $s>s_0$ we have $$\label{eq:test function integral estimate}
\bigg\lvert \int_X \bigg\langle \tau, \frac{\sigma^m_{s,
t(s)}}{{\lVert\sigma^m_{s,t(s)}\rVert}} \bigg \rangle {{\hspace{1mm}}}d(vol)
-
\int_{\mu_0^{-1}(m)} \langle (\phi_1)_* \tau, \delta_{m,0} \rangle
{{\hspace{1mm}}}d\theta_{m,0} \bigg\rvert < \epsilon.$$ First, we know from Lemma \[lemma:estimate on X\_0\] that there exists $s_1$ such that for all $s>s_1$ we have $$\bigg\lvert\int_{X_0} \bigg\langle (\phi_1)_* \tau,
\frac{\tilde{\chi}_s^*(\sigma^{\tilde{m}})}
{{\lVert\tilde{\chi}_s^*(\sigma^{\tilde{m}})\rVert}} \bigg\rangle {{\hspace{1mm}}}d(vol)
- \int_{\mu_0^{-1}(m)} \langle (\phi_1)_* \tau, \delta_{m,0} \rangle
{{\hspace{1mm}}}d\theta_{m,0} \bigg\rvert
< \frac{\epsilon}{4}.$$ Moreover, since $\phi_1 = \phi_t \circ \phi_{1-t}$ and all maps preserve the relevant structures, we have $$\label{eq:one of the pieces of triangle inequality}
\bigg\lvert \int_{X_t} \bigg\langle (\phi_{1-t})_* \tau,
\frac{\tilde{\phi}^*_t \tilde{\chi}^*_s(\sigma^{\tilde{m}})}
{{\lVert\tilde{\phi}^*_t \tilde{\chi}^*_s(\sigma^{\tilde{m}})\rVert}}
\bigg \rangle {{\hspace{1mm}}}d(vol)
- \int_{\mu_0^{-1}(m)} \langle (\phi_1)_* \tau, \delta_{m,0} \rangle
{{\hspace{1mm}}}d\theta_{m,0} \bigg\rvert < \frac{\epsilon}{4}.$$ Next note that $\tau$ is a smooth section on the compact space $X$, so there exists a constant $C>0$ such that ${\lVert\tau\rVert}<C$ for all $x \in X$. Since $\phi_{1-t}$ preserves the Hermitian structure, this also implies that ${\lvert(\tilde{\phi}_{1-t})_* \tau\rvert} <C$ for all $x\in X_t$ and for all $t$. Let $K=K_1$ be the compact subset of $X=X_1$ from Lemma \[lemma:ut-kt\]. Our next step is to approximate the integrals over $X$ and $X_0$ with integrals over $K$ and $K_0$. Specifically, following Lemma \[lemma:ut-kt\] (4) let $\eta>0$ be such that such that $B_\eta = B_{\eta}(m) \subset K$. By Proposition \[prop:support on fibers\], there exists $s_2>0$ such that $$\int_{X\smallsetminus B_{\eta}} \bigg\lvert{
\frac{\sigma^m_{s,t}}{{\lVert\sigma^m_{s,t}\rVert}} \bigg\rvert {{\hspace{1mm}}}d(vol)
< \frac{\epsilon}{4C}}$$ for any $s>s_2$ and $0\leq t\leq t'(s)$ where $t=t'(s)$ is the function constructed in the proof of Proposition \[prop:support on fibers\]. Since $B_\eta \subset K$ and because we have an upper bound ${\lvert\tau\rvert}<C$ on the norm of $\tau$, we conclude $$\int_{X\smallsetminus K} \bigg\lvert{\frac{\sigma^m_{s,t}}{{\lVert\sigma^m_{s,t}\rVert}}} \bigg\rvert {\lvert\tau\rvert} <
\int_{X\smallsetminus B_\eta} \bigg\lvert{\frac{\sigma^m_{s,t}}{{\lVert\sigma^m_{s,t}\rVert}}} \bigg\rvert {\lvert\tau\rvert}
< \frac{\epsilon}{4}$$ for $s>s_2$ and $0\leq t\leq t'(s)$. Thus $$\label{eq:K approximates X}
\begin{split}
{\left\lvert\int_X \bigg\langle \tau, \frac{\sigma^m_{s,t}}{{\lVert\sigma^m_{s,t}\rVert}} \bigg\rangle {{\hspace{1mm}}}d(vol) -
\int_K \bigg\langle \tau, \frac{\sigma^m_{s,t}}{{\lVert\sigma^m_{s,t}\rVert}} \bigg\rangle {{\hspace{1mm}}}d(vol)\right\rvert} & =
{\left\lvert\int_{X\smallsetminus K} \bigg\langle \tau, \frac{\sigma^m_{s,t}}{{\lVert\sigma^m_{s,t}\rVert}} \bigg\rangle {{\hspace{1mm}}}d(vol)\right\rvert} \\
& \leq
\int_{X\smallsetminus K} {\lvert\tau\rvert}\bigg\lvert\frac{\sigma^m_{s,t}}{{\lVert\sigma^m_{s,t}\rVert}} \bigg\rvert d(vol)
< \frac{\epsilon}{4}
\end{split}$$ for these choices of $s$ and $t$, and in this sense, the integral over $X$ is well approximated by one over $K$.
A similar argument, using Lemma \[lemma:support-section-converges-X0\] applied to $X_0$ and $K_0$, gives an $s_3$ such that for $s>s_3$ $$\label{eq:K_0 approximates X_0}
{\left\lvert\int_{X_0} \bigg\langle (\phi_1)_* \tau,
\frac{\tilde{\chi}_s^*(\sigma^{\tilde{m}})}
{{\lVert\tilde{\chi}_s^*(\sigma^{\tilde{m}})\rVert}} \bigg\rangle {{\hspace{1mm}}}d(vol)
- \int_{K_0} \bigg\langle (\phi_1)_* \tau,
\frac{\tilde{\chi}_s^*(\sigma^{\tilde{m}})}
{{\lVert\tilde{\chi}_s^*(\sigma^{\tilde{m}})\rVert}} \bigg\rangle {{\hspace{1mm}}}d(vol) \right\rvert}
< \frac{\epsilon}{4}$$ and so the integral over $X_0$ is well approximated by one over $K_0$.
Next, for any $0<t<1$ we can push forward by $\phi_{1-t}$ to rewrite the integral over $K$ as an integral over $K_t$ as follows. Recalling the definition of $\sigma^m_{s,t}$ from we have $$\int_K \bigg\langle \tau,
\frac{\sigma^m_{s,t(s)}}{{\lVert\sigma^m_{s,t(s)}\rVert}} \bigg \rangle {{\hspace{1mm}}}d(vol)
= \int_{K_t} \bigg\langle (\phi_{1-t})_* \tau,
\frac{\tilde{\rho}_{s,t}^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})}
{{\lVert\tilde{\rho}_{s,t}^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})\rVert}}
\bigg\rangle {{\hspace{1mm}}}d(vol).$$ We then have the following: $$\label{eq:last big sequence}
\begin{split}
\bigg\lvert
\int_{K_t} \bigg\langle (\phi_{1-t})_* \tau,
\frac{\tilde{\rho}_{s,t}^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})}
{{\lVert\tilde{\rho}_{s,t}^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})\rVert}}
\bigg\rangle {{\hspace{1mm}}}d(vol)
& - \int_{K_t} \bigg\langle (\phi_{1-t})_* \tau,
\frac{\phi_t^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})}
{{\lVert\phi_t^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})\rVert}}
\bigg\rangle {{\hspace{1mm}}}d(vol)
\bigg\rvert \\
& \leq
\int_{K_t} \bigg\lvert \bigg\langle (\phi_{1-t})_* \tau,
\frac{\tilde{\rho}_{s,t}^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})}
{{\lVert\tilde{\rho}_{s,t}^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})\rVert}} \bigg\rangle
- \bigg\langle (\phi_{1-t})_* \tau,
\frac{\phi_t^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})}
{{\lVert\phi_t^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})\rVert}}
\bigg\rangle \bigg\rvert {{\hspace{1mm}}}d(vol)
\\
& =
\int_{K_t} \bigg\lvert \bigg\langle (\phi_{1-t})_* \tau,
\frac{\tilde{\rho}_{s,t}^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})}
{{\lVert\tilde{\rho}_{s,t}^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})\rVert}}
- \frac{\phi_t^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})}
{{\lVert\phi_t^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})\rVert}}
\bigg\rangle \bigg\rvert {{\hspace{1mm}}}d(vol)
\\
& \leq
\int_{K_t} {\bigl\lvert(\phi_{1-t})_* \tau\bigr\rvert}
\bigg\lvert \frac{\tilde{\rho}_{s,t}^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})}
{{\lVert\tilde{\rho}_{s,t}^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})\rVert}}
- \frac{\phi_t^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})}
{{\lVert\phi_t^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})\rVert}}
\bigg\rvert {{\hspace{1mm}}}d(vol) \\
& \leq
C \int_{K_t} \bigg\lvert
\frac{\tilde{\rho}_{s,t}^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})}
{{\lVert\tilde{\rho}_{s,t}^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})\rVert}}
- \frac{\phi_t^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})}
{{\lVert\phi_t^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})\rVert}}
\bigg\rvert {{\hspace{1mm}}}d(vol) \\
\end{split}$$ where the last inequality again uses the upper bound ${\lvert(\phi_{1-t})_*\tau\rvert}<C$. Now let $\varepsilon =
\frac{\epsilon}{2C \operatorname{vol}(X_t)}$. (Note that volumes are equal for all fibers, i.e. $\operatorname{vol}(X)=\operatorname{vol}(X_t)$ for all $t$.) Applying Lemma \[last-bloody-technical-argument-left\] to this value of $\varepsilon$, we obtain a continuous monotone (non-increasing) function $t''(s)$ of $s$ such that for all $0 < t < t''(s)$ and all $x
\in K_t$ we have $$\label{eq:estimate from previous lemma}
\bigg\lvert
\frac{\tilde{\rho}_{s,t}^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})}
{{\lVert\tilde{\rho}_{s,t}^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})\rVert}}
- \frac{\phi_t^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})}
{{\lVert\phi_t^*\tilde{\chi}_s^*(\sigma^{\tilde{m}})\rVert}}
\bigg\rvert
< \frac{\epsilon}{4C\operatorname{vol}(X_t)}$$ which implies that the integral in is less than $\epsilon/4$.
Finally, let $t(s) = \min\{t'(s), t''(s)\}$ be the minimum of the two continuous functions $t'(s)$ and $t''(s)$ defined earlier. Then $t(s)$ is a continuous, positive, decreasing function of $s$, and the estimates in and hold for all $0<t<t(s)$. Let $s_0 = \max\{s_1, s_2, s_3\}$. The triangle inequality then implies that the LHS of is less than or equal to the sum of the left-hand sides of , , , and , each of which is less than $\epsilon/4$ for $s>s_0$. This implies that, for this function $t(s)$ and this choice of $s_0$, the inequality holds for $s>s_0$, and we are finished with the proof of Proposition \[proposition:interior\].
In Section \[ss:varying-cplx-str\] we constructed for each $s$ and $t$ a complex structure $J_{s,t}$ on $X$ that is compatible with the symplectic structure. Recall also that the parameter $s \in [0,\infty)$ corresponds to the deformation of complex structures while the $t \in [0,1]$ parameter corresponds to the gradient-Hamiltonian flow from $X_1$ to $X_0$. Our final task is to construct the family $J_s$ of complex structures and the basis of sections $\sigma^m_s$ in the statement of our main Theorem \[theorem:main\]. The basic idea behind the definition below is to let $s$ go to $\infty$ and $t$ go to $0$ simultaneously in such a way that the convergence which is claimed in Theorem \[theorem:main\] occurs. Indeed, in Proposition \[proposition:interior\] we constructed a continuous function $t(s)$ of $s$ which gives rise to certain key estimates for any $s>0$ and $0 \leq t < t(s)$. Thus, setting $J_s := J_{s,t(s)}$ would (nearly) do the job; however, although we need $J_0$ to be the original complex structure (which is $J_{0,1}$) on $X$, our construction of $t(s)$ does not guarantee that $t(0)=1$. The solution to this problem is, roughly speaking, to first move along the gradient-Hamiltonian flow to $t_0$ while keeping $s=0$ before “turning on” the other deformation. More precisely, we make the following definition.
\[def:J\_s\] Let $t(s)$ denote the continuous function constructed in Proposition \[proposition:interior\]. For $s \in [0,\infty)$, we define $$J_s :=
\begin{cases}
J_{0,1+(t_0-1)s} & \textup{ if } 0\leq s \leq 1 \textup{ and } \\
J_{s-1,t(s-1)} & \textup{ if } s > 1.
\end{cases}$$
Note that by construction $J_0 = J_{0,1}$ is the original complex structure on $X$, and as $s\to \infty$, $J_s$ has the same convergence properties as $J_{s,t(s)}$. Moreover, by construction, the family $J_s$ is continuous with respect to the parameter $s$. We are finally ready to prove our main Theorem \[theorem:main\].
Consider the family $\{J_s\}_{s \in [0,\infty)}$ given in Definition \[def:J\_s\]. As already noted above, $\{J_s\}$ is a continuous family and $J_0=J_{0,1}$ is the original complex structure by construction. By Lemma \[lemma:pairs Kahler\], the pair $(\omega=\omega_1, J_s)$ is a Kähler structure on $X$ for each $s
\in [0,\infty)$. Now define $$\label{eq:def sigma m s}
\sigma^m_s :=
\begin{cases}
\sigma^m_{0,1+(t_0-1)s} & \textup{ if } 0 \leq s \leq 1 \\
\sigma^m_{s-1,t(s-1)} & \textup{ if } s > 1.
\end{cases}$$ By construction, $\sigma^m_s$ is an element of $H^0(X, L,
\overline{\partial}_s)$. Moreover, by definition $J_s$ and $\sigma^m_s$ have the same limiting properties as $J_{s,t(s)}$ and $\sigma^m_{s,t(s)}$, and for any interior point $m \in W_0$ it was shown in Proposition \[proposition:interior\] that $\sigma^m_{s,t(s)}$ has the required limiting property of in the statement of Theorem \[theorem:main\]. Thus it remains only to ensure that for each fixed $s$ w have a basis of $H^0(X, L,
\overline{\partial}_s)$. By the limiting properties of the $\sigma^m_s$, we know that as $s$ goes to $\infty$, the supports of the sections $\sigma^m_s$ are increasingly concentrated in pairwise disjoint neighborhoods. It follows that the set $\{\sigma^m_s\}$ must be linearly independent for $s>s_0$ with $s_0$ sufficiently large. Since the complex manifolds $(X,J_s)$ and holomorphic line bundles $(L,
\overline{\partial}_s)$ are isomorphic for all $s$, we also know that $\dim H^0(X, L, \overline{\partial}_s)$ is constant for all $s$, and in particular by assumption (h) in the statement of Theorem \[theorem:main\], we have $\dim
H^0(X,L,\overline{\partial}_s) = |W_0|$ for all $s$. Thus for $s \geq
s_0$ the set $\{\sigma^m_s\}$ is linearly independent and also spans, so it is a basis of $H^0(X,L,\overline{\partial}_s)$ as desired. For $0 \leq s \leq s_0$, following [@HamKon Section 7.2] we extend the basis $\{\sigma^m_s\}_{m \in W_0}$, $s \geq s_0$, to bases of $H^0(X,L,
\overline{\partial}_s)$ for $s$ satisfying $0 \leq s \leq s_0$ in a way that preserves the continuity in the parameter $s$. This family then satisfies all the required properties.
Toric degenerations coming from valuations and Newton-Okounkov bodies {#sec:NOBY}
=====================================================================
In this section we will show that the toric degenerations coming from Newton-Okounkov bodies as in [@HarKav] can be used to create many examples of algebraic varieties $X$ with prequantum data $(\omega, J, L, h, \nabla)$ which satisfy the hypotheses of Theorem \[theorem:main\]. This is the content of the main result of this section, Theorem \[theorem:toric deg from NOBY\].
We first very briefly recall the ingredients in the definition of a Newton-Okounkov body. For details we refer the reader to [@KavKho; @LazMus] and also [@HarKav]. We begin with the definition of a valuation (in our setting). We equip ${{\mathbb{Z}}}^n$ with a group ordering e.g. a lexicographic order.
\[definition:valuation\]
1. Let $A$ be a ${{\mathbb{C}}}$- algebra. A [**valuation**]{} on $A$ is a function $$\nu:A\setminus\{0\}\rightarrow {{\mathbb{Z}}}^n$$ satisfying the following:
1. $ \nu(c f)=\nu(f)$ for all $f\in A\setminus\{0\}$ and $c\in{{\mathbb{C}}}\setminus\{0\}$,
2. $\nu(f+g) \geq \min\{\nu(f),\nu(g)\}$ for all $f, g \in A
\setminus \{0\}$ with $f+g \neq 0$.
3. $\nu(fg)=\nu(f) + \nu(g)$ for all $f,g \in A \setminus \{0\}$.
2. The image $\nu(A\setminus\{0\})\subset{{\mathbb{Z}}}^n$ of a valuation $\nu$ on a ${{\mathbb{C}}}$-algebra $A$ is clearly a semigroup and is called the *value semigroup* of the pair $(A,\nu)$.
3. Moreover, if in addition the valuation has the property that for any pair $f,g\in A\setminus\{0\}$ with same value $\nu(f)=\nu(g)$ there exists a non-zero constant $c \neq 0 \in {{\mathbb{C}}}$ such that either $\nu(g-c f)>\nu(g)$ or else $g-c f = 0$ then we say that the valuation has *one-dimensional leaves*.
If $\nu$ is a valuation with one-dimensional leaves, then the image of $\nu$ is a sublattice of ${{\mathbb{Z}}}^n$ of full rank. Hence, by replacing ${{\mathbb{Z}}}^n$ with this sublattice if necessary, we will always assume without loss of generality that $\nu$ is surjective.
Given a variety $X$, there exist many possible valuations with one-dimensional leaves on its field of rational functions ${{\mathbb{C}}}(X)$. Strictly speaking, we do not need detailed knowledge of the construction in this paper, but we note for the reader’s reference that the following example is the one which arises naturally in geometric contexts: for an $n$-dimensional variety $X$, a choice of an (ordered) coordinate system at a smooth point $p$ on $X$ gives a valuation on ${{\mathbb{C}}}(X)$ with one-dimensional leaves, essentially by computing the order of the zero or pole with respect to the coordinates. See e.g. [@KavKho Examples 2.12 and 2.13 ] or [@LazMus] for details.
The following proposition is simple but fundamental [@KavKho Proposition 2.6]:
\[prop-val-dim\] Let $\nu$ be a valuation on ${{\mathbb{C}}}(X)$ with one-dimensional leaves. Let $V \subset {{\mathbb{C}}}(X)$ be a finite-dimensional subspace of ${{\mathbb{C}}}(X)$. Then $\dim_{{\mathbb{C}}}(V) = {\bigl\lvert\nu(V \setminus \{0\})\bigr\rvert}$.
Let $X$ be a projective variety of dimension $n$ over ${{\mathbb{C}}}$ equipped with a very ample line bundle $L$. Let $E
:= H^0(X, L)$ denote the space of global sections of $L$; it is a finite dimensional vector space over ${{\mathbb{C}}}$. The line bundle $L$ gives rise to the *Kodaira map $\Phi_E$ of $E$*, from $X$ to the projective space ${\mathbb{P}}(E^*)$. The assumption that $L$ is very ample implies that the Kodaira map $\Phi_E$ is an embedding.
Now let $E^k$ denote the image of the $k$-fold product $E \otimes
\cdots \otimes E$ in $H^0(X, L^{\otimes k})$ under the natural map given by taking the product of sections. (In general this map may not be surjective.) The homogeneous coordinate ring of $X$ with respect to the embedding $\Phi_E: X \hookrightarrow {\mathbb{P}}(E^*)$ can be identified with the graded algebra $$R = R(E) = \bigoplus_{k \geq 0} R_k,$$ where $R_k := E^k$. This is a subalgebra of the [*ring of sections*]{} $$R(L) = \bigoplus_{k \geq 0} H^0(X, L^{\otimes k}).$$
For a fixed $\nu$ we now associate a semigroup $S(R) \subset {{\mathbb{N}}}\times {{\mathbb{Z}}}^n$ to $R$. First we identify $E = H^0(X, L)$ with a (finite-dimensional) subspace of ${{\mathbb{C}}}(X)$ by choosing a non-zero element $h \in E$ and mapping $f \in E$ to the rational function $f/h \in {{\mathbb{C}}}(X)$. Similarly, we can associate the rational function $f/h^k$ to an element $f \in R_k := E^k \subseteq H^0(X,
L^{\otimes k})$. We define $$\label{eq:definition S}
S = S(R) = S(R,\nu,h) = \bigcup_{k > 0} \{ (k, \nu(f / h^k)) \mid f \in
E^k \setminus \{0\}\}.$$ If $f \in R_k = E^k$ is a homogeneous element of degree $k$ we also define: $$\tilde{\nu}(f) = (k, \nu(f/h^k)).$$
Now define $C(R) \subseteq {{\mathbb{R}}}\times {{\mathbb{R}}}^n$ to be the cone generated by the semigroup $S(R)$, i.e., it is the smallest closed convex cone centered at the origin containing $S(R)$. We can now define the central object of interest.
\[definition:NO\] Let $\Delta=\Delta(R)=\Delta(R,\nu)$ be the slice of the cone $C(R)$ at level 1, that is, $C(R)\cap (\{1\}\times {{\mathbb{R}}}^n)$, projected to ${{\mathbb{R}}}^n$ via the projection to the second factor ${{\mathbb{R}}}\times {{\mathbb{R}}}^n \to {{\mathbb{R}}}^n$. We have $$\Delta = \overline{ \textup{conv} \left( \bigcup_{k>0}
\left\{\frac{x}{k} : (k,x) \in S(R) \right\} \right) }.$$ The convex body $\Delta$ is called the *Newton-Okounkov body of $R$* with respect to the valuation $\nu$.
From now on, we place the additional assumption that:
*$S$ is finitely generated.*
The above assumption is a rather strong condition on $(X, L, \nu)$ but it holds in many cases of importance. We note that it is possible to have a finitely generated semigroup $S$ for one choice of a valuation $\nu$ and a non-finitely generated one for a different choice of $\nu$. From the above assumption it follows that the Newton-Okounkov body $\Delta(R)$ is a rational polytope. In this context, Anderson proved the following [@Anderson Corollary 5.3].
\[th-toric-degen-NO-body\] There is a flat family $\pi: {{\mathcal X}}\to {{\mathbb{C}}}$ such that:
- For any $z \neq 0$ the fiber $X_z = \pi^{-1}(z)$ is isomorphic to $X$, and $\pi^{-1}({{\mathbb{C}}}^*)$ is isomorphic to $X \times {{\mathbb{C}}}^*$. For the remainder of the discussion we fix an isomorphism $X \times {{\mathbb{C}}}^* \to \pi^{-1}({{\mathbb{C}}}^*) \subset {{\mathcal X}}$.
- The special fiber $X_0 = \pi^{-1}(0)$ is isomorphic to $\textup{Proj}(\textup{gr} R) \cong \textup{Proj}({{\mathbb{C}}}[S])$ and is equipped with an action of ${{\mathbb{T}}}= ({{\mathbb{C}}}^*)^n$, where $n=\dim_{{\mathbb{C}}}X$. The normalization of the variety $\textup{Proj}(\textup{gr} R)$ is the toric variety associated to the rational polytope $\Delta(R)$.
The explicit construction of the family ${{\mathcal X}}$ in [@Anderson] depends on a choice of a so-called Khovanskii basis $\mathcal{B} = \{f_{ij}\}$ (cf. [@HarKav Definition 8.1], and also see [@KavMan] for a general theory of Khovanskii bases). The set $\mathcal{B}$ also allows us to concretely embed ${{\mathcal X}}$ as a subvariety of ${{\mathcal{P}}}\times {{\mathbb{C}}}$ for an appropriate “large” projective space ${{\mathcal{P}}}$. Some of the details are relevant for our later discussions so we briefly recall the construction here; for details we refer to [@HarKav Sections 8 and 9].
By assumption the semigroup $S \subset {{\mathbb{N}}}\times {{\mathbb{Z}}}^n$ is finitely generated. So we can find a finite set consisting of homogeneous elements in $R$ such that their valuations are a set of generators for $S$. More precisely, let $r>0$ be a positive integer and let $\mathcal{B} = \{f_{ij}\}$, for $1 \leq i \leq r$, $1 \leq j \leq n_i = \dim(R_i)$, be a finite set of elements in $R$ satisfying the following properties:
1. the $f_{ij}$ are homogeneous, with $f_{ij} \in R_i$ for all $1 \leq i \leq r, 1 \leq j \leq n_i$, and
2. for each $i$, the collection $\{f_{i1}, f_{i2}, \ldots, f_{i n_i}\}$ is a vector space basis for $R_i$,
3. the set of images $\tilde{\nu}(\mathcal{B}) = \{\tilde{\nu}(f)
\mid f \in \mathcal{B}\}$ generate $S=S(R)$,
For the remainder of this discussion we fix this “Khovanskii basis” $\mathcal{B}$.
We now describe more explicitly, in terms of the Khovanskii basis, the toric degeneration ${{\mathcal X}}$ constructed in [@Anderson] and a concrete embedding of ${{\mathcal X}}$ into ${{\mathcal{P}}}\times {{\mathbb{C}}}$ for a suitable large projective space ${{\mathbb{C}}}$. Let $S_d := S \cap (\{d\} \times {{\mathbb{Z}}}^n)$ denote the level-$d$ piece of the semigroup $S$. By Proposition \[prop-val-dim\], $\dim(E^d) =
|\nu(E^d)| = |S_d|$, and since the $\{f_{ij}\}$ form a Khovanskii basis, for each $s \in S_d$ we know there exists some monomial $f_{11}^{\alpha_{11}} f_{12}^{\alpha_{12}} \cdots
f_{r n_r}^{\alpha_{r n_r}}$ in the $\{f_{ij}\}$’s, where $\sum_{i=1}^r i \sum_{j=1}^{n_i} \alpha_{ij} = d$, such that $\nu(f_{11}^{\alpha_{11}} f_{12}^{\alpha_{12}} \cdots
f_{r n_r}^{\alpha_{r n_r}}) = s$. So for each $s \in S$ we fix a choice of such exponents $\alpha_s := (\alpha_{(ij),s})$ such that the above holds. Then the set $$\label{eq:basis for E^d}
\{ f_{11}^{\alpha_{(11),s}} f_{12}^{\alpha_{(12),s}} \cdots
f_{r n_r}^{\alpha_{(r n_r),s}} \mid s \in S_d\}$$ forms a basis for $E^d$. In [@Anderson], a collection of integers $w_{ij}$ are associated to the $f_{ij}$ in a certain way (for details see [@Anderson] and [@HarKav Section 8]). Using these integers $w_{ij}$ and the above choices we can describe explicitly the toric degeneration ${{\mathcal X}}$ and its embedding as follows. We first define a morphism $X \times {{\mathbb{C}}}^* \to {\mathbb{P}}((E^d)^*) \times {{\mathbb{C}}}^*$ by expressing the Kodaira embedding $X \to {\mathbb{P}}((E^d)^*)$ explicitly using the above basis for $E^d$. In coordinates we have $$\label{eq:embedding of family}
(x,t) \mapsto \bigg( \Big( t^{\sum_{ij}w_{ij} \alpha_{ij}} f_{11}(x)^{\alpha_{11}} \cdots
f_{rn_r}(x)^{\alpha_{rn_r}} \; \bigg\lvert \; \alpha_{ij} = \alpha_{(ij), s} \; , \; s \in S_d \Big), t\bigg)$$ Then the toric degeneration ${{\mathcal X}}\subseteq {{\mathcal{P}}}\times {{\mathbb{C}}}$ is defined to be the closure of the image of . By its construction, $X$ is isomorphic to the fiber $X_1$ and $X_0$ is a toric variety [@Anderson Corollary 5.3].
Note that the pullback to $X$ of the line bundle $L_{{\mathcal{P}}}$ over ${{\mathcal{P}}}$ is $L^{\otimes d}$ by construction. Given any prequantum data $(\omega_{{\mathcal{P}}}, L_{{\mathcal{P}}}, h_{{\mathcal{P}}}, \nabla_{{\mathcal{P}}})$ on ${{\mathcal{P}}}= {\mathbb{P}}((E^d)^*)$, it is clear that this data can be pulled back via the embedding to prequantum data $(\omega, L^{\otimes d}, h, \nabla)$ on the line bundle $L^{\otimes d}$ over $X$.
We have the following, which is the main result of this section.
\[theorem:toric deg from NOBY\] Let $X$ be a smooth, irreducible complex algebraic variety with $\dim_{{{\mathbb{C}}}}(X) = n$, let $L$ be a very ample line bundle on $X$, and $E:=H^0(X,L)$. Then there exists a sufficiently large positive integer $d$ and prequantum data $(\omega_{{\mathcal{P}}}, h_{{\mathcal{P}}}, \nabla_{{\mathcal{P}}})$ on $L_{{\mathcal{P}}}\to
{{\mathcal{P}}}$ such that the family ${{\mathcal X}}\subseteq {{\mathcal{P}}}\times {{\mathbb{C}}}$ constructed above is a toric degeneration of $X$ in the sense of Section \[sec-main-result\], and moreover, with respect to the pullback prequantum data $(\omega, h, \nabla)$ on $L^{\otimes d} \to X$, this toric degeneration satisfies all the required hypotheses (a)-(h), and thus gives ’convergence of polarization’ in these cases.
The fact that ${{\mathcal X}}\subseteq {{\mathcal{P}}}\times {{\mathbb{C}}}$ is a toric degeneration satisfying the hypotheses (a) and (b) of Theorem \[theorem:main\] follows from the construction in [@Anderson] and is shown in [@HarKav]. Moreover, in [@HarKav Section 9] an appropriate Kähler structure $\omega_{{\mathcal{P}}}$ on ${{\mathcal{P}}}$ is constructed which satisfies condition (d). Indeed, the construction of $\omega_{{\mathcal{P}}}$ is by pulling back a Fubini-Study form associated to a hermitian structure on an (even larger) projective space, and in particular – by pulling back the standard prequantum data on a projectivization of a vector space equipped with a hermitian structure – it is clear that we can construct the prequantum data $(\omega_{{\mathcal{P}}}, h_{{\mathcal{P}}},
\nabla_{{\mathcal{P}}})$ compatible with $\omega_{{\mathcal{P}}}$. Now the hypothesis (c) follows by construction, since the relevant prequantum data are defined via pullbacks.
We also claim that hypothesis (e) holds in our situation. Indeed, by our choice of basis of $E^d$, it follows that our embedding has the property that the coordinates of the embedding correspond exactly to elements of $S_d$. In particular, it follows from [@Anderson Proposition 5.1] and [@HarKav Section 8] that the special fiber $X_0$ is exactly the closure of a $\mathbb{T}_0$-orbit through a point of the form $[1:1:\cdots:1]$, where $\mathbb{T}_0 \cong ({{\mathbb{C}}}^*)^n$ acts by weight $s \in S_d
\subseteq {{\mathbb{Z}}}^n$ on the coordinate associated to $s \in S_d$. It also follows that the set $W_0$ defined in is precisely $S_d \subseteq {{\mathbb{Z}}}^n$, and thus hypothesis (h) follows from the fact, already observed above, that $\dim E^d = \dim H^0(X,L^{\otimes d}) =
|S_d|$.
Next, it is well-known [@Hartshorne Chapter II, Section 5, Exercise 5.14(b)] that for sufficiently large $d \gg 0$, we have $E^d = H^0(X,L^{\otimes d})$, so in particular ${\mathbb{P}}((E^{d})^*) = {\mathbb{P}}((H^0(X,L^{\otimes {d}}))^*)$ and the restriction map $H^0({{\mathcal{P}}},L_{{\mathcal{P}}}) = H^0({\mathbb{P}}(H^0(X,L^{\otimes {d}})^*), \mathcal{O}(1)) \to H^0(X,
L^{\otimes {d}})$ is surjective, so ${{\mathcal X}}$ satisfies hypothesis (f) for sufficiently large $d$.
Finally, we claim that for sufficiently large $d \gg 0$ we also have that hypothesis (g) holds. By definition of the torus $\mathbb{T}_0$, the graded semigroup $S$ generates ${{\mathbb{Z}}}\times M$ where $M$ is the character lattice of $\mathbb{T}_0$ (and can be identified with $({\mathfrak{t}}^*_0)_{{{\mathbb{Z}}}}$). It is easy to see that in this situation there exists $d$ sufficiently large such that $S_d = \nu(H^0(X,L^{\otimes d}) \setminus \{0\})$ generates $M$. Since $S_d$ is contained in $\iota^*(({\mathfrak{t}}^*_{{{\mathcal{P}}}})_{{{\mathbb{Z}}}})$ by what we said above, the hypothesis (g) follows for $d$ sufficiently large.
Thus, by taking $d$ large enough so that both of the last two phenomena occur, we obtain the results claimed in the theorem.
Finally, in the situation of an integrable system coming from a toric degeneration arising from a Newton-Okounkov body, the sections $\sigma^m_{s,t}$ we construct in §\[subsec:varying bases\] form a basis of $H^0(X,L,\bar{{\partial}}_{s,t})$ for all values of $s$ and $t$. In particular, in this case we do not need to “extend the basis” non-constructively as in the last sentence of the proof of Theorem \[theorem:main\], at the very end of §\[sec:proof\].
\[thm-sections-lin-ind\] Let the notation be as in Theorem \[theorem:toric deg from NOBY\]. Then for any fixed $s\geq 0$, $t\neq 0$, the set $\{\sigma^m_{s,t} {\mid}m \in W_0\}$ constructed in Definition \[definition:sigma s t\] is linearly independent. In particular, the set $\{ \sigma^m_s {\mid}m\in W_o \}$ constructed in is a basis for $H^0(X,L^d,\bar{{\partial}}^{s})$ for every value of $s\geq 0$.
By Definition \[definition:sigma s t\], and since $\tilde{\phi}_{1-t}$ is an isomorphism of line bundles (see Lemma \[lemma:lift-grH\]), it suffices to show that the $\tilde{\rho}_{s,t}^* \tilde{\chi}^*_{s} \sigma^{\tilde{m}}$ are linearly independent in $H^0(X_t, L_t)$. By [@HamKon Proposition 6.1(2)], for each $s\geq 0$ and each $0 < t \leq 1$ we have a diffeomorphism $\underline{\chi}_{s,t}\colon X_t \to X_t$ such that the following diagram commutes: $$\xymatrix{
{{\mathcal{P}}}\ar[r]^{\chi_s} & {{\mathcal{P}}}\\
X_t \ar@{^{(}->}[u]^{\rho_{s,t}} \ar[r]^{\underline{\chi}_{s,t}} & X_t \ar@{^{(}->}[u]_\rho \\
}$$ where $\rho$ is the standard embedding of $X_t$ into ${{\mathcal{P}}}$ as a complex manifold, and such that $(X_t, \rho^*_{s,t} \omega_{{\mathcal{P}}}, \chi^*_s J_{{\mathcal{P}}})$ is [Kähler]{}. (Note that with our identifications the map $\chi_0$ in [@HamKon] is the identity.) The $\rho_{s,t}$ appearing in the diagram above are the embeddings of Proposition \[prop:embedding of submanifold\]. Similarly, by [@HamKon Proposition 6.3(2)] the map $\underline{\chi}_{s,t}$ lifts to a map $\tilde{\underline{\chi}}_{s,t} \colon L_t \to L_t$ such that $$\xymatrix{
L_{{\mathcal{P}}}\ar[r]^{\tilde{\chi}_{s}} & L_{{\mathcal{P}}}\\
L_{{\mathcal{P}}}\vert_{X_t} \ar[r]^{\tilde{\underline{\chi}}_{s,t}} \ar[u]^{\tilde{\rho}_{s,t}}
& L_{{\mathcal{P}}}\vert_{X_t} \ar[u]_{\tilde{\rho}}
}$$ commutes, where $\tilde{\rho}_{s,t}$ are the maps in Lemma \[lemma:lift-rhos\]. Then $\tilde{\chi}_s^* \sigma^{\tilde{m}}\vert_{X_t} =
\tilde{\underline{\chi}}_{s,t}^* (\sigma^{\tilde{m}}\vert_{X_t})$. Since the $\tilde{\chi}_{s,t}$ are line bundle isomorphisms, it will suffice to show that the $\sigma^{\tilde{m}} \vert_{X_t}$’s are linearly independent.
To see this, recall the construction of the embedding of $X{\times}{{\mathbb{C}}}^*$ into ${{\mathcal{P}}}{\times}{{\mathbb{C}}}$. The sections $$\{f_s = f_{11}^{\alpha_{(11),s}} f_{12}^{\alpha_{(12),s}} \cdots
f_{r n_r}^{\alpha_{(r n_r),s}} \mid s \in S_d\}$$ from form a basis for the global sections of $\mathcal{O}(1) = L_{{\mathcal{P}}}$ on the projective space ${{\mathcal{P}}}=
\mathbb{P}((E^d)^*)$. The embedding of the family $X \times
{{\mathbb{C}}}^* \subset {{\mathcal X}}$ in ${{\mathcal{P}}}\times {{\mathbb{C}}}$ in this basis is given by the map in whose components are $t^{\sum_{ij}w_{ij} \alpha_{ij}} f_s$, $s \in S_d$. Since the line bundle is $\mathcal{O}(1)$, the functions $\sigma^{\tilde{m}}$ in the basis of its holomorphic sections are simply the coordinate functions on ${{\mathcal{P}}}$, and so the section $\sigma^{\tilde{m}}\vert_{X_t}$ corresponds to one of the coordinates $t^{\sum_{ij}w_{ij} \alpha_{ij}} f_s$.
By construction, for fixed $t \neq 0$ the values of the valuation $\tilde{v}$ on the components $t^{\sum_{ij}w_{ij} \alpha_{ij}} f_s$ are distinct. Since elements with distinct values of valuation are linearly independent, it follows that, for any fixed $t \neq 0$, the sections $\{ \sigma^{\tilde{m}} \vert_{X_t} {\mid}m \in W_0 \}$ are linearly independent, as desired.
[10]{}
M. Abreu. Kähler geometry of toric varieties and extremal metrics. , 9(6):641–651, 1998.
D. Anderson. Okounkov bodies and toric degenerations. , 356(3):1183–1202, 2013.
T. Baier, C. Florentino, J. M. Mour[ã]{}o, and J. P. Nunes. Toric [K]{}ähler metrics seen from infinity, quantization and compact tropical amoebas. , 89(3):411–454, 2011.
V. Guillemin. Kaehler structures on toric varieties. , 40(2):285–309, 1994.
V. Guillemin. , volume 122 of [*Progress in Mathematics*]{}. Birkhäuser Boston, Inc., Boston, MA, 1994.
M. D. Hamilton. The quantization of a toric manifold is given by the integer lattice points in the moment polytope. In [*Toric topology*]{}, volume 460 of [*Contemp. Math.*]{}, pages 131–140. Amer. Math. Soc., Providence, RI, 2008.
M. D. Hamilton and H. Konno. Convergence of [K]{}ähler to real polarizations on flag manifolds via toric degenerations. , 12(3):473–509, 2014.
M. Harada and K. Kaveh. Integrable systems, toric degenerations and [O]{}kounkov bodies. , 202(3):927–985, 2015.
R. Hartshorne. . Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52.
K. Kaveh. Crystal bases and [N]{}ewton-[O]{}kounkov bodies. , 164(13):2461–2506, 2015.
K. Kaveh and A. G. Khovanskii. Newton-[O]{}kounkov bodies, semigroups of integral points, graded algebras and intersection theory. , 176(2):925–978, 2012.
K. Kaveh and C. Manon Khvanskii bases, higher rank valuations and tropical geometry. arXiv:1610.00298
V. A. Kiritchenko, E. Yu. Smirnov and V. A. Timorin Schubert calculus and Gelfand-Tsetlin polytopes. 67 (2012), no. 4(406), 89–128; translation in [*Russian Math. Surveys*]{} 67 (2012), no. 4, 685–719.
M. Kogan and E. Miller Toric degeneration of Schubert varieties and Gelfand-Tsetlin polytopes. 193 (2005), no. 1, 1–17.
R. Lazarsfeld and M. Musta[ţ]{}[ă]{}. Convex bodies associated to linear series. , 42(5):783–835, 2009.
T. Nishinou, Y. Nohara, and K. Ueda. Toric degenerations of [G]{}elfand-[C]{}etlin systems and potential functions. , 224(2):648–706, 2010.
J. P. Nunes. Degenerating kähler structures and geometric quantization. , 26, 2014.
A. Okounkov Brunn-Minkowski inequality for multiplicities. 125 (1996), no. 3, 405–411.
A. Okounkov Why would multiplicities be log-concave? The orbit method in geometry and physics (Marseille, 2000), 329–347, [*Progr. Math.*]{}, 213, Birkhäuser Boston, Boston, MA, 2003.
W.-D. Ruan. Lagrangian torus fibration of quintic hypersurfaces. [I]{}. [F]{}ermat quintic case. In [*Winter [S]{}chool on [M]{}irror [S]{}ymmetry, [V]{}ector [B]{}undles and [L]{}agrangian [S]{}ubmanifolds ([C]{}ambridge, [MA]{}, 1999)*]{}, volume 23 of [ *AMS/IP Stud. Adv. Math.*]{}, pages 297–332. Amer. Math. Soc., Providence, RI, 2001.
T. Tao. , volume 126 of [*Graduate Studies in Mathematics*]{}. American Mathematical Society, 2011.
N. M. J. Woodhouse. . Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, second edition, 1992. Oxford Science Publications.
[^1]: The first author is partially supported by an NSERC Discovery Grant.
[^2]: The second author is partially supported by an NSERC Discovery Grant and a Canada Research Chair (Tier 2) Award.
[^3]: The third author is partially supported by a National Science Foundation Grant (Grant ID: DMS-1601303), Simons Foundation Collaboration Grant for Mathematicians, and Simons Fellowship award.
[^4]: Here we use the term “integrable system” in the slightly non-standard sense of [@HarKav Definition 1.1].
| ArXiv |
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abstract: 'The variability inherent in solar wind composition has implications for the variability of the physical conditions in its coronal source regions, providing constraints on models of coronal heating and solar wind generation. We present a generalized prescription for constructing a wavelet power significance measure (confidence level) for the purpose of characterizing the effects of missing data in high cadence solar wind ionic composition measurements. We describe the data gaps present in the 12-minute ACE/SWICS observations of ${\rm O}^{7+}/{\rm O}^{6+}$ during 2008. The decomposition of the in-situ observations into a ‘good measurement’ and a ‘no measurement’ signal allows us to evaluate the performance of a filler signal, i.e., various prescriptions for filling the data gaps. We construct Monte Carlo simulations of synthetic ${\rm O}^{7+}/{\rm O}^{6+}$ composition data and impose the actual data gaps that exist in the observations in order to investigate two different filler signals: one, a linear interpolation between neighboring good data points, and two, the constant mean value of the measured data. Applied to these synthetic data plus filler signal combinations, we quantify the ability of the power spectra significance level procedure to reproduce the ensemble-averaged time-integrated wavelet power per scale of an ideal case, i.e. the synthetic data without imposed data gaps. Finally, we present the wavelet power spectra for the ${\rm O}^{7+}/{\rm O}^{6+}$ data using the confidence levels derived from both the Mean Value and Linear Interpolation data gap filling signals and discuss the results.'
author:
- 'J. K. Edmondson, B. J. Lynch, S. T. Lepri, and T. H. Zurbuchen'
title: 'Analysis of High Cadence In-Situ Solar Wind Ionic Composition Data Using Wavelet Power Spectra Confidence Levels'
---
Introduction
============
Decades of in-situ plasma observations have revealed a rich picture of the solar wind [@Zurbuchen07 and references therein], whose overall structure and magnetic topology follows the solar magnetic activity cycle. Heliospheric solar wind observations reflect the structure of their coronal source regions: a relatively cool, fast solar wind with relatively homogeneous ionic composition and elemental abundances originating from coronal holes [@Geiss95; @McComas02], and a relatively hot, slow solar wind that exhibits considerably more variability in ionic composition and elemental abundances, originating either directly from within the vicinity of coronal streamers [@Gosling97; @Zurbuchen02]. In-situ observations of ionic charge state composition, especially of carbon (${\rm C}^{6+}/{\rm C}^{4+}$) and oxygen (${\rm O}^{7+}/{\rm O}^{6+}$) offer insight into coronal dynamics at temperatures of order one million degrees [e.g., @vonSteiger1997; @Zhao09; @Landi12; @Gilbert12]. Identifiable temporal scales from within the compositional variability may provide insights into the nature of the source regions of the solar wind.
Wavelet transforms are used to identify transient structure coherency as well as global periodicities in time series data [see e.g., @Daubechies1992; @TorrenceCompo1998; @Liu2007]. Wavelet analyses have an advantage over traditional spectral methods by being able to isolate both large timescale and small timescale periodic behavior that occur over only a subset of the time series. Thus, we are able to analyze the frequency decomposition as a function of time. This is extremely useful if we expect the time series to originate from either a time varying source region or, equivalently, to be consecutively sampling many different source regions with varying physical properties, such as in the solar wind.
Recently, @Katsavrias12 used wavelets to examine four solar cycles worth of solar wind plasma, interplanetary magnetic field, and geomagnetic indices to verify intermittent periodicities on timescales shorter than the solar cycle. Common solar timescales ranging from a decade down to hours have been characterized, and timescales of the order of a Carrington Rotation period (approx. 27 days) and shorter (e.g., 14, 9, and 7 days) have been consistently identified in various heliospheric and geomagnetic data [e.g., @Bolzan05; @Fenimore78; @Gonzalez87; @Gonzalez93; @Mursula98; @Nayar01; @Nayar02; @Svalgaard75]. @Temmer07 linked the 9 day timescale to coronal hole variability in the declining phase of solar cycle 23 and @Neugebauer97 used wavelet analyses of [*Ulysses*]{} solar wind speed data to investigate polar microstreams occurring on timescales of 16 hours.
Wavelet power spectra are a powerful tool to identify and characterize structures with specific transient timescales and global periodicities, but all commonly used algorithms require fully populated data-sets. That is inconsistent with solar wind composition data – as well as almost all in situ data-sets – because data gaps occur for a number of reasons. The experiment may undergo maintenance and data may not be available, or the signal to noise of the instrument at a given time may have prevented a valid and accurate measurement. Thus, care must be taken to account for spurious results caused by such data gaps. Thus, to identify characteristic timescales smaller than the largest gap duration, one must either break-up the full data set into disjoint segments of continuous data measurements, or quantify the spurious information introduced into the data set by filling-in the no-measurement times. It is with the latter solution that the methodology described in this paper is concerned.
Our purpose here is to describe a generalized procedure for the construction of wavelet power significance levels that quantify the relative influence of a filler signal of generally arbitrary form interleaved within a measured data signal. The decomposition of the time series allows for a similar decomposition of the total wavelet power spectrum, and thereby quantifying the power spectra associated with the filler signal and nonlinear interference, for comparison against the measured data signal power. Using the decomposition of the signal power spectra, we identify a statistical confidence level against the null hypothesis that a given feature in the total wavelet power spectrum is due to the filler signal and/or interference effects; in other words, we construct a significance measure for the the total wavelet power spectrum that identifies power spectral features resulting from the measured signal.
The structure of the paper is as follows. In Section \[S:WaveletCharacteristics\] we briefly review the wavelet transform, power spectrum, and methods for identifying global periodicities (akin to Fourier modes) as well as transient coherency characteristics. In Section \[S:Data\] we discuss the solar wind ionic composition data obtained by ACE/SWICS during the quiet solar conditions of 2008, and the origin and characteristics of no-measurement data gaps in the context of wavelet analysis. In Section \[S:DataReductionScheme\] we derive the wavelet power statistical confidence level to characterize the effects, and quantify the influence of no-measurement gaps in the data. In Section \[S:MonteCarlo\] we evaluate the performance of two filler signal forms (Linear Interpolation and constant Mean Value) using ensemble-averaged Monte Carlo simulations of a statistical ${\rm O}^{7+}/{\rm O}^{6+}$ ratio model random (1$^{st}$-order Markov) process. In Section \[S:WaveletO7O6\] we examine the wavelet power spectra of actual 12 minute ${\rm O}^{7+}/{\rm O}^{6+}$ data from 2008 with the Linear Interpolation filler signal for the high cadence data gaps, and present our conclusions in Section \[S:Conclusions\].
Rectified Wavelet Power Spectrum Analysis {#S:WaveletCharacteristics}
=========================================
The wavelet transform of a time series $T(t)$ is given by $$\label{E:WaveletTransform}
W_{\rm T}( t , s ) = \int {\rm T} ( t' ) \ \psi^* ( t' , t , s ) \ dt'.$$ In our calculations, the wavelet bases are generated from the Morlet family, though we note all following analysis is valid for any wavelet basis family. The Morlet family is a time-shifted, time-scaled, complex exponential modulated by a Gaussian envelope, $$\psi \left( t' , t , s \right) = \frac{\pi^{1/4}}{\vert s \vert^{1/2}} {\rm exp}\left[ i \omega_{0} \left( \frac{t' - t}{s} \right) \right] {\rm exp}\left[ - \frac{1}{2} \left( \frac{t' - t}{s} \right)^2 \right]$$ where $ \left( t' , t \right) \in I_{T} \times I_{T} \subset \mathbb{R} \times \mathbb{R}$ is the time and time-translation center, respectively, and $s \in I_{S} \subset \mathbb{R}$ is the timescale over which the Gaussian envelope is substantially different from zero. The $\omega_{0} \in \mathbb{R}$ is a non-dimensional frequency parameter defining the number of oscillations of the complex exponential within the Gaussian envelope; we set $\omega_{0} = 6$, yielding approximately three oscillations within the envelope.
The wavelet power spectrum is given by, $\vert W_{\rm T}( t , s ) \vert^2$, for $\psi , T \in L^2 \left( \mathbb{R} \right)$. @TorrenceCompo1998 identify a bias in favor of large timescale features in the canonical power spectrum, which they attribute to the width of the wavelet filter in frequency-space; at large timescales the function is highly compressed yielding sharper peaks of higher amplitude. Equivalently, high frequency peaks tend to be underestimated because the wavelet filter is broad at small timescales. @Liu2007 showed this effect is the difference between the energy and the integration of the energy with respect to time, and thus may be rectified, in practice, by multiplying the wavelet power spectra by the corresponding frequency. Thus, throughout this paper we use the rectified power spectrum given by $$\mathcal{P}_{T} \left( t , s \right) = \vert s \vert^{-1} \vert W_{T} \left( t , s \right) \vert^2.$$
The (rectified) wavelet power spectrum for a general time series and wavelet basis can be highly structured and complex. For solar wind composition data, the time series will likely include characteristic global solar oscillation frequencies, such as the approximate 27-day solar rotation period. In addition to any characteristic global oscillations, the time series will likely be full of transient (non-stationary) ‘coherent structures’ in which variations in the coronal source parameters lead to local variations in the composition ratio data. Thus, we define a localized ‘coherent structure’ as data points that become locally elevated with respect the surrounding measurements, which we note, may spread over a variety of timescales.
Global periodicities feature as horizontal bands of (relatively) strong power in the wavelet power spectrum. We characterize the global periodic behavior by integrating the wavelet power spectrum along each correlation timescale to calculate the energy contained in all wavelet coefficients at that timescale; this is known as the global wavelet power spectrum [see e.g., @LeWang03; @Bolzan05], and for the Morlet family is akin to the Fourier modes. The global periodic frequencies within the time series (e.g., Fourier modes) are identified with the local maxima in the integrated power per timescale. Transient (non-stationary) ‘coherent structures’ feature as localized 2D maxima in the wavelet power spectrum. The timescale corresponding to such local 2D maxima demarcates the coherency size of the transient feature.
[*ACE*]{}/SWICS Measurements of ${\rm O}^{7+}/{\rm O}^{6+}$ Solar Wind Composition {#S:Data}
==================================================================================
The [*Advanced Composition Explorer*]{} spacecraft (ACE) is currently in orbit about the L1 point, $\sim$1.5 million km sunward of Earth [@Stone1998]. Here we analyze data obtained with the Solar Wind Ionic Composition Spectrometer [SWICS; @Gloeckler98]. SWICS is a time-of-flight (TOF) mass spectrometer paired with energy-resolving solid-state detectors (SSDs) and an electrostatic analyzer (ESA) that measures the ionic composition of the solar wind. Ions with the appropriate energy per charge are selected in the ESA. Ion speed is determined in the TOF telescope and the residual energy measured by the SSDs enables particle identification. These measurements allow the independent determination of mass, $M$, charge, $Q$, and energy, $E$, and are virtually free of background contamination [e.g., see @vonSteiger2000].
In order that the wavelet transform defined by equation (\[E:WaveletTransform\]) to be well-defined, and the analysis capable of identifying coherent structure and global oscillation frequencies to the highest time resolution, the (full year) input time series of charge state ratio values must be fully populated at the given cadence. Typically the composition measurements in 1- and 2-hour averages have superb counting statistics, however in the highest time resolution data (12 minute cadence) the flux levels are occasionally too low for a valid measurement to be recorded. For an ionic composition ratio, the presence of a valid data point is subject to the relatively restrictive condition that the ACE/SWICS instrument must have made a measurement for both numerator and denominator with enough counting statistics such that the data reduction algorithm derives a value at the given time resolution, and the denominator must be non-zero. In other words, under low-flux conditions, which occurred throughout 2008, the charge state ratios under consideration occasionally could not be constructed.
The top panel of Figure \[datafig\] shows 12-minute average ${\rm O}^{7+}/{\rm O}^{6+}$ ACE/SWICS measurements for the full 2008 year. Qualitatively, there are no discernible gaps in the time series over the full year. However, upon closer viewing for example of individual Carrington Rotations (bottom panels of Figure \[datafig\]), the ubiquity of such no-measurement data gaps becomes clear. Note the sporadic nature of the gaps between 42 and 49 days within CR2066, as well as the day 121 in CR2069. The frequency of the data gap durations is quantified in Figure \[MissingDataPDF\] which plots the probability distribution function (PDF) of the missing data time durations. From this, we find the vast majority, $90.4\%$, of the no-measurement durations occur on timescales less than 0.1 days (2.4 hours). In addition, $9.4\%$ of the gap durations occur on timescales between 0.1 and 1 days. Only $0.2\%$ of the data gaps have durations greater than 1 day. The single maximum no-measurement duration is 2.5 days (occurring at 214 days into the year).
Any analysis of the wavelet power spectrum of a time series that includes data gaps is only valid to the timescales of the largest data gap (in this case, 2.4 days). Below this timescale, one must break the full year data time series into subsets of continuous measurement durations, and perform similar power spectrum analyses on the individual subset time series. Such a procedure, while valid, leads to a host of issues. For example, any physical global oscillations on timescales below the largest data gap are lost. In addition, the boundary effects associated with the cone of influence within the individual wavelet power spectra [see e.g., @TorrenceCompo1998], become amplified as the size of the data set decreases. In this paper, we take a different approach. We retain as much physical information below the largest data gap timescale as possible by filling the data gaps with a particular signal form and quantifying the propagation of new information introduced into the system throughout the analysis.
Constructing Wavelet Power Confidence Levels to Characterize the Effects of Data Gap {#S:DataReductionScheme}
====================================================================================
In order to attempt to keep any physical information of global oscillation frequencies and coherent structures below the timescale of the largest data gap, we require a fully-populated time series for the full time interval under scrutiny. Therefore, we introduce a particular signal form model to fill the data gaps, and quantify the new information introduced to the system by constructing a statistical confidence level as a measure of the influence of the filler signal on the total wavelet power spectrum.
Wavelet Power Spectrum from a Superposition of Signals
------------------------------------------------------
From a qualitative standpoint, the wavelet power at any given timescale is determined by several factors, the strength of the measured signal, the strength of the filler signal, and interference effects between the data signal and filler signal. To quantify this decomposition of the wavelet power spectrum, we first note that the full time interval, $t \in I_{T} \subset \ \mathbb{R}$, may be decomposed into (discontinuous) interleaved subsets of measurement time, $t \in I_{D} \subset I_{T} \subset \ \mathbb{R}$, and no-measurement time, $t \in I_{F} \subset I_{T} \subset \ \mathbb{R}$. Note, $I_{T} = I_{D} + I_{F}$.
With this decomposition of the time interval, we may then decompose the full time series, $T(t)$, into a linear superposition of two signals over the full duration consisting of the measurement data signal, $D(t)$, such that the values within the no-measurement intervals are set equal to zero; and the no-measurement filler signal, $F(t)$, in which non-zero values fall within the no-measurement intervals. $$\begin{aligned}
D \left( t \right) = & \left\{ \begin{array}{ccr} D \left( t \right) & , & t \in I_{D} \\ 0 & , & t \in I_{F}
\end{array} \right. \\
F \left( t \right) = & \left\{ \begin{array}{ccr} F \left( t \right) & , & t \in I_{F} \\ 0 & , & t \in I_{D}
\end{array} \right.
\label{E:SignalDecomposition}\end{aligned}$$
The full time series is, therefore, $T(t) = D(t) + F(t) \ \forall \ t \in I_{T}$. We note, the full time series may contain zero values, though only where zero measurements were in fact made. On the other hand, the no-measurement intervals are filled by a model of a known functional form.
To demonstrate the procedure, we construct the following example time series shown in Figure \[O7O6model\]. The data signal, $D(t)$, shown in black in the top panel of Figure \[O7O6model\] is a synthetic 1-year time series of ${\rm O}^{7+}/{\rm O}^{6+}$ data (described in further detail in Section \[SS:ModelData\]), into which we introduce the observed 2008 data gaps. The filler signal, $F(t)$, shown in red in the middle panel of Figure \[O7O6model\], is a simple linear interpolation across the data gaps. The filler signal form over an $N$-point gap between good data points ${\rm D}_1$ and ${\rm D}_2$ is given by ${\rm F}_n = {\rm D}_1 + ({\rm D}_2 - {\rm D}_1)(n/(N+1))$ for $n = 1$ to $N$. The bottom panels of Figure \[O7O6model\] plot the composite time series across two Carrington rotations equivalent to those shown in Figure \[datafig\].
The wavelet integral transform is linear in the input signals. For a linear superposition of input signals, $T(t) = D(t) + F(t)$, the wavelet transform in a given basis, $\psi \left( t , t' , s \right)$, of the total signal is simply the linear superposition of the wavelet integral transforms of the component signals. $$\begin{aligned}
\label{E:LinearWaveletIntegral}
\displaystyle
W_\text{T} \left( t , s \right) & = & \int_{I_{T}} \text{T} \left( t' \right) \psi \left( t , t' , s \right) dt' \nonumber \\
\displaystyle
W_\text{T} \left( t , s \right) & = & \int_{I_{T}} \left( \text{D} \left( t' \right) + \text{F} \left( t' \right) \right) \psi \left( t , t' , s \right) dt' \nonumber \\
\displaystyle
W_\text{T} \left( t , s \right) & = & W_\text{D} \left( t , s \right) + W_\text{F} \left( t , s \right)
$$
Where the integration is taken over the entire set, $I_{T} \subset \mathbb{R}$.
In the most general case, the wavelet transform given by equation (\[E:LinearWaveletIntegral\]) is a complex number, $W_{T} : I_{T} \times I_{S} \rightarrow \ \mathbb{C}$, where $t \in I_{F} \subset \ \mathbb{R}$ is the full time interval, and $s \in I_{S} \subset \ \mathbb{R}$ is the timescale interval, and is given by, $$\label{E:ComplexWavelet}
W_{T} \left( t , s \right) = \mathfrak{Re} \lbrace W_{T} \left( t , s \right) \rbrace + i\ \mathfrak{Im} \lbrace W_{T} \left( t , s \right) \rbrace$$
The (rectified) wavelet power signal for the same time and timescale intervals is the square of the amplitude of the wavelet transform, $\mathcal{P}_{T} : I_{T} \times I_{S} \rightarrow \ \mathbb{R}$. $$\label{E:WaveletPower}
\mathcal{P}_{T} \left( t , s \right) = \vert s \vert^{-1} \ \vert W_{T} \left( t , s \right) \vert^2 = \vert s \vert^{-1} \ \left[ \mathfrak{Re}^2 \lbrace W_{T} \left( t , s \right) \rbrace + \mathfrak{Im}^2 \lbrace W_{T} \left( t , s \right) \rbrace \right] \\
$$
In general, the real and imaginary components may take on positive, zero, and negative values, and the square of the real and imaginary components ensures the (rectified) total wavelet power spectrum is positive, semi-definite (i.e., non-negative), for all $\left( t , s \right) \in I_{T} \times I_{S}$.
Substituting the signal decomposition of equations (\[E:LinearWaveletIntegral\]), the amplitude of the wavelet power constructed from a superposition of signals necessarily involves not only the power amplitudes of the individual component signals, but also interference effects between the signals, $$\label{E:NonLinearWaveletPower}
\mathcal{P}_{T} \left( t , s \right) = \mathcal{P}_{D} \left( t , s \right) + \mathcal{P}_{F} \left( t , s \right) + \mathcal{P}_{I} \left( t , s \right)$$ Where we have defined the data signal power, filler signal power, and interference power by, $$\label{E:PowerDecomposition1}
\mathcal{P}_{D} \left( t , s \right) \equiv \ \vert s \vert^{-1} \ \vert W_{D} \left( t , s \right) \vert^2 = \vert s \vert^{-1} \left[ \mathfrak{Re}^2 \lbrace W_{D} \left( t , s \right) \rbrace + \mathfrak{Im}^2 \lbrace W_{D} \left( t , s \right) \rbrace \right]$$ $$\label{E:PowerDecomposition2}
\mathcal{P}_{F} \left( t , s \right) \equiv \ \vert s \vert^{-1} \ \vert W_{F} \left( t , s \right) \vert^2 = \vert s \vert^{-1} \left[ \mathfrak{Re}^2 \lbrace W_{F} \left( t , s \right) \rbrace + \mathfrak{Im}^2 \lbrace W_{F} \left( t , s \right) \rbrace \right]$$ $$\label{E:PowerDecomposition3}
\begin{split}
\mathcal{P}_{I} \left( t , s \right) & \equiv 2 \ \vert s \vert^{-1} \left( \ \mathfrak{Re} \lbrace W_{D} \left( t , s \right) \rbrace \ \mathfrak{Re} \lbrace W_{F} \left( t , s \right) \rbrace \right) \\
& + 2 \ \vert s \vert^{-1} \left( \ \mathfrak{Im} \lbrace W_{D} \left( t , s \right) \rbrace \ \mathfrak{Im} \lbrace W_{F} \left( t , s \right) \rbrace \right)
\end{split}$$
Figure \[WaveletPowerDecomposition1\] plots the wavelet power spectra decomposition for the time series used in Figure \[O7O6model\]: for the data signal $\mathcal{P}_{D}$ (top panel), the filler signal $\mathcal{P}_{F}$ (middle panel), and the interference signal $\mathcal{P}_{I}$ (bottom panel).
From equations (\[E:WaveletPower\]), (\[E:PowerDecomposition1\]) and (\[E:PowerDecomposition2\]), the sets of values realized by the total signal power, the data signal power, and the filler signal power spectrograms are all bounded and non-negative, $\mathcal{P}_{T} \left( t , s \right) \geq 0$, $\mathcal{P}_{D} \left( t , s \right) \geq 0$, and $\mathcal{P}_{F} \left( t , s \right) \geq 0$, for all $\left( t , s \right) \in I_{T} \times I_{S}$ (note, the equality holding if and only if the real and imaginary components of the wavelet transform of the particular time series are simultaneously zero). However, for a given $\left( t , s \right) \in I_{T} \times I_{S}$, the real and imaginary components of the respective data and filler transforms may not be of a similar sign, and thus the respective cross terms may be negative. Therefore, in general, the interference power, $\mathcal{P}_{I} \left( t , s \right)$, of equation (\[E:PowerDecomposition3\]) may realize all real values (positive, zero, and negative).
The negative values of the interference power are interpreted simply as destructive interference, reducing the strictly constructive sum of the individual data and filler signal power spectra such that the total wavelet power spectrum remains a physically meaningful non-negative value. To prove this assertion, we note the decomposition of the time series into measured data and no-measurement filler signals leads to the decomposition of the total wavelet power spectrum given by equation (\[E:NonLinearWaveletPower\]). By equation (\[E:WaveletPower\]), the total wavelet power is positive, semi-definite, $\mathcal{P}_{T} \left( t , s \right) \geq 0$, for all $\left( t , s \right) \in I_{T} \times I_{S}$, thus the decomposition of equation (\[E:NonLinearWaveletPower\]) must be positive, semi-definite for all $\left( t , s \right) \in I_{T} \times I_{S}$, $$\label{E:PositiveSemiDefiniteDecomposition_1}
\mathcal{P}_{D} \left( t , s \right) + \mathcal{P}_{F} \left( t , s \right) + \mathcal{P}_{I} \left( t , s \right) \geq 0 \\$$ It is sufficient to show condition (\[E:PositiveSemiDefiniteDecomposition\_1\]) holds for all $\left( t , s \right) \in I_{T} \times I_{S}$. For any fixed $\left( t_{0} , s_{0} \right) \in I_{T} \times I_{S}$, the values realized by the data and filler power spectra are, by equations (\[E:PowerDecomposition1\]) and (\[E:PowerDecomposition2\]) respectively, $\mathcal{P}_{D} \left( t_{0} , s_{0} \right) = M \geq 0$ and $\mathcal{P}_{F} \left( t_{0} , s_{0} \right) = N \geq 0$. Additionally, their sum is positive, semi-definite, $\mathcal{P}_{D} \left( t_{0} , s_{0} \right) + \mathcal{P}_{F} \left( t_{0} , s_{0} \right) = M + N \geq 0$ (the equality holds if and only if both $M = 0$ and $N = 0$). In the case $\left( t_{0} , s_{0} \right)$ correspond to a positive or zero interference power value, $\mathcal{P}_{I} \left( t_{0} , s_{0} \right) = P \geq 0$, condition (\[E:PositiveSemiDefiniteDecomposition\_1\]) is trivially satisfied. In the case $\left( t_{0} , s_{0} \right)$ correspond to a negative interference power, $\mathcal{P}_{I} \left( t_{0} , s_{0} \right) = P < 0$, condition (\[E:PositiveSemiDefiniteDecomposition\_1\]) may be written, $$\label{E:PositiveSemiDefiniteDecomposition_2}
\begin{array}{c}
\vert M \vert + \vert N \vert - \vert P \vert \geq 0 \\
\vert M \vert + \vert N \vert \geq \vert P \vert \\
\end{array}$$ Since the choice of fixed $\left( t_{0} , s_{0} \right) \in I_{T} \times I_{S}$ is arbitrary, the assertion is proved for all $\left( t , s \right) \in I_{T} \times I_{S}$.
We note, the power decomposition of equation (\[E:NonLinearWaveletPower\]) constrains the form of the filler signal power, and subsequently the interference power, to be comparable with that of the data signal power. For a general signal, the wavelet power amplitude distribution at a given timescale depends on the relative magnitude of the range of values over which the signal is distributed. If a particular filler signal model extends the total signal range too far, then the total wavelet power spectrum will be dominated by filler signal and interference effects, completely saturating the measured signal. This constraint requires the range of values of the filler signal model to be at least of similar order as those of the measured signal (examples include, the mean or RMS values of the measured signal, a bounded linear or spline interpolation between measured data points.). In this paper, we compare Linear Interpolation filler signal and a constant Mean Value filler signal.
Comparison Power Spectrum and Confidence Levels
-----------------------------------------------
We seek to quantify the new information introduced into the total wavelet power spectrum with the choice of filler signal, by constructing a cofidence level against the null-hypothesis that a given feature in the total wavelet power spectrum is the result of the filler signal and/or nonlinear interference effects. In other words, by filling the no-measurement gaps with a filler signal of arbitrary form we are introducing new information into the system. We aim to quantify the influence of the new information in overall the power spectrum, and thereby elucidate the physical information contained in the (incomplete) measured signal to the highest possible cadence.
@TorrenceCompo1998 discuss stationary significance tests for both red-noise and white-noise by equating a weighted local wavelet power spectrum distribution to an assumed (normal) probability distribution, and then calculating the confidence level according to the particular assumed distribution. @Lachowicz09 offered a prescription to construct a significance level for wavelet power spectra against a time series that is the realization of some physical process that generates a signal with an intrinsic power law, $f^{-\alpha}$, variability. The main underlying assumption is that the Fourier power spectra of the comparison signal approximates that for the given signal. In the case of solar wind composition data, we have no *a priori* reason to suspect that a particular ion (or ions in the case of a composition ratio) are generated by a process with an intrinsic power law variability. Thus, we are not interested in comparing against some physical process governed by (say) an intrinsic red-noise power law, but rather simply looking to quantify the effects of both the (arbitrary) filler signal and its associated interference in the total wavelet power spectrum. Thus, the assumption of a comparison of the standard Fourier power spectra between the two signals is no longer physically relevant.
We construct a statistical confidence level, based on the prescription of @Lachowicz09, against the null hypothesis that a particular feature in the total power spectrum is due to either the filler signal, a nonlinear interference effect, or a combination of both; equivalently, that a particular feature in the total signal power spectrum is significant as the result of coherent structures in the measured data signal. Thus, we seek a quantitative comparison measure of the structures of the total signal power against the power spectrum consisting of both the filler signal power and interference power. From equation (\[E:NonLinearWaveletPower\]), we define the comparison power to be, $$\mathcal{P}_{C} \left( t , s \right) \equiv \mathcal{P}_{F} \left( t , s \right) + \mathcal{P}_{I} \left( t , s \right)
\label{E:ComparisonPower}$$
Recall, that while the power of the total power signal is strictly non-negative, $\mathcal{P}_{T} \left( t , s \right) \geq 0$, the range of values of the comparison power spectrum, $\mathcal{P}_{C} \left( t , s \right)$, will, in general, cover some bounded interval that includes zero in the interior, the bounding values of which depend on the relative values of $\mathcal{P}_{F} \left( t , s \right)$ and $\mathcal{P}_{I} \left( t , s \right)$. In other words, the comparison signal includes destructive interference terms of a larger magnitude that the positive filler signal power. That the comparison power may realize negative values requires us to consider the situation in which at a given timescale the comparison power may be so dominated by destructive interference that the resulting confidence level will also realize a negative value. Such an operation is meaningless, since in some sense, it is a comparison between ‘coherent structures’ in the data signal with the process of destructive interference between the data and filler signals.
To rectify this, we use the fact that the comparison power spectrum is bounded from below, $M \equiv \text{inf} \lbrace \mathcal{P}_{C} \left( t , s \right) \rbrace \ \text{for all} \ \left( t , s \right) \in I_{T} \times I_{S}$. We note, in general, $M < 0$, thus, we construct an adjusted total power structure by adding the absolute value of this constant to both sides of equation (\[E:NonLinearWaveletPower\]). $$\mathcal{P}_{T} \left( t , s \right) + \vert M \vert = \mathcal{P}_{D} \left( t , s \right) + \mathcal{P}_{C} \left( t , s \right) + \vert M \vert
\label{E:AdjustedPower}$$
Strictly speaking, we are now constructing a confidence level against the null hypothesis that structures in the adjusted total power, $\mathcal{P}_{T} \left( t , s \right) + \vert M \vert$, are the result of structures in the power spectrum of the adjusted comparison signal, $\mathcal{P}_{C} \left( t , s \right) + \vert M \vert$. We ascribe no physical interpretation to the addition of this constant power value across all timescales. It is required to make the confidence level physically consistent across all possible situations; the idea of comparing physical structures with physical structures by “translating" the process of destructive interference into physically coherent structures. Mathematically, the addition of a constant does not change the relative structure sizes within the power spectra, and thus a significance level constructed on the adjusted spectra retains the physically meaningful information.
For continuous wavelet basis families there is information overlap between timescales (e.g., the basis family is in general not orthonormal), thus we must construct the $p^{th}$ quantile information as a function of timescale. At each fixed timescale, $s_{0} \in I_{S}$, we assume the adjusted comparison power spectrum, $\mathcal{P}_{C} \left( t , s_{0} \right) + \vert M \vert$, is distributed in time as a bounded continuous random variable and construct a probability distribution function, $\rho \left( P ; s_{0} \right)$, from the histogram of the adjusted comparison power values over the full time interval, $t \in I_{T}$; for notational clarity we include the dependence on the given fixed timescale $s_{0}$. There is some ambiguity as to the proper power histogram bin resolution. Under the continuous variable assumption, the bin resolution, $dP$, must be such that all the structures in the adjusted power spectrum are well resolved at the given timescale $s_{0}$. In practice this can be a very small value and therefore computationally expensive. For this study, $dP$ is on the order of 10$^{-5}$.
Physically, the probability distribution function, $\rho \left( P ; s_{0} \right)$, is a measure of the relative influence of the (adjusted) comparison power in the (adjusted) total power spectrum at the given timescale $s_{0}$. The $p^{th}$ quantile significant power level at each timescale is given by the power value, $X_{p} \left( s_{0} \right)$, such that $\text{Prob}\left( \; \rho \left( P ; s_{0} \right) \leq X_{p} \left( s_{0} \right) \; \right)$. Formally, $$\text{Prob}\left( \; \rho \left( P ; s_{0} \right) \leq X_{p} \left( s_{0} \right) \; \right) = \int_{0}^{X_{p} \left( s_{0} \right)} \rho \left( P ; s_{0} \right) \ dP
\label{E:PDF}$$
Note, for each fixed timescale, $s_{0} \in I_{S}$, $X_{p} \left( s_{0} \right)$ is a constant. Therefore, at a given fixed timescale, $s_{0} \in I_{S}$, where the adjusted total power is greater than the power level of the $p^{th}$ quantile, $X_{p} \left( s_{0} \right)$, $$\mathcal{P}_{T} \left( t , s_{0} \right) + \vert M \vert \geq X_{p} \left( s_{0} \right)
\label{E:SignificantCondition}$$
we can say with $p^{th} \%$ confidence that the particular power structure is not due to the filler signal, nor an interference effect between the data signal and the filler signal. There are often timescales in which condition (\[E:SignificantCondition\]) is not satisfied, and thus no (adjusted) total power features are significant with respect to the (adjusted) comparison (filler plus interference) power.
For example, an $80\%$ significance level at each timescale, $s \in I_{S}$, is constructed by (numerically) integrating equation (\[E:PDF\]) until the integral value of exceeds 0.8. The corresponding upper-integration limit, $X_{p} \left( s \right)$, at which this condition is met is the 80$\%$ significant power level at that timescale. Condition (\[E:SignificantCondition\]) then denotes whether the adjusted total power is significant relative to the adjusted comparison power at the coordinates $\left( t , s \right)$.
To illustrate, we choose a fixed timescale, $s_{0} = 2.133$ days, with nice overall variability in the adjusted total power, $\mathcal{P}_{T} \left( t , s_{0} \right)$. Figure \[WaveletPowerDecomposition2\] top panel plots the adjusted comparison wavelet power spectrum, $\mathcal{P}_{C} \left( t , s \right) + |M|$, with a horizontal dashed black line demarcating the (fixed) timescale $s_{0} = 2.133$ days. The bottom panel plots the corresponding PDF, $\rho \left( P ; s_{0} \right)$, of the comparison power signal at the (fixed) timescale $s_{0} = 2.133$ days with the 80%, 90%, and 95% significance power levels, $X_{\lbrace 0.8 , 0.9 , 0.95 \rbrace} \left( s_{0} \right)$, demarcated as vertical red lines. Figure \[WaveletPowerDecomposition3\] top panel shows the adjusted total power, $\mathcal{P}_{T} \left( t , s \right) + |M|$, with (fixed) timescale $s_{0} = 2.133$ day marker. The bottom panel plots the adjusted total power signal at the $s=2.133$ day timescale with the 80%, 90%, and 95% significance power levels, $X_{\lbrace 0.8 , 0.9 , 0.95 \rbrace} \left( s_{0} \right)$, marked respectively with horizontal red lines, corresponding to the power levels calculated from the adjusted comparison signal PDF. For every adjusted total power value greater than the chosen significance level, we can say with 80%, respectively, 90% and 95%, confidence that the power associated with that feature is not due to filler signal or interference effects.
We note, similar effects are seen in the case of the same synthetic time series with the same introduced data gaps, and a constant Mean Value filler signal form (see Appendix \[S:Appendix1\]). Qualitatively, despite the differences between the Linear Interpolation filler signal and constant Mean Value filler signal, the adjusted comparison power spectra share many $0^{th}$-order features (cf. Figure \[WaveletPowerDecomposition2\] and Figure \[fA2\]). Thus, it is the locations and durations of the data gaps, and therefore the interference power $\mathcal{P}_{I} \left( t , s \right)$ that dominates the (adjusted) comparison power spectra, $\mathcal{P}_{C} \left( t , s \right) + |M|$; as opposed to the particular filler power spectrum, $\mathcal{P}_{F} \left( t , s \right)$ associated with a particular form of the $F (t)$ signal.
Evaluating Filler Signal Performance with Monte Carlo Ensemble Modeling {#S:MonteCarlo}
=======================================================================
We have repeated the procedure described in Section \[S:DataReductionScheme\] for an ensemble of 100 different realizations of synthetic ${\rm O}^{7+}/{\rm O}^{6+}$ time series that have the observed 2008 data gaps imposed on each realization. In this section we compare results obtained for the Linear Interpolation filler signal (e.g., Figure \[O7O6model\]) described previously and a constant Mean Value filler signal (e.g., Figure \[fA1\]) where every missing data point is set to the average value of the measurement data points. From these three ensemble sets (the ideal data gap-free model, and the two cases with imposed gaps filled with Linear Interpolation and Mean Value filler forms), we compute the wavelet power spectra for every realization, as well as the 80% confidence level for both filler signal cases (see Appendix \[S:Appendix1\] for representative Figures \[fA1\], \[fA2\], and \[fA3\] corresponding to construction of the power spectra confidence levels for the Mean Value filler signal). From the individual wavelet power spectra for each of the three ensemble sets, we calculate the mean time-integrated power spectra across all (fixed) timescales, as well as the standard deviation. This ensemble-averaged time-integrated power per scale of the ideal set (the synthetic data without the imposed data gaps) is used to compare with the results of the ensemble-averaged time-integrated power per scale *above the 80% significance level* computed for each of the synthetic data sets with gaps and their respective filler signal.
Our application of Monte Carlo modeling can be thought of as a mechanism for investigating the particular frequency response or transfer function of some unknown “black box" in the traditional signal processing sense. The ideal (gap-free) synthetic data corresponds to a set of input waveforms that yield a certain ensemble-averaged, global time-integrated power per scale spectrum. The presence of data gaps, our choice of values to fill those gaps, and our power spectra confidence level threshold condition result in a set of output waveforms which have a well-defined, quantified significance and their own (potentially very different) ensemble-averaged, global time-integrated power per scale spectrum. Understanding and characterizing the influence of missing data on features and properties of the wavelet power spectra is an important and necessary step towards linking those features and properties with the underlying physical processes of their origin.
Modeling Synthetic ${\rm O}^{7+}/{\rm O}^{6+}$ Time Series {#SS:ModelData}
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In order to use Monte Carlo techniques to evaluate the performance of the different filler signals used to replace missing data, we must have a procedure for generating model time series. Obviously, the model time series should be constructed to have statistical properties as similar to the observations as possible, and in our case here, the ${\rm O}^{7+}/{\rm O}^{6+}$ composition ratio data. Due to its intrinsic variability, a number of authors have suggested that solar wind ionic composition measurements can be reasonably approximated by a first-order Markov process [e.g., @Zurbuchen00; @Hefti2000]. Therefore, we construct a random process with the following recursion $$Z_n = Z_{n-1}{\rm exp}\left[ -\Delta t/\tau_{1/e} \right] + G_n$$ where $\Delta t$ in the exponential decay term is the resolution of the data (12 min) and $G_n$ is a random number drawn from a normalized Guassian distribution. For the @Zurbuchen00 $e$-folding time of $\tau_{1/e} = 0.42$ days ($\sim$10 hours), the exponential decay term describing how much memory the process retains is close to unity for the 12-minute data ($e^{-0.02} \sim 0.98$) and slightly less if we were to model the 2-hour averages ($e^{-0.20} \sim 0.82$). The model composition time series is then computed as, $$\label{E:THZModel}
Y_n = {\rm exp}[ \sigma_\ell \hat{Z_n} + \mu_\ell ],$$ where $\hat{Z_n}$ is $Z_n$ normalized to unit variance and $\mu_\ell$, $\sigma_\ell$ are the mean and standard deviation of the natural logarithm of the measured ionic composition ratio. @Zurbuchen00 showed that the ${\rm O}^{7+}/{\rm O}^{6+}$ data had a log-normal distribution with ($\mu_\ell$, $\sigma_\ell$)=($-$1.32, 0.45) and we use those values here. Our model time series reproduces the 10-hour $e$-folding time of the autocorrelation function and has an FFT power spectra that falls off between $f^{-1}$ and $f^{-2}$, consistent with the @Zurbuchen00 analysis.
We note that in the @Edmondson2013 companion paper, we present the results of this modeling tuned to the ${\rm C}^{6+}/{\rm C}^{4+}$ ionic charge state ratio. There we show that, while this type of Markov process modeling produces the log-normal distribution of the in-situ measurements (by construction), the global time-integrated power per scale spectra of the observations contains real information about the physical structure and dynamics of their source region, including properties of the plasma and coronal magnetic field, that are not and cannot be accounted for by a purely random process.
Ensemble-Averaged Integrated Power per Scale {#SS:AvgIPPS}
--------------------------------------------
Figure \[FillerPerformance\] plots the Monte Carlo simulation results for our three ensemble set averages. The ideal case (with no missing data) is shown as the black line; all of the wavelet power of each realization is deemed significant because there are no data gaps. Thus, the ideal integrated power per scale represents the ensemble-average of the global periodicities (Fourier modes) of the ‘input waveforms’. The integrated power per scale for the ensemble-average Linear Interpolation (red asterisks) and ensemble-average Mean Value (blue triangles) cases are the ‘output waveforms’ that result from taking the ideal set of Monte Carlo realizations, adding the 2008 data gaps and a particular filler signal, and applying the 80% power spectra confidence level threshold condition. In other words, the ensemble-average global periodicities (Fourier modes) above the 80% significance level. The error bars in each color represent the statistical uncertainty of one standard deviation in each timescale bin for each of the ensemble sets.
We see that for the Linear Interpolation case, the shape of the ideal cases’ integrated power per scale is well preserved for $s \gtrsim 1$ day but shows increasing departure from the the ideal case at increasingly smaller time scales. The overall relative shape of the global periodicities are qualitatively similar, but the Linear Interpolation case is increasingly attenuating the ‘input waveform’ power for $s < 1$ day. The Mean Value filler ensemble-averaged results however show a very different spectrum response. While there is much more attenuation across all scales, the behavior for scales $s \gtrsim 1$ day retains the shape of both the ideal and Linear Interpolation cases. However, for scales below $\sim$0.1 days (2.4 hrs), the Mean Value ensemble-averaged integrated power per scale rebounds from a local minimum and increases in magnitude through to the Nyquist frequency of the time series.
The Mean Value filler case’s spectral response for $s \lesssim 0.1$ days (2.4 hrs) is primarily the signature of the occurrence frequency of the data gap durations. This small scale (high frequency) amplification is a direct spurious effect following from the interleaving of the constant filler signal within gap durations with relatively high frequency of occurrence, and the measured data signal. The smallest duration data gaps occur ubiquitously throughout the measured data signal, and filling these gaps with any constant value has the effect of creating spurious small scale, pulse-like structures in the full (data plus filler) signal as the wavelet transform passes through the small scales. At wavelet transform scales larger than the majority of gap durations, this effect is mitigated as the gap durations become much smaller than the wavelet filter band pass, hence the integrated power per scale shape reflects that of the ideal gap-free case. In this example, the distribution of 2008 gap durations in the 12 minute ${\rm O}^{7+}/{\rm O}^{6+}$ measurements (Figure \[MissingDataPDF\]) indicates the largest gap is $\sim$2.5 days, the time scale above which the integrated power per scale for all three cases reflect similar trends. Additionally, the vast majority of gap durations, $90.4$%, occur at timescales below 0.1 days in duration, at which point the spurious high-frequency effect dominates. Finally, there is a transition zone between $\sim$0.1 and $\sim$2.5 days in which the slope is much shallower than the ideal gap-free case.
On the other hand, with the Linear Interpolation filler signal, we are able to maintain the relative shape of the ideal data gap-free ensemble-averaged integrated power per scale spectrum over a broader range of scales but with increasing attenuation at progressively smaller scales $s \lesssim 1$ day. This is essentially due to the fact that, in any given data gap, the difference between filler signal values and neighboring synthetic model values are, by construction, much closer (as opposed to the synthetic model values and Mean Value filler).
Figures \[WaveletPowerDecomposition2\] and \[WaveletPowerDecomposition3\], illustrate the construction of significance levels at 80%, 90%, and 95% comparison power. Using this procedure, we calculated the ensemble-averaged integrated power per scale for the Linear Interpolation filler at the 80% (shown as red asterisks in Figure \[FillerPerformance\]), as well as the 90% and 95% significance levels to examine the attenuation due to the significance level threshold conditions. Figure \[HybridFillerPerformance\] plots these results normalized to the ideal gap-free average integrated power per scale spectrum. The ideal case is shown as the black line at unity and the 80%, 90%, and 95% Linear Interpolation cases are shown as red asterisks, green squares, and blue crosses, respectively. Here the scale-dependence of the attenuation with respect to the ideal gap-free ensemble spectrum is readily visible showing a drop from roughly 0.70–0.80 of the ideal average power for $s > 1$ days down to $\sim$0.30 of the ideal power at $s \sim 0.02$ days. For our particular set of model time series and data gap structure, the ensemble-averaged power per scale curves for the different significance levels show very little separation with respect to each other. This could be expected from examination of Figure \[WaveletPowerDecomposition3\] where the time-integrated power for the $s=2.133$ day cut shows only minor differences in the total area under the $\mathcal{P}_T+|M|$ curve and above the various significance level thresholds. Therefore, our selection of the 80% significance level appears reasonable, at least for the time series and data gap properties analyzed here. Mean Value filler comparisons across 80%, 90%, and 95% significance levels exhibit similar trends, albeit with stronger relative attenuation.
Wavelet Analysis of ${\rm O}^{7+}/{\rm O}^{6+}$ 12-Min Data {#S:WaveletO7O6}
===========================================================
We have applied the analysis procedure outlined in Section \[S:DataReductionScheme\] to the ACE/SWICS ${\rm O}^{7+}/{\rm O}^{6+}$ data shown in Figure \[datafig\] using both the Linear Interpolation and constant Mean Value prescriptions for filling the data gaps. The wavelet power spectra for full data plus both filler signal models were calculated, as well as the wavelet power spectra decomposition from the two filler signals and their respective nonlinear interference components. The adjusted comparison power was then used to construct the 80% significance levels for each timescale. The results are shown in Figures \[figO7O6wavelet\], \[figO7O6wavelet2\], and \[figO7O6wavelet3\]. In Figure \[figO7O6wavelet\], the total wavelet power spectra for the ${\rm O}^{7+}/{\rm O}^{6+}$ data with a Linear Interpolation filler and the constant Mean Value filler are shown in the top and bottom panels, respectively. Figure \[figO7O6wavelet2\] plots the corresponding wavelet power that exceeds the 80% confidence level thresholds. Figure \[figO7O6wavelet3\] plots on a linear scale, the normalized time-integrated power per scale for both the overall total power spectra (top row) and 80% significant power (bottom row), for the Linear Interpolation filler signal (right column) and constant Mean Value filler signal (left column).
For the Morlet wavelet family, the time-integrated power per timescale (also known as the global wavelet power) of Figure \[figO7O6wavelet3\] is akin to Fourier mode decomposition. The integrated wavelet power per scales of the ${\rm O}^{7+}/{\rm O}^{6+}$ for both filler signal forms, in both the total and significant power, exhibit a number of well defined peaks corresponding to relatively well defined Fourier modes (globally periodic timescales) in similar timescale neighborhoods.
In the Linear Interpolation filler signal case (left column), there are three strong Fourier modes (peaks) occurring at approximately $\lbrace$ 3, 8–10, 18–28 $\rbrace$ days, in both total and significant power cases. Below $\sim$1 day timescales, it becomes difficult to discern Fourier modes (peaks) from the power law shape. As explained above, the large high-frequency effect (timescales $\lesssim$ 0.1 days) in the Mean Value case (right column) reflects the nature of the gap durations. Outside of this effect, there are well-defined Fourier modes in both the total power and significant power that occur at approximately, $\lbrace$ 2.5, 8–9, 13–17, 25–30, 45–50 $\rbrace$ day timescales.
The smallest identifiable Fourier modes, at 2.5 and 3 days, respectively, may be an artifact of the largest data gap duration in the measured signal. However, @Zhao09 showed that the slow solar wind, as determined by ${\rm O}^{7+}/{\rm O}^{6+} \ge 0.145$, has a mean width centered on the heliospheric current sheet of approximately 20$^\circ$ (40$^\circ$) during solar minimum (maximum). We note that our other significant integrated power peak at the 3–4 day correlation timescale corresponds to an $\sim$45$^\circ$ width given the 13$^\circ$ day$^{-1}$ solar rotation rate. This may reflect crossing the slow solar wind region surrounding the helmet streamer belt in a highly inclined configuration and would be consistent with the width of the slow solar wind distribution observed in the Ulysses fast latitude scan [@McComas00].
@Temmer07 identified 9-day periodicities in ACE solar wind parameters over the 1998-2006 period and showed these likely arose due to the distribution of coronal holes via time series of coronal hole area. @Katsavrias12 likewise identified both the 9-day and 13.5 day peaks in solar wind speed, proton temperature, density, and components of the magnetic field over a four solar cycle interval (1966–2010). The 18–28 day, and 25–30 day timescales are clearly associated with Carrington rotation effects.
Interestingly, the Carrington rotation periodicity is absent from the significant Linear Interpolation filler power spectra. This is largely due to the unusual solar minimum conditions during 2008. In the 12-minute ${\rm O}^{7+}/{\rm O}^{6+}$ data shown in Figure \[datafig\], one may identify a *qualitative* recurrent $\sim$5 day enhancement repeating with a 27-day periodicity for three consecutive Carrington Rotations at the beginning of the year (CR2066–CR2068). However, this enhancement is absent (potentially due to a data gap) in the fourth Carrington Rotation (CR2069) and virtually indistinguishable during the remainder of the year. In fact, the wavelet power in the top panel of Figure \[figO7O6wavelet\] also shows this as a reasonably strong intensity stripe at the 27-day timescale that falls outside of the 80% significance contours for the first 100 days and a more modest intensity signal at that timescale up until 200 days; thus, the signal is present in the total wavelet power, but does not exceed the 80% confidence level threshold derived from the Linear Interpolation filler signal and its interference effects.
On the other hand, while the 27-day periodicity is absent, its first harmonic at approximately 13.5 days is present above the 80% significance level for the entire data set duration. The preference for this periodicity seems likely due to the large-scale coronal magnetic field structure and consequently, the solar wind structure in the heliosphere. During the solar cycle 23 solar minimum, the polar fields were substantially weaker than usual resulting in a more highly warped helmet streamer belt, more pseudostreamers, and more complexity in the mapping of the solar wind source regions to smaller, low latitude coronal holes [e.g., @Lee09; @Riley12]. @MursulaZieger1996 have argued the 13.5 day periodicity could be explained by a two slow-fast stream structure per Carrington Rotation that may result from a highly warped helmet streamer belt, but it is not clear this is universally applicable [e.g., see discussion by @Temmer07].
Finally, the 45–50 day Fourier mode that shows up in the large timescale tail of the Mean Value filler case, is likely a harmonic of the Carrington rotation periodicity. Above this scale, for a single year data set (2008), the total integrated power per scale is primarily bounded by the wavelet cone-of-influence (see Figure \[figO7O6wavelet\]).
Conclusions {#S:Conclusions}
===========
We have presented a generalized procedure for constructing a wavelet power spectrum significance level measure that quantifies the relative influence of two interleaved signals. In this investigation, the total signal is an interleaved combination of measured data and some (general, arbitrary) signal imposed to fill the ‘no measurement’ gaps at the given cadence. We constructed a statistical confidence level against the null-hypothesis that a given feature in the total wavelet power spectrum is strictly due to filler signal and the nonlinear interference effects between the filler signal and measured data signal.
We apply this power spectra confidence procedure on Monte Carlo simulations of synthetic ${\rm O}^{7+}/{\rm O}^{6+}$ ionic composition data to evaluate the performance of the Mean Value and Linear Interpolation filler signals. Using the performance criteria of reproducing the ideal, data gap-free ensemble-averaged time-integrated power per scale, we show that the Linear Interpolation filler signal does a better job than the Mean Value signal across all but the smallest temporal scales and effectively acts as a low-pass filter suppressing the inherent high frequency (small timescale) power that arise from the frequency of the missing data and duration of the data gaps. We show that for our sparsely populated data set, the 80%, 90%, and 95% confidence levels yield almost identical results for the synthetic data ensemble.
We calculated the ${\rm O}^{7+}/{\rm O}^{6+}$ wavelet power during the quiet-sun solar minimum of 2008 using both the Mean Value and Linear Interpolation filler signals, show the structure of their derived 80% power confidence levels, and present the total and $\ge$80% significant time-integrated power per scale spectra. Our analysis using the Linear Interpolation data gap filler signal yields strong Fourier mode harmonics in both the total and significant integrated power per scale spectrum at $\lbrace$ 3, 8–10, 18–28 $\rbrace$ days. Each of these peaks are also visible in the total and significant integrated power per scale when using the Mean Value filler, but the relative magnitude of the spectrum for scales $\gtrsim$0.10 days is dwarfed by the (spurious) power associated with very-high frequencies ($s < 0.10$ days). In a companion publication [@Edmondson2013], we have applied the power spectra confidence analysis presented here to the ${\rm C}^{6+}/{\rm C}^{4+}$ ionic composition ratio and discuss the implications of the coherent structure and variability of ionic composition ratios for current theories of solar wind generation.
The authors would like to thank the reviewer for their comments and suggestions which substantially improved the paper. J.K.E., S.T.L., and T.H.Z. acknowledge support from NASA LWS NNX10AQ61G and NNX07AB99G. B.J.L acknowledges support from AFOSR YIP FA9550-11-1-0048 and NASA HTP NNX11AJ65G.
Power Spectra and Confidence Levels for the Mean Value Filler Signal {#S:Appendix1}
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In Section \[S:DataReductionScheme\] we used one realization of the Zurbuchen et al. inspired Markov process modeling to illustrate the wavelet power spectra confidence procedure. First, we generated a synthetic time series time series from Equations (19) and (20), then imposed data gaps corresponding to the missing data intervals in the 2008 12 minute ${\rm O}^{7+}/{\rm O}^{6+}$ data. The analysis of Section \[S:DataReductionScheme\] used the Linear Interpolation filler signal to populate the missing data intervals and calculate the various wavelet power spectra associated with the ‘good measurement’ data, the filler signal, and their nonlinear interference. A reference comparison power was constructed from the filler signal and interference power contributions and used to quantify the significance levels associated with features in the total wavelet power spectra. In section \[S:MonteCarlo\], we presented the ensemble-averaged results from performing this procedure on a set of 100 realizations of the Markov process using both the Linear Interpolation filler signal and a constant Mean Value filler signal.
Here we present details of our power spectra confidence level construction for the model realization example of section \[S:DataReductionScheme\] with the Mean Value filler signal. Figure \[fA1\] shows time series in two zoomed in views for same synthetic data $D(t)$ shown in Figure \[O7O6model\] but for the constant Mean Value filler signal $F(t)$. Following the decomposition of the time series and calculation the power of the wavelet transforms of the constituent components, the top panel of Figure \[fA2\] plots the comparison power plus offset $\mathcal{P}_{C} + |M|$. The bottom panel of Figure \[fA2\] plots the PDF of the comparison power at the $s = 2.133$ day scale and the 80%, 90%, and 95% levels of the distribution. The similarities and differences in the comparison power between the Mean Value filler signal and the Linear Interpolation case are readily apparent when comparing Figures \[fA2\] and \[WaveletPowerDecomposition2\].
First, we see that the comparison power wavelet has both a similar range in magnitude and qualitative large scale structure in $(t, s)$. For example, the regions of relatively low comparison power levels (saturated as white in the color scale) at the $s\sim10$ day scale features are quite similar in shape and location, the overall trend of comparison power levels at scales $0.1 \lesssim s \lesssim 1.0$ days being elevated with respect to $s \gtrsim 1.0$ days, and the largest power levels (saturated with magenta in the color scale) corresponding to many fine-scale linear striations for $s \lesssim 0.1$ days. From the relative amount of color scale saturation at the smallest scales, the Mean Value comparison power has a broader temporal extent indicating more high frequency interference power throughout the time series. The lower panels of Figures \[fA2\] and \[WaveletPowerDecomposition2\] however, have a very different PDFs for their respective comparison power levels (although the overall range of values are comparable). While the Linear Interpolation PDF is symmetric and centered around a mid-point value of $\sim$44, the Mean Value PDF has more of an exponential fall off from a maximum at $\sim$41.3. Thus, we can see the relative contributions to the comparison power from: (1) the form and values of the $F(t)$ filler signal in shape of the PDF at a given scale, and (2) in the location, duration, and distribution of the data gaps (i.e., where $F(t) \ne 0$) and the resulting interference power in the overall, large scale properties of the comparison wavelet power.
Figure \[fA3\] plots the total power plus offset $\mathcal{P}_{T} + |M|$ in the top panel and the $s = 2.133$ day cut in the lower panel for the Mean Value filler in the same format as and for direct comparison with Linear Interpolation results in Figure \[WaveletPowerDecomposition3\]. Again, there are both important similarities in the overall qualitative properties of the wavelet power and important differences arising from the different filler signals. In the wavelet power, the $s \gtrsim 5.0$ days features are less prominent in the Mean Value case, whereas the smallest scale structures at $s \lesssim 0.1$ days are much more prominent. The lower panel of Figures \[fA3\] shows that the Mean Value case has no total power at the $s=2.133$ day scale that falls above the 95% significance level, and only two temporal locations that exceed the 80% level. The most prominent Mean Value total power feature at $t\sim$DOY 145 is obviously present in the Linear Interpolation wavelet power, but only slightly exceeds the Linear Interpolation 80% significance level at this scale.
The overall, qualitative scale-dependent influence of the Mean Value and Linear Interpolation filler signals to the total wavelet power in the single representative synthetic data example from Section \[S:DataReductionScheme\] are reproduced in the properties of the ensemble-averaged behavior obtained from the Monte Carlo simulations in Section \[S:MonteCarlo\] and in the application to the actual ${\rm O}^{7+}/{\rm O}^{6+}$ measurements presented in Section \[S:WaveletO7O6\].
Bolzan, M., Rosa, R.R., Ramos, F.M., Fagundes, P.R., & Sahai, Y. 2005, J. Atmos. Solar-Terr. Phys. 67, 1843
Daubechies, I. 1992, *Ten Lectures on Wavelets*, Society for Industrial and Applied Mathematics.
Edmondson, J. K., Lynch, B. J., Lepri, S. T., & Zurbuchen, T. H. 2013, “Coherent Structure in Solar Wind ${\rm C}^{6+}/{\rm C}^{4+}$ Ionic Composition Data During the Quiet-Sun Conditions of 2008," , in press
Fenimore, E. E., Asbridge, J. R., Bame, S. J., Feldman, W. C., & Gosling, J. T. 1978, , 83, 4353
Geiss, J., Gloeckler, G., von Steiger, R., Balsiger, H., Fisk, L. A., Galvin, A. B., Ipavich, F. M., Livi, S., McKenzie, J. F., Ogilvie, K. W., & Wilken, B. 1995, Science, 268, 1033
Gilbert, J. A., Lepri, S. T., Landi, E., & Zurbuchen, T. H. 2012, , 751, 20
Gloeckler, G., Cain, J., Ipavich, F. M., Tums, E. O., Bedini, P., Fisk, L. A., Zurbuchen, T. H., Bochsler, P., Fischer, J., Wimmer-Schweingruber, R. F., Geiss, J., & Kallenbach, R. 1998, , 86, 497
Gonzalez A. L. C., & Gonzalez, W. D. 1987, , 92, 4357.
Gonzalez A. L. C., Gonzalez, W. D., Dutra, S. L., & Tsurutani, B. T. 1993, , 98, 9215
Gosling, J. T. 1997, AIP Conf. Ser., 385, 17
Hefti, S., Grünwaldt, H., Bochsler, P., & Aellig, M. R. 2000, , 108, A5, 10527
Katsavrias, Ch., Preka-Papadema, P., & Moussas, X. 2012, , 280, 623 Lachowicz, P. 2009, IEEE Transactions on Signal Processing, arXiv:0906.4176v1 \[astro-ph.HE\]
Landi, E., Alexander, R. L., Gruesbeck, J. R., Gilbert, J. A., Lepri, S. T., Manchester, W. B., & Zurbuchen, T. H. 2012, , 744, 100
Le, G.-M., & Wang, J.-L. 2003, Chinese Journal of Astronomical and Astrophysical, 3, 391Ð394
Lee, C. O., Luhmann, J. G., Zhao, X. P., Liu, Y., Riley, P., Arge, C. N., Russell, C. T., & de Pater, I. 2009, , 256, 345
Liu, Y., San Liang, X., & Weisberg, R. H. 2007, J. Atmos. Oceanic Technol., 24, 2093
McComas, D. J., Barraclough, B. L., Funsten, H. O., et al. 2000, , 105, A5, 10419
McComas, D. J., Elliott, H. A., & von Steiger, R. 2002, , 29, 9, 28-1
Mursula K. & Zieger, B., 1996, , 101, 27077
Mursula K. & Zieger, B., 1998, , 25, 1851
Neugebauer, M., Ruzmaikin, A., & McComas, D. J. 1997, AIP Conf. Proc., 385, 41
Prabhakaran Nayar, S. R., Sanalkumaran Nair, V., Radhika, V. N., & Revathy, K. 2001, , 201, 405
Prabhakaran Nayar, S. R., Radhika, V. N., Revathy, K., & Ramadas, V. 2002, , 208, 359
Riley, P., & Luhmann, J. G. 2012, , 277, 355
Stone, E. C., Frandsen, A. M., Mewaldt, R. A., Christian, E. R., Margolies, D., Ormes, J. F., & Snow, F. 1998, , 86, 1
Svalgaard, L. & Wilcox, J. M. 1975, , 41, 461
Temmer, M., Vrsnak, B., & Veronig, A.M. 2007, , 241, 371
Torrence, C., & Compo, G. P. 1998, “A practical guide to wavelet analysis", Bull. Am. Meteorol. Soc., 61-78
von Steiger, R., Schwadron, N. A., Fisk, L. A., et al. 2000, , 105, 27217
von Steiger, R., Geiss, J., & Gloeckler, G. 1997, Cosmic Winds and the Heliosphere, Space Sci. Ser., edited by J. R. Jokipii, C. P. Sonett, and M. S. Giampapa, Univ. of Ariz. Press, Tucson.
Zhao, L., T. H. Zurbuchen, & L. A. Fisk 2009, , 36, L14104
Zurbuchen, T. H., Hefti, S., Fisk, L. A., Gloeckler, G., & Schwadron, N. A. 2000, , 105, A8, 18327
Zurbuchen, T. H., Fisk, L. A., Gloeckler, G., & von Steiger, R. 2002, ., 29, 9, 66-1
Zurbuchen, T. H. 2007, , 45, 297
| ArXiv |
---
abstract: 'The thermodynamic entropy of an isolated system is given by its von Neumann entropy. Over the last few years, there is an intense activity to understand thermodynamic entropy from the principles of quantum mechanics. More specifically, is there a relation between the (von Neumann) entropy of entanglement between a system and some (separate) environment is related to the thermodynamic entropy? It is difficult to obtain the relation for many body systems, hence, most of the work in the literature has focused on small number systems. In this work, we consider black-holes — that are simple yet macroscopic systems — and show that a direct connection could not be made between the entropy of entanglement and the Hawking temperature. In this work, within the adiabatic approximation, we explicitly show that the Hawking temperature is indeed given by the rate of change of the entropy of entanglement across a black hole’s horizon with regard to the system energy. This is yet another numerical evidence to understand the key features of black hole thermodynamics from the viewpoint of quantum information theory.'
author:
- 'S. Santhosh Kumar'
- 'S. Shankaranarayanan'
title: Quantum entanglement and Hawking temperature
---
Introduction
============
Equilibrium statistical mechanics allows a successful description of the thermodynamic properties of matter . More importantly, it relates entropy, a phenomenological quantity in thermodynamics, to the volume of a certain region in phase space [@Wehrl1978-RMP]. The laws of thermodynamics are also equally applicable to quantum mechanical systems. A lot of progress has been made recently in studying the cold trap atoms that are largely isolated from surroundings [@weiss2006-nature; @gross2008-nature; @smith2013-njp; @yukalov2007-LPL]. Furthermore, the availability of Feshbach resonances is shown to be useful to control the strength of interactions, to realize strongly correlated systems, and to drive these systems between different quantum phases in controlled manner [@Osterloh2002; @wu2004; @rey2010-njp; @santos2010-njp]. These experiments have raised the possibility of understanding the emergence of thermodynamics from principles of quantum mechanics. The fundamental questions that one hopes to answer from these investigations are: How the macroscopic laws of thermodynamics emerge from the reversible quantum dynamics? How to understand the thermalization of a closed quantum systems? What are the relations between information, thermodynamics and quantum mechanics [@2006-Lloyd-NPhys; @2008-Brandao; @horodecki-2008; @popescu97; @vedral98; @plenio98] ? While answer to these questions, for many body system is out of sight, some important progress has been made by considering simple lattice systems (See, for instance, Refs. [@1994-srednicki; @rigol2008-nature; @2012-srednicki; @rahul2015-ARCMP]). In this work, in an attempt to address some of the above questions, our focus is on another simple, yet, macroscopic system — black-holes.
It has long been conjectured that a black hole’s thermodynamic entropy is given by its entropy of entanglement across the horizon [@bombelli86; @srednicki93; @eisert2005; @shanki2006; @shanki-review; @solodukhin2011; @shanki2013]. However, this has never been directly related to the Hawking temperature [@hawking75]. Here we show that:
(i) Hawking temperature is given by the rate of change of the entropy of entanglement across a black hole’s horizon with regard to the system energy.
(ii) The information lost across the horizon is related to black hole entropy and laws of black hole mechanics emerge from entanglement across the horizon.
The model we consider is complementary to other models that investigate the emergence of thermodynamics [@2006-Lloyd-NPhys; @2008-Brandao; @horodecki-2008; @popescu97; @vedral98; @plenio98]: First, we evaluate the entanglement entropy for a relativistic free scalar fields propagating in the black-hole background while the simple lattice models that were considered are non-relativistic. Second, quantum entanglement can be unambiguously quantified only for bipartite systems [@horodecki2009; @eisert2010]. While the bipartite system is an approximation for applications to many body systems, here, the event horizon provides a natural boundary.
Evaluation of the entanglement of a relativistic free scalar field, as always, is the simplest model. However, even for free fields it is difficult to obtain the entanglement entropy. The free fields are Gaussian and these states are entirely characterized by the covariance matrix. It is generally difficult to handle covariance matrices in an infinite dimensional Hilbert space [@eisert2010]. There are two ways to calculate entanglement entropy in the literature. One approach is to use the replica trick which rests on evaluating the partition function on an n-fold cover of the background geometry where a cut is introduced throughout the exterior of the entangling surface [@eisert2010; @cardy2004]. Second is a [*direct approach*]{}, where the Hamiltonian of the field is discretized and the reduced density matrix is evaluated in the real space. We adopt this approach as entanglement entropy may have more symmetries than the Lagrangian of the system [@krishnand2014].
To remove the spurious effects due to the coordinate singularity at the horizon[^1], we consider Lemaître coordinate which is explicitly time-dependent. One of the features that we exploit in our computation is that for a fixed Lemaître time coordinate, Hamiltonian of the scalar field in Schwarzschild space-time reduces to the scalar field Hamiltonian in flat space-time [@shanki-review]. The procedure we adopt is the following:
(i) We perturbatively evolve the Hamiltonian about the fixed Lemaître time.
(ii) We obtain the entanglement entropy at different times. We show that at all times, the entanglement entropy satisfies the area law i. e. $S(\epsilon) = C(\epsilon) A$ where $S(\epsilon)$ is the entanglement entropy evaluated at a given Lemaître time $(\epsilon)$, $C(\epsilon)$ is the proportionality constant that depends on $\epsilon$, and $A$ is the area of black hole horizon. In other words, the value of the entropy is different at different times.
(iii) We calculate the change in entropy as function of $\epsilon$, i. e., $\Delta S/\Delta \epsilon$. Similarly we calculate change in energy $E(\e)$, i.e., $\Delta E/\Delta \epsilon$.
For several black-hole metrics, we explicitly show that ratio of the rate of change of energy and the rate of change of entropy is identical to the Hawking temperature.
The outline of the paper is as follows: In Sec. (\[sec.1\]), we set up our model Hamiltonian to obtain the entanglement entropy in ($D+2$)-dimensional space time. Also, we define [*entanglement temperature*]{}, which had the same structure from the statistical mechanics, that is, ratio of change in total energy to change in entanglement entropy. In Sec. (\[sec.2\]), we numerically show that for different black hole space times, the divergent free [*entanglement temperature*]{} matches approximately with the Hawking temperature obtained from general theory of relativity and its Lovelock generalization. This provides a strong evidence towards the interpretation of entanglement entropy as the Bekenstein-Hawking entropy. Finally in Sec. (\[sec.3\]), we conclude with a discussion to connect our analysis with the eigenstate thermalization hypothesis for the closed quantum systems [@2012-srednicki].
Throughout this work, the metric signature we adopt is $(+,-,-,-)$ and set $\hbar=k_{B} =c=1$.
Model and Setup {#sec.1}
===============
Motivation
----------
Before we go on to evaluating entanglement entropy (EE) of a quantum scalar field propagating in black-hole background, we briefly discuss the motivation for the studying entanglement entropy of a scalar field. Consider the Einstein-Hilbert action with a positive cosmological constant ($|\Lambda|$): \[eq:EHAction\] S\_[\_[EH]{}]{} (|[g]{}) = M\_[\_[Pl]{}]{}\^2 d\^4x . Perturbing the above action w.r.t. the metric $\bar{g}_{\mu\nu} = g_{\mu\nu} + h_{\mu\nu}$, the action up to second order becomes [@shanki-review]: S\_[\_[EH]{}]{}(g, h) &=& - d\^4x . The above action corresponds to massive ($\Lambda$) spin-2 field ($h_{\mu\nu}$) propagating in the background metric $g_{\mu\nu}$. Rewriting, $h_{\mu\nu} = M_{_{\rm Pl}}^{-1} \epsilon_{\mu\nu} \Phi(x^{\mu})$ \[where $\epsilon_{\mu\nu}$ is the constant polarization tensor\], the above action can be written as S\_[\_[EH]{}]{} (g, h) = - 1 2 d\^4x . which is the action for the massive scalar field propagating in the background metric $g_{\mu\nu}$. In this work, we consider massless ($\Lambda = 0$ corresponding to asymptotically flat space-time) scalar field propagating in $(D + 2)-$dimensional spherically symmetric space-time.
Model
-----
The canonical action for the massless, real scalar field $\Phi(x^{\mu})$ propagating in $(D + 2)-$dimensional space-time is \[equ21\] =d\^[D+2]{}[**x**]{}g\^\_([**x**]{})\_([**x**]{}) where $g_{\m\n} $ is the spherically symmetric Lemaître line-element : \[equ22\] ds\^2= d\^2-ł(1-f\[r(,)\])d\^2-r\^2(,)dØ\^2\_D where $\t,\xi$ are the time and radial components in Lemaître coordinates, respectively, $r$ is the radial distance in Schwarzschild coordinate and $d\O_D$ is the $D-$dimensional angular line-element. In order for the line-element (\[equ22\]) to describe a black hole, the space-time must contain a singularity (say at $r = 0$) and have horizons. We assume that the asymptotically flat space-time contains one non-degenerate event-horizon at $r_h$. The specific form of $f(r)$ corresponds to different space-time.
Lemaître coordinate system has the following interesting properties:
(i) The coordinate $\tau$ is time-like all across $0< r < \infty$, similarly $\xi$ is space-like all across $0< r < \infty$.
(ii) Lemaître coordinate system does not have coordinate singularity at the horizon.
(iii) This coordinate system is time-dependent. The test particles at rest relative to the reference system are particles moving freely in the given field .
(iv) Scalar field propagating in this coordinate system is explicitly time-dependent.
The spherical symmetry of the line-element (\[equ22\]) allows us to decompose the normal modes of the scalar field as: \[equ23\] ([**x**]{})=\_[l,m\_i]{}\_[lm\_i]{}(,) Z\_[\_[lm\_i]{}]{}(,\_i), where $i \in \{1,2,\ldots D-1\}$ and $Z_{_{lm_i}}$’s are the real hyper-spherical harmonics. We define the following dimensionless parameters: $\til{r}=r/r_h, \thin \til{\xi}=\xi/r_h,\thin \til{\t}=\t/r_h, \thin\til{\Phi}_{lm}=r_h \, \Phi_{lm}. $ By the substitution of the orthogonal properties of $Z_{_{lm_i}}$, the canonical massless scalar field action becomes, \[equ26\] &=&\_[\_[l,m\_i]{}]{}dd \^D ł\[ (\_[ ]{}\_[lm\_i]{})\^2 -.\
&&ł. (\_\_[lm\_i]{})\^2-\_[lm\_i]{}\^2\] The above action contains non-linear time-dependent terms through $f(\til r)$. Hence, the Hamiltonian obtained from the above action will have non-linear time-dependence. While the full non-linear time-dependence is necessary to understand the small size black-holes, for large size black-holes, it is sufficient to linearize the above action by fixing the time-slice and performing the following infinitesimal transformation about a particular Lemaître time $\til \t$ [@toms]. More specifically,
’=+, ’=,\
(’,’)=(+,),\
\_[lm\_i]{}(,)’\_[lm\_i]{}(’,)=\_[lm\_i]{}(,) \[equ27\]
where $\e$ is the infinitesimal Lemaître time. The functional expansion of $f(\til r)$ about $\e$ and the following relation between the Lemaître coordinates , - =, allow us to perform the perturbative expansion in the above action. After doing the Legendre transformation, the Hamiltonian up to second order in $\e$ is \[Hamilt\_1\] H ()H\_[\_0]{}+ V\_[\_1]{}+\^2 V\_[\_2]{} where $ H_{_0}$ is the unperturbed scalar field Hamiltonian in the flat space-time, $V_{_1} \,\mbox{and}\,V_{_2}$ are the perturbed parts of the Hamiltonian (for details, see Appendix \[app1\]). Physically, the above infinitesimal transformations (\[equ27\]) correspond to perturbatively expanding the scalar field about a particular Lemaître time.
Important observations
----------------------
The Hamiltonian in Eq. (\[Hamilt\_1\]) is key equation regarding which we would like to stress the following points: First, in the limit of $\epsilon \to 0$, the Hamiltonian reduces to that of a free scalar field propagating in flat space-time [@shanki-review]. In other words, the zeroth order Hamiltonian is identical for all the space-times. Higher order $\epsilon$ terms contain information about the global space-time structure and, more importantly, the horizon properties.
Second, the Lemaître coordinate is intrinsically time-dependent; the $\epsilon$ expansion of the Hamiltonian corresponds to the perturbation about the Lemaître time. Here, we assume that the Hamiltonian $H$ undergoes adiabatic evolution and the ground state $\Psi_{GS}$ is the instantaneous ground state at all Lemaître times. This assumption is valid for large black-holes as Hawking evaporation is not significant. Also, since the line-element is time-asymmetric, the vacuum state is Unruh vacuum. Evaluation of the entanglement entropy for different values of $\epsilon$ corresponds to different values of Lemaître time. As we will show explicitly in the next section, entanglement entropy at a given $\epsilon$ satisfies the area law \[$S(\epsilon) \propto A$\] and the proportionality constant depends on $\epsilon$ i. e. $S(\epsilon) = C(\epsilon) A$.
Third, it is not possible to obtain a closed form analytic expression for the density matrix (tracing out the quantum degrees of freedom associated with the scalar field inside a spherical region of radius $r_h$) and hence, we need to resort to numerical methods. In order to do that we take a spatially uniform radial grid, $\{ r_j\}$, with $b = r_{j + 1} - r_j$. We discretize the Hamiltonian $H$ in Eq.(\[Hamilt\_1\]). The procedure to obtain the entanglement entropy for different $\epsilon$ is similar to the one discussed in Refs. [@srednicki93; @shanki-review]. In this work, we assume that the quantum state corresponding to the discretized Hamiltonian is the ground state with wave-function $\Psi_{GS}(x_1,\ldots,x_n;y_1,\ldots,y_{N- n})$. The reduced density matrix $\rho(\vec y,{\vec y\,}')$ is obtained by tracing over the first $n$ of the $N$ oscillators (y,[y]{}’) =(\_[i =1]{}\^[n]{} dx\_i) \_[GS]{}(x\_1,..,x\_n;y) \^[\*]{}\_[GS]{}(x\_1,..,x\_n;[y]{}’)
Fourth, in this work, we use von Neumann entropy \[renyi\] S() = -ł() as the measure of entanglement. In analogy with microcanonical ensemble picture of equilibrium statistical mechanics, evaluation of the Hamiltonian $H$ at different infinitesimal Lemaître time $\epsilon$, corresponds to setting the system at different internal energies. In analogy we define [*entanglement temperature*]{} [@sakaguchi89]: \[temp\_1\] = = The above definition is consistent with the statistical mechanical definition of temperature. In statistical mechanics, temperature is obtained by evaluating change in the entropy and energy w.r.t. thermodynamic quantities. In our case, entanglement entropy and energy depend on the Lemaître time, we have evaluated the change in the entanglement entropy and energy w.r.t. $\epsilon$. In other words, we calculate the change in the ground state energy (entanglement entropy) for different values of $\epsilon$ and find the ratio of the change in the ground state energy and change in the EE. As we will show in the next section, EE and energy goes linearly with $\epsilon$ and hence, the temperature does not depend on $\epsilon$. While the EE and the energy diverge, their ratio is a non-divergent quantity. To understand this, let us do a dimensional analysis
N\^[D+1]{}ł()\^[D+1]{}, \[S\]\
\[T\_[EE]{}\]N\
=\[T\_[EE]{}\] (N/n)\^D \[temper\_1\]
where $A_D$ is the $D+1$ dimensional hyper- surface area. In the thermodynamic limit, by setting $L$ finite with $N \to \infty$ and $b \to 0$, $T_{EE}$ in Eq. (\[temper\_1\]) is finite and independent of $\e$. For large $N$, we show that, in the natural units, the above calculated temperature is identical to Hawking temperature for the corresponding black-hole [@hawking75]: \[eq:HawkingTemp\] T\_[BH]{} = ł. =|\_[\_[ r= r\_h]{} ]{} Fifth, it is important to note the above [*entanglement temperature*]{} is non-zero only for $f(r) \neq 1$. In the case of flat space-time, our analysis shows that the [*entanglement temperature*]{} vanishes, and we obtain $T_{EE}$ numerically for different black hole space-times.
Results and Discussions {#sec.2}
=======================
The Hamiltonian $H$ in Eq. (\[Hamilt\_1\]) is mapped to a system of $N$ coupled time independent harmonic oscillators (HO) with non-periodic boundary conditions. The interaction matrix elements of the Hamiltonian can be found in Ref [@dropbox]. The total internal energy (E) and the entanglement entropy ($S$) for the ground state of the HO’s is computed numerically as a function of $\e$ by using central difference scheme (see Appendix \[app2\]). All the computations are done using MATLAB R$2012$a for the lattice size $N = 600$, $ 10 \leq n \leq 500$ with a minimum accuracy of $10^{-8}$ and a maximum accuracy of $10^{-12}$.
In the following subsections, we compute $T_{EE}$ numerically for two different black-hole space-times, namely, 4 dimensional Schwarzschild and Reissner-Nordström black holes and show that they match with Hawking temperature $T_{BH}$. $T_{EE}$ is calculated by taking the average of [*entanglement temperature*]{} for each $n$’s by fixing $N$.
Schwarzschild (SBH) black holes
-------------------------------
The 4-dimensional Schwarzschild black hole space-time ( put $D=2$) in dimensionless units $\til r$ is given by the line element in Eq.(\[equ22\]) with $f(\til r)$ is given by: f(r)=1-
In Fig.(\[fig1\]), we have plotted total energy (in dimensionless units) and EE versus $\e$ for 4-dimensional Schwarzschild space-time. Following points are important to note regarding the numerical results: First for every $\e$, von Neumann entropy scales approximately as $S \sim (r_h/b)^2$. Second, EE and the total energy increases with $\e$.
Using relation (\[temp\_1\]), we evaluate “entanglement” temperature numerically. In dimensionless units, we get $T_{EE}=0.0793$ which is close to the value of the Hawking temperature $0.079$. However, it is important to note that for different values of $N$, we obtain approximately the same value of entropy. The results are tabulated, see Table(\[table1\]). See Appendix \[app3\], for plots of energy and EE for $n=50,80, 100$ and $130$.
![The plots of total energy (left) and EE (right) as a function of $\e$ for the 4d Schwarzschild black hole. We set $N = 600$ and $n=150$. The cyan coloured dots are the numerical data and the red line is the best linear fit to the data.[]{data-label="fig1"}](fig1)
Reissner-Nordström (RN) black holes
------------------------------------
The 4-dimensional Reissner-Nordström black hole is given by the line element in Eq.(\[equ22\]), where $f(\til
r)$ is \[equ219\] f(r)=1-+ $Q$ is the charge of the black hole. Note that we have rescaled the radius w.r.t the outer horizon ($ r_h=M+\sqrt{M^2-Q^2}$). Choosing $q=Q/r_h$, we get f(r) = 1-+ and the black hole temperature in the unit of $r_h$ is $T_{BH} = (1-q^2)/4\pi$.
Note that we have evaluated the [*entanglement temperature*]{} by fixing the charge $q$. For a fixed charge $q$, the first law of black hole mechanics is given by $ d E=\l(\k/2\pi\r) dA$, where $A$ is the area of the black hole horizon. The energy and EE for different $q$ values have the same profile, which looks exactly like in the previous case and is shown in the middle row in Fig. (\[fig3\]). See Appendix \[app3\], for plots for other values of $n$. As given in the table (\[table1\]), $T_{EE}$ matches with Hawking temperature.
[|C[2cm]{} |C[1.5cm]{} |C[1.5cm]{}|C[1.5cm]{}|]{} **Black hole space time& & $\mathbf{T_{BH}}$ & $\mathbf{T_{EE}}$\
& & 0.07958& 0.07927\
**
& $q=0.1$ & 0.07878 & 0.07836\
& $q=0.2$ & 0.07639 & 0.07507\
&$q=0.3$& 0.07242 & 0.07501\
& $q=0.4$& 0.06685 & 0.06659\
Conclusions and outlook {#sec.3}
========================
In this work, we have given another proof that 4-dimensional black hole entropy can be associated to entropy of entanglement across the horizon by explicitly deriving entanglement temperature. Entanglement temperature is given by the rate of change of the entropy of entanglement across a black hole’s horizon with regard to the system energy. Our new result sheds the light on the interpretation of temperature from entanglement as the Hawking temperature, one more step to understand the black hole thermodynamics from the quantum information theory platform.
Some of the key features of our analysis are: First, while entanglement and energy diverge in the limit of $b \to 0$, the [*entanglement temperature*]{} is a finite quantity. Second, [*entanglement temperature*]{} vanishes for the flat space-time. While the evaluation of the entanglement entropy [*does not*]{} distinguish between the black-hole space-time and flat space-time, entanglement temperature distinguishes the two space-times.
Our analysis also shows that the entanglement entropy satisfies all the properties of the black-hole entropy. First, like the black hole entropy, the entanglement entropy increases and never decreases. Second, the entanglement entropy and the temperature satisfies the first law of black-hole mechanics $dE= T_{EE}
\,dS$. We have shown this explicitly for Schwarzschild black-hole and for Reissner-Norstrom black-hole .
It is quite remarkable that in higher dimensional space time the Rényi entropy provides a convergent alternative to the measure of entanglement [@shanki2013], however, entanglement temperature will depend on the Rényi parameter. While a physical understanding of the Rényi parameter has emerged [@baez_renyi_2011], it is still not clear how to fix the Rényi parameter from first principles [@progress].
Our analysis throws some light on the emergent gravity paradigm [@Sakharov2000; @jacobson95; @padmanabhan2010; @Verlinde2011] where gravity is viewed not as a fundamental force. Here we have shown that the information lost across the horizon is related to the black-hole entropy and the laws of black-hole mechanics emerge from the entanglement across the horizon. Since General Relativity reduces gravity to an effect of the curvature of the space-time, it is thought that the microscopic constituents would be the [*atoms of the space-time*]{} itself. Our analysis shows that entanglement across horizons can be used as building blocks of space-time [@VanRaamsdonk2010-GRG; @VanRaamsdonk2010-IJMP].
One of the unsettling questions in theoretical physics is whether due to Hawking temperature the black-hole has performed a non-unitary transformation on the state of the system aka information loss problem. Our analysis here does not address this for two reasons: (i) Here, we have fixed the radius of the horizon at all times and evaluated the change in the entropy while to address the information loss we need to look at changing horizon radius. (ii) Here, we have used perturbative Hamiltonian, and hence, this analysis fails as the black-hole size shrinks to half-its-size [@Almheiri2013-JHEP]. We hope to report this in future.
While the unitary quantum time-evolution is reversible and retains all information about the initial state, we have shown that the restriction of the degrees of freedom outside the event-horizon at all times leads to temperature analogous to Hawking temperature. Our analysis may have relevance to the eigenstate thermalization hypothesis [@1994-srednicki; @rigol2008-nature; @2012-srednicki; @rahul2015-ARCMP], which we plan to explore.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
Authors wish to thank A. P. Balachandran, Charles Bennett, Samuel Braunstein, Saurya Das and Jens Eisert for discussions and comments. Also, we would like to thank the anonymous referee for the useful comments. All numerical computations were done at the fast computing clusters at IISER-TVM. The work is supported by Max Planck-India Partner Group on Gravity and Cosmology. SSK acknowledges the financial support of the CSIR, Govt. of India through Senior Research Fellowship. SS is partially supported by Ramanujan Fellowship of DST, India.
Calculation of Scalar field Hamiltonian in Lemaître coordinate {#app1}
==============================================================
In this Appendix section, we give details of the derivation of the Hamiltonian (H) upto second order in $\e$. Using the orthogonal properties of the real spherical harmonics $Z_{_{lm_i}}$, the scalar field action reduces to, \[equu26\] S&=&\_[\_[l,m\_i]{}]{}dd \^D ł\[ (\_[ ]{}\_[lm\_i]{})\^2 - -\_[lm\_i]{}\^2\] where $\til{r}=r/r_h, \thin \til{\xi}=\xi/r_h,\thin \til{\t}=\t/r_h, \thin\til{\Phi}_{lm}=r_h \, \Phi_{lm} $ are dimensionless.
Performing the following infinitesimal transformation [@toms] in the above resultant action:
’=+,’=,\
\_[lm\_i]{}(,)’\_[lm\_i]{}(’,)=\_[lm\_i]{}(,),\
(’,’)=(+,)
The action in Eq. (\[equu26\]) becomes, S &&\_[\_[l,m\_i]{}]{}dd ł(+h\_1+\^2 h\_2/2)\^Dł\[ł(1-f-h\_1-ł\[h\_2+h\_1\^2 \])\^[1/2]{} (\_[ ]{}\_[lm\_i]{})\^2.\
& &ł.-ł(1-f-h\_1-ł\[h\_2+h\_1\^2 \])\^[-1/2]{} (\_[ ]{}\_[lm\_i]{})\^2 - .\
& &ł. ł(1-f-h\_1-ł\[h\_2+h\_1\^2 \])\^[1/2]{}\_[lm\_i]{}\^2\] \[equ28\] where $h_1=\dis\frac{\partial\til r}{\pa\til \t} ~~\mbox{and}~~ h_2=\dis\frac{\pa^2\til r}{\pa\til \t^2} $.
Using the relation between the Lemaître coordinates - = gives the following expression, \[equ214\] h\_1=-, h\_2=, |\_= The Hamiltonian $(H)$ corresponding to the above Lagrangian is [ \[equ212\] H\_[\_[l,m\_i]{}]{}dł\[\_[lm\_i]{}\^2+ ł( \_)\^2+\_[\_[lm\_i]{}]{}\^2\] ]{} where \[equ210\] g\_[\_1]{}=+h\_1+\^2 h\_2/2, g\_[\_2]{}= , \_[\_[lm\_i]{}]{}= g\_[\_1]{}\^[D/2]{}\_[lm\_i]{} and $\til{\Pi}_{lm_i}$ is the canonical conjugate momenta corresponding to the field $\til{\chi}_{lm_i}$.
Upon quantization, $\til{\Pi}_{lm_i}$ and $\til{\chi}_{lm_i}$ satisfy the usual canonical commutation relation: \[equ211\] ł\[\_[lm\_i]{}ł([,]{}),\_[l’m’\_i]{}ł([,]{}) \]=i \_[ll’]{}\_[m\_im’\_i]{}ł(-) Using relations (\[equ214\]) and expanding the Hamiltonian up to second order in $\e$, we get, \[equ215\] H&\_[\_[l,m\_i]{}]{}\_\^dł\[\^2\_[lm\_i]{}+\^D ł\[\_[r]{} \]\^2.\
&ł.+\^2\_[lm\_i]{}\]
The Hamiltonian in Eq. (\[equ215\])is of the form \[Hamilt\_11\] HH\_[\_0]{}+ V\_[\_1]{}+\^2 V\_[\_2]{} where $ H_{_0}$ is the unperturbed scalar field Hamiltonian in the flat space-time, $V_{_1} \mbox{and}\,V_{_2}$ are the perturbed parts of the Hamiltonian given by;
H\_[\_0]{}&=&\_[\_[l,m\_i]{}]{}\_\^dł\[\^2\_[lm\_i]{}+ r\^Dł\[\_[r]{} \]\^2 +\^2\_[lm\_i]{}\]\
V\_[\_1]{}&=&\_[\_[l,m\_i]{}]{}\_\^dł\[ .\
& &ł. +\^2\_[lm\_i]{}\]\
V\_[\_2]{}&=&\_[\_[l,m\_i]{}]{}\_\^dł\[(H\^2\_3+H\_4)’\^2\_[lm\_i]{}+ (--+D H\_1 H\_1’-D H\_3 H\_1’+D H\_2’+H\_3 H\_3’ + H\_4’)’\_[lm\_i]{}\_[lm\_i]{}.\
&&ł.+(++-++ D\^2 H\_1’\^2-..\
&&ł.ł.-- D H\_1’ H\_3’+ H\_3’\^2-)\^2\_[lm\_i]{}\] where H\_[\_1]{}= , H\_[\_2]{}= , H\_[\_3]{}=, H\_[\_4]{}=ł()\^2+ and the redefined field operators are \_[lm\_i]{}= \_[lm\_i]{}= such that they satisfy the following canonical commutation relation \[equ216\] ł\[\_[lm\_i]{}(r, ),\_[l’m’\_i]{}(r’,)\]=i \_[ll’]{}\_[m\_im’\_i]{}(r- r’) The Hamiltonian $H$ in Eq. (\[Hamilt\_11\]) is mapped to a system of $N$ coupled time independent harmonic oscillators (HO) with non-periodic boundary conditions. The interaction matrix elements of the Hamiltonian can be found in the Ref.[@dropbox]. The total internal energy (E) and the entanglement entropy ($S_\a$) for the ground state of the HO’s is computed numerically as a function of $\e$ by using central difference scheme.
Central Difference discretization {#app2}
==================================
This is one of the effective method for finding the approximate value for derivative of a function in the neighbourhood of any discrete point, $x_i=x_0+i\;h$,with unit steps of $h$. The Taylor expansion of the function about the point $x_0$ in the forward and backward difference scheme is given respectively by, f(x+h)= f(x) + ++ ......\
f(x-h)=f(x) - +- ....... which implies, f’(x)&= &+ O(h\^2)\
f”(x)&=&+O(h\^2)\
f”’(x)&=&+O(h\^2)
Plots of internal energy and EE as a function of $\e$ for different black hole space-times {#app3}
===========================================================================================
In this section of Appendix, we give plots of EE for different black hole space-times;
![ Plots of the EE as function of $\e$ for the 4-d Schwarzschild black hole with $N=300$, $n=50,80,100$, and $130$, respectively. The blue dots are the numerical data and the red line is the best linear fit to the data. []{data-label="fig2"}](fig2)
![ []{data-label="fig16"}](fig16)
![ []{data-label="fig4"}](fig4)
[10]{}
G. D. [Birkhoff]{}, “[Proof of a Recurrence Theorem for Strongly Transitive Systems]{},” [[*Proceedings of the National Academy of Science*]{} [**17**]{} (Dec., 1931) 650–655](http://dx.doi.org/10.1073/pnas.17.12.650).
J. V. Neumann, “Proof of the quasi-ergodic hypothesis,” [[*Proceedings of the National Academy of Sciences*]{} [**18**]{} no. 1, (1932) 70–82](http://dx.doi.org/10.1073/pnas.18.1.70).
L. Boltzmann, [*Lectures on Gas Theory*]{}. Dover Books on Physics. Dover Publications, 2011.
L. D. Landau and E. M. Lifshitz, [*Statistical Physics*]{}, vol. V of [ *Course of Theoretical Physics*]{}. Elsevier, 3 ed., 1980.
A. Wehrl, “[General properties of entropy]{},” [[*Rev. Mod. Phys.*]{} [**50**]{} no. 2, (Apr., 1978) 221–260](http://dx.doi.org/10.1103/RevModPhys.50.221).
, [Wenger Trevor]{}, and [Weiss David S.]{}, “[A quantum Newton’s cradle]{},” [[*Nature*]{} [**440**]{} no. 7086, (Apr, 2006) 900?903](http://dx.doi.org/http://dx.doi.org/10.1038/nature04693).
, [Gross C.]{}, [Weller A.]{}, [Giovanazzi S.]{}, and [Oberthaler M. K.]{}, “[Squeezing and entanglement in a Bose-Einstein condensate]{},” [[*Nature*]{} [**455**]{} no. 7217, (Oct, 2008) 1216–1219](http://dx.doi.org/http://dx.doi.org/10.1038/nature07332).
D. A. Smith, M. Gring, T. Langen, M. Kuhnert, B. Rauer, R. Geiger, T. Kitagawa, I. Mazets, E. Demler, and J. Schmiedmayer, “Prethermalization revealed by the relaxation dynamics of full distribution functions,” [[*New Journal of Physics*]{} [**15**]{} no. 7, (2013) 075011](http://dx.doi.org/10.1088/1367-2630/15/7/075011).
V. I. Yukalov, “Bose-einstein condensation and gauge symmetry breaking,” [[*Laser Physics Letters*]{} [**4**]{} no. 9, (2007) 632](http://dx.doi.org/10.1002/lapl.200710029).
A. Osterloh, L. Amico, G. Falci, and R. Fazio, “Scaling of entanglement close to a quantum phase transition,” [[*Nature*]{} [**416**]{} (Dec, 2002) 608](http://dx.doi.org/10.1038/416608a).
L.-A. Wu, M. S. Sarandy, and D. A. Lidar, “Quantum phase transitions and bipartite entanglement,” [[*Phys. Rev. Lett.*]{} [**93**]{} (Dec, 2004) 250404](http://dx.doi.org/10.1103/PhysRevLett.93.250404).
J. von Stecher, E. Demler, M. D. Lukin, and A. M. Rey, “Probing interaction-induced ferromagnetism in optical superlattices,” [[*New Journal of Physics*]{} [**12**]{} no. 5, (2010) 055009](http://dx.doi.org/10.1088/1367-2630/12/5/055009).
J. Dinerman and L. F. Santos, “Manipulation of the dynamics of many-body systems via quantum control methods,” [[*New Journal of Physics*]{} [**12**]{} no. 5, (2010) 055025](http://dx.doi.org/10.1088/1367-2630/12/5/055025).
, “[Quantum Thermodynamics: Excuse our ignorance]{},” [[*Nature Physics*]{} [**2**]{} no. 11, (Nov, 2006) 727–728](http://dx.doi.org/http://dx.doi.org/10.1038/nphys456).
and [M. B. Plenio ]{}, “[Entanglement theory and the second law of thermodynamics]{},” [[*Nature Physics*]{} [**4**]{} no. 11, (Nov, 2008) 873–877](http://dx.doi.org/http://dx.doi.org/10.1038/nphys1100).
, “[Quantum entanglement: Reversible path to thermodynamics]{},” [[*Nature Physics*]{} [**4**]{} no. 11, (Nov, 2008) 833–834](http://dx.doi.org/http://dx.doi.org/10.1038/nphys1123).
S. Popescu and D. Rohrlich, “Thermodynamics and the measure of entanglement,” [[*Phys. Rev. A*]{} [**56**]{} (1997) R3319–R3321](http://dx.doi.org/10.1103/PhysRevA.56.R3319).
V. Vedral and M. B. Plenio, “Entanglement measures and purification procedures,” [[*Phys. Rev. A*]{} [**57**]{} (Mar, 1998) 1619–1633](http://dx.doi.org/10.1103/PhysRevA.57.1619).
M. B. Plenio and V. Vedral, “Teleportation, entanglement and thermodynamics in the quantum world,” [[ *Contemporary Physics*]{} [**39**]{} no. 6, (Nov., 1998) 431–446](http://dx.doi.org/10.1080/001075198181766).
M. Srednicki, “Chaos and quantum thermalization,” [[*Phys. Rev. E*]{} [**50**]{} (1994) 888–901](http://dx.doi.org/10.1103/PhysRevE.50.888).
, [Dunjko Vanja]{}, and [Olshanii Maxim]{}, “[Thermalization and its mechanism for generic isolated quantum systems]{},” [[*Nature*]{} [**452**]{} no. 7189, (Apr, 2008) 854?–858](http://dx.doi.org/http://dx.doi.org/10.1038/nature06838).
M. Rigol and M. Srednicki, “Alternatives to eigenstate thermalization,” [[*Phys. Rev. Lett.*]{} [**108**]{} (2012) 110601](http://dx.doi.org/10.1103/PhysRevLett.108.110601).
R. Nandkishore and D. A. Huse, “Many-body localization and thermalization in quantum statistical mechanics,” [[*Annual Review of Condensed Matter Physics*]{} [**6**]{} no. 1, (2015) 15–38](http://dx.doi.org/10.1146/annurev-conmatphys-031214-014726).
L. Bombelli, R. K. Koul, J. Lee, and R. D. Sorkin, “Quantum source of entropy for black holes,” [[ *Phys. Rev. D*]{} [**34**]{} (Jul, 1986) 373–383](http://dx.doi.org/10.1103/PhysRevD.34.373).
M. Srednicki, “Entropy and area,” [[*Phys. Rev. Lett.*]{} [**71**]{} (Aug, 1993) 666–669](http://dx.doi.org/10.1103/PhysRevLett.71.666).
J. Eisert and M. Cramer, “Single-copy entanglement in critical quantum spin chains,” [[*Phys. Rev. A*]{} [**72**]{} (Oct, 2005) 042112](http://dx.doi.org/10.1103/PhysRevA.72.042112).
S. Das and S. Shankaranarayanan, “How robust is the entanglement entropy-area relation?,” [[*Phys. Rev. D*]{} [**73**]{} (Jun, 2006) 121701](http://dx.doi.org/10.1103/PhysRevD.73.121701).
S. Das, S. Shankaranarayanan, and S. Sur, [*Black hole entropy from entanglement: A review* ]{}, vol. 268 of [*Horizons in World Physics*]{}. Nova Science Publishers,, New York, 2009. page 211.
S. N. Solodukhin, “Entanglement [Entropy]{} of [Black]{} [Holes]{},” [[*Living Reviews in Relativity*]{} [**14**]{} (2011) ](http://dx.doi.org/10.12942/lrr-2011-8).
S. L. Braunstein, S. Das, and S. Shankaranarayanan, “Entanglement entropy in all dimensions,” [[ *Journal of High Energy Physics*]{} [**2013**]{} no. 7, (2013) 1–9](http://dx.doi.org/10.1007/JHEP07(2013)130).
S. W. Hawking, “Particle creation by black holes,” [[*Communications in Mathematical Physics*]{} [**43**]{} (1975) 199–220](http://dx.doi.org/10.1007/BF02345020).
R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” [[*Rev. Mod. Phys.*]{} [**81**]{} (Jun, 2009) 865–942](http://dx.doi.org/10.1103/RevModPhys.81.865).
J. Eisert, M. Cramer, and M. B. Plenio, “*Colloquium* : Area laws for the entanglement entropy,” [[*Rev. Mod. Phys.*]{} [**82**]{} (Feb, 2010) 277–306](http://dx.doi.org/10.1103/RevModPhys.82.277).
P. Calabrese and J. L. Cardy, “[Entanglement entropy and quantum field theory]{},” [[ *Journal of Statistical Mechanics: Theory and Experiment*]{} [**0406**]{} (2004) P06002](http://dx.doi.org/10.1088/1742-5468/2004/06/P06002).
K. Mallayya, R. Tibrewala, S. Shankaranarayanan, and T. Padmanabhan, “Zero modes and divergence of entanglement entropy,” [[*Phys. Rev. D*]{} [**90**]{} (Aug, 2014) 044058](http://dx.doi.org/10.1103/PhysRevD.90.044058).
S. Mukohyama, M. Seriu, and H. Kodama, “Thermodynamics of entanglement in schwarzschild spacetime,” [*Phys. Rev. D*]{} [**58**]{} (Jul, 1998) 064001.
L. D. Landau and E. M. Lifshitz, [*The Classical Theory of Fields*]{}, vol. II of [*Course of Theoretical Physics*]{}. Butterworth-Heinemann, 4 ed., 1975.
D. J. Toms, [*[The Schwinger action principle and effective action]{}*]{}. Cambridge monographs on mathematical physics. Cambridge Univ. Press, Cambridge, 2007.
H. Sakaguchi, “Renyi entropy and statistical mechanics,” [[*Progress of Theoretical Physics*]{} [**81**]{} no. 4, (1989) 732–737](http://dx.doi.org/10.1143/PTP.81.732).
Click on the following ‘Dropbox’ link for more details on MATLAB codes that use for the numerical study reported in the paper, <https://www.dropbox.com/sh/wxmq02qxg6m3pbm/AABEE-k4MMow7FJj_JHgfrV_a?dl=0>.
J. C. [Baez]{}, “Rényi entropy and free energy,” [[arXiv:1102.2098 \[quant-ph\]]{}](http://arxiv.org/abs/1102.2098).
.
A. D. Sakharov, “Vacuum quantum fluctuations in curved space and the theory of gravitation,” [[*General Relativity and Gravitation*]{} [**32**]{} no. 2, (2000) 365–367](http://dx.doi.org/10.1023/A:1001947813563).
T. Jacobson, “Thermodynamics of spacetime: The einstein equation of state,” [[*Phys. Rev. Lett.*]{} [**75**]{} (Aug, 1995) 1260–1263](http://dx.doi.org/10.1103/PhysRevLett.75.1260).
T. Padmanabhan, “Thermodynamical aspects of gravity: new insights,” [[*Reports on Progress in Physics*]{} [**73**]{} no. 4, (Apr., 2010) 046901](http://dx.doi.org/10.1088/0034-4885/73/4/046901).
E. Verlinde, “On the origin of gravity and the laws of newton,” [[*Journal of High Energy Physics*]{} [**2011**]{} no. 4, (2011) ](http://dx.doi.org/10.1007/JHEP04(2011)029).
M. Van Raamsdonk, “Building up spacetime with quantum entanglement,” [[*General Relativity and Gravitation*]{} [**42**]{} no. 10, (2010) 2323–2329](http://dx.doi.org/10.1007/s10714-010-1034-0).
M. Van Raamsdonk, “Building up space?time with quantum entanglement,” [[*International Journal of Modern Physics D*]{} [**19**]{} no. 14, (2010) 2429–2435](http://dx.doi.org/10.1142/S0218271810018529).
A. Almheiri, D. Marolf, J. Polchinski, and J. Sully, “Black holes: complementarity or firewalls?,” [[*Journal of High Energy Physics*]{} [**2013**]{} no. 2, (2013) 1–20](http://dx.doi.org/10.1007/JHEP02(2013)062).
[^1]: In Schwarzschild coordinate, $r > 2M$ need to be bipartited [@1998-Mukohyama].
| ArXiv |
---
abstract: 'I show that a particle structure in conformal field theory is incompatible with interactions. As a substitute one has particle-like exitations whose interpolating fields have in addition to their canonical dimension an anomalous contribution. The spectra of anomalous dimension is given in terms of the Lorentz invariant quadratic invariant (compact mass operator) of a conformal generator $R_{\mu }$ with pure discrete spectrum. The perturbative reading of $R_{0\text{ }}$as a Hamiltonian in its own right i.e. associated with an action in a functional integral setting naturally leads to the AdS formulation. The formal service role of AdS in order to access CQFT by a standard perturbative formalism (without being forced to understand first massive theories and then taking their scale-invariant limit) vastly increases the realm of conventionally accessible 4-dim. CQFT beyond those for which one had to use Lagrangians with supersymmetry in order to have a vanishing Beta-function.'
author:
- |
Bert Schroer\
presently CBPF, Rua Dr. Xavier Sigaud, 22290-180 Rio de Janeiro, Brazil\
email: [email protected]\
Prof. emeritus of the Institut für Theoretische Physik\
FU-Berlin, Arnimallee 14, 14195 Berlin, Germany
date: 'May 9, 2000'
title: Particle versus Field Structure in Conformal Quantum Field Theories
---
A few introductory remarks
==========================
Ideas about the use of conformal quantum field theory entered particle physics for the first time at the height of the Kramers-Kronig dispersion relations [@Kastrup]. They were met with reactions ranging from doubts to outright rejection and the subject lay dormant for another 10 years when it reemerged on the statistical mechanics side in connection with second order phase transitions.
In the next section we will show that these early doubts of the old-time particle physicists were partially justified, because the particle structure in CQFT is indeed incompatible with interactions. However far from supplying a coffin nail for its utility in high energy physics, this no-go theorem also contains the message that one must use finer concepts in order preserve the usefulness of conformal quantum field theory as a theoretical laboratory for particle physics. There are massive particle-like objects (“infraparticles” [@Bu]) which have a continuous mass distribution with an accumulation of spectral weight at $p^{2}=m^{2}$ whose generating local fields have an anomalous non-integer (non-semi-integer in the case of Fermion fields) contribution to their long distance behavior. In a CQFT long and short distance behavior coalesce and the accumulation of spectral weight at $p^{2}=0\,$ which becomes related to the anomalous dimension of operators is the vestige of the particle interaction in the massive parent theory from which the CQFT arose by taking the scale-invariant limit. This structure is the collective effect of a total collapse of all multiparticle thresholds on top of each other. The standard LSZ large time scattering limit does not commute with this scaling limit, in fact the LSZ limit of such fields vanishes. It is believed that in order to re-extract from such a situation anything which resembles particle physics one has to apply a more general form of scattering theory [@Bu] which is based on expectation values and probabilities for inclusive cross sections (where outcoming “stuff” below a prescribed energy-momentum resolution is not registered) rather than on amplitudes. But it is presently not clear how one can achieve this. In the case of infraparticles (the electron in QED which is inexorably linked to its photon-cloud) where one also meete a situation of coalescing thresholds, this generalized scattering theory is known to be very useful [@Bu].
Recently there has been a quite different and conceptually[^1] less ambitious but formally quite attractive idea which promises to strengthen the utility of CQFT for particle physics and which is presented in the third section. It basically consists in finding a theory which radically reprocesses the spacetime interpretation and degrees of freedom of CQFT in such a way that now the “energy momentum vector” $R_{\mu }$ of the Dirac-Weyl compactified world $\bar{M}$ becomes the bona fide energy momentum instead of $P_{\mu }$ which in standard canonical or functional terminology means that $R_{\mu }$ is the one related to an action and not $P_{\mu }$. If one insists that this total reshuffling of physical interpretation should leave the basic mathematical building blocks (a certain generating set of algebras and the symmetry group structure) untouched, then there is only one answer: an associated anti De Sitter (AdS) theory [@Wit]. The nontrivial reprocessing leads to a mathematical isomorphism as described in [@Reh1] i.e. it goes far beyond that picture about the AdS-CQFT correspondence which is limited to the (infinitely remote) boundary of AdS (see in particular the remarks at the end of [@Reh2]). The AdS appearance of the AdS structure as a kind of reprocessed CQFT is less surprizing if one recalls the 6-dimensional lightcone formalism which one uses in order to obtain an efficient description of the conformal compactification $\bar{M}$ of Minkowski space $M $ and the construction of its covering $\tilde{M}$ [@Schroer].
In this way one obtains a (perturbative) new constructive non-Lagrangian access to CQFT which opens a new window into the realm of CQFT beyond those few 4-dimensional Lagrangian candidates for which one had to use a combination of gauge theory with supersymmetry. This means that one has no guaranty that the conformal side at all permits a description in terms of an action.
Particle Structure and Triviality
=================================
We start with recalling an old theorem which clarifies the relation between the particle-versus-field content of conformal field theories. To be more precise the following statement is a result of the adaptation of a combination of several theorems [@BF][@Pohl]
The existence of one-particle states in conformally invariant theories forces the associated interpolating fields to be canonical free fields. The only particle-like structures consistent with interactions are hidden in the structure of those interpolating fields which have anomalous dimensions and whose mass spectrum is continuous with an accumulation of weight at $p^{2}=0,\,\,p_{0}>0.$
The easiest way to get a first glimpse at this situation is to look at conformal two-point functions $$\left\langle \psi (x)\psi ^{\ast }(y)\right\rangle =\left\{
\begin{array}{c}
c\frac{1}{-\left( x-y\right) ^{2}},\,\,dim\psi =1 \\
c(\frac{1}{-\left( x-y\right) ^{2}})^{d_{\psi }},\,\,dim\psi =d_{\psi }>1
\end{array}
\right. \label{an}$$ In the first case the application of the LSZ large time scattering limit yields $$\left\langle \psi (x)\psi ^{\ast }(y)\right\rangle =\left\langle \psi
_{in}(x)\psi _{in}^{\ast }(y)\right\rangle$$ which preempts the equality $\psi =\psi ^{in}=\psi ^{out},$ whereas in the anomalous case the large distance fall-off is too strong in order to be reconcilable with the mass shell structure of a zero mass particle which means $$\psi (x)\overset{LSZ}{\rightarrow }0$$ It is worthwhile to reconsider the argument which leads to the absence of interaction in the space created by the interpolating field $\psi .$ The crucial observation is that the presence of a zero mass scalar particle state vector $\left| p\right\rangle $ with $$\left\langle p\left| \psi \right| 0\right\rangle \neq 0$$ forces $\psi $ to have a two-point function with a canonical scale dimension dim$\psi =1.$ The special feature of conformal invariance is that this implies that the two-point function is free i.e. $$\left\langle 0\left| \psi ^{\ast }(x)\psi (y)\right| \right\rangle =c\frac{1}{\left[ -(x-y-i\varepsilon )\right] ^{2}}$$ Such a conclusion relating canonical short distance dimension with absence of interactions cannot be drawn in the massive case. However the following theorem which was proven in the late 50$^{ies}$ by Jost and the present authors, and can be found in [@St-Wi], holds for both cases:
The freeness of the $\psi $ two-point function implies the field $\psi $ to be a free field in Fock space.
The guiding idea is to show that a localized operator or pointlike field which vanishes on the vacuum, vanishes automatically on all states i.e. is the zero operator. This is a consequence of the Reeh-Schlieder theorem [@St-Wi] which in conformal field theory is also known under the name state-field relation). It says that the operators from a region with a nontrivial causal complement (or fields smeared with test functions with support in such a region) act cyclically on the vacuum (and on any other finite energy state). If we denote by $\mathcal{A}(\mathcal{O})$ either the polynomial $^{\ast }$-algebra of unbounded smeared fields with supports of testfunctions in $\mathcal{O}$ or the affiliated bounded operator algebra, this cyclicity property reads $$\overline{\mathcal{A}(\mathcal{O})\Omega }=\mathcal{H}$$ where the bar denotes the closure and $H$ is the Hilbert space generated by all fields (bosonic and fermionic). Since (for fermionic $\psi $ there will be a change of sign) $$\psi (x)\mathcal{A}(\mathcal{O})\Omega =\mathcal{A}(\mathcal{O})\psi
(x)\Omega$$ if we choose $O$ spacelike with respect to $x,$ the vanishing of the “current” $j(x)=(\partial _{\mu }\partial ^{\mu }+m^{2})\psi (x)$ on the vacuum implies the vanishing on the dense set $\mathcal{A}(\mathcal{O})\Omega $ and hence (operators in physics are closable) on all $\mathcal{H}.\,$The next step consists in proving that the commutator of two $\psi s$ on the vacuum is a c-number $$\left( \left[ \psi (x),\psi (y)\right] -i\Delta (x-y)\right) \Omega =0$$ It then follows according to the previous argument that the bracket vanishes identically. We prove this last relation by using the frequency decomposition $\psi =\psi ^{(-)}+\psi ^{(+)}$ (which follows from $j\equiv
0) $ in the commutator $$\left[ \psi (x),\psi (y)\right] \Omega =(\left[ \psi ^{(+)}(x),\psi ^{(+)}(y)\right] +\psi ^{(-)}(x),\psi ^{(+)}(y)-\psi ^{(-)}(y),\psi ^{(+)}(x))\Omega$$ where we omitted all annihilation terms. The on-shell creation with subsequent on-shell annihilation as in the last two terms and the physical spectrum condition only admits the vacuum as its energy momentum content and therefore they yield a c-number which, by a finite renormalization of $\psi $ if necessary, yields $$(\psi ^{(-)}(x),\psi ^{(+)}(y)-\psi ^{(-)}(y),\psi ^{(+)}(x))\Omega =i\Delta
(x-y)\mathbf{1}\Omega$$ Since this and the full commutator is causal, the first term on the right hand side has to vanish all by itself. But on the other hand it is the separate Fouriertransform of momenta which lie on the forward mass shell and hence it is the boundary value of an analytic function in two complex 4-vectors of the form $z=\xi -i\eta ,\eta $ from the forward light cone. However an analytic function which vanish on an open set on its boundary vanished identically (generalized Schwartz reflection principle). The resulting relation on the vacuum holds according to the previous arguments for the operators and therefore we obtained the characterizing relation for a free field. The generalization to any spin including half-integer values is now a routine matter. A closer look at the zero mass situation reveals that contrary to the massive case where the difference of two on-shell vectors is either spacelike or zero, the difference of two lightlike vectors may in addition be lightlike but this only happens for parallel vectors. Since this special configurations should not matter in the sense of L$^{2}$-integrability of zero mass particle wave functions one again expects at least for $d>1+1$ the above result. However a mathematical proof of this result turned out to be quite nontrivial [@Pohl].
It is very helpful to place the above theorem into the setting of a more general theorem relating interactions and particle properties in general local quantum physics which states that operators localized in sub-wedge regions in interacting theories which possess nontrivial matrix elements between vacuum and one-particle states necessarily show the phenomenon of vacuum polarization i.e. operators which create polarization-free one-particle states exist only in interaction free field theories. Polarization-free-generators (PFG) which create pure one-particle states from the vacuum do however exist in any QFT if their localization region is a semi-infinite wedge region or larger [@Essay][@BBS]. Since in conformal theories the wedge region is conformally equivalent to a compact double cone, a conformal one-particle structure according to this more general theorem is only possible in conformal free field theories.
The above argument is typical for a real-time structure which cannot be unraveled in the euclidean formulation.
Trying to make the best out of it
=================================
The negative result on the compatibility of zero mass particle structure with nontriviality of conformal theories should not be misread as an incompatibility with an intuitive idea about what constitutes particle-like excitations. The point here is that conformal theories in particle physics should be considered as the zero mass (scaling) limits of massive theories with mass gaps for which the LSZ scattering theory can be derived. In the scaling limit all the multiparticle thresholds in momentum space coalesce on top of each other and build up the possibly anomalous dimension. In this limit the Wigner particle theory (irreducible representation of the Poincaré group) and with it the prerequisite of the LSZ scattering theory gets lost in the presence of interactions, a fact which we have demonstrated above where it was shown that the field is either free or the LSZ limits are zero. So the right question would be: can one think of a more general scattering theory which may recuperate some of the lost structure in the aforementioned collapse of multiparticle cuts on top of each other? There is indeed another particle concept (“infraparticle”) which goes together with a generalized scattering theory build on inclusive scattering probabilities instead of amplitudes [@Bu]. This concept is expected to distinguish those anomalous dimensional fields which are of relevance in particle physics (which originate from the previous collapse in the scaling limit) from mere mathematical constructs as e.g. generalized free fields with anomalous dimensions. But we think that for the problem at hand, namely the formulation of a theory of anomalous dimension, we do not need to enter this deep and difficult issue of particle-like interpretation since here we restrict our interests in conformal theories as a simplified theoretical laboratory for field- and algebra- aspects and not for the study of particles and their scattering theory. We believe that the setting of local observable algebras which fulfill in addition to Einstein causality also Huygens principle for timelike distances [@Sch] contains all scale limits of theories which are of interest for particle physics and that interaction in this setting is characterized by the appearance of charge-carrying fields with anomalous dimensions. In view of the above No-Go theorem we will consider the noncanonical (anomalous dimension) nature of those fields as our pragmatic definition of interaction in this conformal setting. But we defer this analysis to a following longer paper which contains the relevant mathematical machinery [@Sch].
As a consequence the observable algebra of an interacting conformal field theory (conserved currents etc.) should not have the structure of composites of free fields (e.g. free currents) since otherwise the fields carrying the superselected charges may not have anomalous dimensions. Apart from normalization constants the 2- and 3-point functions of conformal observable fields (currents) are indistinguishable from those formed with free composites with the same integer dimensions. If all correlations would be indistinguishable from those of free composites (total protection) then also the charge-carrying fields associated with such observables can be shown to be free.
A weak form of what in the case of conformal SYM theories has been called (partial) “protection” would be one where the relative normalization between 2-and 3-point functions is that of free composites (partial protection). Apparently perturbative supersymmetry causes partial protections [@prot]. Although such models hardly represent realistic particle physics, they are the only *Lagrangian* candidates for d=1+3 nontrivial conformal field theories and may yet turn out to be the first 4-dimensional mathematically completely controllable models. The interest and fascination in conformal field theories originates to a large part from the well-founded belief that the simplest nontrivial 4-dimensional conformal field theories which will break the age old existence deadlock[^2] for nontrivial quantum field theories in physical spacetime. For this one wants to have as much protection as possible without ending with a free conformal theory.
Instead of entering an ambitious program in order to extract the particle physics “honey” from CQFT which requires a heavy conceptual investment in the area of a generalized scattering theory, there is another way which is more faithful to the formal aspects with which QFT is often identified (erroneously in my opinion, if one uses them for a definition of QFT) namely canonical formalism and/or functional integrals. It starts from the observation that in addition to the translation generator $P_{\mu }$ there is another translation-analogue described by a Lorentz-vector $R_{\mu }.$ It has a timelike purely discrete spectrum and the L-invariant “mass” $m_{c}$ with $m_{c}^{2}=R_{\mu }R^{\mu }$ plays a similar role as the rigid rotation operator $L_{0}$ in chiral theories. In fact it describes a generalized rotation around the Dirac-Weyl compactified Minkowski space $\bar{M}\simeq
S^{3}\times S^{1}.$ Therefore it is not surprising that the bottom of the spectrum of $m_{c}$ is the anomalous part of the scaling dimension common to a whole equivalence class of fields which carry the same superselected charge. But despite all analogies to $P_{\mu }$ this operator is not related to an imagined functional integral action of CQFT. Nevertheless one can ask the question: is there a theory whose Lagrangian can be associated with a Hamiltonian interpretation of $R_{0}?$ In order for this new theory to be useful for particle physics it should keep the same algebraic and group-theoretical building blocks as CQFT i.e. one seeks a mathematical isomorphism which goes hand in hand with that total physical reprocessing which is necessary to accomplish such an impossible looking task. The unique answer is the AdS-CQFT correspondence [@Wit] which was proven to be a such a “radical” isomorphism [@Reh1].
Although this step does not completely answer the question posed at the beginning of how to extract and analyze the particle content of CQFT, it goes a long way to open up conformal field theory as a genuine theoretical laboratory for particle physics. And last not least it facilitates the unsolved problem number one: find a nontrivial physically relevant (i.e. one which fits at least the conceptual framework of local quantum physics, even if it falls short in describing nature) and mathematically controllable model in 4-dimensional QFT.
The presented arguments suggest strongly that there exists a whole world of non-Lagrangian non-supersymmetric CQFT (in the sense that they cannot be accessed in the standard perturbative way) besides the Lagrangian SYM family. In fact the perturbative calculations in the literature already give some support in this direction. This is most visible in [@Ruehl] although these authors, evidently under the strong spell of the string-theoretic origin of the AdS-CQFT, do not interprete their calculations from this viewpoint.
The possible non-Lagrangian nature of most CQFT is in a certain way explained by Rehren’s deep observation [@Reh1][@Reh2] that due to the isomorphic nature of the AdS-CQFT relation there must be degrees of freedom on the conformal side which cannot be described in terms of local fields namely those which originate from the AdS bulk (and not from the boundary) and which are necessary in order to return $CQFT\rightarrow AdS$. This leaves the interesting question of what should one make of the original observation by which the protagonists of the AdS-CQFT correspondence found this relation which is the relation between two Lagrangian field theories namely the conformal SYM model with some form of AdS supergravity [@Wit]. Since this is based on consistency checks within string theory which owes its widespread acceptance to perturbative mathematical consistency and a kind of globalized social contract but certainly not to its harmonious coexistence with the principles underlying particle physics, there is reason for some scepticism; in particular because such degrees of freedom would be easily overlooked in perturbative calculations on the CQFT side. It cannot be overstressed that this correspondence is very different and much more radical then those which arise from a different choice of “field coordinates”. It is impossible to understand its full content in terms of pointlike physical fields.
Some concluding remarks
=======================
If, as argued in this letter, the AdS theories are a useful new calculational tool which open up CQFT to particle physics studies within the standard Lagrangian quantization framework, than perhaps with an additional conceptual investment one could directly understand the structure underlying the anomalous dimension spectra within CQFT i.e. without the described reprocessing on the AdS side. This turns out to be true and will be the subject of a subsequent paper [@Sch] since the necessary conceptual investment does not fit the format of a letter like this.
*Acknowledgements*: I am indebted to Detlev Buchholz and Karl-Henning Rehren for a helpful exchange of emails. Furthermore I would like to thank Francesco Toppan for interesting questions which helped in shaping the presentation.
[99]{} H.A. Kastrup, Ann. Physik 7, (1962) 388
J. Maldacena, Adv. Theor. Math. Phys. **2 (**1998**)**, 231
S.S. Gubser, I. R. Klebanov and A.M. Polyakov, Phys. Lett. **B448**, (1998) 253
E. Witten, Adv. Theor. Math. Phys. **2** (1998) 253
D. Buchholz, “Mathematical Physics Towards the 21st Century”, Proceedings Beer-Sheva 1993, Ben Gurion University Press 1994
K-H Rehren, “Algebraic Holography”, hep-th/9905179
K-H Rehren, “Local Quantum Observables in the Anti-deSitter - Conformal QFT Correspondence”, hepth/0003120
D. Buchholz and K. Fredenhagen, JMP **18**, Vol.5 (1977) 1107
K. Pohlmeyer, Commun. Math. Phys. **12**, (1969) 201
R.F. Streater and A.S. Wightman, *PCT, Spin and Statistics and all That*, Benjamin 1964
B. Schroer, “Facts and Fictions about Anti de Sitter Spacetimes with Local Quantum Matter”, hep-th/9911100
B. Schroer, “Particle Physics and QFT at the Turn of the Century: Old principles with new concepts, (an essay on local quantum physics)”, Invited contribution to the Issue 2000 of JMP, in print, to appear in the June issue
H-J Borchers, D. Buchholz and B. Schroer, “Polarization-Free Generators and the S-Matrix”, hep-th/0003243
B. Schroer, “A Theory of Anomalous Scale-Dimensions”, hep-th/0005134
for example: J. Erdmenger, M. Perez-Victoria, “Non-renormalization of next-to-extremal correlators in N=4 SYM and the AdS/CFT correspondence” and literature quoted therein
L. Hoffmann, A.C. Petkou and W. Ruehl, “Aspects of Conformal Operator Product Expansion in AdS/CFT Correspondence”, hep-th/0002154
[^1]: The attribute “conceptually” here refers to the local quantum physical aspects and not to differential-geometric ones.
[^2]: In any area of Theoretical Physics there always have been plenty of nontrivial mathematically controllable illustrations which demonstrate the nontrivial physical content of the conceptual basis of those areas, not so in 4-dim. QFT. This annoying totally singular situation has been sometimes overemphasized at the cost of practical calculations, but most of the time it went totally ignored.
| ArXiv |
Resonant inelastic x-ray scattering (RIXS) is developing very rapidly into a powerful technique to investigate elementary excitations in the strongly correlated electron systems [@Kao; @Butorin; @Hill; @Kuiper; @Abbamonte]. The application of this technique to insulating copper oxides has made it possible to observe an excitation due to a local charge transfer between copper and oxygen [@Hill] and local $d$-$d$ excitations on copper site [@Kuiper]. In addition, it has been demonstrated that, by using high resolution experiments [@Abbamonte], the momentum-dependent measurement of the charge transfer gap is possible when the incident photon energy $\omega_i$ is tuned through Cu $K$ absorption edge. Thus, the RIXS can be a useful probe to obtain information on the momentum dependence of the elementary excitations.
One of the elementary excitations in the insulating cuprates is the charge-transfer process from the occupied Zhang-Rice singlet band (ZRB) [@Zhang] composed of Cu 3$d_{x^2-y^2}$ and O 2$p_\sigma$ orbitals to the unoccupied upper Hubbard band (UHB). The dispersion of ZRB have been extensively studied by angle-resolved photoemission spectroscopy (ARPES) experiments on the parent compounds of high $T_c$ superconductors [@Wells; @Kim; @Ronning]: A $d$-wave-like dispersion was observed along the (0,$\pi$)-($\pi$,0) line with the minimum of the binding energy at ($\pi$/2,$\pi$/2) [@Ronning]. On the contrary, the dispersion relation and spectral properties of the unoccupied UHB have not been examined and thus remain to be understood. The information of UHB is of crucial importance for the understanding of the motion of electrons in the electron-doped superconductors. In addition, it may be useful to know if the particle-hole symmetry is required for the high temperature superconductivity.
In this Letter, we examine the RIXS spectrum for the Cu $K$-edge, and demonstrate that the characteristic features of the dispersion of UHB can be extracted from the momentum dependence of the spectrum. To see this, we use the half-filled single-band Hubbard model to describe the occupied ZRB and unoccupied UHB by mapping ZRB onto the lower Hubbard band (LHB) in the model. Then, we incorporate Cu 1$s$ and 4$p$ orbitals into the model to include the 1$s$-core hole and excited 4$p$ electron into the intermediate state of the RIXS process. The long-range hoppings are also introduced in the Hubbard model with realistic values obtained from the analysis of ARPES data. We find a characteristic momentum dependence of the Cu $K$-edge RIXS spectrum: The energy of the threshold of the RIXS spectrum at ($\pi$/2,$\pi$/2) is higher than that at (0,0), whereas the energy of the threshold at ($\pi$/2,0) is lower than that at (0,0). This anisotropic dependence is explained by the dispersion of the UHB which has the minimum energy at ($\pi$,0) due to the long-range hoppings. The determination of the UHB will contribute to the understanding of the different behavior of hole- and electron-doped superconductors [@Kim; @Takagi].
We map the ZRB onto the LHB, which is equivalent to the elimination of O $2p$ orbitals. Such mapping was used in the analysis of O $1s$ x-ray absorption spectrum[@Chen]. The Hubbard Hamiltonian with second and third neighbor hoppings for the $3d$ electron system is written as, $$\begin{aligned}
\label{ham3d}
H_{3d} &=& -t\sum_{\langle {\bf i},{\bf j} \rangle_{\rm 1st}, \sigma}
d_{{\bf i},\sigma}^\dag d_{{\bf j},\sigma}
-t'\sum_{\langle {\bf i},{\bf j} \rangle_{\rm 2nd}, \sigma}
d_{{\bf i},\sigma}^\dag d_{{\bf j},\sigma} \nonumber\\
&&-t''\sum_{\langle {\bf i},{\bf j} \rangle_{\rm 3rd}, \sigma}
d_{{\bf i},\sigma}^\dag d_{{\bf j},\sigma} + {\rm H.c.}
+U\sum_{\bf i}
n^d_{{\bf i},\uparrow}n^d_{{\bf i},\downarrow},\end{aligned}$$ where $d_{{\bf i},\sigma}^\dag$ is the creation operator of $3d$ electron with spin $\sigma$ at site ${\bf i}$, $n_{{\bf i},\sigma}^d=d_{{\bf i},\sigma}^\dag
d_{{\bf j},\sigma}$, the summations $\langle {\bf i},{\bf j} \rangle_{\rm 1st}$, $\langle {\bf i},{\bf j} \rangle_{\rm 2nd}$, and $\langle {\bf i},{\bf j} \rangle_{\rm 3rd}$ run over first, second, and third nearest-neighbor pairs, respectively, and the rest of the notation is standard.
Figure \[figpic\] shows the schematic process of Cu $K$-edge RIXS. An absorption of an incident photon with energy $\omega_i$, momentum ${\bf K}_i$, and polarization ${\bf \epsilon}_i$ brings about the dipole transition of an electron from Cu $1s$ to $4p$ orbital \[process (a) in Fig. \[figpic\]\]. In the intermediate states, $3d$ electrons interact with a $1s$-core hole and a photo-excited $4p$ electron via the Coulomb interactions so that the excitations in the $3d$ electron system are evolved \[process (b)\]. The $4p$ electron goes back to the $1s$ orbital again and a photon with energy $\omega_f$, momentum ${\bf K}_f$, and polarization ${\bf\epsilon}_f$ is emitted \[process (c)\]. The differences of the energies and the momenta between incident and emitted photons are transferred to the $3d$ electron system.
In the intermediate state, there are a $1s$-core hole and a $4p$ electron, with which $3d$ electrons interact. Since the 1$s$-core hole is localized because of a small radius of the Cu 1$s$ orbital, the attractive interaction between the 1$s$-core hole and 3$d$ electrons is very strong. The interaction is written as, $$\begin{aligned}
H_{1s\text{-}3d}=-V\sum_{{\bf i},\sigma,\sigma'}
n_{{\bf i},\sigma}^d n_{{\bf i},\sigma'}^s,\end{aligned}$$ where $n_{{\bf i},\sigma}^s$ is the number operator of 1$s$-core hole with spin $\sigma$ at site ${\bf i}$, and $V$ is taken to be positive. On the contrary, since the 4$p$ electron is delocalized, the repulsive interaction between the 4$p$ and 3$d$ electrons as well as the attractive one between the 4$p$ electron and the 1$s$-core hole is small as compared with the 1$s$-3$d$ interaction. In addition, when the core-hole is screened by the 3$d$ electrons through the strong 1$s$-3$d$ interaction, effective charge that acts on the 4$p$ electron at the core-hole site becomes small. Therefore, the interactions related to the 4$p$ electron are neglected for simplicity. Furthermore, we assume that the photo-excited 4$p$ electron enters into the bottom of the 4$p_z$ band with momentum ${\bf k}_0$, where $z$-axis is perpendicular to the CuO$_2$ plane. This assumption is justified as long as the Coulomb interactions associated with the 4$p$ electron are neglected and the resonance condition is set to the threshold of the 1$s$$\rightarrow$4$p_z$ absorption spectrum[@polarization]. Under these assumptions, the RIXS spectrum is expressed as, $$\begin{aligned}
\label{rixs}
I(\Delta {\bf K},\Delta\omega)&=&\sum_\alpha\left|\langle\alpha|
\sum_\sigma s_{{\bf k}_0-{\bf K}_f,\sigma} p_{{\bf k}_0,\sigma}
\right.\nonumber\\&&\times\left.
\frac{1}{H-E_0-\omega_i-i\Gamma}
p_{{\bf k}_0,\sigma}^\dag s_{{\bf k}_0-{\bf K}_i,\sigma}^\dag
|0\rangle\right|^2
\nonumber\\&&\times
\delta(\Delta\omega-E_\alpha+E_0),\end{aligned}$$ where $H=H_{3d}+H_{1s\text{-}3d}+H_{1s,4p}$, $H_{1s,4p}$ being kinetic and on-site energy terms for a 1$s$-core hole and a 4$p$ electron, $\Delta{\bf K}={\bf K}_i-{\bf K}_f$, $\Delta\omega=\omega_i-\omega_f$, $s_{{\bf k},\sigma}^\dag$ ($p_{{\bf k},\sigma}^\dag$) is the creation operator of the 1$s$-core hole (4$p$ electron) with momentum ${\bf k}$ and spin $\sigma$, $|0\rangle$ is the ground state of the half-filled system with energy $E_0$, $|\alpha\rangle$ is the final state of the RIXS process with energy $E_\alpha$, and $\Gamma$ is the inverse of the relaxation time in the intermediate state. In Eq. (\[rixs\]), the terms $H_{1s,4p}$ are replaced by $\varepsilon_{1s\text{-}4p}$ which is the energy difference between the $1s$ level and the bottom of the $4p_z$ band.
The RIXS spectrum of Eq. (\[rixs\]) is calculated on $(\sqrt 8$$\times$$\sqrt 8)$-, $(\sqrt{10}$$\times$$\sqrt{10})$-, and (4$\times$4)-site clusters with periodic boundary condition by using a modified version of the conjugate-gradient method together with the Lanczös technique. We will show the results for the 4$\times$4-site cluster in the following.
The values of the parameters are as follows: $t'/t=-0.34$, $t''/t=0.23$, $U/t=10$, $V/t=15$, and $\Gamma/t=1$ with $t=0.35$ eV. The values of $t$, $t'$, and $t''$ are the same as those used in the analysis of ARPES data of the high $T_c$ superconductors based on the $t$-$t'$-$t''$-$J$ model [@Kim; @Tiny]. The value of $U$ is obtained from the relation $J$=4$t^2$/$U$ and $J$/$t$=0.4. The value of $V$ is set to be larger than that of $U$. The results shown below are insensitive to the magnitude of $V$ as well as of $\Gamma$ [@plat98].
Before going into the RIXS spectrum, we mention the resonance condition in RIXS. To determine the condition, we have to examine the Cu 1$s$ x-ray absorption spectroscopy (XAS) spectrum defined as, $$\begin{aligned}
D(\omega)&=&\frac{1}{\pi}{\rm Im}\langle 0 |
s_{{\bf k}_0-{\bf K}_i,\sigma} p_{{\bf k}_0,\sigma}
\frac{1}{H-E_0-\omega-i\Gamma_{\text{XAS}}}
\nonumber\\&&\times
p_{{\bf k}_0,\sigma}^\dag s_{{\bf k}_0-{\bf K}_i,\sigma}^\dag
|0\rangle,\end{aligned}$$ where $H$ is the same as that in Eq. (\[rixs\]). It is necessary to tune the incident photon energy $\omega_i$ to the energy region where the Cu 1$s$ XAS spectrum appears. The inset in Fig. \[figrixs\] shows $D(\omega)$, where a two-peak structure appears, [*i.e.*]{}, the broad one around $\omega-\varepsilon_{1s\text{-}4p}=-20t$ and the sharp one around $-13t$. The former mainly contains configurations that the core-hole site is doubly occupied by the 3$d$ electrons ($U-2V=-20t$), while the latter dominantly contains configurations that the core-hole site is singly occupied ($-V=-15t$). This means that, when the incident energy is tuned around the former structure, the information about UHB can be extracted from the RIXS spectrum. Thus, we set $\omega_i$ to the threshold of the XAS spectrum denoted by the arrow in the inset.
Figure \[figrixs\] shows the momentum dependence of the RIXS spectrum. The spectra below $\Delta\omega/t\sim2$ and above $\Delta\omega/t\sim5$ have different origins: The former comes from the excitations related to the spin degree of freedom such as two-magnon Raman scattering, the energy scale of which is so small that the spectrum is hard to be observed[@Abbamonte]. On the other hand, the latter is related to the excitations from LHB to UHB. The vertical dotted line in the figure denotes the position of the low-energy peak at $\Delta{\bf K}=(0,0)$ for guide to eyes. The spectra strongly depend on the momentum showing a feature that the weight shifts to higher energy region with increasing $\left|\Delta{\bf K}\right|$. This momentum dependence is also obtained in the 10-site cluster calculations. In addition, the threshold of the spectrum at $\Delta{\bf K}=(\pi/2,0)$ and ($\pi$,$\pi$/2) is lower in energy than that at (0,0). At $\Delta{\bf K}=(\pi/2,\pi/2)$, however, the spectrum appears above the threshold at (0,0), resulting in an anisotropic momentum dependence between the spectra along (0,0) to ($\pi$/2,0) and along (0,0) to ($\pi$/2,$\pi$/2). We note that such an anisotropic feature can not be observed in the RIXS spectrum of the Hubbard model without $t'$ and $t''$ (not shown)[@Typical]. In the 8-site cluster the spectrum at $\Delta{\bf K}=(\pi/2,\pi/2)$ is lower in energy than that at ($\pi$,0), which is consistent with the above feature.
In the intermediate state, the excitations of the 3$d$ electrons from the occupied to unoccupied states are caused by the interaction with the 1$s$-core hole, [*i.e.*]{}, $H_{1s\text{-}3d}=-V/N\sum_{{\bf k}_1\sim{\bf k}_4,\sigma,\sigma'}
\delta_{{\bf k}_2-{\bf k}_1,{\bf k}_3-{\bf k}_4}
d_{{\bf k}_1,\sigma}^\dag d_{{\bf k}_2,\sigma}
s_{{\bf k}_3,\sigma'}^\dag s_{{\bf k}_4,\sigma'}$. The operator $d_{{\bf k}_1,\sigma}^\dag d_{{\bf k}_2,\sigma}$ represents a particle-hole excitation. Therefore, we analyze the RIXS process by decomposing into such a particle-hole excitation in order to understand the anisotropic momentum dependence of the threshold along (0,0) to ($\pi$/2,0) and along (0,0) to ($\pi$/2,$\pi$/2) in Fig. \[figrixs\].
As a first step, we consider the particle-hole excitation as the convolution of the single-particle excitation spectra $A({\bf k},\omega)$ between occupied LHB and unoccupied UHB. Figure \[figakw\] shows $A({\bf k},\omega)$ in the half-filled Hubbard model with $t'$ and $t''$ terms[@hubakw]. Below the chemical potential denoted by the dotted line, a sharp peak appears at ($\pi$/2,$\pi/$2) with the lowest-binding energy. In contrast, the spectrum at ($\pi$,0) is very broad and deep in energy. These features are consistent with the ARPES data for Sr$_2$CuO$_2$Cl$_2$[@Wells; @Kim]. Above the chemical potential, the dispersion of UHB has the minimum of the energy at ${\bf k}$=($\pi$,0)[@SCB]. We show below that the ($\pi$,0) spectrum in UHB plays a crucial role in the RIXS spectrum with $\Delta{\bf K}$=($\pi$/2,0).
Now, we examine the lowest-energy excitations with $\Delta{\bf K}$=(0,0), ($\pi$/2,0), and ($\pi/$2,$\pi$/2) in the convoluted spectrum $\int_{{\cal E}\le\mu} d{\cal E}A({\bf k}+\Delta{\bf K},{\cal E}+\omega)
A({\bf k},{\cal E})$, where $\mu$ is the chemical potential. For the case that $\Delta{\bf K}$=(0,0), the minimum excitation energy in the convoluted spectrum is $\sim$5$t$ at ${\bf k}$=($\pi$,0). In the same way, the lowest-energy excitation with $\Delta{\bf K}$=($\pi$/2,0) in the convoluted spectrum is that from ${\bf k}$=($\pi$/2,0) of LHB to ($\pi$,0) of UHB with the energy of $\sim$4$t$. This value is smaller than that for the $\Delta{\bf K}$=(0,0) case, being consistent with the relation of the thresholds of the RIXS spectra between $\Delta{\bf K}$=(0,0) and ($\pi$/2,0). In contrast, the lowest-energy excitation with $\Delta{\bf K}$=($\pi$/2,$\pi$/2) in the convoluted spectrum is inconsistent with that in the RIXS spectrum, because the excitation energy, which is determined by the peaks at ${\bf k}$=($\pi$/2,$-\pi$/2) of LHB and at ($\pi$,0) of UHB, is almost the same as that for $\Delta{\bf K}$=($\pi$/2,0). This means that the argument based on the convoluted spectrum of $A({\bf k},\omega)$ is insufficient to understand the anisotropic behavior of the threshold, and thus we need to treat the process of the particle-hole excitation exactly. Therefore, we introduce the spectral function $B({\bf k},\Delta{\bf K};\omega)$ of the two-body Green’s function, which can describe the particle-hole excitation, defined as $$\begin{aligned}
B({\bf k},\Delta{\bf K};\omega)&=&
\sum_\alpha\left|\langle\alpha|\sum_\sigma
d_{{\bf k}+\Delta{\bf K},\sigma}^\dag d_{{\bf k},\sigma}|0\rangle\right|^2
\nonumber\\&&\times
\delta(\omega-E_\alpha+E_0),\end{aligned}$$ where the states $|\alpha\rangle$ have the same point-group symmetry as that of the final states of the RIXS process. Figure \[figtwo\] shows $B({\bf k},\Delta{\bf K};\omega)$ with $[{\bf k},\Delta{\bf K}]=[(\pi/2,0),(\pi/2,0)]$ and $[(\pi/2,-\pi/2),(\pi/2,\pi/2)]$. Note that ${\bf k}+\Delta{\bf K}=(\pi,0)$ in both cases. The spectrum with $[{\bf k}, \Delta{\bf K}]=[(\pi/2,0),(\pi/2,0)]$ reproduces very well the RIXS spectrum near the threshold. The intensity of $B({\bf k},\Delta{\bf K};\omega)$ with $\Delta{\bf K}=(\pi/2,\pi/2)$ is very small, implying that the process $\sum_\sigma d_{(\pi,0),\sigma}^\dag d_{(\pi/2,-\pi/2),\sigma}$ is almost forbidden[@SDW]. This is the origin of the small weight of $I(\Delta{\bf K},\omega)$ with $\Delta{\bf K}=(\pi/2,\pi/2)$ around $\omega/t\sim 5$. We note that this behavior can not be obtained by the convolution of $A({\bf k},\omega)$ mentioned above. At the higher energy region, $\omega/t\sim 7$, different processes keeping $\Delta{\bf K}=(\pi/2,\pi/2)$, for example, the annihilation of the $(\pi/2,0)$ electron and the creation of the $(\pi,\pi/2)$ one, dominate the excitations. These processes induce the large weight above $\omega/t\sim 7$.
In summary, we have examined the momentum dependence of the Cu $K$-edge RIXS spectrum by using numerically exact diagonalization technique on small clusters. Regarding the ZRB as the LHB, we have adopted the Hubbard model with Cu $1s$- and $4p$-bands. We have also introduced the long-range hoppings, $t'$ and $t''$, of the ZR singlet, and found that the threshold of the spectrum at $\Delta{\bf K}=(\pi/2,0)$ is small compared with that at (0,0), whereas the threshold at $(\pi/2,\pi/2)$ is larger than that at (0,0). This dependence is caused by the ($\pi$,0) state in UHB. Thus, by examining the RIXS spectrum, we can extract the property of the unoccupied states that is crucially important for the electron-doped superconductors and also for the understanding of the behavior different from the hole-doped superconductors [@Kim; @Takagi]. Very recently, Stanford’s group has reported [@Hasan] that interesting data with momentum dependent inelastic scattering signal has been seen in the 2 eV range, which reveals the property of unoccupied states right above the charge-transfer gap. These progress in both theory and experiment will open a new prospect of the physics of cuprates.
The authors thank Z.-X. Shen for valuable discussions. This work was supported by Priority-Areas Grants from the Ministry of Education, Science, Culture and Sport of Japan, CREST, and NEDO. Computations were carried out in ISSP, Univ. of Tokyo; IMR, Tohoku Univ.; and Tohoku Univ..
C.-C. Kao [*et al.*]{}, Phys. Rev. B [**54**]{}, 16361 (1996). S. M. Butorin [*et al.*]{}, Phys. Rev. B [**55**]{}, 4242 (1997). J. P. Hill [*et al.*]{}, Phys. Rev. Lett. [**80**]{}, 4967 (1998). P. Kuiper [*et al.*]{}, Phys. Rev. Lett. [**80**]{}, 5204 (1998). P. Abbamonte [*et al.*]{}, cond-mat/9810095. The best resolution used was 0.45 eV. F. C. Zhang and T. M. Rice, Phys. Rev. B [**37**]{}, 3759 (1988). B. O. Wells [*et al.*]{}, Phys. Rev. Lett. [**74**]{}, 964 (1995). C. Kim [*et al.*]{}, Phys. Rev. Lett. [**80**]{}, 4245 (1998). F. Ronning [*et al.*]{}, Science, [**282**]{}, 2067 (1998). H. Takagi [*et al.*]{}, Physica C [**162-164**]{}, 1001 (1989). C. T. Chen [*et al.*]{}, Phys. Rev. Lett. [**66**]{}, 104 (1991). The experimental data for the Cu $K$-edge absorption spectrum show that the out-of-plane ($z$) polarized peaks are lower in energy than in-plane polarized peaks[@Kosugi]. In order to examine the RIXS spectrum with the single component of the 4$p$ bands, we consider the case of $z$-polarized photon. In 8- and 10-site clusters, only $t$ and $t'$ are included for hopping parameters due to their small system sizes. P. M. Platzman and E. D. Isaacs, Phys. Rev. B [**57**]{}, 11107 (1998). K. Tsutsui, T. Tohyama, and S. Maekawa (unpublished). For $A({\bf k},\omega)$ in the Hubbard model without $t'$ and $t''$, see P. W. Leung [*et al.*]{}, Phys. Rev. B [**46**]{}, 11779 (1992). In the large $U$ limit, the dispersion of the UHB is described by the single-hole dispersion of the $t$-$t'$-$t''$-$J$ model with $t$$<$0, $t'$$>$0, and $t''$$<$0. The sign difference comes from the fact that the carrier in UHB is an electron \[see T. Tohyama. and S. Maekawa. Phys. Rev. B [**49**]{}, 3596 (1994)\]. Although the energy at ($\pi$,0) is almost the same as that at ($\pi$/2,$\pi$/2) in the $t$-$J$ model, the former decreases when $t'$($>$0) and $t''$($<$0) are introduced. By defining $\epsilon$(${\bf k}$)=4$t'$cos$k_x$cos$k_y$+2$t''$(cos$k_x$+cos$k_y$), the energy difference between at ($\pi$/2,$\pi$/2) and ($\pi$,0) is proportional to $\epsilon$($\pi$/2,$\pi$/2)$-$$\epsilon$($\pi$,0)=$-$8$t''$+4$t'$. In the spin-density wave mean-field approximation, $B({\bf k},\Delta{\bf K};\omega)$ with \[${\bf k}$, $\Delta{\bf K}$\]= \[($\pi$/2,$-\pi$/2),($\pi$/2,$\pi$/2)\] is rigorously zero. This is due to the effect of the coherence factor arising from the antiferromagnetic long-range order as is the case of the BCS superconductivity. The detail will be shown elsewhere. Z. Hasan, E. Issacs, and Z.-X. Shen (unpublished). N. Kosugi [*et al.*]{}, Chem. Phys. [**135**]{}, 149 (1989)
| ArXiv |
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abstract: |
The ad-trading desks of media-buying agencies are increasingly relying on complex algorithms for purchasing advertising inventory. In particular, Real-Time Bidding (RTB) algorithms respond to many auctions – usually Vickrey auctions – throughout the day for buying ad-inventory with the aim of maximizing one or several key performance indicators (KPI). The optimization problems faced by companies building bidding strategies are new and interesting for the community of applied mathematicians. In this article, we introduce a stochastic optimal control model that addresses the question of the optimal bidding strategy in various realistic contexts: the maximization of the inventory bought with a given amount of cash in the framework of audience strategies, the maximization of the number of conversions/acquisitions with a given amount of cash, etc. In our model, the sequence of auctions is modeled by a Poisson process and the *price to beat* for each auction is modeled by a random variable following almost any probability distribution. We show that the optimal bids are characterized by a Hamilton-Jacobi-Bellman equation, and that almost-closed-form solutions can be found by using a fluid limit. Numerical examples are also provided.
**Keywords:** Real-Time Bidding, Vickrey auctions, Stochastic optimal control, Convex analysis, Fluid limit approximation.
author:
- 'Joaquin Fernandez-Tapia[^1], Olivier Guéant[^2], Jean-Michel Lasry[^3]'
nocite: '[@*]'
title: 'Optimal Real-Time Bidding Strategies[^4]'
---
Introduction
============
From the viewpoint of a company launching an advertising campaign, the goal of digital advertising is to increase its return on investment by leveraging the different channels enabling an interaction with its potential customers: desktop display, mobile, social media, e-mailing, etc. Usually, this is achieved via branding campaigns, by prospecting individuals who are likely to be in affinity with a given product/campaign, or by driving those who have already shown some interest into a final conversion (*e.g.* a purchase).\
In recent years, the advertising industry has gone through a lot of upheavals: numerous technological changes, a deluge of newly available data, the emergence of a huge number of ad-tech startups entering the market, etc. In particular, new mechanisms have emerged and have completely changed the way digital ad inventory is purchased. In practice, the inventory is often purchased programmatically, and it is possible to algorithmically buy it unit by unit, with the hope of making real the original promise of the advertising and media buying industry: *targeting the right person, at the right time, and in the right context.*\
Programmatic media buying has skyrocketed over the last five years. Although these figures can only be rough approximations, it is estimated that the total net advertising revenue linked to programmatic desktop display in Europe was around bn in 2014. For programmatic mobile display and video display the figures were respectively m and m – see [@iab]. Overall, the total growth in net advertising revenue related to programmatic media buying was around 70% in Europe between 2013 and 2014. IAB Europe estimates in [@iab] that the percentage of revenue coming from programmatic media buying is a two-digit number for all formats: 39% for desktop display, 27% for mobile display, and 12% for video display. In the US, the figures are even more staggering with \$5.89bn spent programmatically on desktop/laptop display, and \$4.44bn on mobile/tablet display, in 2014 (source: eMarketer.com).\
One of the main and most exciting developments in programmatic media buying is Real-Time Bidding (or RTB). RTB is a new paradigm in the way digital inventory is purchased: advertisers[^5] can buy online inventory through real-time auctions for displaying a banner (or a short video). These real-time auctions make it possible for advertisers to target individual users on a per-access basis.\
In a nutshell, each time a user visits a website, the publisher – the supply side – connects to a virtual marketplace, called an *ad exchange*, in order to trigger an auction for each available slot that can be allocated to advertising. On the demand side, ad trading desks receive auction requests (sometimes through a Demand-Side Platform – DSP), together with information about the user, the type of website, etc., and choose the bid level that best suits their strategy. Once the different bids are processed, the slot is attributed to the bidder who has proposed the highest bid and the price paid depends on the type of auction. The entire process, from the user entering the website to the display of the banner, takes around 100 milliseconds.\
RTB auctions are usually of the Vickrey type, also known as “second-price auctions”. In short, the mechanism is the following: first, the participants send their bids in a sealed way, then, the item (here the slot) is sold to the participant who has proposed the highest bid, and the price paid by this participant corresponds to the second best bid (or to a minimum price if there is only one participant). Structurally, Vickrey auctions give participants an incentive to reveal their true valuation for the item – see [@vickrey].\
The problem faced by ad-trading desks is to choose the optimal bid level each time they receive a request to participate in a Vickrey auction. Here, optimality may have different meanings, depending on the considered key performance indicator (KPI). In all cases, the complexity of the problem arises from the need of optimizing a functional depending on *macroscopic* quantities at an hourly, daily or weekly timescale, by interacting with the system at the *microscopic* scale of each auction, *i.e.* through a high-frequency/low-latency bidding algorithm participating in thousands of auctions per second. This multi-scale feature leads to the need of mathematical models that are both realistic and tractable, because numerical methods are often cumbersome and time-consuming in the case of multi-scale problems. In this article, we rely on methods coming from stochastic optimal control and we show that the optimal bidding strategy can be approximated very precisely (and almost in closed form) by using classical tools of convex optimization.\
Besides the classical literature on Vickrey auctions (see for instance [@vickrey2; @vickrey3; @vickrey]) – which is related to auction theory and more generally to game theory –, the academic literature on this new kind of problems is really scarce. General approaches for Real-Time Bidding optimization from a buyer’s perspective can be found mostly in conference proceedings from the computer-science community (*e.g.* [@lee; @zhang]). Our approach is similar to the one presented in the work of Amin *et al.* [@amin]: both are Markov Decision Process (MDP) approaches[^6] and dealing with similar auction problems. However, besides the originality of their model, Amin *et al.* do not extend their mathematical development beyond the baseline discrete case. Another author introduced an MDP approach in the conference paper [@yuan], but he focused on the problem from a publisher perspective. In general, the supply-side perspective has generated more academic research than the demand-side one (see Yuan’s PhD dissertation [@yuan2] and the articles by Balseiro *et al.* [@balseiro1; @balseiro2]). A recent study of RTB auctions from a buyer’s perspective is Stavrogianni’s PhD dissertation [@stravrogiannis].\
Our stochastic optimal control approach is inspired by the academic literature in algorithmic trading [@hft1; @hft2; @hft3], where, similarly to our problem, the goal is to optimize a macroscopic functional depending on the terminal state of the algorithm (*e.g.* at the end of the day) by continuously making decisions on a high-frequency basis (*i.e.* milliseconds). Moreover, like in high-frequency trading models involving limit orders, the algorithm should react to a system driven by one or several controlled Poisson processes.\
In this paper – the first of a series on Real-Time Bidding, see [@fglpricing] and [@fgllearning] –, we model by a marked Poisson process the sequence of auction requests received by an ad-trading desk: the Poisson process models the arrival times of the requests, and the marks correspond to independent random variables $(p_n)_{n \in \mathbb{N}^*}$ modeling the *price to beat*, *i.e.* the highest bid proposed by the other participants’ in the auction.[^7] Every time an auction is received, the algorithm sends a bid $b$ to the auction server. For the $n^{\text{th}}$ auction, the inventory is purchased by the algorithm if and only if the bid sent by the algorithm is greater than the price to beat $p_n$ (and in that case the price paid for the slot is $p_n$). The rationale for considering this *statistical model*, rather than a more complicated game-theoretical one, comes from: (i) the large number of auction requests (several hundreds per second) for most segments of audience, and (ii) our assumption that the algorithm is restricted to an homogeneous subset of the inventory (*i.e.* we assume that a segmentation of the different audiences and contexts has been carried out beforehand, or, in other words, that the problem we consider is at the tactical “execution” level – see also [@f1; @f2]).\
In Section 2, we introduce the main notations of our modeling framework, and we focus on a stochastic optimal control problem where an ad trader aims at maximizing the total number of banners displayed, for a given spending (audience strategy). We exhibit the characterization of the optimal bidding strategy with a Hamilton-Jacobi-Bellman equation, and show that the optimal bidding strategy can be obtained in almost-closed form by using a fluid-limit approximation. Moreover, this approximation leads to a new characterization of the bidding strategy in the form of an optimal scheduling, which in practice can be tracked by a feedback-control mechanism. The main result is indeed that the budget should be spent evenly over the considered time window.\
In Section 3, we propose several extensions of our model. In particular, we generalize the initial model by considering several sources and types of inventory, and we consider another objective function for taking account of a very important KPI: the number of conversions. In these extensions, as in the initial model with only one source of inventory, we find that the total budget should be spent evenly over the considered time window – see [@fgllearning] for a different conclusion when on-line learning is considered. However, we find that the optimal bidding strategies are not uniform across sources, but instead proportional to an index which is, in the general case with multiple sources and conversions, a simple function of (i) the *a priori* interest for each source/type of inventory, (ii) the *a priori* interest for each source of conversion, and (iii) the probability of conversion associated to each source. In Section 3, we also discuss the difference between first-price and second-price auctions, the impact of floor prices, and more general – nonlinear – objective criterions.
A model for audience strategies
===============================
In this section, we present a model where an ad trader wishes to spend a given amount of money $\bar{S}$ over a time window $[0,T]$, in order to display a maximum number of banners to a given population. He receives auction requests at random times from a single ad exchange – and does not know in advance how many auction requests he will receive. The model we propose is naturally written in continuous time and the (random) occurrences of auction requests are modeled by the jumps of a Poisson process. A simplified discrete-time modeling framework is presented in the appendix for readers who are more used to discrete-time Markov decision processes.\
The modeling framework in continuous time
-----------------------------------------
Let us fix a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ equipped with a filtration $(\mathcal{F}_t)_{t\in \mathbb{R}_+}$ satisfying the usual conditions. We assume that all stochastic processes are defined on $(\Omega, \mathcal{F},(\mathcal{F}_t)_{t\in \mathbb{R}_+}, \mathbb{P})$.\
*Auctions:*\
We consider an ad trader connected to an ad exchange.[^8] He receives requests to participate in auctions in order to purchase inventory and display some banners to the specific population he wants to target. Requests are modeled with a marked Poisson process: the arrival of requests is triggered by the jumps of a Poisson process $(N_t)_t$ with intensity $\lambda > 0$,[^9] and the marks $(p_n)_{n \in \mathbb{N}^*}$ correspond, for each auction, to the highest bid among the other participants’ bids.\
Every time he receives a request to participate in an auction, the ad trader can bid a price: at time $t$, if he receives a request, then we denote his bid by $b_t$. We assume that the ad trader stands ready to bid (possibly a bid equal to $0$ or $+\infty$) at all times. In particular, we assume that $(b_t)_t$ is a predictable process with values in $\mathbb{R}_+ \cup \{+\infty\}$.\
If at time $t$ the $n^{\text{th}}$ auction occurs, the outcome of this auction is the following:
- If $b_t > p_{n}$, then the ad trader wins the auction: he pays the price $p_{n}$ and his banner is displayed.
- If $b_t \le p_{n}$, then[^10] the ad trader does not win the auction. In particular, the trader’s banner is not displayed and he pays nothing.
An important assumption of our model is that $(p_n)_{n \in \mathbb{N}^*}$ are *i.i.d.* random variables distributed according to an absolutely continuous distribution. We denote by $F$ the associated cumulative distribution function and by $f$ the associated probability density function. Our assumptions are the following:
- $\forall n\in \mathbb N^*$, $p_n$ is almost surely positive. In particular, $F(0) = 0$.
- $\forall p > 0, f(p) > 0$.[^11]
- $\lim_{p \to +\infty} p^3 f(p) = 0$.\
*Remaining cash process:*\
We denote by $(S_t)_t$ the process modeling the amount of cash to be spent. Its dynamics is:[^12] $$\label{dynS}
dS_t = - p_{N_t} \mathbf{1}_{\{b_t> p_{N_t}\}}dN_t, \quad S_0 = \bar{S}.$$
*Inventory process:*\
The number of impressions, *i.e.* the number of banners displayed, is modeled by the inventory process $(I_t)_t$. Its dynamics is: $$dI_t = \mathbf{1}_{\{b_t> p_{N_t}\}}dN_t, \quad I_0 = 0.$$
*Stochastic optimal control problem:*\
In the model of this section, the trader aims at maximizing the expected number of banners displayed over $[0,T]$. We should impose not to spend more than $\bar{S}$, but, for technical reasons, we do prefer to consider a relaxed problem, and to penalize extra-spending. We consider therefore the maximization of the following criterion: $$\mathbb E \left[I_T - K \min\left(S_T,0\right)^2\right],$$ where $K$ measures the importance of the penalty for extra-spending.[^13]\
For mathematical reasons, we do prefer to write the stochastic optimal control problem as a minimization problem: $$\inf_{(b_t)_t \in \mathcal{A}}\mathbb E \left[-I_T + K \min\left(S_T,0\right)^2\right],$$ where $\mathcal{A}$ is the set of predictable processes with values in $\mathbb{R}_+ \cup \{+\infty\}$.\
*Value function:*\
To tackle this stochastic optimal control model, we introduce the value function $$u: (t,I,S) \in [0,T]\times \mathbb{N}\times (-\infty, \bar{S}] \mapsto \inf_{(b_s)_{s\ge t} \in \mathcal{A}_{t}} \mathbb E\left[-I^{b,t,I}_T + K \min\left(S^{b,t,S}_T,0\right)^2\right],$$ where $\mathcal{A}_{t}$ is the set of predictable processes on $[t,T]$ with values in $\mathbb{R}_+ \cup \{+\infty\}$, and $$dS^{b,t,S}_s = - p_{N_s} \mathbf{1}_{\{b_s> p_{N_s}\}}dN_s, \quad S^{b,t,S}_t = S,$$ $$dI^{b,t,I}_s = \mathbf{1}_{\{b_s> p_{N_s}\}}dN_s, \quad I^{b,t,I}_t = I.$$
Solution of the stochastic optimal control problem
--------------------------------------------------
### A characterization with a non-standard HJB equation
To solve the above stochastic optimal control problem, we introduce the associated Hamilton-Jacobi-Bellman (HJB) equation which corresponds to the continuous-time equivalent of the Bellman equation (A.2) derived in discrete time from the dynamic programming principle – see the appendix. Here the HJB equation is:[^14]
$$\label{m1:HJB}
-\partial_t u(t,I,S) - \lambda \inf_{b \in \mathbb R_+}\int_0^b f(p) (u(t,I+1,S-p) - u(t,I,S)) dp = 0,$$
with terminal condition $u(T,I,S) = - I + K \min\left(S,0\right)^2$.\
Eq. (\[m1:HJB\]) is a non-standard integro-differential HJB equation – which is similar to Eq. (A.2). Because the objective function is affine in the variable $I$, the dimensionality of the problem can be reduced by considering the ansatz $u(t,I,S) = - I + v(t,S)$.\
With this ansatz, Eq. (\[m1:HJB\]) becomes indeed: $$\label{m1:HJB2}
-\partial_t v(t,S) - \lambda \inf_{b \in \mathbb R_+} \int_0^b f(p) (v(t,S-p) - v(t,S)-1) dp = 0,$$ with terminal condition $v(T,S) = K \min\left(S,0\right)^2$.[^15]\
Eq. (\[m1:HJB2\]) is another non-standard integro-differential Hamilton-Jacobi-Bellman, but with only one spatial dimension instead of two. It is straightforward to verify that it corresponds to the following stochastic optimal control problem: $$\label{m1:obj2}
\inf_{(b_t)_t \in \mathcal{A}}\mathbb E \left[-\lambda \int_0^T F(b_t) dt + K \min\left(S_T,0\right)^2\right],$$ where $\mathcal{A}$ is the set of predictable processes with values in $\mathbb{R}_+ \cup \{+\infty\}$.\
The change of variables $u(t,I,S) = - I + v(t,S)$ is key to reduce the dimensionality of the problem. For understanding the underlying rationale, let us notice that $-u(t,I,S)$ is (up to the penalization term) the number of impressions an ad trader should expect to purchase over the time interval $[0,T]$, if, at time $t$, he has already purchased a number of impressions equal to $I$, and has an amount $S$ to be spent optimally over $[t,T]$. Because the number of impressions that will be purchased over $[t,T]$ does not depend on the number of impressions already purchased, except through the budget that has been spent – or, equivalently, that remains to be spent –, the quantity $-u(t,I,S) - I$ should be independent of $I$. Hence the ansatz $u(t,I,S) = - I + v(t,S)$. In particular, the quantity $-v(t,S)$ represents (up to the penalization term) the number of impressions an ad trader should expect to purchase over the time interval $[t,T]$, if he has, at time $t$, an amount $S$ to be spent optimally over $[t,T]$.\
### The reasons why we need to go beyond Eq. (\[m1:HJB2\])
From the previous paragraphs, one could consider that the problem is almost entirely solved. Because Eq. (\[m1:HJB2\]) characterizes the optimal bidding strategy, finding the optimal bid in each state $(t,S)$ of the system simply boils down indeed to implementing a backward (monotone) numerical scheme to approximate the solution of Eq. (\[m1:HJB2\]).\
However, it is noteworthy that the problem faced by the ad trader is a two-scale one:
- The macroscopic scale is the tactical scale.[^16] This scale is related to the time horizon $T$ and to the amount of cash $\bar{S}$ the ad trader has to spend over the time interval $[0,T]$. In practice, for intraday tactics, the order of magnitude is in the range $10^{2}-10^{4}$ s (seconds) for $T$, and in the range for $\bar{S}$.
- The microscopic scale is the scale of each auction, related to the intensity $\lambda$, and to the variables $b$ and $p$ (in the integral term of Eq. (\[m1:HJB2\])). The order of magnitude for these variables is around $10^3$ s$^{-1}$ for $\lambda$, and in the range for each bid.[^17]
If one wants to approximate the solution of Eq. (\[m1:HJB2\]) on a grid in $(t,S)$, the time step $\Delta t$ and the cash/spatial step $\Delta S$ should be compatible with the microscopic scale. To take account of the fast arrival of auction requests, a natural time step is $\Delta t$ around $10^{-4}$ second. As far as the cash to spend is concerned, if one wants to be precise when computing the optimal bid for each auction, a natural value for $\Delta S$ is around , given the previously discussed orders of magnitude for the bids. Given the orders of magnitude considered at the macroscopic scale, we need to consider a numerical scheme on a grid with $10^6-10^8$ points in time and $10^{8}-10^{11}$ points in space, which is computationally extremely costly.[^18] Even though smart numerical methods can be built to avoid computations at each point of the grid, the multi-scale nature of the problem is an important issue in practice.[^19]\
A way to avoid using a computer-intensive numerical method is to look for almost-closed-form approximations. As we will show below, such an approximation can be found by using a first-order Taylor expansion in the integral term of Eq. (\[m1:HJB2\]).
Almost-closed-form approximation of the solution
------------------------------------------------
In this subsection, we aim at approximating the value function $v$ and the resulting optimal control function $(t,S) \mapsto b^*(t,S)$ corresponding to the optimal bidding strategy – the latter is characterized either by $$\label{sol}v(t,S_{t-} - b^*(t,S_{t-})) = v(t,S_{t-}) + 1,$$ if this equation has a solution, or by $b^*(t,S_{t-}) = +\infty$ otherwise, *i.e.* if we are in the case where $\lim_{S \to -\infty} v(t,S) \le v(t,S_{t-}) + 1$.\
The main idea underlying the approximation we obtain in the following paragraphs is to replace the term $ v(t,S-p) - v(t,S)$ in Eq. (\[m1:HJB2\]) by its first-order Taylor expansion $-p \partial_S v(t,S)$. We therefore replace Eq. (\[m1:HJB2\]) by:
$$\label{m1:HJBc}-\partial_t v(t,S) - \lambda \inf_{b \in \mathbb R_+} - \int_0^b f(p)\left(1+ p \partial_S v(t,S)\right) dp = 0.$$
with terminal condition $v(T,S) = K \min\left(S,0\right)^2$.\
Intuitively, this approximation is relevant because of the multi-scale nature of the problem: the range of values for $b$ should be several orders of magnitude smaller than the values taken by $S$.[^20]
### A fluid-limit approximation
In the following paragraphs, we show that Eq. (\[m1:HJBc\]) is the Hamilton-Jacobi equation associated with a variational problem which can be regarded as the fluid limit of the stochastic optimal control problem we considered in the previous subsection.\
For that purpose, let us introduce the following optimization problem: $$\inf_{(\tilde{b}_t)_t \in \mathcal{A}_{\text{det}}} -\lambda \int_0^T F(\tilde{b}_t) dt + K \min\left(\tilde{S}^{\tilde{b}}_T,0\right)^2,$$ where $$d\tilde{S}^{\tilde{b}}_t = - \lambda G(\tilde{b}_t)dt, \quad G: x \in \mathbb{R}_+ \cup \{+\infty\} \mapsto \int_0^{x} p f(p) dp,$$ and where $\mathcal{A}_{\text{det}}$ is the set of $\mathcal{F}_0$-measurable processes with values in $\mathbb{R}_+ \cup \{+\infty\}$.\
The value function $\tilde{v}$ associated with this problem is defined as: $$\tilde{v}(t,S) = \inf_{(\tilde{b}_s)_{s\ge t} \in \mathcal{A}_{\text{det}}} -\lambda \int_t^T F(\tilde{b}_s) ds + K \min\left(\tilde{S}^{\tilde{b},t,S}_T,0\right)^2,$$ where $$d\tilde{S}^{\tilde{b},t,S}_s = - \lambda G(\tilde{b}_s)ds, \quad \tilde{S}^{\tilde{b},t,S}_t = S.$$
We have the following Theorem:
\[theo\] Let us define $$\begin{aligned}
H(x) &=& \lambda \sup_{b \in \mathbb R_+} \int_0^b f(p)\left(1+ xp\right) dp.\\
&=& \left\{
\begin{array}{ll}
- \lambda x \int_0^{-\frac{1}{x}} F(p) dp, & x <0 \\
\lambda\left(1+x\int_0^{\infty} pf(p) dp\right), & x \ge 0.
\end{array}
\right.\\\end{aligned}$$ The value function $\tilde{v}$ is given by: $$\tilde{v}(t,S) = \sup_{x \le 0} \left(Sx - (T-t) H(x) - \frac{x^2}{4K}\right).$$ It is the unique weak semi-concave solution of the Hamilton-Jacobi equation (\[m1:HJBc\]).\
Furthermore, the optimal control function $(t,S) \mapsto \tilde{b}^*(t,S)$ is given by the following:
- If $S \ge \lambda (T-t) \int_0^{\infty} pf(p) dp$, then $\tilde{b}^*(t,S) = +\infty$.
- If $S < \lambda (T-t) \int_0^{\infty} pf(p) dp$, then $\tilde{b}^*(t,S) = -\frac{1}{x^*}$, where $x^*$ is characterized by $$\label{m1:characy}
S = (T-t) H'(x^*) + \frac{x^*}{2K}.$$
By using the change of variables $a_s = \lambda G(\tilde{b}_s) \in \mathcal{I} = [0,\lambda \int_0^\infty p f(p) dp]$, we have:
$$\label{lfa}
\tilde{v}(t,S) = \inf_{(a_s)_{s\ge t} \in \mathcal{A}'_{\text{det}}} \int_t^T L\left(a_s\right) ds + K \min\left(\widehat{S}^{a,t,S}_T,0\right)^2,$$
where $\mathcal{A}'_{\text{det}}$ is the set of $\mathcal{F}_0$-measurable processes with values in $\mathcal{I}$, where $$d\widehat{S}^{a,t,S}_s = -a_s ds, \quad \widehat{S}^{a,t,S}_t = S,$$ and where the function $L$ is defined by: $$L : a \in \mathcal{I} \mapsto -\lambda F\left(G^{-1}\left(\frac{a}{\lambda}\right)\right).$$
$L$ is continuously differentiable on the interior of $\mathcal{I}$, with: $$\begin{aligned}
L'(a) &=& \left(G^{-1}\right)'\left(\frac{a}{\lambda}\right) F'\left(G^{-1}\left(\frac{a}{\lambda}\right)\right)\\
&=& -\frac{f\left(G^{-1}\left(\frac{a}{\lambda}\right)\right)}{G'\left(G^{-1}\left(\frac{a}{\lambda}\right)\right)}\\
&=& -\frac{1}{G^{-1}\left(\frac{a}{\lambda}\right)}.\\\end{aligned}$$
In particular, since $G$ is an increasing function, $L'$ is increasing, and, therefore, $L$ is strictly convex.\
Let us now compute the Legendre-Fenchel transform of $L$: $$\begin{aligned}
\nonumber L^*(x) &=& \sup_{a \in \mathcal{I} } x a - L(a)\\
\nonumber &=& \sup_{b \in \mathbb{R}_+ } \lambda x G(b) + \lambda F(b)\\
\label{m1:supb}&=& \lambda \sup_{b \in \mathbb{R}_+ } \int_0^b (1+ xp) f(p) dp\\
\nonumber &=& H(x).\end{aligned}$$
If $x < 0$, then the first order condition associated with the supremum in Eq. (\[m1:supb\]) is $f(b^*) \left(1+ b^*x\right) = 0$. In other words:
$$x < 0 \Rightarrow b^* = - \frac{1}{x}.$$
If $x$ is nonnegative, then $b^* = +\infty$.\
In the former case, we can write (by using an integration by parts): $$\begin{aligned}
H(x) &=& \lambda \sup_{b \in \mathbb R_+} \int_0^b \left(1+ xp\right) f(p) dp\\
&=& \lambda \int_0^{b^*} f(p)\left(1+ xp\right) dp \\
&=& -\lambda x \int_0^{-\frac 1x} F(p) dp. \\\end{aligned}$$
The Legendre-Fenchel transform of $L$ is therefore $H$, which can indeed be written as: $$H(x) =\left\{
\begin{array}{ll}
- \lambda x \int_0^{-\frac{1}{x}} F(p) dp, & x <0 \\
\lambda\left(1+x\int_0^{\infty} pf(p) dp\right), & x \ge 0.
\end{array}
\right.$$
It is straightforward to check that $H$ is a twice continuously differentiable convex function.[^21]\
We know, from classical variational calculus, that $\tilde{v}$ is given by the Hopf-Lax formula: $$\begin{aligned}
\label{m1:hl} \tilde{v}(t,S) &=& \inf_{y \in [S -\lambda \int_0^\infty pf(p) dp(T-t),S]} \left((T-t) L\left(\frac{S-y}{T-t}\right) + K\min(y,0)^2\right).\end{aligned}$$
By using an infimal convolution, we can transform Eq. (\[m1:hl\]) into: $$\begin{aligned}
\nonumber \tilde{v}(t,S) &=& \sup_{x} \left(Sx - (T-t) H(x) - \sup_y \left(yx - K\min(y,0)^2\right)\right)\\
\nonumber &=& \sup_{x} \left(Sx - (T-t) H(x) - \chi_{\mathbb R_-}(x) - \frac{x^2}{4K}\right)\\
\label{m1:vcff} &=& \sup_{x \le 0} \left(Sx - (T-t) H(x) - \frac{x^2}{4K}\right).\end{aligned}$$
This is the first result of the Theorem.\
Furthermore, we know that it is the unique weak semi-concave solution of the following Hamilton-Jacobi equation: $$-\partial_t \tilde{v}(t,S) + H(\partial_S \tilde{v}(t,S)) = 0,$$ with terminal condition $\tilde{v}(T,S) = K\min(S,0)^2$.\
This equation is exactly the same as Eq. (\[m1:HJBc\]), hence the second assertion of the Theorem.\
Moreover the optimal control in the modified problem (\[lfa\]) is given by $$a^*(t,S) = H'(\partial_S \tilde{v}(t,S)).$$
Therefore, the optimal control function in the initial problem is: $$\tilde{b}^*(t,S) = \left\{
\begin{array}{ll}
-\frac{1}{\partial_S \tilde{v}(t,S)}, & \partial_S \tilde{v}(t,S) <0 \\
+\infty, & \partial_S \tilde{v}(t,S) \ge 0.
\end{array}
\right.$$
If $S \ge (T-t) H'(0) = \lambda (T-t) \int_0^{\infty} pf(p) dp$, then the supremum in Eq. (\[m1:vcff\]) is reached at $x^*=0$. Therefore, $\partial_S \tilde{v}(t,S) = 0$ and $\tilde{b}^*(t,S) = +\infty$.\
Otherwise, the supremum in Eq. (\[m1:vcff\]) is reached at $x^*\le 0$ uniquely characterized by: $$S = (T-t) H'(x^*) + \frac{x^*}{2K}.$$
In particular, $\partial_S \tilde{v}(t,S) = x^*$ and $\tilde{b}^*(t,S) = -\frac{1}{x^*}$ where $x^*$ is characterized by Eq. (\[m1:characy\]).\
\
### The approximation of the optimal bidding strategy and its interpretation
Theorem \[theo\] invites to use $\tilde{v}$ as an approximation for $v$. Then the optimal bidding strategy $(b^*_t)_t$ can be approximated either by $\tilde{b}^*(t,S_{t-})$, or by solving Eq. (\[sol\]) in which $v$ is replaced by $\tilde{v}$. In what follows, we consider the former approach. Furthermore, we consider the limit case where $K\to +\infty$. Subsequently, we approximate the optimal bid $b^*_t$ at time $t$ by $\tilde{b}^*_\infty(t,S_{t-})$, where the function $\tilde{b}^*_\infty$ is defined by:\
- If $S \ge \lambda (T-t) \int_0^{\infty} pf(p) dp$, then $\tilde{b}^*_\infty(t,S) = +\infty$.
- If $S < \lambda (T-t) \int_0^{\infty} pf(p) dp$, then $\tilde{b}^*_\infty(t,S)$ is given by: $$\tilde{b}^*_\infty(t,S) = - \frac{1}{{H'}^{-1}\left(\frac{S}{T-t}\right)}.$$
In order to understand the above results, it is worth recalling that $\int_0^{\infty} pf(p) dp$ is the average price to beat, and $\lambda (T-t)$ is the expected number of auctions the ad trader will participate in between the time $t$ and the terminal time $T$. Therefore, $\lambda (T-t) \int_0^{\infty} pf(p) dp$ is the expected amount of cash the ad trader will spend if he bids an infinite price in order to win all the auctions for which he receives a request. It is therefore natural, in our risk-neutral setting, to bid $\tilde{b}^*_\infty(t,S_{t-}) = +\infty$ at time $t$ if the remaining amount to spend $S_{t-}$ is greater than $\lambda (T-t) \int_0^{\infty} pf(p) dp$.[^22]\
If $S_{t-} < \lambda (T-t) \int_0^{\infty} pf(p) dp$, then the approximation $\tilde{b}^*_\infty(t,S_{t-})$ of the optimal bid $b^*_t$ verifies:
$$\lambda (T-t) \int_0^{\tilde{b}^*_\infty(t,S_{t-})} pf(p) dp = S_{t-}.$$
To understand this bidding strategy, let us consider the situation from time $t$, in the interesting case, *i.e.* when $S_{t-} < \lambda (T-t) \int_0^{\infty} pf(p) dp$. If we use the above approximation of the optimal bidding strategy, the dynamics of the remaining cash is (for $0 \le t \le s \le T$):
$$dS_s = - p_{N_s} \mathbf{1}_{\{\tilde{b}^*_\infty(s,S_{s-})> p_{N_s}\}}dN_s, \qquad S_t = S.$$
Therefore:
$$d\mathbb{E}[S_s] = - \mathbb{E}\left[\lambda\int_0^{\tilde{b}^*_\infty(s,S_{s-})} p f(p) dp\right]ds = - \frac{\mathbb{E}[S_{s}]}{T-s} ds.$$
This gives $\mathbb{E}[S_s] = S \frac{T-s}{T-t}$. In other words, the approximation of the optimal bidding strategy is such that, on average, the remaining cash is spent evenly across $[t,T]$.[^23]\
One of the main consequences of the previous analysis is that optimality is not only characterized by the optimal bid, but also by the speed at which the budget is spent. The latter is very helpful in terms of implementation. Indeed, defining the solution by an *optimal scheduling curve* (also known as budget-pacing, see [@f2]) makes it possible to dynamically control the bid by a feedback-control system tracking a target curve which does not depend on the form of the function $f$. With this point of view, the exact knowledge of the optimal bidding function is no more necessary. However, this analysis is based on the fluid-limit approximation. Subsequently, it is valid as far as the number of auction requests is large enough. For audience strategies focusing on small subsets of the inventory (*e.g.* targeting only a few customers, restricting the strategy to *premium inventory* only, purchasing slots for a very specific banner format, etc.), a numerical approximation of the solution of the initial stochastic control problem is needed.[^24]\
Numerical examples {#num}
------------------
We present a numerical example for an algorithm spending during a period of 100 seconds. Due to the high-frequency nature of the algorithm, considering a larger time frame and a larger budget does not make an important difference from a mathematical standpoint. Notice that the spending rate ($\sim$ /day) – even though it may seem small given the sizes of overall advertizing budgets – is realistic in the light of our problem: we are focusing on an algorithm which is already restricted to a particular channel, audience, and subset of inventory. In practice, ad trading desks dynamically allocate the overall budget throughout the day across a large array of algorithms, such as the one presented here.[^25]
### Numerical approximations of $v$ and $b^*$
For our numerical experiments, we assume that the distribution of the price to beat is an exponential distribution[^26] with parameter $\mu$, *i.e.* $f(p) = \mu e^{-\mu p}$. We chose $\mu = 2\cdot 10^{3}$ $^{-1}$, which is consistent with a CPM of the order of $0.5$.\
We represent in Figure \[fig:valuef\] a numerical approximation of the solution $v$ of the HJB equation .[^27] As stated above, the quantity $-v(t,S)$ can be interpreted (up to the penalty term) as the expected number of impressions that will be purchased from time $t$ by an algorithm following the optimal strategy, when the remaining budget at time $t$ is $S$.\
When solving the HJB equation , the most important quantity to compute is the optimal control, *i.e.* the optimal bid function $(t,S) \mapsto b^*(t,S)$. This function is plotted in Figure \[fig:logbid\]. Because the values of the optimal bid increase abruptly as $t\rightarrow T$, we prefer to plot the logarithm in basis $10$ of $b^*(t,S)$.\
The optimal bids plotted in Figure \[fig:logbid\] are consistent with intuition: the higher the remaining budget and the closer to the terminal time, the higher the optimal bid.
### Simulations
One of the consequences of the fluid-limit approximation is that the average spending rate should be constant. An important question is whether or not this feature is also true when one uses the optimal bidding strategy obtained numerically by solving the HJB equation . To answer this question, we carried out simulations by drawing the times at which auction requests occur and the values of the price to beat for each auction (using the same set of parameters as above).\
Figures \[fig:app1\], \[fig:app2\] and \[fig:app3\] show the result for different time windows (we zoom in from 100 s, to 10 s, to 1 s): the remaining-cash process is plotted in the left panels and the number of impressions is plotted in the right panels. For the (realistic) values of the parameters we considered, it is clear that the budget is spent linearly.\
We also carried out simulations for a smaller value of $\lambda$ ($\lambda = 2 \text{ s}^{-1}$) – *i.e.* in a less liquid market, with a smaller number of auction requests – in order to test the robustness of the qualitative result obtained with the fluid-limit approximation. In Figure \[fig:low2\], we plot the solution $v$ to the HJB equation and the associated optimal bids (in log). In Figure \[fig:low1\], we plot the outcome of the simulations corresponding to the new values of the parameters.\
As above, we see that spending rate fluctuates but remains around a constant corresponding to even spending over the trading period.\
Extensions and discussion about the model
=========================================
In this section, we present two important extensions of the model introduced in Section 2. The first one is a multi-source extension: we consider the case where auction requests arise from several ad exchanges (or from the same ad exchange but from different types of audience), with different frequencies, and different laws for the price to beat. The second extension is related to conversions: in addition to the number of impressions they buy, ad trading desks often consider the number of conversions as a KPI, which is more in line with the return-on-investment goals of the brand or company launching the advertising campaign. For both extensions, we show that the problem remains in dimension $2$ (after a change of variables), even though the dimension of the state space increases proportionally with the number of sources and almost doubles when conversions are taken into account.\
In addition to these two extensions, we discuss many features of the model. In particular, the difference between Vickrey auctions and first-price auctions is addressed, along with the existence of floor prices of several kinds. KPIs are finally discussed because the dimensionality reduction at work in the model of Section 2 and in the two extensions presented below is associated with the linearity of the objective function (up to the penalty term).\
Introducing heterogeneity in auctions requests
----------------------------------------------
The first extension we consider focuses on the situation where the inventory can be purchased from different sources, *i.e.* different ad exchanges, or simply different types of audience. The heterogeneity comes from differences in the frequency of arrival of auction requests, in the distribution of the price to beat associated with each source, or in the relative importance given to each type of inventory in the objective function. An important situation is when the inventory is homogeneous across different ad exchanges and the differences only lie in the statistical properties of the auctions. In that case, we show – in the fluid-limit case – that the optimal bid are the same across venues, which is consistent with the result obtained in [@f1]. However, when audiences are heterogenous, we show that the optimal bids differ from one source to the next as a function of the relative importance given to the different types of audience in the objective function.\
### Setup of the model
In what follows, we still consider an ad trader who wishes to spend a given amount of money $\bar{S}$ over a time window $[0,T]$.\
*Auctions:*\
This ad trader is connected to $J>1$ sources from which he receives requests to participate in auctions in order to purchase inventory – we assume that the trader knows from which source each auction request arises. Requests are modeled with $J$ marked Poisson processes: the arrival of requests from the source $j \in \lbrace 1, \ldots, J \rbrace$ is triggered by the jumps of the Poisson process $(N^j_t)_t$ with intensity $\lambda^j > 0$, and the marks $(p^j_n)_{n \in \mathbb{N}^*}$ correspond, for each auction request sent by the source $j$, to the highest bid sent by the other participants.[^28]\
Every time he receives from the source $j$ a request to participate in an auction, the ad trader can bid a price: at time $t$, if he receives a request from the source $j$, then we denote his bid by $b^j_t$. As in the model with only one source, we assume that the ad trader stands ready to bid (possibly a bid equal to $0$ or $+\infty$) at all times. In particular, we assume that for each $j \in \lbrace 1, \ldots, J \rbrace$, the process $(b^j_t)_t$ is a predictable process with values in $\mathbb{R}_+ \cup \{+\infty\}$.\
If at time $t$ the $n^{\text{th}}$ auction associated with the source $j$ occurs, the outcome of this auction is the following:
- If $b^j_t > p^j_{n}$, then the ad trader wins the auction: he pays the price $p^j_{n}$ and his banner is displayed.
- If $b^j_t \le p^j_{n}$, then another trader wins the auction.
Like in the model with only one source, we assume that, for each $j \in \lbrace 1, \ldots, J \rbrace$, $(p^j_n)_{n \in \mathbb{N}^*}$ are *i.i.d.* random variables distributed according to an absolutely continuous distribution. We denote by $F^j$ the cumulative distribution function and by $f^j$ the probability density function associated with the source $j$. As above, we assume, for each $j \in \lbrace 1, \ldots, J \rbrace$, that:
- $\forall n\in \mathbb N^*$, $p^j_n$ is almost surely positive. In particular, $F^j(0) = 0$.
- $\forall p > 0, f^j(p) > 0$.
- $\lim_{p \to +\infty} p^3 f^j(p) = 0$.\
We also assume that the random variables $(p^j_n)_{j \in \lbrace 1, \ldots, J \rbrace, n \in \mathbb{N}^*}$ are all independent.\
*Remaining cash process:*\
As above, we denote by $(S_t)_t$ the process modeling the amount of cash to be spent. Its dynamics is: $$dS_t = - \sum_{j=1}^J p^j_{N^j_t} \mathbf{1}_{\{b^j_t> p^j_{N^j_t}\}}dN^j_t, \quad S_0 = \bar{S}.$$
*Inventory processes:*\
For each $j \in \lbrace 1, \ldots, J \rbrace$, the number of impressions associated with the auction requests coming for the source $j$ is modeled by an inventory process $(I^j_t)_t$. For each $j \in \lbrace 1, \ldots, J \rbrace$, the dynamics of $(I^j_t)_t$ is: $$dI^j_t = \mathbf{1}_{\{b^j_t> p^j_{N^j_t}\}}dN^j_t, \quad I^j_0 = 0.$$
To simplify exposition, we write $I_t = (I^1_t, \ldots, I^J_t) \in \mathbb{N}^J$, and we write the dynamics of the process $(I_t)_t$:
$$dI_t = \sum_{j=1}^J \mathbf{1}_{\{b^j_t> p^j_{N^j_t}\}}dN^j_t e^j,$$ where $(e^1, \ldots, e^J)$ is the canonical basis of $\mathbb{R}^J$.\
*Stochastic optimal control problem:*\
In this first extension of the model, the trader aims at maximizing the expected value of an indicator of the form $$\alpha^1 I^1_T + \ldots + \alpha^J I^J_T, \quad \alpha^1,\ldots, \alpha^J \ge 0.$$ As in Section 2, we consider a relaxed problem with a penalty for extra-spending: $$\inf_{(b^1_t, \ldots, b^J_t)_t \in \mathcal{A}^J}\mathbb E \left[-\sum_{j=1}^J \alpha^j I^j_T + K \min\left(S_T,0\right)^2\right].$$
### HJB equations: from dimension $J+2$ to dimension $2$
The value function associated with this problem is: $$u: (t,I,S) \in [0,T]\times \mathbb{N}^J\times (-\infty, \bar{S}] \mapsto \inf_{(b^1_s, \ldots, b^J_s)_{s\ge t} \in \mathcal{A}_{t}^J} \mathbb E\left[-\sum_{j=1}^J \alpha^j {I_T^j}^{b,t,I^j} + K \min\left(S^{b,t,S}_T,0\right)^2\right],$$ where $$dS^{b,t,S}_s = - \sum_{j=1}^J p^j_{N^j_s} \mathbf{1}_{\{b^j_s> p^j_{N^j_s}\}}dN^j_s, \quad S^{b,t,S}_t = S,$$ $$d{I^j_s}^{b,t,I^j} = \mathbf{1}_{\{b^j_s> p^j_{N^j_s}\}}dN^j_s, \quad {I^j_t}^{b,t,I^j} = I^j, \quad \forall j \in \lbrace 1, \ldots, J \rbrace.$$
The associated Hamilton-Jacobi-Bellman equation is:
$$\label{m1:HJBmulti}
-\partial_t u(t,I,S) - \sum_{j=1}^J \lambda^j \inf_{b^j \in \mathbb R_+}\int_0^{b^j} f^j(p) (u(t,I+e^j,S-p) - u(t,I,S)) dp = 0,$$
with terminal condition $u(T,I^1,\ldots,I^J,S) = - \sum_{j=1}^J \alpha^j I^j + K \min\left(S,0\right)^2$.\
Eq. (\[m1:HJBmulti\]) is a non-standard integro-differential HJB equation in dimension $J+2$ which generalizes Eq. (\[m1:HJB\]). As in the case with only one source, we can consider an ansatz of the form $$u(t,I^1,\ldots,I^J,S) = - \sum_{j=1}^J \alpha^j I^j + v(t,S).$$
With this ansatz, Eq. (\[m1:HJBmulti\]) becomes indeed: $$\label{m1:HJBmulti2}
-\partial_t v(t,S) - \sum_{j=1}^J \lambda^j \inf_{b^j \in \mathbb R_+} \int_0^{b^j} f^j(p) (v(t,S-p) - v(t,S)-\alpha^j) dp = 0,$$ with terminal condition $v(T,S) = K \min\left(S,0\right)^2$.\
Eq. (\[m1:HJBmulti2\]) is another HJB equation, but in dimension $2$, which generalize Eq. (\[m1:HJB2\]).
### Fluid limit approximation
As in the model with only one source, one could numerically approximate the solution to Eq. (\[m1:HJBmulti2\]) and subsequently find the optimal bidding strategy. However, it is interesting to consider the approximation arising from the replacement of the term $ v(t,S-p) - v(t,S)$ in Eq. (\[m1:HJBmulti2\]) by its first-order Taylor expansion $-p \partial_S v(t,S)$. We therefore replace Eq. (\[m1:HJBmulti2\]) by:
$$\label{m1:HJBmultic}-\partial_t v(t,S) - \sum_{j=1}^J \lambda^j \inf_{b^j \in \mathbb R_+} - \int_0^{b^j} f^j(p) (\alpha^j + p \partial_S v(t,S)) dp = 0,$$
with terminal condition $v(T,S) = K \min\left(S,0\right)^2$.\
Eq. (\[m1:HJBmultic\]) is related to the fluid limit of the above stochastic optimal control problem which is defined by:
$$\inf_{(\tilde{b}^1_t,\ldots, \tilde{b}^J_t)_t \in \mathcal{A}_{\text{det}}^J} -\sum_{j=1}^J\lambda^j \alpha^j \int_0^T F^j(\tilde{b}^j_t) dt + K \min\left(\tilde{S}^{\tilde{b}}_T,0\right)^2,$$
where $$d\tilde{S}^{\tilde{b}}_t = - \sum_{j=1}^J\lambda^j G^j(\tilde{b}^j_t)dt, \quad G^j: x \in \mathbb{R}_+ \cup \{+\infty\} \mapsto \int_0^{x} p f^j(p) dp.$$
The value function $\tilde{v}$ associated with this problem is:
$$\tilde{v}(t,S) = \inf_{(\tilde{b}^1_s, \ldots, \tilde{b}^J_s )_{s\ge t} \in \mathcal{A}_{\text{det}}} -\sum_{j=1}^J\lambda^j \alpha^j \int_t^T F^j(\tilde{b}^j_s) ds + K \min\left(\tilde{S}^{\tilde{b},t,S}_T,0\right)^2,$$
where $$d\tilde{S}^{\tilde{b},t,S}_s = - \sum_{j=1}^J\lambda^j G^j(\tilde{b}^j_s)ds, \quad \tilde{S}^{\tilde{b},t,S}_t = S.$$
Using the same techniques as for Theorem \[theo\], we can prove the following Theorem:
\[theomulti\] Let us define $$\begin{aligned}
\forall j \in \lbrace 1, \ldots, J \rbrace, \quad H^j(x) &=& \lambda^j \sup_{b^j \in \mathbb R_+} \int_0^{b^j} f^j(p)\left(\alpha^j+ xp\right) dp,\end{aligned}$$ and $$H = H^1 + \ldots + H^J.$$
The value function $\tilde{v}$ is given by: $$\tilde{v}(t,S) = \sup_{x \le 0} \left(Sx - (T-t) H(x) - \frac{x^2}{4K}\right).$$ It is the unique weak semi-concave solution of the Hamilton-Jacobi equation (\[m1:HJBmultic\]).\
Furthermore, the optimal control function $(t,S) \mapsto \tilde{b}^*(t,S)$ is given by the following:
- If $S \ge \sum_{j=1}^J\lambda^j (T-t) \int_0^{\infty} pf^j(p) dp$, then: $$\forall j \in \lbrace 1, \ldots, J \rbrace, \quad \tilde{b}^{j*}(t,S) = +\infty.$$
- If $S < \sum_{j=1}^J\lambda^j (T-t) \int_0^{\infty} pf^j(p) dp$, then: $$\forall j \in \lbrace 1, \ldots, J \rbrace, \quad \tilde{b}^{j*}(t,S) = -\frac{\alpha^j}{x^*},$$ where $x^*$ is characterized by $$\label{m1:characmultiy}
S = (T-t) H'(x^*) + \frac{x^*}{2K}.$$
By using the change of variables $a^j_s = \lambda^j G^j(\tilde{b}^j_s) \in \mathcal{I}^j = [0,\lambda^j \int_0^\infty p f^j(p) dp]$, we have: $$\label{lfamulti}
\tilde{v}(t,S) = \inf_{(a^1_s, \ldots, a^J_s)_{s\ge t} \in \mathcal{A}'_{\text{det},\mathcal{I}^1, \ldots, \mathcal{I}^J}} \int_t^T \sum_{j=1}^J L^j\left(a^j_s\right) ds + K \min\left(\widehat{S}^{a,t,S}_T,0\right)^2,$$ where $\mathcal{A}'_{\text{det},\mathcal{I}^1, \ldots, \mathcal{I}^J}$ is the set of $\mathcal{F}_0$-measurable processes with values in $\mathcal{I}^1 \times \ldots \times \mathcal{I}^J$, where $$d\widehat{S}^{a,t,S}_s = -\left(a^1_s + \ldots + a^J_s\right) ds, \quad \widehat{S}^{a,t,S}_t = S,$$ and where, for all $j \in \lbrace 1, \ldots, J \rbrace$, the function $L^j$ is defined by: $$L^j : a^j \in \mathcal{I}^j \mapsto -\lambda^j \alpha^j F^j\left({G^j}^{-1}\left(\frac{a^j}{\lambda^j}\right)\right).$$
An equivalent way to write Eq. (\[lfamulti\]) is: $$\label{lfamulti2}
\tilde{v}(t,S) = \inf_{(\mathfrak{a}_s)_{s\ge t} \in \mathcal{A}''_{\text{det}}} \int_t^T L(\mathfrak{a}_s) ds + K \min\left(\check{S}^{\mathfrak{a},t,S}_T,0\right)^2,$$ where $\mathcal{A}''_{\text{det}}$ is the set of $\mathcal{F}_0$-measurable processes with values in $\mathcal{I}=[0,\sum_{j=1}^J\lambda^j \int_0^\infty p f^j(p) dp]$, where $$d\check{S}^{\mathfrak{a},t,S}_s = -\mathfrak{a}_s ds, \quad \check{S}^{\mathfrak{a},t,S}_t = S,$$ and where $$L: \mathfrak{a} \in \mathcal{I} \mapsto \inf_{(a^1, \ldots, a^J) \in \mathcal{I}^1 \times \ldots \times \mathcal{I}^J, a^1+ \ldots + a^J = \mathfrak{a} } \sum_{j=1}^J L^j\left(a^j\right).$$
As in the proof of Theorem \[theo\], $L^j$ is a convex function, continuously differentiable on the interior of $\mathcal{I}^j$.\
Let us now compute the Legendre-Fenchel transform of $L^j$: $$\begin{aligned}
\nonumber {L^j}^*(x) &=& \sup_{a^j \in \mathcal{I}^j } x a^j - L^j(a^j)\\
\nonumber &=& \sup_{b^j \in \mathbb{R}_+ } \lambda^j x G^j(b^j) + \lambda^j \alpha^j F^j(b^j)\\
\label{m1:supbmulti}&=& \lambda^j \sup_{b^j \in \mathbb{R}_+ } \int_0^{b^j} (\alpha^j+ xp) f^j(p) dp\\
\nonumber &=& H^j(x).\end{aligned}$$
As in the proof of Theorem \[theo\], we can easily find the supremum in Eq. (\[m1:supbmulti\]):
- If $x < 0$, then $b^{j*} = - \frac{\alpha^j}{x}$, and $$\begin{aligned}
H^j(x) &=& \lambda^j \int_0^{- \frac{\alpha^j}{x}} (\alpha^j+ xp) f^j(p) dp\\
&=& -\lambda^j x \int_0^{-\frac{\alpha^j}{x}} F^j(p) dp. \\\end{aligned}$$
- If $x$ is nonnegative, then $b^{j*} = +\infty$, and $$H^j(x) = \lambda^j\left(\alpha^j+ x \int_0^{\infty} p f^j(p) dp\right).$$
In particular, $H^j$ is a $C^2$ function.\
By using an infimal convolution, the function $L$ is convex, and its Legendre transform is $H = H^1 + \ldots + H^J$.\
From Eq. (\[lfamulti2\]), $\tilde{v}$ is given by the Hopf-Lax formula: $$\begin{aligned}
\tilde{v}(t,S) &=& \inf_{y \in [S - \sum_{j=1}^J \lambda^j \int_0^\infty pf^j(p) dp (T-t),S]} \left((T-t) L\left(\frac{S-y}{T-t}\right) + K\min(y,0)^2\right).\end{aligned}$$
Equivalently, by infimal convolution, we have: $$\begin{aligned}
\nonumber \tilde{v}(t,S) &=& \sup_{x} \left(Sx - (T-t) H(x) - \sup_y \left(yx - K\min(y,0)^2\right)\right)\\
\label{m1:vcffmulti} &=& \sup_{x \le 0} \left(Sx - (T-t) H(x) - \frac{x^2}{4K}\right).\end{aligned}$$
This is the first result of the Theorem.\
Furthermore, we know that $\tilde{v}$ is the unique weak semi-concave solution of the following Hamilton-Jacobi equation: $$-\partial_t \tilde{v}(t,S) + H(\partial_S \tilde{v}(t,S)) = 0,$$ with terminal condition $\tilde{v}(T,S) = K\min(S,0)^2$.\
This equation is exactly the same as Eq. (\[m1:HJBmultic\]), hence the second assertion of the Theorem.\
Moreover the optimal control in the modified problem (\[lfamulti2\]) is given by $\mathfrak{a}^*(t,S) = H'(\partial_S \tilde{v}(t,S))$. Therefore, by classical results on infimal convolutions, the optimal controls in the modified problem (\[lfamulti\]) are given by: $$\forall j \in \lbrace 1, \ldots, J \rbrace,\quad a^{j*}(t,S) = {H^j}'(\partial_S \tilde{v}(t,S)).$$
We conclude that the optimal control function in the initial problem is: $$\tilde{b}^{j*}(t,S) = \left\{
\begin{array}{ll}
-\frac{\alpha^j}{\partial_S \tilde{v}(t,S)}, & \partial_S \tilde{v}(t,S) <0 \\
+\infty, & \partial_S \tilde{v}(t,S) \ge 0.
\end{array}
\right.$$
If $S \ge (T-t) H'(0) = \sum_{j=1}^J\lambda^j (T-t) \int_0^{\infty} pf^j(p) dp$, then the supremum in Eq. (\[m1:vcffmulti\]) is reached at $x^*=0$. Therefore, $\partial_S \tilde{v}(t,S) = 0$ and $\forall j \in \lbrace 1, \ldots, J \rbrace, \tilde{b}^{j*}(t,S) = +\infty$.\
Otherwise, the supremum in Eq. (\[m1:vcffmulti\]) is reached at $x^*\le 0$ uniquely characterized by: $$S = (T-t) H'(x^*) + \frac{x^*}{2K}.$$
In particular, $\partial_S \tilde{v}(t,S) = x^*$ and $\forall j \in \lbrace 1, \ldots, J \rbrace, \tilde{b}^{j*}(t,S) = -\frac{\alpha^j}{x^*}$ where $x^*$ is characterized by Eq. (\[m1:characmultiy\]).\
\
Following the same reasoning as in Section 2, we approximate the optimal bids $b^{1*}_t, \ldots, b^{J*}_t$ at time $t$ by using the optimal bids in the fluid-limit approximation and in the limit case $K \to +\infty$. In other words, we approximate the optimal bids $b^{1*}_t, \ldots, b^{J*}_t$ at time $t$ by $\tilde{b}^{1*}_\infty(t,S_{t-}), \ldots, \tilde{b}^{J*}_\infty(t,S_{t-})$, where the functions $(\tilde{b}^{j*}_\infty)_j$ are defined by:\
- If $S \ge \sum_{j=1}^J\lambda^j (T-t) \int_0^{\infty} pf^j(p) dp$, then: $$\forall j \in \lbrace 1, \ldots, J \rbrace, \quad \tilde{b}^{j*}_\infty(t,S) = +\infty.$$
- If $S < \sum_{j=1}^J\lambda^j (T-t) \int_0^{\infty} pf^j(p) dp$, then: $$\label{multichar}\forall j \in \lbrace 1, \ldots, J \rbrace, \quad \tilde{b}^{j*}_\infty(t,S) = - \frac{\alpha^j}{{H'}^{-1}\left(\frac{S}{T-t}\right)}.$$
This approximation of the optimal bidding strategy deserves several remarks.\
First, the analysis for the case $S_{t-} \ge \sum_{j=1}^J\lambda^j (T-t) \int_0^{\infty} pf^j(p) dp$ is the same as in the one-source case. The value $\sum_{j=1}^J\lambda^j (T-t) \int_0^{\infty} pf^j(p) dp$ represents the expected amount that will be spent between time $t$ and time $T$ if the ad trader wins all the auctions he participates in. Therefore, it is natural to send a very high bid to be sure to win all the auctions if the amount to spend $S_{t-}$ is greater than that value.\
Second, if $S_{t-} < \sum_{j=1}^J\lambda^j (T-t) \int_0^{\infty} pf^j(p) dp$, then we see from Eq. (\[multichar\]) that the bids differ from one source to the other according to the weights $(\alpha^j)_j$: $$\forall j,j' \in \lbrace 1, \ldots, J \rbrace,\quad \frac{\tilde{b}^{j*}_\infty(t,S)}{\alpha^j} = \frac{\tilde{b}^{j'*}_\infty(t,S)}{\alpha^{j'}}.$$ In particular, the bids should be the same across sources at a given point in time if one is only interested in the total number of impressions, *i.e.* if $\alpha^1 = \ldots = \alpha^J$.\
In the case $S_{t-} < \sum_{j=1}^J\lambda^j (T-t) \int_0^{\infty} pf^j(p) dp$, it is also interesting to notice that Eq. (\[multichar\]) implies that
$$\forall t \in [0,T], \quad \sum_{j=1}^J\lambda^j (T-t) \int_0^{\tilde{b}^{j*}_\infty(t,S_{t-})} pf^j(p) dp = S_{t-}.$$
If we use the above approximation of the optimal bidding strategy, the dynamics of the remaining cash is (for $0 \le t \le s \le T$):
$$dS_s = - \sum_{j=1}^J p^j_{N^j_s} \mathbf{1}_{\{b^{j*}_\infty(s,S_{s-})> p^j_{N^j_s}\}}dN^j_s, \qquad S_t = S.$$
Therefore:
$$d\mathbb{E}[S_s] = - \mathbb{E}\left[\sum_{j=1}^J \lambda^j \int_0^{\tilde{b}^{j*}_\infty(s,S_{s-})} p f^j(p) dp ds \right] = - \frac{\mathbb{E}[S_{s}]}{T-s} ds.$$
This gives $\mathbb{E}[S_s] = S \frac{T-s}{T-t}$. As in the one-source case, the approximation of the optimal bidding strategy is such that, on average, the remaining cash is spent evenly across $[t,T]$.
What about conversions?
-----------------------
So far, we have only considered problems where the KPI maximized by the ad trader was linked to the number of impressions he purchases. Practitioners are interested in other KPIs than the number of impressions (related to the CPM). In particular, the number of clicks, or the number of acquisitions of a product (following a click on a banner) are very important indicators of the success of a campaign – the KPIs used in practice are the CPC and the CPA.\
In what follows, we consider a model where two variables are optimized upon: the number of impressions, and the number of conversions (which can be regarded as clicks or acquisitions, depending on the considered applications). In fact, we generalize the previous model by considering more general marked Poisson processes, where the marks are not limited to the prices to beat, but also model the occurrence of a conversion with a random variable following a Bernoulli distribution (the parameter of this Bernoulli distribution is known as the *conversion rate*, *i.e.* the probability to turn an impression into a conversion).\
### Setup of the model
As above, we still consider an ad trader who wishes to spend a given amount of money $\bar{S}$ over a time window $[0,T]$.\
*Auctions:*\
This ad trader is connected to $J>1$ sources from which he receives requests to participate in auctions in order to purchase inventory – we assume that the trader knows from which source each auction request arises. Requests are modeled with $J$ marked Poisson processes: the arrival of requests from the source $j \in \lbrace 1, \ldots, J \rbrace$ is triggered by the jumps of the Poisson process $(N^j_t)_t$ with intensity $\lambda^j > 0$, and the marks $(p^j_n)_{n \in \mathbb{N}^*}$ and $(\xi^j_n)_{n \in \mathbb{N}^*}$ correspond, for each auction request sent by the source $j$, respectively to the highest bid sent by the other participants, and to the occurrence of a conversion – $\xi^j_n \in \{0,1\}$ only makes sense if the auction is won by the ad trader.\
Every time he receives from the source $j$ a request to participate in an auction, the ad trader can bid a price: at time $t$, if he receives a request from the source $j$, then we denote his bid by $b^j_t$. As in the above model, we assume that for each $j \in \lbrace 1, \ldots, J \rbrace$, the process $(b^j_t)_t$ is a predictable process with values in $\mathbb{R}_+ \cup \{+\infty\}$.\
If at time $t$ the $n^{\text{th}}$ auction associated with the source $j$ occurs, the outcome of this auction is the following:
- If $b^j_t > p^j_{n}$, then the ad trader wins the auction: he pays the price $p^j_{n}$ and his banner is displayed. Moreover, a conversion occurs if and only if $\xi^j_n = 1$.
- If $b^j_t \le p^j_{n}$, then another trader wins the auction.
As above, we assume that for each $j \in \lbrace 1, \ldots, J \rbrace$, $(p^j_n)_{n \in \mathbb{N}^*}$ are *i.i.d.* random variables distributed according to an absolutely continuous distribution. We denote by $F^j$ the cumulative distribution function and by $f^j$ the probability density function associated with the source $j$. As above, we assume, for each $j \in \lbrace 1, \ldots, J \rbrace$, that:
- $\forall n\in \mathbb N^*$, $p^j_n$ is almost surely positive. In particular, $F^j(0) = 0$.
- $\forall p > 0, f^j(p) > 0$.
- $\lim_{p \to +\infty} p^3 f^j(p) = 0$.\
We also assume that the random variables $(p^j_n)_{j \in \lbrace 1, \ldots, J \rbrace, n \in \mathbb{N}^*}$ are all independent.\
As far as the variables $(\xi^j_n)_{j \in \lbrace 1, \ldots, J \rbrace, n \in \mathbb{N}^*}$ are concerned, we assume that they are all independent and independent from the variables $(p^j_n)_{j \in \lbrace 1, \ldots, J \rbrace, n \in \mathbb{N}^*}$. Moreover, we assume that for each $j \in \lbrace 1, \ldots, J \rbrace$, $(\xi^j_n)_{n \in \mathbb{N}^*}$ are *i.i.d.* random variables distributed according to a Bernoulli distribution with parameter $\nu^j \in [0,1]$.\
*Remaining cash process:*\
As above, we denote by $(S_t)_t$ the process modeling the amount of cash to be spent. Its dynamics is:
$$dS_t = - \sum_{j=1}^J p^j_{N^j_t} \mathbf{1}_{\{b^j_t> p^j_{N^j_t}\}}dN^j_t, \quad S_0 = \bar{S}.$$
*Inventory processes:*\
For each $j \in \lbrace 1, \ldots, J \rbrace$, the number of impressions associated with the auction requests coming for the source $j$ is modeled by an inventory process $(I^j_t)_t$. For each $j \in \lbrace 1, \ldots, J \rbrace$, the dynamics of $(I^j_t)_t$ is: $$dI^j_t =\mathbf{1}_{\{b^j_t> p^j_{N^j_t}\}}dN^j_t, \quad I^j_0 = 0.$$
*Processes for the number of conversions:*\
For each $j \in \lbrace 1, \ldots, J \rbrace$, the number of conversions associated with the auction requests coming for the source $j$ is modeled by a new process $(C^j_t)_t$. For each $j \in \lbrace 1, \ldots, J \rbrace$, the dynamics of $(C^j_t)_t$ is: $$dC^j_t = \xi^j_{N^j_t} \mathbf{1}_{\{b^j_t> p^j_{N^j_t}\}}dN^j_t, \quad C^j_0 = 0.$$
*Stochastic optimal control problem:*\
In this second extension of the model, the trader aims at maximizing the expected value of an indicator of the form $$\left(\alpha^1 I^1_T + \ldots + \alpha^J I^J_T\right) + \left(\delta^1 C^1_T + \ldots + \delta^J C^J_T\right) , \quad \alpha^1,\ldots, \alpha^J \ge 0,\quad \delta^1,\ldots, \delta^J \ge 0.$$
The relaxed problem we consider is: $$\inf_{(b^1_t, \ldots, b^J_t)_t \in \mathcal{A}^J}\mathbb E \left[-\sum_{j=1}^J \alpha^j I^j_T - \sum_{j=1}^J \delta^j C^j_T + K \min\left(S_T,0\right)^2\right].$$
### HJB equations: from dimension $2J+2$ to dimension $2$
The value function associated with this problem is: $$u: (t,I,C,S) \in [0,T]\times \mathbb{N}^J\times \mathbb{N}^J\times (-\infty, \bar{S}]$$$$\mapsto \inf_{(b^1_s, \ldots, b^J_s)_{s\ge t} \in \mathcal{A}_{t}^J} \mathbb E\left[-\sum_{j=1}^J \alpha^j {I_T^j}^{b,t,I^j} - \sum_{j=1}^J \delta^j {C^j_T}^{b,t,C^j} + K \min\left(S^{b,t,S}_T,0\right)^2\right],$$ where $$dS^{b,t,S}_s = - \sum_{j=1}^J p^j_{N^j_s} \mathbf{1}_{\{b^j_s> p^j_{N^j_s}\}}dN^j_s, \quad S^{b,t,S}_t = S,$$ $$d{I^j_s}^{b,t,I^j} = \mathbf{1}_{\{b^j_s> p^j_{N^j_s}\}}dN^j_s, \quad {I^j_t}^{b,t,I^j} = I^j, \quad \forall j \in \lbrace 1, \ldots, J \rbrace,$$ and $$d{C^j_s}^{b,t,C^j} = \xi^j_{N^j_s} \mathbf{1}_{\{b^j_s> p^j_{N^j_s}\}}dN^j_s, \quad {C^j_t}^{b,t,C^j} = C^j, \quad \forall j \in \lbrace 1, \ldots, J \rbrace.$$
In particular, there are $J$ new state variables corresponding to the number of conversions associated with each of the $J$ sources.\
The associated Hamilton-Jacobi-Bellman equation is:
$$-\partial_t u(t,I,C,S) - \sum_{j=1}^J \lambda^j \inf_{b^j \in \mathbb R_+}\int_0^{b^j} f^j(p) \left[(1-\nu^j)(u(t,I+e^j,C,S-p) - u(t,I,C,S))\right.$$
$$\label{m1:HJBconv}
\left.+ \nu^j (u(t,I+e^j,C+e^j,S-p) - u(t,I,C,S)) \right] dp = 0,$$
with terminal condition $$u(T,I^1,\ldots,I^J, C^1,\ldots,C^J,S) = - \sum_{j=1}^J \alpha^j I^j - \sum_{j=1}^J \delta^j C^j + K \min\left(S,0\right)^2.$$
Eq. (\[m1:HJBconv\]) is a non-standard integro-differential HJB equation in dimension $2J+2$ which generalizes Eq. (\[m1:HJBmulti\]). In this extension, we consider an ansatz of the form $$u(t,I^1,\ldots,I^J, C^1,\ldots,C^J,S) = - \sum_{j=1}^J \alpha^j I^j - \sum_{j=1}^J \delta^j C^j + v(t,S).$$
With this ansatz, Eq. (\[m1:HJBconv\]) becomes another HJB equation, but in dimension $2$, which generalizes Eq. (\[m1:HJBmulti2\]) to the case where conversions are taken into account: $$\label{m1:HJBconv2}
-\partial_t v(t,S) - \sum_{j=1}^J \lambda^j \inf_{b^j \in \mathbb R_+} \int_0^{b^j} f^j(p) (v(t,S-p) - v(t,S)-\alpha^j - \nu^j \delta^j) dp = 0,$$ with terminal condition $v(T,S) = K \min\left(S,0\right)^2$.
### Fluid limit approximation
Eq. (\[m1:HJBconv2\]) is the same as Eq. (\[m1:HJBmulti2\]), except that $\alpha^j$ is replaced by $\alpha^j + \nu^j \delta^j$. In particular we can use similar rules as in the previous extension to approximate the optimal bidding strategy.\
We introduce the function $H$ defined by:
$$\begin{aligned}
H: x \in \mathbb{R} \mapsto H(x) &=& \sum_{j=1}^J \lambda^j \sup_{b^j \in \mathbb R_+} \int_0^{b^j} f^j(p)\left(\alpha^j + \nu^j \delta^j + xp\right) dp\\
&=& \left\{
\begin{array}{ll}
\sum_{j=1}^J -\lambda^j x \int_0^{-\frac{\alpha^j + \nu^j \delta^j}{x}} F^j(p) dp , & x <0 \\
\sum_{j=1}^J \lambda^j \left(\alpha^j + \nu^j \delta^j+x\int_0^{\infty} pf(p) dp\right), & x \ge 0.
\end{array}
\right.\end{aligned}$$
In the case of conversions, we approximate the optimal bids $b^{1*}_t, \ldots, b^{J*}_t$ at time $t$ by $\tilde{b}^{1*}_\infty(t,S_{t-}), \ldots, \tilde{b}^{J*}_\infty(t,S_{t-})$, where the functions $(\tilde{b}^{j*}_\infty)_j$ are defined by:\
- If $S \ge \sum_{j=1}^J\lambda^j (T-t) \int_0^{\infty} pf^j(p) dp$, then: $$\forall j \in \lbrace 1, \ldots, J \rbrace, \quad \tilde{b}^{j*}_\infty(t,S) = +\infty.$$
- If $S < \sum_{j=1}^J\lambda^j (T-t) \int_0^{\infty} pf^j(p) dp$, then: $$\forall j \in \lbrace 1, \ldots, J \rbrace, \quad \tilde{b}^{j*}_\infty(t,S) = - \frac{\alpha^j + \nu^j \delta^j}{{H'}^{-1}\left(\frac{S}{T-t}\right)}.$$
As in the previous extension of the model, if $S_{t-} \ge \sum_{j=1}^J\lambda^j (T-t) \int_0^{\infty} pf^j(p) dp$, then it is natural to send a very high bid to be sure to win all the auctions.\
If $S_{t-} < \sum_{j=1}^J\lambda^j (T-t) \int_0^{\infty} pf^j(p) dp$, then the bids differ from one source to the other according to the following rule: $$\forall j,j' \in \lbrace 1, \ldots, J \rbrace,\quad \frac{\tilde{b}^{j*}_\infty(t,S)}{\alpha^j+\nu^j \delta^j} = \frac{\tilde{b}^{j'*}_\infty(t,S)}{\alpha^{j'}+\nu^{j'} \delta^{j'}}.$$ In particular, if the ad trader only cares about the total number of conversions, *i.e.* if $\alpha^1 = \ldots = \alpha^J = 0$ and $\delta^1 = \ldots = \delta^J = 1$, then the bids are the same across sources, up to a multiplicative factor which corresponds to the probability of conversion associated with each source.\
Finally, in this extension as in the previous one, if one uses the fluid-limit approximation, then, the budget is expected to be spent evenly.
Discussion about the models
---------------------------
We have presented a first model in Section 2 and two extensions of that first model earlier in this section. In the following paragraphs, we aim at challenging the assumptions underlying these models. In particular, we discuss the specificities associated with second-price auctions, and the existence of floor prices, which goes against the assumptions on the distribution of the price to beat in the models presented above. We also discuss nonlinear KPIs for which there is no dimensionality reduction.
### First-price auctions vs. second-price auctions
In both the initial model of Section 2 and the two extensions presented in this section, we considered Vickrey auctions. In other words, the price paid by the ad trader when he wins the auction is not the price he has bid (the first price), but a lower price corresponding to the highest bid that has been proposed by other participants (the second price). In the case of first-price auctions, the dynamics of the remaining budget (in the single-source model of Section 2) is not anymore given by Eq. (\[dynS\]), but instead by: $$dS_t = - b_t \mathbf{1}_{\{b_t> p_{N_t}\}}dN_t, \quad S_0 = \bar{S}.$$
If we consider the same objective function as in Section 2, then the new value function $$u: (t,I,S) \in [0,T]\times \mathbb{N}\times (-\infty, \bar{S}] \mapsto \inf_{(b_s)_{s\ge t} \in \mathcal{A}_{t}} \mathbb E\left[-I^{b,t,I}_T + K \min\left(S^{b,t,S}_T,0\right)^2\right],$$ where $$dS^{b,t,S}_s = - b_s \mathbf{1}_{\{b_s> p_{N_s}\}}dN_s, \quad S^{b,t,S}_t = S,$$ and $$dI^{b,t,I}_s = \mathbf{1}_{\{b_s> p_{N_s}\}}dN_s, \quad I^{b,t,I}_t = I,$$ is associated with the following HJB equation: $$-\partial_t u(t,I,S) - \lambda \inf_{b \in \mathbb R_+}F(b) (u(t,I+1,S-b) - u(t,I,S)) = 0,$$ with terminal condition $u(T,I,S) = - I + K \min\left(S,0\right)^2$.\
With the ansatz $u(t,I,S) = - I + v(t,S)$, this equation becomes: $$\label{m1:HJB2first}
-\partial_t v(t,S) - \lambda \inf_{b \in \mathbb R_+} F(b) (v(t,S-b) - v(t,S)-1) = 0,$$ with terminal condition $v(T,S) = K \min\left(S,0\right)^2$.\
Eq. (\[m1:HJB2first\]) replaces Eq. (\[m1:HJB2\]). In particular, the first order condition is not anymore $$v(t,S-b) - v(t,S)-1 = 0,$$ but instead $$v(t,S-b) - v(t,S)-1 = \frac{F(b)}{f(b)} \partial_S v(t,S-b).$$
In particular, because $v$ is nonincreasing with respect to $S$, the optimal bid should always be lower in the case of first-price auctions than in the case of second-price auctions.\
It is also noteworthy that a fluid-limit approximation can be considered in the case of Eq. (\[m1:HJB2first\]) by solving
$$\label{m1:HJB2firsttilde}
-\partial_t \tilde{v}(t,S) + \lambda \sup_{b \in \mathbb R_+} (F(b) b \partial_S \tilde{v}(t,S) + F(b)) = 0,$$
with terminal condition $\tilde{v}(T,S) = K \min\left(S,0\right)^2$.\
In particular, Eq. (\[m1:HJB2firsttilde\]) is a first-order Hamilton-Jacobi equation, and by using the same techniques as in the case of second-price auctions, one also finds that the optimal strategy consists of spending the remaining budget evenly.\
### Floor prices
In order to improve performances, publishers can set floor prices which sometimes modify the underlying nature of the auction. Floor prices come indeed in two flavors:
- Hard floors (or reserve prices): the bid of the ad trader is only taken into account if it is higher than the floor price level $\phi$ set by the publisher (the supply side). When a hard floor level $\phi$ is set, everything works as if a “ghost player” was always bidding $\phi$. In particular, the cumulative distribution function $F$ may be discontinuous, with a jump at price $\phi$.[^29]
- Soft floors: if the best bid is below the soft floor level $\phi$, then the winner pays its own bid, and not the second price; otherwise, the price paid is the maximum between the second price and $\phi$. In other words, the auction is no more of the Vickrey type: it is a second-price auction with a hard floor for large bids and it becomes a first-price auction for small bids.\
Our model can be generalized to tackle the case of hard floors by generalizing our approach to the case of a discontinuous cumulative distribution function $F$ – in that case $f$ can be regarded as a distribution. The result is still that the budget should be spent evenly, but the optimal bidding strategy is more cumbersome to write – and the exact expression has no theoretical interest. The case of soft floors can also be addressed, but it is really cumbersome. In particular, in the case of a uniform soft floor $\phi$ for all the auctions, Eq. (\[m1:HJB2\]) is replaced by an equation of the form
$$-\partial_t v(t,S) - \lambda \inf\left\{\inf_{b \in [0,\phi]} F(b) (v(t,S-b) - v(t,S)-1),\right.$$ $$\left. \inf_{b>\phi} \int_0^b f(p) (v(t,S-\max(p,\phi)) - v(t,S)-1) dp \right\} = 0,$$ with terminal condition $v(T,S) = K \min\left(S,0\right)^2$.\
In practice, it is also noteworthy that our bid may impact the strategic behavior of other users and in particular the behavior of the publishers who may set *dynamic floor prices*. From a modeling point of view, it means that the function $f$ is impacted by the strategy itself, and it opens a vast field where (mean-field) game theory could be very useful.
### Nonlinear KPIs
It is customary in the advertising industry to work with different metrics or KPIs. Practitioners usually aim indeed at minimizing the CPM, the CPC (cost per click) or the CPA (cost per acquisition).\
With the notations of our model, the CPM at time $T$ is naturally defined by: $$\text{CPM}_T = \frac{\bar{S} - S_T}{I_T}.$$ If the total spending is imposed to be $\bar{S}$, minimizing the expected value of $\text{CPM}_T$ boils down to minimizing $\mathbb{E}\left[\frac 1{I_T}\right]$, and not maximizing $\mathbb{E}[I_T]$ – or, equivalently, minimizing $\mathbb{E}[-I_T]$, as in the model of Section 2.\
It is noteworthy that our model somehow ignores the risk-aversion effect induced by the nonlinearity of traditional KPIs, but this is only a side-effect of convexity, as nonlinear KPIs have not been built to capture any form of risk aversion. In fact, the real issue is that the change of variables (from $u$ to $v$) at the heart of the dimensionality reduction used in our model does not extend to nonlinear KPIs. Nevertheless, from a mathematical perspective, our approach based on the dynamic programming principle (and HJB equations ) remains valid for any KPI.[^30]\
The above analysis for the CPM is also valid in the case of the CPC and the CPA. In the extension of the model involving conversions – which may be regarded as clicks or acquisitions – we considered a linear function of the number of conversions $\sum_{j=1}^J \delta^j C^j_T$, whereas practitioners would rather consider nonlinear KPIs related to the average amount paid to obtain the different types of conversion.\
Overall, we think that linear KPIs are as relevant as the traditional nonlinear ones currently used by marketers (when the total budget is fixed), and that linear KPIs should be preferred, from a mathematical perspective, for solving the problems faced by ad trading desks.
Conclusion {#conclusion .unnumbered}
==========
In this research paper, we have addressed several problems faced by media trading desks buying ad inventory through real-time auctions. These problems are new for the community of applied mathematicians, as the industry of programmatic advertising is itself quite recent. However, many ideas coming from the works of applied mathematicians on algorithmic trading in quantitative finance are inspiring for developing new models and subsequently lighting the way to innovation in a new and fast-growing field which is in demand for mathematical methods (as Finance was 25 years ago).\
Our contribution is multifold. First of all, from a modeling perspective, we model auctions arriving at random times by marked Poisson processes, where the jumps of the Poisson processes stand for the occurrence of auction requests and the marks for several variables such as the best bid proposed by other participants, and the occurrence of a conversion. This approach makes it possible to use the dynamic programming principle and derive a simple characterization of the optimal bidding strategy – through a Hamilton-Jacobi-Bellman equation. Furthermore, it allows to consider multiple sources and types of inventory in a very simple way.\
Secondly, by considering linear objective criterions (or KPIs), we manage to reduce the dimensionality of the problem. In particular, whatever the initial number of inventory sources and types, and whether or not we consider conversions, we show that the problem always boils down to solving a Hamilton-Jacobi-Bellman PDE in dimension 2 (one time dimension and one spatial dimension).\
Thirdly, by considering a fluid-limit approximation of the problem, we obtain almost-closed-form solutions for the optimal bidding strategy. Moreover, the results in the fluid-limit approximation allow to characterize the optimal strategy not only in terms of the optimal bids, but in terms of an optimal scheduling for the budget that remains to be spent. The latter simplifies the implementation, as the optimal bidding strategy can be dynamically approximated by a feedback-control tracking mechanism.\
Eventually, the modeling approach we have proposed in this article opens the door to future research where the optimal control of the bidding strategy is coupled with on-line learning (treated in a companion article – see [@fgllearning]), where the different participants in the auctions – including the publisher – could adopt strategic behaviors, where the information about the outcomes of auctions is subject to an important latency, etc.\
Appendix: A simple model in discrete time {#appendix-a-simple-model-in-discrete-time .unnumbered}
=========================================
In this appendix, we present a simplified and discrete-time version of the model introduced in Section 2 for providing the readers – especially those who are not used with continuous-time models – with intuition about the optimization approach used throughout the paper.\
Let us introduce a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ equipped with a discrete filtration $(\mathcal{F}_n)_{n \in \mathbb{N}}$ satisfying the usual conditions. The variables we define in this subsection are considered over this filtered probability space.\
Let us consider a bidding system participating in a sequence of $N$ Vickrey auctions. We assume that the state of the system is described by three variables: (i) the number $n$ of auction requests received by the ad trading desk, (ii) the number $I_n$ of impressions purchased after the $n^{\text{th}}$ auction (we assume that $I_0 = 0$), and (iii) the remaining budget $S_n$ after the $n^{\text{th}}$ auction – we assume that $S_0 = \bar{S}$ is the maximal total amount to be spent. The goal of the algorithm is to maximize an objective function of the form $\mathbb{E}\left[g(N,I_N,S_N)\right]$, for some function $g$. In particular, $g(N,I_N,S_N)$ can be equal to $I_N$ if we want to maximize the number of impressions.\
For each auction request, the algorithm chooses a bid level. For the $n^{\text{th}}$ auction, we denote here by $b_n$ the bid sent to the auction server ($b_n$ has to be $\mathcal{F}_{n-1}$-measurable). When the $n^{\text{th}}$ auction occurs, a random variable $p_n$ is drawn with the probability density function $f$. This random variable represents the price to beat.\
The resulting dynamics of the system is the following:
- If $b_n > p_n$, then the algorithm wins the auction, the price paid is $p_n$, and the system evolves from the state $(n-1,I,S)$ to the state $(n,I+1,S-p_n)$.
- If $b_n \leq p_n$, then the algorithm does not win the auction and the system evolves from the state $(n-1,I,S)$ to the state $(n,I,S)$.
Let us define the *value function*: $$u(n,I,S) = \max_{(b_k)_{k>n} \in \mathcal{A}_n} \mathbb{E}\left[g(N,I_N,S_N)|\mathcal{F}_n\right],$$ where $(b_k)_{k>n} \in \mathcal{A}_n$ if and only if $(b_k)_{k>n}$ is a predictable process such that $S_N \ge 0$, almost surely.\
In order to solve this optimization problem, we simply use the dynamic programming principle which yields the following Bellman equation: $$u(n-1,I,S) = \max_{b_{n} \in [0,S]}\mathbb{E}\left[u(n,I+1,S-p_n)\mathbf{1}_{b_n > p_n} + u(n,I,S)\mathbf{1}_{b_n\leq p_n}\right]. \qquad \text{(A.1)}$$
Eq. (A.1) can also be written as the following difference equation $$u(n,I,S) - u(n-1,I,S) + \max_{b_n \in [0,S]}\mathbb{E}\left[(u(n,I+1,S-p_n) - u(n,I,S))\mathbf{1}_{b_n > p_n}\right] = 0,$$ *i.e.*: $$u(n,I,S) - u(n-1,I,S) + \max_{b_n \in [0,S]}\int_0^{b_n}(u(n,I+1,S-p) - u(n,I,S)) f(p) dp = 0. \qquad \text{(A.2)}$$
From the terminal condition $u(N,I,S) = g(N,I,S)$, the value function $u$ can easily be approximated numerically by backward induction on a grid.\
The optimal bid for the $n^{\text{th}}$ auction is given by the optimality condition $$u(n,I_{n-1}+1,S_{n-1}-b^*_n) = u(n,I_{n-1},S_{n-1})$$ or by $b^*_n = S_{n-1}$ if $u(n,I_{n-1}+1,0) > u(n,I_{n-1},S_{n-1})$. In particular, when the constraint is not binding, the optimal bid is consistent with the fact that, in the case of Vickrey auctions, participants have the incentive to bid their true valuation for the item.\
It is noteworthy that a similar change of variables as in the continuous-time model of Section 2 can be used in the special case where $g(N,I,S) = I$. In that case, if we write $u(n,I,S) = I + v(n,S)$, then the Bellman equation (A.2) can be simplified into: $$v(n,S) - v(n-1,S) + \max_{b_n \in [0,S]}\int_0^{b_n}(v(n,S-p) - v(n,S) + 1) f(p) dp = 0,$$ and the terminal condition is $v(N,S) = 0$.\
This model in discrete-time is useful to understand the general modeling framework we use, but it is limited. First, in practice, auctions arrive at random times and we do not know in advance how many auction requests the algorithm will receive. Moreover, for problems where different sources of auction requests have to be treated in parallel, the above discrete modeling approach is not convenient. Continuous-time models, where the (random) occurrence of auction requests are modeled by the jumps of a Poisson process, are definitely more realistic and flexible.\
[10]{} Amin, K., Kearns, M., Key, P., & Schwaighofer, A. (2012). Budget optimization for sponsored search: Censored learning in MDPS. arXiv preprint arXiv:1210.4847. Avellaneda, M., & Stoikov, S. (2008). High-frequency trading in a limit order book. Quantitative Finance, 8(3), 217-224. Balseiro, S., & Candogan, O. (2015). Optimal Contracts for Intermediaries in Online Advertising. Available at SSRN 2546609. Balseiro, S. R., Feldman, J., Mirrokni, V., & Muthukrishnan, S. (2014). Yield optimization of display advertising with ad exchange. Management Science, 60(12), 2886-2907. Barles, G., & Imbert, C. (2008). Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited. In Annales de l’IHP Analyse non linéaire, 25(3), 567-585. Cannarsa, P., & Sinestrari, C. (2004). Semiconcave functions, Hamilton-Jacobi equations, and optimal control (Vol. 58). Springer Science & Business Media. Engelbrecht-Wiggans, R. (1993). Optimal auctions revisited. Games and Economic Behavior, 5(2), 227-239. Evans, L. C. (1999). Partial differential equations. American Mathematical Society Fernandez-Tapia, J. (2016). Statistical modeling of Vickrey auctions and applications to automated bidding strategies. Optimization Letters. Fernandez-Tapia, J. (2015). An analytical solution to the budget-pacing problem in programmatic advertising. Technical report. Fernandez-Tapia, J., Guéant, O., & Lasry, J. M. (2016). On the pricing of performance-based programmatic ad buying contracts. Working paper. Fernandez-Tapia, J., Guéant, O., & Lasry, J. M. (2016). Real-time bidding strategies with on-line learning. Working paper. Guéant, O., Lehalle, C. A., & Fernandez-Tapia, J. (2013). Dealing with the inventory risk: a solution to the market making problem. Mathematics and financial economics, 7(4), 477-507. Guéant, O., Lehalle, C. A., & Fernandez-Tapia, J. (2012). Optimal portfolio liquidation with limit orders. SIAM Journal on Financial Mathematics, 3(1), 740-764. IAB Europe (2015), European Programmatic Market Sizing 2014 Karlsson, N., & Zhang, J. (2013, June). Applications of feedback control in online advertising. In American Control Conference (ACC), 2013 (pp. 6008-6013). IEEE. Laffont, J. J., & Maskin, E. (1980). Optimal reservation price in the Vickrey auction. Economics Letters, 6(4), 309-313. Lamberton, D., Pagès, G., & Tarrès, P. (2004). When can the two-armed bandit algorithm be trusted?. Annals of Applied Probability, 1424-1454. Lamberton, D., & Pagès, G. (2008). How fast is the bandit?. Stochastic Analysis and Applications, 26(3), 603-623. Lamberton, D., & Pagès, G. (2008). A penalized bandit algorithm. Electron. J. Probab, 13, 341-373. Lee, K. C., Jalali, A., & Dasdan, A. (2013, August). Real time bid optimization with smooth budget delivery in online advertising. In Proceedings of the Seventh International Workshop on Data Mining for Online Advertising (p. 1). ACM. Stavrogiannis, L. C. (2014). Competition between demand-side intermediaries in ad exchanges (Doctoral dissertation, University of Southampton). Vickrey, W. (1961). Counterspeculation, auctions, and competitive sealed tenders. The Journal of finance, 16(1), 8-37. Yuan, S. (2015). Supply Side Optimisation in Online Display Advertising (Doctoral dissertation, UCL (University College London)). Yuan, S., & Wang, J. (2012, October). Sequential selection of correlated ads by POMDPs. In Proceedings of the 21st ACM international conference on Information and knowledge management (pp. 515-524). ACM. Zhang, W., Yuan, S., & Wang, J. (2014, August). Optimal real-time bidding for display advertising. In Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining (pp. 1077-1086). ACM. Zhang, W., Rong, Y., Wang, J., Zhu, T. & Wang, X. (2016). Feedback control of real-time display advertising. Working paper.
[^1]: Université Pierre et Marie Curie, Laboratoire de Probabilités et Modèles Aléatoires (LPMA). 4 Place Jussieu, 75005 Paris. Joaquin acknowledges support in 2015 from the Chair “Économie et gestion des nouvelles données”.
[^2]: Ensae-Crest, 3 avenue Pierre Larousse, 92245 Malakoff Cedex, France. Member of the Scientific Advisory Board of Havas Media. The content of this article does not reflect the official opinion or the practices of Havas Media. Corresponding author: `[email protected]`.
[^3]: Institut Louis Bachelier. Member of the Scientific Advisory Board of Havas Media. The content of this article does not reflect the official opinion or the practices of Havas Media.
[^4]: The authors would like to thank Dominique Delport (Havas Media), Julien Laugel (MFG Labs), Pierre-Louis Lions (Collège de France), and Arnaud Parent (Havas Media) for the discussions they had on the topic. Two anonymous referees also deserve to be thanked.
[^5]: Mostly through intermediaries such as media agencies, but sometimes also by themselves when they set up trading desks.
[^6]: Even though ours is in continuous time.
[^7]: In particular, by assuming that the best price of other participants is an exogenous variable, we ignore all game-theoretical effects, or, equivalently, all strategic interactions between market participants. Although we do not consider these aspects in the present paper, it would be very interesting to consider several agents – or continuums of agents in a mean field game setup – with different goals and to look for the Nash equilibriums in the repeated auction game at the heart of our model. This would be particularly relevant for auctions on small and specific parts of the audience.
[^8]: The ad trader may be connected to several ad exchanges, but we assume in this first model that he does not make any difference between auction requests coming from the different ad exchanges.
[^9]: In this paper, we only consider the case of a constant intensity $\lambda$. A time-varying auction-request intensity can be handled via a change of time, similar as in [@f2], where the main idea is to switch from *physical time* to *trading time*.
[^10]: The equality case is not considered separately because we assume that the distribution of the variables $(p_n)_n$ is absolutely continuous – see below.
[^11]: The model can easily be (slightly) modified to cover the case of truncated functions $f$, *i.e.* $f$ equal to 0 above a given price level.
[^12]: This process is well defined because $\forall n\in \mathbb{N}^*, p_n \in L^1(\Omega)$.
[^13]: In particular, we will consider the limit case $K \to +\infty$ as a way to see what happens when the maximal total amount to spend is imposed.
[^14]: For not making this paper a too technical one, we focus on applications and not on the mathematical analysis of the integro-differential Bellman equations we derive. The interested reader can derive existence and uniqueness results by using for instance the advanced viscosity techniques presented in [@bi]. It is noteworthy that we are going to approximate the problem by its fluid limit for which the classical approach with weak semi-concave solutions applies – see [@can; @evans].
[^15]: Equivalently, Eq. (\[m1:HJB2\]) can be written $$-\partial_t v(t,S) - \lambda \inf_{b \in \mathbb R_+} \left(-F(b) + \int_0^b f(p) (v(t,S-p) - v(t,S)) dp\right) = 0.$$
[^16]: This scale may instead be regarded as mesoscopic, if we consider that the advertising campaign constitutes the macroscopic scale.
[^17]: These values may seem quite small at first reading. We remind the reader that the industry standard is to measure bid levels and prices for bulks of one thousand impressions (CPM, or cost-per-mille). For instance, if for a given auction a practitioner talks about a bid of , then in reality that is .
[^18]: It is not an issue that the numerical solution of Eq. (\[m1:HJB2\]) takes far more time (seconds, minutes, ...) to be computed than the maximal time authorized to send a bid to the auction servers (milliseconds). In practice, the solution can be computed in advance, and then be embedded in an in-memory look-up table allowing low-latency requests.
[^19]: This problem is even more critical in the case of the on-line estimation of the parameters – see [@fgllearning].
[^20]: This point will be discussed below.
[^21]: Straightforward computations give: $$H'(x) =\left\{
\begin{array}{ll}
\lambda \int_0^{-\frac{1}{x}} pf(p) dp = \lambda G\left(-\frac{1}{x}\right), & x <0 \\
\lambda \int_0^{\infty} pf(p) dp, & x \ge 0.\\
\end{array}
\right.$$
In particular, $H'(0) = \lambda \int_0^{\infty} pf(p) dp$.\
Because $\lim_{p \to +\infty} p^3 f(p) = 0$, the function $H'$ is continuously differentiable with: $$H''(x) =\left\{
\begin{array}{ll}
-\lambda \frac{1}{x^3} f\left(-\frac 1x\right), & x <0 \\
0, & x \ge 0.
\end{array}
\right.$$
[^22]: Considering very large values of $b$ may seem to go against the fluid-limit approximation. However, this is not really the case because the price eventually paid for each auction won remains really small compared to the budget $S$, except perhaps near the terminal time $T$.
[^23]: In the case of a time-varying arrival rate of auction requests $\lambda_t$, a similar variational problem, solved in [@f2], shows that the optimal strategy is to spend proportionally to $\lambda_t$.
[^24]: In that case, the size of the grid can be reduced, because $\lambda$ is small.
[^25]: The optimal allocation of the overall budget across several bidding algorithms goes beyond the scope of this work. However, we can see the extensions of Section 3 as a way of addressing this issue. A natural alternative is to use multi-armed bandit algorithms such as those proposed in [@pa1; @pa2; @pa3].
[^26]: A Pareto distribution or an extreme-value distribution are also relevant choices.
[^27]: For approximating numerically the solution $v$ of the HJB equation , we consider a backward-in-time explicit scheme on a $(t,S)$ grid. The minimizer $b^*$ at each point of the grid is found by using the first order condition (\[sol\]) (and an affine approximation between successive points). The integral is then approximated by a closed form formula by using an affine approximation of $v$ between successive points of the grid and the exponential form of $f$.
[^28]: Throughout this extension and the next one, the superscripts $j \in \lbrace 1, \ldots, J\rbrace$ designate the sources and not exponents.
[^29]: It is noteworthy that different publishers may set different hard floors. In other words, there may be several jumps in $F$.
[^30]: The KPI only impacts the terminal condition on $u$.
| ArXiv |
---
abstract: 'The $\overline{B}_{q}^{\ast}$ ${\to}$ $DP$, $DV$ weak decays are studied with the perturbative QCD approach, where $q$ $=$ $u$, $d$ and $s$; $P$ and $V$ denote the ground $SU(3)$ pseudoscalar and vector meson nonet. It is found that the branching ratios for the color-allowed $\overline{B}_{q}^{\ast}$ ${\to}$ $D_{q}{\rho}^{-}$ decays can reach up to $10^{-9}$ or more, and should be promisingly measurable at the running LHC and forthcoming SuperKEKB experiments in the near future.'
author:
- Junfeng Sun
- Jie Gao
- Yueling Yang
- Qin Chang
- Na Wang
- Gongru Lu
- Jinshu Huang
title: 'Study of the $\overline{B}_{q}^{\ast}$ ${\to}$ $DM$ decays with perturbative QCD approach'
---
Introduction {#sec01}
============
In accordance with the conventional quark model assignments, the ground spin-singlet pseudoscalar $B_{q}$ mesons and spin-triplet vector $B^{\ast}_{q}$ mesons have the same flavor components, and consist of one valence heavy antiquark $\bar{b}$ and one light quark $q$, i.e., $\bar{b}q$, with $q$ $=$ $u$, $d$, $s$ [@pdg]. With the two $e^{+}e^{-}$ $B$-factory BaBar and Belle experiments, there is a combined data sample of over $1\,ab^{-1}$ at the ${\Upsilon}(4S)$ resonance. The $B_{u,d}$ meson weak decay modes with branching ratio of over $10^{-6}$ have been well measured [@epjc74]. The $B_{s}$ meson, which can be produced in hadron collisions or at/over the resonance ${\Upsilon}(5S)$ in $e^{+}e^{-}$ collisions, is being carefully scrutinized. However, the study of the $B_{q}^{\ast}$ mesons has not actually attracted much attention yet, subject to the relatively inadequate statistics. Because the mass of the $B_{q}^{\ast}$ mesons is a bit larger than that of the $B_{q}$ mesons, the $B_{q}^{\ast}$ meson should be produced at the relatively higher energy rather than at the resonance ${\Upsilon}(4S)$ in $e^{+}e^{-}$ collisions. With the high luminosities and large production cross section at the running LHC, the forthcoming SuperKEKB and future [*Super proton proton Collider*]{} (SppC, which is still in the preliminary discussion and research stage up to now), more and more $B_{q}^{\ast}$ mesons will be accumulated in the future, which makes the $B_{q}^{\ast}$ mesons another research laboratory for testing the Cabibbo-Kobayashi-Maskawa (CKM) picture for $CP$-violating phenomena, examining our comprehension of the underlying dynamical mechanism for the weak decays of the heavy flavor hadrons.
Having the same valence quark components and approximately an equal mass, both the $B^{\ast}_{q}$ and $B_{q}$ mesons can decay via weak interactions into the same final states. On the one hand, the $B^{\ast}_{q}$ and $B_{q}$ meson weak decays would provide each other with a spurious background; on the other hand, the interplay between the $B_{q}^{\ast}$ and $B_{q}$ weak decays could offer some potential useful information to constrain parameters within the standard model, and might shed some fresh light on various intriguing puzzles in the $B_{q}$ meson decays. The $B_{q}$ meson decays are well described by the bottom quark decay with the light spectator quark $q$ in the spectator model. At the quark level, most of the hadronic $B_{q}$ meson decays involve the $b$ ${\to}$ $c$ transition due to the hierarchy relation among the CKM matrix elements. As is well known, there is a more than $3\,{\sigma}$ discrepancy between the value of ${\vert}V_{cb}{\vert}$ obtained from inclusive determinations, ${\vert}V_{cb}{\vert}$ $=$ $(42.2{\pm}0.8){\times}10^{-3}$, and from exclusive ones, ${\vert}V_{cb}{\vert}$ $=$ $(39.2{\pm}0.7){\times}10^{-3}$ [@pdg]. Besides the semileptonic $\overline{B}_{q}^{(\ast)}$ ${\to}$ $D^{(\ast)}{\ell}\bar{\nu}$ decays, the nonleptonic $\overline{B}_{q}^{(\ast)}$ ${\to}$ $DM$ decays, with $M$ representing the ground $SU(3)$ pseudoscalar $P$ and the vector $V$ meson nonet, are also induced by the $b$ ${\to}$ $c$ transition, and hence could be used to extract/constrain the CKM matrix element ${\vert}V_{cb}{\vert}$.
From the dynamical point of view, the phenomenological models used for the $\overline{B}_{q}$ ${\to}$ $DM$ decays might, in principle, be extended and applied to the $\overline{B}_{q}^{\ast}$ ${\to}$ $DM$ decays. The practical applicability and reliability of these models could be reevaluated with the $\overline{B}_{q}^{\ast}$ ${\to}$ $DM$ decays. Recently, some attractive QCD-inspired methods, such as the perturbative QCD (pQCD) approach [@prd52.3958; @prd55.5577; @prd56.1615; @plb504.6; @prd63.054008; @prd63.074006; @prd63.074009; @epjc23.275], the QCD factorization (QCDF) approach [@prl83.1914; @npb591.313; @npb606.245; @plb488.46; @plb509.263; @prd64.014036; @npb774.64; @prd77.074013], soft and collinear effective theory [@prd63.014006; @prd63.114020; @plb516.134; @prd65.054022; @prd66.014017; @npb643.431; @plb553.267; @npb685.249] and so on, have been developed vigorously and employed widely to explain measurements on the $B_{q}$ meson decays. The $\overline{B}_{q}$ ${\to}$ $DM$ decays have been studied with the QCDF [@npb591.313; @plb476.339] and pQCD [@prd69.094018; @prd78.014018] approaches, but there are few research works on the $B_{q}^{\ast}$ meson weak decays. Recently, the $\overline{B}_{q}^{\ast}$ ${\to}$ $D_{q}V$ decays have been investigated with the QCDF approach [@epjc76.523], and it is shown that the $\overline{B}_{q}^{{\ast}0}$ ${\to}$ $D_{q}^{+}{\rho}^{-}$ decays with branching ratios of ${\cal O}(10^{-8})$ might be accessible to the existing and future heavy flavor experiments. In this paper, we will give a comprehensive investigation into the two-body nonleptonic $\overline{B}_{q}^{\ast}$ ${\to}$ $DM$ decays with the pQCD approach in order to provide the future experimental research with an available reference.
As is well known, the $B^{\ast}_{q}$ meson decays are dominated by the electromagnetic interactions rather than the weak interactions, which differs significantly from the $B_{q}$ meson decays. One can easily expect that the branching ratios for the $\overline{B}_{q}^{\ast}$ ${\to}$ $DM$ weak decays should be very small due to the short electromagnetic lifetimes of the $B_{q}^{\ast}$ mesons [@epja52.90], although these processes are favored by the CKM matrix element ${\vert}V_{cb}{\vert}$. Of course, an abnormal large branching ratio might be a possible hint of new physics beyond the standard model. There is still no experimental report on the $\overline{B}_{q}^{\ast}$ ${\to}$ $DM$ weak decays so far. Furthermore, the $\overline{B}_{q}^{\ast}$ ${\to}$ $DM$ weak decays offer the unique opportunity of observing the weak decay of a vector meson, where polarization effects could be explored.
This paper is organized as follows. In section \[sec02\], we present the theoretical framework, the conventions and notations, together with amplitudes for the $\overline{B}_{q}^{\ast}$ ${\to}$ $DM$ decays. Section \[sec03\] is devoted to the numerical results and discussion. The final section is a summary.
theoretical framework {#sec02}
=====================
The effective Hamiltonian {#sec0201}
-------------------------
As is well known, the weak decays of the $B_{q}^{(\ast)}$ mesons inevitably involve multiple length scales, including the mass of $m_{W}$ for the virtual gauge boson $W$, the mass of $m_{b}$ for the decaying bottom quark, the infrared confinement scale ${\Lambda}_{\rm QCD}$ of the strong interactions, and $m_{W}$ ${\gg}$ $m_{b}$ ${\gg}$ ${\Lambda}_{\rm QCD}$. So, one usually has to resort to the effective theory approximation scheme. With the operator product expansion and the renormalization group (RG) method, the effective Hamiltonian for the $\overline{B}_{q}^{\ast}$ ${\to}$ $DM$ decays can be written as [@9512380], $${\cal H}_{\rm eff}\, =\, \frac{G_{F}}{\sqrt{2}}\,
\sum\limits_{q^{\prime}=d,s} V_{cb}\,V_{uq^{\prime}}^{\ast}
\Big\{ C_{1}({\mu})\,Q_{1}({\mu})+C_{2}({\mu})\,Q_{2}({\mu})\Big\}
+{\rm h.c.}
\label{hamilton},$$ where $G_{F}$ ${\simeq}$ $1.166{\times}10^{-5}\,{\rm GeV}^{-2}$ [@pdg] is the Fermi coupling constant.
Using the Wolfenstein parametrization, the CKM factor $V_{cb}V_{uq^{\prime}}^{\ast}$ are expressed as a series expansion of the small Wolfenstein parameter ${\lambda}$ ${\approx}$ $0.2$ [@pdg]. Up to the order of ${\cal O}({\lambda}^{7})$, they can be written as follows: $$\begin{aligned}
V_{cb}\,V_{ud}^{\ast}
&=& A\,{\lambda}^{2}\,( 1 -{\lambda}^{2}/2-{\lambda}^{4}/8 )
+{\cal O}({\lambda}^{7})
\label{vcbvud}, \\
%---------------------------------------------------------
V_{cb}\,V_{us}^{\ast}
&=& A\,{\lambda}^{3}+{\cal O}({\lambda}^{7})
\label{vcbvus}.
\end{aligned}$$ It is very clearly seen that the both $V_{cb}\,V_{ud}^{\ast}$ and $V_{cb}\,V_{us}^{\ast}$ are real-valued, i.e., there is no weak phase difference. However, nonzero weak phase difference is necessary and indispensable for the direct $CP$ violation. Therefore, none of direct $CP$ violation should be expected for the $\overline{B}_{q}^{\ast}$ ${\to}$ $DM$ decays.
The renormalization scale ${\mu}$ separates the physical contributions into the short- and long-distance parts. The Wilson coefficients $C_{1,2}$ summarize the physical contributions above the scale ${\mu}$. They, in principle, are calculable order by order in the strong coupling ${\alpha}_{s}$ at the scale $m_{W}$ with the ordinary perturbation theory, and then evolved with the RG equation to the characteristic scale ${\mu}$ ${\sim}$ ${\cal O}(m_{b})$ for the bottom quark decay [@9512380]. The Wilson coefficients at the scale $m_{W}$ are determined at the quark level rather than the hadron level, so they are regarded as process-independent couplings of the local operators $Q_{i}$. Their explicit analytical expressions, including the next-to-leading order corrections, have been given in Ref.[@9512380].
The physical contributions from the scales lower than ${\mu}$ are contained in the hadronic matrix elements (HME) where the local four-quark operators are sandwiched between the initial and final hadron states. The local six-dimension operators arising from the $W$-boson exchange are defined as follows: $$\begin{aligned}
Q_{1} &=& [\,\bar{c}_{\alpha}\,{\gamma}_{\mu}\,(1-{\gamma}_{5})\,b_{\alpha}]\,\
[\,\bar{q}^{\prime}_{\beta}\,{\gamma}^{\mu}\,(1-{\gamma}_{5})\,u_{\beta}]
\label{operator:q1},
\\
Q_{2} &=& [\,\bar{c}_{\alpha}\,{\gamma}_{\mu}\,(1-{\gamma}_{5})\,b_{\beta}]\,\
[\,\bar{q}^{\prime}_{\beta}\,{\gamma}^{\mu}\,(1-{\gamma}_{5})\,u_{\alpha}]
\label{operator:q2}.
\end{aligned}$$ where ${\alpha}$ and ${\beta}$ are color indices, i.e., the gluonic corrections are included. The operator $Q_{1}$ ($Q_{2}$) consists of two color-singlet (color-octet) currents. The operators $Q_{1}$ and $Q_{2}$, called current-current operators or tree operators, have the same flavor form and a different color structure. It is obvious that the $\overline{B}_{q}^{\ast}$ ${\to}$ $DM$ decays are uncontaminated by the contributions from the penguin operators, which is positive to extract the CKM matrix element ${\vert}V_{cb}{\vert}$.
Because of the participation of the strong interaction, especially, the long-distance effects in the conversion from the quarks of the local operators to the initial and final hadrons, barricades are still erected on the approaches of nonleptonic $\overline{B}_{q}^{(\ast)}$ weak decays, which complicates the calculation. HME of the local operators are the most intricate part for theoretical calculation, where the perturbative and nonperturbative contributions entangle with each other. To evaluate the HME amplitudes, one usually has to resort to some plausible approximations and assumptions, which results in the model-dependence of theoretical predictions. It is obvious that a large part of the uncertainties does come from the practical treatment of HME, due to our inadequate understanding of the hadronization mechanism and the low-energy QCD behavior. For the phenomenology of the $\overline{B}_{q}^{\ast}$ ${\to}$ $DM$ decays, one of the main tasks at this stage is how to effectively factorize HME of the local operators into hard and soft parts, and how to evaluate HME properly.
Hadronic matrix elements {#sec0202}
------------------------
One of the phenomenological schemes for the HME calculation is the factorization approximation based on Bjorken’s [*a priori*]{} color transparency hypothesis, which says that the color singlet energetic hadron would have flown rapidly away from the color fields existing in the neighborhood of the interaction point before the soft gluons are exchanged among hadrons [@npb11.325]. Modeled on the amplitudes for exclusive processes with the Lepage-Brodsky approach [@prd22.2157], HME are usually written as the convolution integral of the hard kernels and the hadron distribution amplitudes (DAs). Hard kernels are expressed as the scattering amplitudes for the transition of the heavy bottom quark into light quarks. They are generally computable at the quark level with the perturbation theory as a series of expansion in the parameter $1/m_{b}$ and the strong coupling constant ${\alpha}_{s}$ in the heavy quark limit. It is assumed that the soft and nonperturbative contributions of HME could be absorbed into hadron DAs. The distribution amplitudes are functions of parton momentum fractions. They, although not calculable, are regarded as universal and can be determined by nonperturbative means or extracted from data. With the traits of universality and determinability of hadron DAs, HME have a sample structure and can be evaluated to make predictions.
Besides the factorizable contributions to HME, the nonfactorizable corrections to HME also play an important role in commenting on the experimental measurements and solving the so-called puzzles and anomalies, and hence should be carefully considered, as commonly recognized by theoretical physicists. In order to regulate the endpoint singularities which appear in the spectator scattering and annihilation amplitudes with the QCDF approach and spoil the perturbative calculation with the collinear approximation [@npb591.313; @npb606.245; @plb488.46; @plb509.263; @prd64.014036], it is suggested by the pQCD approach [@prd52.3958; @prd55.5577; @prd56.1615; @plb504.6; @prd63.054008; @prd63.074006; @prd63.074009; @epjc23.275] that the transverse momentum of quarks should be conserved and, additionally, that a Sudakov factor should be introduced to DAs for all the participant hadrons to further suppress the long-distance and soft contributions. The basic pQCD formula for nonleptonic weak decay amplitudes could be factorized into three parts: the hard effects enclosed by the Wilson coefficients $C_{i}$, hard scattering kernels ${\cal H}_{i}$, and the universal wave functions ${\Phi}_{j}$. The general form is a multidimensional integral [@prd52.3958; @prd55.5577; @prd56.1615; @plb504.6; @prd63.054008; @prd63.074006; @prd63.074009; @epjc23.275], $${\cal A}_{i}\ {\propto}\ {\int}\, {\prod_j}dx_{j}\,db_{j}\,
C_{i}(t)\,{\cal H}_{i}(t_{i},x_{j},b_{j})\,{\Phi}_{j}(x_{j},b_{j})\,e^{-S_{j}}
\label{hadronic},$$ where $x_{j}$ is the longitudinal momentum fraction of the valence quarks. $b_{j}$ is the conjugate variable of the transverse momentum $k_{jT}$. The scale $t_{i}$ is preferably chosen to be the maximum virtuality of all the internal particles. The Sudakov factor $e^{-S_{j}}$, together with the particular scale $t_{i}$, will ensure the perturbative calculation is feasible and reliable.
Kinematic variables {#sec0203}
-------------------
The $\overline{B}_{q}^{(\ast)}$ weak decays are actually dominated by the $b$ quark weak decay. In the heavy quark limit, the light quark originating from the heavy bottom quark decay is assumed to be energetic and race quickly away from the weak interaction point. If the velocity $v$ ${\sim}$ $c$ (the speed of light), the light quarks move near the light-cone line. The light-cone dynamics can be used to describe the relativistic system along the light-front direction. The light-cone coordinates $(x^{+},x^{-},x_{\perp})$ of space-time are defined as $x^{\pm}$ $=$ $(x^{0}{\pm}x^{3})/\sqrt{2}$ (or $(t{\pm}x^{3})/\sqrt{2}$) and $x_{\perp}$ $=$ $x^{i}$ with $i$ $=$ $1$ and $2$. $x^{\pm}$ $=$ $0$ is called the light-front. The scalar product of any two four-dimensional vectors is given by $a{\cdot}b$ $=$ $a_{\mu}b^{\mu}$ $=$ $a^{+}b^{-}$ $+$ $a^{-}b^{+}$ $-$ $a_{\perp}{\cdot}b_{\perp}$. In the rest frame of the $\overline{B}_{q}^{\ast}$ meson, the final $D$ and $M$ mesons are back-to-back. The light-cone kinematic variables are defined as follows: $$p_{\overline{B}_{q}^{\ast}}\, =\, p_{1}\, =\, \frac{m_{1}}{\sqrt{2}}(1,1,0)
\label{kine-p1},$$ $$p_{D}\, =\, p_{2}\, =\, (p_{2}^{+},p_{2}^{-},0)
\label{kine-p2},$$ $$p_{M}\, =\, p_{3}\, =\, (p_{3}^{-},p_{3}^{+},0)
\label{kine-p3},$$ $$k_{i}\, =\, x_{i}\,p_{i}+(0,0,k_{iT})
\label{kine-ki},$$ $$p_{i}^{\pm}\, =\, (E_{i}\,{\pm}\,p)/\sqrt{2}
\label{kine-pipm},$$ $$t\, =\, 2\,p_{1}{\cdot}p_{2}\, =\, m_{1}^{2}+m_{2}^{2}-m_{3}^{2}\, =\,2\,m_{1}\,E_{2}
\label{kine-t},$$ $$u\, =\, 2\,p_{1}{\cdot}p_{3}\, =\, m_{1}^{2}-m_{2}^{2}+m_{3}^{2}\, =\,2\,m_{1}\,E_{3}
\label{kine-u},$$ $$s\, =\, 2\,p_{2}{\cdot}p_{3}\, =\, m_{1}^{2}-m_{2}^{2}-m_{3}^{2}
\label{kine-s},$$ $$s\,t +s\,u-t\,u \,=\, 4\,m_{1}^{2}\,p^{2}
\label{kine-pcm},$$ where the subscripts $i$ $=$ $1$, $2$ and $3$ of the variables (such as, four-dimensional momentum $p_{i}$, energy $E_{i}$, and mass $m_{i}$) correspond to the $\overline{B}_{q}^{\ast}$, $D$ and $M$ mesons, respectively. $k_{i}$ is the momentum of the light antiquark carrying the longitudinal momentum fraction $x_{i}$. $k_{iT}$ is the transverse momentum. $t$, $u$ and $s$ are the Lorentz scalar variables. $p$ is the common momentum of the final states. These momenta are shown in Fig.\[fig:fey-t\](a), Fig.\[fig:fey-c\](a) and Fig.\[fig:fey-a\](a).
Wave functions {#sec0204}
--------------
As aforementioned, wave functions are the essential input parameters in the master pQCD formula for the HME calculation. Following the notations in Refs. [@npb529.323; @prd65.014007; @prd92.074028; @plb751.171; @plb752.322; @jhep9901.010; @jhep0703.069; @jhep0605.004], the wave functions of the participating meson are defined as the meson-to-vacuum HME. $${\langle}0{\vert}\bar{q}_{i}(z)b_{j}(0){\vert}
\overline{B}_{q}^{\ast}(p,{\epsilon}^{\parallel}){\rangle}\,
=\, \frac{f_{B_{q}^{\ast}}}{4} {\int}d^{4}k\,e^{-ik{\cdot}z}
\Big\{ \!\!\not{\!\epsilon}^{\parallel}\, \Big[
m_{B_{q}^{\ast}}\, {\Phi}_{B_{q}^{\ast}}^{v}(k)\, -
\!\not{p}\, {\Phi}_{B_{q}^{\ast}}^{t}(k) \Big] \Big\}_{ji}
\label{wf-bq01},$$ $${\langle}0{\vert}\bar{q}_{i}(z)b_{j}(0){\vert}
\overline{B}_{q}^{\ast}(p,{\epsilon}^{\perp}){\rangle}\,
=\, \frac{f_{B_{q}^{\ast}}}{4} {\int}d^{4}k\,e^{-ik{\cdot}z}
\Big\{ \!\!\not{\!\epsilon}^{\perp}\, \Big[
m_{B_{q}^{\ast}}\, {\Phi}_{B_{q}^{\ast}}^{V}(k)\, -
\!\not{p}\, {\Phi}_{B_{q}^{\ast}}^{T}(k) \Big] \Big\}_{ji}
\label{wf-bc02},$$ $${\langle}D_{q}(p){\vert}\bar{c}_{i}(0)q_{j}(z){\vert}0{\rangle}\,
=\, \frac{i\,f_{D_{q}}}{4}{\int}d^{4}k\,e^{+ik{\cdot}z}\,
\Big\{ {\gamma}_{5}\Big[ \!\!\not{p}\,{\Phi}_{D_{q}}^{a}(k)
+m_{D_{q}}\,{\Phi}_{D_{q}}^{p}(k) \Big] \Big\}_{ji}
\label{wf-cq01},$$ $$\begin{aligned}
{\langle}P(p){\vert}\bar{q}_{i}(0)q^{\prime}_{j}(z){\vert}0{\rangle}\,
&=& \frac{1}{4}{\int}d^{4}k\,e^{+ik{\cdot}z}\,
\Big\{ {\gamma}_{5}\Big[ \!\!\not{p}\,{\Phi}_{P}^{a}(k) +
{\mu}_{P}\,{\Phi}_{P}^{p}(k)
\nonumber \\ & & \qquad \qquad \qquad +
{\mu}_{P}\,(\not{n}_{+}\!\!\not{n}_{-}-1)\,{\Phi}_{P}^{t}(k)
\Big] \Big\}_{ji}
\label{wf-p},
\end{aligned}$$ $${\langle}V(p,{\epsilon}^{\parallel}){\vert}\bar{q}_{i}(0)q^{\prime}_{j}(z){\vert}0{\rangle}\,
=\, \frac{1}{4}{\int}d^{4}k\,e^{+ik{\cdot}z}\,
\Big\{ \!\!\not{\epsilon}^{\parallel}\,m_{V}\,{\Phi}_{V}^{v}(k)
+\!\!\not{\epsilon}^{\parallel}\!\!\not{p}\,{\Phi}_{V}^{t}(k)
-m_{V}\,{\Phi}_{V}^{s}(k) \Big\}_{ji}
\label{wf-v-el},$$ $$\begin{aligned}
{\langle}V(p,{\epsilon}^{\perp}){\vert}\bar{q}_{i}(0)q^{\prime}_{j}(z){\vert}0{\rangle}\,
&=& \frac{1}{4}{\int}d^{4}k\,e^{+ik{\cdot}z}\,
\Big\{ \!\!\not{\epsilon}^{\perp}\,m_{V}\,{\Phi}_{V}^{V}(k)
+\!\!\not{\epsilon}^{\perp}\!\!\not{p}\,{\Phi}_{V}^{T}(k)
\nonumber \\ & & \qquad +
\frac{i\,m_{V}}{p{\cdot}n_{+}}\,{\gamma}_{5}\,{\varepsilon}_{{\mu}{\nu}{\alpha}{\beta}}
{\gamma}^{\mu}\,{\epsilon}^{{\perp}{\nu}}\,p^{\alpha}\,
n_{+}^{\beta}\,{\Phi}_{V}^{A}(k) \Big\}_{ji}
\label{wfv-et},
\end{aligned}$$ where $f_{B_{q}^{\ast}}$ and $f_{D_{q}}$ are the decay constants of the $\overline{B}_{q}^{\ast}$ meson and the $D_{q}$ meson, respectively. ${\epsilon}^{\parallel}$ and ${\epsilon}^{\perp}$ are the longitudinal and transverse polarization vectors. $n_{+}$ $=$ $(1,0,0)$ and $n_{-}$ $=$ $(0,1,0)$ are the positive and negative null vectors, i.e., $n_{\pm}^{2}$ $=$ $0$. The chiral factor ${\mu}_{P}$ relates the pseudoscalar meson mass to the quark mass through the following way [@jhep9901.010], $${\mu}_{P}\, =\, \frac{m_{\pi}^{2}}{m_{u}+m_{d}}
\, =\, \frac{m_{K}^{2}}{m_{u,d}+m_{s}}
\, {\approx}\, (1.6{\pm}0.2)\,\text{GeV}
\label{up}.$$ With the twist classification based on the power counting rule in the infinite momentum frame [@npb529.323; @prd65.014007], the wave functions ${\Phi}_{B_{q}^{\ast},V}^{v,T}$ and ${\Phi}_{D_{q},P}^{a}$ are twist-2 (the leading twist), while the wave functions ${\Phi}_{B_{q}^{\ast},V}^{t,V,s,A}$ and ${\Phi}_{D_{q},P}^{p,t}$ are twist-3. By integrating out the transverse momentum from the wave functions, one can obtain the corresponding distribution amplitudes. In our calculation, the expressions of the DAs for the heavy-flavored mesons are [@prd92.074028; @plb751.171; @plb752.322] $${\phi}_{B_{q}^{\ast}}^{v,T}(x) = A\, x\,\bar{x}\,
{\exp}\Big\{ -\frac{1}{8\,{\omega}_{B_{q}^{\ast}}^{2}}\,
\Big( \frac{m_{q}^{2}}{x}+\frac{m_{b}^{2}}{\bar{x}} \Big) \Big\}
\label{da-bqlv},$$ $${\phi}_{B_{q}^{\ast}}^{t}(x) = B\, (\bar{x}-x)^{2}\,
{\exp}\Big\{ -\frac{1}{8\,{\omega}_{B_{q}^{\ast}}^{2}}\,
\Big( \frac{m_{q}^{2}}{x}+\frac{m_{b}^{2}}{\bar{x}} \Big) \Big\}
\label{da-bqlt},$$ $${\phi}_{B_{q}^{\ast}}^{V}(x) = C\, \{1+(\bar{x}-x)^{2}\}\,
{\exp}\Big\{ -\frac{1}{8\,{\omega}_{B_{q}^{\ast}}^{2}}\,
\Big( \frac{m_{q}^{2}}{x}+\frac{m_{b}^{2}}{\bar{x}} \Big) \Big\}
\label{da-bqtt},$$ $${\phi}_{D_{q}}^{a}(x) = D\, x\,\bar{x}\,
{\exp}\Big\{ -\frac{1}{8\,{\omega}_{D_{q}}^{2}}\,
\Big( \frac{m_{q}^{2}}{x}+\frac{m_{c}^{2}}{\bar{x}} \Big) \Big\}
\label{da-cqa},$$ $${\phi}_{D_{q}}^{p}(x) = E\,
{\exp}\Big\{ -\frac{1}{8\,{\omega}_{D_{q}}^{2}}\,
\Big( \frac{m_{q}^{2}}{x}+\frac{m_{c}^{2}}{\bar{x}} \Big) \Big\}
\label{da-cqp},$$ where $x$ and $\bar{x}$ (${\equiv}$ $1$ $-$ $x$) are the longitudinal momentum fractions of the light and heavy partons; $m_{b}$, $m_{c}$ and $m_{q}$ are the mass of the valence $b$, $c$ and $q$ quarks. The parameter ${\omega}_{i}$ determines the average transverse momentum of the partons, and ${\omega}_{i}$ ${\approx}$ $m_{i}\,{\alpha}_{s}(m_{i})$. The parameters $A$, $B$, $C$, $D$ and $E$ are the normalization coefficients to satisfy the conditions, $${\int}_{0}^{1}dx\,{\phi}_{B_{q}^{\ast}}^{v,t,V,T}(x)=1
\label{wave-nb},$$ $${\int}_{0}^{1}dx\,{\phi}_{D_{q}}^{a,p}(x) =1
\label{wave-nd}.$$ The main distinguishing feature of the above DAs in Eqs.(\[da-bqlv\]-\[da-cqp\]) is the exponential functions, where the exponential factors are proportional to the ratio of the parton mass squared $m_{i}^{2}$ to the momentum fraction $x_{i}$, i.e., $m_{i}^{2}/x_{i}$. Hence, the DAs of Eqs.(\[da-bqlv\]-\[da-cqp\]) are generally consistent with the ansatz that the momentum fractions are shared among the valence quarks according to the quark mass, i.e., a light quark will carry a smaller fraction of the parton momentum than a heavy quark in a heavy-light system. In addition, the exponential functions strongly suppress the contributions from the endpoint of $x$, $\bar{x}$ ${\to}$ $0$, and naturally provide the effective truncation for the endpoint and soft contributions.
As is well known, there are many phenomenological DA models for the charmed mesons. Some have been recited by Eq.(30) in Ref.[@prd78.014018]. One of the favorable DA models from the experimental data, without the distinction between the twist-2 and twist-3, has the common expression as below, $${\phi}_{D_{q}}(x) = 6\,x\,\bar{x}\,\big\{1+C_{D_{q}}\,(\bar{x}-x) \big\}
\label{wave-d-xb},$$ where the parameter $C_{D_{u,d}}$ $=$ $0.5$ for the $D_{u,d}$ meson, and $C_{D_{s}}$ $=$ $0.4$ for the $D_{s}$ meson.
The expressions of the twist-2 quark-antiquark DAs for the light pseudoscalar and vector mesons have the expansion [@jhep9901.010; @jhep0703.069; @jhep0605.004], $${\phi}_{P}^{a}(x)\, =\, i\,f_{P}\,6\,x\,\bar{x}\,
\sum\limits_{i=0} a^{P}_{i}\,C_{i}^{3/2}({\xi})
\label{da-pa},$$ $${\phi}_{V}^{v}(x) \, =\, f_{V}\,6\,x\,\bar{x}\,
\sum\limits_{i=0} a^{\parallel}_{i}\,C_{i}^{3/2}({\xi})
\label{da-rho-v},$$ $${\phi}_{V}^{T}(x) \, =\, f_{V}^{T}\,6\,x\,\bar{x}\,
\sum\limits_{i=0} a^{\perp}_{i}\,C_{i}^{3/2}({\xi})
\label{da-rho-T},$$ where $f_{P}$ is the decay constant for the pseudoscalar meson $P$; $f_{V}$ and $f_{V}^{T}$ are the vector and tensor (also called the longitudinal and transverse) decay constants for the vector meson $V$. The nonperturbative parameters of $a_{i}^{P,{\parallel},{\perp}}$ are called the Gegenbauer moments, and $a_{0}^{P,{\parallel},{\perp}}$ $=$ $1$ for the asymptotic forms, $a_{{\rm odd}~i}^{P,{\parallel},{\perp}}$ $=$ $0$ for the DAs of the $G$-parity eigenstates, such as the unflavored ${\pi}$, ${\eta}$, ${\eta}^{\prime}$, ${\rho}$, ${\omega}$, ${\phi}$ mesons. The short-hand notation ${\xi}$ $=$ $x$ $-$ $\bar{x}$ $=$ $2\,x$ $-$ $1$. The analytical expressions of the Gegenbauer polynomials $C_{i}^{j}({\xi})$ are as below, $$\begin{aligned}
& & C_{0}^{j}({\xi})\, =\, 1
\label{eq-c0}, \\
& & C_{1}^{j}({\xi})\, =\, 2\,j\,{\xi}
\label{eq-c1}, \\
& & C_{2}^{j}({\xi})\, =\, 2\,j\,(j+1)\,{\xi}^{2}-j
\label{eq-c2}, \\
& & ...... \nonumber
\end{aligned}$$ As for the twist-3 DAs for the light pseudoscalar and vector mesons, their asymptotic forms will be employed in this paper for the simplification [@prd78.014018; @jhep9901.010; @jhep0703.069; @jhep0605.004], i.e., $$\begin{aligned}
{\phi}_{P}^{p}(x)&=& +i\,f_{P}\, C_{0}^{1/2}({\xi}) % + a_{2}^{p}\,C_{2}^{1/2}(t)
\label{da-pp}, \\
%-----------------------------------------------------
{\phi}_{P}^{t}(x)&=& -i\,f_{P}\, C_{1}^{1/2}({\xi})
\label{da-pt}, \\
%-----------------------------------------------------
{\phi}_{V}^{t}(x) &=& +3\, f_{V}^{T}\,{\xi}^{2}
\label{da-rho-t}, \\
%-----------------------------------------------------
{\phi}_{V}^{s}(x) &=& -3\, f_{V}^{T}\,{\xi}
\label{da-rho-s}, \\
%-----------------------------------------------------
{\phi}_{V}^{V}(x) &=& +\frac{3}{4}\,f_{V}\,(1+{\xi}^{2})
\label{da-rho-V}, \\
%-----------------------------------------------------
{\phi}_{V}^{A}(x) &=& -\frac{3}{2}\,f_{V}\,{\xi}
\label{da-rho-A}.
\end{aligned}$$
Decay amplitudes {#sec0205}
----------------
As aforementioned, the $\overline{B}_{q}^{\ast}$ ${\to}$ $DM$ weak decays are induced practically by the $b$ quark decay at the quark level. There are three possible types of Feynman diagrams for the $\overline{B}_{q}^{\ast}$ ${\to}$ $DM$ decays with the pQCD approach, i.e., the color-allowed topologies of Fig.\[fig:fey-t\] induced by the external $W$-emission interactions, the color-suppressed topologies of Fig.\[fig:fey-c\] induced by the internal $W$-emission interactions, and the annihilation topologies of Fig.\[fig:fey-a\] induced by the $W$-exchange interactions. In the emission topologies of Fig.\[fig:fey-t\] (Fig.\[fig:fey-c\]), the light spectator quark in the $\overline{B}_{q}^{\ast}$ meson is absorbed by the recoiled $D_{q}$ ($M_{q}$) meson, and the exchanged gluons are space-like. In the annihilation topologies of Fig.\[fig:fey-a\], the exchanged gluons are time-like, which then split into the light quark-antiquark pair.
![The color-allowed diagrams for the $\overline{B}_{q}^{\ast}$ ${\to}$ $D_{q}M$ decays with the pQCD approach, where (a,b) and (c,d) are factorizable and nonfactorizable emission topologies, respectively.[]{data-label="fig:fey-t"}](t.ps){width="90.00000%"}
![The color-suppressed diagrams for the $\overline{B}^{\ast}_{q}$ ${\to}$ $D^{0}M_{q}$ decays with the pQCD approach.[]{data-label="fig:fey-c"}](c.ps){width="90.00000%"}
![The annihilation diagrams for the $\overline{B}_{q}^{{\ast}0}$ ${\to}$ $DM$ decays with the pQCD approach.[]{data-label="fig:fey-a"}](a.ps){width="90.00000%"}
The first two diagrams of Fig.\[fig:fey-t\], Fig.\[fig:fey-c\], and Fig.\[fig:fey-a\] are usually called the factorizable topologies. In the color-allowed (color-suppressed) factorizable emission topologies, the gluons are exchanged only between the initial $\overline{B}_{q}^{\ast}$ and the recoil $D_{q}$ ($M_{q}$) meson pair, and the emission $M$ ($D^{0}$) meson could be completely parted from the $\overline{B}_{q}^{\ast}D_{q}$ ($\overline{B}_{q}^{\ast}M_{q}$) system. In the factorizable annihilation topologies, the gluons are exchanged only between the final $DM$ meson pair, and the initial $\overline{B}_{q}^{\ast}$ meson could be directly separated from the $DM$ meson pair. Hence, in the factorizable emission (annihilation) topologies, the integral of the wave functions for the emission (initial) mesons reduces to the corresponding decay constant. For the factorizable topologies, the decay amplitudes will have the relatively simple structures, and can be written as the product of the decay constants and the hadron transition form factors. With the pQCD approach, the form factors can be written as the convolution integral of the hard scattering amplitudes and the hadron DAs.
The last two diagrams of Fig.\[fig:fey-t\], Fig.\[fig:fey-c\], and Fig.\[fig:fey-a\] are usually called the nonfactorizable topologies. In the nonfactorizable topologies, the emission meson is entangled with the gluons that radiated from the spectator quark, and hence on meson can be separated clearly from the other mesons. Hence, the decay amplitudes for the nonfactorizable topologies have quite complicated structures, and the amplitude convolution integral involve the wave functions of all the participating mesons. The nonfactorizable emission topologies within the pQCD framework are also called the spectator scattering topologies with the QCDF approach. Especially for the color-suppressed emission topologies, the factorizable contributions are proportional to the small parameter $a_{2}$, hence, the nonfactorizable contributions, being proportional to the large Wilson coefficient $C_{1}$, should be significant. As widely recognized, the nonfactorizable contributions play an important role in clarifying or reducing some discrepancies between the theoretical results and the experimental data on the nonleptonic $B$ meson weak decays.
Among the three possible types of Feynman diagrams (Fig.\[fig:fey-t\], Fig.\[fig:fey-c\], and Fig.\[fig:fey-a\]), only one or two of them will contribute to the specific $\overline{B}_{q}^{\ast}$ ${\to}$ $DM$ decays. The explicit amplitudes for the concrete $\overline{B}_{q}^{\ast}$ ${\to}$ $DP$, $DV$ decays have been collected in the Appendixes \[amp-dp\] and \[amp-dv\], and the building blocks in the Appendixes \[block-t\], \[block-c\] and \[block-a\]. According to the polarization relations between the initial and final vector mesons, the amplitudes for the $\overline{B}_{q}^{\ast}$ ${\to}$ $DV$ decays can generally be decomposed into the following structures [@prd66.054013; @ijmpa31.1650146; @npb911.890; @prd95.036024], $${\cal A}(\overline{B}_{q}^{\ast}{\to}DV)\, =\,
{\cal A}_{L}({\epsilon}_{B_{q}^{\ast}}^{\parallel},{\epsilon}_{V}^{\parallel})
+{\cal A}_{N}({\epsilon}_{B_{q}^{\ast}}^{\perp}{\cdot}{\epsilon}_{V}^{\perp})
+i\,{\cal A}_{T}\,{\varepsilon}_{{\mu}{\nu}{\alpha}{\beta}}\,
{\epsilon}_{B_{q}^{\ast}}^{\mu}\,{\epsilon}_{V}^{\nu}\,
p_{B_{q}^{\ast}}^{\alpha}\,p_{V}^{\beta}
\label{eq:amp01}.$$ which is conventionally written as the helicity amplitudes, $$H_{0}\ =\ {\cal A}_{L}({\epsilon}_{B_{q}^{\ast}}^{\parallel},{\epsilon}_{V}^{\parallel})
\label{eq:amp02},$$ $$H_{\parallel}\ =\ \sqrt{2}\,{\cal A}_{N}
\label{eq:amp03},$$ $$H_{\perp}\ =\ \sqrt{2}\,m_{B_{q}^{\ast}}\,p\, {\cal A}_{T}
\label{eq:amp04}.$$ As is well known, it is commonly assumed that the $SU(3)$ symmetry breaking interactions mixes the isospin-singlet neutral members of the octet with the singlet states. The ideal mixing angle ${\theta}_{V}$ (with ${\sin}{\theta}_{V}$ $=$ $1/\sqrt{3}$) between the octet and the singlet states is almost true in practice for the physical ${\omega}$ and ${\phi}$ mesons, i.e., the valence quark components are ${\omega}$ $=$ $(u\bar{u}+d\bar{d})/\sqrt{2}$ and ${\phi}$ $=$ $s\bar{s}$. As for the mixing among the light pseudoscalar mesons, the notations known as the quark-flavor basis description [@prd58.114006] is adopted here, and for simplicity, the possible gluonium and charmonium compositions are neglected for the time being, i.e., $$\left(\begin{array}{c}
{\eta} \\ {\eta}^{\prime}
\end{array}\right) =
\left(\begin{array}{cc}
{\cos}{\theta}_{P} & -{\sin}{\theta}_{P} \\
{\sin}{\theta}_{P} & {\cos}{\theta}_{P}
\end{array}\right)
\left(\begin{array}{c}
{\eta}_{q} \\ {\eta}_{s}
\end{array}\right)
\label{mixing01},$$ where the flavor states ${\eta}_{q}$ $=$ $(u\bar{u}+d\bar{d})/{\sqrt{2}}$ and ${\eta}_{s}$ $=$ $s\bar{s}$. The mixing angle determined from experimental data is ${\theta}_{P}$ $=$ $(39.3{\pm}1.0)^{\circ}$ [@prd58.114006]. The mass relations between the physical states (${\eta}$ and ${\eta}^{\prime}$) and the flavor states (${\eta}_{q}$ and ${\eta}_{s}$) are $$\begin{aligned}
m_{{\eta}_{q}}^{2}&=& \displaystyle
m_{\eta}^{2}\,{\cos}^{2}{\theta}_{P}
+m_{{\eta}^{\prime}}^{2}\,{\sin}^{2}{\theta}_{P}
-\frac{\sqrt{2}\,f_{{\eta}_{s}}}{f_{{\eta}_{q}}}\,
(m_{{\eta}^{\prime}}^{2}- m_{\eta}^{2})\,
{\cos}{\theta}_{P}\,{\sin}{\theta}_{P}
\label{ss12}, \\
%-----------------------------------------------------
m_{{\eta}_{s}}^{2}&=& \displaystyle
m_{\eta}^{2}\,{\sin}^{2}{\theta}_{P}
+m_{{\eta}^{\prime}}^{2}\,{\cos}^{2}{\theta}_{P}
-\frac{f_{{\eta}_{q}}}{\sqrt{2}\,f_{{\eta}_{s}}}
(m_{{\eta}^{\prime}}^{2}- m_{\eta}^{2})\,
{\cos}{\theta}_{P}\,{\sin}{\theta}_{P}
\label{ss13},
\end{aligned}$$ where $f_{{\eta}_{q}}$ and $f_{{\eta}_{s}}$ are the decay constants.
The amplitudes for the $\overline{B}_{q}^{\ast}$ ${\to}$ $D{\eta}$, $D{\eta}^{\prime}$ decays can be written as $$\begin{aligned}
{\cal A}(\overline{B}_{q}^{\ast}{\to}D{\eta})
&=&
{\cos}{\theta}_{P}\,{\cal A}(\overline{B}_{q}^{\ast}{\to}D{\eta}_{q})
-{\sin}{\theta}_{P}\,{\cal A}(\overline{B}_{q}^{\ast}{\to}D{\eta}_{s})
\label{amp-bu-eta}, \\
%------------------------ D eta^prime
{\cal A}(\overline{B}_{q}^{\ast}{\to}D{\eta}^{\prime})
&=&
{\sin}{\theta}_{P}\,{\cal A}(\overline{B}_{q}^{\ast}{\to}D{\eta}_{q})
+{\cos}{\theta}_{P}\,{\cal A}(\overline{B}_{q}^{\ast}{\to}D{\eta}_{s})
\label{amp-bu-etap}.
\end{aligned}$$
Numerical results and discussion {#sec03}
================================
In the rest frame of the $\overline{B}_{q}^{\ast}$ meson, the branching ratio is defined as $${\cal B}r(\overline{B}_{q}^{\ast}{\to}DV)\, =\, \frac{1}{24{\pi}}\,
\frac{p}{m_{B^{\ast}_{q}}^{2}{\Gamma}_{B^{\ast}_{q}}}\, \Big\{
{\vert}H_{0}{\vert}^{2}+{\vert}H_{\parallel}{\vert}^{2}
+{\vert}H_{\perp}{\vert}^{2} \Big\}
\label{br-dv},$$ $${\cal B}r(\overline{B}_{q}^{\ast}{\to}DP)\, =\, \frac{1}{24{\pi}}\,
\frac{p}{m_{B^{\ast}_{q}}^{2}{\Gamma}_{B^{\ast}_{q}}}\,
{\vert}{\cal A}(\overline{B}_{q}^{\ast}{\to}DP){\vert}^{2}
\label{br-dp},$$ where ${\Gamma}_{B^{\ast}_{q}}$ is the full decay width of the $\overline{B}_{q}^{\ast}$ meson.
Unfortunately, the experimental data on ${\Gamma}_{B^{\ast}_{q}}$ are still unavailable until now. As is generally known, the electromagnetic radiation processes $\overline{B}^{\ast}_{q}$ ${\to}$ $\overline{B}_{q}{\gamma}$ dominate the $\overline{B}^{\ast}_{q}$ meson decays, and the mass differences between the $\overline{B}^{\ast}_{q}$ and $\overline{B}_{q}$ mesons are very small, $m_{B_{q}^{\ast}}$ $-$ $m_{B_{q}}$ ${\lesssim}$ 50 MeV [@pdg], which results in the fact that the photons from the $\overline{B}^{\ast}_{q}$ ${\to}$ $\overline{B}_{q}{\gamma}$ process are too soft to be easily identified by the detectors at the existing experiments. A good approximation for the decay width is ${\Gamma}_{B_{q}^{\ast}}$ ${\approx}$ ${\Gamma}(\overline{B}_{q}^{\ast}{\to}\overline{B}_{q}{\gamma})$. Theoretically, there is the close relation between the partial decay width for the $\overline{B}_{q}^{\ast}$ ${\to}$ $\overline{B}_{q}{\gamma}$ decay and the magnetic dipole (M1) moment of the $\overline{B}_{q}^{\ast}$ meson [@epja52.90], i.e., $${\Gamma}(\overline{B}_{q}^{\ast}{\to}\overline{B}_{q}{\gamma})\, =\,
\frac{4}{3}\,{\alpha}_{\rm em}\, k_{\gamma}^{3}\, {\mu}^{2}_{h}
\label{m1-width},$$ where ${\alpha}_{\rm em}$ is the fine structure constant; $k_{\gamma}$ $=$ $(m_{B_{q}^{\ast}}^{2}-m_{B_{q}}^{2})/2m_{B_{q}^{\ast}}$ is the photon momentum in the rest frame of the $\overline{B}_{q}^{\ast}$ meson; ${\mu}_{h}$ is the M1 moment of the $\overline{B}_{q}^{\ast}$ meson. There are a large number of theoretical predictions on the partial decay width ${\Gamma}(\overline{B}_{q}^{\ast}{\to}\overline{B}_{q}{\gamma})$. Many of these have been collected into Table 7 of Ref.[@jhep1404.177] and Tables 3 and 4 of Ref.[@epja52.90]. However, there are big differences among these estimations with various models, due to our inaccurate information about the M1 moments of mesons. In principle, the M1 moment of a hadron should be the sum of the M1 moments of its constituent quarks. As is well known, for an elementary particle, the M1 moment is proportional to the charge and inversely proportional to the mass. Hence, the M1 moment of the heavy-light $\overline{B}_{q}^{\ast}$ meson should be mainly affected by the M1 moment of the light quark rather than the bottom quark. With the M1 moments of the light $u$, $d$ and $s$ quarks in the terms of the nuclear magnetons ${\mu}_{N}$, i.e., ${\mu}_{u}$ ${\simeq}$ $1.85\,{\mu}_{N}$, ${\mu}_{d}$ ${\simeq}$ $-0.97\,{\mu}_{N}$, and ${\mu}_{s}$ ${\simeq}$ $-0.61\,{\mu}_{N}$ [@uds], it is expected to have the relations ${\Gamma}(\overline{B}_{u}^{\ast}{\to}\overline{B}_{u}{\gamma})$ $>$ ${\Gamma}(\overline{B}_{d}^{\ast}{\to}\overline{B}_{d}{\gamma})$ $>$ ${\Gamma}(\overline{B}_{s}^{\ast}{\to}\overline{B}_{s}{\gamma})$, and therefore the relations ${\Gamma}_{B_{u}^{\ast}}$ $>$ ${\Gamma}_{B_{d}^{\ast}}$ $>$ ${\Gamma}_{B_{s}^{\ast}}$. It is far beyond the scope of this paper to elaborate more on the details of the decay width ${\Gamma}_{B_{q}^{\ast}}$. In our calculation, in order to give a quantitative estimation of the branching ratios for the $\overline{B}_{q}^{\ast}$ ${\to}$ $DM$ decays, we will use the following values of the decay widths, $${\Gamma}_{B_{u}^{\ast}}\ {\sim}\
{\Gamma}(\overline{B}_{u}^{\ast}{\to}\overline{B}_{u}{\gamma})\ {\sim}\
450\,\text{eV}
\label{m1-width-u},$$ $${\Gamma}_{B_{d}^{\ast}}\ {\sim}\
{\Gamma}(\overline{B}_{d}^{\ast}{\to}\overline{B}_{d}{\gamma})\ {\sim}\
150\,\text{eV}
\label{m1-width-d},$$ $${\Gamma}_{B_{s}^{\ast}}\ {\sim}\
{\Gamma}(\overline{B}_{s}^{\ast}{\to}\overline{B}_{s}{\gamma})\ {\sim}\
100\,\text{eV}
\label{m1-width-s},$$ which is basically consistent with the recent results of Ref.[@epja52.90].
CKM parameter $A$ $=$ $0.811{\pm}0.026$ [@pdg], ${\lambda}$ $=$ $0.22506{\pm}0.00050$ [@pdg],
---------------------------------------------------------------------- --------------------------------------------------------------------- ----------------------------------------------------------------------------
mass of the particles $m_{{\pi}^{\pm}}$ $=$ $139.57$ MeV [@pdg], $m_{K^{\pm}}$ $=$ $493.677{\pm}0.016$ MeV [@pdg],
$m_{b}$ $=$ $4.78{\pm}0.06$ GeV [@pdg], $m_{{\pi}^{0}}$ $=$ $134.98$ MeV [@pdg], $m_{K^{0}}$ $=$ $497.611{\pm}0.013$ MeV [@pdg],
$m_{c}$ $=$ $1.67{\pm}0.07$ GeV [@pdg], $m_{{\eta}^{\prime}}$ $=$ $957.78{\pm}0.06$ MeV [@pdg], $m_{\eta}$ $=$ $547.862{\pm}0.017$ MeV [@pdg],
$m_{s}$ ${\simeq}$ $0.51$ GeV [@uds], $m_{\rho}$ $=$ $775.26{\pm}0.25$ MeV [@pdg], $m_{K^{{\ast}0}}$ $=$ $895.81{\pm}0.19$ MeV [@pdg],
$m_{u,d}$ ${\simeq}$ $0.31$ GeV [@uds], $m_{\omega}$ $=$ $782.62{\pm}0.12$ MeV [@pdg], $m_{K^{{\ast}{\pm}}}$ $=$ $891.66{\pm}0.26$ MeV [@pdg],
$m_{B_{s}^{\ast}}$ $=$ $5415.4^{+1.8}_{-1.5}$ MeV [@pdg], $m_{B_{u,d}^{\ast}}$ $=$ $5324.65{\pm}0.25$ MeV [@pdg], $m_{\phi}$ $=$ $1019.461{\pm}0.019$ MeV [@pdg],
$m_{D_{s}}$ $=$ $1968.27{\pm}0.10$ MeV [@pdg], $m_{D_{d}}$ $=$ $1869.58{\pm}0.09$ MeV [@pdg], $m_{D_{u}}$ $=$ $1864.83{\pm}0.05$ MeV [@pdg],
decay constant $f_{\pi}$ $=$ $130.2{\pm}1.7$ MeV [@pdg], $f_{K}$ $=$ $155.6{\pm}0.4$ MeV [@pdg],
$f_{{\eta}_{q}}$ $=$ $(1.07{\pm}0.02)\,f_{\pi}$ [@prd58.114006], $f_{K^{\ast}}$ $=$ $220{\pm}5$ MeV [@jhep0703.069], $f_{K^{\ast}}^{T}$ $=$ $185{\pm}10$ MeV [@jhep0703.069],
$f_{{\eta}_{s}}$ $=$ $(1.34{\pm}0.06)\,f_{\pi}$ [@prd58.114006], $f_{\rho}$ $=$ $216{\pm}3$ MeV [@jhep0703.069], $f_{\rho}^{T}$ $=$ $165{\pm}9$ MeV [@jhep0703.069],
$f_{D_{s}}$ $=$ $249.0{\pm}1.2$ MeV [@pdg], $f_{\omega}$ $=$ $187{\pm}5$ MeV [@jhep0703.069], $f_{\omega}^{T}$ $=$ $151{\pm}9$ MeV [@jhep0703.069],
$f_{D_{u,d}}$ $=$ $211.9{\pm}1.1$ MeV [@pdg], $f_{\phi}$ $=$ $215{\pm}5$ MeV [@jhep0703.069], $f_{\phi}^{T}$ $=$ $186{\pm}9$ MeV [@jhep0703.069],
$f_{B_{s}^{\ast}}$ $=$ $213{\pm}7$ MeV [@prd91.114509], $f_{B_{u,d}^{\ast}}$ $=$ $175{\pm}6$ MeV [@prd91.114509],
Gegenbauer moment $a_{2}^{{\pi},{\eta}_{q,s}}$ $=$ $0.25{\pm}0.15$ [@jhep0605.004], $a_{2}^{{\parallel},{\rho},{\omega}}$ $=$ $0.15{\pm}0.07$ [@jhep0703.069],
$a_{1}^{K}$ $=$ $-0.06{\pm}0.03$ [@jhep0605.004], $a_{2}^{K}$ $=$ $0.25{\pm}0.15$ [@jhep0605.004], $a_{2}^{{\perp},{\rho},{\omega}}$ $=$ $0.14{\pm}0.06$ [@jhep0703.069],
$a_{1}^{{\parallel},K^{\ast}}$ $=$ $-0.03{\pm}0.02$ [@jhep0703.069], $a_{2}^{{\parallel},K^{\ast}}$ $=$ $0.11{\pm}0.09$ [@jhep0703.069], $a_{2}^{{\parallel},{\phi}}$ $=$ $0.18{\pm}0.08$ [@jhep0703.069],
$a_{1}^{{\perp},K^{\ast}}$ $=$ $-0.04{\pm}0.03$ [@jhep0703.069], $a_{2}^{{\perp},K^{\ast}}$ $=$ $0.10{\pm}0.08$ [@jhep0703.069], $a_{2}^{{\perp},{\phi}}$ $=$ $0.14{\pm}0.07$ [@jhep0703.069].
: The numerical values of input parameters.[]{data-label="tab:input"}
decay mode class unit I II III
------------------------------------------------------------------ ------- ------------ --------------------------------------------------- --------------------------------------------------- ---------------------------------------------------
$B_{u}^{{\ast}-}$ ${\to}$ $D_{u}^{0}{\pi}^{-}$ T-I $10^{-10}$ $ 6.61^{+ 2.12+ 0.07+ 0.75}_{- 0.92- 0.79- 0.69}$ $ 1.25^{+ 0.25+ 0.11+ 0.16}_{- 0.13- 0.13- 0.15}$ $ 0.56^{+ 0.13+ 0.09+ 0.08}_{- 0.07- 0.09- 0.07}$
$B_{u}^{{\ast}-}$ ${\to}$ $D_{u}^{0}K^{-}$ T-II $10^{-11}$ $ 5.38^{+ 1.97+ 0.05+ 0.55}_{- 0.85- 0.63- 0.51}$ $ 0.95^{+ 0.20+ 0.09+ 0.11}_{- 0.10- 0.10- 0.10}$ $ 0.42^{+ 0.10+ 0.07+ 0.05}_{- 0.05- 0.07- 0.05}$
$\overline{B}_{d}^{{\ast}0}$ ${\to}$ $D_{d}^{+}{\pi}^{-}$ T-I $10^{ -9}$ $ 2.22^{+ 0.52+ 0.02+ 0.27}_{- 0.20- 0.23- 0.24}$ $ 0.51^{+ 0.08+ 0.04+ 0.07}_{- 0.03- 0.05- 0.06}$ $ 0.28^{+ 0.04+ 0.03+ 0.04}_{- 0.02- 0.04- 0.03}$
$\overline{B}_{d}^{{\ast}0}$ ${\to}$ $D_{d}^{+}K^{-}$ T-II $10^{-10}$ $ 1.69^{+ 0.38+ 0.01+ 0.15}_{- 0.15- 0.18- 0.14}$ $ 0.38^{+ 0.06+ 0.03+ 0.03}_{- 0.02- 0.03- 0.03}$ $ 0.20^{+ 0.03+ 0.03+ 0.02}_{- 0.01- 0.03- 0.02}$
$\overline{B}_{d}^{{\ast}0}$ ${\to}$ $D_{u}^{0}{\pi}^{0}$ C-I $10^{-12}$ $ 4.04^{+ 3.19+ 0.38+ 1.32}_{- 2.47- 0.39- 1.05}$ $ 6.11^{+ 0.81+ 0.32+ 1.72}_{- 0.44- 0.29- 1.28}$ $ 7.21^{+ 1.02+ 0.13+ 1.66}_{- 0.88- 0.09- 1.35}$
$\overline{B}_{d}^{{\ast}0}$ ${\to}$ $D_{u}^{0}{\eta}$ C-I $10^{-12}$ $ 3.48^{+ 2.78+ 0.21+ 0.80}_{- 2.29- 0.11- 1.28}$ $ 4.49^{+ 0.63+ 0.36+ 0.98}_{- 0.43- 0.25- 0.84}$ $ 3.60^{+ 0.73+ 0.22+ 0.68}_{- 0.71- 0.16- 0.60}$
$\overline{B}_{d}^{{\ast}0}$ ${\to}$ $D_{u}^{0}{\eta}^{\prime}$ C-I $10^{-12}$ $ 2.26^{+ 1.80+ 0.13+ 0.67}_{- 1.49- 0.07- 0.72}$ $ 2.91^{+ 0.41+ 0.23+ 0.68}_{- 0.28- 0.16- 0.56}$ $ 2.33^{+ 0.47+ 0.14+ 0.49}_{- 0.46- 0.10- 0.41}$
$\overline{B}_{d}^{{\ast}0}$ ${\to}$ $D_{u}^{0}\overline{K}^{0}$ C-II $10^{-13}$ $ 2.19^{+ 4.55+ 0.72+ 2.30}_{- 0.44- 0.51- 1.40}$ $ 9.66^{+ 1.59+ 0.72+ 2.86}_{- 1.16- 0.50- 2.14}$ $ 9.54^{+ 1.94+ 0.39+ 1.96}_{- 1.92- 0.27- 1.72}$
$\overline{B}_{d}^{{\ast}0}$ ${\to}$ $D_{s}^{+}K^{-}$ A-I $10^{-12}$ $ 0.64^{+ 0.05+ 0.17+ 1.09}_{- 0.04- 0.16- 0.42}$ $ 1.30^{+ 0.06+ 0.10+ 0.89}_{- 0.01- 0.09- 0.40}$ $ 0.40^{+ 0.05+ 0.03+ 0.20}_{- 0.01- 0.03- 0.08}$
$\overline{B}_{s}^{{\ast}0}$ ${\to}$ $D_{s}^{+}{\pi}^{-}$ T-I $10^{ -9}$ $ 5.68^{+ 1.17+ 0.09+ 0.60}_{- 0.46- 0.50- 0.56}$ $ 1.48^{+ 0.21+ 0.07+ 0.16}_{- 0.09- 0.12- 0.15}$ $ 0.81^{+ 0.11+ 0.07+ 0.09}_{- 0.05- 0.09- 0.08}$
$\overline{B}_{s}^{{\ast}0}$ ${\to}$ $D_{s}^{+}K^{-}$ T-II $10^{-10}$ $ 4.30^{+ 0.89+ 0.09+ 0.40}_{- 0.35- 0.37- 0.38}$ $ 1.17^{+ 0.17+ 0.06+ 0.12}_{- 0.07- 0.09- 0.11}$ $ 0.64^{+ 0.09+ 0.06+ 0.06}_{- 0.04- 0.07- 0.06}$
$\overline{B}_{s}^{{\ast}0}$ ${\to}$ $D_{d}^{+}{\pi}^{-}$ A-II $10^{-14}$ $ 0.16^{+ 0.17+ 0.75+ 2.36}_{- 0.02- 0.15- 0.08}$ $ 1.71^{+ 0.31+ 0.48+ 2.44}_{- 0.05- 0.46- 0.40}$ $ 1.80^{+ 0.58+ 0.10+ 2.08}_{- 0.24- 0.10- 0.58}$
$\overline{B}_{s}^{{\ast}0}$ ${\to}$ $D_{u}^{0}{\pi}^{0}$ A-II $10^{-14}$ $ 0.08^{+ 0.09+ 0.38+ 1.19}_{- 0.01- 0.08- 0.04}$ $ 0.86^{+ 0.16+ 0.24+ 1.23}_{- 0.03- 0.23- 0.20}$ $ 0.90^{+ 0.29+ 0.05+ 1.05}_{- 0.12- 0.05- 0.29}$
$\overline{B}_{s}^{{\ast}0}$ ${\to}$ $D_{u}^{0}{\eta}$ C-II $10^{-12}$ $ 0.20^{+ 0.38+ 0.06+ 0.15}_{- 0.02- 0.06- 0.09}$ $ 0.90^{+ 0.16+ 0.05+ 0.26}_{- 0.12- 0.07- 0.20}$ $ 1.01^{+ 0.21+ 0.01+ 0.27}_{- 0.21- 0.03- 0.22}$
$\overline{B}_{s}^{{\ast}0}$ ${\to}$ $D_{u}^{0}{\eta}^{\prime}$ C-II $10^{-12}$ $ 0.36^{+ 0.72+ 0.10+ 0.21}_{- 0.08- 0.08- 0.16}$ $ 1.38^{+ 0.24+ 0.08+ 0.31}_{- 0.19- 0.09- 0.31}$ $ 1.22^{+ 0.30+ 0.02+ 0.23}_{- 0.30- 0.06- 0.26}$
$\overline{B}_{s}^{{\ast}0}$ ${\to}$ $D_{u}^{0}K^{0}$ C-I $10^{-11}$ $ 3.31^{+ 2.07+ 0.12+ 1.09}_{- 1.88- 0.24- 0.77}$ $ 4.31^{+ 0.51+ 0.16+ 1.22}_{- 0.28- 0.25- 0.93}$ $ 4.21^{+ 0.61+ 0.02+ 0.94}_{- 0.53- 0.16- 0.80}$
: The branching ratios for the $\overline{B}_{q}^{\ast}$ ${\to}$ $DP$ decays with the different DA scenarios, where the theoretical uncertainties come from the scale $(1{\pm}0.1)t_{i}$, the mass $m_{c}$ and $m_{b}$, and the hadronic parameters (including the decay constants, Gegenbauer moments, and so on).[]{data-label="tab:br-dp"}
decay mode class unit Scenario I Scenario II Scenario III
------------------------------------------------------------------------ ------- ------------ --------------------------------------------------- --------------------------------------------------- ---------------------------------------------------
$B_{u}^{{\ast}-}$ ${\to}$ $D_{u}^{0}{\rho}^{-}$ T-I $10^{ -9}$ $ 2.02^{+ 0.55+ 0.02+ 0.23}_{- 0.23- 0.22- 0.21}$ $ 0.43^{+ 0.08+ 0.04+ 0.05}_{- 0.04- 0.04- 0.05}$ $ 0.19^{+ 0.04+ 0.03+ 0.02}_{- 0.02- 0.03- 0.02}$
$B_{u}^{{\ast}-}$ ${\to}$ $D_{u}^{0}K^{{\ast}-}$ T-II $10^{-10}$ $ 1.14^{+ 0.32+ 0.01+ 0.16}_{- 0.13- 0.13- 0.14}$ $ 0.25^{+ 0.05+ 0.02+ 0.04}_{- 0.02- 0.03- 0.03}$ $ 0.11^{+ 0.02+ 0.02+ 0.02}_{- 0.01- 0.02- 0.01}$
$\overline{B}_{d}^{{\ast}0}$ ${\to}$ $D_{d}^{+}{\rho}^{-}$ T-I $10^{ -9}$ $ 6.80^{+ 1.55+ 0.03+ 0.79}_{- 0.60- 0.66- 0.72}$ $ 1.72^{+ 0.26+ 0.11+ 0.21}_{- 0.11- 0.15- 0.19}$ $ 0.95^{+ 0.13+ 0.11+ 0.11}_{- 0.06- 0.11- 0.10}$
$\overline{B}_{d}^{{\ast}0}$ ${\to}$ $D_{d}^{+}K^{{\ast}-}$ T-II $10^{-10}$ $ 3.87^{+ 0.87+ 0.01+ 0.51}_{- 0.33- 0.36- 0.46}$ $ 0.99^{+ 0.15+ 0.07+ 0.13}_{- 0.06- 0.09- 0.12}$ $ 0.53^{+ 0.07+ 0.07+ 0.07}_{- 0.03- 0.07- 0.07}$
$\overline{B}_{d}^{{\ast}0}$ ${\to}$ $D_{u}^{0}{\rho}^{0}$ C-I $10^{-11}$ $ 2.55^{+ 1.26+ 0.15+ 0.57}_{- 1.06- 0.09- 0.47}$ $ 4.60^{+ 0.61+ 0.15+ 0.84}_{- 0.44- 0.09- 0.72}$ $ 5.95^{+ 0.79+ 0.23+ 1.14}_{- 0.76- 0.19- 0.98}$
$\overline{B}_{d}^{{\ast}0}$ ${\to}$ $D_{u}^{0}{\omega}$ C-I $10^{-11}$ $ 2.43^{+ 1.00+ 0.17+ 0.75}_{- 0.79- 0.11- 0.60}$ $ 4.32^{+ 0.51+ 0.21+ 1.11}_{- 0.35- 0.15- 0.91}$ $ 4.71^{+ 0.64+ 0.16+ 1.14}_{- 0.62- 0.14- 0.94}$
$\overline{B}_{d}^{{\ast}0}$ ${\to}$ $D_{u}^{0}\overline{K}^{{\ast}0}$ C-II $10^{-12}$ $ 3.58^{+ 1.77+ 0.22+ 1.14}_{- 1.52- 0.16- 0.89}$ $ 6.26^{+ 0.90+ 0.32+ 1.70}_{- 0.70- 0.25- 1.40}$ $ 7.83^{+ 1.16+ 0.28+ 2.17}_{- 1.16- 0.29- 1.76}$
$\overline{B}_{d}^{{\ast}0}$ ${\to}$ $D_{s}^{+}K^{{\ast}-}$ A-I $10^{-12}$ $ 5.55^{+ 0.94+ 0.50+ 2.57}_{- 0.46- 0.48- 1.68}$ $ 6.82^{+ 0.58+ 0.54+ 2.09}_{- 0.13- 0.49- 1.43}$ $ 3.98^{+ 0.54+ 0.10+ 1.19}_{- 0.18- 0.08- 0.78}$
$\overline{B}_{s}^{{\ast}0}$ ${\to}$ $D_{s}^{+}{\rho}^{-}$ T-I $10^{ -8}$ $ 1.72^{+ 0.35+ 0.03+ 0.19}_{- 0.14- 0.13- 0.17}$ $ 0.50^{+ 0.07+ 0.03+ 0.05}_{- 0.03- 0.04- 0.05}$ $ 0.27^{+ 0.04+ 0.02+ 0.03}_{- 0.02- 0.03- 0.03}$
$\overline{B}_{s}^{{\ast}0}$ ${\to}$ $D_{s}^{+}K^{{\ast}-}$ T-II $10^{ -9}$ $ 1.00^{+ 0.21+ 0.02+ 0.13}_{- 0.08- 0.09- 0.12}$ $ 0.31^{+ 0.05+ 0.02+ 0.04}_{- 0.02- 0.03- 0.04}$ $ 0.17^{+ 0.02+ 0.01+ 0.02}_{- 0.01- 0.02- 0.02}$
$\overline{B}_{s}^{{\ast}0}$ ${\to}$ $D_{d}^{+}{\rho}^{-}$ A-II $10^{-13}$ $ 2.39^{+ 0.79+ 0.47+ 0.90}_{- 0.31- 0.20- 0.53}$ $ 6.23^{+ 0.26+ 0.86+ 1.27}_{- 0.00- 0.72- 0.91}$ $ 2.88^{+ 0.31+ 0.13+ 0.50}_{- 0.04- 0.11- 0.36}$
$\overline{B}_{s}^{{\ast}0}$ ${\to}$ $D_{u}^{0}{\rho}^{0}$ A-II $10^{-13}$ $ 1.19^{+ 0.40+ 0.24+ 0.45}_{- 0.15- 0.10- 0.27}$ $ 3.12^{+ 0.13+ 0.43+ 0.63}_{- 0.00- 0.36- 0.45}$ $ 1.44^{+ 0.15+ 0.06+ 0.25}_{- 0.02- 0.05- 0.18}$
$\overline{B}_{s}^{{\ast}0}$ ${\to}$ $D_{u}^{0}{\omega}$ A-II $10^{-13}$ $ 0.97^{+ 0.33+ 0.19+ 0.37}_{- 0.13- 0.08- 0.22}$ $ 2.42^{+ 0.11+ 0.34+ 0.55}_{- 0.00- 0.28- 0.40}$ $ 1.12^{+ 0.12+ 0.05+ 0.23}_{- 0.02- 0.04- 0.17}$
$\overline{B}_{s}^{{\ast}0}$ ${\to}$ $D_{u}^{0}{\phi}$ C-II $10^{-11}$ $ 0.79^{+ 0.32+ 0.03+ 0.18}_{- 0.30- 0.04- 0.15}$ $ 1.20^{+ 0.17+ 0.04+ 0.26}_{- 0.13- 0.04- 0.22}$ $ 1.50^{+ 0.22+ 0.02+ 0.31}_{- 0.21- 0.04- 0.27}$
$\overline{B}_{s}^{{\ast}0}$ ${\to}$ $D_{u}^{0}K^{{\ast}0}$ C-I $10^{-10}$ $ 1.69^{+ 0.67+ 0.07+ 0.51}_{- 0.61- 0.09- 0.41}$ $ 2.58^{+ 0.32+ 0.07+ 0.70}_{- 0.23- 0.09- 0.57}$ $ 3.19^{+ 0.42+ 0.05+ 0.87}_{- 0.40- 0.11- 0.71}$
: The branching ratios for the $\overline{B}_{q}^{\ast}$ ${\to}$ $DV$ decays with the different DA scenarios, where the theoretical uncertainties arise from the scale $(1{\pm}0.1)t_{i}$, the mass $m_{c}$ and $m_{b}$, and the hadronic parameters (including the decay constants, Gegenbauer moments,and so on).[]{data-label="tab:br-dv"}
The numerical values of other input parameters are collected in Table \[tab:input\], where their central values will be fixed as the default inputs unless otherwise specified. In addition, in order to investigate the effects from different DA models, we explore three scenarios,
- Scenario I: Eqs.(\[da-bqlv\]-\[da-cqp\]) for the DAs of ${\phi}_{B^{\ast}}^{v,t,V,T}$ and ${\phi}_{D_{q}}^{a,p}$;
- Scenario II: ${\phi}_{B^{\ast}}^{v,t,V,T}$ $=$ Eq.(\[da-bqlv\]), and ${\phi}_{D_{q}}^{a,p}$ $=$ Eq.(\[da-cqa\]);
- Scenario III: ${\phi}_{B^{\ast}}^{v,t,V,T}$ $=$ Eq.(\[da-bqlv\]), and ${\phi}_{D_{q}}^{a,p}$ $=$ Eq.(\[wave-d-xb\]).
Our numerical results on the branching ratios are presented in Tables \[tab:br-dp\] and \[tab:br-dv\], where the uncertainties come from the typical scale $(1{\pm}0.1)t_{i}$, the mass $m_{c}$ and $m_{b}$, and the hadronic parameters (including the decay constants, Gegenbauer moments, and so on), respectively. The following are some comments.
\(1) Generally, the $\overline{B}_{q}^{\ast}$ ${\to}$ $DP$ decay modes could be divided into three categories, i.e. the “T”, “C”, and “A” types are dominated by contributions from the color-allowed emission topologies of Fig.\[fig:fey-t\], the color-suppressed emission topologies of Fig.\[fig:fey-c\], and the pure annihilation topologies of Fig.\[fig:fey-a\], respectively. And each category could be further divided into two classes, i.e., the decay amplitudes of the classes “I” and “II” are proportional to the CKM factors of $V_{cb}\,V_{ud}^{\ast}$ ${\sim}$ $A{\lambda}^{2}$ and $V_{cb}\,V_{us}^{\ast}$ ${\sim}$ $A{\lambda}^{3}$, respectively. There are many hierarchical relations among the branching ratios, such as, $${\cal B}r(\text{class T-I}) >
{\cal B}r(\text{class C-I}) >
{\cal B}r(\text{class A-I})
\label{r-01-01},$$ $${\cal B}r(\text{class T-II}) >
{\cal B}r(\text{class C-II}) >
{\cal B}r(\text{class A-II})
\label{r-01-02},$$ $${\cal B}r(\text{class X-I}) >
{\cal B}r(\text{class X-II}), \quad
\text{for X\,=\,T,\,C,\,A}
\label{r-01-03}.$$ These categories and relations also happen to hold true for the $\overline{B}_{q}^{\ast}$ ${\to}$ $DV$ decays.
For the “T” and “C” types of the $\overline{B}_{q}^{\ast}$ ${\to}$ $DM$ decays, the annihilation contributions have a negligible impact on the branching ratios, and they are strongly suppressed relative to the emission contributions, as is stated by the QCDF approach [@npb591.313].
For the “T” types of the $\overline{B}_{q}^{\ast}$ ${\to}$ $DM$ decays, the factorizable contributions from the emission topologies to the branching ratios are dominant over other contributions. However, for the “C” types of the $\overline{B}_{q}^{\ast}$ ${\to}$ $DM$ decays, the nonfactorizable contributions to the branching ratios become very important, and sometimes even dominant.
\(2) With the law of conservation of angular momentum, three partial wave amplitudes, including the $s$-, $p$-, and $d$-wave amplitudes, all will contribute to the $\overline{B}_{q}^{\ast}$ ${\to}$ $DV$ decays, while only the $p$-wave amplitude will contribute to the $\overline{B}_{q}^{\ast}$ ${\to}$ $DP$ decays. Besides, the branching ratios are proportional to the squares of the decay constants with the pQCD approach. With the magnitude relations between the decay constants $f_{V}$ $>$ $f_{P}$, one should expect to have the general relation of the branching ratios, $${\cal B}r(\overline{B}_{q}^{\ast}{\to}DV) >
{\cal B}r(\overline{B}_{q}^{\ast}{\to}DP)
\label{r-02-01},$$ for the final vector $V$ and pseudoscalar $P$ mesons carrying the same flavor, azimuthal and magnetic isospin quantum numbers. And due to the relations between the decay constants $f_{B_{s}^{\ast}}$ $>$ $f_{B_{u,d}^{\ast}}$ and $f_{D_{s}}$ $>$ $f_{D_{u,d}}$, and the relations between the decay widths ${\Gamma}_{B_{s}^{\ast}}$ $<$ ${\Gamma}_{B_{u,d}^{\ast}}$, the color-allowed $\overline{B}_{s}^{{\ast}0}$ ${\to}$ $D_{s}^{+}{\rho}^{-}$ decay has a relatively large branching ratio.
Furthermore, our study results show that for the “T” types of the $\overline{B}_{q}^{\ast}$ ${\to}$ $DV$ decays, the contributions of the longitudinal polarization part are dominant. Take the $\overline{B}_{s}^{{\ast}0}$ ${\to}$ $D_{s}^{+}{\rho}^{-}$ decay for example, the longitudinal polarization fraction $f_{0}$ ${\equiv}$ $\frac{{\vert}H_{0}{\vert}^{2}}{{\vert}H_{0}{\vert}^{2}
+{\vert}H_{\parallel}{\vert}^{2}+{\vert}H_{\perp}{\vert}^{2}}$ ${\approx}$ $90\%$ ($85\%$), the parallel polarization fraction $f_{\parallel}$ ${\equiv}$ $\frac{{\vert}H_{0}{\vert}^{2}}{{\vert}H_{\parallel}{\vert}^{2}
+{\vert}H_{\parallel}{\vert}^{2}+{\vert}H_{\perp}{\vert}^{2}}$ ${\approx}$ $9\%$ ($12\%$), and the perpendicular polarization fraction $f_{\perp}$ ${\equiv}$ $\frac{{\vert}H_{0}{\vert}^{2}}{{\vert}H_{\perp}{\vert}^{2}
+{\vert}H_{\parallel}{\vert}^{2}+{\vert}H_{\perp}{\vert}^{2}}$ ${\approx}$ $1\%$ ($3\%$) with the DA scenarios I (II and III), which generally agree with those obtained by the QCDF approach [@epjc76.523].
\(3) As is well known, the theoretical results depend on the values of the input parameters. From the numbers in Tables \[tab:br-dp\] and \[tab:br-dv\], it is clearly seen that the main uncertainty is due to the limited knowledge of the hadron DAs, for example, the large discrepancy among the different DA scenarios. Besides the theoretical uncertainties listed in Tables \[tab:br-dp\] and \[tab:br-dv\], the CKM parameters will bring some 6% uncertainties. With a different value of the decay width ${\Gamma}_{B_{q}^{\ast}}$, the branching ratios in Tables \[tab:br-dp\] and \[tab:br-dv\] should be multiplied by the factors of ${450\,{\rm eV}}/{{\Gamma}_{B_{u}^{\ast}}}$, ${150\,{\rm eV}}/{{\Gamma}_{B_{d}^{\ast}}}$, ${100\,{\rm eV}}/{{\Gamma}_{B_{s}^{\ast}}}$ for the $B_{u}^{\ast}$, $B_{d}^{\ast}$, $B_{s}^{\ast}$ weak decays, respectively. To reduce the theoretical uncertainties, one of the commonly used methods is to exploit the rate of the branching ratios, such as, $$\frac{{\cal B}r(\overline{B}_{u}^{{\ast}-}{\to}D_{u}^{0}{\pi}^{-})}
{{\cal B}r(\overline{B}_{u}^{{\ast}-}{\to}D_{u}^{0}K^{-})}
\ {\approx}\ \frac{f_{\pi}^{2}}{{\lambda}^{2}\,f_{K}^{2}}
\label{r-03-01},$$ $$\frac{{\cal B}r(\overline{B}_{u}^{{\ast}-}{\to}D_{u}^{0}{\rho}^{-})}
{{\cal B}r(\overline{B}_{u}^{{\ast}-}{\to}D_{u}^{0}K^{{\ast}-})}
\ {\approx}\ \frac{f_{\rho}^{2}}{{\lambda}^{2}\,f_{K^{\ast}}^{2}}
\label{r-03-02},$$ $$\frac{{\cal B}r(\overline{B}_{s}^{{\ast}0}{\to}D_{u}^{0}{\phi})}
{{\cal B}r(\overline{B}_{s}^{{\ast}0}{\to}D_{u}^{0}K^{{\ast}0})}
\ {\approx}\ \frac{{\lambda}^{2}\,f_{\phi}^{2}}{f_{K^{\ast}}^{2}}
\label{r-03-03}.$$ (4) The branching ratios for the $\overline{B}_{q}^{\ast}$ ${\to}$ $DM$ decays are smaller by at least five orders of magnitude than the branching ratios for the $\overline{B}_{q}$ ${\to}$ $DM$ decays [@prd78.014018]. This fact implies that the possible background from the $\overline{B}_{q}^{\ast}$ ${\to}$ $DM$ decays could be safely neglected when the $\overline{B}_{q}$ ${\to}$ $DM$ decays were analyzed, but not vice versa, i.e., one of main pollution backgrounds for the $\overline{B}_{q}^{\ast}$ ${\to}$ $DM$ decays would come from the $\overline{B}_{q}$ ${\to}$ $DM$ decays, even if the invariant mass of the $DM$ meson pair could be used to distinguish the $\overline{B}_{q}^{\ast}$ meson from the $\overline{B}_{q}$ meson experimentally.
channels %/$b\bar{b}$ event %/$B_{s}$ event
-------------------------------------------------- ------------------------------ ------------------------------
All $B_{s}$ events $19.5^{+3.0}_{-2.3}$
$B_{s}^{{\ast}0}\overline{B}_{s}^{{\ast}0}$ $90.1^{+3.8}_{-4.0}{\pm}0.2$
$B_{s}^{{\ast}0}\overline{B}_{s}^{0}+{\rm c.c.}$ $7.3^{+3.3}_{-3.0}{\pm}0.1$
$B^{\ast}\overline{B}^{\ast}$ $37.5^{+2.1}_{-1.9}{\pm}3.0$
$B^{\ast}\overline{B}+{\rm c.c.}$ $13.7{\pm}1.3{\pm}1.1$
$B^{\ast}\overline{B}{\pi}+{\rm c.c.}$ $7.3^{+2.3}_{-2.1}{\pm}0.8$
$B^{\ast}\overline{B}^{\ast}{\pi}$ $1.0^{+1.4}_{-1.3}{\pm}0.4$
: The channel fractions at the ${\Upsilon}(5S)$ resonance [@epjc74.3026].[]{data-label="tab:bb-fr"}
\(5) The event numbers of the $B_{q}^{\ast}$ meson in a data sample can be calculated by the following formula, $$N(B_{q}^{\ast})\ =\ {\cal L}_{\rm int}\,
{\times}\, {\sigma}_{b\bar{b}}\,{\times}\,f_{B_{q}}\, {\times}\,
\frac{ f_{B_{q}^{\ast}} } { f_{B_{q}} }
\label{r-05-01}.$$ $$f_{B_{q}^{\ast}}\ =\ 2\,{\times}\,f_{B_{q}^{\ast}\overline{B}_{q}^{\ast}}
+2\,{\times}\,f_{B_{q}^{\ast}\overline{B}_{q}^{\ast}{\pi}}
+f_{B_{q}^{\ast}\overline{B}_{q}+{\rm c.c.}}
+f_{B_{q}^{\ast}\overline{B}_{q}{\pi}+{\rm c.c}}
+{\cdots}
\label{r-02-02},$$ where ${\cal L}_{\rm int}$ is the integrated luminosity, ${\sigma}_{b\bar{b}}$ denotes the $b\bar{b}$ pair production cross section, $f_{B_{q}}$, $f_{B_{q}^{\ast}\overline{B}_{q}^{\ast}}$, ${\cdots}$ refer to the production fraction of all the $B_{q}$ meson, the $B_{q}^{\ast}\overline{B}_{q}^{\ast}$ meson pair, ${\cdots}$. The production fractions of specific modes at the center-of-mass of the ${\Upsilon}(5S)$ resonance [@epjc74.3026] are listed in Table \[tab:bb-fr\]. With a large production cross section of the process $e^{+}e^{-}$ ${\to}$ $b\bar{b}$ at the ${\Upsilon}(5S)$ peak ${\sigma}_{b\bar{b}}$ $=$ $(0.340{\pm}0.016)\,{\rm nb}$ [@epjc74.3026], it is expected that some $3.3{\times}10^{9}$ $B_{u,d}^{\ast}$ and $1.2{\times}10^{9}$ $B_{s}^{\ast}$ mesons could be available per $10\,{\rm ab}^{-1}$ ${\Upsilon}(5S)$ dataset. The branching ratios of the color-allowed “T-I” class $\overline{B}_{q}^{\ast}$ ${\to}$ $DM$ decays can reach up to ${\cal O}(10^{-9}$) or more, which are essentially coincident with those obtained by the QCDF approach [@epjc76.523]. Hence, a few events of the $\overline{B}_{q}^{\ast}$ ${\to}$ $D_{q}{\pi}^{-}$ and $\overline{B}_{u,d}^{\ast}$ ${\to}$ $D_{u,d}{\rho}^{-}$ decays, and dozens of the $\overline{B}_{s}^{{\ast}0}$ ${\to}$ $D_{s}^{+}{\rho}^{-}$ decay, might be available at the forthcoming SuperKEKB. At high energy hadron colliders, for example, given with the cross section at the LHCb ${\sigma}_{b\bar{b}}$ ${\approx}$ $100\,{\rm {\mu}b}$ [@pdg; @crp16.435; @prl118.052002], with a similar ratio $f_{B_{u}}$ $=$ $f_{B_{d}}$ $=$ $0.344{\pm}0.021$ and $f_{B_{s}}$ $=$ $0.115{\pm}0.013$ at Tevatron [@pdg; @1002.5012] and a similar ratio $f_{B_{q}^{\ast}}/f_{B_{q}}$ at the ${\Upsilon}(5S)$ meson [@epjc74.3026], some $9.8{\times}10^{13}$ $B_{u,d}^{\ast}$ events and $2.2{\times}10^{13}$ $B_{s}^{\ast}$ events per ${\rm ab}^{-1}$ dataset could be available at the LHCb, corresponding to more than $10^{5}$ of the $\overline{B}_{s}^{{\ast}0}$ ${\to}$ $D_{s}^{+}{\rho}^{-}$ decay events and over $10^{4}$ of the $\overline{B}_{q}^{\ast}$ ${\to}$ $D_{q}{\pi}^{-}$ and $\overline{B}_{u,d}^{\ast}$ ${\to}$ $D_{u,d}{\rho}^{-}$ decay events, which should be easily measured by the future LHCb experiments.
Summary {#sec04}
=======
Besides the dominant electromagnetic decay mode, the ground vector $B_{q}^{\ast}$ meson ($q$ $=$ $u$, $d$ and $s$) can also decay via the weak interactions within the standard model. A large amount of the $B_{q}^{\ast}$ mesons are expected to be accumulated with the running LHC and the forthcoming SuperKEKB, which makes it seemingly possible to explore the $B_{q}^{\ast}$ meson weak decays experimentally. The theoretical study is necessary to offer a ready reference. In this paper, we investigated the $\overline{B}_{q}^{\ast}$ ${\to}$ $DP$, $DV$ decays with the phenomenological pQCD approach. It is found that the color-allowed $\overline{B}_{q}^{\ast}$ ${\to}$ $D_{q}{\rho}^{-}$ decays have branching ratios ${\gtrsim}$ $10^{-9}$, and should be promisingly accessible at the high luminosity experiments in the future.
Acknowledgments {#acknowledgments .unnumbered}
===============
The work is supported by the National Natural Science Foundation of China (Grant Nos. U1632109, 11547014 and 11475055).
The amplitude for the $\overline{B}_{q}^{\ast}$ ${\to}$ $DP$ decays {#amp-dp}
===================================================================
$${\cal A}(B_{u}^{{\ast}-}{\to}D_{u}^{0}{\pi}^{-})\, =\,
{\cal F}\, V_{cb}\,V_{ud}^{\ast}\, \Big\{
\sum\limits_{i}\,{\cal M}^{T}_{i,P}
+\sum\limits_{j}\,{\cal M}^{C}_{j,P} \Big\}
\label{amp:bu-d0-pim},$$
$${\cal A}(B_{u}^{{\ast}-}{\to}D_{u}^{0}K^{-}) \, =\,
{\cal F}\, V_{cb}\,V_{us}^{\ast}\, \Big\{
\sum\limits_{i}\,{\cal M}^{T}_{i,P}
+\sum\limits_{j}\,{\cal M}^{C}_{j,P} \Big\}
\label{amp:bu-d0-km},$$
$${\cal A}(\overline{B}_{d}^{{\ast}0}{\to}D_{d}^{+}{\pi}^{-}) \, =\,
{\cal F}\, V_{cb}\,V_{ud}^{\ast}\, \Big\{
\sum\limits_{i}\,{\cal M}^{T}_{i,P}
+\sum\limits_{j}\,{\cal M}^{A}_{j,P} \Big\}
\label{amp:bd-dp-pim},$$
$${\cal A}(\overline{B}_{d}^{{\ast}0}{\to}D_{d}^{+}K^{-}) \, =\,
{\cal F}\, V_{cb}\,V_{us}^{\ast}\,
\sum\limits_{i}\,{\cal M}^{T}_{i,P}
\label{amp:bd-dp-km},$$
$$\sqrt{2}\,{\cal A}(\overline{B}_{d}^{{\ast}0}{\to}D_{u}^{0}{\pi}^{0}) \, =\,
{\cal F}\, V_{cb}\,V_{ud}^{\ast}\, \Big\{
-\sum\limits_{i}\,{\cal M}^{C}_{i,P}
+\sum\limits_{j}\,{\cal M}^{A}_{j,P} \Big\}
\label{amp:bd-du-pi0},$$
$$\sqrt{2}\, {\cal A}(\overline{B}_{d}^{{\ast}0}{\to}D_{u}^{0}{\eta}_{q}) \, =\,
{\cal F}\, V_{cb}\,V_{ud}^{\ast}\, \Big\{
\sum\limits_{i}\,{\cal M}^{C}_{i,P}
+\sum\limits_{j}\,{\cal M}^{A}_{j,P} \Big\}
\label{amp:bd-du-etaq},$$
$${\cal A}(\overline{B}_{d}^{{\ast}0}{\to}D_{u}^{0}\overline{K}^{0}) \, =\,
{\cal F}\, V_{cb}\,V_{us}^{\ast}\,
\sum\limits_{i}\,{\cal M}^{C}_{i,P}
\label{amp:bd-du-k0},$$
$${\cal A}(\overline{B}_{d}^{{\ast}0}{\to}D_{s}^{+}K^{-}) \, =\,
{\cal F}\, V_{cb}\,V_{ud}^{\ast}\,
\sum\limits_{i}\,{\cal M}^{A}_{i,P}
\label{amp:bd-ds-km},$$
$${\cal A}(\overline{B}_{s}^{{\ast}0}{\to}D_{s}^{+}{\pi}^{-}) \, =\,
{\cal F}\, V_{cb}\,V_{ud}^{\ast}\,
\sum\limits_{i}\,{\cal M}^{T}_{i,P}
\label{amp:bs-ds-pim},$$
$${\cal A}(\overline{B}_{s}^{{\ast}0}{\to}D_{s}^{+}K^{-}) \, =\,
{\cal F}\, V_{cb}\,V_{us}^{\ast}\, \Big\{
\sum\limits_{i}\,{\cal M}^{T}_{i,P}
+\sum\limits_{j}\,{\cal M}^{A}_{j,P} \Big\}
\label{amp:bs-ds-km},$$
$${\cal A}(\overline{B}_{s}^{{\ast}0}{\to}D_{d}^{+}{\pi}^{-}) \, =\,
{\cal F}\, V_{cb}\,V_{us}^{\ast}\,
\sum\limits_{i}\,{\cal M}^{A}_{i,P}
\label{amp:bs-dd-pim},$$
$$\sqrt{2}\,{\cal A}(\overline{B}_{s}^{{\ast}0}{\to}D_{u}^{0}{\pi}^{0}) \, =\,
{\cal F}\, V_{cb}\,V_{us}^{\ast}\,
\sum\limits_{i}\,{\cal M}^{A}_{i,P}
\label{amp:bs-du-piz},$$
$$\sqrt{2}\,{\cal A}(\overline{B}_{s}^{{\ast}0}{\to}D_{u}^{0}{\eta}_{q}) \, =\,
{\cal F}\, V_{cb}\,V_{us}^{\ast}\,
\sum\limits_{i}\,{\cal M}^{A}_{i,P}
\label{amp:bs-du-etaq},$$
$${\cal A}(\overline{B}_{s}^{{\ast}0}{\to}D_{u}^{0}{\eta}_{s}) \, =\,
{\cal F}\, V_{cb}\,V_{us}^{\ast}\,
\sum\limits_{i}\,{\cal M}^{C}_{i,P}
\label{amp:bs-du-etas},$$
$${\cal A}(\overline{B}_{s}^{{\ast}0}{\to}D_{u}^{0}K^{0}) \, =\,
{\cal F}\, V_{cb}\,V_{ud}^{\ast}\,
\sum\limits_{i}\,{\cal M}^{C}_{i,P}
\label{amp:bs-du-kz},$$
$${\cal F}\, =\, \frac{G_{F}}{\sqrt{2}}\, \frac{{\pi}\,C_{F}}{N_{c}}\,
f_{{B}_{q}^{\ast}}\, f_{D}
\label{eq:amp-coe},$$
where ${\cal M}^{k}_{i,j}$ is the amplitude building blocks. The superscripts $k$ $=$ $T$, $C$, $A$ correspond to the color-allowed emission topologies of Fig.\[fig:fey-t\], the color-suppressed emission topologies of Fig.\[fig:fey-c\], the annihilation topologies of Fig.\[fig:fey-a\]. The subscripts $i$ $=$ $a$, $b$, $c$, $d$ correspond to the diagram indices. The subscripts $j$ $=$ $P$, $L$, $N$, $T$ correspond to the different helicity amplitudes. The analytical expressions of the amplitude building blocks ${\cal M}^{k}_{i,j}$ are given in the Appendix \[block-t\], \[block-c\], \[block-a\].
The amplitude for the $\overline{B}_{q}^{\ast}$ ${\to}$ $DV$ decays {#amp-dv}
===================================================================
$$i\,{\cal A}_{\lambda}(B_{u}^{{\ast}-}{\to}D_{u}^{0}{\rho}^{-})\, =\,
{\cal F}\, V_{cb}\,V_{ud}^{\ast}\, \Big\{
\sum\limits_{i}\,{\cal M}^{T}_{i,{\lambda}}
+\sum\limits_{j}\,{\cal M}^{C}_{j,{\lambda}} \Big\}
\label{amp:bu-d0-rhom},$$
$$i\,{\cal A}_{\lambda}(B_{u}^{{\ast}-}{\to}D_{u}^{0}K^{{\ast}-})\, =\,
{\cal F}\, V_{cb}\,V_{us}^{\ast}\, \Big\{
\sum\limits_{i}\,{\cal M}^{T}_{i,{\lambda}}
+\sum\limits_{j}\,{\cal M}^{C}_{j,{\lambda}} \Big\}
\label{amp:bu-d0-kvm},$$
$$i\,{\cal A}_{\lambda}(\overline{B}_{d}^{{\ast}0}{\to}D_{d}^{+}{\rho}^{-}) \, =\,
{\cal F}\, V_{cb}\,V_{ud}^{\ast}\, \Big\{
\sum\limits_{i}\,{\cal M}^{T}_{i,{\lambda}}
+\sum\limits_{j}\,{\cal M}^{A}_{j,{\lambda}} \Big\}
\label{amp:bd-dp-rhom},$$
$$i{\cal A}_{\lambda}(\overline{B}_{d}^{{\ast}0}{\to}D_{d}^{+}K^{{\ast}-}) \, =\,
{\cal F}\, V_{cb}\,V_{us}^{\ast}\,
\sum\limits_{i}\,{\cal M}^{T}_{i,{\lambda}}
\label{amp:bd-dp-kvm},$$
$$i\,\sqrt{2}\,{\cal A}_{\lambda}(\overline{B}_{d}^{{\ast}0}{\to}D_{u}^{0}{\rho}^{0}) \, =\,
{\cal F}\, V_{cb}\,V_{ud}^{\ast}\, \Big\{
-\sum\limits_{i}\,{\cal M}^{C}_{i,{\lambda}}
+\sum\limits_{j}\,{\cal M}^{A}_{j,{\lambda}} \Big\}
\label{amp:bd-du-rhoz},$$
$$i\,\sqrt{2}\,{\cal A}_{\lambda}(\overline{B}_{d}^{{\ast}0}{\to}D_{u}^{0}{\omega}) \, =\,
{\cal F}\, V_{cb}\,V_{ud}^{\ast}\, \Big\{
\sum\limits_{i}\,{\cal M}^{C}_{i,{\lambda}}
+\sum\limits_{j}\,{\cal M}^{A}_{j,{\lambda}} \Big\}
\label{amp:bd-du-w},$$
$$i{\cal A}_{\lambda}(\overline{B}_{d}^{{\ast}0}{\to}D_{u}^{0}\overline{K}^{{\ast}0}) \, =\,
{\cal F}\, V_{cb}\,V_{us}^{\ast}\,
\sum\limits_{i}\,{\cal M}^{C}_{i,{\lambda}}
\label{amp:bd-du-kvz},$$
$$i{\cal A}_{\lambda}(\overline{B}_{d}^{{\ast}0}{\to}D_{s}^{+}K^{{\ast}-}) \, =\,
{\cal F}\, V_{cb}\,V_{ud}^{\ast}\,
\sum\limits_{i}\,{\cal M}^{A}_{i,{\lambda}}
\label{amp:bd-ds-kvm},$$
$$i{\cal A}_{\lambda}(\overline{B}_{s}^{{\ast}0}{\to}D_{s}^{+}{\rho}^{-}) \, =\,
{\cal F}\, V_{cb}\,V_{ud}^{\ast}\,
\sum\limits_{i}\,{\cal M}^{T}_{i,{\lambda}}
\label{amp:bs-ds-rhom},$$
$$i{\cal A}_{\lambda}(\overline{B}_{s}^{{\ast}0}{\to}D_{s}^{+}K^{{\ast}-}) \, =\,
{\cal F}\, V_{cb}\,V_{us}^{\ast}\, \Big\{
\sum\limits_{i}\,{\cal M}^{T}_{i,{\lambda}}
+\sum\limits_{j}\,{\cal M}^{A}_{j,{\lambda}} \Big\}
\label{amp:bs-ds-kvm},$$
$$i{\cal A}_{\lambda}(\overline{B}_{s}^{{\ast}0}{\to}D_{d}^{+}{\rho}^{-}) \, =\,
{\cal F}\, V_{cb}\,V_{us}^{\ast}\,
\sum\limits_{i}\,{\cal M}^{A}_{i,{\lambda}}
\label{amp:bs-dd-rhom},$$
$$i\,\sqrt{2}\,{\cal A}_{\lambda}(\overline{B}_{s}^{{\ast}0}{\to}D_{u}^{0}{\rho}^{0}) \, =\,
{\cal F}\, V_{cb}\,V_{us}^{\ast}\,
\sum\limits_{i}\,{\cal M}^{A}_{i,{\lambda}}
\label{amp:bs-du-rhoz},$$
$$i\,\sqrt{2}\,{\cal A}_{\lambda}(\overline{B}_{s}^{{\ast}0}{\to}D_{u}^{0}{\omega}) \, =\,
{\cal F}\, V_{cb}\,V_{us}^{\ast}\,
\sum\limits_{i}\,{\cal M}^{A}_{i,{\lambda}}
\label{amp:bs-du-w},$$
$$i{\cal A}_{\lambda}(\overline{B}_{s}^{{\ast}0}{\to}D_{u}^{0}{\phi}) \, =\,
{\cal F}\, V_{cb}\,V_{us}^{\ast}\,
\sum\limits_{i}\,{\cal M}^{C}_{i,{\lambda}}
\label{amp:bs-du-phi},$$
$$i{\cal A}_{\lambda}(\overline{B}_{s}^{{\ast}0}{\to}D_{u}^{0}K^{{\ast}0}) \, =\,
{\cal F}\, V_{cb}\,V_{ud}^{\ast}\,
\sum\limits_{i}\,{\cal M}^{C}_{i,{\lambda}}
\label{amp:bs-du-kvz},$$
where the index ${\lambda}$ corresponds to three different helicity amplitudes, i.e., ${\lambda}$ $=$ $L$, $N$, $T$.
Amplitude building blocks for the color-allowed $\overline{B}_{q}^{\ast}$ ${\to}$ $D_{q}M$ decays {#block-t}
=================================================================================================
The expressions of the amplitude building blocks ${\cal M}^{T}_{i,j}$ for the color-allowed topologies are presented as follows, where the subscript $i$ corresponds to the diagram indices of Fig.\[fig:fey-t\]; and $j$ corresponds to the different helicity amplitudes. $$\begin{aligned}
{\cal M}^{T}_{a,P} &=&
2\,m_{1}\,p\,
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{\infty}b_{1}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}\,
H^{T}_{f}({\alpha}^{T},{\beta}^{T}_{a},b_{1},b_{2})\,
E^{T}_{f}(t^{T}_{a})
\nonumber \\ &{\times}&
{\alpha}_{s}(t^{T}_{a})\, a_{1}(t^{T}_{a})\,
{\phi}_{B_{q}^{\ast}}^{v}(x_{1})\,
\Big\{ {\phi}_{D}^{a}(x_{2})\,
( m_{1}^{2}\,\bar{x}_{2}+m_{3}^{2}\,x_{2} )
+ {\phi}_{D}^{p}(x_{2})\, m_{2}\,m_{b} \Big\}
\label{amp:t-a-p},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{T}_{a,L} &=&
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{\infty}b_{1}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}\,
H^{T}_{f}({\alpha}^{T},{\beta}^{T}_{a},b_{1},b_{2})\,
E^{T}_{f}(t^{T}_{a})\, {\alpha}_{s}(t^{T}_{a})
\nonumber \\ &{\times}&
a_{1}(t^{T}_{a})\, {\phi}_{B_{q}^{\ast}}^{v}(x_{1})\,
\Big\{ {\phi}_{D}^{a}(x_{2})\,
( m_{1}^{2}\,s\,\bar{x}_{2}+m_{3}^{2}\,t\,x_{2} )
+ {\phi}_{D}^{p}(x_{2})\, m_{2}\,m_{b}\,u \Big\}
\label{amp:t-a-l},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{T}_{a,N} &=& m_{1}\,m_{3}\,
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{\infty}b_{1}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}\,
H^{T}_{f}({\alpha}^{T},{\beta}^{T}_{a},b_{1},b_{2})\,
E^{T}_{f}(t^{T}_{a})
\nonumber \\ &{\times}&
{\alpha}_{s}(t^{T}_{a})\, a_{1}(t^{T}_{a})\,
{\phi}_{B_{q}^{\ast}}^{V}(x_{1})\,
\Big\{ {\phi}_{D}^{a}(x_{2})\, (2\,m_{2}^{2}\,x_{2} -t)
-2\, m_{2}\,m_{b}\, {\phi}_{D}^{p}(x_{2}) \Big\}
\label{amp:t-a-n},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{T}_{a,T} &=& 2\,m_{1}\,m_{3}\,
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{\infty}b_{1}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}\,
H^{T}_{f}({\alpha}^{T},{\beta}^{T}_{a},b_{1},b_{2})\,
E^{T}_{f}(t^{T}_{a})
\nonumber \\ &{\times}&
{\alpha}_{s}(t^{T}_{a})\, a_{1}(t^{T}_{a})\,
{\phi}_{B_{q}^{\ast}}^{V}(x_{1})\, {\phi}_{D}^{a}(x_{2})
\label{amp:t-a-t},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{T}_{b,P} &=&
2\,m_{1}\,p\,
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{\infty}b_{1}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}\,
H^{T}_{f}({\alpha}^{T},{\beta}^{T}_{b},b_{2},b_{1})\,
E^{T}_{f}(t^{T}_{b})
\nonumber \\ &{\times}&
{\alpha}_{s}(t^{T}_{b})\, \Big\{
{\phi}_{B_{q}^{\ast}}^{v}(x_{1})\, \Big[
2\,m_{2}\,m_{c}\,{\phi}_{D}^{p}(x_{2})
-{\phi}_{D}^{a}(x_{2})\, (m_{2}^{2}\,
\bar{x}_{1}+m_{3}^{2}\,x_{1}) \Big]
\nonumber \\ &+&
{\phi}_{B_{q}^{\ast}}^{t}(x_{1})\, \Big[
2\,m_{1}\,m_{2}\,{\phi}_{D}^{p}(x_{2})\, \bar{x}_{1}
-m_{1}\, m_{c}\, {\phi}_{D}^{a}(x_{2}) \Big] \Big\}\, a_{1}(t^{T}_{b})
\label{amp:t-b-p},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{T}_{b,L} &=&
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{\infty}b_{1}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}\,
H^{T}_{f}({\alpha}^{T},{\beta}^{T}_{b},b_{2},b_{1})\,
E^{T}_{f}(t^{T}_{b})\, {\alpha}_{s}(t^{T}_{b})
\nonumber \\ &{\times}&
a_{1}(t^{T}_{b})\, \Big\{
{\phi}_{B_{q}^{\ast}}^{t}(x_{1})\, \Big[
2\,m_{1}\,m_{2}\, {\phi}_{D}^{p}(x_{2})\,(s-u\,x_{1})
-m_{1}\,m_{c}\,s\,{\phi}_{D}^{a}(x_{2}) \Big]
\nonumber \\ &+&
{\phi}_{B_{q}^{\ast}}^{v}(x_{1})\, \Big[
{\phi}_{D}^{a}(x_{2})\,( m_{3}^{2}\,t\,\,x_{1}
-m_{2}^{2}\,u\,\bar{x}_{1} )
+2\,m_{2}\,m_{c}\,u\,{\phi}_{D}^{p}(x_{2}) \Big] \Big\}
\label{amp:t-b-l},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{T}_{b,N} &=& m_{3}\,
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{\infty}b_{1}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}\,
H^{T}_{f}({\alpha}^{T},{\beta}^{T}_{b},b_{2},b_{1})\,
E^{T}_{f}(t^{T}_{b})
\nonumber \\ &{\times}&
{\alpha}_{s}(t^{T}_{b})\, \Big\{
{\phi}_{B_{q}^{\ast}}^{V}(x_{1})\, m_{1}\,\Big[
{\phi}_{D}^{a}(x_{2})\,(2\,m_{2}^{2}-t\,x_{1})
-4\,m_{2}\,m_{c}\,{\phi}_{D}^{p}(x_{2}) \Big]
\nonumber \\ &+&
{\phi}_{B_{q}^{\ast}}^{T}(x_{1})\, \Big[
{\phi}_{D}^{a}(x_{2})\,t\,m_{c}
+{\phi}_{D}^{p}(x_{2})\,2\,m_{2}\,(2\,m_{1}^{2}\,x_{1}-t)
\Big] \Big\}\, a_{1}(t^{T}_{b})
\label{amp:t-b-n},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{T}_{b,T} &=& 2\,m_{3}\,
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{\infty}b_{1}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}\,
H^{T}_{f}({\alpha}^{T},{\beta}^{T}_{b},b_{2},b_{1})\,
E^{T}_{f}(t^{T}_{b})\, {\alpha}_{s}(t^{T}_{b})
\nonumber \\ &{\times}&
a_{1}(t^{T}_{b})\, \Big\{
{\phi}_{B_{q}^{\ast}}^{T}(x_{1})\, \Big[
{\phi}_{D}^{p}(x_{2})\,2\,m_{2}
-{\phi}_{D}^{a}(x_{2})\,m_{c} \Big]
-{\phi}_{B_{q}^{\ast}}^{V}(x_{1})\,
{\phi}_{D}^{a}(x_{2})\,m_{1}\,x_{1} \Big\}
\label{amp:t-b-t},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{T}_{c,P} &=&
\frac{2\,m_{1}\,p}{N_{c}}\,
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}b_{3}db_{3}\,
{\delta}(b_{1}-b_{2})
\nonumber \\ &{\times}&
{\phi}_{P}^{a}(x_{3})\, {\alpha}_{s}(t^{T}_{c})\,
C_{2}(t^{T}_{c})\,
\Big\{ {\phi}_{B_{q}^{\ast}}^{v}(x_{1})\, {\phi}_{D}^{a}(x_{2})\,
(2\,m_{2}^{2}\,x_{2}+s\,\bar{x}_{3}-t\,x_{1})
\nonumber \\ &+&
{\phi}_{B_{q}^{\ast}}^{t}(x_{1})\, {\phi}_{D}^{p}(x_{2})\,
m_{1}\,m_{2}\,(x_{1}-x_{2}) \Big\}\,
H^{T}_{n}({\alpha}^{T},{\beta}^{T}_{c},b_{3},b_{2})\,
E_{n}(t^{T}_{c})
\label{amp:t-c-p},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{T}_{c,L} &=&
\frac{1}{N_{c}}\,
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}b_{3}db_{3}\,
H^{T}_{n}({\alpha}^{T},{\beta}^{T}_{c},b_{3},b_{2})
\nonumber \\ &{\times}&
{\delta}(b_{1}-b_{2})\,
E_{n}(t^{T}_{c})\, {\phi}_{V}^{v}(x_{3})\,
\Big\{ {\phi}_{B_{q}^{\ast}}^{v}(x_{1})\,
{\phi}_{D}^{a}(x_{2})\, u\,
(2\,m_{2}^{2}\,x_{2}+s\,\bar{x}_{3}-t\,x_{1})
\nonumber \\ &+&
{\phi}_{B_{q}^{\ast}}^{t}(x_{1})\,
{\phi}_{D}^{p}(x_{2})\, m_{1}\,m_{2}\,
(u\,x_{1}-s\,x_{2}-2\,m_{3}^{2}\,\bar{x}_{3})
\Big\}\,
{\alpha}_{s}(t^{T}_{c})\,
C_{2}(t^{T}_{c})
\label{amp:t-c-l},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{T}_{c,N} &=&
\frac{m_{3}}{N_{c}}\,
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}b_{3}db_{3}
\nonumber \\ &{\times}&
H^{T}_{n}({\alpha}^{T},{\beta}^{T}_{c},b_{3},b_{2})\,
E_{n}(t^{T}_{c})\, {\alpha}_{s}(t^{T}_{c})\,
C_{2}(t^{T}_{c})\, {\delta}(b_{1}-b_{2})
\nonumber \\ &{\times}&
\Big\{ {\phi}_{B_{q}^{\ast}}^{V}(x_{1})\, {\phi}_{D}^{a}(x_{2})\,
{\phi}_{V}^{V}(x_{3})\, 2\,m_{1}\,
(t\,x_{1}-2\,m_{2}^{2}\,x_{2}-s\,\bar{x}_{3})
\nonumber \\ & & +
{\phi}_{B_{q}^{\ast}}^{T}(x_{1})\, {\phi}_{D}^{p}(x_{2})\,
{\phi}_{V}^{V}(x_{3})\, m_{2}\,
(t\,x_{2}+u\,\bar{x}_{3}-2\,m_{1}^{2}\,x_{1})
\nonumber \\ & & +
{\phi}_{B_{q}^{\ast}}^{T}(x_{1})\, {\phi}_{D}^{p}(x_{2})\,
{\phi}_{V}^{A}(x_{3})\, 2\,m_{1}\,m_{2}\,p\,
(x_{2}-\bar{x}_{3}) \Big\}
\label{amp:a-03},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{T}_{c,T} &=&
\frac{m_{3}}{N_{c}\,p}\,
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}b_{3}db_{3}
\nonumber \\ &{\times}&
H^{T}_{n}({\alpha}^{T},{\beta}^{T}_{c},b_{3},b_{2})\,
E_{n}(t^{T}_{c})\, {\alpha}_{s}(t^{T}_{c})\,
C_{2}(t^{T}_{c})\, {\delta}(b_{1}-b_{2})
\nonumber \\ &{\times}&
\Big\{ {\phi}_{B_{q}^{\ast}}^{V}(x_{1})\, {\phi}_{D}^{a}(x_{2})\,
{\phi}_{V}^{A}(x_{3})\, 2\, (2\,m_{2}^{2}\,x_{2}+s\,\bar{x}_{3}-t\,x_{1})
\nonumber \\ & & +
{\phi}_{B_{q}^{\ast}}^{T}(x_{1})\, {\phi}_{D}^{p}(x_{2})\,
{\phi}_{V}^{A}(x_{3})\, r_{2}\,
(2\,m_{1}^{2}\,x_{1}-t\,x_{2}-u\,\bar{x}_{3})
\nonumber \\ & & +
{\phi}_{B_{q}^{\ast}}^{T}(x_{1})\, {\phi}_{D}^{p}(x_{2})\,
{\phi}_{V}^{V}(x_{3})\, 2\,m_{2}\,p\, (\bar{x}_{3}-x_{2}) \Big\}
\label{amp:t-c-t},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{T}_{d,P} &=&
\frac{2\,m_{1}\,p}{N_{c}}\,
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}b_{3}db_{3}\,
{\phi}_{P}^{a}(x_{3})
\nonumber \\ &{\times}&
{\delta}(b_{1}-b_{2})\, {\alpha}_{s}(t^{T}_{d})\,
C_{2}(t^{T}_{d})\, E_{n}(t^{T}_{d})\, \Big\{
{\phi}_{B_{q}^{\ast}}^{v}(x_{1})\,
{\phi}_{D}^{a}(x_{2})\, s\,(x_{2}-x_{3})
\nonumber \\ &+&
{\phi}_{B_{q}^{\ast}}^{t}(x_{1})\, {\phi}_{D}^{p}(x_{2})\,
m_{1}\,m_{2}\,(x_{1}-x_{2}) \Big\}\,
H^{T}_{n}({\alpha}^{T},{\beta}^{T}_{d},b_{3},b_{2})
\label{amp:t-d-p},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{T}_{d,L} &=&
\frac{1}{N_{c}}\,
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}b_{3}db_{3}\,
{\delta}(b_{1}-b_{2})
\nonumber \\ &{\times}&
E_{n}(t^{T}_{d})\, {\alpha}_{s}(t^{T}_{d})\,
C_{2}(t^{T}_{d})\, {\phi}_{V}^{v}(x_{3})\,
\Big\{ {\phi}_{B_{q}^{\ast}}^{v}(x_{1})\, {\phi}_{D}^{a}(x_{2})\,
4\,m_{1}^{2}\,p^{2}\,(x_{2}-x_{3})
\nonumber \\ &+&
{\phi}_{B_{q}^{\ast}}^{t}(x_{1})\, {\phi}_{D}^{p}(x_{2})\,
m_{1}\,m_{2}\,( u\,x_{1}-s\,x_{2}-2\,m_{3}^{2}\,x_{3}) \Big\}\,
H^{T}_{n}({\alpha}^{T},{\beta}^{T}_{d},b_{3},b_{2})
\label{amp:t-d-l},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{T}_{d,N} &=&
\frac{m_{2}\,m_{3}}{N_{c}}\,
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}b_{3}db_{3}\,
{\delta}(b_{1}-b_{2})
\nonumber \\ &{\times}&
{\phi}_{B_{q}^{\ast}}^{T}(x_{1})\,
{\phi}_{D}^{p}(x_{2})\, {\alpha}_{s}(t^{T}_{d})\,
C_{2}(t^{T}_{d})\, \Big\{ {\phi}_{V}^{V}(x_{3})\,
( t\,x_{2}+u\,x_{3}-2\,m_{1}^{2}\,x_{1})
\nonumber \\ &+&
{\phi}_{V}^{A}(x_{3})\,2\,m_{1}\,p\,(x_{2}-x_{3})
\Big\}\, H^{T}_{n}({\alpha}^{T},{\beta}^{T}_{d},b_{3},b_{2})\,
E_{n}(t^{T}_{d})
\label{amp:t-d-n},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{T}_{d,T} &=&
\frac{m_{2}\,m_{3}}{N_{c}\,m_{1}\,p}\,
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}b_{3}db_{3}\,
{\delta}(b_{1}-b_{2})
\nonumber \\ &{\times}&
{\phi}_{B_{q}^{\ast}}^{T}(x_{1})\,
{\phi}_{D}^{p}(x_{2})\, {\alpha}_{s}(t^{T}_{d})\,
C_{2}(t^{T}_{d})\, \Big\{ {\phi}_{V}^{A}(x_{3})\,
( 2\,m_{1}^{2}\,x_{1}-t\,x_{2}-u\,x_{3})
\nonumber \\ &+&
{\phi}_{V}^{V}(x_{3})\,2\,m_{1}\,p\,(x_{3}-x_{2})
\Big\}\, H^{T}_{n}({\alpha}^{T},{\beta}^{T}_{d},b_{3},b_{2})\,
E_{n}(t^{T}_{d})
\label{amp:t-d-t},
\end{aligned}$$ where $N_{c}$ $=$ $3$ is the color number. ${\alpha}_{s}$ is the strong coupling constant. $C_{1,2}$ are the Wilson coefficients. The parameter $a_{i}$ is defined as $$\begin{aligned}
a_{1} &=& C_{1}+\frac{1}{N_{c}}\,C_{2}
\label{coe-a1}, \\
a_{2} &=& C_{2}+\frac{1}{N_{c}}\,C_{1}
\label{coe-a2}.
\end{aligned}$$ The functions $H_{f,n}^{T}$ and the Sudakov factors $E_{f,n}^{T}$ are defined as follows, where the subscripts $f$ and $n$ correspond to the factorizable and nonfactorizable topologies. $$H_{f}^{T}({\alpha},{\beta},b_{i},b_{j})\, =\,
K_{0}(b_{i}\sqrt{-{\alpha}})\, \Big\{
{\theta}(b_{i}-b_{j}) K_{0}(b_{i}\sqrt{-{\beta}})\,
I_{0}(b_{j}\sqrt{-{\beta}})
+ (b_{i} {\leftrightarrow} b_{j}) \Big\}
\label{amp:hft},$$ $$\begin{aligned}
H_{n}^{T}({\alpha},{\beta},b_{i},b_{j}) &=&
\Big\{ {\theta}(b_{i}-b_{j})\, K_{0}(b_{i}\sqrt{-{\alpha}})\,
I_{0}(b_{j}\sqrt{-{\alpha}}) + (b_{i} {\leftrightarrow} b_{j})
\Big\}
\nonumber \\ & & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! {\times}\,
\Big\{ {\theta}(-{\beta})\, K_{0}(b_{i}\sqrt{-{\beta}})
+\frac{{\pi}}{2}\,
{\theta}(+{\beta})\, \Big[ i\,J_{0}(b_{i}\sqrt{{\beta}})
- Y_{0}(b_{i}\sqrt{{\beta}}) \Big] \Big\}
\label{amp:hnt},
\end{aligned}$$ $$E_{f}^{T}(t)\ =\ {\exp}\{ -S_{B_{q}^{\ast}}(t)-S_{D}(t) \}
\label{sudakov-ft},$$ $$E_{n}(t)\ =\ {\exp}\{ -S_{B_{q}^{\ast}}(t)-S_{D}(t)-S_{M}(t) \}
\label{sudakov-nt},$$ $$S_{B_{q}^{\ast}}(t)\, =\, s(x_{1},b_{1},p_{1}^{+})
+2{\int}_{1/b_{1}}^{t}\frac{d{\mu}}{\mu}{\gamma}_{q}
\label{sudakov-bq},$$ $$S_{D}(t)\, =\, s(x_{2},b_{2},p_{2}^{+}) + s(\bar{x}_{2},b_{2},p_{2}^{+})
+2{\int}_{1/b_{2}}^{t}\frac{d{\mu}}{\mu}{\gamma}_{q}
\label{sudakov-cq},$$ $$S_{M}(t)\, =\, s(x_{3},b_{3},p_{3}^{+}) + s(\bar{x}_{3},b_{3},p_{3}^{+})
+2{\int}_{1/b_{3}}^{t}\frac{d{\mu}}{\mu}{\gamma}_{q}
\label{sudakov-m},$$ where $I_{0}$, $J_{0}$, $K_{0}$ and $Y_{0}$ are the Bessel functions; ${\gamma}_{q}$ $=$ $-{\alpha}_{s}/{\pi}$ is the quark anomalous dimension; the expression of $s(x,b,Q)$ can be found in the appendix of Ref.[@prd52.3958]; ${\alpha}^{T}$ and ${\beta}_{i}^{T}$ are the virtualities of the gluon and quark propagators; the subscripts of the quark virtuality ${\beta}_{i}^{T}$ and the typical scale $t_{i}^{T}$ correspond to the diagram indices of Fig.\[fig:fey-t\]. $$\begin{aligned}
{\alpha}^{T} &=& x_{1}^{2}\,m_{1}^{2}+x_{2}^{2}\,m_{2}^{2}-x_{1}\,x_{2}\,t
\label{gluon-t}, \\
%-----------------------------------------------------
{\beta}_{a}^{T} &=& x_{2}^{2}\,m_{2}^{2}-x_{2}\,t+m_{1}^{2}-m_{b}^{2}
\label{beta-ta}, \\
%-----------------------------------------------------
{\beta}_{b}^{T} &=& x_{1}^{2}\,m_{1}^{2}-x_{1}\,t+m_{2}^{2}-m_{c}^{2}
\label{beta-tb}, \\
%-----------------------------------------------------
{\beta}_{c}^{T} &=& {\alpha}^{T}+\bar{x}_{3}^{2}\,m_{3}^{2}
-x_{1}\,\bar{x}_{3}\,u+x_{2}\,\bar{x}_{3}\,s
\label{beta-tc}, \\
%-----------------------------------------------------
{\beta}_{d}^{T} &=& {\alpha}^{T}+x_{3}^{2}\,m_{3}^{2}
-x_{1}\,x_{3}\,u+x_{2}\,x_{3}\,s
\label{beta-td}, \\
t_{a(b)}^{T} &=&
{\max}(\sqrt{-{\alpha}^{T}},\sqrt{{\vert}{\beta}_{a(b)}^{T}{\vert}},1/b_{1},1/b_{2})
\label{t-tab}, \\
t_{c(d)}^{T} &=&
{\max}(\sqrt{-{\alpha}^{T}},\sqrt{{\vert}{\beta}_{c(d)}^{T}{\vert}},1/b_{2},1/b_{3})
\label{t-tcd}.
\end{aligned}$$
Amplitude building blocks for the color-suppressed $\overline{B}_{q}^{\ast}$ ${\to}$ $DM_{q}$ decays {#block-c}
====================================================================================================
The expressions of the amplitude building blocks ${\cal M}^{C}_{i,j}$ for the color-suppressed topologies are displayed as follows, where the subscript $i$ corresponds to the diagram indices of Fig.\[fig:fey-c\]; and $j$ corresponds to the different helicity amplitudes. $$\begin{aligned}
{\cal M}^{C}_{a,P} &=&
-{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}b_{1}db_{1}
{\int}_{0}^{\infty}b_{3}db_{3}\,
{\phi}_{B_{q}^{\ast}}^{v}(x_{1})\,
{\alpha}_{s}(t^{C}_{a})\, a_{2}(t^{C}_{a})
\nonumber \\ &{\times}&
H^{C}_{f}({\alpha}^{C},{\beta}^{C}_{a},b_{1},b_{3})\,
\Big\{ 2\,m_{1}\,p\, {\phi}_{P}^{a}(x_{3})\,
(m_{1}^{2}\,\bar{x}_{3}+m_{2}^{2}\,x_{3})
\nonumber \\ &+&
2\,m_{1}\,p\, {\mu}_{P}\,m_{b}\, {\phi}_{P}^{p}(x_{3})
+ {\mu}_{P}\,m_{b}\, t\, {\phi}_{P}^{t}(x_{3}) \Big\}\,
E^{C}_{f}(t^{C}_{a})
\label{amp:c-a-p},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{C}_{a,L} &=&
-{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}b_{1}db_{1}
{\int}_{0}^{\infty}b_{3}db_{3}\,
H^{C}_{f}({\alpha}^{C},{\beta}^{C}_{a},b_{1},b_{3})
\nonumber \\ &{\times}&
{\alpha}_{s}(t^{C}_{a})\, a_{2}(t^{C}_{a})\,
{\phi}_{B_{q}^{\ast}}^{v}(x_{1})\,
\Big\{ {\phi}_{V}^{v}(x_{3})\,
(m_{1}^{2}\,s\,\bar{x}_{3}+m_{2}^{2}\,u\,x_{3})
\nonumber \\ &+&
m_{3}\,m_{b}\,t\, {\phi}_{V}^{t}(x_{3})
+ 2\,m_{1}\,p\, m_{3}\,m_{b}\, {\phi}_{V}^{s}(x_{3})
\Big\}\, E^{C}_{f}(t^{C}_{a})
\label{amp:c-a-l},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{C}_{a,N} &=&
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}b_{1}db_{1}
{\int}_{0}^{\infty}b_{3}db_{3}\,
H^{C}_{f}({\alpha}^{C},{\beta}^{C}_{a},b_{1},b_{3})
\nonumber \\ &{\times}&
{\alpha}_{s}(t^{C}_{a})\, a_{2}(t^{C}_{a})\,
{\phi}_{B_{q}^{\ast}}^{V}(x_{1})\,
\Big\{ {\phi}_{V}^{V}(x_{3})\,m_{1}\,m_{3}\,(t-s\,x_{3})
\nonumber \\ &+&
m_{1}\,m_{b}\,s\, {\phi}_{V}^{T}(x_{3})
+ 2\,m_{3}\,p\, m_{1}^{2}\,\bar{x}_{3}\, {\phi}_{V}^{A}(x_{3})
\Big\}\, E^{C}_{f}(t^{C}_{a})
\label{amp:c-a-n},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{C}_{a,T} &=&
-{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}b_{1}db_{1}
{\int}_{0}^{\infty}b_{3}db_{3}\,
H^{C}_{f}({\alpha}^{C},{\beta}^{C}_{a},b_{1},b_{3})
\nonumber \\ &{\times}&
{\alpha}_{s}(t^{C}_{a})\, a_{2}(t^{C}_{a})\,
{\phi}_{B_{q}^{\ast}}^{V}(x_{1})\,
\Big\{ (m_{3}/p)\, {\phi}_{V}^{A}(x_{3})\,(t-s\,x_{3})
\nonumber \\ &+&
{\phi}_{V}^{V}(x_{3})\,2\,m_{1}\,m_{3}\,\bar{x}_{3}
+ {\phi}_{V}^{T}(x_{3})\,2\,m_{1}\,m_{b}
\Big\}\, E^{C}_{f}(t^{C}_{a})
\label{amp:c-a-t},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{C}_{b,P} &=&
2\,m_{1}\,p\,
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}b_{1}db_{1}
{\int}_{0}^{\infty}b_{3}db_{3}\,
H^{C}_{f}({\alpha}^{C},{\beta}^{C}_{b},b_{3},b_{1})\,
E^{C}_{f}(t^{C}_{b})\, {\alpha}_{s}(t^{C}_{b})
\nonumber \\ &{\times}&
a_{2}(t^{C}_{b})\, \Big\{ {\phi}_{B_{q}^{\ast}}^{v}(x_{1})\,
{\phi}_{P}^{a}(x_{3})\, (m_{3}^{2}\,\bar{x}_{1}+m_{2}^{2}\,x_{1})
-{\phi}_{B_{q}^{\ast}}^{t}(x_{1})\, {\phi}_{P}^{p}(x_{3})\,
2\,m_{1}\,{\mu}_{P}\,\bar{x}_{1} \Big\}
\label{amp:c-b-p},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{C}_{b,L} &=&
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}b_{1}db_{1}
{\int}_{0}^{\infty}b_{3}db_{3}\,
H^{C}_{f}({\alpha}^{C},{\beta}^{C}_{b},b_{3},b_{1})\,
E^{C}_{f}(t^{C}_{b})\, {\alpha}_{s}(t^{C}_{b})\,
a_{2}(t^{C}_{b})
\nonumber \\ &{\times}&
\Big\{ {\phi}_{B_{q}^{\ast}}^{v}(x_{1})\,
{\phi}_{V}^{v}(x_{3})\, (m_{2}^{2}\,u\,x_{1}-m_{3}^{2}\,t\,\bar{x}_{1})
-{\phi}_{B_{q}^{\ast}}^{t}(x_{1})\, {\phi}_{V}^{s}(x_{3})\,
4\,m_{1}^{2}\,m_{3}\,p\,\bar{x}_{1} \Big\}
\label{amp:c-b-l},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{C}_{b,N} &=&
m_{1}\,m_{3}\, {\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}b_{1}db_{1}
{\int}_{0}^{\infty}b_{3}db_{3}\,
H^{C}_{f}({\alpha}^{C},{\beta}^{C}_{b},b_{3},b_{1})\,
E^{C}_{f}(t^{C}_{b})
\nonumber \\ &{\times}&
{\alpha}_{s}(t^{C}_{b})\, a_{2}(t^{C}_{b})\,
{\phi}_{B_{q}^{\ast}}^{V}(x_{1})\, \Big\{
{\phi}_{V}^{V}(x_{3})\, (s-t\,x_{1})
+{\phi}_{V}^{A}(x_{3})\, 2\,m_{1}\,p\,\bar{x}_{1} \Big\}
\label{amp:c-b-n},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{C}_{b,T} &=&
\frac{-m_{3}}{p}\, {\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}b_{1}db_{1}
{\int}_{0}^{\infty}b_{3}db_{3}\,
H^{C}_{f}({\alpha}^{C},{\beta}^{C}_{b},b_{3},b_{1})\,
E^{C}_{f}(t^{C}_{b})
\nonumber \\ &{\times}&
{\alpha}_{s}(t^{C}_{b})\, a_{2}(t^{C}_{b})\,
{\phi}_{B_{q}^{\ast}}^{V}(x_{1})\, \Big\{
{\phi}_{V}^{A}(x_{3})\, (s-t\,x_{1})
+{\phi}_{V}^{V}(x_{3})\, 2\,m_{1}\,p\,\bar{x}_{1} \Big\}
\label{amp:c-b-t},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{C}_{c,P} &=&
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}b_{3}db_{3}\,
H^{C}_{n}({\alpha}^{C},{\beta}^{C}_{c},b_{2},b_{3})
\nonumber \\ &{\times}&
{\delta}(b_{1}-b_{3})\, \Big\{
{\phi}_{B_{q}^{\ast}}^{t}(x_{1})\, {\phi}_{D}^{a}(x_{2})\,
m_{1}\, {\mu}_{P}\, \Big[ {\phi}_{P}^{t}(x_{3})\,
(t\,x_{1}-2\,m_{2}^{2}\,\bar{x}_{2}-s\,x_{3})
\nonumber \\ &+&
{\phi}_{P}^{p}(x_{3})\, 2\,m_{1}\,p\,(x_{3}-x_{1})
\Big] -
{\phi}_{B_{q}^{\ast}}^{v}(x_{1})\, {\phi}_{P}^{a}(x_{3})\,
2\,m_{1}\,p\,\Big[ {\phi}_{D}^{p}(x_{2})\, m_{2}\,m_{c}
\nonumber \\ &+&
{\phi}_{D}^{a}(x_{2})\, (s\,\bar{x}_{2}+2\,m_{3}^{2}\,x_{3}-u\,x_{1})
\Big] \Big\}\, E_{n}(t^{C}_{c})\, {\alpha}_{s}(t^{C}_{c})\,
C_{1}(t^{C}_{c})/N_{c}
\label{amp:c-c-p},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{C}_{c,L} &=&
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}b_{3}db_{3}\,
H^{C}_{n}({\alpha}^{C},{\beta}^{C}_{c},b_{2},b_{3})
\nonumber \\ &{\times}&
{\delta}(b_{1}-b_{3})\, \Big\{
{\phi}_{B_{q}^{\ast}}^{t}(x_{1})\, {\phi}_{D}^{a}(x_{2})\,
m_{1}\, m_{3}\, \Big[ {\phi}_{V}^{t}(x_{3})\,
(t\,x_{1}-2\,m_{2}^{2}\,\bar{x}_{2}-s\,x_{3})
\nonumber \\ &+&
{\phi}_{V}^{s}(x_{3})\, 2\,m_{1}\,p\,(x_{3}-x_{1})
\Big] + {\phi}_{B_{q}^{\ast}}^{v}(x_{1})\,
{\phi}_{V}^{v}(x_{3})\,
\Big[ -{\phi}_{D}^{p}(x_{2})\, m_{2}\,m_{c}\,u
\nonumber \\ &+&
{\phi}_{D}^{a}(x_{2})\, 4\,m_{1}^{2}\,p^{2}\,
(x_{1}-\bar{x}_{2})
\Big] \Big\}\, E_{n}(t^{C}_{c})\, {\alpha}_{s}(t^{C}_{c})\,
C_{1}(t^{C}_{c})/N_{c}
\label{amp:c-c-l},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{C}_{c,N} &=&
\frac{1}{N_{c}}\,
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}b_{3}db_{3}\,
H^{C}_{n}({\alpha}^{C},{\beta}^{C}_{c},b_{2},b_{3})
\nonumber \\ &{\times}&
{\delta}(b_{1}-b_{3})\, E_{n}(t^{C}_{c})\,
{\alpha}_{s}(t^{C}_{c}) \, \Big\{
{\phi}_{B_{q}^{\ast}}^{V}(x_{1})\, {\phi}_{D}^{p}(x_{2})\,
{\phi}_{V}^{V}(x_{3})\,2\,m_{1}\,m_{2}\,m_{3}\,m_{c}
\nonumber \\ &+&
{\phi}_{B_{q}^{\ast}}^{T}(x_{1})\, {\phi}_{D}^{a}(x_{2})\,
{\phi}_{V}^{T}(x_{3})\,\Big[
m_{1}^{2}\,s\,(\bar{x}_{2}-x_{1})
+m_{3}^{2}\,t\,(x_{3}-\bar{x}_{2}) \Big] \Big\}\,
C_{1}(t^{C}_{c})
\label{amp:c-c-n},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{C}_{c,T} &=&
\frac{2}{N_{c}}\,
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}b_{3}db_{3}\,
H^{C}_{n}({\alpha}^{C},{\beta}^{C}_{c},b_{2},b_{3})
\nonumber \\ &{\times}&
C_{1}(t^{C}_{c})\, \Big\{
{\phi}_{B_{q}^{\ast}}^{T}(x_{1})\, {\phi}_{D}^{a}(x_{2})\,
{\phi}_{V}^{T}(x_{3})\, \Big[ m_{1}^{2}\,(x_{1}-\bar{x}_{2})
+ m_{3}^{2}\,(\bar{x}_{2}-x_{3}) \Big]
\nonumber \\ &-&
{\phi}_{B_{q}^{\ast}}^{V}(x_{1})\, {\phi}_{D}^{p}(x_{2})\,
{\phi}_{V}^{A}(x_{3})\,m_{2}\,m_{3}\,m_{c}/p
\Big\}\, E_{n}(t^{C}_{c})\, {\alpha}_{s}(t^{C}_{c})\,
{\delta}(b_{1}-b_{3})
\label{amp:c-c-t},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{C}_{d,P} &=&
\frac{1}{N_{c}}\,
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}b_{3}db_{3}\,
H^{C}_{n}({\alpha}^{C},{\beta}^{C}_{d},b_{2},b_{3})\,
E_{n}(t^{C}_{d})
\nonumber \\ &{\times}&
{\delta}(b_{1}-b_{3})\, {\alpha}_{s}(t^{C}_{d})\,
C_{1}(t^{C}_{d})\, {\phi}_{D}^{a}(x_{2})\, \Big\{
{\phi}_{B_{q}^{\ast}}^{v}(x_{1})\, {\phi}_{P}^{a}(x_{3})\,
2\,m_{1}\,p\,s\,(x_{2}-x_{3})
\nonumber \\ &+&
{\phi}_{B_{q}^{\ast}}^{t}(x_{1})\,m_{1}\,{\mu}_{P}\,
\Big[ {\phi}_{P}^{p}(x_{3})\, 2\,m_{1}\,p\, (x_{3}-x_{1})
+ {\phi}_{P}^{t}(x_{3})\,(2\,m_{2}^{2}\,x_{2}+s\,x_{3}-t\,x_{1})
\Big] \Big\}
\label{amp:c-d-p},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{C}_{d,L} &=&
\frac{1}{N_{c}}\,
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}b_{3}db_{3}\,
H^{C}_{n}({\alpha}^{C},{\beta}^{C}_{d},b_{2},b_{3})\,
E_{n}(t^{C}_{d})
\nonumber \\ &{\times}&
{\delta}(b_{1}-b_{3})\, {\alpha}_{s}(t^{C}_{d})\,
C_{1}(t^{C}_{d})\, {\phi}_{D}^{a}(x_{2})\, \Big\{
{\phi}_{B_{q}^{\ast}}^{v}(x_{1})\, {\phi}_{V}^{v}(x_{3})\,
4\,m_{1}^{2}\,p^{2}\,(x_{2}-x_{3})
\nonumber \\ &+&
{\phi}_{B_{q}^{\ast}}^{t}(x_{1})\,m_{1}\,m_{3}\,
\Big[ {\phi}_{V}^{s}(x_{3})\, 2\,m_{1}\,p\, (x_{3}-x_{1})
+ {\phi}_{V}^{t}(x_{3})\,(2\,m_{2}^{2}\,x_{2}+s\,x_{3}-t\,x_{1})
\Big] \Big\}
\label{amp:c-d-l},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{C}_{d,N} &=&
\frac{1}{N_{c}}\,
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}b_{3}db_{3}\,
H^{C}_{n}({\alpha}^{C},{\beta}^{C}_{d},b_{2},b_{3})\,
E_{n}(t^{C}_{d})
\nonumber \\ & & \!\!\!\! \!\!\!\! \!\!\!\!
\!\!\!\! \!\!\!\! \!\!\!\! {\times}\,
{\delta}(b_{1}-b_{3})\,{\alpha}_{s}(t^{C}_{d})\,
C_{1}(t^{C}_{d})\, {\phi}_{B_{q}^{\ast}}^{T}(x_{1})\,
{\phi}_{D}^{a}(x_{2})\, {\phi}_{V}^{T}(x_{3})\,
\Big\{ m_{1}^{2}\,s\,(x_{1}-x_{2})
+m_{3}^{2}\,t\,(x_{2}-x_{3}) \Big\}
\label{amp:c-d-n},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{C}_{d,T} &=&
\frac{2}{N_{c}}\,
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}b_{3}db_{3}\,
H^{C}_{n}({\alpha}^{C},{\beta}^{C}_{d},b_{2},b_{3})\,
E_{n}(t^{C}_{d})
\nonumber \\ & & \!\!\!\! \!\!\!\! \!\!\!\!
\!\!\!\! {\times}\,
{\delta}(b_{1}-b_{3})\,{\alpha}_{s}(t^{C}_{d})\,
C_{1}(t^{C}_{d})\, {\phi}_{B_{q}^{\ast}}^{T}(x_{1})\,
{\phi}_{D}^{a}(x_{2})\, {\phi}_{V}^{T}(x_{3})\,
\Big\{ m_{1}^{2}\,(x_{2}-x_{1})
+m_{3}^{2}\,(x_{3}-x_{2}) \Big\}
\label{amp:c-d-t}.
\end{aligned}$$ The functions $H_{f,n}^{C}$ have the similar expressions for $H_{f,n}^{T}$, i.e., $$H_{f}^{C}({\alpha},{\beta},b_{i},b_{j})\, =\,
H_{f}^{T}({\alpha},{\beta},b_{i},b_{j})
\label{amp:hfc},$$ $$H_{n}^{C}({\alpha},{\beta},b_{i},b_{j})\, =\,
H_{n}^{T}({\alpha},{\beta},b_{i},b_{j})
\label{amp:hnc}.$$ The Sudakov factor $E_{f}^{C}$ are defined as $$E_{f}^{C}(t)\ =\ {\exp}\{ -S_{B_{q}^{\ast}}(t)-S_{M}(t) \}
\label{sudakov-fc},$$ and the expressions for $E_{n}(t)$, $S_{B_{q}^{\ast}}(t)$, $S_{D}(t)$ and $S_{M}(t)$ are the same as those given in the Appendix \[block-t\]. ${\alpha}^{C}$ and ${\beta}_{i}^{C}$ are the gluon and quark virtualities; the subscripts of ${\beta}_{i}^{C}$ and $t_{i}^{C}$ correspond to the diagram indices of Fig.\[fig:fey-c\]. $$\begin{aligned}
{\alpha}^{C} &=& x_{1}^{2}\,m_{1}^{2}+x_{3}^{2}\,m_{3}^{2}-x_{1}\,x_{3}\,u
\label{gluon-c}, \\
%-----------------------------------------------------
{\beta}_{a}^{C} &=& x_{3}^{2}\,m_{3}^{2}-x_{3}\,u+m_{1}^{2}-m_{b}^{2}
\label{beta-ca}, \\
%-----------------------------------------------------
{\beta}_{b}^{C} &=& x_{1}^{2}\,m_{1}^{2}-x_{1}\,u+m_{3}^{2}
\label{beta-cb}, \\
%-----------------------------------------------------
{\beta}_{c}^{C} &=& {\alpha}^{C}+\bar{x}_{2}^{2}\,m_{2}^{2}
-x_{1}\,\bar{x}_{2}\,t+x_{3}\,\bar{x}_{2}\,s-m_{c}^{2}
\label{beta-cc}, \\
%-----------------------------------------------------
{\beta}_{d}^{C} &=& {\alpha}^{C}+x_{2}^{2}\,m_{2}^{2}
-x_{1}\,x_{2}\,t+x_{2}\,x_{3}\,s
\label{beta-cd}, \\
t_{a(b)}^{C} &=&
{\max}(\sqrt{-{\alpha}^{C}},\sqrt{{\vert}{\beta}_{a(b)}^{C}{\vert}},1/b_{1},1/b_{3})
\label{t-cab}, \\
t_{c(d)}^{C} &=&
{\max}(\sqrt{-{\alpha}^{C}},\sqrt{{\vert}{\beta}_{c(d)}^{C}{\vert}},1/b_{2},1/b_{3})
\label{t-ccd}.
\end{aligned}$$
Amplitude building blocks for the annihilation $\overline{B}^{{\ast}0}$ ${\to}$ $DM$ decays {#block-a}
===========================================================================================
The expressions of the amplitude building blocks ${\cal M}^{A}_{i,j}$ for the annihilation topologies are listed as follows, where the subscript $i$ corresponds to the diagram indices of Fig.\[fig:fey-a\]; and $j$ corresponds to different helicity amplitudes. $$\begin{aligned}
{\cal M}^{A}_{a,P} &=&
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}b_{3}db_{3}\,
H^{A}_{f}({\alpha}^{A},{\beta}^{A}_{a},b_{2},b_{3})\,
E^{A}_{f}(t^{A}_{a})\, {\alpha}_{s}(t^{A}_{a})
\nonumber \\ &{\times}&
a_{2}(t^{A}_{a})\,
\Big\{ {\phi}_{D}^{p}(x_{2})\, \Big[
{\phi}_{P}^{a}(x_{3})\,4\,m_{1}\,m_{2}\,m_{c}\,p
+{\phi}_{P}^{p}(x_{3})\,4\,m_{1}\,m_{2}\,{\mu}_{P}\,p\,x_{3}
\nonumber \\ &+&
{\phi}_{P}^{t}(x_{3})\,2\,m_{2}\, {\mu}_{P}\,(t+u\,\bar{x}_{3})
\Big] - {\phi}_{D}^{a}(x_{2})\, \Big[
{\phi}_{P}^{p}(x_{3})\,\,2\,m_{1}\,m_{c}\,{\mu}_{P}\,p
\nonumber \\ &+&
{\phi}_{P}^{a}(x_{3})\,2\,m_{1}\,p\,
(m_{1}^{2}\,\bar{x}_{3}+m_{2}^{2}\,x_{3})
+{\phi}_{P}^{t}(x_{3})\,m_{c}\,{\mu}_{P}\,t \Big] \Big\}
\label{amp:a-a-p},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{A}_{a,L} &=&
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}b_{3}db_{3}\,
H^{A}_{f}({\alpha}^{A},{\beta}^{A}_{a},b_{2},b_{3})\,
E^{A}_{f}(t^{A}_{a})\, {\alpha}_{s}(t^{A}_{a})
\nonumber \\ &{\times}&
a_{2}(t^{A}_{a})\,
\Big\{ {\phi}_{D}^{p}(x_{2})\, \Big[
{\phi}_{V}^{v}(x_{3})\,2\,m_{2}\,m_{c}\,u
-{\phi}_{V}^{t}(x_{3})\,2\,m_{2}\,m_{3}\,(t+u\,\bar{x}_{3})
\nonumber \\ &-&
{\phi}_{V}^{s}(x_{3})\,4\,m_{1}\, m_{2}\,m_{3}\,p\,x_{3}
\Big] + {\phi}_{D}^{a}(x_{2})\, \Big[
{\phi}_{V}^{s}(x_{3})\,2\,m_{1}\,m_{3}\,m_{c}\,p
\nonumber \\ &-&
{\phi}_{V}^{v}(x_{3})\,
(m_{2}^{2}\,u\,x_{3}+m_{1}^{2}\,s\,\bar{x}_{3})
+{\phi}_{V}^{t}(x_{3})\,m_{3}\,m_{c}\,t \Big] \Big\}
\label{amp:a-a-l},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{A}_{a,N} &=&
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}b_{3}db_{3}\,
H^{A}_{f}({\alpha}^{A},{\beta}^{A}_{a},b_{2},b_{3})\,
E^{A}_{f}(t^{A}_{a})
\nonumber \\ &{\times}&
\Big\{ {\phi}_{D}^{a}(x_{2})\, \Big[
{\phi}_{V}^{V}(x_{3})\,m_{1}\,m_{3}\,(s\,\bar{x}_{3}+2\,m_{2}^{2})
-{\phi}_{V}^{T}(x_{3})\,m_{1}\,m_{c}\,s
\nonumber \\ &+&
{\phi}_{V}^{A}(x_{3})\,2\,m_{1}^{2}\, m_{3}\,p\,\bar{x}_{3}
\Big] - {\phi}_{D}^{p}(x_{2})\, \Big[
{\phi}_{V}^{V}(x_{3})\,4\,m_{1}\,m_{2}\,m_{3}\,m_{c}
\nonumber \\ &-&
{\phi}_{V}^{T}(x_{3})\,2\,m_{1}\,m_{2}\,(s+2\,m_{3}^{2}\,\bar{x}_{3})
\Big] \Big\}\, {\alpha}_{s}(t^{A}_{a})\, a_{2}(t^{A}_{a})
\label{amp:a-a-n},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{A}_{a,T} &=&
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}b_{3}db_{3}\,
H^{A}_{f}({\alpha}^{A},{\beta}^{A}_{a},b_{2},b_{3})\,
E^{A}_{f}(t^{A}_{a})
\nonumber \\ &{\times}&
\Big\{ {\phi}_{D}^{p}(x_{2})\, 4\,m_{2}\Big[
{\phi}_{V}^{T}(x_{3})\,m_{1}
+{\phi}_{V}^{A}(x_{3})\,m_{3}\,m_{c}/p \Big]
\nonumber \\ &-&
{\phi}_{D}^{a}(x_{2})\, \Big[
{\phi}_{V}^{V}(x_{3})\,2\,m_{1}\,m_{3}\,\bar{x}_{3}
+{\phi}_{V}^{T}(x_{3})\,2\,m_{1}\,m_{c}
\nonumber \\ &+&
{\phi}_{V}^{A}(x_{3})\,(m_{3}/p)\,(s\,\bar{x}_{3}+2\,m_{2}^{2})
\Big] \Big\}\, {\alpha}_{s}(t^{A}_{a}) \, a_{2}(t^{A}_{a})
\label{amp:a-a-t},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{A}_{b,P} &=&
2\,m_{1}\,p\, {\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}b_{3}db_{3}\,
H^{A}_{f}({\alpha}^{A},{\beta}^{A}_{b},b_{3},b_{2})\,
E^{A}_{f}(t^{A}_{b})\, {\alpha}_{s}(t^{A}_{b})
\nonumber \\ &{\times}&
a_{2}(t^{A}_{b})\,
\Big\{ {\phi}_{D}^{p}(x_{2})\, {\phi}_{P}^{p}(x_{3})\,
2\,m_{2}\,{\mu}_{P}\,\bar{x}_{2}
-{\phi}_{D}^{a}(x_{2})\, {\phi}_{P}^{a}(x_{3})\,
(m_{1}^{2}\,x_{2}+m_{3}^{2}\,\bar{x}_{2}) \Big\}
\label{amp:a-b-p},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{A}_{b,L} &=&
-{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}b_{3}db_{3}\,
H^{A}_{f}({\alpha}^{A},{\beta}^{A}_{b},b_{3},b_{2})\,
E^{A}_{f}(t^{A}_{b})\, {\alpha}_{s}(t^{A}_{b})\, a_{2}(t^{A}_{b})
\nonumber \\ &{\times}&
\Big\{ {\phi}_{D}^{p}(x_{2})\, {\phi}_{V}^{s}(x_{3})\,
4\,m_{1}\,m_{2}\,m_{3}\,p\,\bar{x}_{2}
+{\phi}_{D}^{a}(x_{2})\, {\phi}_{V}^{v}(x_{3})\,
(m_{1}^{2}\,s\,x_{2}+m_{3}^{2}\,t\,\bar{x}_{2}) \Big\}
\label{amp:a-b-l},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{A}_{b,N} &=&
m_{1}\,m_{3}\,{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}b_{3}db_{3}\,
H^{A}_{f}({\alpha}^{A},{\beta}^{A}_{b},b_{3},b_{2})\,
E^{A}_{f}(t^{A}_{b})
\nonumber \\ &{\times}&
{\alpha}_{s}(t^{A}_{b})\, a_{2}(t^{A}_{b})\,
{\phi}_{D}^{a}(x_{2})\, \Big\{ {\phi}_{V}^{V}(x_{3})\,
(s+2\,m_{2}^{2}\,x_{2})- {\phi}_{V}^{A}(x_{3})\,2\,m_{1}\,p \Big\}
\label{amp:a-b-n},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{A}_{b,T} &=&
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}b_{3}db_{3}\,
H^{A}_{f}({\alpha}^{A},{\beta}^{A}_{b},b_{3},b_{2})\,
E^{A}_{f}(t^{A}_{b})\, {\alpha}_{s}(t^{A}_{b})
\nonumber \\ &{\times}&
a_{2}(t^{A}_{b})\,
{\phi}_{D}^{a}(x_{2})\, \Big\{ {\phi}_{V}^{V}(x_{3})\,
2\,m_{1}\,m_{3}- {\phi}_{V}^{A}(x_{3})\,(m_{3}/p)\,
(s+2\,m_{2}^{2}\,x_{2}) \Big\}
\label{amp:a-b-t},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{A}_{c,P} &=&
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}b_{1}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}db_{3}\,
H^{A}_{n}({\alpha}^{A},{\beta}^{A}_{c},b_{1},b_{2})
\nonumber \\ &{\times}&
{\delta}(b_{2}-b_{3}) \,
\Big\{ {\phi}_{D}^{a}(x_{2})\,
{\phi}_{P}^{a}(x_{3})\, 2\,m_{1}\,p \Big[
{\phi}_{B_{q}^{\ast}}^{v}(x_{1})\,
(s\,x_{2}+2\,m_{3}^{2}\,\bar{x}_{3}-u\,\bar{x}_{1})
\nonumber \\ &+&
{\phi}_{B_{q}^{\ast}}^{t}(x_{1})\,m_{1}\,m_{b} \Big] +
{\phi}_{B_{q}^{\ast}}^{v}(x_{1})\, {\phi}_{D}^{p}(x_{2})\,
m_{2}\,{\mu}_{p}\,\Big[ {\phi}_{P}^{p}(x_{3})\,
2\,m_{1}\,p\,(x_{2}-\bar{x}_{3})
\nonumber \\ &+&
{\phi}_{P}^{t}(x_{3})\,(2\,m_{1}^{2}\,\bar{x}_{1}
-t\,x_{2}-u\,\bar{x}_{3}) \Big] \Big\}\, E_{n}(t^{A}_{c})\,
{\alpha}_{s}(t^{A}_{c})\, C_{1}(t^{A}_{c})/N_{c}
\label{amp:a-c-p},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{A}_{c,L} &=&
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}b_{1}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}db_{3}\,
H^{A}_{n}({\alpha}^{A},{\beta}^{A}_{c},b_{1},b_{2})
\nonumber \\ &{\times}&
{\delta}(b_{2}-b_{3})\,
\Big\{ {\phi}_{B_{q}^{\ast}}^{v}(x_{1})\,
{\phi}_{D}^{p}(x_{2})\, m_{2}\,m_{3} \Big[
{\phi}_{V}^{t}(x_{3})\,
(t\,x_{2}+u\,\bar{x}_{3}-2\,m_{1}^{2}\,\bar{x}_{1})
\nonumber \\ &+&
{\phi}_{V}^{s}(x_{3})\, 2\,m_{1}\,p\,(\bar{x}_{3}-x_{2})
\Big] + {\phi}_{D}^{a}(x_{2})\, {\phi}_{V}^{v}(x_{3})\,
\Big[ {\phi}_{B_{q}^{\ast}}^{t}(x_{1})\, m_{1}\,m_{b}\,s
\nonumber \\ &+&
{\phi}_{B_{q}^{\ast}}^{v}(x_{1})\,
4\,m_{1}^{2}\,p^{2}\,(x_{2}-\bar{x}_{1}) \Big]
\Big\}\, E_{n}(t^{A}_{c})\,
{\alpha}_{s}(t^{A}_{c})\, C_{1}(t^{A}_{c})/N_{c}
\label{amp:a-c-l},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{A}_{c,N} &=&
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}b_{1}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}db_{3}\,
H^{A}_{n}({\alpha}^{A},{\beta}^{A}_{c},b_{1},b_{2})\,
E_{n}(t^{A}_{c})
\nonumber \\ &{\times}&
{\delta}(b_{2}-b_{3})\, {\alpha}_{s}(t^{A}_{c})\,
\Big\{ {\phi}_{B_{q}^{\ast}}^{V}(x_{1})\,
{\phi}_{D}^{p}(x_{2})\, {\phi}_{V}^{T}(x_{3})\,
m_{1}\,m_{2}\,(u\,\bar{x}_{1}-s\,x_{2}-2\,m_{3}^{2}\,\bar{x}_{3})
\nonumber \\ &+&
{\phi}_{B_{q}^{\ast}}^{T}(x_{1})\, {\phi}_{D}^{a}(x_{2})\,
m_{3}\,m_{b}\, \Big[ {\phi}_{V}^{A}(x_{3})\,2\,m_{1}\,p
-{\phi}_{V}^{V}(x_{3})\,t \Big]
\Big\}\, C_{1}(t^{A}_{c})/N_{c}
\label{amp:a-c-n},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{A}_{c,T} &=&
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}b_{1}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}db_{3}\,
H^{A}_{n}({\alpha}^{A},{\beta}^{A}_{c},b_{1},b_{2})
\nonumber \\ &{\times}&
{\delta}(b_{2}-b_{3}) \,
E_{n}(t^{A}_{c})\, {\alpha}_{s}(t^{A}_{c})\,
\Big\{ {\phi}_{B_{q}^{\ast}}^{V}(x_{1})\,
{\phi}_{D}^{p}(x_{2})\, {\phi}_{V}^{T}(x_{3})\,
2\,m_{1}\,m_{2}\,(\bar{x}_{1}-x_{2})
\nonumber \\ &+&
{\phi}_{B_{q}^{\ast}}^{T}(x_{1})\, {\phi}_{D}^{a}(x_{2})\,
m_{3}\,m_{b}\, \Big[ {\phi}_{V}^{A}(x_{3})\,t/(m_{1}\,p)
-2\,{\phi}_{V}^{V}(x_{3}) \Big]
\Big\}\, C_{1}(t^{A}_{c})/N_{c}
\label{amp:a-c-t},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{A}_{d,P} &=&
\frac{1}{N_{c}}\,
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}b_{1}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}db_{3}\,
H^{A}_{n}({\alpha}^{A},{\beta}^{A}_{d},b_{1},b_{2})\, E_{n}(t^{A}_{d})
\nonumber \\ &{\times}&
{\delta}(b_{2}-b_{3})\, {\alpha}_{s}(t^{A}_{d})\,
C_{1}(t^{A}_{d})\, {\phi}_{B_{q}^{\ast}}^{v}(x_{1})\, \Big\{
{\phi}_{D}^{a}(x_{2})\, {\phi}_{P}^{a}(x_{3})\,
2\,m_{1}\,p\,( 2\,m_{2}^{2}\,x_{2}+s\,\bar{x}_{3}-t\,x_{1})
\nonumber \\ &+&
{\phi}_{D}^{p}(x_{2})\,
m_{2}\,{\mu}_{P}\, \Big[ {\phi}_{P}^{t}(x_{3})\,
(2\,m_{1}^{2}\,x_{1}-t\,x_{2}-u\,\bar{x}_{3})
+{\phi}_{P}^{p}(x_{3})\,2\,m_{1}\,p\,(\bar{x}_{3}-x_{2})
\Big] \Big\}
\label{amp:a-d-p},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{A}_{d,L} &=&
\frac{1}{N_{c}}\,
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}b_{1}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}db_{3}\,
H^{A}_{n}({\alpha}^{A},{\beta}^{A}_{d},b_{1},b_{2})\, E_{n}(t^{A}_{d})
\nonumber \\ &{\times}&
{\delta}(b_{2}-b_{3})\, {\alpha}_{s}(t^{A}_{d})\,
C_{1}(t^{A}_{d})\, {\phi}_{B_{q}^{\ast}}^{v}(x_{1})\, \Big\{
{\phi}_{D}^{a}(x_{2})\, {\phi}_{V}^{v}(x_{3})\,
u\,( 2\,m_{2}^{2}\,x_{2}+s\,\bar{x}_{3}-t\,x_{1})
\nonumber \\ &-&
{\phi}_{D}^{p}(x_{2})\,
m_{2}\,m_{3}\, \Big[ {\phi}_{V}^{t}(x_{3})\,
(2\,m_{1}^{2}\,x_{1}-t\,x_{2}-u\,\bar{x}_{3})
+{\phi}_{V}^{s}(x_{3})\,2\,m_{1}\,p\,(\bar{x}_{3}-x_{2})
\Big] \Big\}
\label{amp:a-d-l},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{A}_{d,N} &=&
\frac{1}{N_{c}}\,
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}b_{1}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}db_{3}\,
H^{A}_{n}({\alpha}^{A},{\beta}^{A}_{d},b_{1},b_{2})
\nonumber \\ &{\times}&
E_{n}(t^{A}_{d})\, {\alpha}_{s}(t^{A}_{d})\,
{\phi}_{B_{q}^{\ast}}^{V}(x_{1})\, \Big\{
{\phi}_{D}^{a}(x_{2})\, {\phi}_{V}^{V}(x_{3})\,
2\,m_{1}\,m_{3}\,( t\,x_{1}-2\,m_{2}^{2}\,x_{2}-s\,\bar{x}_{3})
\nonumber \\ &+&
{\phi}_{D}^{p}(x_{2})\,{\phi}_{V}^{T}(x_{3})\,
m_{1}\,m_{2}\,(u\,x_{1}-s\,x_{2}-2\,m_{3}^{2}\,\bar{x}_{3})
\Big] \Big\}\, C_{1}(t^{A}_{d})\, {\delta}(b_{2}-b_{3})
\label{amp:a-d-n},
\end{aligned}$$ $$\begin{aligned}
{\cal M}^{A}_{d,T} &=&
\frac{2}{N_{c}}\,
{\int}_{0}^{1}dx_{1}
{\int}_{0}^{1}dx_{2}
{\int}_{0}^{1}dx_{3}
{\int}_{0}^{\infty}b_{1}db_{1}
{\int}_{0}^{\infty}b_{2}db_{2}
{\int}_{0}^{\infty}db_{3}\,
H^{A}_{n}({\alpha}^{A},{\beta}^{A}_{d},b_{1},b_{2})
\nonumber \\ &{\times}&
E_{n}(t^{A}_{d})\, {\alpha}_{s}(t^{A}_{d})\,
{\phi}_{B_{q}^{\ast}}^{V}(x_{1})\, \Big\{
{\phi}_{D}^{a}(x_{2})\, {\phi}_{V}^{A}(x_{3})\,
(m_{3}/p)\,( 2\,m_{2}^{2}\,x_{2}+s\,\bar{x}_{3}-t\,x_{1})
\nonumber \\ &+&
{\phi}_{D}^{p}(x_{2})\,{\phi}_{V}^{T}(x_{3})\,
m_{1}\,m_{2}\,(x_{1}-x_{2}) \Big] \Big\}\,
C_{1}(t^{A}_{d})\, {\delta}(b_{2}-b_{3})
\label{amp:a-d-t}.
\end{aligned}$$ The functions $H_{f,n}^{A}$ and the Sudakov factor $E_{f}^{A}$ are defined as follows. $$\begin{aligned}
H_{f}^{A}({\alpha},{\beta},b_{i},b_{j}) &=&
\frac{{\pi}^{2}}{4} \, \Big\{ i\,J_{0}(b_{i}\sqrt{{\alpha}})
- Y_{0}(b_{i}\sqrt{{\alpha}}) \Big\}
\nonumber \\ & & \!\!\!\! \!\!\!\! \!\!\!\! {\times}\,
\Big\{ {\theta}(b_{i}-b_{j}) \Big[ i\,J_{0}(b_{i}\sqrt{{\beta}})
- Y_{0}(b_{i}\sqrt{{\beta}}) \Big] J_{0}(b_{j}\sqrt{{\beta}})
+\, (b_{i} {\leftrightarrow} b_{j}) \Big\}
\label{amp:hfa},
\end{aligned}$$ $$\begin{aligned}
H_{n}^{A}({\alpha},{\beta},b_{i},b_{j}) &=&
\Big\{ {\theta}(-{\beta})\, K_{0}(b_{i}\sqrt{-{\beta}})
+ \frac{\pi}{2}{\theta}(+{\beta})\,
\Big[ i\,J_{0}(b_{i}\sqrt{{\beta}})
- Y_{0}(b_{i}\sqrt{{\beta}}) \Big] \Big\}
\nonumber \\ & & \!\!\!\! \!\!\!\! \!\!\!
{\times}\, \frac{\pi}{2} \,
\Big\{ {\theta}(b_{i}-b_{j}) \Big[ i\,J_{0}(b_{i}\sqrt{{\alpha}})
- Y_{0}(b_{i}\sqrt{{\alpha}}) \Big]
J_{0}(b_{j}\sqrt{{\alpha}})
+ (b_{i} {\leftrightarrow} b_{j}) \Big\}
\label{amp:hna},
\end{aligned}$$ $$E_{f}^{A}(t)\ =\ {\exp}\{ -S_{D}(t)-S_{M}(t) \}
\label{sudakov-fa},$$ and the expressions for $E_{n}(t)$, $S_{B_{q}^{\ast}}(t)$, $S_{D}(t)$ and $S_{M}(t)$ are the same as those given in the Appendix \[block-t\]. ${\alpha}^{A}$ and ${\beta}_{i}^{A}$ are the gluon and quark virtualities; the subscripts of ${\beta}_{i}^{A}$ and $t_{i}^{A}$ correspond to the diagram indices of Fig.\[fig:fey-a\]. $$\begin{aligned}
{\alpha}^{A} &=& x_{2}^{2}\,m_{2}^{2}+\bar{x}_{3}^{2}\,m_{3}^{2}+x_{2}\,\bar{x}_{3}\,s
\label{gluon-a}, \\
%-----------------------------------------------------
{\beta}_{a}^{A} &=& \bar{x}_{3}^{2}\,m_{3}^{2}+\bar{x}_{3}\,s+m_{2}^{2}-m_{c}^{2}
\label{beta-aa}, \\
%-----------------------------------------------------
{\beta}_{b}^{A} &=& x_{2}^{2}\,m_{2}^{2}+x_{2}\,s+m_{3}^{2}
\label{beta-ab}, \\
%-----------------------------------------------------
{\beta}_{c}^{A} &=& {\alpha}^{A}+\bar{x}_{1}^{2}\,m_{1}^{2}
-\bar{x}_{1}\,x_{2}\,t-\bar{x}_{1}\,\bar{x}_{3}\,u-m_{b}^{2}
\label{beta-ac}, \\
%-----------------------------------------------------
{\beta}_{d}^{A} &=& {\alpha}^{A}+x_{1}^{2}\,m_{1}^{2}
-x_{1}\,x_{2}\,t-x_{1}\,\bar{x}_{3}\,u
\label{beta-ad}, \\
t_{a(b)}^{A} &=&
{\max}(\sqrt{{\alpha}^{A}},\sqrt{{\vert}{\beta}_{a(b)}^{A}{\vert}},1/b_{2},1/b_{3})
\label{t-aab}, \\
t_{c(d)}^{A} &=&
{\max}(\sqrt{{\alpha}^{A}},\sqrt{{\vert}{\beta}_{c(d)}^{A}{\vert}},1/b_{1},1/b_{2})
\label{t-acd}.
\end{aligned}$$
[99]{} C. Patrignani [*et al.*]{} (Particle Data Group), Chin. Phys. C 40, 100001 (2016). Ed. A. Bevan [*et al.*]{}, Eur. Phys. J. C 74, 3026 (2014). http://lhcb-operationsplots.web.cern.ch/lhcb-operationsplots/index.htm. H. Li, Phys. Rev. D 52, 3958 (1995). C. Chang, H. Li, Phys. Rev. D 55, 5577 (1997). T. Yeh, H. Li, Phys. Rev. D 56, 1615 (1997). Y. Keum, H. Li, A. Sanda, Phys. Lett. B 504, 6 (2001). Y. Keum, H. Li, A. Sanda, Phys. Rev. D 63, 054008 (2001). Y. Keum, H. Li, Phys. Rev. D 63, 074006 (2001). C. Lü, K. Ukai, M. Yang, Phys. Rev. D 63, 074009 (2001). C. Lü, M. Yang, Eur. Phys. J. C 23, 275 (2002). M. Beneke [*et al.*]{}, Phys. Rev. Lett. 83, 1914 (1999). M. Beneke [*et al.*]{}, Nucl. Phys. B 591, 313 (2000). M. Beneke [*et al.*]{}, Nucl. Phys. B 606, 245 (2001). D. Du, D. Yang, G. Zhu, Phys. Lett. B 488, 46 (2000). D. Du, D. Yang, G. Zhu, Phys. Lett. B 509, 263 (2001). D. Du, D. Yang, G. Zhu, Phys. Rev. D 64, 014036 (2001). M. Beneke, J. Rohrer, D. Yang, Nucl. Phys. B 774, 64 (2007). J. Sun [*et al.*]{}, Phys. Rev. D 77, 074013 (2008). C. Bauer, S. Fleming, M. Luke, Phys. Rev. D 63, 014006 (2000). C. Bauer [*et al.*]{}, Phys. Rev. D 63, 114020 (2001). C. Bauer, I. Stewart, Phys. Lett. B 516, 134 (2001). C. Bauer, D. Pirjol, I. Stewart, Phys. Rev. D 65, 054022 (2002). C. Bauer [*et al.*]{}, Phys. Rev. D 66, 014017 (2002). M. Beneke [*et al.*]{}, Nucl. Phys. B 643, 431 (2002). M. Beneke, T. Feldmann, Phys. Lett. B 553, 267 (2003). M. Beneke, T. Feldmann, Nucl. Phys. B 685, 249 (2004). J. Chay, Phys. Lett. B 476, 339 (2000). Y. Keum [*et al.*]{}, Phys. Rev. D 69, 094018 (2004). R. Li, C. Lü, H. Zou, Phys. Rev. D 78, 014018 (2008). Q. Chang [*et al.*]{}, Eur. Phys. J. C 76, 523 (2016). V. Šimonis, Eur. Phys. J. A 52, 90 (2016). G. Buchalla, A. Buras, M. Lautenbacher, Rev. Mod. Phys. 68, 1125, (1996). J. Bjorken, Nucl. Phys. B (Proc. Suppl.) 11, 325 (1989). G. Lepage, S. Brodsky, Phys. Rev. D 22, 2157 (1980). P. Ball [*et al.*]{}, Nucl. Phys. B 529, 323 (1998). T. Kurimoto, H. Li, A. Sanda, Phys. Rev. D 65, 014007 (2001). J. Sun [*et al.*]{}, Phys. Rev. D 92, 074028 (2015). Y. Yang [*et al.*]{}, Phys. Lett. B 751, 171 (2015). J. Sun [*et al.*]{}, Phys. Lett. B 752, 322 (2016). P. Ball, JHEP, 9901, 010 (1999). P. Ball, G. Jones, JHEP, 0703, 069 (2007). P. Ball, V. Braun, A. Lenz, JHEP, 0605, 004 (2006). C. Chen, Y. Keum, H. Li, Phys. Rev. D 66, 054013 (2002). Y. Yang [*et al.*]{}, Int. J. Mod. Phys. A 31, 1650146 (2016). J. Sun [*et al.*]{}, Nucl. Phys. B 911, 890 (2016). J. Sun [*et al.*]{}, Phys. Rev. D 95, 036024 (2017). Th. Feldmann, P. Kroll and B. Stech, Phys. Rev. D 58, 114006 (1998). C. Cheung, C. Hwang, JHEP, 1404, 177 (2014). A. Kamal, Particle Physics, Springer-Verlag Berlin Heidelberg, 2014, p. 297-298. B. Colquhoun [*et al.*]{} (HPQCD Collaboration), Phys. Rev. D 91, 114509 (2015). Ed. A. Bevan [*et al.*]{}, Eur. Phys. J. C 74, 3026 (2014). T. Gershon, M. Needham, Comptes Rendus Physique 16, 435 (2015). R. Aaij . [*et al.*]{} (LHCb Collaboration), Phys. Rev. Lett. 118, 052002 (2017). A. Akeroyd [*et al.*]{}, arXiv:1002.5012\[hep-ex\].
| ArXiv |
---
author:
- 'I. Platais'
- 'C. Melo'
- 'J.-C. Mermilliod'
- 'V. Kozhurina-Platais'
- 'J. P. Fulbright'
- 'R. A. Méndez'
- 'M. Altmann'
- 'J. Sperauskas'
bibliography:
- '5756bib.bib'
date: 'Received 2 June 2006/ Accepted 21 August 2006'
title: 'WIYN Open Cluster Study. XXVI. Improved kinematic membership and spectroscopy of IC 2391 [^1] [^2] '
---
Introduction
============
IC 2391 is a young ($\sim$35 Myr) and nearby ($d\sim$150 pc) open cluster located in Vela ($\ell=270\degr$, $b=-7\degr$). Its proximity is very appealing for any detailed studies of intrinsically faint low-mass stars and brown dwarfs [@bar04]. The significance of IC 2391 is clearly demonstrated by a large number of literature references over the last decade in the SIMBAD Astronomical Database: $\sim$200 publications where the cluster has been mentioned or studied. One problem facing the researchers of IC 2391 is the scarcity of confirmed cluster members. For a long time, the known cluster membership was confined to merely $\sim$20 stars, all brighter than $V$$\sim$11 [@hog60]. Then, @sta89 reported on the proper-motion study of 883 stars over a $48\arcmin\times41\arcmin$ area. Proper motions, $BV\!RI$ photometry, and high-resolution spectroscopy together yielded a list of ten additional probable cluster members down to $V=14$. This list was substantially extended by using the ROSAT imaging data [@pat93; @pat96; @sim98] to take advantage of the known strong X-ray activity among the young G-K-M spectral type stars. In the follow-up spectroscopic study, @sta97 confirmed the cluster membership of 23 X-ray selected stars down to $V$$\sim$15, using the radial velocity, Li line, and H$_\alpha$ appearance as membership criteria. @dod04 attempted to identify more cluster members by mining USNO-B and 2MASS catalogs. From 185 astrometrically selected possible cluster members, a total of 35 stars are brighter than $R=15$. However, a disturbingly small fraction of these stars ($\sim$20%) are common with the Patten & Simon (1993) list in the same magnitude range and spatial coverage. The latest search for cluster members in the central $30\arcmin\times30\arcmin$ region of IC 2391 by the XMM-Newton X-ray observatory resulted in nine relatively faint possible new members [@mar05].
As indicated above, proper motions have been used as a kinematic membership discriminator for IC 2391. However, only the study by @kin79 provides precise relative proper motions ($\sigma=0.9$ mas yr$^{-1}$) down to $V$$\sim$12 over a $1\fdg7\times0\fdg9$ area. In this study, from a total of 232 stars about 40 have proper motions consistent with membership in IC 2391. No formal membership probabilities are calculated, apparently owing to the sparseness of the cluster. Another way to ascertain the membership status, independent of any assumptions on the astrophysical properties of probable cluster members, is to use radial velocities. There is a rich literature on this subject for IC 2391, e.g., @fei61, @bus65, @per69a, @vhoo72, @lev88, @sta97, @bar99. Nearly 100 stars have had their radial velocities measured, many of them several times. In many cases, however, the precision of the radial velocities was low, especially for the early and very late type stars, thereby largely precluding assignment of a reliable membership status to them.
Often IC 2391 is considered along with IC 2602, because both have very similar properties and are separated spatially only by $\sim$50 pc, thus suggesting a common origin. Their absolute proper motions, however, differ significantly. The projected total velocity in the tangential plane for IC 2391 is 33.8 mas yr$^{-1}$, whereas for IC 2602 it is only 20.5 mas yr$^{-1}$ [@rob99]. A much larger tangential velocity of IC 2391 considerably increases the reliability of membership probabilities drawn from proper motions, since a smaller fraction of field stars are expected to share the motion of the cluster. This and the limited precision of our proper motions are one of the main reasons for selecting IC 2391 as the subject of this study.
@ran01 spectroscopically analyzed $\sim$50 X-ray selected candidate members in IC 2391 and IC 2602. From the analysis of 8 Fe I lines in four stars, the mean metallicity of IC 2391 was derived to be \[Fe/H\]=$-0.03\pm0.07$. In this study, Li abundance was obtained for 32 possible members of IC 2391, covering a wide range of $T_{\rm eff}$ – from 3500 to 6600 K. It was found that stars warmer than $\sim$5800 K or more massive than $\sim$1 M$_\odot$ show no significant signs of Li depletion. For cooler late-G to early-K stars, the pattern of Li abundances in IC 2391 and the Pleiades is similar, although hinting that in this $T_{\rm eff}$ range Li is less depleted in IC 2391, as one would expect from the age difference. A more detailed analysis of Li abundance in IC 2391 is hindered by the small number of stars in the @ran01 sample.
Young open clusters appear to have stars with a broad range of rotational rates [@her05]. That is also confirmed by the observed rotational rates for late-type stars [@pat96] and the $v\sin i$ distribution in IC 2391 [@sta97]. From the standpoint of Li abundances, X-ray luminosities, and stellar evolution, it is vital to identify fast rotators in the enlarged sample of cluster stars.
In some aspects, the level of our understanding of the open cluster IC 2391 is similar to NGC 2451A, which was recently studied by @pla01 as one of the WIYN Open Cluster Study (WOCS) targets. The lack of comprehensive astrometric cluster membership prompted us to include IC 2391 among the WOCS clusters. Following the WOCS strategy [@mat00], we derived new proper motions and calculated the cluster membership probabilities. For many probable cluster members, high-resolution spectroscopy served to measure the radial velocities, projected rotational velocities $v\sin i$, Li abundance, and equivalent width of H$_\alpha$. A few carefully selected cluster stars are used to obtain metallicity \[Fe/H\]. New CCD photometry is used to construct reliable color-magnitude diagram and perform the isochrone fit.
Astrometric reductions and cluster membership
=============================================
A total of four 8$\times$10 inch photographic plates (scale $=55\farcs1$ mm$^{-1}$), taken with the 51 cm double astrograph of Cesco Observatory in El Leoncito, Argentina, were used for astrometry. Two of these visual-bandpass plates (103a-G emulsion and OG-515 filter) were obtained in 1967.29, the other two in 1996.14. An objective wire-grating was used to produce diffraction images for all stars brighter than $V$$\sim$13. Each first-epoch plate contains two exposures: a 30 min and an offset 1 min exposure.
Our target stars were drawn from the COSMOS/UKST Object Catalog [@yen92]. In this catalog the object brightness is given in $B_{\rm J}$ magnitudes as derived in the natural photographic system (IIa-J emulsion and GG-395 filter) of the UK 1.2 m Schmidt Telescope at Siding Spring, Australia [@bla82]. Due to the scan-time limitations set by the measuring machine, the sample selection required an optimization. All stars down to $B_{\rm J}=13.0$ were chosen in a $3\fdg5\times2\fdg7$ rectangle centered on $\alpha=8^{\rm h} 40^{\rm m}$ and $\delta=-52\degr 53\arcmin$ (equinox J2000.0). A sub-sample of fainter stars at $B_{\rm J}=14.6$ in the same area served as anonymous astrometric reference stars. Then, within this rectangle all additional stars down to $B_{\rm J}=16.2$ were selected in a circle with the radius of $0\fdg8$ centered on $\alpha=8^{\rm h} 42\fm5$ and $\delta=-53\degr$.
Altogether, our initial sample included over 7,000 stars. All measurable images of these stars were digitized with the Yale 2020G PDS microdensitometer in a fine-raster, object-by-object mode. The image positions were determined using the Yale Image Centering routine [@lee83], which includes a two-dimensional Gaussian fit.
The positions and proper motions were calculated using the standard SPM (Southern Proper Motion program) astrometric reductions, described in detail by @gir98 and @pla98. Owing to the relatively small 9-deg$^2$ field, only linear and quadratic plate-tilt terms were used in the proper-motion plate model. The standard error of proper motions was estimated to range from 1.4 to 2.1 mas yr$^{-1}$, depending on the star’s magnitude and hence on the number of available grating images. The calculated relative proper motions are free of apparent systematic errors. The distribution of proper motions or a vector-point diagram (VPD) is shown in Fig. 1. A visible clumping of proper motions at $\mu_x=-20$, $\mu_y=+18$ mas yr$^{-1}$ indicates the presence of IC 2391 members.
The local sample method [@koz95] was used to calculate the cluster membership. In this method, for each target star a representative sub-sample (a bin) of other stars is formed that shares the properties of the target, such as the brightness. We used a wide 10-magnitude sliding brightness bin, which for the brightest and faintest stars narrows down to 5 magnitudes. No spatial window was used for this sample of proper motions. Similar to the case of NGC 2451A [@pla01], a flat distribution of field stars in VPD was adopted in the vicinity of the cluster centroid. The resulting membership probability, $P_\mu$, is defined as
$$P_\mu=\frac{\Phi_c}{\Phi_c+\Phi_f},$$
where, $\Phi_c$ is a Gaussian representing the cluster star distribution in the VPD and $\Phi_f$ is the distribution of field stars, both defined within the same magnitude bin. The following parameters were adopted for the cluster star distribution: the cluster center in the VPD at $\mu_{x}^{c}=-20.1$ and $\mu_{y}^{c}=+17.6$ mas yr$^{-1}$; the Gaussian width $\sigma_{c}=1.7$ mas yr$^{-1}$. A total of 115 stars, all having membership probability $P_{\mu}\geq5\%$, are listed in Table 1. A formal sum of membership probabilities indicates that in our sample of 6991 stars, only $\sim$65 are members of IC 2391.
Finally, precise equatorial coordinates were calculated for all stars, choosing UCAC2 stars [@zac04] as a reference frame. The coordinate transformation required a quadratic plate model supplemented with two main cubic distortion terms. The standard error of that transformation via the least-squares formalism is $\sim$60 mas. At the epoch and equinox of J2000, the estimated average accuracy of the catalog[^3] positions is about 30 mas. We note that this study provides precise coordinates for many X-ray selected cluster members, which so far have had only approximate coordinates from @pat96 and @mar05.
Photometry and CCD reductions
=============================
A few sources providing $U\!BV$ photometry cover mainly the inner area of IC 2391, e.g., @hog60, @lyn60, @per69b. In January 1997 we obtained new CCD $BV$ photometry for the majority of the possible proper-motion members. The observations were made at the Cerro Tololo Inter-American Observatory (CTIO) 0.9 m telescope with the Tektronics $2048\times2048$ CCD chip, which covers $13\farcm5\times13\farcm5$ on the sky.
As in the case of NGC 2451A [@pla01], each probable cluster member or else a group of members with mutual separations less than $\sim$10$\arcmin$ was observed individually. Only a small fraction of these stars could be identified on more than one CCD frame in each filter. In total, 114 CCD frames were obtained in $B$ and $V$ filters with exposure time varied from 1 to 150 s, depending on the star’s magnitude. A set of @lan92 standards was taken three times a night over 22-24 January, 1997 – spanning our observations of IC 2391. The twilight sky frames were used to correct the pixel-to-pixel sensitivity variations.
All CCD frames were reduced using the IRAF DAOPHOTX photometry package. The aperture photometry routine PHOT was applied because the target stars were optimally exposed and well-isolated. The details of transformation of the instrumental magnitudes into the standard $BV$ system for this observing run are given in @pla01. The final $BV$ magnitudes are believed to be on the standard system to within 0.02 mag and have a standard error of 0.03 mag in $V$ and 0.02 in $B-V$. It should be noted that the reddest stars may have a slightly less accurate photometry since the reddest standard star has only $B-V=1.18$.
Table \[tab:phot\] shows the comparison of our CCD $BV$ photometry with seven other sources of $BV$ photometry, mainly photoelectric. This table contains the literature reference, the number of common stars $(n)$, mean $\Delta V$ and $\Delta (B-V)$ in the sense of our CCD photometry minus the published one. The errors are standard deviations from the mean calculated difference in magnitude or color. All the sources of photometry are consistent, though on average our CCD $V$-magnitudes appear to be fainter by $\sim$0.03 than those from the other sources.
<span style="font-variant:small-caps;">Coravel</span> radial velocities and $v\sin i$
=====================================================================================
The radial-velocity observations were made with the photoelectric spectrometer <span style="font-variant:small-caps;">Coravel</span> [@bar79; @may85] on the Danish 1.54 m telescope at ESO, La Silla, Chile. They were obtained starting in March 1984 through April 1996 (when the <span style="font-variant:small-caps;">Coravel</span> was retired) during the course of regular ESO and Danish time runs allocated to the open cluster studies. In the <span style="font-variant:small-caps;">Coravel</span>-type instruments, the spectrum of a star is electro-mechanically correlated, i.e., scanned with an appropriate spectral mask in the focal plane. The output correlation profile can be described with a Gaussian (position, depth, width) plus a continuum level, readily providing an estimate of radial velocity and $v\sin i$. The latter is computed following the techniques described by @ben81 [@ben84]. The radial velocities are on the system defined by @udr99, calibrated with high-precision data from the <span style="font-variant:small-caps;">ELODIE</span> spectrograph [@bar96].
The initial sample consisted of only 8 stars, all brighter than $B=11.6$ and already thought to be possible members in 1983. Three of them (3664=SHJM 7, 3722, 5382) were found to be SB1 spectroscopic binaries and two were found to be SB2 (389, 4413). It should be noted that <span style="font-variant:small-caps;">Coravel</span> can detect a secondary if the magnitude difference is smaller than $\sim$1.5-1.8 mag.
The list of candidate cluster members was greatly enlarged by the ROSAT X-ray detections in the area of IC 2391 [@pat93; @pat96]. A total of 17 such stars ($B<13.8$) were observed once or twice in February 1995 and/or January-April 1996. Among them, two additional likely spectroscopic binaries (5768, 5859) were detected.
Two presumably constant stars are common to the <span style="font-variant:small-caps;">Coravel</span> and FEROS (see Sect. 5) samples. The radial velocity difference, in the sense <span style="font-variant:small-caps;">Coravel</span>$-$FEROS, for star 4362 is +0.09 km s$^{-1}$ and for star 4809 is $-0.10$ km s$^{-1}$. This indicates very good agreement between the two systems. There are 13 stars in common with the radial-velocity data obtained in 1995 at CTIO by @sta97. One of them, 5859=VXR 67a, appears to be a spectroscopic binary judging from two <span style="font-variant:small-caps;">Coravel</span> observations (Table 3). The spectroscopic binary, suspected by @sta97, 4549=VXR 30, is a definite SB1 from the <span style="font-variant:small-caps;">Coravel</span> data. For the remaining 11 stars, the radial velocity difference ‘<span style="font-variant:small-caps;">Coravel</span>$-$CTIO’ is $-0.6\pm0.3$ km s$^{-1}$, which is in good agreement with the listed internal and external errors by @sta97.
Table 3 contains all <span style="font-variant:small-caps;">Coravel</span> heliocentric radial velocities and their estimated standard errors. The last four entries were obtained with the Lithuanian <span style="font-variant:small-caps;">Coravel</span>-type spectrometer [@upg02] at the CASLEO 2.2 m telescope in El Leoncito, Argentina, in February 2002.
Table \[tab:meanrv\] lists the mean radial velocities, $v\sin i$, the associated errors and other parameters. For a star, depending on the structure of the correlation profile, up to three sets of parameters (for components A and B, and a blend) can be given. The estimated formal standard deviation of the mean radial velocity is denoted by $\sigma$. The estimated mean internal uncertainty of a radial velocity measurement is denoted by $\epsilon=$ max$(\sigma, mean~error)/\sqrt{n}$, where $n$ is the number of measurements. The internal error of an individual observation consists of three components as described by @bar79 and @mer89. We note that the uncertainty, $\epsilon$, is progressively underestimated at increasing $v\sin i$ values [@nor96]. The ratio, standard deviation $\sigma$ vs. the mean internal uncertainty of individual measurements, denoted by the E/I, is used to calculate the probability $P(\chi^2)$ that the scatter is due to random noise. A star is considered a spectroscopic binary, if $P(\chi^2)\leq0.001$. In the case of a single measurement, the $P(\chi^2)$ is meaningless and marked by 9.999. Additional columns 6, 7, 8, and 9 contain the number of measurements, $n$, the time span in days covered by the observations, $\Delta T$, measured $v\sin i$ and its standard error. Binarity status and the component are indicated in last column.
Finally, we note that the systems of $v\sin i$ from <span style="font-variant:small-caps;">Coravel</span>, FEROS, and @ran01 are all in excellent agreement for $v\sin i\leq 20$ km s$^{-1}$ to within $\sim$1 km s$^{-1}$, but can differ by up to $\sim$10 km s$^{-1}$ for large $v\sin i$ and/or in some cases of spectroscopic binaries.
Spectroscopy with FEROS
=======================
The spectroscopic observations were carried out in two observing runs in February 2004 and February 2005 using the high-resolution, two-fiber ($R\sim$50,000) FEROS echelle spectrograph [@kaufer99] at the ESO 2.2 m telescope in La Silla, Chile. This spectrograph provides a wavelength range from 360 to 920 nm, covered by 39 echelle orders. Such a wide wavelength range is essential in the chemical abundance studies allowing one to select appropriate unblended metal lines. The observations were taken in the Object-Sky mode in which the target is centered onto the Object fiber, whereas the Sky fiber collects the light from the sky background. The reductions of spectra were performed using the standard FEROS pipeline, which includes flat-fielding, sky-background subtraction, removal of cosmic rays, wavelength calibration, and barycentric velocity correction. The pipeline yields a 1-D re-binned spectrum evenly sampled at 0.03Å steps.
Radial velocities are derived by cross-correlation techniques using a K0 III spectral type digital binary mask as the template [@bar79; @queloz95]. The resulting cross-correlation function (CCF) can in most cases be approximated by a Gaussian function whose center readily gives the radial velocity and the width (Gaussian $\sigma$) related to broadening mechanisms such as turbulent motion, gravity pressure, and rotation.
Owing to a relatively high $S/N$, normally higher than 50 (see Table 5), the photon noise errors in our radial velocity measurements typically range between 5-15 m s$^{-1}$. The final uncertainties in derived radial velocities are a combination of the photon noise errors and the overnight spectrograph drift due to the changes in the index of air refractivity and atmospheric pressure. Usually, these shifts are on the order of a few hundred m s$^{-1}$ per night. One way to correct for this drift is to have the calibration lamp illuminate a second fiber while the first fiber receives the stellar light. Although the FEROS two-fiber configuration allows for this option [@setiawan00], the subsequent data reduction and analysis is complicated and not warranted for young stars where the radial-velocity jitter exceeds the internal precision of measurement by a lot. For example, for very active and young T Tauri type stars, this jitter is not lower than $\sim$0.6 km s$^{-1}$ and can be as high as $\sim$2 km s$^{-1}$ [@melo03]. With age these effects gradually abate, though the rotational modulation of stellar active regions can generate a radial velocity scatter up to 50 m s$^{-1}$ even at the age of Hyades at $\sim$600 Myr [@pau04]. We therefore opted for a simpler approach, as described below.
In order to correct for this drift, 1-3 radial velocity standards from the CORALIE extra-solar planet survey were observed a few times during the night. For each standard star, the velocity drift was computed as the difference between the CORALIE radial velocity and the observed radial velocity. In the case of more than one standard star, the mean drift was calculated. The radial-velocity corrections for the program stars were computed using a linear interpolation between each two drift points. A typical drift correction is shown in Fig. \[fig:drift\]. Over two observing runs, the r.m.s. of the differences between the corrected radial velocities of standard stars and the CORALIE radial velocities is below 20 m s$^{-1}$, indicating that the procedure works well. One should keep in mind, however, that the final uncertainty of radial velocities could be higher, since at some level a linear interpolation might not approximate the actual pattern of a drift accurately. We note that the r.m.s. scatter of the drift correction between the FEROS and CORALIE radial velocities is $\sim$150 m s${^-1}$, similar to the value found by [@melo01] using $\tau$ Cet as a standard.
A few double-lined spectroscopic binaries were found among our target stars (Table 5, Fig. \[fig:sbs\]). Their cluster membership status is briefly discussed in Sect. 6.4.
FEROS $v\sin i$
---------------
The width, $\sigma $, of the cross-correlation function (CCF) is a product of several broadening mechanisms related to gravity, turbulence, magnetic fields, effective temperature, metallicity, and rotation. In addition, the instrumental profile also contributes to the broadening of spectral lines and, therefore, to the CCF. Thus, in order to correctly measure the contribution of rotation to the width of the CCF, we should model all sources of broadening, except what is due to rotation. The width of FEROS CCF was calibrated as described in [@melo01]. For fast rotators ($v\sin i \ga 30$ km s$^{-1}$), the final $v\sin i$ was derived as follows. The CCF is fitted by a family of functions $CCF_{v\sin i}=C-D[g_0\otimes G(v\sin i)]$ which is a convolution of the CCF of a non-rotating star, $g_0$, in turn approximated by a Gaussian and the [@Gray92] rotational profile computed for a set of discrete rotational velocities, $G(v\sin i)$. For each function $CCF_{v\sin i}$, we can find the radial velocity $V_{\rm r}$, the depth D, and the continuum C by minimizing the quantity $\chi^2_{v\sin i}$. Figure 1 in @melo03 illustrates this method. The uncertainty of the $v\sin i$ measurement is $\sim$1.5 km s$^{-1}$ for $v\sin \la 30$ km s$^{-1}$ and $\sim$10% of the $v\sin i$ value for $v\sin \ga 30$ km s$^{-1}$. The FEROS measurements of $v\sin i$ are listed in Table 5. The averaged $v\sin i$ values are given in Table \[tab:meanli\]. In the case of SB2 spectroscopic binaries, an average $v\sin i$ is given only for the A-component.
\[Fe/H\] determination
----------------------
The FEROS high-resolution spectra are used to determine the \[Fe/H\] abundance of IC 2391. However, other abundance studies of relatively young open clusters, such as the Hyades, Pleiades, and the Ursa Major moving group [@sch06; @yon04; @pau03], have found that the atmospheres of the G- and K-dwarfs show deviations from the simple, plane-parallel atmosphere models used in most abundance studies. High excitation lines and lines of ionized species are stronger than predicted by simple model atmospheres, even though the same models perform adequately for older G- and K- dwarfs like the Sun. @sch06 suggest that photospheric spots and faculae in young dwarfs can possibly produce such deviations.
Until the solution to this problem is found, we only perform a simple abundance analysis on a few stars in IC 2391. The atmosphere’s problem is minimized in warmer G-dwarfs, hence we chose four slowly-rotating G-dwarfs with $T_{\rm eff}$ values ranging between 5200 K and 5900 K in the sample.
We adopted the list of iron lines from @ful06. The lines in this list were selected to be those least affected by blending for use in the study of metal-rich bulge giants, where the differential analysis was done relative to Arcturus. For this study, we used the Sun as the differential standard and adopt a solar iron abundance of $\log \epsilon$(Fe)$=$7.45. The lines were measured manually using the IRAF [*splot*]{} package, and the measured equivalent widths are given in Table 6. We measured the solar line equivalent widths from the Solar Atlas by @kur84 and a high $S/N$ ($>$200) FEROS sky spectrum. We did not use lines stronger than $\sim$120 mÅ.
There is good agreement between the results from the Solar Atlas and the FEROS solar spectrum. If the $gf$-values of the Fe lines are adjusted to yield a solar Fe abundance of $\log \epsilon$(Fe)$=$7.45 for the equivalent widths measured in the Solar Atlas, the FEROS sky spectrum yields an Fe I abundance of $7.44 \pm 0.06$ (97 lines) and an Fe II abundance of $7.42 \pm 0.07$ (5 lines). Therefore, we believe that the instrumental effects in our differential analysis method are negligible.
Due to the aforementioned potential problems with the atmospheres, we did not use spectroscopic indicators to set the stellar parameters, with the exception of setting the model atmosphere \[m/H\] value to match the derived \[Fe/H\] value. We used a grid of solar-abundance ratio atmospheres by Fiorella Castelli[^4] that include updated opacity distribution functions and the 2002 version of the MOOG spectrum synthesis program [@moog].
We set the $T_{\rm eff}$ values as an average from three color-to-$T_{\rm eff}$ calibrations for dwarf stars [@ram05], adopting the initial \[Fe/H\]=0, very close to the estimate by @ran01. We used $B-V$, $V-J$, and $V-K_s$ color indices to obtain $T_{\rm eff}$, where $J$ and $K_s$ are the 2MASS magnitudes [@skr06]. In the temperature range 4400-5700 K, the common stars between our study and @ran01 indicate a nearly perfect match of temperature scales. If all nine common stars are considered, the r.m.s. scatter of differences approaches 140 K. It should not be overlooked, however, that such photometrically-calibrated effective temperatures may be biased (e.g., cooler), if a star is a binary and/or still on the pre-main-sequence evolutionary tracks. The @ram05 calibrations do not consider such effects though their presence in IC 2391 is undeniable (see Sect. 6.5). We derived the surface gravities $\log g$ by interpolation of the 35-Myr, solar-metallicity isochrone by @gir02. Finally, we used the relationship provided by @all04 to set the microturbulence parameter $v_{\rm t}$.
The derived atmosphere parameters and Fe abundances are given in Table \[tab:abund\]. The line-by-line abundances show slight trends with respect to excitation potential and line strength, but we did not adjust the stellar parameters or remove any high-excitation lines from the analysis. We point out an especially large difference (mean of $0.15 \pm 0.06$ dex) between the abundances derived from Fe I and Fe II lines, noting that by definition these same lines give the same abundance for both species in the Sun. This is similar to what was seen in studies of other young open clusters, so we only used the results for the Fe I lines to derive the mean cluster abundance.
The weighted mean \[Fe I/H\] value for the four IC 2391 stars is $+0.06 \pm 0.06$ (s.d.). When adopting the solar abundance of $\log \epsilon$(Fe)$=$7.45, the mean \[Fe/H\] reported by @ran01 translates into \[Fe/H\]=$+0.04\pm0.07$, a metallicity estimate nearly identical to our value. Recently, in the framework of UVES Paranal Observatory Project, @stu06 obtained elemental abundances for five early-type stars in IC 2391. The weighted mean \[Fe/H\] value for the two bona fide cluster members (HD 74275=4522 and SHJM 2=3722) from this study is $+0.10\pm0.07$, again in good agreement with our value of \[Fe/H\]. A third bona fide member HD 74044=7027 was also included in the @stu06 analysis; however, it shows signs of being chemically peculiar and, hence, is not suitable for deriving average elemental abundances in cluster stars.
Abundances derived from Fe I lines are sensitive to the adopted $T_{\rm eff}$ values: a $+100$ K increase in $T_{\rm eff}$ will raise the \[Fe/H\] value by about $+0.10$ dex. Our temperature scale comes only from photometric colors, so it is sensitive to errors in the photometry, to color shifts due to unresolved binaries, and to the intrinsic uncertainties of the fitting functions used to derive the $T_{\rm eff}$ scales (the $\sigma T_{\rm eff}$ values the T(B-V), T(V-J), and T(V-K) calibrations are 88 K, 62 K, and 50 K, respectively). For example, we note that star 3359 is slightly above the isochrone fitting the main sequence in two out of the three color-magnitude diagrams (see Sect. 6.5). If this is due to the problem in photometry, and we have adopted a $T_{\rm eff}$ value that could be lower than true $T_{\rm eff}$ by $\sim$50 K, then our final \[Fe/H\] value for star 3359 is too low by $\sim$0.05 dex.
Li abundance
------------
The equivalent width of Li 6708Å is known to be a good youth indicator for cool stars (late G to mid-M type stars) whose interiors are fully convective during the pre-main sequence phase [e.g., @martin97]. For hotter stars (F to mid-G corresponding to $B-V=0.4-0.7$), the development of a radiative core in the pre-main-sequence phase prevents a rapid Li depletion in these stars. Thus, for hotter stars, the changes in Li abundance are insignificant between a few Myr and the Pleiades age [e.g., @sod93].
The equivalent width of Li 6708Å and H$_\alpha$ was measured manually using the IRAF [*splot*]{} task. Because in fast rotating stars this Li feature is blended with an Fe I line at 6707.441 Å, we applied an empirical correction [Sect. 2.2 @sod93] to the equivalent width of the Li feature for all stars with $v\sin i>10$ km s$^{-1}$. Then, the equivalent width was converted into Li abundance, $\log$ N(Li), via the grid of curves of growth tabulated by @sod93. Effective temperatures were derived as described in Sect. 5.2. The initial analysis of the $\log$ N(Li) distribution as a function of $T_{\rm eff}$ showed a number of stars with abnormally high Li abundance, exceeding $\log$ N(Li)$=3.5$ dex. In our FEROS spectra these stars have $v\sin i \geq40$ km s$^{-1}$ and a poorly-defined continuum around the Li feature. For such stars, it was decided to adopt the smoothed adjacent spectrum just outside the expected width of a Li feature for a continuum. The measured equivalent widths, EW(Li), are listed in Table \[tab:meanli\]. Ambiguous lithium non-detections due to the high rotational velocity or binarity are marked by $-99$. For stars with $v\sin i<15$ km s$^{-1}$, the r.m.s. scatter of measured EWs from two independent measurements is estimated to be $\sim$6 mÅ. In this range of $v\sin i$, there are only three stars, 4362, 4413, and 5859, in common with @ran01. The measurements of EW in both studies agree to within $2\sigma$ of the errors quoted by @ran01. For bona fide cluster members of IC 2391, the distribution of lithium abundance as a function of $T_{\rm eff}$ is shown in Fig. \[fig:li\].
From the FEROS spectra we also estimated an equivalent width of H$_{\alpha}$. Eight relatively cool stars in Table \[tab:meanli\] show H$_{\alpha}$ in emission and therefore have negative EW values.
Results
=======
One of the goals of this paper is to assemble a list of bona fide members of IC 2391 and their basic properties down to $V\sim15$. Such a list is expected to be useful for subsequent future studies of this sparse open cluster.
Bona fide cluster members
-------------------------
We used the following criteria to decide whether or not to assign the status of bona fide member to a probable cluster member:
1. Proper motion membership probability $P_{\mu}\geq5\%$, or all 115 stars from Table 1. This low $P_{\mu}$ threshold is chosen in order to essentially preserve all possible cluster members.
2. Within the photometric errors a star is on the zero-age main sequence (ZAMS) or above it due to the binarity and/or the pre-main sequence status at $V>12$.
3. The radial velocity is within $\sim$3 km s$^{-1}$ around the cluster’s mean $<$RV$>=$$+$14.8 km s$^{-1}$. Since most of our radial velocity measurements were obtained at a single epoch, this limit works against the single-lined binary cluster members that may have their instantaneous radial velocity outside the 3 km s$^{-1}$ window. If a star is a known or suspected binary, this criterion is relaxed.
4. At the age of IC 2391 ($\sim$40 Myr), the stars cooler than $T_{\rm eff}\sim$ 7000 K must have a Li I feature in their spectra at 6708Å. The lack of lithium indicates that a star is a much older field star. The Li test could not be applied to several fainter stars due to the lack of high-resolution spectra.
If any of the criteria listed above fails, such a star may not qualify for a list of bona fide cluster members. In the case of double-lined spectroscopic binaries their barycentric velocity should be within $\sim$5 km s$^{-1}$ around the cluster’s mean velocity. Thus, a total of 66 bona fide cluster members are selected and denoted by an asterisk in the last column of Table 1. For convenience, we also provide cross-identifications with HD stars (numbers greater than 70000) in Table 1 and the VXR numbers from @pat96. It is still possible that a few stars among these stars may not survive further scrutiny for cluster membership, and some rejected stars may turn out to be genuine cluster members.
A list of bona fide cluster members was used to find the cluster center, previously not very well known for IC 2391. We assumed a Gaussian profile to model marginal distributions in right ascension and declination of the observed star spatial density. A fit to these distributions yields a cluster center, equal to $\alpha$=8$^{\rm h}41^{\rm m}00\fs3$ and $\delta$=$-53\degr00\arcmin36\arcsec$ (J2000.0). Due to a highly asymmetric distribution of cluster members, the uncertainty in the cluster center is about $1\arcmin$.
The sample of bona fide cluster members allowed us to re-examine the distribution of radial velocities drawn nearly exclusively from this study only, i.e., the merged list of <span style="font-variant:small-caps;">Coravel</span> and FEROS radial velocities. We selected 42 stars showing no apparent signs of duplicity. The distribution of radial velocities was binned in 0.4 km s$^{-1}$ increments and then fitted with a Gaussian (Fig. \[fig:rv\_hist\]). The best fit yields a mean radial velocity of 14.80$\pm0.69$ km s$^{-1}$ (s.d.). This is very close to the mean radial velocity of 14.6 km s$^{-1}$ estimated from the @sta97 data. The distribution of radial velocities appears to be slightly skewed. We believe that small number statistics is a primary source of this skewness, although stars with $v\sin i>90$ km $^{-1}$ may also contribute to this effect by having their radial velocity slightly increased.
A few stars in Tables \[tab:meanrv\] and \[tab:meanli\] that have their radial velocities very close to the cluster’s mean velocity fail one or two additional criteria. Two of them, 2717 and 5314, are located below the main sequence but otherwise would qualify for cluster membership and probably deserve further scrutiny. The third star, 5376, is almost a perfect cluster member, notwithstanding the noted absence of lithium. The available spectrum is rather noisy; therefore, we cannot be confident about the lack of Li feature, so additional checks are required. According to @ran01, the cluster member 4636=SHJM 9, only by $\sim$100 K hotter than star 5376, has a fairly prominent Li 6708 feature with EW(Li)=100 mÅ. Hence, there is no reason to assume that Li would be depleted below detection in star 5376, if it were a cluster member.
[c]{}\
\
Membership status of Patten & Simon and SHJM stars
--------------------------------------------------
Many of the recent advances in the understanding of IC 2391 rest on the X-ray source list from ROSAT observations [@pat93; @pat96]. A total of 80 sources have been identified from these observations. Owing to a rather low spatial resolution ($\sim$5-30$\arcsec$ FWHM) for ROSAT detectors, identification of optical counterparts in many cases is uncertain. To ensure that none of the possible counterparts with $V<19$ is missing, @pat96 provide a total of 184 possible identifications and then try to narrow down the list using available proper motions, photometry, and spectroscopy. In essence, the papers by @sta97, analyzing 26 counterparts, and @ran01, re-analyzing a subsample from the Stauffer et al. paper, is a continuation of this effort. We have astrometric data for 49 optical counterparts, all brighter than $V\sim15$ (see cross-identifications in Table \[tab:patten\]). Among these stars, 12 are field stars with the membership probability, $P_{\mu}$=0. A formal sum of probabilities indicates that the expected number of cluster members is $\sim$25. This estimate is lower than the actual number of members because we could not incorporate radial distribution in the membership calculation (see Sect. 2). Indeed, there are 34 bona fide cluster members among the likely optical counterparts of X-ray sources. We conclude that, while proven successful in finding many low-mass cluster members, the X-ray activity alone is not a decisive and comprehensive membership criterion.
To illustrate how deceptive corroborative evidence can sometimes be, consider star 6576=WXR 50a. It is listed as a “suspected cluster member based on photometry and/or spectroscopy” by @pat96. Its radial velocity from our study is +15.13 km s$^{-1}$ and +17 km s$^{-1}$ [@sta97], both very close to the cluster’s mean radial velocity. For this star, the Li abundance is 2.7 (our study) and 2.6 according to @ran01, tightly following the $\log$ N(Li) vs. $T_{\rm eff}$ trend for IC 2391 (Fig. \[fig:li\]). A closer inspection of color-magnitude diagrams $V,(V-I)$ and $J,(J-K)$ shows that a star is located $\sim$0.5 mag [*below*]{} the main sequence. However, it is the proper motion of 6576 ($\mu_x=-8.8$ and $\mu_y=+2.6$ mas yr$^{-1}$) that, at $11\sigma_c$ apart from the cluster’s mean motion, rules out the cluster membership.
One star, 4658=VXR 45a, warrants a special statement. While it is not listed by us as a bona fide member because it is located relatively far above the main sequence in the color-magnitude diagram, its high proper-motion membership probability of $P_{\mu}=75\%$ strongly suggests the cluster membership. This G9 spectral type star, also a BY Dra type variable V370 Vel, stands out by its huge rotational velocity of $v \sin i =$238 km s$^{-1}$ [@mars04], by far exceeding any other $v \sin i$ measurement in IC 2391 [@sta97 see also Tables \[tab:meanrv\] and \[tab:meanli\]]. The short photometric rotational period equal to $P_{\rm rot}=0.223$ days [@pat96] is also clearly visible in the X-ray time series observations [@mar03]. If this is a genuine cluster member, then the question is what mechanism spun it up or prevented the dissipation of its primordial angular momentum, while most cluster members similar to star 4658 are slow rotators ($v \sin i<20$ km s$^{-1}$). It should be noted that such ultrafast rotators can be found in other young open clusters, e.g., in the Pleiades [@vlee87], $\alpha$ Per [@pro92], IC 2602 [@sta97]. One possible explanation of this phenomenon is offered by @bars96, invoking a paradigm of the angular momentum loss saturation.
There is another fast rotator VXR 80a with $v \sin i \sim$150 km s$^{-1}$ [@sta97] among the suspected cluster members by @pat96. Since we have not measured this star, the only source of proper motion for VXR 80a=King 391 is a study by @kin79. In this paper the cluster’s proper motion centroid is at $\mu_{x}=-17.0$ and $\mu_{y}=+16.3$ mas yr$^{-1}$, derived by using the counterparts of our bona fide cluster members in @kin79. If we adopt the standard error of proper motions $\sigma$=1.5 mas yr$^{-1}$, then the proper motion of King 391 is more than 6$\sigma$ apart from the cluster’s mean motion which rules out the cluster membership.
The star SHJM 5 also shows $v \sin i \sim$150 km s$^{-1}$ [@sta89]. This star is a member of a visual binary (our star 4757) for which we have the membership probability $P_{\mu}=28\%$, consistent with cluster membership. Finally, star 2457 with $v \sin i =$177 km s$^{-1}$ is another possible cluster member despite its measured radial velocity at $-7.25$ km s$^{-1}$, since its astrometric membership probability is $P_{\mu}=71\%$ and it is located just slightly above the main sequence in both $BV$ and $JK$ color-magnitude diagrams.
Prior to ROSAT measurements, the so-called SHJM list of 10 relatively faint cluster members was published by @sta89. Cross-identifications of SHJM stars are provided by @pat96. Based on our astrometry and criteria listed in Sect. 6.1, stars SHJM 1,2,6,9,10 appear to be bona fide cluster members, while SHJM 8 was not measured but SHJM 7=3664 has a membership probability of zero. The membership for SHJM 3=1820 and (SHJM 4 + SHJM 5)=4757 is not definitely constrained by our data. For SHJM 3 this is because its radial velocity from our data is +12.76 km s$^{-1}$, while @sta89 provide +19.5 km s$^{-1}$, thus indicating a possible spectroscopic binary. Its astrometric membership probability is only 9%, which is more a characteristic of a field star. Our conservative lower limit of equivalent width for the Li feature for SHJM 3 is 2.5 times larger than the measurement by @sta89, thus implying unusually high abundance of Li at $T_{\rm eff}$=4480 K in the context of the overall $\log$ N(Li) vs. $T_{\rm eff}$ curve for IC 2391 (see Fig. \[fig:li\]).
Membership of some bright stars
-------------------------------
[**3518=HIP 42504**]{}: a 5th magnitude star whose Hipparcos astrometry is consistent with cluster membership. Its redward position from the main sequence in the $BV$ and $JK$ CMDs (see Sect. 6.5) indicates duplicity, which is confirmed by radial velocity measurements [@fei61; @lev88]. The star is an SB1 spectroscopic binary with P=3.2d, eccentricity $e=0.05$, and $\gamma$-velocity$=14.5$ km s$^{-1}$.
[**3658=HD 74438**]{}: this star is $\sim$0.9 magnitudes above the main sequence in the color-magnitude diagram, indicating a potential triple system. Its proper-motion membership probability $P_{\mu}=88$% is high and reliable. @fei61 lists two measurements of radial velocity: $+7.5$ and $+17.5$ km s$^{-1}$, whereas @bus65 derives $+$21: km s$^{-1}$ from five spectrograms. Thus, the scatter of radial velocities is indicative of a non-single status of the star. A formal mean velocity matches the cluster’s radial velocity. We note that at the XMM-Newton observatory @mar05 were able to obtain only the upper limit of X-ray luminosity for this star, which is typical of A spectral type stars. On the grounds of the available data, we consider star 3658 a bona fide cluster member, albeit one requiring more spectroscopic studies.
[**4484=HIP 42536=$o$ Velorum**]{}: the brightest star in IC 2391, also a $\beta$ CMa type variable. Its Hipparcos parallax $\pi=6.59\pm0.51$ mas is very close to the mean parallax of IC 2391 equal to 6.85$\pm$0.22 [@rob99]; however, Hipparcos proper motion in declination $\mu_{\delta}=+35.1\pm0.5$ mas yr$^{-1}$ is 20$\sigma$ (!) apart from the cluster’s mean proper motion $\mu_{\delta}=+22.7\pm0.2$ [@rob99]. @gon01 report a significantly lower absolute proper motion $\mu_{\delta}=+23.6\pm2.0$ consistent with @rob99. Our membership probability, $P_{\mu}=81$%, and the mean radial velocity of $+15.2$ km s$^{-1}$ by @vhoo72 together indicate that the star is a bona fide cluster member. The discrepant Hipparcos proper motion can be explained as the effect of unresolved binarity [@wie99].
[**5459=HD 74009**]{}: a star with lower membership probability ($P_{\mu}=25\%$) than the other relatively bright stars. However, the mean radial velocity of $+14$ km s$^{-1}$ by @lev88 strongly supports cluster membership. Apparent variability of the observed radial velocity [@lev88] indicates a possible spectroscopic binary.
[**7847=HIP 42823**]{}: another bright star in IC 2391 first considered to be a member by @egg91. Its Hipparcos parallax is only $0.7\sigma$ apart from the mean parallax of IC 2391, while Hipparcos proper motion deviates from the mean by $\sim$3 mas yr$^{-1}$ in both coordinates. Our membership probability of 78% supports the association of star 7847 with IC 2391, however, the lack of radial velocity measurements prevents us from assigning it the status of a definite cluster member.
Spectroscopic binaries
----------------------
Identifying spectroscopic binaries in star clusters has a dual purpose. First, the evolutionary paths of stars are reasonably well-understood only for single stars; therefore, it is critical that we can identify these stars in the color-magnitude diagram with high confidence. Radial velocities from high-resolution spectroscopy is a powerful tool for detecting close binaries in a wide range of mass ratios. Second, the binaries themselves are very important to astronomy, especially those in the star clusters that provide a coeval sample of stars in a wide range of masses, all having the same initial composition.
The early spectroscopic work in IC 2391 was focussed on bright, early type stars [@fei61; @bus65; @per69a; @vhoo72; @lev88]. In Table 1 – last column – we list only double-lined spectroscopic binaries (SB2), single-lined spectroscopic binaries (SB1 or SB as listed in the source), and suspected spectroscopic binaries (SB:), if a star shows a variable radial velocity or the mean radial velocity in different studies is significantly variant. The binaries with orbital solutions are marked with a letter ‘o’. Since the internal mean error in these studies can reach up to 3-5 km s$^{-1}$, not all cases of suspected SB may be real.
Among the IC 2391 F-K type main sequence stars, the long-term <span style="font-variant:small-caps;">Coravel</span> observations have revealed two SB2 and four SB1 spectroscopic binaries (see Tables 3 and \[tab:meanrv\]). An SB2 spectroscopic binary 4413 has a sufficient number of observations to calculate the orbital elements listed in Table \[tab:param\]. This binary system (Fig. \[fig:orb\]) consists of two nearly equal-mass components on a fairly eccentric orbit ($e\sim0.3$). Its $\gamma$-velocity of 14.35 km s$^{-1}$ clearly supports the cluster membership.
According to its proper motion, $\mu_x=-4.0$ and $\mu_y=+11.8$ mas yr$^{-1}$, the SB2 spectroscopic binary 389 is a field star. As indicated by <span style="font-variant:small-caps;">Coravel</span> observations in Table 3, this binary was not resolved over four epochs, and these observations yield an estimate of the $\gamma$-velocity at 9.27 km s$^{-1}$ (Table \[tab:meanrv\]). The @wil41 method allows us to determine the $\gamma$-velocity more precisely and to obtain the mass ratio in this binary system. Thus, the refined $\gamma$-velocity is $+10.24$ km s$^{-1}$ (K$_{1}=19.36$ and K$_{2}=21.75$ km s$^{-1}$) and the mass ratio $M_a/M_b=0.89$. Clearly, the $\gamma$-velocity of star 389 also rules out cluster membership.
Finally, the FEROS high-resolution spectra obtained in one or two epochs (Table 5) indicate five SB2 binaries, four of them (819, 2540, 3497, 7372) are bona fide cluster members. For a set of four FEROS double-lined spectroscopic binaries, the observed cross-correlation functions are shown in Fig. \[fig:sbs\]. One of them (star 7711) has a barycentric velocity clearly incompatible with the mean radial velocity of the cluster.
Altogether, among 66 bona fide members, there are seven SB2, six SB1, and nine suspected spectroscopic binaries. If all spectroscopic binary categories are considered, then the binary frequency in IC 2391 is at least $\sim$30%. The location of all 22 spectroscopic binaries in $BV$ color-magnitude diagram is shown in Fig. \[fig:cmd\_zams\].
Color-Magnitude Diagram
-----------------------
There are two sources of complete photometry for the bona fide cluster members: our CCD $BV$ and 2MASS $J\!H\!K_s$ [@skr06] photometry. In the case of a missing CCD $BV$ photometry value, we used photoelectric $BV$ photometry from the literature. Such values in Table 1 are recognizeable by having only two decimal digits. The best previous CMD in $U\!BV$ is presented by @per69b. These authors conclude that IC 2391 is unreddened, i.e, $E(B-V)=0.00$. It was later confirmed by measurements in the Vilnius seven-color photometric system [@for98] and is also adopted in the present paper. The true reddening probably is not exactly zero; however, the limited precision of existing $U\!BV$ photometry and still unclear binarity status of some early spectral type cluster members preclude us from deriving a better reddening estimate. The following fits to the CMD, however, do not support the high value of $E(B-V)=0.06$ reported by @bar04. The studies by @sta89 [@sta97] and @pat96 mainly employ $V\!I$ photometry in constructing the CMD of IC 2391, assuming a distance modulus (m-M)=6.05. We note that the latter is 0.23 mag larger than the distance modulus from Hipparcos parallaxes [@rob99]. The color-magnitude diagrams in these studies indicate that the stars at $V\geq11$ are located above the main sequence, presumably still being on the pre-main-sequence tracks. Our list of bona fide cluster members allows us to explore the properties of the $BV$ CMD (Fig. \[fig:cmd\_zams\]) in more detail.
We made a trial fit of ZAMS [@lan82] to the $BV$ CMD, assuming zero reddening, $E(B-V)=0.0$, and a distance modulus of $V_{0}-M_{V}$=5.95. It appears that bright stars are essentially unevolved, with the exception of the brightest member, star 4484 = $o$ Velorum, whose color is suspect due to possible duplicity (see Sect. 6.3). The lower main sequence seems to end at $V\ga12$ giving rise to pre-main sequence stars at fainter magnitudes. An exact interpretation of the $BV$ color-magnitude diagram for young open cluster clusters is complicated by the fact that K dwarfs in the Pleiades are either subluminous and/or have abnormally blue $B-V$ color [@sta03], hence falling below the ZAMS. These authors suggest that all young K dwarfs may show a similar anomaly, thereby limiting the ability to obtain an estimate of the so-called PMS isochrone age. We examine this effect in the following section.
### Main-sequence fittting, distance modulus, age
One of the goals of this study is to obtain a reliable photometric distance from the main sequence fitting. The empirical ZAMS used in Fig. \[fig:cmd\_zams\] is not adequate for at least two reasons. First, the ZAMS from @lan82 is based on the Hyades distance modulus of $m-M=3.28$ mag , while Hipparcos parallaxes for the Hyades members yield 3.33 mag. Second, at the time of compiling this ZAMS, the metallicities of open clusters were poorly known. The uncertainty in \[Fe/H\] is directly related to the uncertainty of the amount of the metal-line blanketing effect on colors and magnitudes – a major source of systematic errors in photometric distances.
Recently, @pin03 [@pin04] and @an06 derived new isochrones specifically targeted to measuring the distances to open clusters and other parameters from the main-sequence fitting. They used the Yale Rotating Evolution Code (YREC) to construct stellar evolution tracks for masses $0.2\le~M_{\sun}\le8$ and metallicities $-0.3\le[{\rm Fe/H}]\le+0.2$. The tracks were interpolated to provide theoretical isochrones for stellar ages from 20 Myr to 4 Gyr, with the pre-main sequence phase included where appropriate. Then, the authors used the empirical $T_{\rm eff}$-color transformations from @lej98 and applied small additional corrections to isochrones in order to match the photometry for the Hyades, adopting its metallicity at \[Fe/H\]=+0.13 dex.
We used these isochrones to derive a precise distance modulus and to probe the range of possible cluster ages. The key parameter for assuring reliability of main-sequence fitting is metallicity. Our new spectroscopic estimate of \[Fe/H\] for IC 2391 has an accuracy similar to the metallicities used by @an06 in their analysis of four open clusters. In previous studies the isochrone age of IC 2391 is estimated to be $\sim$30-35 Myr [@sta97; @mer81], while the location of the lithium-depletion boundary indicates an age of $50\pm5$ Myr [@bar04]. We chose 30, 40, and 55 Myr isochrones from @an06 and interpolated them to match the cluster’s metallicity of \[Fe/H\]=+0.06 dex. These isochrones were fitted to the $BV$ CMD (Fig. \[fig:cmd\_iso\]) by adjusting them to the lower envelope of main sequence in the color range of $0.2<B-V<0.7$ and by assuming zero reddening, $E(B-V)=0.00$ mag. The resulting distance modulus is $V_0-M_V=6.01$ mag, which is larger by 0.19 mag or $2.7\sigma$ than the distance modulus inferred from the mean parallax via the Hipparcos combined abscissae solution by @rob99 and @vlee99. Following the argumentation by @pin98 and the study of NGC 2451A [@pla01], we adopted the uncertainty in our distance modulus to be 0.05 mag.
This result should be considered in the context of a trigonometric distance to the Pleiades from Hipparcos measurements vs. the Pleiades distance estimate from the main-sequence fitting. The Hipparcos measurements yielded a significantly smaller distance modulus for the Pleiades (i.e., $m-M=5.37\pm0.07$ mag, equivalent to $118.3\pm3.5$ pc in van Leeuwen 1999) than the distance modulus from the main-sequence fitting, e.g., $m-M=5.60\pm0.04$ [@pin98].
It is distressing to find another open cluster whose Hipparcos parallax distance is significantly shorter than the photometric distance. We point out that among the eleven IC 2391 members used by @rob99 to derive the mean cluster parallax, only one star (7027=HIP 42450) has a smaller Hipparcos parallax than our photometric parallax. Incidently, this star is the second faintest in the sample. What kind of systematic errors could possibly bias our photometric distance? The first is a small uncertainty in the reddening of IC 2391. If the true reddening is as high as $E(B-V)=0.01$ mag, that alone would reduce the distance modulus by 0.03 mag. Second, if the true metallicity is solar, i.e., \[Fe/H\]=0.0 dex, it would also reduce the distance modulus by an additional $\sim$0.08 mag. However, it is unlikely (although not imposssible) that either parameter is off by this much. Also, a major change in the stellar parameters would be required to lower the \[Fe/H\] value enough to account for the 0.19 mag discrepancy in the distance modulus.
In the light of another looming Hipparcos distance problem, it is instructive to have a look of how it is resolved for the Pleiades open cluster. The following diverse and independent studies actually all support the long distance to the Pleiades. Thus, @gat00 measured trigonometric parallaxes of seven members of the Pleiades clusters and obtained an equivalent to the distance modulus $m-M=5.59\pm0.12$ mag. Using the Hubble Space Telescope’s Fine Guidance Sensor, @sod05 obtained absolute trigonometric parallaxes for three Pleiades members leading to the distance modulus $m-M=5.63\pm0.02$ mag. When @pan04 applied Kepler’s third law and the mass-luminosity relation to their interfermetric measurements, they found that the Pleiades visual binary star Atlas has $m-M=5.66\pm0.04$ mag. A different set of interferometric observations for Atlas, combined with the measurements of radial velocities, produced better-constrained orbital elements and, consequently, a purely geometric distance or $m-M=5.60\pm0.07$ mag [@zwa04]. More strong evidence comes from the combined orbital solution of the eclipsing binary HD 23642, yielding the distance modulus $m-M=5.60\pm0.03$ mag [@mun04]. Finally, the frequency observations in six $\delta$ Scuti stars in the Pleiades match the eigenfrequencies of rotating stellar models best, when assuming $m-M=5.60-5.70$ [@fox06] and using the other generally adopted parameters of the Pleiades.
Regarding Hipparcos parallaxes, @vlee05a indicates that correlations in the abscissa residuals for bright stars and weight disparities between each two fields-of-view in the attitude reconstruction process can be attributed to the problematic parallaxes for some open clusters, particularly the Pleiades. @mak02 has shown that by accounting for the average residuals around the so-called reference circle helps to reduce the Pleiades Hipparcos parallax to an equivalent of $m-M=5.55\pm0.06$. This, however, is not the final word since the raw Hipparcos data are under complete re-reduction [@vlee05b].
Our $BV$ color-magnitude diagram (Fig. \[fig:cmd\_iso\]) conclusively shows that the main sequence extends as deep as $V$$\sim$12 or $M_v=+6$ and appears to populate the pre-main-sequence tracks at fainter magnitudes. If we adopt an age of IC 2391 equal to 40 Myr, our 40-Myr-old pre-main sequence isochrone (see Fig. \[fig:cmd\_iso\]) is located above the bulk of presumably pre-main-sequence stars. That could be explained by the observed ‘blueshift’ of K dwarfs in the $BV$ CMD of Pleiades [@sta03], noting that this effect in IC 2391 starts at $B-V\sim0.7$ or at a spectral type G5. A 55-Myr old isochrone significantly mitigates this effect but, on the other hand, it does not fit the upper main sequence well, suggesting that the isochrone (nuclear) age is less than 55 Myr. In fact, the age estimate is constrained just by the single brightest cluster member 4484 = $o$ Velorum. Because of its suspected duplicity (see Sect. 6.3) and a $\beta$CMa type variability [@vhoo72], this star is not a typical B3 spectral type subgiant – its color $B-V=-0.18$ could be biased and, subsequently, affect the isochrone age. Summarizing, the age estimate from the $BV$ CMD could be anywhere between 30 and 50 Myr.
Recently @dod04 attempted to use the 2MASS photometry to construct a CMD of IC 2391. One lesson learned from this exercise is that the accuracy of 2MASS photometry is inadequate at $H>14$ ($J>15$). Our bona fide cluster members are not fainter than $J=11.5$, so they can serve as building blocks in constructing an empirical ZAMS from the 2MASS data. The $JK$ color-magnitude diagram of IC 2391 (Fig. \[fig:cmd\_2mass\]) has a considerably larger scatter across the main sequence than does the $BV$ CMD (Fig. \[fig:cmd\_iso\]), possibly due to the higher sensitivity to a binary presence. It is quite possible that several undetected spectroscopic binaries are still hidden in our sample of bona fide cluster members. Some other reasons for this scatter may include source confusion and different photometric measurement reductions for stars brighter and fainter than $J\sim9$ [@cut03].
In the $J\!K$ CMD (Fig. \[fig:cmd\_2mass\]), we overplotted 40 and 100 Myr solar metallicity isochrones from @bon04 using the same distance modulus as for the initial ZAMS fit in Fig. \[fig:cmd\_zams\], i.e., $V_{0}-M_{V}$=5.95. Only a handful of stars are located right on the 40 Myr isochrone, which itself poorly fits the upper main sequence. The best fit with a 100 Myr isochrone is most likely compromised by some bias in the 2MASS photometry at bright magnitudes and/or by using a $T_{\rm eff}$-color transformation that is somewhat uncertain in this magnitude range. Thus, choosing a 100 Myr isochrone helps to eliminate the mismatch at the bright end of main sequence; however, it is very unlikely that IC 2391 is indeed that old.
Following the referee’s suggestion we investigated the $V\!K$ color-magnitude diagram. In these colors it is expected that the K dwarf ‘blueshift’ in the CMD would disappear [@sta03]. Our $V\!K$ CMD for bona fide cluster members is given in Fig. \[fig:cmd\_vk\_iso\]. We attempted to fit this CMD with 30,35,40,45,50,55-Myr isochrones; however, a relatively poor match to the essential part of main sequence comprising F-G spectral type stars ($0.3\le~V-K_s\le1.5$) prevented us from improving the distance modulus over a similar fit of the $BV$ CMD. Hence, we adopted the distance modulus of $V_0-M_V=6.01$ obtained from the $BV$ CMD. It is difficult to judge what causes the scatter across the main sequence and the offset of its upper portion. Possibly, the 2MASS photometry contributes to these effects, but we also cannot entirely rule out small isochrone calibration problems for the $V-K_s$ color index. On the other hand, the lower part of color-magnitude diagram is indeed well-defined and allows us to confirm the best estimate of isochrone age for IC 2391 at 40 Myr.
The so-called lithium-depletion boundary (LDB) age method yields a higher age for IC 2391 at 50-53 Myr [@bar99; @bar04] and 50 Myr by @jef05 using five different models. It should be noted that this is in line with similarly higher ages by this method for the $\alpha$ Per cluster [@sta99] and the Pleiades [@sta98]. Since our data on IC 2391 do not reach the lithium-depletion boundary, we are not in a position to resolve the age differences.
Rotational velocities $v\sin i$
-------------------------------
The distribution of $v\sin i$ for cluster members ranges from $\sim$2 to 240 km s$^{-1}$ (Fig. \[fig:vsini\]). Such a spread has already been observed by [@sta97] for the IC 2391+IC 2602 members and can be interpreted as a result of the early angular momentum evolution of low-mass stars, which appears to be regulated by the disk-locking mechanism [e.g., @edw93]. According to this scenario, a gradual dissipation of the disk weakens the magnetic coupling between the star and its circumstellar disk, thereby releasing the star to spin up as it contracts during its pre-main-sequence phase.
[@her05] analyzed a large data set of rotational periods for 500 low-mass stars in five nearby young open clusters (Orion nebula cluster, NGC 2264, $\alpha$ Per, IC 2602, and the Pleiades). They show that 50-60% of the stars still on the convective tracks, i.e., the vertical part of the pre-main-sequence tracks, appear to be released from the locking mechanism early and thus account for the fast rotators. Conversely, the remaining stars lose a considerable amount of their angular momentum in the first few million years and enter onto the ZAMS as slow rotators. The distribution of $v\sin i$ in IC 2391 (Fig. \[fig:vsini\]) seemingly follows this pattern. However, a relatively short empirical life-time of circumstellar disks of $\sim$6 Myr [@hai01] makes IC 2391, at its age of 40 Myr, an improper environment for testing the disk-locking mechanism. Recently, @bar03 put forward a new interpretation of the observed pattern of rotational velocities (periods) in star clusters like IC 2391 and older. In this interpretive paradigm, the observed rotational morphology in G, K, and M stars along the so-called $I$ and $C$ sequences is driven by dynamos of two types that evolve in synchrony with the extent of the convective zone in stars. For IC 2391 the only source of rotational periods for 16 stars is the study by @pat96. We hope that the much larger sample of $v\sin i$ obtained in this study will help to advance the understanding of rotational evolution.
Conclusions
===========
A decade ago @sta97 commented that IC 2391, as well as IC 2602, are not giving up their secrets easily. Placed at an optimum age, these two clusters are ideal laboratories for studies of interface between the main sequence and late pre-main-sequence phases and their timescales. However, the intrinsic paucity of cluster members in both IC clusters is the main factor limiting the scientific return in the sense that even a single peculiar cluster member or an interloper may distort the observable trends and relationships. One such sensitive case is star 1820 (see Sect. 6.2) and its location on the Li abundance curve.
In this study we have significantly advanced the knowledge of IC 2391 in the range $-2<M_v<+8.5$ or down to $\sim$0.5M$_{\sun}$. This is a first extensive proper-motion study in the region of IC 2391 providing membership probabilities over a 9 deg$^2$ area, in effect assuring a high-degree of completeness in the chosen ranges of magnitudes, as explained in Sect. 2. We have measured the radial velocity for most of them. The new kinematic members mainly fill in the F2-K5 spectral region. The <span style="font-variant:small-caps;">Coravel</span> measurements of 31 stars considerably strengthen the membership status for most of them. Several spectroscopic binaries were detected; among them seven are double-lined SB2, with one yielding an orbital solution. The FEROS high-dispersion spectra served to measure radial velocities and projected rotational velocities and to estimate Li and Fe abundance. The latter is found to be \[Fe/H\]$=+0.06$ on the scale of solar abundance $\log \epsilon$(Fe)=7.45 dex.
A total of 66 bona fide cluster members are selected by combining kinematic, spectroscopic, and photometric membership criteria. The $BV$, $J\!K$, and $V\!K$ color-magnitude diagrams were constructed using the bona fide cluster members. The main sequence fit yields a distance modulus of $V_{0}-M_{V}$=6.01 mag or 159.2 pc that is significantly larger than the distance modulus and distance (5.82 mag and 146 pc) from the Hipparcos mean parallax for IC 2391.
The problem with Hipparcos mean cluster parallaxes could be quite complicated. For example, another nearby open cluster NGC 2451A, which is located only $18\degr$ away from IC 2391, does not show any discrepancy between the Hipparcos parallax and the distance modulus derived from the main sequence fitting [@pla01], while the number of measured stars and their properties in both clusters are nearly identical. Both clusters are located near the ecliptic latitude of $\beta\sim-60\degr$ and had been observed by Hipparcos relatively frequently and close in time. Comparative studies of these two open clusters may reveal clues to the cause of a $\sim$0.5 mas offset in the Hipparcos mean parallax for IC 2391.
Among the most important tasks to advance the knowledge about IC 2391 are 1) extending reliable kinematic membership to fainter magnitudes and 2) accumulating high-resolution spectroscopy for all possible cluster members, especially for known or suspected spectroscopic binaries. The cluster population should be cleanly separated into single and binaries stars prior to extensive studies of individual stars and establishing a reliable distance to IC 2391 by other means.
We thank Hugo Levato for organizing the observing run at the CASLEO 2.2 m telescope. We appreciate helpful comments by Heiner Schwan and Peter De Cat about the properties of $o$ Velorum. We also thank Jeremy King and Simon Schuler for stimulating discussions on elemental abundances. We thank the referee, John Stauffer, for detailed and thoughtful comments that greatly helped to improve the interpretation of the results. We also thank Deokkeun An, Don Terndrup, and Marc Pinsonneault for their help with the new YREC isochrones generated at Ohio State University. This research made use of the SIMBAD database operated at the CDS, Strasbourg, France. This publication made use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. I. Platais gratefully acknowledges support from the National Science Foundation through grant AST 04-06689 to Johns Hopkins University. J. Fulbright acknowledges support through grants from the W.M. Keck Foundation and the Gordon and Betty Moore Foundation, to establish a program of data-intensive science at to the Johns Hopkins University. M. Altmann and R. Mendez acknowledge the support by the Chilean Centro de Astrofísica FONDAP (15010003). The travel by J. Sperauskas to El Leoncito, Argentina was in part supported by NSF supplemental funding AST 01-39797. We heartily thank Lois J. Evans for meticulous editing of this paper.
5756tb2.tex
5756tb4.tex
5756tb7.tex
5756tb8.tex
5756tb9.tex
5756tb10.tex
[^1]: Based on observations collected at the European Southern Observatory, Chile (Program IDs: 072.D-0107 and 074.D-0096)
[^2]: Tables 1, 3, 5, and 6 are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/
[^3]: The catalog is available at http://www.astro.yale.edu/astrom/
[^4]: http://wwwuser.oat.ts.astro.it/castelli/
| ArXiv |
---
abstract: |
Abstract
========
The studies of the electroweak symmetry breaking sector (EWSBS) at $\gamma\gamma$ colliders were considered previously in the loop processes of $\gamma\gamma \to w_Lw_L,\,z_Lz_L$, but they are suffered from the huge $W_T W_T$ and $Z_TZ_T$ backgrounds. Here we present another possible process that involves spectator $W$’s and $W_L$’s, the latter of which are scattered strongly by the interactions of the EWSBS. We also show that this process should be safe from the transverse backgrounds and it can probe the structure of the EWSBS.
author:
- Kingman Cheung
---
[**Possibility of Studying Electroweak Symmetry Breaking at [$\gamma \gamma$]{} Colliders**]{}
Dept. of Physics & Astronomy, Northwestern University, Evanston, Illinois 60208, USA\
Introduction {#intro}
============
So far very little is known about the electroweak symmetry-breaking-sector (EWSBS), except it gives masses to the vector bosons via the spontaneous symmetry breaking, and masses to fermions via the Yukawa couplings. In the minimal standard model (SM) one scalar Higgs boson is responsible for the electroweak symmetry breaking but its mass is not determined by the model. If in the future no Higgs boson is found below 800 GeV, the heavy Higgs scenario ($\approx 1$ TeV) will imply a strongly interacting Higgs sector because the Higgs self-coupling $\lambda\sim m_H^2$ becomes strong [@quigg]. However, there is no evidence to favor the model with a scalar Higgs, and so any models that can break the electroweak symmetry the same way as the single Higgs does can be a candidate for the EWSBS.
One of the best ways to uncover the underlying dynamics of the EWSBS is to study the longitudinal vector boson scattering [@quigg; @chano]. The Equivalence Theorem (ET) recalls, at high energy, the equivalence between the longitudinal part $(W_L)$ of the vector bosons to the corresponding Goldstone bosons ($w_L$) that were “eaten" in the Higgs mechanism. These Goldstone bosons originate from the EWSBS so that their scattering must be via the interactions of the EWSBS, and therefore the $W_L W_L$ scattering can reveal the dynamics of the EWSBS.
The strong $W_L W_L$ scattering have been studied quite seriously at the hadronic supercolliders [@bagger], but less at the $e^+e^-$ colliders, and very little at the $\gamma\gamma$ colliders. In hadronic colliders, only the “gold-plated" modes, the leptonic decays of the $W$ and $Z$ bosons, have been considered due to the messy hadronic backgrounds; whereas in $e^+e^-$ and $\gamma\gamma$ colliders one can make use of the hadronic decay mode or mixed decay mode of the final state $W$’s or $Z$’s. With the advance in the photon collider designs it is possible to construct an almost monochromatic $\gamma\gamma$ collider based on the next generation linear $e^+e^-$ colliders using the laser backscattering method [@teln]. The monochromaticity of the photon beams depends on the polarizations of the initial electron and the laser photon. The polarizations of the initial electron and the laser photon can be adjusted to maximize the monochromaticity of the photon beam [@teln] with a center-of-mass energy about 0.8 of the parent $e^+e^-$ collider. Hence, a 2 TeV $e^+e^-$ collider will give a 1.6 TeV $\gamma\gamma$ collider by the laser backscattering method. For the following we will assume a monochromatic $\gamma\gamma$ collider of energy 1.5 TeV with an integrated luminosity of 100 fb$^{-1}$.
Studies of the strongly interacting EWSBS in $\gamma\gamma$ collision have been considered previously in Refs. [@previous]. They all concentrate on $\gamma\gamma \rightarrow W_L W_L$ or $Z_L Z_L$. Unfortunately, the $\gamma\gamma\to W_T W_T$ is almost three orders of magnitude larger than the $W_LW_L$ signal. Although we can improve the signal-to-background ratio by requiring the final state $W$’s away from the beam, it hardly reduces the $W_TW_T$ background to the level of the $W_LW_L$ signal. On the other hand, both the $\gamma\gamma\to Z Z$ signal and background are absent on tree level. But the box diagram contribution to $Z_TZ_T$ has been shown to be very large at high $m(ZZ)$ region, and so the $Z_T Z_T$ background is dominant over the $Z_L Z_L$ signal in the search of the SM Higgs with $m_H\agt 300$ GeV and in probing the other strong EWSB signals [@ZTZT]. As illustrated in Refs. [@previous], the central part of interest is the $w_L w_L \to w_L w_L$ or $z_Lz_L$, but the effects of the strong EWSBS only come in on loop level in these processes so that the effects might not be so significant. In the following we present a new type of processes involving $W_L W_L\to W_L
W_L,\,Z_LZ_L$ at $\gamma\gamma$ colliders, schematically shown in Fig. \[one\] [@brodsky]. These $W_LW_L$ scattering processes will be in analogy to the $W_L W_L$ scattering considered at the hadronic supercolliders and $e^+e^-$ colliders. The advantages of the processes in Fig. \[one\] are that the $W_LW_L$ scattering is no longer on loop level, and additional vector bosons in the final state can be tagged on to eliminate the large $W_T W_T$ and $Z_T Z_T$ backgrounds. In addition, both the $W_L^+
W_L^-$ and $W_L^\pm W_L^\pm$ scattering can be studied in $\gamma\gamma$ collision but only one of them can be studied in the $e^+e^-$ or $e^-e^-$ collisions. Also any $Z_LZ_L$ pair in the final state must come from the $W_LW_L$ fusion because photon will not couple to $Z$ on tree level. Totally, we can study four scattering processes, $W_L^\pm W_L^\pm \to W^\pm_L W^\pm_L$, $W^+_L W^-_L \to W^+_L W^-_L,\, Z_L Z_L$.
For simplification we will use the effective $W_L$ luminosity inside a photon in analogy to the effective $W$ approximation. This approximation will suffice for the purpose here for we will consider the kinematic region where the EWSBS will interact strongly, or in another words, in the large invariant mass region of the vector boson pair. The luminosity function of a $W_L$ inside a photon in the asymtotic energy limit is given by [@zerwas] $$\label{lum}
f_{W_L/\gamma}(x) = \frac{\alpha}{\pi} \left [ \frac{1-x}{x} +
\frac{x(1-x)}{2}\; \left ( \log \frac{s(1-x)^2}{m_W^2} - 2 \right ) \right
]\,,$$ which is in analogy to the luminosity function $f_{W_L/e}(x) = \frac{\alpha}{4\pi x_{\rm w}} \frac{1-x}{x}$ of $W_L$ inside an electron. The first term in Eq. (\[lum\]) is approximately equal to the luminosity of $W_L$ inside an electron, and the logarithm factor will enhance the luminosity at high energy. This is the reason why the signal rates can be achieved higher than those in the $e^+e^-$ colliders at the same energy.
Models & Predictions
====================
In this section, we will calculate the number of signal events predicted by some of the models that have been proposed for the EWSBS. In $\gamma\gamma$ collision we can study the following subprocesses $$\begin{aligned}
W_L^+ W_L^- &\rightarrow & W_L^+ W_L^- \,, Z_L Z_L \;, \\
W_L^\pm W_L^\pm & \rightarrow & W_L^\pm W_L^\pm \,.\end{aligned}$$ In analogy to the pion scattering in QCD, the scattering amplitudes of these processes can be expressed in terms of an amplitude function $A(s,t,u)$. Their scattering amplitudes are then expressed as $$\begin{aligned}
%
{\cal M}(W_L^\pm W_L^\pm \rightarrow W_L^\pm W_L^\pm)&=& A(t,s,u)+A(u,t,s)
\,,\\
%
{\cal M}( W_L^+ W_L^- \rightarrow W_L^+ W_L^- ) & = & A(s,t,u) + A(t,s,u)
\,, \\
%
{\cal M}( W_L^+ W_L^- \rightarrow Z_L Z_L ) & = & A(s,t,u) \,,
%\end{aligned}$$ up to the symmetry factor of identical particles in the final state. The details of each model and the invariant amplitudes predicted by each model are summarized in Ref. [@bagger]. Here we only give a brief account of these models. The models can be classified according to the spin and isospin of the resonance fields, and there are scalar-like, vector-like, and nonresonant models. For scalar-like models we will employ the standard model with a 1 TeV Higgs, $O(2N)$ model with the cutoff $\Lambda=2$ TeV, and the model with a chirally-coupled scalar of mass $m_S=1$ TeV and width $\Gamma_S=350$ GeV. For the vector-like models we choose the chirally-coupled vector field (technirho) of masses $m_\rho=1$, 1.2, and 1.5 TeV, and $\Gamma_\rho=0.4$, 0.5, and 0.6 TeV respectively. In the case of no light resonances we use the amplitudes predicted by the Low Energy Theorem (LET) and extrapolate them to high energies.
Each of the $W_L W_L$ scattering amplitudes grows with energy until reaching the resonances, [*e.g.*]{} SM Higgs boson of the minimal SM. The presence of the resonances (scalar or vector) is the natural unitarization to the scattering amplitudes, except there might be slight violation of unitarity around the resonance peak. After the resonance, the scattering amplitudes will stay below the unitarity limit. But for the nonresonant models the unitarity is likely to be saturated before reaching the lightest resonance. Here we employ the LET amplitude function, $A(s,t,u)=s/v^2$, for the nonresonant models. From the partial wave analysis, the only nonzero partial wave coefficients $a^I_J$ are $a^0_0$, $a^1_1$, and $a^2_0$. Among the nonzero $a^I_J$’s, $a^0_0$ saturates the unitarity ($|a^I_J|<1$) at the lowest energy $4\sqrt{\pi}v\approx 1.7$ TeV. So for the $\gamma\gamma$ colliders of 1.5 TeV, unitarity violation should not be a problem, therefore, we simply extrapolate the LET amplitudes without any unitarization.
We show the number of signal events predicted by these models for each scattering channel in Table \[table1\], with $\sqrt{s_{\gamma\gamma}}=1.5$ TeV and integrated luminosity of 100 fb$^{-1}$, and under the acceptance cuts of $$M_{WW}\;{\rm or}\; M_{ZZ} > 500\;{\rm GeV}\quad {\rm and}\quad |y(W,Z)|<1.5\,.$$ One interesting thing to note here is that different channel is sensitive to different new physics. If the underlying dynamics of the EWSBS is scalar-like the signal is more likely to be found in the $W_L^+ W_L^-$ channel, and next at the $Z_LZ_L$ channel, due to the presence of $I=0,J=0$ scalars. But if the underlying dynamics is vector-like the signal in the $W_L^+W_L^-$ channel will be far more important that the $Z_L Z_L$ channel. On the other hand, if no light resonances are within reach the $Z_LZ_L$ channel has the largest signal rate, and next is the $W^\pm_L W^\pm_L$ channels. So by counting the number of $W^\pm_L
W^\pm_L$, $W^+_L W^-_L$, and $Z_L Z_L$ pairs in the final state one can tell the different structure of the EWSBS [@han]. But to distinguish a $W$ from a $Z$ by the dijet mass measurement is not a trivial issue, though we can use the $B$-tagging to distinguish a $W$ from a $Z$ somehow. For a discussion on this subject please see, [*e.g.*]{}, Ref. [@han].
The number of signal events in Table \[table1\] does not include any detection efficiencies of the $W_L$’s coming out from the strong scattering region, nor the tagging efficiencies for the spectator $W$’s. The tagging efficiencies for the spectator $W$’s will be dealt with in the next section. The detection efficiencies of the $W_L$’s consist of the branching ratios of the $W_L$ into jets or leptons, and the tagging efficiencies of these decay products. The branching ratio BR($W\to jj)\approx$BR$(Z\to jj)
\approx 0.7$. Assuming a 30% (reasonable to pessimistic) tagging efficiencies for the decay products, we have about 15% overall detection efficiencies for the $W_LW_L$ coming out from the strong scattering region.
Tagging the Spectator $W$’s
===========================
So far we have not considered any backgrounds nor background suppression techniques. In our calculation, we use the effective $W_L$ luminosity which does not predict the correct kinematics for the spectator $W$’s, and therefore any acceptance cuts on the spectator $W$’s will be unrealistic. However, we need to tag at least one or both of these spectator $W$’s in order to eliminate the enormous $\gamma\gamma\to W_T W_T,\,Z_T Z_T$ backgrounds. One way to remedy is to carry out an exact SM calculation of $\gamma\gamma\to WWWW$ or $WWZZ$ with a heavy Higgs boson, and estimate the acceptance efficiencies on tagging the spectator $W$’s, and then apply these efficiencies to the other models which can only be calculated using the effective $W_L$ luminosity.
However, the calculations of the processes $\gamma\gamma\to WWWW$ or $WWZZ$ are non-trivial. Instead, we can obtain the tagging efficiencies by calculating a simpler process $\gamma\gamma \to WWH$ for $m_H\approx $1 TeV, with and without imposing acceptance cuts on the final state $W$’s. We will calculate the total cross section for $\gamma\gamma\to WWH$ without any cuts, and also the cross section with the acceptance cuts $$p_T(W) > 25\;{\rm GeV},\qquad |y(W)| < 1.5\;{\rm or}\;2$$ on either one or both of the $W$’s. The cross sections are presented in Table \[table2\] for $m_H=1$ TeV. There are two tagging efficiencies corresponding to tagging at least one or both of the spectator $W$’s. From Table \[table2\], if we require the spectator $W$’s within a rapidity of $|y(W)|<1.5$ the tagging efficiencies are 91% and 42% for tagging at least one or both the $W$’s respectively. To eliminate the $W_T W_T$ or $Z_T
Z_T$ backgrounds we need only tag one of the spectator $W$’s plus the $W_L$’s from the strong scattering. A further confirmation by tagging two spectator $W$’s will result in an efficiency of only 42%. But if we tag both spectator $W$’s within the rapidity $|y(W)|<2$ the double-tag efficiency increases to 82%. This drastic difference of the double-tag efficiencies between rapidity cut of 1.5 and 2 demonstrates that it is likely (40% chance) to have at least one spectator $W$ in the forward rapidity region $1.5<|y(W)|<2$. Next we can multiply these efficiencies to the numbers in Table \[table1\] to get a more reliable number of signal events when the spectator $W$’s are tagged. Taking into account of the 15% (from the last section) detection efficiency for the $W_L W_L$ plus the tagging efficiency of at least one or both of the spectator $W$’s, we still have at least 10% overall efficiency. With 10% efficiency we still have a sizeable number of signal events. Scalar-type models will be shown up in the $W^+W^-\to W^+W^-$ channel with at least 47 events. The vector-like models will also be shown up in the $W^+W^-\to W^+W^-$ channel if the vector resonance is within reach of the energy of the $\gamma\gamma$ collider. For nonresonant models we have about 15 events for the $W^\pm W^\pm \to W^\pm W^\pm$ channels and 17 events for $W^+W^-\to ZZ$ channel.
Background Discussions
======================
The continuum productions of $\gamma\gamma\to WWWW$ and $WWZZ$, together with the heavy quark production of $\gamma\gamma\to t\bar t t\bar t$ followed by the top decays into $W$’s, form the irreducible set of backgrounds. They are the SM predictions that any significant excess of $W_L W_L$ or $Z_L Z_L$ events will indicate some kinds of new physics for the EWSBS. The other reducible backgrounds include the productions of $W$’s with jets, $Z$’s with jets, and multi-jet.
The $WWWW$ and $WWZZ$ productions are of order $\alpha_w^4$, and so should be at most the same level as our strong $W_L W_L$ signal. Although the $t\bar t t\bar t$ background is ${\cal O}(\alpha_s^2/\alpha_w^2)$ larger than the $WWWW$ background, we can to certain extent reduce it by reconstructing the top and by imposing the top-mass constraints. The other QCD backgrounds of $W$’s or $Z$’s with jets are reducible by the $W$ or $Z$ mass constraints.
In addition, we can make use of the kinematics of the spectator $W$’s and the strongly scattered $W_L$’s [@kingman]. The $p_T$ of the spectator $W$’s should be of order $m_W/2$ after the photon emits an almost on-shell $W_L$, which then participates in the strong scattering. Also, as mentioned in the last section, at least one of the spectator $W$’s tend to go forward in the rapidity region $|y(W)|>1.5$. On the other hand, the $W_L W_L$ after the strong scattering come out in the central rapidity region with large $p_T$ and large invariant mass, and back-to-back in the transverse plane, which are all due to the strong interaction of the EWSBS. But it is hardly true for the backgrounds. Acceptance cuts can be formulated based on the above arguments to substantially reduce the backgrounds [@future].
In conclusions, we have demonstrated another type of processes in $\gamma\gamma$ collision that can probe the strongly interacting EWSBS scenario. The processes do not involve the indirect loop effects, and also are safe from the huge $W_T W_T$ or $Z_T Z_T$ backgrounds due to the presence of the spectator $W$’s. Even with only 10% overall efficiency we still have enough signal events with 100 fb$^{-1}$ luminosities. Irreducible backgrounds from $WWWW$, $WWZZ$, and $t\bar t t\bar t$ can be reduced by considering the special kinematics of the strongly scattered $W_L$’s and the spectator $W$’s. Other reducible backgrounds are reduced by the mass constraints.
We are grateful to V. Barger, D. Bowser-Chao, and T. Han for useful discussions. This work was supported by the U. S. Department of Energy, Division of High Energy Physics, under Grant DE-FG02-91-ER40684.
B. W. Lee, C. Quigg and H. Thacker, Phys. Rev. [**D16**]{}, 1519 (1977). M. S. Chanowitz and M. K. Gaillard, Nucl. Phys. [**B261**]{}, 379 (1985). J. Bagger, [*et al.*]{}, Fermilab Report number FERMILAB-PUB-93-040-T (to appear in Phys. Rev. D), and reference therein. V. Telnov, Nucl. Instr. & Methods [**A294**]{}, 72 (1990); I. Ginzburg, G. Kotkin, V. Serbo and V. Telnov, Nucl. Instr. & Methods [**205**]{}, 47 (1983); [*idem*]{} [**219**]{}, 5 (1984). E. Boos and G. Jikia, Phys. Lett. [**B275**]{}, 164 (1992); A. Abbasabadi, D. Bowser-Chao, D. Dicus, and W. Repko, Michigan State Univ. preprint MSUTH-92-03 (1992); R. Rosenfeld, Northeastern Univ. preprint NUB-3074/93-Th (1993). G. Jikia, Phys. Lett. [**B298**]{}, 224 (1993); M. S. Berger, UW-Madison preprint MAD/PH/771; C. Kao and D. Dicus, UT-Austin preprint DOE-ER-40757-024. This type of processes was pointed out briefly by S. Brodsky in his talk at the “2nd International Workshop on Physics and Experiments at $e^+e^-$ colliders", Waikoloa, Hawaii (April 1993), see also SLAC-PUB-6314. K. Hagiwara, I. Watanabe, and P. Zerwas, Phys. Lett. [**B278**]{}, 187 (1992). Plenary Talk by T. Han at the same workshop in Ref. [@brodsky], see also Fermilab-Conf-93/217-T. Talk by K. Cheung at the same workshop in Ref. [@brodsky] and at the “Workshop on Physics at Current Accelerator and the Supercolliders", Argonne, Illinois (June 1993), see also NUHEP-TH-93-16. K.Cheung, in preparation.
$W^+_L W^+_L \to W^+_L W^+_L$ $W^+_L W^-_L \to W^+_L W^-_L$ $W^+_L W^-_L \to Z_L Z_L$
--------------------------------------- ------------------------------- ------------------------------- ---------------------------
\(1) SM Higgs
$m_H=1$ TeV 88 1600 760
\(2) chirally-coupled scalar
$m_S=1$ TeV, $\Gamma_S=350$ GeV 100 570 430
\(3) O(2N) 90 470 350
\(4) chirally-coupled vector
a\. $m_V=1$ TeV, $\Gamma_V=0.4$ TeV 180 2400 280
b\. $m_V=1.2$ TeV, $\Gamma_V=0.5$ TeV 52 590 29
c\. $m_V=1.5$ TeV, $\Gamma_V=0.6$ TeV 88 120 40
LET 150 110 170
: \[table1\] The number of the signal events for the strong $W_L W_L$ scattering predicted by various models at $\gamma\gamma$ collider of $\sqrt{s}=1.5$ TeV. The acceptance cuts on the final $W_LW_L$ or $Z_LZ_L$ are: $m(WW,ZZ)>500$ GeV and $|y(W,Z)|<1.5$. The luminosity is assumed 100 fb$^{-1}$. No efficiencies are included here.
$|y(W)|<$ No cuts Tagging at least one $W$ Tagging both $W$’s
----------- --------- -------------------------- --------------------
- 14.7 - -
1.5 - 13.4 (91%) 6.16 (42%)
2.0 - 14.5 (98.5%) 12.1 (82%)
: \[table2\] Table showing the cross sections (fb) for the process $\gamma\gamma\to WWH$ with a SM Higgs boson of mass $m_H=1$ TeV at $\sqrt{s_{\gamma\gamma}}=1.5$ TeV, with and without imposing acceptance cuts on the final state $W$’s. The acceptance cuts are $p_T(W)>25$ GeV and $|y(W)|<1.5$ or 2. The second column shows the total cross section without cuts. The third column corresponds to tagging at least one of the $W$’s, and the last column corresponds to tagging both. The percentages in the parentheses are the efficiencies.
| ArXiv |
---
abstract: 'A *binary frame template* is a device for creating binary matroids from graphic or cographic matroids. Such matroids are said to *conform* or *coconform* to the template. We introduce a preorder on these templates and determine the nontrivial templates that are minimal with respect to this order. As an application of our main result, we determine the eventual growth rates of certain minor-closed classes of binary matroids, including the class of binary matroids with no minor isomorphic to $PG(3,2)$. Our main result applies to all highly-connected matroids in a class, not just those of maximum size. As a second application, we characterize the highly-connected 1-flowing matroids.'
address:
- |
Department of Mathematics\
Louisiana State University\
Baton Rouge, Louisiana
- |
Department of Mathematics\
Louisiana State University\
Baton Rouge, Louisiana
author:
- Kevin Grace
- 'Stefan H. M. van Zwam'
title: Templates for Binary Matroids
---
[^1]
Introduction
============
Geelen, Gerards, and Whittle [@ggw15] recently announced a structure theorem describing the highly connected members of any proper minor-closed class of matroids representable over a given finite field. In this paper we study some consequences of their result. To state a first, rough version of their result, we need the following definitions.
A matroid $M$ is *vertically $k$-connected* if, for each partition $(X,Y)$ of the ground set of $M$ with $r(X)+r(Y)-r(M)<k-1$, either $X$ or $Y$ is spanning. We denote the unique prime subfield of $\mathbb{F}$ by $\mathbb{F}_{\textnormal{prime}}$. We say that a matroid $M_2$ is a *rank-$(\leq t)$ perturbation* of a matroid $M_1$ if there exist matrices $A_1$ and $A_2$ over $\mathbb{F}$ such that $r(M(A_1-A_2))\leq t$ and such that $M_1\cong M(A_1)$ and $M_2\cong M(A_2)$.
We now restate [@ggw15 Theorem 3.3]. Its proof is forthcoming in a future paper by Geelen, Gerards, and Whittle.
\[ggw3.3\] Let $\mathbb{F}$ be a finite field and let $m_0$ be a positive integer. Then there exist $k,n,t\in\mathbb{Z}_+$ such that, if $M$ is a matroid representable over $\mathbb{F}$ such that $M$ or $M^*$ is vertically $k$-connected and such that $M$ has an $M(K_n)$-minor but no $PG(m_0-1,\mathbb{F}_{\textnormal{prime}})$-minor, then $M$ is a rank-$(\leq t)$ perturbation of a frame matroid representable over $\mathbb{F}$.
Let us consider a very simple example of a rank-1 perturbation. Let $A_1$ be the binary matrix $$\begin{bmatrix}
1&0&0&0&1&1&1&0&0&0\\
0&1&0&0&1&0&0&1&1&0\\
0&0&1&0&0&1&0&1&0&1\\
0&0&0&1&0&0&1&0&1&1\\
\end{bmatrix},$$ and let $A_2$ be the binary matrix $$\begin{bmatrix}
0&1&1&1&1&1&1&0&0&0\\
1&0&1&1&1&0&0&1&1&0\\
0&0&1&0&0&1&0&1&0&1\\
0&0&0&1&0&0&1&0&1&1\\
\end{bmatrix}.$$ Note that $A_2$ is the result of adding the rank-1 matrix $$\begin{bmatrix}
1&1&1&1&0&0&0&0&0&0\\
1&1&1&1&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0\\
\end{bmatrix}$$ to $A_1$. Therefore, the vector matroid $M(A_2)$ is a rank-1 perturbation of $M(A_1)$.
Theorem \[ggw3.3\] is essentially a simplified version of a much more complex structure theorem [@ggw15 Theorem 4.2]. Geelen, Gerards, and Whittle introduced the concept of a *template* as a tool to capture much of this complexity.
Our focus in this paper is on the binary case. Roughly speaking, a binary frame template can be thought of as a recipe for constructing a representable matroid from a graphic or cographic matroid. A matroid constructed in this way is said to *conform* or *coconform* to the template.
In the example above, we may think of $M(A_2)$ as the matroid obtained from the vector matroid of the following matrix by contracting the element indexing the final column. Note that the large submatrix on the bottom left is $A_1$: $$\left[
\begin{array}{@{}cccccccccc|c@{}}
1&1&1&1&0&0&0&0&0&0&1\\
\hline
1&0&0&0&1&1&1&0&0&0&1\\
0&1&0&0&1&0&0&1&1&0&1\\
0&0&1&0&0&1&0&1&0&1&0\\
0&0&0&1&0&0&1&0&1&1&0\\
\end{array}
\right]$$ In fact, for any matrix $A$ of the following form, where $v$ and $w$ are arbitrary binary vectors, the matroid $M(A)/c$ conforms to the template $\Phi_{CX}$, which we will define in Section \[Reducing a Template\]:
----------------------------- -----
$v$ 1
incidence matrix of a graph $w$
----------------------------- -----
Let $\mathcal{M}(\Phi)$ denote the set of matroids representable over a field $\mathbb{F}$ that conform to a frame template $\Phi$. Theorem \[ggwframe\] below is a slight modification of [@ggw15 Theorem 4.2]. The modification is explained in Section \[Preliminaries\].
\[ggwframe\] Let $\mathbb{F}$ be a finite field, let $m$ be a positive integer, and let $\mathcal{M}$ be a minor-closed class of matroids representable over $\mathbb{F}$. Then there exist $k,l\in \mathbb{Z}_+$ and frame templates $\Phi_1,\dots, \Phi_s, \Psi_1,\dots, \Psi_t$ such that
- $\mathcal{M}$ contains each of the classes $\mathcal{M}(\Phi_1),\dots,\mathcal{M}(\Phi_s)$,
- $\mathcal{M}$ contains the duals of the matroids in each of the classes $\mathcal{M}(\Psi_1),\dots,\mathcal{M}(\Psi_t)$, and
- if $M$ is a simple vertically $k$-connected member of $\mathcal{M}$ with at least $l$ elements and with no $PG(m-1,\mathbb{F}_{\textnormal{prime}})$ minor, then either $M$ is a member of at least one of the classes $\mathcal{M}(\Phi_1),\dots,\mathcal{M}(\Phi_s)$, or $M^*$ is a member of at least one of the classes $\mathcal{M}(\Psi_1),\dots,\mathcal{M}(\Psi_t)$.
Our contribution is to shed some light on how these templates are related to each other. We define a preorder on the set of frame templates. Our main result, Theorem \[minimal\], is a list of nontrivial binary frame templates that are minimal with respect to this preorder.
One application of this result involves growth rates of minor-closed classes of binary matroids. The *growth rate function* of a minor-closed class $\mathcal{M}$ is the function whose value at an integer $r\geq0$ is given by the maximum number of elements in a simple matroid in $\mathcal{M}$ of rank at most $r$. We prove that a minor-closed class of binary matroids has a growth rate that is eventually equal to the growth rate of the class of graphic matroids if and only if it contains all graphic matroids but does not contain the class of matroids conforming to a certain template. The class of matroids conforming to this template is exactly the class of matroids having an even-cycle representation with a blocking pair. Geelen and Nelson also proved this result in [@gn15]. We also prove the following theorem. Here, $\mathcal{EX}(F)$ denotes the class of binary matroids with no $F$-minor. If $f$ and $g$ are functions, we write $f(r)\approx g(r)$ if $f(r)=g(r)$ for all but finitely many $r$.
\[EXPG32\] The growth rate function for $\mathcal{EX}(PG(3,2))$ is $$h_{\mathcal{EX}(PG(3,2))}\approx r^2-r+1.$$
Note that $r^2-r+1$ is the growth rate of the class of even-cycle matroids.
Our main result goes beyond growth rates because it gives information about all highly-connected matroids in a minor-closed class, not just the maximum-sized matroids. This is illustrated by our second application, involving 1-flowing matroids. The 1-flowing property is a generalization of the max-flow min-cut property of graphs. We prove the following.
\[1flowing\] There exist $k,l\in\mathbb{Z}_+$ such that every simple, vertically $k$-connected, 1-flowing matroid with at least $l$ elements is either graphic or cographic.
We use templates to study a minor-closed class $\mathcal{M}$ by describing the highly-connected matroids in the class. This analysis follows a certain pattern:
1. Find a matroid $N$ not in $\mathcal{M}$.
2. Find all templates such that $N$ is not a minor of any matroid conforming to that template.
3. If all matroids conforming to these templates are in $\mathcal{M}$, then the analysis is complete.
4. Otherwise, repeat Step (1).
From the definition of conforming to a template, which we will give in Section \[Preliminaries\], it will not be difficult to see that for each binary frame template $\Phi$, there are integers $t_1$ and $t_2$ such that every matroid conforming to $\Phi$ is a rank-$(\leq t_1)$ perturbation of a graphic matroid and every matroid coconforming to $\Phi$ is a rank-$(\leq t_2)$ perturbation of a cographic matroid. Thus, by Theorem \[ggwframe\], the highly connected matroids in a minor-closed class of binary matroids are “close” to being graphic or cographic. In this regard, the work regarding templates resembles work done by Robertson and Seymour concerning minor-closed classes of graphs. In Theorem 1.3 of [@rs03], Robertson and Seymour showed that highly-connected graphs in a minor-closed class are in some sense “close” to being embeddable in some surface.
Section \[Preliminaries\] of this paper repeats the necessary definitions found in [@ggw15]. In Section \[Reducing a Template\], we prove our main result, as well as giving some machinery leading up to it. Section \[Growth Rates\] applies our result to growth rates of minor-closed classes of binary matroids, and in Section \[1-flowing Matroids\], we prove Theorem \[1flowing\].
Preliminaries {#Preliminaries}
=============
We repeat here several definitions concerning highly connected matroids which can be found in Geelen, Gerards, and Whittle [@ggw15]. Although the results found in [@ggw15] are technically about matrices rather than matroids, it suffices for our purposes to state the results in terms of their immediate matroid consequences.
Let $A$ be a matrix over a field $\mathbb{F}$. Then $A$ is a *frame matrix* if each column of $A$ has at most two nonzero entries. We let $\mathbb{F}^{\times}$ denote the multiplicative group of $\mathbb{F}$. Let $\Gamma$ be a subgroup of $\mathbb{F}^{\times}$. A $\Gamma$-frame matrix is a frame matrix $A$ such that:
- Each column of $A$ with a nonzero entry contains a 1.
- If a column of $A$ has a second nonzero entry, then that entry is $-\gamma$ for some $\gamma\in\Gamma$.
In the case where $\Gamma$ is the multiplicative group of one element, a matrix is a $\Gamma$-frame matrix if and only if it is the signed incidence matrix of a graph, with possibly a row removed. In particular, a binary matroid is graphic if and only if it can be represented by a matrix over $\mathrm{GF}(2)$ in which no column has more than two nonzero entries.
To facilitate the description of their structure theorem, Geelen, Gerards, and Whittle capture capture much of the complexity with the concept of a “template.” Let $\mathbb{F}$ be a finite field. A *frame template* over $\mathbb{F}$ is a tuple $\Phi=(\Gamma,C,X,Y_0,Y_1,A_1,\Delta,\Lambda)$ such that the following hold[^2]:
- $\Gamma$ is a subgroup of $\mathbb{F}^{\times}$.
- $C$, $X$, $Y_0$ and $Y_1$ are disjoint finite sets.
- $A_1\in \mathbb{F}^{X\times (C\cup Y_0\cup Y_1)}$.
- $\Lambda$ is a subgroup of the additive group of $\mathbb{F}^X$ and is closed under scaling by elements of $\Gamma$.
- $\Delta$ is a subgroup of the additive group of $\mathbb{F}^{C\cup Y_0 \cup Y_1}$ and is closed under scaling by elements of $\Gamma$.
Let $\Phi=(\Gamma,C,X,Y_0,Y_1,A_1,\Delta,\Lambda)$ be a frame template. Let $B$ and $E$ be finite sets, and let $A'\in\mathbb{F}^{B\times E}$. We say that $A'$ *respects* $\Phi$ if the following hold:
- $X\subseteq B$ and $C, Y_0, Y_1\subseteq E$.
- $A'[X, C\cup Y_0\cup Y_1]=A_1$.
- There exists a set $Z\subseteq E-(C\cup Y_0\cup Y_1)$ such that $A'[X,Z]=0$, each column of $A'[B-X,Z]$ is a unit vector, and $A'[B-X, E-(C\cup Y_0\cup Y_1\cup Z)]$ is a $\Gamma$-frame matrix.
- Each column of $A'[X,E-(C\cup Y_0\cup Y_1\cup Z)]$ is contained in $\Lambda$.
- Each row of $A'[B-X, C\cup Y_0\cup Y_1]$ is contained in $\Delta$.
Figure \[fig:A’\] shows the structure of $A'$.
[ r|c|c|ccc| ]{} &&&&\
&&&&&\
$X$&columns from $\Lambda$&$0$&&$A_1$&\
&&&&&\
&&&\
&&&&&\
&&&&&\
&&&&&\
&&&&&\
Suppose that $A'$ respects $\Phi$ and that $Z$ satisfies (iii) above. Now suppose that $A\in \mathbb{F}^{B\times E}$ satisfies the following conditions:
- $A[B,E-Z]=A'[B,E-Z]$
- For each $i\in Z$ there exists $j\in Y_1$ such that the $i$-th column of $A$ is the sum of the $i$-th and the $j$-th columns of $A'$.
We say that any such matrix *conforms* to $\Phi$.
Let $M$ be a matroid representable over $\mathbb{F}$. We say that $M$ *conforms* to $\Phi$ if there is a matrix $A$ that conforms to $\Phi$ such that $M$ is isomorphic to $M(A)/C\backslash Y_1$.
Let $\mathcal{M}(\Phi)$ denote the set of matroids representable over $\mathbb{F}$ that conform to $\Phi$. Recall that a matroid $M$ is *vertically $k$-connected* if, for each partition $(X,Y)$ of the ground set of $M$ with $r(X)+r(Y)-r(M)<k-1$, either $X$ or $Y$ is spanning. We denote the unique prime subfield of $\mathbb{F}$ by $\mathbb{F}_{\textnormal{prime}}$. Geelen, Gerards, and Whittle will prove Theorem \[ggwframe\] in a future paper. This theorem is actually a slight modification of the theorem found in [@ggw15]. In that paper, there is no mention of the requirement that a matroid have size at least $l$. However, Geelen (personal communication) has stated that this is necessary to ensure that adding a finite number of matroids to the class $\mathcal{M}$ does not add any templates to the list $\Phi_1,\dots, \Phi_s, \Psi_1,\dots, \Psi_t$.
Although the term *coconform* does not appear in [@ggw15], we define it in the following obvious way.
A matroid $M$ *coconforms* to a template $\Phi$ if its dual $M^*$ conforms to $\Phi$.
To simplify the proofs in this paper, it will be helpful to expand the concept of conforming slightly.
\[virtual\] Let $A'$ be a matrix that respects $\Phi$, as defined above, except that we allow columns of $A'[B-X,Z]$ to be either unit columns or zero columns. Let $A$ be a matrix that is constructed from $A'$ as described above. Thus, $A[B,E-Z]=A'[B,E-Z]$, and for each $i\in Z$ there exists $j\in Y_1$ such that the $i$-th column of $A$ is the sum of the $i$-th and the $j$-th columns of $A'$. Let $M$ be isomorphic to $M(A)/C\backslash Y_1$. We say that $A$ and $M$ *virtually conform* to $\Phi$ and that $A'$ *virtually respects* $\Phi$. If $M^*$ virtually conforms to $\Phi$, we say that $M$ *virtually coconforms* to $\Phi$.
We will denote the set of matroids representable over $\mathbb{F}$ that virtually conform to $\Phi$ by $\mathcal{M}_v(\Phi)$ and the set of matroids representable over $\mathbb{F}$ that virtually coconform to $\Phi$ by $\mathcal{M}^*_v(\Phi)$.
The following notation will be used throughout this paper. We denote an empty matrix by $[\emptyset]$. We denote a group of one element by $\{0\}$ or $\{1\}$, if it is an additive or multiplicative group, respectively. If $S'$ is a subset of a set $S$ and $G$ is a subgroup of the additive group $\mathbb{F}^S$, we denote by $G|S'$ the projection of $G$ into $\mathbb{F}^{S'}$. Similarly, if $\bar{x}\in G$, we denote the projection of $\bar{x}$ into $\mathbb{F}^{S'}$ by $\bar{x}|S'$.
Unexplained notation and terminology will generally follow Oxley [@o11]. One exception is that we denote the vector matroid of a matrix $A$ by $M(A)$, rather than $M[A]$.
Reducing a Template {#Reducing a Template}
===================
In this section, we will introduce reductions and show that every template reduces to one of several basic templates.
Since templates are used to study minor-closed classes of matroids, a natural question to ask is whether the set of matroids conforming to a particular template is minor-closed. The answer is no, in general. For example, if $|Y_0|=1$, then a matroid conforms to the following binary frame template if and only if it is a graphic matroid with a loop: $$(\{1\},\emptyset,\emptyset,Y_0,\emptyset,[\emptyset],\{0\},\{0\}).$$ Clearly, this is not a minor-closed class.
Another question to ask is whether there might be some sort of minor relationship between a pair of templates, where every matroid conforming to one template is a minor of a matroid conforming to the other. These questions motivate the following discussion.
A *reduction* is an operation on a frame template $\Phi$ that produces a frame template $\Phi'$ such that $\mathcal{M}(\Phi')\subseteq \mathcal{M}(\Phi)$.
\[reductions\] The following operations are reductions on a frame template $\Phi$:
- Replace $\Gamma$ with a proper subgroup.
- Replace $\Lambda$ with a proper subgroup closed under multiplication by elements from $\Gamma$.
- Replace $\Delta$ with a proper subgroup closed under multiplication by elements from $\Gamma$.
- Remove an element $y$ from $Y_1$. (More precisely, replace $A_1$ with $A_1[X, Y_0\cup (Y_1-y)\cup C]$ and replace $\Delta$ with $\Delta|(Y_0\cup (Y_1-y)\cup C)$.
- For all matrices $A'$ respecting $\Phi$, perform an elementary row operation on $A'[X, E]$. (Note that this alters the matrix $A_1$ and performs a change of basis on $\Lambda$.)
- If there is some element $x\in X$ such that, for every matrix $A'$ respecting $\Phi$, we have that $A'[\{x\},E]$ is a zero row vector, remove $x$ from $X$. (This simply has the effect of removing a zero row from every matrix conforming to $\Phi$.)
- Let $c\in C$ be such that $A_1[X,\{c\}]$ is a unit column whose nonzero entry is in the row indexed by $x\in X$, and let either $\lambda_x=0$ for each $\lambda\in\Lambda$ or $\delta_c=0$ for each $\delta\in\Delta$. Let $\Delta'$ be the result of adding $-\delta_cA_1[\{x\},Y_0\cup Y_1\cup C]$ to each element $\delta\in\Delta$. Replace $\Delta$ with $\Delta'$, and then remove $c$ from $C$ and $d$ from $D$. (More precisely, replace $A_1$ with $A_1[X-x, Y_0\cup Y_1\cup (C-c)]$, replace $\Lambda$ with $\Lambda|(X-x)$, and replace $\Delta$ with $\Delta'|(Y_0\cup Y_1\cup (C-c))$.)
- Let $c\in C$ be such that $A_1[X,\{c\}]$ is a zero column and $\delta_c=0$ for all $\delta\in\Delta$. Then remove $c$ from $C$. (More precisely, replace $A_1$ with $A_1[X, Y_0\cup Y_1\cup (C-c)]$, and replace $\Delta$ with $\Delta|(Y_0\cup Y_1\cup (C-c))$.)
Let $\Phi'$ be the template that results from performing one of operations (1)-(8) on $\Phi$.
For (1)-(3), every matrix $A'$ respecting $\Phi'$ also respects $\Phi$.
For (4), let $A'$ be a matrix respecting $\Phi'$, and let $M$ be the matroid $M(A)/C\backslash Y_1$, where $A$ is a matrix conforming to $\Phi'$ that has been constructed from $A'$ respecting $\Phi'$ as described in Section \[Preliminaries\]. Since $Y_1$ is deleted to produce $M$, the only effect of $Y_1$ on $M$ is that for each $i\in Z$ there exists $j\in Y_1$ such that the $i$-th column of $A$ is the sum of the $i$-th and the $j$-th columns of $A'$. But each $j\in Y_1$ in the template $\Phi'$ is also contained in $Y_1$ in the template $\Phi$. Therefore, $A$ conforms to $\Phi$, as does $M$.
For (5) and (6), elementary row operations and removing zero rows produce isomorphic matroids.
Operations (7) and (8) have the effect of contracting $c$ from $M(A)\backslash Y_1$ for every matrix $A$ conforming to $\Phi$. Since all of $C$ is contracted to produce a matroid $M$ conforming to $\Phi$, the matroids we produce by performing either of these operations still conform to $\Phi$.
For $i\in\{1,\dots, 8\}$, we call operation $(i)$ above a *reduction of type $i$*.
The operations listed in the definition below are not reductions as defined above, but we continue the numbering from Proposition \[reductions\] for ease of reference.
\[weaklyconforming\] A template $\Phi'$ is a *template minor* of $\Phi$ if $\Phi'$ is obtained from $\Phi$ by repeatedly performing the following operations:
- Performing a reduction of type 1-8 on $\Phi$.
- Removing an element $y$ from $Y_0$, replacing $A_1$ with $A_1[X,(Y_0-y)\cup Y_1\cup C]$, and replacing $\Delta$ with $\Delta|((Y_0-y)\cup Y_1\cup C)$. (This has the effect of deleting $y$ from every matroid conforming to $\Phi$.)
- Let $x\in X$ with $\lambda_x=0$ for every $\lambda\in\Lambda$, and let $y\in Y_0$ be such that $(A_1)_{x,y}\neq0$. Then contract $y$ from every matroid conforming to $\Phi$. (More precisely, perform row operations on $A_1$ so that $A_1[X, \{y\}]$ is a unit column with $(A_1)_{x,y}=1$. Then replace every element $\delta\in\Delta$ with the row vector $-\delta_y A_1[\{x\}, Y_0\cup Y_1\cup C]+\delta$. This induces a group homomorphism $\Delta\rightarrow\Delta'$, where $\Delta'$ is also a subgroup of the additive group of $\mathbb{F}^{C\cup Y_0 \cup Y_1}$ and is closed under scaling by elements of $\Gamma$. Finally, replace $A_1$ with $A_1[X-x,(Y_0-y)\cup Y_1\cup C]$, project $\Lambda$ into $\mathbb{F}^{X-x}$, and project $\Delta'$ into $\mathbb{F}^{(Y_0-y)\cup Y_1\cup C}$. The resulting groups play the roles of $\Lambda$ and $\Delta$, respectively in $\Phi'$.)
- Let $y\in Y_0$ with $\delta_y=0$ for every $\delta\in\Delta$. Then contract $y$ from every matroid conforming to $\Phi$. (More precisely, if $A_1[X, \{y\}]$ is a zero vector, this is the same as simply removing $y$ from $Y_0$. Otherwise, choose some $x\in X$ such that $(A_1)_{x,y}\neq0$. Then for every matrix $A'$ that respects $\Phi$, perform row operations so that $A_1[X,\{y\}]$ is a unit column with $(A_1)_{x,y}=1$. This induces a group isomorphism $\Lambda\rightarrow\Lambda'$ where $\Lambda'$ is also a subgroup of the additive group of $\mathbb{F}^X$ and is closed under scaling by elements of $\Gamma$. Finally, replace $A_1$ with $A_1[X-x,(Y_0-y)\cup Y_1\cup C]$, project $\Lambda'$ into $\mathbb{F}^{X-x}$, and project $\Delta$ into $\mathbb{F}^{(Y_0-y)\cup Y_1\cup C}$. The resulting groups play the roles of $\Lambda$ and $\Delta$, respectively in $\Phi'$.)
Let $\Phi'$ be a template minor of $\Phi$, and let $A'$ be a matrix that virtually respects $\Phi'$. Let $A$ be a matrix that virtually conforms to $\Phi'$, and let $M$ be a matroid that virtually conforms to $\Phi'$. We say that $A'$ *weakly respects* $\Phi$ and that $A$ and $M$ *weakly conform* to $\Phi$. Let $\mathcal{M}_w(\Phi)$ denote the set of matroids representable over $\mathbb{F}$ that weakly conform to $\Phi$, and let $\mathcal{M}^*_w(\Phi)$ denote the set of matroids representable over $\mathbb{F}$ whose duals weakly conform to $\Phi$. If $M\in\mathcal{M}^*_w(\Phi)$, we say that $M$ *weakly coconforms* to $\Phi$.
\[minor\] If a matroid $M$ weakly conforms to a template $\Phi$, then $M$ is a minor of a matroid that conforms to $\Phi$.
Let $\Phi'$ be a template minor of $\Phi$. As we can see from Definition \[weaklyconforming\], every matroid $M$ weakly conforming to $\Phi'$ is a minor of a matroid virtually conforming to $\Phi$. It remains to analyze the case where $M$ virtually conforms to $\Phi$; so $M$ is isomorphic to $M(K)/C\backslash Y_1$, where $K$ is built from a matrix $K'$ that virtually respects $\Phi$. Consider the following matrix $A'$ obtained from $K'$ by adding a row $r$ and a column $c$.
[ r|c|c|c|c|ccc| ]{} &&&&&&\
&&&&&&\
$X$&0&columns from $\Lambda$&&&$A_1$&\
&&&&&&\
&&&&&\
&&&&&&&\
&&&&&&&\
&&&&&&&\
&&&&&&&\
$r$&1&0&$1\cdots1$&0&\
From $A'$, we can obtain a matrix $A$ conforming to $\Phi$ such that $M$ is isomorphic to $M(A)/C\backslash Y_1/c$. So $M$ is a minor of a matroid conforming to $\Phi$.
An easy consequence of Lemma \[minor\] is that Theorem \[ggwframe\], which deals with minor-closed classes, can be restated in terms of weak conforming.
\[weakframe\] Let $\mathbb{F}$ be a finite field, let $m$ be a positive integer, and let $\mathcal{M}$ be a minor-closed class of matroids representable over $\mathbb{F}$. Then there exist $k,l\in \mathbb{Z}_+$ and frame templates $\Phi_1,\dots, \Phi_s, \Psi_1,\dots, \Psi_t$ such that
- $\mathcal{M}$ contains each of the classes $\mathcal{M}_w(\Phi_1),\dots,\mathcal{M}_w(\Phi_s)$,
- $\mathcal{M}$ contains the duals of the matroids in each of the classes $\mathcal{M}_w(\Psi_1)$,$\dots$,$\mathcal{M}_w(\Psi_t)$, and
- if $M$ is a simple vertically $k$-connected member of $\mathcal{M}$ with at least $l$ elements and with no $PG(m-1,\mathbb{F}_{\textnormal{prime}})$ minor, then either $M$ is a member of at least one of the classes $\mathcal{M}_v(\Phi_1),\dots,\mathcal{M}_v(\Phi_s)$ or $M^*$ is a member of at least one of the classes $\mathcal{M}_v(\Psi_1),\dots,\mathcal{M}_v(\Psi_t)$.
Let $\Phi_1,\dots, \Phi_s, \Psi_1,\dots, \Psi_t$ be the templates whose existence is implied by Theorem \[ggwframe\]. For $\Phi\in\{\Phi_1,\dots, \Phi_s\}$, Lemma \[minor\] implies that any matroid $M\in \mathcal{M}_w(\Phi)$ is a minor of a matroid $N\in\mathcal{M}(\Phi)$. Since $\mathcal{M}$ contains $\mathcal{M}(\Phi)$ and is minor-closed, $\mathcal{M}$ contains $\mathcal{M}_w(\Phi)$ as well. Similarly, $\mathcal{M}$ contains the duals of the matroids in each of the classes $\mathcal{M}_w(\Psi_1),\dots,\mathcal{M}_w(\Psi_t)$. The third condition above holds since every matroid conforming to a template also virtually conforms to it.
If $\mathcal{M}_{w}(\Phi)=\mathcal{M}_{w}(\Phi')$, we say that $\Phi$ is *equivalent* to $\Phi'$ and write $\Phi\sim\Phi'$. It is clear that $\sim$ is indeed an equivalence relation.
Let $T_{\mathbb{F}}$ be the set of all frame templates over $\mathbb{F}$. We define a preorder $\preceq$ on $T_{\mathbb{F}}$ as follows. We say $\Phi\preceq\Phi'$ if $\mathcal{M}_w(\Phi)\subseteq\mathcal{M}_w(\Phi')$. This is indeed a preorder since reflexivity and transitivity follow from the subset relation. We may obtain a partial order by considering equivalence classes of templates, with equivalence as defined above. However, the templates themselves, rather than equivalence classes, are the objects we work with in this paper.
Let $\Phi_0$ be the frame template with all groups trivial and all sets empty. We call this template the *trivial template*. In general, we say that a template $\Phi$ is *trivial* if $\Phi\preceq\Phi_0$. It is easy to see that for any template $\Phi$, we have $\Phi_0\preceq \Phi$. Therefore, if $\Phi\preceq\Phi_0$, then actually $\Phi\sim\Phi_0$.
Our desire is to find a collection of minimal nontrivial templates. For the remainder of this paper, we restrict our attention to binary frame templates: those frame templates where $\mathbb{F}=\mathrm{GF}(2)$ and $\Gamma$ is the group of one element.
- Let $\Phi_C$ be the template with all groups trivial and all sets empty except that $|C|=1$ and $\Delta\cong\mathbb{Z}/2\mathbb{Z}$.
- Let $\Phi_X$ be the template with all groups trivial and all sets empty except that $|X|=1$ and $\Lambda\cong\mathbb{Z}/2\mathbb{Z}$.
- Let $\Phi_{Y_0}$ be the template with all groups trivial and all sets empty except that $|Y_0|=1$ and $\Delta\cong\mathbb{Z}/2\mathbb{Z}$.
- Let $\Phi_{CX}$ be the template with $Y_0=Y_1=\emptyset$, with $|C|=|X|=1$, with $\Delta\cong\Lambda\cong\mathbb{Z}/2\mathbb{Z}$, with $\Gamma$ trivial, and with $A_1=[1]$.
- Let $\Phi_{Y_1}$ be the template with all groups trivial, with $C=Y_0=\emptyset$, with $|Y_1|=3$ and $|X|=2$, and with $A_1=
\begin{bmatrix}
1& 0 &1\\
0& 1 & 1
\end{bmatrix}$.
It is not too difficult to see that the Fano matroid $F_7$ virtually conforms to each of $\Phi_C$, $\Phi_X$, $\Phi_{CX}$, $\Phi_{Y_0}$, and $\Phi_{Y_1}$. Therefore, these templates are nontrivial. In fact, one can see that $\mathcal{M}(\Phi_{Y_0})$ is the set of graft matroids, that $\mathcal{M}(\Phi_C)$ is the class of matroids obtained by closing the set of graft matroids under minors, and that $\mathcal{M}(\Phi_X)$ is the class of even-cycle matroids. In Lemma \[Y1minors\], we will show that $\mathcal{M}_v(\Phi_{Y_1})$ is the class of matroids having an even-cycle representation with a blocking pair.
Our goal in defining reductions and weak conforming was essentially to perform operations on matrices while leaving the $\Gamma$-frame submatrix intact. The following lemma does not contribute to that goal; so we will only make occasional use of it.
\[YCD\] The following relations hold:
- $\Phi_{Y_1}\preceq\Phi_X$
- $\Phi_{Y_1}\preceq\Phi_C$
- $\Phi_{Y_0}\preceq\Phi_C$
- $\Phi_C\preceq\Phi_{CX}$
- $\Phi_X\preceq\Phi_{CX}$
For (1), note that any simple matroid $M$ of rank $r$ virtually conforming to $\Phi_{Y_1}$ is a restriction of the vector matroid of a matrix $A$ of the following form:
----------------------- --- ----- ----- ------------ ------------ ------------
1 0 1 $1\cdots1$ $0\cdots0$ $1\cdots1$
0 1 1 $0\cdots0$ $1\cdots1$ $1\cdots1$
$\Gamma$-frame matrix $I$ $I$ $I$
----------------------- --- ----- ----- ------------ ------------ ------------
If we label the sets of rows and columns of $A$ as $B$ and $E$ respectively, and the first row as $x$, then we see that $A[B-x,E]$ is a $\Gamma$-frame matrix. If we let $X=\{x\}$, then we see that $M$ conforms to $\Phi_X$.
For (2), consider the matrix $A$ above. Note that it is obtained by contracting $c$ in the following matrix:
----------------------- --- ----- ----- ------------ ------------ ------------ ---
0 0 1 0$\cdots$0 0$\cdots$0 1$\cdots1$ 1
1 0 0 $1\cdots1$ $0\cdots0$ $0\cdots0$ 1
0 1 0 $0\cdots0$ $1\cdots1$ $0\cdots0$ 1
$\Gamma$-frame matrix $I$ $I$ $I$ 0
----------------------- --- ----- ----- ------------ ------------ ------------ ---
Removing $c$ from this matrix, we obtain a $\Gamma$-frame matrix. Therefore, $M$ conforms to $\Phi_C$.
For (3), any matroid $M$ conforming to $\Phi_{Y_0}$ is the vector matroid of a matrix of the following form, where $v$ is an arbitrary column vector:
----------------------- -----
$\Gamma$-frame matrix $v$
----------------------- -----
Let $A$ be the matrix below. Label its sets of rows and columns as $B$ and $E$ respectively, and let $c$ be the last column, with $C=\{c\}$.
0 1 1
----------------------- --- -----
$\Gamma$-frame matrix 0 $v$
Note that $M$ is isomorphic to $M(A)/C$. Since $A[B,E-C]$ is a $\Gamma$-frame matrix, we see that $M$ conforms to $\Phi_C$.
For (4), let $A$ be a matrix conforming to $\Phi_C$ and let $M=M(A)/C$ be the corresponding matroid conforming to $\Phi_C$. If the column of $A$ indexed by $C$ is a zero column, then construct the matrix $\bar{A}$ by adding a unit row, indexed by $X$, whose nonzero entry is in the column indexed by $C$. One readily sees that $\bar{A}$ conforms to $\Phi_{CX}$ and that the corresponding matroid $M(\bar{A})/C$ is equal to $M$. Otherwise, if the column of $A$ indexed by $C$ has a nonzero entry, then one readily sees that $A$ conforms to $\Phi_{CX}$ by considering the row containing the nonzero entry to be indexed by $X$.
For (5), any matroid $M$ conforming to $\Phi_D$ is the vector matroid of a matrix of the following form, where $v$ is an arbitrary row vector:
[|c|]{} $v$\
\
$\Gamma$-frame matrix\
\
Consider the following matrix $A$, whose last column is indexed by $\{c\}=C$:
$v$ 1
----------------------- ---
$0$ 1
$\Gamma$-frame matrix 0
The matroid $M$ is isomorphic to $M(A)/c$, which conforms to $\Phi_{CX}$.
\[yshift\] Let $\Phi$ be a template with $y\in Y_1$. Let $\Phi'$ be the template obtained from $\Phi$ by removing $y$ from $Y_1$ and placing it in $Y_0$. Then $\Phi'\preceq \Phi$.
Any matrix respecting $\Phi'$ virtually respects $\Phi$ by adding column $y$ only to the zero $Z$ column. Thus, any matroid conforming to $\Phi'$ weakly conforms to $\Phi$.
We call the operation described in Lemma \[yshift\] a *$y$-shift*.
Let $\Phi=(\Gamma,C,X,Y_0,Y_1,A_1,\Delta,\Lambda)$ be a frame template over a finite field $\mathbb{F}$. We say that $\Phi$ is in *standard form* if there are disjoint sets $C_0,C_1,X_0,$ and $X_1$ such that $C=C_0\cup C_1$, such that $X=X_0\cup X_1$, such that $A_1[X_0,C_0]$ is an identity matrix, and such that $A_1[X_1,C]$ is a zero matrix.
Figure \[fig:A’ standard\], with the stars representing arbitrary matrices, shows a matrix that virtually respects a template in standard form. Note that if $\Phi$ is in standard form, $|C_0|=|X_0|$. Also note that any of $C_0,C_1,X_0,$ or $X_1$ may be empty. Finally, note that we have defined standard form for frame templates over any finite field, not just binary frame templates.
[ r|c|c|cccc| ]{} &&&&&\
$X_0$&columns from $\Lambda|X_0$&0&&&$*$\
$X_1$&columns from $\Lambda|X_1$&0&&&\
&&&\
&&&&&&\
&&&&&&\
&&&&&&\
&&&&&&\
\[standard\] Every frame template $\Phi=(\Gamma,C,X,Y_0,Y_1,A_1,\Delta,\Lambda)$ is equivalent to a frame template in standard form.
Choose a basis $C_0$ for $M(A_1[X,C])$, and let $C_1=C-C_0$. Repeatedly perform operation (5) to obtain a template $\Phi'$ where $A_1[X,C_0]$ consists of an identity matrix on top of a zero matrix. Each use of operation (5) results in an equivalent template; therefore, $\Phi\sim\Phi'$. Let $X_0\subseteq X$ index the rows of the identity matrix, and let $X_1\subseteq X$ index the rows of the zero matrix. Since $C_0$ is a basis for $M(A_1[X,C])$, the matrix $A_1[X,C_1]$ must be a zero matrix as well. Thus, $\Phi'$ is in standard form.
Throughout the rest of this paper, we will implicitly use Lemma \[standard\] to assume that all templates are in standard form. Also, the operations (1)-(12) to which we will refer throughout the rest of this paper are the operations (1)-(8) from Proposition \[reductions\] and (9)-(12) from Definition \[weaklyconforming\].
\[PhiD\] If $\Phi=(\{1\},C,X,Y_0,Y_1,A_1,\Delta,\Lambda)$ is a binary frame template with $\Lambda|X_1$ nontrivial, then $\Phi_X\preceq\Phi$.
Perform operations (2) and (3) on $\Phi$ to obtain the following template, where $\lambda$ is an element of $\Lambda$ with $\lambda_x\neq0$ for some $x\in X_1$: $$(\{1\},C,X,Y_0,Y_1,A_1,\{0\},\{\mathbf{0}, \lambda\}).$$ On this template, repeatedly perform operation (7), then (8), then (4), and then (10) until the following template is obtained: $$(\{1\},\emptyset,X_1,\emptyset,\emptyset,[\emptyset],\{0\},\{\mathbf{0}, \lambda\}).$$ On this template, repeatedly perform operation (5) to obtain a template that is identical to the previous one except that the support of $\lambda$ contains only one element of $X_1$. On this template, repeatedly perform operation (6) to obtain the following template, where $x\in X_1$: $$(\{1\},\emptyset,\{x\},\emptyset,\emptyset,[\emptyset],\{0\},\mathbb{Z}/2\mathbb{Z}).$$ This template is $\Phi_X$.
\[PhiC\] If $\Phi=(\{1\},C,X,Y_0,Y_1,A_1,\Delta,\Lambda)$ is a binary frame template, then either $\Phi_C\preceq\Phi$ or $\Phi$ is equivalent to a template with $C_1=\emptyset$.
Suppose there is an element $\delta\in\Delta|C$ that is not in the row space of $A_1[X,C]$. Repeatedly perform operations (4) and (10) on $\Phi$ until the following template is obtained: $$(\{1\},C,X,\emptyset,\emptyset,A_1[X,C],\Delta|C,\Lambda).$$ On this template, perform operations (2) and (3) to obtain the following template: $$(\{1\},C,X,\emptyset,\emptyset,A_1[X,C],\{\mathbf{0}, \delta\},\{0\}).$$ Every matrix virtually respecting this template is row equivalent to a matrix virtually respecting a template that is identical to the previous template except that there is the additional condition that $\delta|C_0$ is a zero vector. Note that $\delta|C_1$ is nonzero since, in the previous template, $\delta$ was not in the row space of $A_1[X,C]$. Now, on the current template, repeatedly perform operation (7) and then operation (6) to obtain the following template: $$\Phi'=(\{1\},C_1,\emptyset,\emptyset,\emptyset,[\emptyset],\{\mathbf{0}, \delta|C_1\},\{0\}).$$
Now, any matroid $M$ conforming to $\Phi'$ is obtained by contracting $C_1$ from $M(A)$, where $A$ is a matrix conforming to $\Phi'$. By contracting any single element $c\in C_1$, where $\delta_c=1$, we turn the rest of the elements of $C_1$ into loops. So $C_1-c$ is deleted to obtain $M$. Thus, $M$ conforms to the template $$(\{1\},\{c\},\emptyset,\emptyset,\emptyset,[\emptyset],\mathbb{Z}/2\mathbb{Z},\{0\}),$$ which is $\Phi_C$. Similarly, the converse is true that any matroid conforming to $\Phi_C$ conforms to $\Phi'$. Thus, $\Phi_C\sim\Phi'\preceq\Phi$.
Now suppose that every element of $\Delta|C$ is in the row space of $A_1[X,C]$. Thus, contraction of $C_0$ turns the elements of $C_1$ into loops, and contraction of $C_1$ is the same as deletion of $C_1$. By deleting $C_1$ from every matrix virtually conforming to $\Phi$, we see that $\Phi$ is equivalent to a template with $C_1=\emptyset$.
\[PhiCD\] If $\Phi=(\{1\},C,X,Y_0,Y_1,A_1,\Delta,\Lambda)$ is a binary frame template, then one of the following is true:
- $\Phi_C\preceq\Phi$
- $\Phi$ is equivalent to a template with $\Lambda|X_1$ nontrivial and $\Phi_X\preceq\Phi$
- $\Phi$ is equivalent to a template with $\Lambda|X_0$ nontrivial and $\Phi_{CX}\preceq\Phi$
- $\Phi$ is equivalent to a template with $\Lambda$ trivial and $C=\emptyset$.
By Lemmas \[PhiD\] and \[PhiC\], we may assume that $\Lambda|X_1$ is trivial and that $C_1=\emptyset$.
First, suppose there exist elements $\delta\in\Delta|C_0$ and $\lambda\in\Lambda|X_0$ such that there are an odd number of natural numbers $i$ with $\delta_i=\lambda_i=1$. Thus, $\Lambda|X_0$ is nontrivial. Repeatedly perform operations (4) and (10) on $\Phi$ until the following template is obtained: $$(\{1\},C_0,X,\emptyset,\emptyset,A_1[X,C_0],\Delta|C_0,\Lambda).$$ On this template, repeatedly perform operation (6) to obtain the following template: $$\Phi'=(\{1\},C_0,X_0,\emptyset,\emptyset,A_1[X_0,C_0],\Delta|C_0,\Lambda|X_0).$$ Perform operations (2) and (3) on $\Phi'$ to obtain the following template: $$(\{1\},C_0,X_0,\emptyset,\emptyset,A_1[X_0,C_0],\{\mathbf{0}, \delta\},\{\mathbf{0}, \lambda\}).$$ Any matroid conforming to this template is obtained by contracting $C_0$. If $\delta$ is in the row labeled by $r$ and $\lambda$ is in the column labeled by $c$, then when $C_0$ is contracted, 1 is added to the entry of the $\Gamma$-frame matrix in row $r$ and column $c$. Otherwise, the entry remains unchanged when $C$ is contracted. We see then that this template is equivalent to $\Phi_{CX}$, where 1s are used to replace $\delta$ and $\lambda$.
Thus, we may assume that for every element $\delta\in\Delta|C_0$ and $\lambda\in\Lambda|X_0$, there are an even number of natural numbers $i$ such that $\delta_i=\lambda_i=1$. This implies that contraction of $C$ has no effect on the $\Gamma$-frame matrix. So $\Phi$ is equivalent to a template with $\Lambda|X_0$ trivial. Therefore, since $\Lambda|X_1$ is trivial, we see that $\Lambda$ is trivial. Note that operation (7) is a reduction that produces an equivalent template, since $C$ must be contracted to produce a matroid that conforms to a template. By repeatedly performing operation (7), we obtain a template equivalent to $\Phi$ with $C=\emptyset$.
\[PhiY0\] If $\Phi=(\{1\},C,X,Y_0,Y_1,A_1,\Delta,\Lambda)$ is a binary frame template with $\Lambda$ trivial and with $C=\emptyset$, then either $\Phi_{Y_0}\preceq\Phi$ or $\Phi$ is equivalent to a template with $\Delta$ trivial.
First, suppose there is an element $\delta\in\Delta$ that is not in the row space of $A_1=A_1[X_1,(Y_0\cup Y_1)]$. Recall that a $y$-shift is the operation described in Lemma \[yshift\]. Repeatedly perform $y$-shifts to obtain the following template, where $Y'_0=Y_0\cup Y_1$: $$(\{1\},\emptyset,X,Y'_0,\emptyset,A_1,\Delta,\{0\}).$$ On this template, perform operation (3) to obtain the following template: $$(\{1\},\emptyset,X,Y'_0,\emptyset,A_1,\{\mathbf{0}, \delta\},\{0\}).$$
Choose a basis $B'$ for $M(A_1)$. By performing elementary row operations on every matrix virtually respecting $\Phi$, we may assume that $A_1[X,B']$ consists of an identity matrix with zero rows below it and that $\delta|B'$ is the zero vector. By assumption, there is some element $y\in (Y'_0-B')$ such that $\delta_y$ is nonzero. Thus, we can repeatedly perform operation (10) to obtain the following template: $$(\{1\},\emptyset,X,B'\cup y,\emptyset,A_1[X,B'\cup y],\{\mathbf{0}, \delta|(B'\cup y)\},\{0\}).$$ Now, we can repeatedly perform operation (6) and then operation (12) to obtain the following template: $$(\{1\},\emptyset,\emptyset,\{y\},\emptyset,[\emptyset],\mathbb{Z}/2\mathbb{Z},\{0\}),$$ which is $\Phi_{Y_0}$.
Now suppose that every element $\delta\in\Delta$ is in the row space of $A_1=A_1[X,(Y_0\cup Y_1)]$. Since $\Lambda$ is trivial, by performing elementary row operations on every matrix virtually respecting $\Phi$, we obtain a template equivalent to $\Phi$ with $\Delta$ trivial.
\[PhiY1\] Let $\Phi=(\{1\},C,X,Y_0,Y_1,A_1,\Delta,\Lambda)$ be a binary frame template with $\Lambda$ and $\Delta$ trivial. If $M(A_1[X_1,(Y_0\cup Y_1)])$ has a circuit $Y'$ with $|Y'\cap Y_1|\geq3$, then $\Phi_{Y_1}\preceq\Phi$.
Any matroid conforming to $\Phi$ is obtained by contracting $C$. Since $\Lambda$ and $\Delta$ are trivial, we may assume that $C=X_0=\emptyset$ and therefore that $X=X_1$. Repeatedly perform operation (4) and then operation (10) on $\Phi$ to obtain the following template: $$(\{1\},\emptyset,X,Y_0\cap Y',Y_1\cap Y',A_1[X,Y'],\{0\},\{0\}).$$ Choose any 3-element subset of $Y'\cap Y_1$ and call it $Y''$. Repeatedly perform $y$-shifts to obtain the following template: $$(\{1\},\emptyset,X,Y'-Y'',Y'',A_1[X,Y'],\{0\},\{0\}).$$ On this template, repeatedly perform operation (11) to obtain the following template: $$(\{1\},\emptyset,X',\emptyset,Y'',A_1[X',Y''],\{0\},\{0\}),$$ where $X'$ is the subset of $X$ that remains after $Y'-Y''$ is contracted. On this template, repeatedly perform operations (5) and (6) to obtain the following template, where $X''$ is a 2-element subset of $X'$: $$(\{1\},\emptyset,X'',\emptyset,Y'',\begin{bmatrix}
1& 0 &1\\
0& 1 & 1
\end{bmatrix},\{0\},\{0\}).$$ This template is $\Phi_{Y_1}$.
\[simpleY1\] If $\Phi$ is a frame template with $\Delta$ trivial, then $\Phi$ is equivalent to a template $\Phi'$ where $A_1[X,Y_1]$ is a matrix with every column nonzero and where no column is a copy of another. Moreover, if $\Phi$ is a binary frame template, then $M(A_1[X,Y_1])$ is simple.
Let $A$ be a matrix that virtually conforms to $\Phi$. Since $\Delta$ is trivial, the columns of $A$ indexed by elements of $Z$ are formed by placing a column of $A_1[X,Y_1]$ on top of a unit column or a zero column. These columns can be made using any copy of the same column of $A_1[X,Y_1]$; so only one copy is needed. If any column of $A_1[X,Y_1]$ is a zero column, then any column indexed by an element of $Z$ that is made with this zero column can also be made as a column indexed by an element of $E-(Z\cup Y_0\cup Y_1\cup C)$ and choosing for the element of $\Lambda$ the zero vector. Thus, no zero columns of $A_1[X,Y_1]$ are needed.
In the binary case, $M(A_1[X,Y_1])$ has no parallel elements because any such elements index copies of the same column. Also, $M(A_1[X,Y_1])$ has no loops because every column of $A_1[X,Y_1]$ is nonzero. Therefore, $M(A_1[X,Y_1])$ is simple.
\[HL\] Let $\Phi$ be a binary frame template. Then at least one of the following is true:
- $\Phi_0\sim\Phi$
- $\Phi'\preceq \Phi$ for some $\Phi'\in\{\Phi_X,\Phi_C,\Phi_{CX},\Phi_{Y_0},\Phi_{Y_1}\}$
- $\Phi$ is equivalent to a template where $C=\emptyset$, where $\Lambda$ and $\Delta$ are trivial, and where $A_1$ is of the following form, with $Y_0=V_0\cup V_1$, with $L$ an arbitrary binary matrix, and with each column of $H$ containing at most two nonzero entries:
----- ----- -----
$I$ 0 $H$
0 $I$ $L$
----- ----- -----
.
Suppose neither (i) nor (ii) holds. By Lemma \[PhiCD\], we may assume that $\Lambda$ is trivial and $C=\emptyset$. By Lemma \[PhiY0\], we may assume that $\Delta$ is trivial. By Lemma \[PhiY1\], every dependent set of $M(A)=M(A_1[X_1,(Y_0\cup Y_1)])$ has an intersection with $Y_1$ with size at most 2. So by elementary row operations, we may assume that $A_1$ is of the following form, where $Y_0=V_0\cup V_1$, where $L$ is an arbitrary binary matrix, where $K$ consists of unit and zero columns, and where each column of $H$ contains at most two nonzero entries:
----- ----- ----- -----
$I$ $K$ 0 $H$
0 0 $I$ $L$
----- ----- ----- -----
.
However, by Lemma \[simpleY1\], we may assume that $K$ is an empty matrix. Thus, (iii) holds.
\[minimal\] Let $\Phi$ be a binary frame template. Then at least one of the following is true:
- $\Phi_0\sim\Phi$
- $\Phi'\preceq \Phi$ for some $\Phi'\in\{\Phi_X,\Phi_C,\Phi_{CX},\Phi_{Y_0},\Phi_{Y_1}\}$
- There exist $k,l\in\mathbb{Z}_+$ such that no simple, vertically $k$-connected matroid with at least $l$ elements either virtually conforms or virtually coconforms to $\Phi$.
Suppose for contradiction that none of outcomes (i)-(iii) hold for $\Phi$. By Lemma \[HL\], outcome (iii) of that lemma holds. Note that any simple matroid $N$ virtually conforming to $\Phi$ is a restriction of a matroid $M$ represented by a matrix of the following form, where $Z=Z_0\cup Z_1$, where $Y_0=V_0\cup V_1$, and where the $\Gamma$-frame matrix has $n$ rows and has a vector matroid isomorphic to the cycle matroid of the graph $K_{n+1}$:
[ r|c|cccc|c|c|c| ]{} &&&&\
&&$1\cdots1$&&&&&&\
&&&$1\cdots1$&&&&&\
&&&&$\cdots$&&&&\
&&&&&$1\cdots1$&&&\
&&&0&$I$&$L$\
&$\Gamma$-frame matrix&$I$&$I$&$\cdots$&$I$&0&0&0\
Also recall from the definition of conforming to a template that $Y_0\subseteq E(N)$.
We see that $$\begin{aligned}
\lambda_N(Y_0\cup (Z_1\cap E(N)))
&\leq \lambda_M(Y_0\cup Z_1)\\
&= r_M(Y_0\cup Z_1)+r_M(E-(Y_0\cup Z_1))-r(M)\\
&= |V_0|+|Y_1| + |Y_1|+n-(|Y_1|+|V_0|+n)\\
&= |Y_1|.\end{aligned}$$
Note that each column of the above matrix, except possibly those columns indexed by $V_1$, has at most two nonzero entries. Thus, $M$ is graphic and $\Phi$ is trivial if $V_0=\emptyset$. Since (i) does not hold, $\Phi$ is nontrivial. Therefore, $V_0\neq\emptyset$, and $E(N)-(Y_0\cup Z_1)$ is not spanning. Thus, if $k>|Y_1|+1$, then $N$ is not vertically $k$-connected unless $Y_0\cup(Z_1\cap E(N)) $ is spanning in $N$. This implies that $n=0$; in that case, $N$ is only simple if the $\Gamma$-frame matrix is a $0\times0$ matrix. This implies that $|E(N)|\leq |Y_0\cup Y_1|$. So if $l>|Y_0\cup Y_1|$, then no simple, vertically $k$-connected matroid with at least $l$ elements virtually conforms to $\Phi$.
Now, consider a simple matroid $N^*$ which virtually coconforms to $\Phi$. Then $N$ is a restriction of $M$ with $Y_0\subseteq E(N)$. Since a matroid and its dual have the same connectivity function, we have $\lambda_{N^*}(Y_0\cup (Z_1\cap E(N))\leq |Y_1|$. So if $k>|Y_1|+1$, then $N^*$ is not vertically $k$-connected unless either $Y_0\cup (Z_1\cap E(N))$ or $E(N)-(Z_1\cup Y_0)$ is spanning in $N^*$, implying that either $E(N)-(Z_1\cup Y_0)$ or $Y_0\cup (Z_1\cap E(N))$ is independent in $N$. If $E(N)-(Z_1\cup Y_0)$ is independent in $N$, then $$\begin{aligned}
|E(N)-(Z_1\cup Y_0)|&=r_N(E(N)-(Z_1\cup Y_0))\\
&\leq r_M(E(M)-(Z_1\cup Y_0))\\
&=|Y_1|+n.\end{aligned}$$ By the formula for corank, we have $$\begin{aligned}
r_{N^*}(E(N)-(Z_1\cup Y_0))&\leq r_{M^*}(E(N)-(Z_1\cup Y_0))\\
&= |E(N)-(Z_1\cup Y_0)|+r_M(Z_1\cup Y_0)-r(M)\\
&\leq |Y_1|+n+|Y_1|+|V_0|-(|Y_1|+|V_0|+n)\\
&=|Y_1|.\end{aligned}$$ Since $N^*$ is simple and binary, we have $|E(N)-(Z_1\cup Y_0)|\leq 2^{|Y_1|}-1$. This implies that $|E(N)|\leq 2^{|Y_1|}-1+|Y_1|+|Y_0|$. Thus, if we set $l$ greater than this value, then no simple, vertically $k$-connected matroid with at least $l$ elements virtually coconforms to $\Phi$ unless $Y_0\cup (Z_1\cap E(N))$ is independent in $N$. Since (iii) does not hold, this must be true for some matroid $N$. In particular, $Y_0=V_0\cup V_1$ is independent in $N$, implying that $H$ is a linearly independent matrix.
Let $P$ denote the matrix $$P=
\left[
\begin{array}{cc}
1& 0\\
0& 1\\
0&1\\
\hline
1&1\\
\end{array}
\right].$$ Suppose $A_1[X,V_1]$ has $P$ as a submatrix, with the first three rows of $P$ contained in $H$ and the last row of $P$ contained in $L$. Then $A_1$ contains the following submatrix, with the first three columns contained in $A_1[X,Y_1]$ and the last two contained in $A_1[X,V_1]$:
$$\left[
\begin{array}{ccc|cc}
1&0&0&1&0\\
0&1&0&0&1\\
0&0&1&0&1\\
\hline
0&0&0&1&1\\
\end{array}
\right].$$ After contracting all other elements of $Y_1$ by repeatedly performing $y$-shifts and operation (12), the columns of this submatrix form a circuit in $M(A_1)$ whose intersection with $Y_1$ has size 3. However, we have already deduced by Lemma \[PhiY1\] that this is impossible. Therefore, $A_1$ does not contain $P$ as a submatrix, with the first three rows of $P$ contained in $H$ and the last row of $P$ contained in $L$. We will refer to this fact by saying that $A_1$ has no *$P$-configuration*.
Let $\{1,2,\dots, m\}$ be the rows of $L$. (So $|V_0|=m$.) Let $S_i$ be the submatrix of $H$ obtained by restricting $H$ to the columns $j$ such that $L_{i,j}=1$. Recall that $H$, and therefore $S_i$, contain at most two nonzero entries per column. Also, since $H$ is linearly independent, each column has at least one nonzero entry, and no column is a copy of another. Suppose a column $e$ of $S_i$ contains exactly two nonzero entries. Since $A_1$ has no $P$-configuration, all other columns of $S_i$ must contain a nonzero entry in exactly one of the same rows as $e$. Suppose that there are columns $f$ and $g$ in $S_i$ such that $f$ contains a nonzero entry in one of the same rows as $e$, but $g$ contains a nonzero entry in the other row. Then $S_i$ contains the following submatrix: $$\begin{blockarray}{ccc}
e & f & g \\
\begin{block}{[ccc]}
1 & 1 & 0 \\
1 & 0 & 1 \\
\end{block}
\end{blockarray}.$$ Since $H$ is a linearly independent matrix, $f$ or $g$ (say $f$) must have an additional nonzero entry in $H$. To avoid $f$ and $g$ forming a $P$-configuration, $g$ must have an additional nonzero entry in the same row as $f$. Therefore, $S_i$ contains the following submatrix: $$\begin{blockarray}{ccc}
e & f & g \\
\begin{block}{[ccc]}
1 & 1 & 0 \\
1 & 0 & 1 \\
0 & 1 & 1\\
\end{block}
\end{blockarray}.$$ Since each column of $H$ contains at most two nonzero entries, $\{e,f,g\}$ is a dependent set of columns, contradicting the assumption that $H$ is linearly independent.
Therefore, we deduce that each $S_i$ either consists entirely of unit columns or contains a row $s_i$ consisting entirely of 1s. Note that each $S_i$ is the incidence matrix of a star, with possibly one row removed. We will call $s_i$ the *star center* of row $i$. If $S_i$ consists entirely of unit columns, then we define its star center to be $s_i=\emptyset$.
If the sets of columns of all the $S_i$ are pairwise disjoint, then by adding each row $i$ to its star center $s_i$, we see that every matroid virtually conforming to $\Phi$ can be represented by a matrix with at most two nonzero entries per column. Thus, $\Phi$ is trivial, contradicting the assumption that (i) does not hold. Also, if $i$ and $j$ are distinct rows of $L$ with distinct star centers $s_i$ and $s_j$, then $S_i$ and $S_j$ can have at most one column in common because otherwise, the columns they have in common form a linearly dependent set in $H$.
Now suppose there are $S_i$ and $S_j$ with $s_i=s_j$. Also, suppose that neither $S_i$ nor $S_j$ is a submatrix of the other. Then $A_1$ contains the following submatrix. In fact, after repeatedly performing $y$-shifts, operation (11), and operation (10), we may assume that $A_1$ is the following matrix, with the first three columns indexed by $Y_1$, the next two indexed by $V_0$, and the last three by $V_1$: $$\left[
\begin{array}{ccc|cc|ccc}
1&0&0&0&0&1&1&1\\
0&1&0&0&0&0&1&0\\
0&0&1&0&0&0&0&1\\
\hline
0&0&0&1&0&1&1&0\\
0&0&0&0&1&0&1&1\\
\end{array}
\right].$$ Add the fourth row to the first, and swap the fourth and sixth columns to obtain the following matrix: $$\left[
\begin{array}{ccc|cc|ccc}
1&0&0&0&0&1&0&1\\
0&1&0&0&0&0&1&0\\
0&0&1&0&0&0&0&1\\
\hline
0&0&0&1&0&1&1&0\\
0&0&0&0&1&0&1&1\\
\end{array}
\right].$$ The last two columns of this matrix contain a $P$-configuration.
Now suppose there are matrices $S_i$ and $S_j$ so that $S_j$ is a submatrix of $S_i$. Then $A_1$ contains a submatrix obtained by deleting columns from a matrix of the following form, where the left portion comes from the set $V_0$, the upper-right portion comes from the matrix $H$, the lower-left portion comes from the matrix $L$, and $x$ is 1 or 0 depending on whether or not the last column is contained in $S_j$: $$\left[
\begin{array}{cc|ccccccc}
0&0&1&\cdots&1&1&\cdots&1&1\\
0&0&1&&&&&&0\\
\vdots&\vdots&&\ddots&&&&&\vdots\\
0&0&&&1&&&&0\\
0&0&&&&1&&&0\\
\vdots&\vdots&&&&&\ddots&&\vdots\\
0&0&&&&&&1&0\\
\hline
1&0&1&\cdots&1&1&\cdots&1&1\\
0&1&0&\cdots&0&1&\cdots&1&x
\end{array}
\right].$$
Choose any column contained in $S_j$ and perform row operations so that this column becomes a unit column with nonzero entry in $L$. Then we obtain the following matrix: $$\left[
\begin{array}{cc|cccccccc}
0&1&1&\cdots&1&0&0&\cdots&0&x+1\\
0&0&1&&&&&&&0\\
\vdots&\vdots&&\ddots&&&&&&\vdots\\
0&0&&&1&&&&&0\\
0&1&&&&0&1&\cdots&1&x\\
0&0&&&&&1&&&0\\
\vdots&\vdots&&&&&&\ddots&&\vdots\\
0&0&&&&&&&1&0\\
\hline
1&1&1&\cdots&1&0&\cdots&\cdots&0&x+1\\
0&1&0&\cdots&0&1&\cdots&\cdots&1&x
\end{array}
\right].$$
Now, by swapping the appropriate columns, we obtain the following: $$\left[
\begin{array}{cc|cccccccc}
0&0&1&\cdots&1&1&0&\cdots&0&x+1\\
0&0&1&&&&&&&0\\
\vdots&\vdots&&\ddots&&&&&&\vdots\\
0&0&&&1&&&&&0\\
0&0&&&&1&1&\cdots&1&x\\
0&0&&&&&1&&&0\\
\vdots&\vdots&&&&&&\ddots&&\vdots\\
0&0&&&&&&&1&0\\
\hline
1&0&1&\cdots&1&1&0&\cdots&0&x+1\\
0&1&0&\cdots&0&1&\cdots&\cdots&1&x
\end{array}
\right].$$
We see that in this new matrix, $S_i$ and $S_j$ have only one column in common and $s_i\neq s_j$. The last column is in $S_i$ if $x=0$ and $S_j$ if $x=1$. Thus, this case reduces to the final case that remains to be checked: for all $i$ and $j$, we have $s_i\neq s_j$ and $S_i$ and $S_j$ have at most one column in common. Since each column of $H$ contains at most two nonzero entries, and since all $S_i$ have distinct star centers, we see that a column of $H$ can be contained in at most two $S_i$. By adding each row $i$ to its star center $s_i$, one can see that every matrix virtually conforming to $\Phi$ can be rewritten so that every column contains at most two nonzero entries. Therefore, $\Phi$ is trivial, and (i) holds.
This completes the contradiction and proves the result.
Outcome (iii) of Theorem \[minimal\] only occurs in very specific situations. In fact, due to connectivity considerations, it is not needed in order to use Corollary \[weakframe\].
\[describes\] Let $\mathcal{M}$ be a minor-closed class of binary matroids, and suppose there exist $k,l,m\in \mathbb{Z}_+$ and a set $\mathcal{T}_{\mathcal{M}}=\{\Phi_1,\dots, \Phi_s, \Psi_1,\dots, \Psi_t\}$ of binary frame templates such that
- $\mathcal{M}$ contains each of the classes $\mathcal{M}_w(\Phi_1),\dots,\mathcal{M}_w(\Phi_s)$,
- $\mathcal{M}$ contains the duals of the matroids in each of the classes $\mathcal{M}_w(\Psi_1)$,$\dots$,$\mathcal{M}_w(\Psi_t)$,
- if $M$ is a simple vertically $k$-connected member of $\mathcal{M}$ with at least $l$ elements and with no $PG(m-1,2)$ minor, then either $M$ is a member of at least one of the classes $\mathcal{M}_v(\Phi_1),\dots,\mathcal{M}_v(\Phi_s)$ or $M^*$ is a member of at least one of the classes $\mathcal{M}_v(\Psi_1),\dots,\mathcal{M}_v(\Psi_t)$, and
- for each template $\Phi\in\mathcal{T}_{\mathcal{M}}$, either $\Phi$ is trivial or $\Phi'\preceq \Phi$ for some $\Phi'\in\{\Phi_X,\Phi_C,\Phi_{CX},\Phi_{Y_0},\Phi_{Y_1}\}$.
We say that $\mathcal{T}_{\mathcal{M}}$ *describes* $\mathcal{M}$.
By combining Corollary \[weakframe\] with Theorem \[minimal\], one can observe that every proper minor-closed class $\mathcal{M}$ of binary matroids can be described by a set of templates. Moreover, that set is nonempty if and only if $\mathcal{M}$ contains all graphic matroids or all cographic matroids.
\[Y0Y1\] Let $\mathcal{M}$ be a minor-closed class of binary matroids, and let $\{\Phi_1,\dots, \Phi_s, \Psi_1,\dots, \Psi_t\}$ be a set of templates describing $\mathcal{M}$. If any of these templates is nontrivial, then $\mathcal{M}$ contains $\mathcal{M}(\Phi_{Y_0})$, $\mathcal{M}(\Phi_{Y_1})$, $\mathcal{M}^*(\Phi_{Y_0})$, or $\mathcal{M}^*(\Phi_{Y_1})$.
Let $\Phi$ be a nontrivial template in the set $\{\Phi_1,\dots, \Phi_s\}$. By Definition \[describes\] and Lemma \[YCD\], either $\Phi_{Y_0}\preceq\Phi$ or $\Phi_{Y_1}\preceq\Phi$. If $\Phi_{Y_0}\preceq\Phi$, then $$\mathcal{M}(\Phi_{Y_0})\subseteq\mathcal{M}_v(\Phi_{Y_0})\subseteq\mathcal{M}_v(\Phi)\subseteq\mathcal{M},$$ where the first containment holds because every matroid conforming to a template also virtually conforms to it, the second containment holds by definition of $\preceq$, and the third containment holds by Definition \[describes\]. In the case where $\Phi_{Y_1}\preceq\Phi$, a similar argument shows that $\mathcal{M}(\Phi_{Y_1})\subseteq\mathcal{M}$.
If $\Psi$ is a nontrivial template in the set $\{\Psi_1,\dots, \Psi_s\}$, a similar argument shows that either $\mathcal{M}^*(\Phi_{Y_0})\subseteq\mathcal{M}$, or $\mathcal{M}^*(\Phi_{Y_1})\subseteq\mathcal{M}$.
Growth Rates {#Growth Rates}
============
Let $\mathcal{M}$ be a minor-closed class of matroids. Let $h_{\mathcal{M}}(r)$ denote the *growth rate function* of $\mathcal{M}$: the function whose value at an integer $r\geq0$ is given by the maximum number of elements in a simple matroid in $\mathcal{M}$ of rank at most $r$. For a matroid $M$, we denote by $\varepsilon(M)$ the size of the simplification of $M$, that is the number of rank-1 flats of $M$. By combining the main result in [@gkw09] with earlier results of Geelen and Whittle [@gw03] and Geelen and Kabell [@gk09], Geelen, Kung, and Whittle proved the following:
\[growthrate\]
If $\mathcal{M}$ is a nonempty minor-closed class of matroids, then there exists $c\in\mathbb{R}$ such that either:
- $h_{\mathcal{M}}(r)\leq cr$ for all $r$,
- $\binom{r+1}{2}\leq h_{\mathcal{M}}(r)\leq cr^2$ for all $r$ and $\mathcal{M}$ contains all graphic matroids,
- there is a prime-power $q$ such that $\frac{q^r-1}{q-1}\leq h_{\mathcal{M}}(r)\leq cq^r$ for all $r$ and $\mathcal{M}$ contains all $\mathrm{GF}(q)$-representable matroids, or
- $h_{\mathcal{M}}$ is infinite and $\mathcal{M}$ contains all simple rank-2 matroids.
If outcome (2) of the Growth Rate Theorem holds for a minor-closed class $\mathcal{M}$, then $\mathcal{M}$ is said to be *quadratically dense*. In this section, we will consider growth rates of some quadratically dense classes of binary matroids. Let $\mathcal{EX}(F)$ denote the class of binary matroids with no $F$-minor. If $f$ and $g$ are functions, we write $f(r)\approx g(r)$ if $f(r)=g(r)$ for all but finitely many $r$.
Since the growth rate function for the class of graphic matroids is $\binom{r+1}{2}$, the Growth Rate Theorem implies that, if $F$ is a nongraphic binary matroid, $$h_{\mathcal{EX}(F)}(r)\geq\binom{r+1}{2}.$$ Kung et. al. [@kmpr14] pose the following question: For which nongraphic binary matroids $F$ of rank 4 does equality hold above for all but finitely many $r$? Geelen and Nelson answer this question in [@gn15]. Let $N_{12}$ be the matroid formed by deleting a three-element independent set from $PG(3,2)$. The nongraphic binary matroids $F$ of rank 4 for which $h_{\mathcal{EX}(F)}(r)\approx\binom{r+1}{2}$ are exactly the nongraphic restrictions of $N_{12}$. We present here an alternate proof. Both proofs allow us to answer the question when $F$ is a matroid of any rank, not just rank 4. We will prove the following theorem after proving several lemmas.
\[quadgrowth\] Let $\mathcal{M}$ be a minor-closed class of binary matroids. Then $h_{\mathcal{M}}(r)\approx\binom{r+1}{2}$ if and only if $\mathcal{M}$ contains all graphic matroids but does not contain $\mathcal{M}_v(\Phi_{Y_1})$.
Our proof of Theorem \[quadgrowth\] will depend on the following theorem, proved by Geelen and Nelson in [@gn15]:
\[gn51\] Let $\mathcal{M}$ be a quadratically dense minor-closed class of matroids and let $p(x)$ be a real quadratic polynomial with positive leading coefficient. If $h_{\mathcal{M}}(n)>p(n)$ for infinitely many $n\in\mathbb{Z}^+$, then for all integers $r,s\geq1$ there exists a vertically $s$-connected matroid $M\in\mathcal{M}$ satisfying $\varepsilon(M)>p(r(M))$ and $r(M)\geq r$.
An *even-cycle matroid* is a binary matroid of the form $M=M\binom{w}{D}$, where $D\in\mathrm{GF}(2)^{V\times E}$ is the vertex-edge incidence matrix of a graph $G=(V,E)$ and $w\in\mathrm{GF}(2)^E$ is the characteristic vector of a set $W\subseteq E$. The pair $(G,W)$ is an *even-cycle representation* of $M$. The edges in $W$ are called *odd* edges, and the other edges are *even* edges. An *odd cycle* of $(G,W)$ is a cycle of $G$ with an odd number of odd edges. A *blocking pair* of $(G,W)$ is a pair of vertices $u,v$ of $G$ so that every odd cycle passes through at least one of these vertices. *Resigning* at a vertex $u$ of $G$ occurs when all the edges incident with $u$ are changed from even to odd and vice-versa. It is easy to see that this corresponds to adding the row of the matrix corresponding to $u$ to the characteristic vector of $W$. Therefore, resigning at a vertex does not change an even-cycle matroid. It is also easy to see that if an even-cycle representation has a blocking pair, then we can resign so that every odd edge is incident with at least one vertex in the blocking pair. For our purposes, it will be convenient to think of a blocking pair in this way.
For $r\geq2$, let $A_r$ be the following binary matrix, where we choose for the $\Gamma$-frame matrix the matrix representation of $M(K_{r-1})$, so that the identity matrices are $(r-2)\times(r-2)$ matrices.
----------------------- --- ----- ----- ------------ ------------ ------------
1 0 1 $1\cdots1$ $0\cdots0$ $1\cdots1$
0 1 1 $0\cdots0$ $1\cdots1$ $1\cdots1$
$\Gamma$-frame matrix $I$ $I$ $I$
----------------------- --- ----- ----- ------------ ------------ ------------
Note that $M(A_r)$ is the largest simple matroid of rank $r$ that virtually conforms to $\Phi_{Y_1}$.
Let $X_r$ be the largest simple matroid of rank $r$ that virtually conforms to $\Phi_{Y_1}$. Equivalently, $X_1=U_{1,1}$, and for $r\geq2$, we have $X_r=M(A_r)$.
\[Y1minors\] The class $\mathcal{M}_v(\Phi_{Y_1})$ is the class of matroids having an even-cycle representation with a blocking pair. This class is minor-closed.
Any simple matroid $M$ virtually conforming to $\Phi_{Y_1}$ is a restriction of $X_r$ for some $r$.
Label the rows of $A_r$ as $1,\dots,r$. Add to the matrix row $r+1$, which is the sum of rows $2,\dots, r$. This does not change the matroid $X_r$. We see that $X_r$ is an even-cycle matroid $(G,W)$, where row 1 is the characteristic vector of $W$ and rows $2,\dots, r+1$ form the incidence matrix of $G$. Moreover, every edge in $W$ is incident with the vertex corresponding to either row 2 or row $r+1$. Thus, every matroid virtually conforming to $\Phi_{Y_1}$ has an even-cycle representation with a blocking pair. Conversely, every matroid that has an even-cycle representation with a blocking pair $\{u,v\}$ virtually conforms to $\Phi_{Y_1}$, by making $u$ correspond to the second row and making $v$ correspond to row $r+1$, which can be removed without changing the matroid.
By resigning whenever we wish to contract an element represented by an odd edge, it is not difficult to see that the class of matroids having an even-cycle representation with a blocking pair is minor-closed.
\[restriction\] Any simple, rank-$r$ matroid $M$ that is a minor of a matroid virtually conforming to $\Phi_{Y_1}$ is a restriction of $X_r$.
From the preceding lemma, $M$ is a restriction of some $X_{r'}$. So $M$ has an even-cycle representation $(G,W)$ with a blocking pair $\{u,v\}$. Let $w$ be the characteristic vector of $W$. There are $r'-r$ rows in the matrix $A_{r'}[(V\cup w)-{v}, E(M)]$ whose deletion does not alter the matroid $M$. After these rows are deleted, the resulting matrix is a submatrix of $A_r$.
\[Y0minorY1\] Every matroid virtually conforming to $\Phi_{Y_1}$ is a minor of a matroid conforming to $\Phi_{Y_0}$.
By Lemma \[YCD\], we have $\Phi_{Y_1}\preceq\Phi_C$. Every matroid conforming to $\Phi_C$ is obtained by contracting an element from a matroid conforming to $\Phi_{Y_0}$.
\[graphicvscographic\] Let $k$ be a positive integer. Then there are at most finitely many integers $r$ such that the complete graphic matroid $M(K_{r+1})$ is a rank-($\leq k$) perturbation of a cographic matroid.
Let $N$ be a cographic matroid. Observe that adding a rank-1 matrix to a matrix representation of a binary matroid $N$ changes $\varepsilon(N)$ by a factor of at most 2. This occurs when, in every rank-1 flat of $N$, there is at least one nonloop element indexing a column that is changed by adding the rank-1 matrix and at least one nonloop element indexing a column that remains unchanged when the rank-1 matrix is added. Thus, if $M$ is a rank-$(\leq t)$ perturbation of $N$, we have $\varepsilon(M)\leq2^t\varepsilon(N)$.
Let $r=r(M)$. Recall that a cographic matroid $N$ has $\varepsilon(N)\leq3r(N)-3$. Therefore, $\varepsilon(M)\leq2^t(3r(N)-3)\leq2^t(3(r+t)-3)$. For fixed $t$ and sufficiently large $r$, this expression is less than $\binom{r+1}{2}=\varepsilon(M(K_{r+1}))$.
\[conformonly\] Let $\mathcal{M}$ be a quadratically dense minor-closed class of matroids representable over a given field $\mathbb{F}$. Let $\{\Phi_1,\dots,\Phi_s,\Psi_1,\dots,\Psi_t\}$ be a set of templates describing $\mathcal{M}$. For sufficiently large $r$, the growth rate $h_{\mathcal{M}}(r)$ is equal to the size of the largest simple matroid of rank $r$ that virtually conforms to any template in $\{\Phi_1,\dots,\Phi_s\}$.
Let $h'_{\mathcal{M}}(r)$ denote the size of the largest simple matroid of rank $r$ that virtually conforms to any template in $\{\Phi_1,\dots,\Phi_s\}$. So $h_{\mathcal{M}}(r)\geq h'_{\mathcal{M}}(r)$. The size of the largest simple matroid of rank $r$ that virtually conforms to any particular template is a quadratic polynomial in $r$. Thus, for sufficiently large $r$, the function $h'_{\mathcal{M}}(r)$ is a quadratic polynomial as well.
By Definition \[describes\], there exist $k,l\in \mathbb{Z}_+$ so that every simple vertically $k$-connected member of $\mathcal{M}$ with at least $l$ elements either weakly conforms to a template in $\{\Phi_1,\dots,\Phi_s\}$ or weakly coconforms to some template in $\{\Psi_1,\dots,\Psi_t\}$. Suppose, for contradiction, that $h_{\mathcal{M}}(r)>h'_{\mathcal{M}}(r)$ for infinitely many $r$. Theorem \[gn51\], with $h'_{\mathcal{M}}(r)$ playing the role of $p(r)$, implies that there is a sequence $M_1, M_2,\dots$ of vertically $k$-connected matroids in $\mathcal{M}$ such that $\varepsilon(M_i)>h'_{\mathcal{M}}(i)$ and $r(M_i)\geq i$. Thus, in this sequence, there are infinitely many matroids that are vertically $k$-connected and have size at least $l$. Since these matroids are too large to virtually conform to any template in $\{\Phi_1,\dots,\Phi_t\}$, there is at least one nontrivial template $\Psi\in\{\Psi_1,\dots,\Psi_t\}$ such that infinitely many vertically $k$-connected matroids in $\mathcal{M}$ coconform to $\Psi$. However, since $\mathcal{M}$ contains all graphic matroids and since every complete graphic matroid has infinite vertical connectivity (hence vertical $k$-connectivity), we have that infinitely many complete graphic matroids coconform to $\Psi$. For some $t$ depending on $\Psi$, every matroid coconforming to $\Psi$ is a rank-$(\leq t)$ perturbation of a cographic matroid. This contradicts Lemma \[graphicvscographic\]. By contradiction, the result holds.
\[proof\]
First, suppose $h_{\mathcal{M}}(r)\approx\binom{r+1}{2}$. By the Growth Rate Theorem, $\mathcal{M}$ contains all graphic matroids. For $r\geq1$, we have $|X_r|=\binom{r-1}{2}+3r-3$, which for $r>2$ is greater than $\binom{r+1}{2}$. Thus, $\mathcal{M}$ does not contain $\mathcal{M}_v(\Phi_{Y_1})$.
Now, suppose $\mathcal{M}$ contains all graphic matroids but does not contain $\mathcal{M}_v(\Phi_{Y_1})$. Since $\mathcal{M}$ contains all graphic matroids, there is a nonempty set $\{\Phi_1,\dots,\Phi_s,\Psi_1,\dots,\Psi_t\}$ of binary frame templates describing $\mathcal{M}$. By Lemma \[conformonly\], $h_{\mathcal{M}}(r)$ is equal to the size of the largest simple matroid of rank $r$ that conforms to any template in $\{\Phi_1,\dots,\Phi_s\}$. Suppose $\Phi$ is a nontrivial template in $\{\Phi_1,\dots,\Phi_s\}$. By Corollary \[Y0Y1\], either $\Phi_{Y_0}\preceq\Phi$ or $\Phi_{Y_1}\preceq\Phi$. Since $\mathcal{M}$ does not contain $\mathcal{M}_v(\Phi_{Y_1})$, we must have $\Phi_{Y_0}\preceq\Phi$. However, by Lemma \[Y0minorY1\], this implies $\mathcal{M}_v(\Phi_{Y_1})\subseteq\mathcal{M}$. Therefore, we conclude that $h_{\mathcal{M}}(r)\approx\binom{r+1}{2}$, completing the proof.
\[EXF\] Let $F$ be a simple, binary matroid of rank $r$. Then $h_{\mathcal{EX}(F)}\approx\binom{r+1}{2}$ if and only if $F$ is a nongraphic restriction of $X_r$.
By Theorem \[quadgrowth\], $h_{\mathcal{EX}(F)}\approx\binom{r+1}{2}$ if and only if $\mathcal{EX}(F)$ contains all graphic matroids but does not contain $\mathcal{M}_v(\Phi_{Y_1})$. The condition that $\mathcal{EX}(F)$ contains all graphic matroids is equivalent to the condition that $F$ is nongraphic. By Lemma \[restriction\], the condition that $\mathcal{EX}(F)$ does not contain $\mathcal{M}_v(\Phi_{Y_1})$ is equivalent to the condition that $F$ is a restriction of $X_r$.
Note that $X_4=N_{12}$; so this answers the question posed in [@kmpr14].
We now consider the growth rate of $\mathcal{EX}(PG(3,2))$. We will prove Theorem \[EXPG32\], which we restate below.
The growth rate function for $\mathcal{EX}(PG(3,2))$ is $$h_{\mathcal{EX}(PG(3,2))}\approx r^2-r+1.$$
We will use the following.
\[PG32Phi\] Let $\mathcal{T}_{\mathcal{EX}(PG(3,2))}=\{\Phi_1,\dots\Phi_s,\Psi_1,\dots,\Psi_t\}$. If $\Phi\in\{\Phi_1,\dots\Phi_s\}$, then either $\Phi=\Phi_X$ or $\Phi$ is a template with $C=\emptyset$ and with $\Lambda$ and $\Delta$ trivial.
The class of matroids conforming to $\Phi_X$ is exactly the class of even-cycle matroids. This class is minor-closed. The largest simple, even-cycle matroid of rank $r$ has an even-cycle representation obtained from the graph $K_r$ by adding to each even edge an odd edge in parallel as well as adding one odd loop to the graph. Therefore, the class of even-cycle matroids has growth rate $2\binom{r}{2}+1=r^2-r+1$. So the largest simple, even-cycle matroid of rank 4 has size 13. Since $PG(3,2)$ has size $15$, we have $\mathcal{M}(\Phi_X)\subseteq\mathcal{EX}(PG(3,2))$. Therefore, we may assume that $\Phi_X\in\mathcal{T}_{\mathcal{EX}(PG(3,2))}$.
Since $\Phi_0\preceq\Phi_X$, we may assume that $\Phi_0\notin\{\Phi_1,\dots\Phi_s\}$. Let $$\Phi=(\{1\},C,X,Y_0,Y_1,A_1,\Delta,\Lambda)$$ be a nontrivial template such that $\Phi\neq\Phi_X$ and $\Phi\in\{\Phi_1,\dots\Phi_s\}$. Consider the graft matroid $M(K_6,V(K_6))$. A straightforward computation shows that, by contracting the nongraphic element, we obtain $PG(3,2)$. Therefore, $\Phi_{Y_0}\npreceq\Phi$. By Lemma \[YCD\], we also have $\Phi_C\npreceq\Phi$ and $\Phi_{CX}\npreceq\Phi$.
Now, we may assume that $\Phi$ is in standard form. Since $\Phi_C\npreceq\Phi$, by Lemma \[PhiC\] we may assume that $C_1=\emptyset$. Also, by Lemma \[PhiCD\], since $\Phi_{CX}\npreceq\Phi$ and $\Phi_C\npreceq\Phi$, either $\Lambda|X_1$ is nontrivial and $\Phi_X\preceq\Phi$ or $\Lambda$ is trivial and $C=\emptyset$.
First, suppose that $\Lambda$ is trivial and $C=\emptyset$. Since $\Phi_{Y_0}\npreceq\Phi$, Lemma \[PhiY0\] implies that $\Phi$ is equivalent to a template with $\Delta$ trivial. So we may assume $$\Phi=(\{1\},\emptyset,X,Y_0,Y_1,A_1,\{0\},\{0\}),$$ which is one of the possible conclusions of the lemma.
Thus, we may assume that $\Lambda|X_1$ is nontrivial and $\Phi_X\preceq\Phi$. Suppose $|\Lambda|X_1|>2$. On the template $$\Phi=(\{1\},C_0,Y_0,Y_1,A_1,\Delta,\Lambda),$$ perform operation (3) and then repeatedly perform operations (4) and (10) to obtain the template $$(\{1\},C_0,X,\emptyset,\emptyset,A_1[X,C_0],\{0\},\Lambda).$$ Then repeatedly perform operation (7) to obtain $$(\{1\},\emptyset,X_1,\emptyset,\emptyset,[\emptyset],\{0\},\Lambda|X_1).$$
Since $\Lambda|X_1$ has characteristic 2 and size greater than 2, it contains a subgroup $\Lambda'$ isomorphic to $(\mathbb{Z}/2\mathbb{Z})\times(\mathbb{Z}/2\mathbb{Z})$. Perform operation (2) to obtain the template $$(\{1\},\emptyset,X_1,\emptyset,\emptyset,[\emptyset],\{0\},\Lambda');$$ then repeatedly perform operations (5) and (6) to obtain $$(\{1\},\emptyset,X',\emptyset,\emptyset,[\emptyset],\{0\},\Lambda''),$$ where $|X'|=2$ and $\Lambda''$ is the additive group generated by $\begin{bmatrix}
1 \\
0
\end{bmatrix}$ and $\begin{bmatrix}
0 \\
1
\end{bmatrix}$. One readily sees that $PG(3,2)$ conforms to this template. Therefore, $|\Lambda|=2$. We may perform row operations so that $\Lambda$ is generated by $[1,0\ldots,0]^T$. Let $\Sigma$ be the element of $X$ such that $\Lambda|\{\Sigma\}$ is nonzero.
Now, suppose there is an element $\bar{x}\in\Delta$ that is not in the row space of $A_1$. Perform operations (2) and (3) on $\Phi$ to obtain $$(\{1\},C_0,X,Y_0,Y_1,A_1,\{0,\bar{x}\},\{0\}).$$ Now, by a similar argument to the one used in the proof of Lemma \[PhiY0\], we have $\Phi_{Y_0}\preceq\Phi$. Since we already know this is not the case, we deduce that every element of $\Delta$ is in the row space of $A_1$.
Let $\bar{x}\in\Delta|C_0$ and $\bar{y}\in\Lambda$ be such that there are an odd number of natural numbers $i$ such that $\bar{x}_i=\bar{y}_i=1$. Then we call the ordered pair $(\bar{x},\bar{y})$ a *pair of odd type*. Otherwise, $(\bar{x},\bar{y})$ is a *pair of even type*. Suppose $(\bar{x},\bar{y})$ is a pair of odd type with $\bar{y}|X_1$ a zero vector. By performing operations (2) and (3) and repeatedly performing operations (4) and (10), we obtain $$(\{1\},C_0,X,\emptyset,\emptyset,A_1[X,C],\{0,\bar{x}\},\{0,\bar{y}\}),$$ which is equivalent to $\Phi_{CX}$. We already know this is not the case. Therefore, for every pair $(\bar{x},\bar{y})$ of odd type, $\bar{y}|X_1=[1,0,\dots,0]^T$.
Suppose $\bar{x}\in\Delta|C$ and $\bar{y}_1,\bar{y}_2\in\Lambda$ are such that $\bar{y}_1|X_1=\bar{y}_2|X_1=[1,0,\dots,0]^T$, such that $(\bar{x},\bar{y}_1)$ is a pair of odd type, and such that $(\bar{x},\bar{y}_2)$ is a pair of even type. Then $(\bar{y}_1+\bar{y}_2)|X_1$ is a zero vector, and $(\bar{x},\bar{y}_1+\bar{y}_2)$ is a pair of odd type. Therefore, either all pairs $(\bar{x},\bar{y})\in\Delta|C\times\Lambda$ are of even type, in which case $\Phi$ is equivalent to a template with $\Lambda|X_0$ trivial and $C=\emptyset$, or if $(\bar{x},\bar{y})$ is a pair of odd type, then $(\bar{x},\bar{z})$ is of odd type for every $\bar{z}\in\Lambda$ with $\bar{z}|X_1$ nonzero. In this case, consider any matrix virtually conforming to $\Phi$. After contracting $C$, we can restore the $\Gamma$-frame matrix by adding $\Sigma$ to each row where the $\Gamma$-frame matrix has been altered. Therefore, $\Phi$ is equivalent to a template with $\Lambda|X_0$ trivial and $C=\emptyset$.
So we now have that $$\Phi=(\{1\},\emptyset,X,Y_0,Y_1,A_1,\Delta,\Lambda),$$ with $\Lambda$ generated by $[1,0\ldots,0]^T$ and with every element of $\Delta$ in the row space of $A_1$. We will now show that, in fact, $\Phi$ is equivalent to a template with $\Delta$ trivial. On $\Phi$, perform $y$-shifts to obtain the following template, where $Y'_0=Y_0\cup Y_1$: $$\Phi'=(\{1\},\emptyset,X,Y'_0,\emptyset,A_1,\Delta,\Lambda).$$ By repeatedly performing operation (5) and then operation (6) on this template, we may assume that $A_1$ has the following form, with the star representing an arbitrary binary matrix and $\bar{v}$ representing an arbitrary row vector: $$\left[
\begin{array}{c|c}
0\cdots0&\bar{v}\\
\hline
I_{|X|-1}&*
\end{array}
\right].$$ Also, since $\Lambda|(D-\{\Sigma\})$ is trivial, we may perform row operations on every matrix conforming to $\Phi'$ to obtain a template $$\Phi''=(\{1\},\emptyset,X,Y'_0,\emptyset,A_1,\Delta'',\Lambda),$$ so that every element of $\Delta''$ has 0 for its first $|X|-1$ entries. Since every element of $\Delta$ was in the row space of $A_1$, the only possible nonzero element of $\Delta''$ is the row vector with 0 for its first $|X|-1$ entries and whose last $|Y'_0|-|X|+1$ entries form the row vector $\bar{v}$. Note that operations (5) and (6) and the row operations we performed on every matrix conforming to $\Phi'$ each changes a template to an equivalent template. Thus, we may assume that $\bar{v}$ is nonzero and that $\Delta''=\{\bf{0},$$\bar{v}\}$ because otherwise, $\Phi$ is equivalent to a template with $\Delta$ trivial. So, for some $y\in Y'_0$, we have $\bar{v}_y=1$. On the template $\Phi''$, repeatedly perform operation (11) and then operation (10) to obtain the following template: $$\Phi'''=(\{1\},\emptyset,\{\Sigma\},\{y\},\emptyset,[1],\mathbb{Z}/2\mathbb{Z},\mathbb{Z}/2\mathbb{Z}).$$
The following matrix conforms to $\Phi'''$: $$\left[
\begin{array}{ccccccccccccccc|c}
0&0&0&0&0&0&0&0&0&0&1&1&1&1&1&1\\
\hline
1&0&0&0&1&1&1&0&0&0&0&0&0&1&0&1\\
0&1&0&0&1&0&0&1&1&0&0&0&1&0&0&1\\
0&0&1&0&0&1&0&1&0&1&0&1&0&0&0&1\\
0&0&0&1&0&0&1&0&1&1&1&0&0&0&0&1
\end{array}
\right].$$ By contracting $y$, we obtain $PG(3,2)$. Thus, we have shown that $\Phi$ must be equivalent to a template with $\Delta$ trivial. So we may assume $$\Phi=(\{1\},\emptyset,X,Y_0,Y_1,A_1,\{0\},\Lambda),$$ with $\Lambda$ generated by $[1,0,\ldots,0]^T$.
Now, let us consider the structure of the matrix $A_1$. By repeated use of operation (5), we may assume that $A_1$ is of the following form, with the top row indexed by $\Sigma$, with $*$ representing an arbitrary row vector, with $Y_0=V_0\cup V_1$, and with each $L_i$ representing an arbitrary binary matrix:
------------ ------------ ------------ ------------ -------
$0\cdots0$ $0\cdots0$ $1\cdots1$ $0\cdots0$ $*$
$I$ $L_0$ $L_1$ 0 $L_2$
0 0 0 $I$ $L_3$
------------ ------------ ------------ ------------ -------
Suppose either $L_0$ or $L_1$ has a column with two or more nonzero entries. Let $y$ be the element of $Y_1$ that indexes that column, and let $Y'$ be the union of $\{y\}$ with the subset of $Y_1$ that indexes the columns of the identity submatrix of $A_1[X,Y_1]$. Repeatedly perform operations (4) and (10) on $\Phi$ to obtain $$(\{1\},\emptyset,X,\emptyset,Y',A_1,\{0\},\Lambda).$$ On this template, repeatedly perform $y$-shifts, operation (11), and operation (6) to obtain $$(\{1\},\emptyset,X',\emptyset,Y'',\begin{bmatrix}
0&0&x\\
1& 0 &1\\
0& 1 & 1
\end{bmatrix},\{0\},\Lambda),$$ where $x=i$ if $y$ indexes a column of $L_i$ and where $X'$ and $Y''$ index the set of rows and columns, respectively, of the matrix $\begin{bmatrix}
0&0&x\\
1& 0 &1\\
0& 1 & 1
\end{bmatrix}$.
The following matrix conforms to this template. By contracting the columns printed in bold, we obtain $PG(3,2)$. $$\left[
\begin{array}{cccccccccccccc|cccc}
0&0&0&0&0&0&0&0&0&0&1&1&1&\bf{1}&\bf{0}&\bf{0}&x&x\\
0&0&0&0&0&0&0&0&0&0&0&0&0&\bf{0}&\bf{1}&\bf{0}&1&1\\
0&0&0&0&0&0&0&0&0&0&0&0&0&\bf{0}&\bf{0}&\bf{1}&1&1\\
\hline
1&0&0&0&1&1&1&0&0&0&0&0&0&\bf{1}&\bf{0}&\bf{0}&1&0\\
0&1&0&0&1&0&0&1&1&0&0&0&0&\bf{1}&\bf{0}&\bf{0}&0&1\\
0&0&1&0&0&1&0&1&0&1&1&0&1&\bf{0}&\bf{1}&\bf{0}&0&0\\
0&0&0&1&0&0&1&0&1&1&0&1&1&\bf{0}&\bf{0}&\bf{1}&0&0\\
\end{array}
\right].$$ This shows that $L_0$ and $L_1$ consist entirely of unit and zero columns. Thus, by Lemma \[simpleY1\], $L_0$ is an empty matrix and $L_1$ consists entirely of distinct unit columns. Therefore, $A_1$ is of the following form:
------------ ------------ ------------ ------------ -------
$0\cdots0$ $0\cdots0$ $1\cdots1$ $0\cdots0$ $*$
$I$ 0 $I$ 0 $Q_1$
0 $I$ 0 0 $Q_2$
0 0 0 $I$ $Q_3$
------------ ------------ ------------ ------------ -------
with each $Q_i$ representing an arbitrary binary matrix.
Let $M$ be any matroid conforming to $\Phi$ with rank and connectivity functions $r$ and $\lambda$, respectively. Let $r'$ be the rank of the submatrix of $A_1$ consisting of $Q_1$, $Q_2$, and the row vector we have denoted with a star. Then $r(Y_0)=|V_0|+r'$ and $r(E(M)-Y_0)=r(M)-|V_0|$. Thus, $\lambda(Y_0)=r'$. So if $k>r'+1$, then $M$ is not vertically $k$-connected unless $Y_0$ or $E(M)-Y_0$ is spanning. If $Y_0$ is spanning in $M$, then the $\Gamma$-frame matrix used to construct $M$ has 0 rows. Thus, $M$ is not simple unless $|E(M)|\leq |Y_0|+|Y_1|+1$, with the 1 coming from the element $[1,0\cdots,0]^T$ of $\Lambda$. Thus, if we set $l>|Y_0|+|Y_1|+1$, then no simple, vertically $k$-connected matroid with at least $l$ elements conforms to $\Phi$ unless $E(M)-V_0$ is spanning in $M$. Therefore, we have $V_0=\emptyset$.
Let $Q$ be the submatrix of $A_1$ consisting of $Q_1$ and $Q_2$. If every column of $Q$ has at most two nonzero entries, then $\Phi\preceq\Phi_X$, and as we deduced above, we may assume $\Phi=\Phi_D$. Therefore, we assume that $Q$ has a column $c$, indexed by the element $y\in Y_0$ with three or more nonzero entries.
Repeatedly perform operation (10) on $\Phi$ to obtain the template $$\Phi'=(\{1\},\emptyset,X,\{y\},Y_1,A_1[D,Y_1\cup\{y\}],\{0\},\Lambda).$$ Let $c=\left[\begin{array}{c}
c_1\\
\hline
c_2
\end{array}\right]$, with $c_1$ a column of $Q_1$ and $c_2$ a column of $Q_2$. Consider the following cases:
1. The vector $c_1$ has three nonzero entries.
2. The vector $c_1$ has two nonzero entries, and $c_2$ has one nonzero entry.
3. The vector $c_1$ has one nonzero entry, and $c_2$ has two nonzero entries.
4. The vector $c_2$ has three nonzero entries.
In Case $i$, repeatedly perform $y$-shifts and operation (11) to obtain the template $$\Phi''_i=(\{1\},\emptyset,X',\{y\},Y'_1,A_{1,i},\{0\},\Lambda),$$ where $A_{1,i}$ is the matrix defined below with rows indexed by $X'$ and columns indexed by $Y'_1\cup\{y\}$. In each case, the last column is indexed by $y$, and it turns out that the value of $x$ does not matter. $$A_{1,1}=\left[
\begin{array}{ccc|ccc|c}
0&0&0&1&1&1&x\\
\hline
1&0&0&1&0&0&1\\
0&1&0&0&1&0&1\\
0&0&1&0&0&1&1\\
\end{array}
\right]
A_{1,2}=\left[
\begin{array}{ccc|cc|c}
0&0&0&1&1&x\\
\hline
1&0&0&1&0&1\\
0&1&0&0&1&1\\
0&0&1&0&0&1\\
\end{array}
\right]$$
$$A_{1,3}=\left[
\begin{array}{ccc|c|c}
0&0&0&1&x\\
\hline
1&0&0&1&1\\
0&1&0&0&1\\
0&0&1&0&1\\
\end{array}
\right]
A_{1,4}=\left[
\begin{array}{ccc|c}
0&0&0&x\\
\hline
1&0&0&1\\
0&1&0&1\\
0&0&1&1\\
\end{array}
\right]$$
In Case $i$, the matrix below virtually conforms to $\Phi''_i$. By contracting the columns printed in bold, we obtain $PG(3,2)$.
1. $$\left[
\begin{array}{ccc|cccccccccccc|c}
1&1&0&0&0&0&1&1&1&0&0&0&1&1&1&\textbf{\textit{x}}\\
0&0&0&1&0&0&1&0&0&1&0&0&1&0&0&\bf{1}\\
0&0&0&0&1&0&0&1&0&0&1&0&0&1&0&\bf{1}\\
0&0&0&0&0&1&0&0&1&0&0&1&0&0&1&\bf{1}\\
\hline
1&0&1&1&1&1&1&1&1&0&0&0&0&0&0&\bf{0}\\
\end{array}
\right]$$
2. $$\left[
\begin{array}{ccccccc|ccccccccc|c}
1&0&0&0&1&1&1&0&0&\bf{0}&1&1&0&0&1&1&\textbf{\textit{x}}\\
0&0&0&0&0&0&0&1&0&\bf{0}&1&0&1&0&1&0&\bf{1}\\
0&0&0&0&0&0&0&0&1&\bf{0}&0&1&0&1&0&1&\bf{1}\\
0&0&0&0&0&0&0&0&0&\bf{1}&0&0&0&0&0&0&\bf{1}\\
\hline
0&1&0&1&1&0&1&1&1&\bf{1}&1&1&0&0&0&0&\bf{0}\\
0&0&1&1&0&1&1&0&0&\bf{0}&0&0&1&1&1&1&\bf{0}\\
\end{array}
\right]$$
3. $$\left[
\begin{array}{ccccc|cccccccccccc|c}
0&0&0&\bf{1}&\bf{1}&0&0&0&0&0&0&1&0&0&0&0&0&\textbf{\textit{x}}\\
0&0&0&\bf{0}&\bf{0}&1&0&0&1&0&0&1&0&0&1&0&0&\bf{1}\\
0&0&0&\bf{0}&\bf{0}&0&1&0&0&1&0&0&1&0&0&1&0&\bf{1}\\
0&0&0&\bf{0}&\bf{0}&0&0&1&0&0&1&0&0&1&0&0&1&\bf{1}\\
\hline
1&0&1&\bf{1}&\bf{0}&1&1&1&0&0&0&0&0&0&0&0&0&\bf{0}\\
0&1&1&\bf{0}&\bf{1}&0&0&0&1&1&1&1&0&0&0&0&0&\bf{0}\\
0&0&0&\bf{0}&\bf{1}&0&0&0&0&0&0&0&1&1&0&0&0&\bf{0}\\
\end{array}
\right]$$
4. $$\left[
\begin{array}{ccccc|cccccccccccc|c}
1&0&\bf{1}&1&\bf{1}&0&0&0&0&0&0&0&0&0&0&0&0&\textbf{\textit{x}}\\
0&0&\bf{0}&0&\bf{0}&1&0&0&1&0&0&1&0&0&1&0&0&\bf{1}\\
0&0&\bf{0}&0&\bf{0}&0&1&0&0&1&0&0&1&0&0&1&0&\bf{1}\\
0&0&\bf{0}&0&\bf{0}&0&0&1&0&0&1&0&0&1&0&0&1&\bf{1}\\
\hline
0&0&\bf{1}&0&\bf{0}&1&1&1&0&0&0&0&0&0&0&0&0&\bf{0}\\
0&0&\bf{0}&0&\bf{1}&0&0&0&1&1&1&0&0&0&0&0&0&\bf{0}\\
0&1&\bf{0}&1&\bf{1}&0&0&0&0&0&0&1&1&1&0&0&0&\bf{0}\\
\end{array}
\right]$$
By contradiction, this completes the proof.
\[proof\]
Let $\mathcal{M}=\mathcal{EX}(PG(3,2))$, and let $\mathcal{T}_{\mathcal{M}}=\linebreak\{\Phi_1,\dots\Phi_s,\Psi_1,\dots,\Psi_t\}$. By Lemma \[conformonly\], for sufficiently large $r$, we have $h_{\mathcal{M}}(r)$ equal to the size of the largest simple matroid of rank $r$ that virtually conforms to any template in $\Phi\in\{\Phi_1,\dots\Phi_s\}$. If $\Phi\in\{\Phi_1,\dots\Phi_s\}$, then by Lemma \[PG32Phi\] either $\Phi=\Phi_X$ or $\Phi$ is of the form $(\{1\},\emptyset,X,Y_0,Y_1,A_1,\{0\},\{0\})$, for some matrix $A_1$ and some sets $X$, $Y_0$, and $Y_1$. Moreover, by operation (5), we may assume that $A_1$ is of the following form, with $Y_0=V_0\cup V_1$ and with the stars representing arbitrary binary matrices:
----- ----- ----- -----
$I$ $*$ 0 $*$
0 0 $I$ $*$
----- ----- ----- -----
.
The largest simple matroid of rank $r$ that virtually conforms to $\Phi$ is obtained by taking for the $\Gamma$-frame matrix a matrix representation of $M(K_{n+1})$, where $n=r-r(M(A_1[X,Y_1]))-|V_0|$. Thus, the largest simple matroid of rank $r$ that virtually conforms to $\Phi$ has size $\binom{n+1}{2}+|Y_1|n+|Y_1|+|Y_0|$. Substituting $r-r(M(A_1[X,Y_1]))-|V_0|$ for $n$, one sees that for sufficiently large $r$, this expression is less than $r^2-r+1$. Since the class of matroids virtually conforming to $\Phi_X$ is the class of even-cycle matroids, which has growth rate $r^2-r+1$, the result holds.
1-flowing Matroids {#1-flowing Matroids}
==================
In this section, we prove Theorem \[1flowing\]. The 1-flowing property is a generalization of the max-flow min-cut property of graphs. See Seymour [@s81] or Mayhew [@m15] for more of the background and motivation concerning 1-flowing matroids. We follow the notation and exposition of [@m15].
Let $e$ be an element of a matroid $M$. Let $c_x$ be a non-negative integral capacity assigned to each element $x\in E(M)-e$. A flow is a function $f$ that assigns to each circuit $C$ containing $e$ a non-negative real number $f_C$ with the constraint that for each $x\in E-e$, the sum of $f_C$ over all circuits containing both $e$ and $x$ is at most $c_x$. We say that $M$ is *$e$-flowing* if, for every assignment of capacities, there is a flow whose sum over all circuits containing $e$ is equal to $$\min\{\sum_{x\in C^*-e}c_x | C^* \textnormal{ is a cocircuit containing }e \}.$$ If $M$ is $e$-flowing for each $e\in E(M)$, then $M$ is *1-flowing*.
The matroid $T_{11}$ is the even-cycle matroid obtained from $K_5$ by adding a loop and making every edge odd, including the loop. In [@s81], Seymour showed the following.
The The class of 1-flowing matroids is minor-closed. Moreover, $AG(3,2)$, $U_{2,4}$, $T_{11}$, and $T^*_{11}$ are excluded minors for the class of 1-flowing matroids.
Seymour [@s81] conjectured that these are the only excluded minors.
The set of excluded minors for the class of 1-flowing matroids consists of $AG(3,2)$, $U_{2,4}$, $T_{11}$, and $T^*_{11}$.
Since $U_{2,4}$ is an excluded minor for the class of 1-flowing matroids, all such matroids are binary. Therefore, the results in this paper apply to 1-flowing matroids. We will now prove Theorem \[1flowing\], which we restate below.
There exist $k,l\in\mathbb{Z}_+$ such that every simple, vertically $k$-connected, 1-flowing matroid with at least $l$ elements is either graphic or cographic.
The matroid $AG(3,2)$ conforms to $\Phi_{Y_1}$ since it is a restriction of $N_{12}$. Indeed, consider the matrix representing $N_{12}$ that virtually conforms to $\Phi_{Y_1}$. Add the rows labeled by $X$ in this matrix to one of the other rows. Then we can see the matrix representation $[I_4|J_4-I_4]$ of $AG(3,2)$ as a restriction of $N_{12}$. Also, it is not difficult to see that $AG(3,2)$ can be obtained from a matroid conforming to $\Phi_{Y_0}$ by contracting $Y_0$. Thus, $\mathcal{EX}(AG(3,2))$ contains neither $\mathcal{M}(\Phi_{Y_0})$ nor $\mathcal{M}(\Phi_{Y_1})$. Since $AG(3,2)$ is self-dual, $\mathcal{EX}(AG(3,2))$ does not contain $\mathcal{M}^*(\Phi_{Y_0})$, or $\mathcal{M}^*(\Phi_{Y_1})$ either. Therefore, by Corollary \[Y0Y1\], $\mathcal{EX}(AG(3,2))$ is described by the trivial template. Thus, since $AG(3,2)$ is an excluded minor for the class of 1-flowing matroids, there exist $k,l\in\mathbb{Z}_+$ such that every simple, vertically $k$-connected, 1-flowing matroid with at least $l$ elements either conforms or coconforms to the trivial template. The result follows.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank the two anonymous referees for carefully reading the manuscript. In particular, we thank the first referee for giving many suggestions that improved the manuscript, including one that greatly simplified the proof of Lemma \[graphicvscographic\].
[99]{}
Jim Geelen, Bert Gerards, and Geoff Whittle, The highly connected matroids in minor-closed classes, [*Annals of Combinatorics*]{} [**19**]{} (2015), 107–123.
Jim Geelen, and Kasper Kabell, Projective geometries in dense matroids, [*Journal of Combinatorial Theory, Series B*]{} [**99**]{} (2009), 1–8.
Jim Geelen, and Geoff Whittle, Cliques in dense $\mathrm{GF}(q)$-representable matroids, [*Journal of Combinatorial Theory, Series B*]{} [**87**]{} (2003), 264–269.
Jim Geelen, Joseph P.S. Kung, and Geoff Whittle, Growth rates of minor-closed classes of matroids, [*Journal of Combinatorial Theory, Series B*]{} [**99**]{} (2009), 420–427.
Jim Geelen and Peter Nelson, Matroids denser than a clique, [*Journal of Combinatorial Theory, Series B*]{} [**114**]{} (2015), 51–69.
Joseph P. S. Kung, Dillon Mayhew, Irene Pivotto, and Gordon F. Royle, Maximum size binary matroids with no $AG(3,2)$-minor Are graphic, [*SIAM Journal on Discrete Mathematics*]{} [**28**]{} (2014), 1559–1577.
Dillon Mayhew, Seymour’s 1-flowing Conjecture III, The Matroid Union, WordPress, 19 Mar. 2015, Web, 19 May 2016.
James Oxley, [*Matroid Theory*]{}, Second Edition, Oxford University Press, New York, 2011.
Neil Robertson and P.D. Seymour, Graph Minors. XVI. Excluding a non-planar graph, [*Journal of Combinatorial Theory, Series B*]{} [**89**]{} (2003), 43–75.
P.D. Seymour, Matroids and Multicommodity Flows. European Journal of Combinatorics [**2**]{} (1981), 257-290.
[^1]: The first author was supported by a Huel D. Perkins Fellowship from the Louisiana State University Graduate School. The second author was supported by the National Science Foundation, grant 1500343.
[^2]: The authors of [@ggw15] divided our set $X$ into two separate sets which they called $X$ and $D$. Their set $X$ can be absorbed into $Y_0$, therefore we omit it.
| ArXiv |
---
abstract: 'We present a way to include non local potentials in the standard Diffusion Monte Carlo method without using the locality approximation. We define a stochastic projection based on a fixed node effective Hamiltonian, whose lowest energy is an upper bound of the true ground state energy, even in the presence of non local operators in the Hamiltonian. The variational property of the resulting algorithm provides a stable diffusion process, even in the case of divergent non local potentials, like the hard-core pseudopotentials. It turns out that the modification required to improve the standard Diffusion Monte Carlo algorithm is simple.'
author:
- Michele Casula
title: Beyond the locality approximation in the standard diffusion Monte Carlo method
---
Diffusion Monte Carlo (DMC) is one of the most successful methods to compute the ground state properties of quantum systems. Although the fixed node (FN) approximation is needed to cure the infamous sign problem for fermions, the accuracy of the DMC framework has yielded many benchmark results[@foulkesreview]. However, when the DMC method is applied to “ab initio” realistic Hamiltonians, its computational cost scales $\propto Z^{6.5}$, where $Z$ is the atomic number[@ceperley86]. Therefore, the use of pseudopotentials is necessary to make those calculations feasible.
Since the pseudopotentials are usually non local, the “locality approximation” is made besides the FN, by replacing the true Hamiltonian $H$ with an *effective* one $H^{\mathrm{eff}}$, which reads[@mitas]: $$H^{\mathrm{eff}} = K + V_{\mathrm{loc}} + \frac{\int dx^\prime \langle
x^\prime | V_{\mathrm{non~loc}} | x \rangle \Psi_T(x^\prime)}{ \Psi_T(x)},
\label{H_locality}$$ where $K$ is the kinetic operator, $V_{\mathrm{loc}}$ is the local potential, and the last term in Eq. \[H\_locality\] is the non local potential localized by means of the trial wave function $\Psi_T$. The projection is then realized by iteratively applying the operator $G=\exp(-\tau (H^{\mathrm{eff}} - E_{\mathrm{eff}}))$ to $\Psi_T$ in order to filter out its high energy components. The localized potential enters in the branching part (birth and death process) of the algorithm, while the usual FN constraint is employed to limit the diffusion process within the nodal pockets of $\Psi_T$, and avoid the fermionic sign problem. Thus $E_{\mathrm{eff}}$ is the FN ground state energy of $H^{\mathrm{eff}}$, computed during the sampling of the mixed distribution $\Psi_{\mathrm{eff}} \Psi_T$: $$E_{\mathrm{eff}}=\frac{\langle \Psi_{\mathrm{eff}} | H^{\mathrm{eff}} | \Psi_T
\rangle}{\langle \Psi_{\mathrm{eff}} | \Psi_T \rangle} =
\frac{\langle \Psi_{\mathrm{eff}} | H | \Psi_T
\rangle}{\langle \Psi_{\mathrm{eff}} | \Psi_T \rangle} = E_{MA}.
\label{E_MA_identity}$$ $E_{MA}$ is the *mixed average* of $H$, and the above identity holds because $H^{\mathrm{eff}} \Psi_T/\Psi_T = H \Psi_T/\Psi_T$. Since $\Psi_{\mathrm{eff}}$ is the FN ground state of $H^{\mathrm{eff}}$, which differs from $H$, $E_{MA}$ is no longer equal to the variational FN energy of $H$, defined as: $$E_{FN}=\langle \Psi_{\mathrm{eff}} | H | \Psi_{\mathrm{eff}} \rangle /
\langle \Psi_{\mathrm{eff}} | \Psi_{\mathrm{eff}} \rangle.
\label{E_FN}$$ Therefore, in contrast with the case of local Hamiltonians, $E_{MA}$ calculated with the locality approximation does not in general give an upper bound to the ground state energy of $H$ (variational principle).
In a previous work[@lrdmc], we introduced the Lattice Regularized Diffusion Monte Carlo algorithm (LRDMC), which provides an upper bound for the true ground state energy and allows estimate $E_{FN}$, even in the case of non local potentials. In this paper we propose an extension of the standard DMC framework that gives the same results as the LRDMC method, after a proper modification of the DMC propagator.
We start by considering the importance sampling Green function $$G(x^\prime \leftarrow x, \tau) = \frac{\Psi_T(x^\prime)}{\Psi_T(x)} \langle
x^\prime | e^{-\tau (H - E_T)} | x \rangle,
\label{green_dmc}$$ where $E_T$ is an energy offset, $\tau$ the time step, and $x$ a vector of particle coordinates. In the diffusion Monte Carlo method, $G(x^\prime
\leftarrow x, \tau)$ is iteratively applied to $\Psi_T^2$, in order to sample stochastically the mixed distribution $\Phi(x,t)=\Psi_T(x)
\Psi(x,t)$, $\Psi(x,t)$ converging to the lowest possible state in energy. To rewrite $G(x^\prime \leftarrow x,\tau)$ (Eq. \[green\_dmc\]) in a practical way, it is necessary to resort to the Trotter break up, which is exact in the limit of $\tau
\rightarrow 0$. Here we split the Hamiltonian into local and non local operators, and we end up with the following expression for the Green function: $$G(x^\prime \leftarrow x,\tau) \simeq \int dx'' ~ T_{x^\prime,x''}(\tau) ~
G_{DMC}(x'' \leftarrow x, \tau),
\label{new_green_function}$$ where $G_{DMC}(x^\prime \leftarrow x, \tau)$ is the usual DMC propagator[@foulkesreview], $$\frac{1}{(2 \pi \tau)^{\frac{3N}{2}}}
\exp\left[-\frac{(x^\prime - x - \tau v(x))^2}{2 \tau}\right]
e^{-\tau(E^{\mathrm{loc}}_L(x^\prime)- E_T)},
\label{DMC_green_function}$$ and $T_{x^\prime,x}(\tau)$ is the matrix containing the non local potential, $$\frac{\Psi_T(x^\prime)}{\Psi_T(x)} \langle x^\prime | e^{-\tau
V_{\mathrm{non~loc}}}| x \rangle
\simeq \delta_{x^\prime,x} - \tau V_{x^\prime,x}.$$ In the above Eqs. $N$ is the total number of particles, $v(x)=\nabla \ln
|\Psi_T(x)| $ the drift velocity, $E^{\mathrm{loc}}_L(x)=(K+V_{\mathrm{loc}})\Psi_T(x)/\Psi_T(x)$ the contribution to the local energy coming from the local operators, and $V_{x^\prime,x}=\frac{\Psi_T(x^\prime)}{\Psi_T(x)} \langle
x^\prime | V_{\mathrm{non~loc}} | x \rangle$. The final form of $G_{DMC}$ has been obtained by further splitting the Hamiltonian into the kinetic and potential part, while the exponential of the non local potential in $T$ has been linearized up to order $\tau$.
If the case of pseudopotentials, the number of non-zero matrix elements $V_{x^\prime,x}$ will be *finite*, once a quadrature rule with a discrete mesh of points is applied to evaluate the projection over the angular components of the pseudopotential[@fahy; @mitas]. Therefore, the process in $G(x^\prime \leftarrow x,\tau)$ driven by $T_{x^\prime,x}(\tau)$ can be calculated using a heat bath algorithm, since $T_{x^\prime,x}(\tau)
/ \sum_{x''} T_{x'',x}(\tau)$ can be seen as a transition probability, and it can be computed *a priori* for all possible new coordinates $x^\prime$. We notice that the matrix elements $T_{x^\prime,x}(\tau)$ are easily evaluated in a standard DMC algorithm, since $V_{x^\prime,x}$ are already computed to calculate the localized pseudopotential in Eq. \[H\_locality\]: $$\frac{\int dx^\prime \langle x^\prime | V_{\mathrm{non~
loc}} | x \rangle \Psi_T(x^\prime)}{ \Psi_T(x)} = \sum_{x^\prime} V_{x^\prime,x}.
\label{potential_locality}$$ At variance with the locality approximation, $ V_{x^\prime,x}$ contribute now to move the particles, according to the transition matrix $T$ ($T$-moves).
An important limitation of this idea is given by the sign problem. Indeed both $\frac{\Psi_T(x^\prime)}{\Psi_T(x)}$ and $\langle
x^\prime | V_{\mathrm{non~loc}} | x \rangle$ can change sign, which should be included in the weights, but this yields averages with exponentially increasing noise. A solution is to apply the FN approximation not only to $G_{DMC}$ but also to $T$, which becomes: $$T^{FN}_{x^\prime,x}(\tau) = \delta_{x,x^\prime} - \tau V^-_{x^\prime,x},
\label{T_FN}$$ where we defined $V^\pm_{x^\prime,x}=1/2 (V_{x^\prime,x} \pm | V_{x^\prime,x} |)$. In practice, we keep only those matrix elements which give a positive $T_{x^\prime,x}(\tau)$. Moreover, we add to the *diagonal* potential the so called “sign flip term”, i.e. the sum over the discarded matrix elements $V^+_{x^\prime,x}$. Therefore, the local potential becomes $$V^{\textrm{eff}}(x)=V_{\mathrm{loc}}(x) + \sum_{x^\prime} V^+_{x^\prime,x}.$$ This is equivalent to work with a new effective FN Hamiltonian $$\begin{aligned}
\label{H_effective}
H^{\textrm{eff}}_{x,x} & = & K + V^{\textrm{eff}}(x) \\
H^{\textrm{eff}}_{x^\prime,x} & = & \langle x^\prime | V_{\mathrm{non~loc}} | x \rangle
\textrm{~~~if $V_{x^\prime,x}<0$}.\nonumber \end{aligned}$$
In contrast to the effective Hamiltonian of the locality approximation written in Eq. \[H\_locality\], the ground state energy $E_{\mathrm{eff}}(=E_{MA})$ of the above $H^{\textrm{eff}}$ is an upper bound for the ground state energy of the true $H$. As shown in Ref. for the Lattice Green function Monte Carlo, this variational property is due to the sign flip term (*positive* contribution) added to the local potential, *and* the $T$-moves driven by the off diagonal matrix elements $V^-_{x^\prime,x}$. Instead, in the locality approximation also $V^-_{x^\prime,x}$ is summed in the diagonal part (Eq. \[potential\_locality\]), and this leads to an *attractive* potential, which cannot provide a variational property for $E_{MA}$. Moreover, we found that the negative divergences of the fully localized potential on the nodes of $\Psi_T$ are responsible in some case (e.g. see Fig. \[noise\]) for numerical instabilities in the locality approximation, which disappear once $H^{\textrm{eff}}$ in Eq. \[H\_effective\] is used together with the $T^{FN}$-moves. Indeed, whenever $V^-_{x^\prime,x}$ is large, it pushes the walker away from the attractive regions of the localized potential, and protects the sampling from divergences in the weights.
Once a $T^{FN}$-move is generated according to the transition probability $T^{FN}_{x^\prime,x}(\tau) / \sum_{x''}
T^{FN}_{x'',x}(\tau)$, the walker should acquire the weight $w_T(x,\tau)=\sum_{x''}
T^{FN}_{x'',x}(\tau)$ due to the normalization of the $T^{FN}$ matrix. This weight can be recast as an exponential form valid up to order $\tau$, $$w_T=1-\tau \sum_{x^\prime} V^-_{x^\prime,x} \simeq \exp\left[-\tau \sum_{x^\prime}
V^-_{x^\prime,x}\right].
\label{T_weight}$$ Thus the overall weight $w(x,\tau)$ of $G(x^\prime \leftarrow x,\tau)$ will be $$w(x,\tau) = w_{DMC}~ w_T = \exp \left[-\tau (E_L(x) - E_T)
\right],
\label{final_weight}$$ where $w_{DMC}$ is the weight of $G_{DMC}$ for the effective Hamiltonian ($V_{\mathrm{loc}}$ replaced by $V^{\textrm{eff}}$), and $E_L(x)=H^{\mathrm{eff}} \Psi_T/\Psi_T = H \Psi_T/\Psi_T$ is the local energy. Notice that a non-symmetric branching factor has been included in $G_{DMC}$ (Eq. \[DMC\_green\_function\]). When we use the exponential form in Eq. \[T\_weight\], and consequently the weight in Eq. \[final\_weight\], the time step error is usually smaller than that obtained with the linear form. This can be understood in the limit of perfect importance sampling. Indeed, if $\Psi_T$ is close to the ground state of $H$, the weight in Eq. \[final\_weight\] is almost constant, since the variance of $E_L(x)$ is small, and the time step bias is reduced.
The proposed DMC scheme for fixed node Hamiltonians with non local potentials is the following: (i) perform a diffusion-drift move according to $G_\textrm{diff}(x^\prime \leftarrow
x, \tau)= \exp\left[-(x^\prime - x - \tau v(x))^2 /2 \tau\right]/(2 \pi
\tau)^{\frac{3N}{2}} $ *as is done* in the standard DMC algorithm, and accept or reject this move according to the probability $$min\left[1,\frac{G_\textrm{diff}(x \leftarrow x^\prime, \tau)
\Psi_T^2(x^\prime)}{G_\textrm{diff}(x^\prime \leftarrow x, \tau)
\Psi_T^2(x)}\right];
\label{acc_rej}$$ (ii) weight the walker with the factor $\exp \left[-\tau (E_L(x^\prime) - E_T) \right]$; (iii) displace the walker a second time, with a $T$-move selected according to the transition probability $p(x'' \leftarrow x^\prime, \tau)=
T^{FN}_{x'',x^\prime}(\tau) / \sum_y T^{FN}_{y,x^\prime}(\tau)$, computed *a priori* for all possible new $x''$. The branching process will be the same as in the usual DMC algorithm. In practice, only the $T$-move is the new step, which is performed *after* weighting the walker[@time_step_error]. Although we perform an acceptance/rejection step (Eq. \[acc\_rej\]), which has been shown to reduce the time step error[@cyrus_dmc] in $G_\textrm{DMC}$, the algorithm does not satisfy exactly the detailed balance except in the limit of $\tau \rightarrow 0$, due to the break up of $G$ into $G_\textrm{DMC}$ and $T^{FN}$ (Eq. \[new\_green\_function\]), and the use of a non symmetric branching factor in Eq. \[DMC\_green\_function\].
In order to estimate the variational FN energy $E_{FN}$ (Eq. \[E\_FN\]), and study the quality of the locality approximation, we introduce a more general effective Hamiltonian[@lrdmc] $H^{\alpha,\gamma}$, $$\begin{aligned}
\label{H_last}
H^{\alpha,\gamma}_{x,x} & = & K + V_{\mathrm{loc}}(x) + (1+\gamma) \sum_{x^\prime}
V^+_{x^\prime,x} \nonumber \\
& & + \alpha (1+\gamma) \sum_{x^\prime} V^-_{x^\prime,x} \\
H^{\alpha,\gamma}_{x^\prime,x} & = & -\gamma ~ \langle x^\prime |
V_{\mathrm{non~loc}} | x \rangle \textrm{~~~~~~~~~~~~~~~~~if $V_{x^\prime,x}>0$} \nonumber \\
H^{\alpha,\gamma}_{x^\prime,x} & = & (1-\alpha(1+\gamma))~ \langle x^\prime |
V_{\mathrm{non~loc}} | x \rangle \textrm{~~if $V_{x^\prime,x}<0$}, \nonumber \end{aligned}$$ where $0 \le \alpha \le 1$ and $0 \le \gamma \le 1/\alpha -1$ are two external parameters. In order to sample the Green function $G(x^\prime \leftarrow x, \tau)$ for $H^{\alpha,\gamma}$, it is sufficient to modify the matrix $T_{x^\prime,x}(\tau)$, which becomes $$T^{\alpha,\gamma}_{x^\prime,x} = \left \{
\begin{array}{ll}
1 & \textrm{if $x = x^\prime$} \label{T_matrix_new} \\
\tau ~ \gamma ~ V^+_{x^\prime,x} & \textrm{if $V_{x^\prime,x}>0$} \\
- \tau ~ (1-\alpha(1+\gamma)) ~ V^-_{x^\prime,x} & \textrm{if $V_{x^\prime,x}<0$}.
\end{array}
\right.$$ The ground state $E(\alpha,\gamma)$ of $H^{\alpha,\gamma}$ is equal to $E_{MA}(\alpha,\gamma)$ (Eq. \[E\_MA\_identity\]), since $H^{\alpha,\gamma} \Psi_T/\Psi_T = H\Psi_T/\Psi_T$ by construction. The Hamiltonian in Eq. \[H\_effective\] is recovered with $\alpha=0$ and $\gamma=0$, while the Hamiltonian of the locality approximation (Eq. \[H\_locality\]) is obtained with $\alpha=1$ and $\gamma=0$. Therefore, $H^{\alpha,\gamma}$ can interpolate between these two extremes, but the variational principle for $E_{MA}(\alpha,\gamma)$ is not guaranteed as soon as $\alpha \ne 0$, since the attractive term $\alpha (1+\gamma) \sum_{x^\prime} V^-_{x^\prime,x}$ is added to the diagonal potential. However by means of $H^{\alpha,\gamma}$ one can estimate the value of $E_{FN}(\alpha,\gamma)$ (Eq. \[E\_FN\]), which is *variational* for every $\alpha$ and $\gamma$, since it is the expectation value of the true $H$ on the ground state of $H^{\alpha,\gamma}$. Indeed $H=H^{\alpha,\gamma}-(1+\gamma) \partial_\gamma H^{\alpha,\gamma}$, and the Hellmann-Feynman theorem leads to the relation $$E_{FN}(\alpha,\gamma)=E(\alpha,\gamma)-(1+\gamma)~ \partial_\gamma E(\alpha,\gamma).
\label{evaluate_E_FN}$$ One can show [@sandro_effective] that, for a given value of $\alpha$, the lowest $E_{FN}(\alpha,\gamma)$ is obtained for $\gamma=0$. Therefore, in order to find the best variational estimate of the ground state of $H$, it is enough to calculate the expression in Eq. \[evaluate\_E\_FN\] with $\gamma=0$. In this way one can check which $\alpha$ provides the best variational state for $H$. The derivative $\partial_\gamma E(\alpha,0)$ can be computed with either finite differences or correlated sampling. In both cases, one should keep in mind that $\gamma < 1/\alpha -1$, to guarantee the positivity of the $T^{\alpha,\gamma}$ matrix (Eq. \[T\_matrix\_new\]), and so calculating $E_{FN}(\alpha,0)$ becomes harder as $\alpha$ gets closer to $1$.
Here we present the application of the method to the $Si$ and $C$ pseudoatoms. We computed $E_{MA}(\alpha,0)$ and $E_{FN}(\alpha,0)$ for $\alpha=0,0.5,0.9$, and the DMC energy with the locality approximation, which corresponds to $E_{MA}(1,0)$. With an aim to quantify the locality error, and the correction provided by the effective Hamiltonian $H^{\alpha,\gamma}$, we used three different trial wave functions (with no Jastrow, a two-body, and a three-body (electron-electron-ion) Jastrow factor respectively), sharing the same determinantal part, and hence the same nodes. In this way, the FN error can be separated from the effect of the locality approximation, which causes a dependence of the DMC energy on the *shape* of $\Psi_T$.
For the $Si$ atom we used an $s-p$ norm-conserving Hartree-Fock (HF) pseudopotential, which is soft and has been generated using the Vanderbilt construction[@vanderbilt]. The determinantal part of $\Psi_T$ is a HF wave function with a $6s6p/1s1p$ Gaussian basis set. The 2-body Jastrow is from Ref. , while the 3-body Jastrow factor is from Ref. . Both of the Jastrow factors have been optimized using an energy minimization procedure[@sorella_hessian]. The results are reported in Fig. \[Si\_data\_alpha\]. The variational $E_{FN}(\alpha,0)$ improves going from $\alpha=0$ to $\alpha=0.9$, i.e. approaching the locality approximation. It means that at least for this soft pseudopotential the locality approximation ($\alpha=1,\gamma=0$) gives a ground state which is a good variational wave function for $H$. Notice however that the standard DMC energies $E_{MA}(1,0)$ have a sizable locality error, while $E_{FN}$ with $\alpha=0.9$ depends only slightly on the shape of the trial wave function. A similar result was obtained with the LRDMC method for the same pseudoatom[@lrdmc].
For the $C$ atom we chose to work with an SBK pseudopotential[@SBK], which is extremely hard, since it diverges like $1/r^2$ in the $s$ channel, and $1/r$ in its local component. The Slater part of $\Psi_T$ is an antisymmetrized geminal power (AGP) wave function[@casula_atoms; @casula_wf] with a $2s2p$ Gaussian basis set, optimized in the presence of the 3-body Jastrow factor by minimizing its variational energy[@sorella_hessian]. The determinantal part has been kept fixed in the other two $\Psi_T$’s, which differ only by their Jastrow factors. The results are plotted in Fig. \[C\_data\_alpha\]. Here the locality approximation is very poor, as it leads to non variational $E_{MA}$. The spikes in Fig. \[noise\], coming from regions of the configuration space where the *effective* potential is attractive, are surely responsible of the non variational results. Surprisingly, $\Psi_T$ without Jastrow, which has a higher energy, leads to much more stable DMC simulations. The locality approximation, which relies on the quality of the shape of $\Psi_T$ in the core, performs poorly with this hard-core pseudopotential, since it is difficult to find the optimal shape of $\Psi_T$ in the core region, due to the divergence of the non local pseudopotential. Indeed $E_{FN}(\alpha,0)$ is higher for $\alpha=0.9$, being the worst for the 3-body Jastrow factor. On the other hand, the best variational $E_{FN}(\alpha,0)$ is obtained for $\alpha=0$, irrespective of the form of the Jastrow factor.
To summarize, we have described a scheme to treat non local potentials within the standard DMC method. We have extended the DMC formalism to handle a generic Hamiltonian with discrete off-diagonal matrix elements and the fixed node approximation. Only a simple modification of the standard algorithm is required to include the $T$-moves generated according to the non local potentials. By using an effective Hamiltonian approach, we showed that it is possible to have stable simulations, even in the case of divergent hard-core pseudopotentials, and obtain variational results. A similar effective Hamiltonian has been successfully used in the LRDMC method. The difference is in the kinetic part, which is discretized in the lattice regularized approach. The LRDMC and the DMC methods have the same efficiency for small $Z$, although it is possible to have a gain in the LRDMC efficiency by an *ad hoc* choice of the kinetic parameters, particularly for heavier elements. We conclude, by noting that the same Green function presented here can be used in the Reptation Quantum Monte Carlo method[@baroni99].
This work was supported by the NSF grant DMR-0404853. We thank S. Sorella, C. Umrigar, D. M. Ceperley, S. Chiesa, and J. Kim for useful discussions.
[99]{} W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Rajagopal, Rev. Mod. Phys. [**73**]{}, 33 (2001). D. M. Ceperley, J. Stat. Phys. [**43**]{}, 815 (1986); A. Ma, N. D. Drummond, M. D. Towler, and R. J. Needs, Phys. Rev. E [**71**]{}, 066704 (2005). L. Mitáš, E. L. Shirley and D. M. Ceperley, J. Chem. Phys. [**95**]{}, 3467 (1991). M. Casula, C. Filippi, and S. Sorella, Phys. Rev. Lett. [ **95**]{}, 100201 (2005). S. Fahy, X. W. Wang and Steven G. Louie, Phys. Rev. B [**42**]{}, 3503 (1990). D. F. B. ten Haaf, H. J. M. van Bemmel, J. M. J. van Leeuwen, W. van Saarloos, and D. M. Ceperley, , 13039 (1995). Since the normalization of $T^\textrm{FN}_{x',x}$ (Eq. \[T\_weight\]) depends on $x$, the best choice is to weight the walker before the $T$-move. P. J. Reynolds, *et al.*, J. Chem. Phys. [**77**]{}, 5593 (1982); C. J. Umrigar, *et al.*, J. Chem. Phys. [**99**]{}, 2865 (1993). S. Sorella, cond-mat/0201388. D. Vanderbilt, Phys. Rev. B [**32**]{}, 8412 (1985). C. Filippi and C. J. Umrigar, J. Chem. Phys. [**105**]{}, 213 (1996). M. Casula, C. Attaccalite, and S. Sorella, J. Chem. Phys. [**121**]{}, 7710 (2004). S. Sorella, Phys. Rev. B [**71**]{}, 241103 (2005). W. J. Stevens, H. Basch, and M. Krauss, J. Chem. Phys. [**81**]{}, 6026 (1984). M. Casula, and S. Sorella, J. Chem. Phys. [**119**]{}, 6500 (2003). S. Baroni, and S. Moroni, Phys. Rev. Lett. [**82**]{}, 4745 (1999).
| ArXiv |
---
abstract: 'Dalitz-plot analyses of $B\rightarrow K\pi\pi$ decays provide direct access to decay amplitudes, and thereby weak and strong phases can be disentangled by resolving the interference patterns in phase space between intermediate resonant states. A phenomenological isospin analysis of $B\rightarrow K^*(\rightarrow K\pi)\pi$ decay amplitudes is presented exploiting available amplitude analyses performed at the , Belle and LHCb experiments. A first application consists in constraining the CKM parameters thanks to an external hadronic input. A method, proposed some time ago by two different groups and relying on a bound on the electroweak penguin contribution, is shown to lack the desired robustness and accuracy, and we propose a more alluring alternative using a bound on the annihilation contribution. A second application consists in extracting information on hadronic amplitudes assuming the values of the CKM parameters from a global fit to quark flavour data. The current data yields several solutions, which do not fully support the hierarchy of hadronic amplitudes usually expected from theoretical arguments (colour suppression, suppression of electroweak penguins), as illustrated from computations within QCD factorisation. Some prospects concerning the impact of future measurements at LHCb and Belle II are also presented. Results are obtained with the [[CKMfitter]{}]{} analysis package, featuring the frequentist statistical approach and using the Rfit scheme to handle theoretical uncertainties.'
author:
- |
J. Charles$^{\,a}$, S. Descotes-Genon$^{\,b}$, J. Ocariz$^{\,c,d}$, A. Pérez Pérez$^{\,e}$\
for the [[CKMfitter]{}]{} Group
title: 'Disentangling weak and strong interactions in $B\to K^*(\to K\pi)\pi$ Dalitz-plot analyses'
---
Introduction {#sec:Introduction}
============
Non-leptonic $B$ decays have been extensively studied at the $B$-factories and Belle [@Bevan:2014iga], as well at the LHCb experiment [@Bediaga:2012py]. Within the Standard Model (SM) some of these modes provide valuable information on the Cabibbo-Kobayashi-Maskawa (CKM) matrix and the structure of $CP$ violation [@Cabibbo:1963yz; @Kobayashi:1973fv], entangled with hadronic amplitudes describing processes either at the tree level or the loop level (the so-called penguin contributions). Depending on the transition considered, one may or may not get rid of hadronic contributions which are notoriously difficult to assess. For instance, in $b\rightarrow c\bar{c}s$ processes, the CKM phase in the dominant tree amplitude is the same as that of the Cabibbo-suppressed penguin one, so the only relevant weak phase is the $B_d$-mixing phase $2\beta$ (up to a very high accuracy) and it can be extracted from a $CP$ asymmetry out of which QCD contributions drop to a very high accuracy. For charmless $B$ decays, the two leading amplitudes often carry different CKM and strong phases, and thus the extraction of CKM couplings can be more challenging. In some cases, for instance the determination of $\alpha$ from $B\to\pi\pi$ [@Olivier], one can use flavour symmetries such as isospin in order to extract all hadronic contributions from experimental measurements, while constraining CKM parameters. This has provided many useful constraints for the global analysis of the CKM matrix within the Standard Model and the accurate determination of its parameters [@Charles:2004jd; @Charles:2015gya; @CKMfitterwebsite; @Koppenburg:2017mad], as well as inputs for some models of New Physics [@Deschamps:2009rh; @Lenz:2010gu; @Lenz:2012az; @Charles:2013aka].
The constraints obtained from some of the non-leptonic two-body $B$ decays can be contrasted with the unclear situation of the theoretical computations for these processes. Several methods (QCD factorisation [@Beneke:1999br; @Beneke:2000ry; @Beneke:2003zv; @Beneke:2006hg], perturbative QCD approach [@Li:2001ay; @Li:2002mi; @Li:2003yj; @Ali:2007ff; @Li:2014rwa; @Wang:2016rlo], Soft-Collinear Effective Theory [@Bauer:2002aj; @Beneke:2003pa; @Bauer:2004tj; @Bauer:2005kd; @Becher:2014oda]) were devised more than a decade ago to compute hadronic contributions for non-leptonic decays. However, some of their aspects remain debated at the conceptual level [@DescotesGenon:2001hm; @Ciuchini:2001gv; @Beneke:2004bn; @Manohar:2006nz; @Li:2009wba; @Feng:2009rp; @Beneke:2009az; @Becher:2011dz; @Beneke:2015wfa], and they struggle to reproduce some data on $B$ decays into two mesons, especially $\pi^0\pi^0$, $\rho^0\rho^0$, $K\pi$, $\phi K^*$, $\rho K^*$ [@Beneke:2015wfa]. Considering the progress performed meanwhile in the determination of the CKM matrix, it is clear that by now, most of these non-leptonic modes provide more a test of our understanding of hadronic process rather than competitive constraints on the values of the CKM parameters, even though it can be interesting to consider them from one point of view or the other.
Our analysis is focused on the study of $B\rightarrow K^*(\rightarrow K\pi)\pi$ decay amplitudes, with the help of isospin symmetry. Among the various $b\rightarrow u\bar{u}s$ processes, the choice of $B\rightarrow K^*\pi$ system is motivated by the fact that an amplitude (Dalitz-plot) analysis of the three-body final state $K\pi\pi$ provides access to several interference phases among different intermediate $K^*\pi$ states. The information provided by these physical observables highlights the potential of the $B\rightarrow K^*\pi$ system $(VP)$ compared with $B\rightarrow K\pi$ $(PP)$ where only branching ratios and $CP$ asymmetries are accessible. Similarly, the $B\rightarrow K^*\pi$ system leads to the final $K\pi\pi$ state with a richer pattern of interferences and thus a larger set of observables than other pseudoscalar-vector states, like, say, $B\to K\rho$ (indeed, $K\pi\pi$ exhibits $K^*$ resonances from either of the two combinations of $K\pi$ pairs, whereas the $\rho$ meson comes from the only $\pi\pi$ pair available). In addition, the study of these modes provides experimental information on the dynamics of pseudoscalar-vector modes, which is less known and more challenging from the theoretical point of view. Finally, this system has been studied extensively at the [@Aubert:2008bj; @Aubert:2009me; @BABAR:2011ae; @Lees:2015uun] and Belle [@Garmash:2006bj; @Dalseno:2008wwa] experiments, and a large set of observables is readily available.
Let us mention that other approaches, going beyond isospin symmetry, have been proposed to study this system. For instance, one can use $SU(3)$ symmetry and $SU(3)$-related channels in addition to the ones that we consider in this paper [@Bhattacharya:2013boa; @Bhattacharya:2015uua]. Another proposal is the construction of the fully SU(3)-symmetric amplitude [@Bhattacharya:2014eca] to which the spin-one intermediate resonances that we consider here do not contribute.
The rest of this article is organised in the following way. In Sec. \[sec:Dalitz\], we discuss the observables provided by the analysis of the $K\pi\pi$ Dalitz plot analysis. In Sec. \[sec:Isospin\], we recall how isospin symmetry is used to reduce the set of hadronic amplitudes and their connection with diagram topologies. In Sec. \[sec:CKM\], we discuss two methods to exploit these decays in order to extract information on the CKM matrix, making some assumptions about the size of specific contributions (either electroweak penguins or annihilation). In Sec. \[sec:Hadronic\], we take the opposite point of view. Taking into account our current knowledge of the CKM matrix from global analysis, we set constraints on the hadronic amplitudes used to describe these decays, and we make a brief comparison with theoretical estimates based on QCD factorisation. In Sec. \[sec:prospect\], we perform a brief prospective study, determining how the improved measurements expected from LHCb and Belle II may modify the determination of the hadronic amplitudes before concluding. In the Appendices, we discuss various technical aspects concerning the inputs and the fits presented in the paper.
Dalitz-plot amplitudes {#sec:Dalitz}
======================
Charmless hadronic $B$ decays are a particularly rich source of experimental information [@Bevan:2014iga; @Bediaga:2012py]. For $B$ decays into three light mesons (pions and kaons), the kinematics of the three-body final state can be completely determined experimentally, thus allowing for a complete characterisation of the Dalitz-plot (DP) phase space. In addition to quasi-two-body event-counting observables, the interference phases between pairs of resonances can also be accessed, and $CP$-odd (weak) phases can be disentangled from $CP$-even (strong) ones. Let us however stress that the extraction of the experimental information relies heavily on the so-called isobar approximation, widely used in experimental analyses because of its simplicity, and in spite of its known shortcomings [@Amato:2016xjv].
The $B\rightarrow K\pi\pi$ system is particularly interesting, as the decay amplitudes from intermediate $B\rightarrow PV$ resonances ($K^\star(892)$ and $\rho(770)$) receive sizable contributions from both tree-level and loop diagrams, and interfere directly in the common phase-space regions (namely the “corners” of the DP). The presence of additional resonant intermediate states further constrain the interference patterns and help resolving potential phase ambiguities. In the case of $B^0\rightarrow K^+\pi^-\pi^0$ and $B^+\rightarrow K^0_S\pi^+\pi^0$, two different $K^\star(892)$ states contribute to the decay amplitude, and their interference phases can be directly measured. For $B^0\rightarrow K^0_S\pi^+\pi^-$, the time-dependent evolution of the decay amplitudes for $B^0$ and $\overline{B^0}$ provides (indirect) access to the relative phase between the $B^0\rightarrow K^{\star+}\pi^-$ and $\overline{B^0}\rightarrow K^{\star-}\pi^+$ amplitudes.
In the isobar approximation [@Amato:2016xjv], the total decay amplitude for a given mode is a sum of intermediate resonant contributions, and each of these is a complex function of phase-space: ${\cal A}(DP)= \sum_i A_iF_i(DP)$, where the sum rolls over all the intermediate resonances providing sizable contributions, the $F_i$ functions are the “lineshapes” of each resonance, and the isobar parameters $A_i$ are complex coefficients indicating the strength of each intermediate amplitude. The corresponding relation is $\overline{{\cal A}}(DP)=\sum_i \overline{A_i}~\overline{F_i}(DP)$ for $CP$-conjugate amplitudes.
Any convention-independent function of isobar parameters is a physical observable. For instance, for a given resonance “$i$”, its direct $CP$ asymmetry $A_{CP}$ is expressed as $$A_{CP}^i = \frac{|\overline{A_i}|^2-|A_i|^2}{|\overline{A_i}|^2+|A_i|^2} ,$$ and its partial fit fraction $FF^i$ is $$FF^i = \frac{(|A_i|^2+|\overline{A_i}|^2) \int_{DP} |F_i(DP)|^2 d(DP)}
{\sum_{jk} (A_j A^*_k + \overline{A_j}~\overline{A^*_k}) \int_{DP} F_j(DP) F^*_k(DP) d(DP)}.$$ To obtain the partial branching fraction ${\cal B}^i$, the fit fraction has to be multiplied by the total branching fraction of the final state (e.g., $B^0\to K^0_S\pi^+\pi^-$), $${\cal B}^i = {\it FF}^i \times {\cal B}_{incl} .$$ A phase difference $\varphi_{ij}$ between two resonances “$i$” and “$j$” contributing to the same total decay amplitude (i.e., between resonances in the same DP) is $$\label{eq:phasediff1}
\varphi^{ij} = \arg(A_i/A_j) ,\qquad \overline{\varphi}_{ij} = \arg\left(\overline{A_i}/\overline{A_j}\right)\,,$$ and a phase difference between the two $CP$-conjugate amplitudes for resonance “$i$” is $$\label{eq:phasediff2}
\Delta\varphi^{i} = \arg\left(\frac{q}{p}\frac{\overline{A_i}}{A_i}\right)\,,$$ where $q/p$ is the $B^0-\overline{B^0}$ oscillation parameter.
For $B\rightarrow K^\star\pi$ modes, there are in total 13 physical observables. These can be classified as four branching fractions, four direct $CP$ asymmetries and five phase differences:
- The $CP$-averaged ${\cal B}^{+-}=BR(B^0\rightarrow K^{\star+}\pi^{-})$ branching fraction and its corresponding $CP$ asymmetry $A_{CP}^{+-}$. These observables can be measured independently in the $B^0\rightarrow K^0_S\pi^+\pi^-$ and $B^0\rightarrow K^+\pi^-\pi^0$ Dalitz planes.
- The $CP$-averaged ${\cal B}^{00}=BR(B^0\rightarrow K^{\star 0}\pi^{0})$ branching fraction and its corresponding $CP$ asymmetry $A_{CP}^{00}$. These observables can be accessed both in the $B^0\rightarrow K^+\pi^-\pi^0$ and $B^0\rightarrow K^0_S\pi^0\pi^0$ Dalitz planes.
- The $CP$-averaged ${\cal B}^{+0}=BR(B^+\rightarrow K^{\star+}\pi^{0})$ branching fraction and its corresponding $CP$ asymmetry $A_{CP}^{+0}$. These observables can be measured both in the $B^+\rightarrow K^0_S\pi^+\pi^0$ and $B^+\rightarrow K^+\pi^0\pi^0$ Dalitz planes.
- The $CP$-averaged ${\cal B}^{0+}=BR(B^+\rightarrow K^{\star 0}\pi^{+})$ branching fraction and its corresponding $CP$ asymmetry $A_{CP}^{0+}$. They can be measured both in the $B^+\rightarrow K^+\pi^+\pi^-$ and $B^+\rightarrow K^0_S\pi^0\pi^+$ Dalitz planes.
- The phase difference $\varphi^{00,+-}$ between $B^0\rightarrow K^{\star+}\pi^{-}$ and $B^0\rightarrow K^{\star 0}\pi^{0}$, and its corresponding $CP$ conjugate $\overline\varphi^{{00},-+}$. They can be measured in the $B^0\rightarrow K^+\pi^-\pi^0$ Dalitz plane and in its $CP$ conjugate DP $\overline{B^0}\rightarrow K^-\pi^+\pi^0$, respectively.
- The phase difference $\varphi^{+0,0+}$ between $B^+\rightarrow K^{\star+}\pi^{0}$ and $B^+\rightarrow K^{\star 0}\pi^{+}$, and its corresponding $CP$ conjugate $\overline\varphi^{{-0},0-}$. They can be measured in the $B^+\rightarrow K^0_S\pi^+\pi^0$ Dalitz plane and in its $CP$ conjugate DP $B^-\rightarrow K^0_S\pi^-\pi^0$, respectively.
- The phase difference $\Delta\varphi^{+-}$ between $B^0\rightarrow K^{\star+}\pi^-$ and its $CP$ conjugate $\overline{B^0}\rightarrow K^{\star -}\pi^+$. This phase difference can only be measured in a time-dependent analysis of the $K^0_S\pi^+\pi^-$ DP. As $K^{\star +}\pi^-$ is only accessible for $B^0$ and $K^{\star -}\pi^+$ to $\overline{B^0}$ only, the $B^0\rightarrow K^{\star+}\pi^-$ and $\overline{B^0}\rightarrow K^{\star -}\pi^+$ amplitudes do not interfere directly (they contribute to different DPs). But they do interfere with intermediate resonant amplitudes that are accessible to both $B^0$ and $\overline{B^0}$, like $\rho^0(770)K^0_S$ or $f_0(980)K^0_S$, and thus the time-dependent oscillation is sensitive to the combined phases from mixing and decay amplitudes.
Real-valued physical observables {#sec:Dalitz_new_Observbles}
--------------------------------
The set of physical observables described in the previous paragraph (branching fractions, $CP$ asymmetries and phase differences) has the advantage of providing straightforward physical interpretations. From a technical point of view though, the phase differences suffer from the drawback of their definition with a $2\pi$ periodicity. This feature becomes an issue when the experimental uncertainties on the phases are large and the correlations between observables are significant, since there is no straightforward way to properly implement their covariance into a fit algorithm. Moreover the uncertainties on the phases are related to the moduli of the corresponding amplitudes, leading to problems when the latter are not known precisely and can reach values compatible with zero. As a solution to this issue, a set of real-valued Cartesian physical observables is defined, in which the $CP$ asymmetries and phase differences are expressed in terms of the real and imaginary parts of ratios of isobar amplitudes scaled by the ratios of the corresponding branching fractions and $CP$ asymmetries. The new observables are functions of branching fractions, $CP$ asymmetries and phase differences, and are thus physical observables. The new set of observables, similar to the $U$ and $I$ observables defined in $B\to \rho\pi$ [@Olivier], are expressed as the real and imaginary parts of ratios of amplitudes as follows,
$$\begin{aligned}
{\mathcal Re}\left(A_i/A_j\right) = \sqrt{\frac{{\cal B}^i}{{\cal B}^j}\frac{A_{CP}^i - 1}{A_{CP}^j - 1}} \cos(\varphi_{ij})\,, \\
{\mathcal Im}\left(A_i/A_j\right) = \sqrt{\frac{{\cal B}^i}{{\cal B}^j}\frac{A_{CP}^i - 1}{A_{CP}^j - 1}} \sin(\varphi_{ij})\,, \\
{\mathcal Re}\left(\overline{A}_i/\overline{A}_j\right) = \sqrt{\frac{{\cal B}^i}{{\cal B}^j}\frac{A_{CP}^i + 1}{A_{CP}^j + 1}} \cos(\overline{\varphi}_{ij})\,, \\
{\mathcal Im}\left(\overline{A}_i/\overline{A}_j\right) = \sqrt{\frac{{\cal B}^i}{{\cal B}^j}\frac{A_{CP}^i + 1}{A_{CP}^j + 1}} \sin(\overline{\varphi}_{ij})\,.\end{aligned}$$
We see that some observables are not defined in the case $A_{CP}^j=\pm 1$, as could be expected from the following argument. Let us suppose that $A_{CP}^j=+1$ for the $j$-th resonance, i.e., we have the amplitude $A_j=0$: the quantities ${\mathcal Re}(A_i/A_j)$ and ${\mathcal Im}(A_i/A_j)$ are not defined, but neither is the phase difference between $A_i$ and $A_j$. Therefore, in both parametrisations (real and imaginary part of ratios, or branching ratios, $CP$ asymmetries and phase differences), the singular case $A_{CP}^j=\pm 1$ leads to some undefined observables. Let us add that this case does not occur in practice for our analysis.
For each $B\rightarrow K\pi\pi$ mode considered in this paper, the real and imaginary parts of amplitude ratios used as inputs are the following:
$$\begin{aligned}
&& B^0\rightarrow K^0_S\pi^+\pi^- : \label{eq:KsPiPi_inputs} \\ && \nonumber \qquad
{\mathcal B}(K^{*+}\pi^-)
\ ; \ \\&&\nonumber \qquad
{\mathcal Re}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right]
\ ; \
{\mathcal Im}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right]\ , \
\\
&& B^0\rightarrow K^+\pi^-\pi^0 : \label{eq:KPiPi0_inputs} \\ && \nonumber \qquad
\left\{ \begin{array}{l}\displaystyle {\mathcal B}(K^{*0}\pi^0)
\ ; \ \qquad
\left| \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right| \ ; \ \\
\displaystyle
{\mathcal Re}\left[ \frac{A(K^{*0}\pi^0)}{A(K^{*+}\pi^-)} \right]
\ ; \
{\mathcal Im}\left[ \frac{A(K^{*0}\pi^0)}{A(K^{*+}\pi^-)} \right]
\ ; \ \\
\displaystyle
{\mathcal Re}\left[ \frac{\overline{A}(\overline{K}^{*0}\pi^0)}{\overline{A}(K^{*-}\pi^+)} \right]
\ ; \
{\mathcal Im}\left[ \frac{\overline{A}(\overline{K}^{*0}\pi^0)}{\overline{A}(K^{*-}\pi^+)} \right]\ , \
\end{array}\right.
\\
&& B^+\rightarrow K^+\pi^-\pi^+ : \label{eq:KPiPi_inputs}\\&& \nonumber\qquad
{\mathcal B}(K^{*0}\pi^+)
\ ; \
\left| \frac{\overline{A}(\overline{K}^{*0}\pi^-)}{A(K^{*0}\pi^+)} \right|\ , \\
&& B^+\rightarrow K^0_S\pi^+\pi^0 : \label{eq:K0PiPi0_inputs}\\&& \nonumber\qquad
\left\{ \begin{array}{l}\displaystyle
{\mathcal B}(K^{*+}\pi^0)
\ ; \
\left| \frac{\overline{A}(K^{*-}\pi^0)}{A(K^{*+}\pi^0)} \right| \ ; \ \\
\displaystyle
{\mathcal Re}\left[ \frac{A(K^{*+}\pi^0)}{A(K^{*0}\pi^+)} \right]
\ ; \
{\mathcal Im}\left[ \frac{A(K^{*+}\pi^0)}{A(K^{*0}\pi^+)} \right]
\ ; \ \\ \displaystyle
{\mathcal Re}\left[ \frac{\overline{A}(K^{*-}\pi^0)}{\overline{A}(\overline{K}^{*0}\pi^-)} \right]
\ ; \
{\mathcal Im}\left[ \frac{\overline{A}(K^{*-}\pi^0)}{\overline{A}(\overline{K}^{*0}\pi^-)} \right]\ .
\end{array}\right.\end{aligned}$$
This choice of inputs is motivated by the fact that amplitude analyses are sensitive to ratios of isobar amplitudes. The sensitivity to phase differences leads to a sensitivity to the real and imaginary part of these ratios. It has to be said that the set of inputs listed previously is just one of the possible sets of independent observables that can be extracted from this set of amplitude analyses. In order to combine and Belle results, it is straightforward to express the experimental results in the above format, and then combine them as is done for independent measurements. Furthermore, experimental information from other analyses which are not amplitude and/or time-dependent, i.e., which are only sensitive to ${\mathcal B}$ and $A_{CP}$, can be also added in a straightforward fashion.
In order to properly use the experimental information in the above format it will be necessary to use the full covariance matrix, both statistical and systematic, of the isobar amplitudes. This will allow us to properly propagate the uncertainties as well as the correlations of the experimental inputs to the ones exploited in the phenomenological fit.
Isospin analysis of $B\rightarrow K^*\pi$ decays {#sec:Isospin}
================================================
The isospin formalism used in this work is described in detail in Ref. [@PerezPerez:2008gna]. Only the main ingredients are summarised below.
Without any loss of generality, exploiting the unitarity of the CKM matrix, the $B^0\rightarrow K^{*+}\pi^-$ decay amplitude $A^{+-}$ can be parametrised as $$\label{eq:BtoKstarPlusPiMinusAmplitude}
A^{+-} = V_{ub}^*V_{us} T^{+-} + V_{tb}^*V_{ts} P^{+-} ,$$ with similar expressions for the $CP$-conjugate amplitude $\bar{A}^{-+}$ (the CKM factors appearing as complex conjugates), and for the remaining three amplitudes $A^{ij}=A(B^{i+j}\rightarrow K^{*i}\pi^j)$, corresponding to the $(i,j)=(0,+)$, $(+,0)$, $(00)$ modes. The tree and penguin contributions are now defined through their CKM factors rather than their diagrammatic structure: they can include contributions from additional $c$-quark penguin diagrams due to the re-expression of $V_{cb}^*V_{cs}$ in Eq. (\[eq:BtoKstarPlusPiMinusAmplitude\]). In the following, $T^{ij}$ and $P^{ij}$ will be called hadronic amplitudes.
Note that the relative CKM matrix elements in Eq. (\[eq:BtoKstarPlusPiMinusAmplitude\]) significantly enhance the penguin contributions with respect to the tree ones, providing an improved sensitivity to the former. The isospin invariance imposes a quadrilateral relation among these four decay amplitudes, derived in Ref. [@Nir:1991cu] for $B\to K\pi$, but equivalently applicable in the $K^*\pi$ case: $$\label{eq:isospinRelations}
A^{0+} + \sqrt{2} A^{+0} = A^{+-} + \sqrt{2} A^{00},$$ and a similar expression for the $CP$-conjugate amplitudes. These can be used to rewrite the decay amplitudes in the “canonical” parametrisation, $$\begin{aligned}
\label{eq:canonicalParametrisation}
\begin{array}{cclclc}
A^{+-} & = & V_{us}V_{ub}^*T^{+-} & + & V_{ts}V_{tb}^*P^{+-} & ,
\\
A^{0+} & = & V_{us}V_{ub}^*N^{0+} & + & V_{ts}V_{tb}^*(-P^{+-}+P_{\rm EW}^{\rm C}) & ,
\\
\sqrt{2}A^{+0} & = & V_{us}V_{ub}^*T^{+0} & + & V_{ts}V_{tb}^*P^{+0} & ,
\\
\sqrt{2}A^{00} & = & V_{us}V_{ub}^*T^{00}_{\rm C} & + & V_{ts}V_{tb}^*(-P^{+-}+P_{\rm EW}) & ,
\end{array}\end{aligned}$$ with $$\begin{aligned}
T^{+0}&=&T^{+-}+T_{\rm C}^{00}-N^{0+}\,,\\
P^{+0}&=&P^{+-}+P_{\rm EW}-P_{\rm EW}^{\rm C}\,.\end{aligned}$$ This parametrisation is frequently used in the literature with various slightly different conventions, and is expected to hold up to a very high accuracy (see Refs. [@Gronau:2005pq; @Botella:2006zi] for isospin-breaking contributions to $B\to\pi\pi$ decays). The notation is chosen to illustrate the main diagram topologies contributing to the decay amplitude under consideration. $N^{0+}$ makes reference to the fact that the contribution to $B^+\rightarrow K^{*0}\pi^+$ with a $V_{us}V_{ub}^*$ term corresponds to an annihilation/exchange topology; $T^{00}_{\rm C}$ denotes the colour-suppressed $B^0\rightarrow K^{*0}\pi^0$ tree amplitude; the EW subscript in the $P_{\rm EW}$ and $P_{\rm EW}^{\rm C}$ terms refers to the $\Delta I=1$ electroweak penguin contributions to the decay amplitudes. We can also introduce the $\Delta I=3/2$ combination $T_{3/2}=T^{+-}+T^{00}_{\rm C}$.
One naively expects that colour-suppressed contributions will indeed be suppressed compared to their colour-allowed partner, and that electroweak penguins and annihilation contributions will be much smaller than tree and QCD penguins. These expectations can be expressed quantitatively using theoretical approaches like QCD factorisation [@Beneke:1999br; @Beneke:2000ry; @Beneke:2003zv; @Beneke:2006hg]. Some of these assumptions have been challenged by the experimental data gathered, in particular the mechanism of colour suppression in $B\to\pi\pi$ and the smallness of the annihilation part for $B\to K\pi$ [@Beneke:2006mk; @Bell:2009fm; @Bell:2015koa; @Beneke:2015wfa; @Li:2014rwa; @Olivier].
The complete set of $B\rightarrow K^*\pi$ decay amplitudes, constrained by the isospin relations described in Eq. (\[eq:isospinRelations\]) are fully described by 13 parameters, which can be classified as 11 hadronic and 2 CKM parameters following Eq. (\[eq:canonicalParametrisation\]). A unique feature of the $B\rightarrow K^*\pi$ system is that this number of unknowns matches the total number of physical observables discussed in Sec. \[sec:Dalitz\]. One could thus expect that all parameters (hadronic and CKM) could be fixed from the data. However, it turns out that the weak and strong phases can be redefined in such a way as to absorb in the CKM parameters any constraints on the hadronic ones. This property, known as [*reparametrisation invariance*]{}, is derived in detail in Refs. [@Botella:2005ks; @PerezPerez:2008gna] and we recall its essential aspects here. The decay amplitude of a $B$ meson into a final state can be written as: $$\begin{aligned}
\label{eq:Af1}
A_f&=&m_1 e^{i\phi_1}e^{i\delta_1}+m_2 e^{i\phi_2}e^{i\delta_2} \ , \\
\bar{A}_{\bar{f}}&=&m_1 e^{-i\phi_1}e^{i\delta_1}+m_2 e^{-i\phi_2}e^{i\delta_2} \ , \label{eq:Abarf1}\end{aligned}$$ where $\phi_i$ are $CP$-odd (weak) phases, $\delta_i$ are $CP$-even (strong) phases, and $m$ are real magnitudes. Any additional term $M_3e^{i\phi_3}e^{i\delta_3}$ can be expressed as a linear combination of $e^{i\phi_1}$ and $e^{i\phi_2}$ (with the appropriate properties under $CP$ violation), leading to the fact that the decay amplitudes can be written in terms of any other pair of weak phases $\{\varphi_1, \varphi_2\}$ as long as $\varphi_1\neq \varphi_2$ (mod $\pi$): $$\begin{aligned}
\label{eq:Af2}
A_f&=&M_1 e^{i\varphi_1}e^{i\Delta_1}+M_2 e^{i\varphi_2}e^{i\Delta_2} \ , \\
\bar{A}_{\bar{f}}&=&M_1 e^{-i\varphi_1}e^{i\Delta_1}+M_2 e^{-i\varphi_2}e^{i\Delta_2} \ , \label{eq:Abarf2}\end{aligned}$$ with $$\begin{aligned}
M_1e^{i\Delta_1} &=&[m_1e^{i\delta _1}\sin(\phi_1-\varphi_2)+m_2e^{i\delta_2}\sin(\phi_2-\varphi_2)]
\nonumber\\ &&\qquad\qquad/\sin(\varphi_2-\varphi_1) \ , \label{eq:m1}\\
M_2e^{i\Delta_2} &=&[m_1e^{i\delta _1}\sin(\phi_1-\varphi_1)+m_2e^{i\delta_2}\sin(\phi_2-\varphi_1)]
\nonumber\\ &&\qquad\qquad/\sin(\varphi_2-\varphi_1)\ . \label{eq:m2}\end{aligned}$$
This change in the set of weak basis does not have any physical implications, hence the name of re-parameterisation invariance. We can now take two different sets of weak phases $\{\phi_1, \phi_2\}$ and $\{\varphi_1, \varphi_2\}$ with $\phi_1=\varphi_1$ but $\phi_2\neq\varphi_2$. If an algorithm existed to extract $\phi_2$ as a function of physical observables related to these decay amplitudes, the similarity of Eqs. (\[eq:Af1\])-(\[eq:Abarf1\]) and Eqs. (\[eq:Af2\])-(\[eq:Abarf2\]) indicate that $\varphi_2$ would be extracted exactly using the same function with the same measurements as input, leading to $\varphi_2=\phi_2$, in contradiction with the original statement that we are free to express the physical observables using an arbitrary choice for the weak basis.
We have thus to abandon the idea of an algorithm allowing one to extract both CKM and hadronic parameters from a set of physical observables. The weak phases in the parameterisation of the decay amplitudes cannot be extracted without additional hadronic hypothesis. This discussion holds if the two weak phases used to describe the decay amplitudes are different (modulo $\phi$). The argument does not apply when only one weak phase can be used to describe the decay amplitude: setting one of the amplitudes to zero, say $m_2=0$, breaks reparametrisation invariance, as can be seen easily in Eqs. (\[eq:m1\])-(\[eq:m2\]). In such cases, weak phases can be extracted from experiment, e.g., the extraction of $\alpha$ from $B\to \pi\pi$, the extraction of $\beta$ from $J/\psi K_S$ or $\gamma$ from $B\to DK$. In each case, an amplitude is assumed to vanish, either approximately (extraction of $\alpha$ and $\beta$) or exactly (extraction of $\gamma$) [@Olivier; @Bevan:2014iga; @Bediaga:2012py].
In view of this limitation, two main strategies can be considered for the system considered here: either implementing additional constraints on some hadronic parameters in order to extract the CKM phases using the $B \rightarrow K^*\pi$ observables, or fix the CKM parameters to their known values from a global fit and use the $B \rightarrow K^*\pi$ observables to extract information on the hadronic contributions to the decay amplitudes. Both approaches are described below.
Constraints on CKM phases {#sec:CKM}
=========================
We illustrate the first strategy using two specific examples. The first example is similar in spirit to the Gronau-London method for extracting the CKM angle $\alpha$ [@Gronau:1990ka], which relies on neglecting the contributions of electroweak penguins to the $B\rightarrow\pi\pi$ decay amplitudes. The second example assumes that upper bounds on annihilation/exchange contributions can be estimated from external information.
The CPS/GPSZ method: setting a bound on electroweak penguins {#subsec:CPS}
------------------------------------------------------------
In $B\rightarrow\pi\pi$ decays, the electroweak penguin contribution can be related to the tree amplitude in a model-independent way using Fierz transformations of the relevant current-current operators in the effective Hamiltonian for $B\to \pi\pi$ decays [@Buras:1998rb; @Neubert:1998pt; @Neubert:1998jq; @Charles:2004jd]. One can predict the ratio $R=P_{\rm EW}/T_{3/2}\simeq -3/2 (C_9+C_{10})/(C_1+C_2)=(1.35\pm 0.12) \%$ only in terms of short-distance Wilson Coefficients, since long-distance hadronic matrix elements drop from the ratio (neglecting the operators $O_7$ and $O_8$ due to their small Wilson coefficients compared to $O_9$ and $O_{10}$). This leads to the prediction that there is no strong phase difference between $P_{\rm EW}$ and $T_{3/2}$ so that electroweak penguins do not generate a charge asymmetry in $B^+\to \pi^+\pi^0$ if this picture holds: this prediction is in agreement with the present experimental average of the corresponding asymmetry. Moreover, this assumption is crucial to ensure the usefulness of the Gronau-London method to extract the CKM angle $\alpha$ from an isospin analysis of $B\rightarrow\pi\pi$ decay amplitudes [@Charles:2004jd; @Olivier]: setting the electroweak penguin to zero in the Gronau-London breaks the reparametrisation invariance described in Sec. \[sec:Isospin\] and opens the possibility of extracting weak phases.
One may want to follow a similar approach and use some knowledge or assumptions on the electroweak penguin in the case of $B\to K\pi$ or $B\to K^*\pi$ in order to constrain the CKM factors. This approach is sometimes referred to as the CPS/GPSZ method [@Ciuchini:2006kv; @Gronau:2006qn]. Indeed, as shown in Eq. (\[eq:canonicalParametrisation\]), the penguins in $A^{00}$ and $A^{+-}$ differ only by the $P_{\rm EW}$ term. By neglecting its contribution to $A^{00}$, these two decay amplitudes can be combined so that their (now identical) penguin terms can be eliminated, $$\begin{aligned}
A^0 = A^{+-} + \sqrt{2}A^{00} = V_{us}V_{ub}^*(T^{+-}+T^{00}_{\rm C}),\end{aligned}$$ and then, together with its $CP$-conjugate amplitude $\bar{A}^0$, a convention-independent amplitude ratio $R^0$ can be defined as $$\begin{aligned}
\label{eq:Eq4}
R^0 = \frac{q}{p}\frac{\bar{A}^0}{A^0} = e^{-2i\beta}e^{-2i\gamma} = e^{2i\alpha}.\end{aligned}$$ The $A^0$ amplitude can be extracted using the decay chains $B^0\rightarrow K^{*+}(\rightarrow K^+\pi^0)\pi^-$ and $B^0\rightarrow K^{*0}(\rightarrow K^+\pi^-)\pi^0$ contributing to the same $B^0\rightarrow K^+\pi^-\pi^0$ Dalitz plot, so that both the partial decay rates and their interference phase can be measured in an amplitude analysis. Similarly, $\bar{A}^0$ can be extracted from the $CP$-conjugate $\bar{B}^0\rightarrow K^-\pi^+\pi^0$ DP using the same procedure. Then, the phase difference between $A^{+-}$ and $\bar{A}^{-+}$ can be extracted from the $B^0\rightarrow K^0_{\rm S}\pi^+\pi^-$ DP, considering the $B^0\rightarrow K^{*+}(\rightarrow K^0\pi^+)\pi^-$ decay chain, and its $CP$-conjugate $\bar{B}^0\rightarrow K^{*-}(\rightarrow \bar{K}^0\pi^-)\pi^+$, which do interfere through mixing. Let us stress that this method is a measurement of $\alpha$ rather than a measurement of $\gamma$, in contrast with the claims in Refs. [@Ciuchini:2006kv; @Gronau:2006qn].
However, the method used to bound $P_{\rm EW}$ for the $\pi\pi$ system cannot be used directly in the $K^*\pi$ case. In the $\pi\pi$ case, $SU(2)$ symmetry guarantees that the matrix element with the combination of operators $O_1-O_2$ vanishes, so that it does not enter tree amplitudes. A similar argument would hold for $SU(3)$ symmetry in the case of the $K\pi$ system, but it does not for the vector-pseudoscalar $K^*\pi$ system. It is thus not possible to cancel hadronic matrix elements when considering $P_{\rm EW}/T_{3/2}$, which becomes a complex quantity suffering from (potentially large) hadronic uncertainties [@Gronau:2003yf; @Ciuchini:2006kv]. The size of the electroweak penguin (relative to the tree contributions), is parametrised as $$\frac{P_{\rm EW}}{T_{3/2}} = R \frac{1-r_{\rm VP}}{1+r_{\rm VP}} ,
\label{eq:PEWfromCPS}$$ where $R\simeq (1.35\pm 0.12)\%$ is the value obtained in the $SU(3)$ limit for $B\to \pi K$ (and identical to the one obtained from $B\to\pi\pi$ using the arguments in Refs. [@Buras:1998rb; @Neubert:1998pt; @Neubert:1998jq]), and $r_{\rm VP}$ is a complex parameter measuring the deviation of $P/T_{3/2}$ from this value corresponding to $$r_{\rm VP}=\frac{\langle K^*\pi(I=3/2)|Q_1-Q_2|B\rangle}{\langle K^*\pi(I=3/2)|Q_1+Q_2|B\rangle}\,.$$ Estimates on factorisation and/or $SU(3)$ flavour relations suggest $|r_{\rm VP}|\leq 0.05$ [@Ciuchini:2006kv; @Gronau:2006qn]. However it is clear that both approximations can easily be broken, suggesting a more conservative upper bound $|r_{\rm VP}|\leq 0.30$.
The presence of these hadronic uncertainties have important consequences for the method. Indeed, it turns out that including a non-vanishing $P_{\rm EW}$ completely disturbs the extraction of $\alpha$. The electroweak penguin can provide a ${\cal O}(1)$ contribution to $CP$-violating effects in charmless $b\rightarrow s$ processes, as its CKM coupling amplifies its contribution to the decay amplitude: $P_{\rm EW}$ is multiplied by a large CKM factor $V_{ts}V_{tb}^*=O(\lambda^2)$ compared to the tree-level amplitudes multiplied by a CKM factor $V_{us}V_{ub}^*=O(\lambda^4)$. Therefore, unless $P_{\rm EW}$ is particularly suppressed due to some specific hadronic dynamics, its presence modifies the CKM constraint obtained following this method in a very significant way.
It would be difficult to illustrate this point using the current data, due to the experimental uncertainties described in the next sections. We choose thus to discuss this problem using a reference scenario described in Tab. \[tab:IdealCase\], where the hadronic amplitudes have been assigned arbitrary (but realistic) values and they are used to derive a complete set of experimental inputs with arbitrary (and much more precise than currently available) uncertainties. As shown in App. \[App:Exp\_inputs\] (cf. Tab. \[tab:IdealCase\]), the current world averages for branching ratios and $CP$ asymmetries in $B^0\rightarrow K^{*+}\pi^-$ and $B^0\rightarrow K^{*0}\pi^0$ agree broadly with these values, which also reproduce the expected hierarchies among hadronic amplitudes, if we set the CKM parameters to their current values from our global fit [@Charles:2004jd; @Charles:2015gya; @CKMfitterwebsite]. We choose a penguin parameter $P^{+-}$ with a magnitude $28$ times smaller than the tree parameter $T^{+-}$, and a phase fixed at $-7^\circ$. The electroweak $P_{\rm EW}$ parameter has a value $66$ times smaller in magnitude than the tree parameter $T^{+-}$, and its phase is arbitrarily fixed to $+15^\circ$ in order to get a good agreement with the current central values. Our results do not depend significantly on this phase, and a similar outcome occurs if we choose sets with a vanishing phase for $P_{\rm EW}$ (though the agreement with the current data will be less good).
We use the values of the observables derived with this set of hadronic parameters, and we perform a CPS/ GPSZ analysis to extract a constraint on the CKM parameters. Fig. \[fig:RhoEta\_CPS\] shows the constraints derived in the $\bar{\rho}-\bar{\eta}$ plane. If we assume $P_{\rm EW}=0$ (upper panel), the extracted constraint is equivalent to a constraint on the CKM angle $\alpha$, as expected from Eq. (\[eq:Eq4\]). However, the confidence regions in the $\bar{\rho}-\bar{\eta}$ plane are very strongly biased, and the true value of the parameters are far from belonging to the 95% confidence regions. On the other hand, if we fix $P_{\rm EW}$ to its true value (with a magnitude of $0.038$), the bias is removed but the constraint deviates from a pure $\alpha$-like shape (for instance, it does not include the origin point $\bar\rho=\bar\eta=0$). We notice that the uncertainties on $R$ and, more significantly, $r_{VP}$, have an important impact on the precision of the constraint on $(\bar\rho,\bar\eta)$.
![ Constraints in the $\bar{\rho}-\bar{\eta}$ plane from the amplitude ratio $R^0$ method, using the arbitrary but realistic numerical values for the input parameters, detailed in the text. In the top panel, the $P_{\rm EW}$ hadronic parameter is set to zero. In the bottom panel, the $P_{\rm EW}$ hadronic parameter is set to its true generation value with different theoretical errors on $R$ and $r_{VP}$ parameters (defined in Eq. (\[eq:PEWfromCPS\])), either zero (green solid-line contour), 10% and 5% (blue dashed-line contour), and 10% and 30% (red solid-dashed-line contour). The parameters $\bar{\rho}$ and $\bar{\eta}$ are fixed to their current values from the global CKM fit [@Charles:2004jd; @Charles:2015gya; @CKMfitterwebsite], indicated by the magenta point. []{data-label="fig:RhoEta_CPS"}](RhoEta_CPS_ZeroPew.eps "fig:"){width="7cm"} ![ Constraints in the $\bar{\rho}-\bar{\eta}$ plane from the amplitude ratio $R^0$ method, using the arbitrary but realistic numerical values for the input parameters, detailed in the text. In the top panel, the $P_{\rm EW}$ hadronic parameter is set to zero. In the bottom panel, the $P_{\rm EW}$ hadronic parameter is set to its true generation value with different theoretical errors on $R$ and $r_{VP}$ parameters (defined in Eq. (\[eq:PEWfromCPS\])), either zero (green solid-line contour), 10% and 5% (blue dashed-line contour), and 10% and 30% (red solid-dashed-line contour). The parameters $\bar{\rho}$ and $\bar{\eta}$ are fixed to their current values from the global CKM fit [@Charles:2004jd; @Charles:2015gya; @CKMfitterwebsite], indicated by the magenta point. []{data-label="fig:RhoEta_CPS"}](RhoEta_CPS_FixPew_and_Vary_Rreal_and_rVP.eps "fig:"){width="7cm"}
This simple illustration with our reference scenario shows that the CPS/GPSZ method is limited both in robustness and accuracy due to the assumption on a negligible $P_{\rm EW}$: a small non-vanishing value breaks the relation between the phase of $R^0$ and the CKM angle $\alpha$, and therefore, even a small uncertainty on the $P_{\rm EW}$ value would translate into large biases on the CKM constraints. It shows that this method would require a very accurate understanding of hadronic amplitudes in order to extract a meaningful constraint on the unitarity triangle, and the presence of non-vanishing electroweak penguins dilutes the potential of this method significantly.
![ Top: constraints in the $\bar{\rho}-\bar{\eta}$ plane from the annihilation/exchange method, using the arbitrary but realistic numerical values for the input parameters detailed in the text. The green solid-line contour is the constraint obtained by fixing the $N^{0+}$ hadronic parameter to its generation value; the blue dotted-line contour is the constraint obtained by setting an upper bound on the $\left|N^{0+}/T^{+-}\right|$ ratio at twice its generation value. The parameters $\bar{\rho}$ and $\bar{\eta}$ are fixed to their current values from the global CKM fit [@Charles:2004jd; @Charles:2015gya; @CKMfitterwebsite], indicated by the magenta point. Bottom: size of the $\beta - \beta_{\rm gen}$ 68% confidence interval vs the upper-bound on $|N^{0+}/T^{+-}|$ in units of its generation value. []{data-label="fig:RhoEta_All"}](RhoEta_All_N0pBound20Times.eps "fig:"){width="7cm"} ![ Top: constraints in the $\bar{\rho}-\bar{\eta}$ plane from the annihilation/exchange method, using the arbitrary but realistic numerical values for the input parameters detailed in the text. The green solid-line contour is the constraint obtained by fixing the $N^{0+}$ hadronic parameter to its generation value; the blue dotted-line contour is the constraint obtained by setting an upper bound on the $\left|N^{0+}/T^{+-}\right|$ ratio at twice its generation value. The parameters $\bar{\rho}$ and $\bar{\eta}$ are fixed to their current values from the global CKM fit [@Charles:2004jd; @Charles:2015gya; @CKMfitterwebsite], indicated by the magenta point. Bottom: size of the $\beta - \beta_{\rm gen}$ 68% confidence interval vs the upper-bound on $|N^{0+}/T^{+-}|$ in units of its generation value. []{data-label="fig:RhoEta_All"}](beta_voverage_vs_N0pOTpmBound.eps "fig:"){width="8cm"}
Setting bounds on annihilation/exchange contributions {#subsec:N}
-----------------------------------------------------
As discussed in the previous paragraphs, the penguin contributions for $B\rightarrow K^*\pi$ decays are strongly CKM-enhanced, impacting the CPS/GPSZ method based on neglecting a penguin amplitude $P_{\rm EW}$. This method exhibits a strong sensitivity to small changes or uncertainties in values assigned to the electroweak penguin contribution. An alternative and safer approach consists in constraining a tree amplitude, with a CKM-suppressed contribution. Among the various hadronic amplitudes introduced, it seems appropriate to choose the annihilation amplitude $N^{0+}$, which is expected to be smaller than $T^{+-}$, and which could even be smaller than the colour-suppressed $T^{00}_{\rm C}$. Unfortunately, no direct, clean constraints on $N^{0+}$ can be extracted from data and from the theoretical point of view, $N^{0+}$ is dominated by incalculable non-factorisable contributions in QCD factorisation [@Beneke:1999br; @Beneke:2000ry; @Beneke:2003zv; @Beneke:2006hg]. On the other hand, indirect upper bounds on $N^{0+}$ may be inferred from either the $B^+\to K^{*0} \pi^+$ decay rate or from the $U$-spin related mode $B^+\to K^{*0}K^+$.
This method, like the previous one, hinges on a specific assumption on hadronic amplitudes. Fixing $N^{0+}$ breaks the reparametrisation invariance in Sec. \[sec:Isospin\], and thus provides a way of measuring weak phases. We can compare the two approaches by using the same reference scenario as in Sec. \[subsec:CPS\], i.e., the values gathered in Tab. \[tab:IdealCase\]. We have an annihilation parameter $N^{0+}$ with a magnitude $18$ times smaller than the tree parameter $T^{+-}$, and a phase fixed at $108^\circ$. All $B\rightarrow K^*\pi$ physical observables are used as inputs. This time, all hadronic parameters are free to vary in the fits, except for the annihilation/exchange parameter $N^{0+}$, which is subject to two different hypotheses: either its value is fixed to its generation value, or the ratio $\left|N^{0+}/T^{+-}\right|$ is constrained in a range (up to twice its generation value).
The resulting constraints on the $\bar{\rho}-\bar{\eta}$ are shown on the upper plot of Fig. \[fig:RhoEta\_All\]. We stress that in this fit, the value of $N^{0+}$ is bound, but the other amplitudes (including $P_{\rm EW}$) are left free to vary. Using a loose bound on $\left|N^{0+}/T^{+-}\right|$ yields a less tight constraint, but in contrast with the CPS/GPSZ method, the CKM generation value is here included. One may notice that the resulting constraint is similar to the one corresponding to the CKM angle $\beta$. This can be understood in the following way. Let us assume that we neglect the contribution from $N^{0+}$. We obtain the following amplitude to be considered $$A'=A^{0+}=V_{ts}V_{tb}^*(-P^{+-}+P_{\rm EW}^{\rm C}),$$ and then, together with its $CP$-conjugate amplitude $\bar{A}'$, a convention-independent amplitude ratio $R'$ can be defined as $$R' = \frac{q}{p}\frac{\bar{A}'}{A} = e^{-2i\beta}\,,$$ in agreement with the convention used to fix the phase of the $B$-meson state. This justifies the $\beta$-like shape of the constraint obtained when fixing the value of the annihilation parameter. The presence of the oscillation phase $q/p$ here, starting from a decay of a charged $B$, may seem surprising. However, one should keep in mind that the measurement of $B^+\to K^{*0}\pi^+$ and its $CP$-conjugate amplitude are not sufficient to determine the relative phase between $A'$ and $\bar{A}'$: this requires one to reconstruct the whole quadrilateral equation Eq. (\[eq:isospinRelations\]), where the phases are provided by interferences between mixing and decay amplitudes in $B_0$ and $\bar{B}_0$ decays. In other words, the phase observables obtained from the Dalitz plot are always of the form Eq. (\[eq:phasediff1\])-(\[eq:phasediff2\]): their combination can only lead to a ratio of $CP$-conjugate amplitudes multiplied by the oscillation parameter $q/p$.
The lower plot of Fig. \[fig:RhoEta\_All\] describes how the constraint on $\beta$ loosens around its true value when the range allowed for $\left|N^{0+}/T^{+-}\right|$ is increased compared to its initial value ($0.143$). We see that the method is stable and keeps on including the true value for $\beta$ even in the case of a mild constraint on $\left|N^{0+}/T^{+-}\right|$.
Constraints on hadronic parameters using current data {#sec:Hadronic}
=====================================================
As already anticipated in Sec. \[sec:Isospin\], a second strategy to exploit the data consists in assuming that the CKM matrix is already well determined from the CKM global fit [@Charles:2004jd; @Charles:2015gya; @CKMfitterwebsite]. The measurements of $B\rightarrow K^\star\pi$ observables (isobar parameters) can then be used to extract constraints on the hadronic parameters in Eq. (\[eq:canonicalParametrisation\]).
Experimental inputs {#subsec:exp_inputs}
-------------------
For this study, the complete set of available results from the and Belle experiments is used. The level of detail for the publicly available results varies according to the decay mode in consideration. In most cases, at least one amplitude DP analysis of $B^0$ and $B^+$ decays is public [@Amhis:2016xyh], and at least one input from each physical observable is available. In addition, the conventions used in the various DP analyses are usually different. Ideally, one would like to have access to the complete covariance matrix, including statistical and systematic uncertainties, for all isobar parameters, as done for instance in Ref. [@Aubert:2009me]. Since such information is not always available, the published results are used in order to derive ad-hoc approximate covariance matrices, implementing all the available information (central values, total uncertainties, correlations among parameters). The inputs for this study are the following:
- Two three-dimensional covariance matrices, cf. Eq. (\[eq:KsPiPi\_inputs\]), from the time-dependent DP analysis of $B^0\rightarrow K^0_S\pi^+\pi^-$ in Ref. [@Aubert:2009me], and two three-dimensional covariance matrices from the Belle time-dependent DP analysis of $B^0\rightarrow K^0_S\pi^+\pi^-$ in Ref. [@Dalseno:2008wwa]. Both the and Belle analyses found two quasi-degenerate solutions each, with very similar goodness-of-fit merits. The combination of these solutions is described in App. \[App:comb\_inputs\], and is taken as input for this study.
- A five-dimensional covariance matrix, cf. Eq. (\[eq:KPiPi0\_inputs\]), from the $B^0\rightarrow K^+\pi^-\pi^0$ DP analysis [@BABAR:2011ae].
- A two-dimensional covariance matrix, cf. Eq. (\[eq:KPiPi\_inputs\]), from the $B^+\rightarrow K^+\pi^+\pi^-$ DP analysis [@Aubert:2008bj], and a two-dimensional covariance matrix from the Belle $B^+\rightarrow K^+\pi^+\pi^-$ DP analysis [@Garmash:2006bj].
- A simplified uncorrelated four-dimensional input, cf. Eq. (\[eq:K0PiPi0\_inputs\]), from the $B^+\rightarrow K^0_S\pi^+\pi^0$ preliminary DP analysis [@Lees:2015uun].
Besides the inputs described previously, there are other experimental measurements on different three-body final states performed in the quasi-two-body approach, which provide measurements of branching ratios and $CP$ asymmetries only. Such is the case of the result on the $B^+\to K^+\pi^0\pi^0$ final state [@Lees:2011aaa], where the branching ratio and the $CP$ asymmetry of the $B^+\to K^{*}(892)^+\pi^0$ contribution are measured. In this study, these two measurements are treated as uncorrelated, and they are combined with the inputs from the DP analyses mentioned previously.
These sets of experimental central values and covariance matrices are described in App. \[App:Exp\_inputs\], where the combinations of the results from and Belle are also described.
Finally, we notice that the time-dependent asymmetry in $B\to K_S\pi^0\pi^0$ has been measured [@Abe:2007xd; @Aubert:2007ub]. As these are global analyses integrated over the whole DP, we cannot take these measurements into account. In principle a time-dependent isobar analysis of the $K_S\pi^0\pi^0$ DP could be performed and it could bring some independent information on $B\to K^{*0}\pi^0$ intermediate amplitudes. Since this more challenging analysis has not been done yet, we will not consider this channel for the time being.
Selected results for $CP$ asymmetries and hadronic amplitudes
-------------------------------------------------------------
Using the experimental inputs described in Sec. \[subsec:exp\_inputs\], a fit to the complete set of hadronic parameters is performed. We discuss the fit results focusing on three aspects: the most significant direct $CP$ asymmetries, the significance of electroweak penguins, and the relative hierarchies of hadronic contributions to the tree amplitudes. As will be seen in the following, the fit results can be interpreted in terms of two sets of local minima, out of which one yields constraints on the hadronic parameters in better agreement with the expectations from CPS/GPSZ, the measured direct $CP$ asymmetries and the expected relative hierarchies of hadronic contributions.
### Direct $CP$ violation in $B^0\rightarrow K^{\star+}\pi^-$
The $B^0\rightarrow K^{\star+}\pi^-$ amplitude can be accessed both in the $B^0\rightarrow K^0_{\rm S}\pi^+\pi^-$ and $B^0\rightarrow K^+\pi^-\pi^0$ Dalitz-plot analyses. The direct $CP$ asymmetry $A_{\rm CP}(B^0\rightarrow K^{\star+}\pi^-)$ has been measured by in both modes [@BABAR:2011ae; @Aubert:2009me] and by Belle in the $B^0\rightarrow K^0_{\rm S}\pi^+\pi^-$ mode [@Dalseno:2008wwa]. All three measurements yield a negative value: incidentally, this matches also the sign of the two-body $B^0\rightarrow K^+\pi^-$ $CP$ asymmetry, for which direct $CP$ violation is clearly established.
Using the amplitude DP analysis results from these three measurements as inputs, the combined constraint on $A_{\rm CP}(B^0\rightarrow K^{\star+}\pi^-)$ is shown in Fig. \[fig:ACP\_KstpPim\]. The combined value is 3.0 $\sigma$ away from zero, and the 68% confidence interval on this $CP$ asymmetry is $0.21\pm 0.07$ approximately. This result is to be compared with the $0.23\pm 0.06$ value provided by HFLAV [@Amhis:2016xyh]. The difference is likely to come from the fact that HFLAV performs an average of the $CP$ asymmetries extracted from individual experiments, while this analysis uses isobar values as inputs which are averaged over the various experiments before being translated into values for the $CP$ parameters: since the relationships between these two sets of quantities are non-linear, the two steps (averaging over experiments and translating from one type of observables to another) yield the same central values only in the case of very small uncertainties. In the current situation, where sizeable uncertainties affect the determinations from individual experiments, it is not surprising that minor discrepancies arise between our approach and the HFLAV result.
As can be readily seen from Eq. (\[eq:BtoKstarPlusPiMinusAmplitude\]), a non-vanishing asymmetry in this mode requires a strong phase difference between the tree $T^{+-}$ and penguin $P^{+-}$ hadronic parameters that is strictly different from zero. Fig. \[fig:PpmOTpm\_BtoKstPi\] shows the two-dimensional constraint on the modulus and phase of the $P^{+-}/T^{+-}$ ratio. Two solutions with very similar $\chi^2$ are found, both incompatible with a vanishing phase difference. The first solution corresponds to a small (but non-vanishing) positive strong phase, with similar $\left|V_{ts}V_{tb}^\star P^{+-}\right|$ and $\left|V_{us}V_{ub}^\star T^{+-}\right|$ contributions to the total decay amplitude, and is called Solution I in the following. The other solution, denoted Solution II, corresponds to a larger, negative, strong phase, with a significantly larger penguin contribution. We notice that Solution I is closer to usual theoretical expectations concerning the relative size of penguin and tree contributions.
Let us stress that the presence of two solutions for $P^{+-}/T^{+-}$ is not related to the presence of ambiguities in the individual and Belle measurements for $B^+\to K^+\pi^+\pi^-$ and $B^0\to K^0_S\pi^+\pi^-$, since we have performed their combinations in order to select a single solution for each process. Therefore, the presence of two solutions in Fig. \[fig:PpmOTpm\_BtoKstPi\] is a global feature of our non-linear fit, arising from the overall structure of the current combined measurements (central values and uncertainties) that we use as inputs.
![ Constraint on the direct $CP$ asymmetry parameter $C(B^0\rightarrow K^{\star+}\pi^-) = -A_{\rm CP}(B^0\rightarrow K^{\star+}\pi^-)$ from data on $B^0\to K^0_S\pi^+\pi^-$ (red curve), Belle data on $B^0\to K^0_S\pi^+\pi^-$ (blue curve), data on $B^0\to K^+\pi^-\pi^0$ (green curve) and the combination of all these measurements (green shaded curve). The constraints are obtained using the observables described in the text. []{data-label="fig:ACP_KstpPim"}](Acp_B0toKstpPim_all.eps){width="8cm"}
![ Two-dimensional constraint on the modulus and phase of the $P^{+-}/T^{+-}$ ratio. For convenience, the modulus is multiplied by the ratio of CKM factors appearing in the tree and penguin contributions to the $B^0\rightarrow K^{\star+}\pi^-$ decay amplitude. []{data-label="fig:PpmOTpm_BtoKstPi"}](PpmOTpm_BtoKstPi_all_BaBarAndBelle.eps){width="8cm"}
### Direct $CP$ violation in $B^+\rightarrow K^{\star+}\pi^0$
The $B^+\rightarrow K^{\star+}\pi^0$ amplitude can be accessed in a $B^+\rightarrow K^0_{\rm S}\pi^+\pi^0$ Dalitz-plot analysis, for which only a preliminary result from is available [@Lees:2015uun]. A large, negative $CP$ asymmetry $A_{\rm CP}(B^+\rightarrow K^{\star+}\pi^0) = -0.52\pm 0.14\pm 0.04 ^{+0.04}_{-0.02}$ is reported there with a 3.4 $\sigma$ significance. This $CP$ asymmetry has also been measured by through a quasi-two-body analysis of the $B^+\rightarrow K^+\pi^0\pi^0$ final state [@Lees:2011aaa], obtaining $A_{\rm CP}(B^+\rightarrow K^{\star+}\pi^0) = -0.06\pm 0.24\pm 0.04$. The combination of these two measurement yields $A_{\rm CP}(B^+\rightarrow K^{\star+}\pi^0) = -0.39\pm 0.12\pm 0.03$, with a 3.2 $\sigma$ significance.
In contrast with the $B^0\rightarrow K^{\star+}\pi^-$ case, in the canonical parametrisation Eq. (\[eq:canonicalParametrisation\]), the decay amplitude for $B^+\rightarrow K^{\star+}\pi^0$ includes several hadronic contributions both to the total tree and penguin terms, namely $$\begin{aligned}
\sqrt{2}A^{+0} & = & V_{us}V_{ub}^*T^{+0} + V_{ts}V_{tb}^*P^{+0} \\ \nonumber
& = & V_{us}V_{ub}^*(T^{+-}+T_{\rm C}^{00}-N^{0+}) \\ \nonumber
& &+ V_{ts}V_{tb}^*(P^{+-}+P_{\rm EW}-P_{\rm EW}^{\rm C}) \ , \end{aligned}$$ and therefore no straightforward constraint on a single pair of hadronic parameters can be extracted, as several degenerate combinations can reproduce the observed value of the $CP$ asymmetry $A_{\rm CP}(B^+\rightarrow K^{\star+}\pi^0)$. This is illustrated in Fig. \[fig:POTp0\_BtoKstPi\], where six different local minima are found in the fit, all with similar $\chi^2$ values. The three minima with positive strong phases correspond to Solution I, while the three minima with negative strong phases correspond to Solution II. The relative size of the total tree and penguin contributions is bound within a relatively narrow range: we get $|P^{+0}/T^{+0}| \in (0.018,0.126)$ at $68\%$ C.L.
![ Constraint on the direct $CP$ asymmetry parameter $C(B^+\rightarrow K^{\star+}\pi^0) = -A_{\rm CP}(B^+\rightarrow K^{\star+}\pi^0)$ from data on $B^+\to K^0_S\pi^+\pi^0$ (red curve), data on $B^+\to K^+\pi^0\pi^0$ (blue curve) and the combination (green shaded curve). The constraints are obtained using the observables described in the text. []{data-label="fig:ACP_KstpPi0"}](Acp_BptoKstpPi0_all.eps){width="8cm"}
![ Top: two-dimensional constraint on the modulus and phase of the $(P^{+-}+P_{\rm EW}-P_{\rm EW}^{\rm C})/(T^{+-}+T_{\rm C}^{00}-N^{0+})$ ratio. For convenience, the modulus is multiplied by the ratio of CKM factors appearing in the tree and penguin contributions to the $B^+\rightarrow K^{\star+}\pi^0$ decay amplitude. Bottom: one-dimensional constraint on the modulus of the $(P^{+-}+P_{\rm EW}-P_{\rm EW}^{\rm C})/(T^{+-}+T_{\rm C}^{00}-N^{0+})$ ratio. []{data-label="fig:POTp0_BtoKstPi"}](POTp0_BtoKstPi_all_BaBarAndBelle.eps "fig:"){width="8cm"} ![ Top: two-dimensional constraint on the modulus and phase of the $(P^{+-}+P_{\rm EW}-P_{\rm EW}^{\rm C})/(T^{+-}+T_{\rm C}^{00}-N^{0+})$ ratio. For convenience, the modulus is multiplied by the ratio of CKM factors appearing in the tree and penguin contributions to the $B^+\rightarrow K^{\star+}\pi^0$ decay amplitude. Bottom: one-dimensional constraint on the modulus of the $(P^{+-}+P_{\rm EW}-P_{\rm EW}^{\rm C})/(T^{+-}+T_{\rm C}^{00}-N^{0+})$ ratio. []{data-label="fig:POTp0_BtoKstPi"}](ModPOTp0_BtoKstPi_all_BaBarAndBelle.eps "fig:"){width="8cm"}
### Hierarchy among penguins: electroweak penguins {#sec:hierarchypenguins}
In Sec. \[subsec:CPS\], we described the CPS/GPSZ method designed to extract weak phases from $B\to\pi K$ assuming some control on the size of the electroweak penguin. According to this method, the electroweak penguin is expected to yield a small contribution to the decay amplitudes, with no significant phase difference. We are actually in a position to test this expectation by fitting the hadronic parameters using the and Belle data as inputs. Fig. \[fig:PewOT3o2\] shows the two-dimensional constraint on $r_{VP}$, in other words, the ratio $P_{\rm EW}/T_{3/2}$ ratio, showing two local minima. The CPS/GPSZ prediction is also indicated in this figure. In Fig. \[fig:POTvsrVP\], we provide the regions allowed for $|r_{VP}|$ and the modulus of the ratio $|P^{+-}/T^{+-}|$, exhibiting two favoured values, the smaller one being associated with Solution I and the larger one with Solution II. The latter one corresponds to a significantly large electroweak penguin amplitude and it is clearly incompatible with the CPS/GPSZ prediction by more than one order of magnitude. A better agreement, yet still marginal, is found for the smaller minimum that corresponds to Solution I: the central value for the ratio is about a factor of three larger than CPS/GPSZ, and a small, positive phase is preferred. For this minimum, an inflation of the uncertainty on $\left| r_{\rm VP}\right|$ up to $30\%$ would be needed to ensure proper agreement. In any case, it is clear that the data prefers a larger value of $|r_{\rm VP}|$ than the estimates originally proposed.
![ Two-dimensional constraint on real and imaginary parts on the $r_{VP}$ parameter defined in Eq. (\[eq:PEWfromCPS\]). The area encircled with the solid (dashed) red line corresponds to the CPS/GPSZ prediction, with a $5\%$ ($30\%$) uncertainty on the $r_{\rm VP}$ parameter. \[fig:PewOT3o2\]](ReImrVP_BtoKstPi_all_BaBarAndBelle_v2.eps){width="8cm"}
![ Two-dimensional constraint on $|r_{VP}|$ defined in Eq. (\[eq:PEWfromCPS\]) and ${\rm Log}_{10}\left(|P^{+-}/T^{+-}|\right)$. The vertical solid (dashed) red line corresponds to the CPS/GPSZ prediction, with a $5\%$ ($30\%$) uncertainty. \[fig:POTvsrVP\] ](ModrVP_vs_Log10ModPpmOTpm_BtoKstPi_all_BaBarAndBelle.eps){width="8cm"}
Moreover, the contribution from the electroweak penguin is found to be about twice larger than the main penguin contribution $P^{+-}$. This is illustrated in Fig. \[fig:PewOP\], where only one narrow solution is found in the $P_{\rm EW}/P^{+-}$ plane, as both solutions I and II provide essentially the same constraint. The relative phase between these two parameters is bound to the interval $(-25,+10)^\circ$ at $95\%$ C.L. Additional tests allow us to demonstrate that this strong constraint on the relative $P_{\rm EW}/P^{+-}$ penguin contributions is predominantly driven by the $\varphi^{00,+-}$ phase differences measured in the Dalitz-plot analysis of $B^0\rightarrow K^+\pi^+\pi^0$ decays. The strong constraint on the $P_{\rm EW}/P^{+-}$ ratio is turned into a mild upper bound when removing the $\varphi^{00,+-}$ phase differences from the experimental inputs. The addition of these two observables as fit inputs increases the minimal $\chi^2$ by 7.7 units, which corresponds to a 2.6 $\sigma$ discrepancy. Since the latter is driven by a measurement from a single experiment, additional experimental results are needed to confirm such a large value for the electroweak penguin parameter.
![ Top: two-dimensional constraint on the modulus and phase of the complex $P_{\rm EW}/P^{+-}$ ratio. Bottom: constraint on the $\left|P_{\rm EW}/P^{+-}\right|$ ratio, using the complete set of experimental inputs (red curve), and removing the measurement of the $\varphi^{00,+-}$ phases from the $B^0\rightarrow K^+\pi^+\pi^0$ Dalitz-plot analysis (green shaded curve). \[fig:PewOP\]](PewOPpm_BtoKstPi_all_BaBarAndBelle.eps "fig:"){width="8cm"} ![ Top: two-dimensional constraint on the modulus and phase of the complex $P_{\rm EW}/P^{+-}$ ratio. Bottom: constraint on the $\left|P_{\rm EW}/P^{+-}\right|$ ratio, using the complete set of experimental inputs (red curve), and removing the measurement of the $\varphi^{00,+-}$ phases from the $B^0\rightarrow K^+\pi^+\pi^0$ Dalitz-plot analysis (green shaded curve). \[fig:PewOP\]](ModPewOPpm_BtoKstPi_all_BaBarAndBelle.eps "fig:"){width="8cm"}
In view of colour suppression, the electroweak penguin $P_{\rm EW}^{\rm C}$ is expected to yield a smaller contribution than $P_{\rm EW}$ to the decay amplitudes. This hypothesis is tested in Fig. \[fig:PewCOPew\], which shows that current data favours a similar size for the two contributions, and a small relative phase (up to $40^{\circ}$) between the colour-allowed and the colour-suppressed electroweak penguins. Both Solutions I and II show the same structure with four different local minima.
![ Top: two-dimensional constraint on the modulus and phase of the $P_{\rm EW}^{\rm C}/P_{\rm EW}$ ratio. Bottom: one-dimensional constraint on ${\rm Log}_{10}\left(\left|P_{\rm EW}^{\rm C}/P_{\rm EW}\right|\right)$, using the complete set of experimental inputs (red curve), and removing the measurement of the $\varphi^{00,+-}$ phases from the $B^0\rightarrow K^+\pi^+\pi^0$ Dalitz-plot analysis (green shaded curve). []{data-label="fig:PewCOPew"}](PewCOPew_BtoKstPi_all_BaBarAndBelle.eps "fig:"){width="8cm"} ![ Top: two-dimensional constraint on the modulus and phase of the $P_{\rm EW}^{\rm C}/P_{\rm EW}$ ratio. Bottom: one-dimensional constraint on ${\rm Log}_{10}\left(\left|P_{\rm EW}^{\rm C}/P_{\rm EW}\right|\right)$, using the complete set of experimental inputs (red curve), and removing the measurement of the $\varphi^{00,+-}$ phases from the $B^0\rightarrow K^+\pi^+\pi^0$ Dalitz-plot analysis (green shaded curve). []{data-label="fig:PewCOPew"}](Log10ModPewCOPew_BtoKstPi_all_BaBarAndBelle.eps "fig:"){width="8cm"}
### Hierarchy among tree amplitudes: colour suppression and annihilation
![ Two-dimensional constraint on the modulus and phase of the $T^{00}_{\rm C}/T^{+-}$ (top) and $N^{0+}/T^{+-}$ (bottom) ratios. []{data-label="fig:T00OTpm_N0pOTpm"}](T00OTpm_BtoKstPi_all_BaBarAndBelle.eps "fig:"){width="7cm"} ![ Two-dimensional constraint on the modulus and phase of the $T^{00}_{\rm C}/T^{+-}$ (top) and $N^{0+}/T^{+-}$ (bottom) ratios. []{data-label="fig:T00OTpm_N0pOTpm"}](N0pOTpm_BtoKstPi_all_BaBarAndBelle.eps "fig:"){width="7cm"}
As already discussed, the hadronic parameter $T^{00}_{\rm C}$ is expected to be suppressed with respect to the main tree parameter $T^{+-}$. Also, the annihilation topology is expected to provide negligible contributions to the decay amplitudes. These expectations can be compared with the extraction of these hadronic parameters from data in Fig. \[fig:T00OTpm\_N0pOTpm\].
For colour suppression, the current data provides no constraint on the relative phase between the $T^{00}_{\rm C}$ and $T^{+-}$ tree parameters, and only a mild upper bound on the modulus can be inferred; the tighter constraint is provided by Solution I that excludes values of $|T^{00}_{\rm C}/T^{+-}|$ larger than $1.6$ at $95\%$ C.L. The constraint from Solution II is more than one order of magnitude looser.
Similarly, for annihilation, Solution I provides slightly tighter constraints on its contribution to the total tree amplitude with the bound $|N^{0+}/T^{+-}|<2.5$ at $95\%$ C.L., while the bound from Solution II is much looser.
Comparison with theoretical expectations {#QCDFcomparison}
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Quantity Fit result QCDF
-------------------------------------------------------------------- ----------------------------------------- -------------------------------
$\displaystyle {\mathcal Re}\frac{N^{0+}}{T^{+-}}$ $(-5.31, 4.73)$ $0.011 \pm 0.027$
$\displaystyle {\mathcal Im}\frac{N^{0+}}{T^{+-}}$ $(-9.59, 7.73)$ $0.003\pm 0.028$
$\displaystyle {\mathcal Re}\frac{P_{\rm EW}^{\rm C}}{P_{\rm EW}}$ $(0.69,1.14)$ $0.17\pm 0.19$
$\displaystyle {\mathcal Im}\frac{P_{\rm EW}^{\rm C}}{P_{\rm EW}}$ $(-0.48,-0.28)~\cup~(-0.13,0.22)~\cup$ $-0.08\pm0.14$
$(0.34,0.60)$
$\displaystyle {\mathcal Re}\frac{P_{\rm EW}^{\rm C}}{P^{+-}}$ $(1.29,2.08)$ $-$
$\displaystyle {\mathcal Im}\frac{P_{\rm EW}^{\rm C}}{P^{+-}}$ $(-1.09,-0.75)~\cup~(-0.51,-0.10)~\cup$ $-$
$(-0.08,0.16)~\cup~(0.47,0.83)$
$\displaystyle {\mathcal Re}\frac{P_{\rm EW}^{\rm C}}{T^{+-}}$ $(-0.12,0.34)$ $0.0027\pm 0.0031$
$\displaystyle {\mathcal Im}\frac{P_{\rm EW}^{\rm C}}{T^{+-}}$ $(-0.42,0.05)$ $-0.0015^{+0.0024}_{-0.0025}$
$\displaystyle {\mathcal Re}\frac{P^{+-}}{P_{\rm EW}}$ $(0.49,0.56)$ $3.9^{+3.2}_{-3.3}$
$\displaystyle {\mathcal Im}\frac{P^{+-}}{P_{\rm EW}}$ $(-0.03,0.16)$ $1.8\pm 3.3$
$\displaystyle {\mathcal Re}\frac{P_{\rm EW}}{T^{+-}}$ $(0.0, 0.25)$ $0.0154^{+0.0059}_{-0.0060}$
$\displaystyle {\mathcal Im}\frac{P_{\rm EW}}{T^{+-}}$ $(-0.40,-0.09)~\cup~(-0.02,0.02)$ $-0.0014^{+0.0023}_{-0.0022}$
$\displaystyle {\mathcal Re}\frac{P^{+-}}{T^{+-}}$ $( 0.023,0.140)$ $0.053\pm0.039$
$\displaystyle {\mathcal Im}\frac{P^{+-}}{T^{+-}}$ $(-0.20,-0.04)~\cup~(0.0, 0.01)$ $0.016\pm0.044$
$\displaystyle {\mathcal Re}\frac{T^{00}_{\rm C}}{T^{+-}}$ $(-0.26,2.24)$ $0.13\pm0.17$
$\displaystyle {\mathcal Im}\frac{T^{00}_{\rm C}}{T^{+-}}$ $(-3.28,0.74)$ $-0.11\pm0.15$
We have extracted the values of the hadronic amplitudes from the data currently available. It may prove interesting to compare these results with theoretical expectations. For this exercise, we use QCD factorisation [@Beneke:1999br; @Beneke:2000ry; @Beneke:2003zv; @Beneke:2006hg] as a benchmark point, keeping in mind that other approaches (discussed in the introduction) are available. In order to keep the comparison simple and meaningful, we consider the real and imaginary part of several ratios of hadronic amplitudes.
We obtain our theoretical values in the following way. We follow Ref. [@Beneke:2003zv] for the expressions within QCD factorisation, and we use the same model for the power-suppressed and infrared-divergent contributions coming from hard scattering and weak annihilation: these contributions are formally $1/m_b$-suppressed but numerically non negligible, and play a crucial role in some of the amplitudes. On the other hand, we update the hadronic parameters in order to take into account more recent determinations of these quantities, see App. \[app:QCDFinputs\]. We use the Rfit scheme to handle theoretical uncertainties [@Charles:2004jd; @Charles:2015gya; @CKMfitterwebsite; @Charles:2016qtt] (in particular for the hadronic parameters and the $1/m_b$ power-suppressed contributions), and we compute only ratios of hadronic amplitudes using QCD factorisation. We stress that we provide the estimates within QCD factorisation simply to compare the results of our experimental fit for the hadronic amplitudes with typical theoretical expectations concerning the same quantities. In particular we neglect Next-Next-to-Leading Order corrections that have been partially computed in Refs. [@BHWS; @Bell:2007tv; @Bell:2009nk; @Beneke:2009ek; @Bell:2015koa], and we do not attempt to perform a fully combined fit of the theoretical predictions with the experimental data, as the large uncertainties would make the interpretation difficult.
Our results for the ratios of hadronic amplitudes are shown in Fig. \[fig:QCDFcomparison2D\] and in Tab. \[tab:QCDFcomparison1D\]. We notice that for most of the ratios, a good agreement is found. The global fit to the experimental data has often much larger uncertainties than theoretical predictions: with better data in the future, we may be able to perform very non trivial tests of the non-leptonic dynamics and the isobar approximation. The situation for $P_{\rm EW}^{\rm C}/P_{\rm EW}$ is slightly different, since the two determinations (experiment and theory) exhibit similar uncertainties and disagree with each other, providing an interesting test for QCD factorisation, which however goes beyond the scope of this study.
There are two cases where the theoretical output from QCD factorisation is significantly less precise than the constraints from the combined fit. For $P_{\rm EW}^C/P^{+-}$, both numerator and denominator can be (independently) very small in QCD factorisation, and numerical instabilities in this ratio prevent us from having a precise prediction. For $P^{+-}/P_{\rm EW}$, the impressively accurate experimental determination, as discussed in Sec. \[sec:hierarchypenguins\], is predominantly driven by the $\varphi^{00,+-}$ phase differences measured in the Dalitz-plot analysis of $B^0\rightarrow K^+\pi^+\pi^0$ decays. Removing this input yields a much milder constraint on $P^{+-}/P_{\rm EW}$. On the other hand in QCD factorisation, the formally leading contributions to the $P^{+-}$ penguin amplitude are somewhat numerically suppressed, and compete with the model estimate of power corrections: due to the Rfit treatment used, the two contributions can either compensate each other almost exactly or add up coherently, leading to a $\sim\pm 100\%$ relative uncertainty, which is only in marginal agreement with the fit output. Thus we conclude that the $P^{+-}/P_{\rm EW}$ ratio is both particularly sensitive to the power corrections to QCD factorisation and experimentally well constrained, so that it can be used to provide an insight on non factorisable contributions, provided one assumes negligible effects from New Physics.
Prospects for LHCb and Belle II {#sec:prospect}
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In this section, we study the impact of improved measurements of $K\pi\pi$ modes from the LHCb and Belle II experiments. During the first run of the LHC, the LHCb experiment has collected large datasets of B-hadron decays, including charmless $B^0,B^+,B_s$ meson decays into tree-body modes. LHCb is currently collecting additional data in Run-2. In particular, due to the excellent performances of the LHCb detector for identifying charged long-lived mesons, the experiment has the potential for producing the most accurate charmless three-body results in the $B^+\rightarrow K^+\pi^-\pi^+$ mode, owing to high-purity event samples much larger than the ones collected by and Belle. Using $3.0\ {\rm fb}^{-1}$ of data recorded during the LHC Run 1, first results on this mode are already available [@Aaij:2014iva], and a complete amplitude analysis is expected to be produced in the short-term future. For the $B^0\rightarrow K^0_S\pi^+\pi^-$ mode, the event-collection efficiency is challenged by the combined requirements on reconstructing the $K^0_S\rightarrow \pi^+\pi^-$ decay and tagging the $B$ meson flavour, but nonetheless the $B^0\rightarrow K^0_S\pi^+\pi^-$ data samples collected by LHCb are already larger than the ones from and Belle. As it is more difficult to anticipate the reach of LHCb Dalitz-plot analyses for modes including $\pi^0$ mesons in the final state, the $B^0\rightarrow K^+\pi^+\pi^0$, $B^+\rightarrow K^0_S\pi^+\pi^0$ $B^+\rightarrow K^+\pi^0\pi^0$ and $B^0\rightarrow K_S^0\pi^0\pi^0$ channels are not considered here. In addition, LHCb has also the potential for studying $B_s$ decay modes, and LHCb can reach $B\to KK\pi$ modes with branching ratios out of reach for $B$-factories.
The Belle II experiment [@Urquijo:2015qsa], currently in the stages of construction and commissioning, will operate in an experimental environment very similar to the one of the and Belle experiments. Therefore Belle II has the potential for studying all modes accessed by the $B$-factories, with expected sensitivities that should scale in proportion to its expected total luminosity (i.e., $50\ {\rm ab}^{-1}$). In addition, Belle II has the potential for accessing the $B^+\rightarrow K^+\pi^0\pi^0$ and $B^0\rightarrow K_S^0\pi^0\pi^0$ modes (for which the $B$-factories could not produce Dalitz-plot results) but these modes will provide low-accuracy information, redundant with some of the modes considered in this paper: therefore they are not included here.
Since both the LHCb and Belle II have the potential for studying large, high-quality samples of $B^+\rightarrow K^+\pi^-\pi^+$, it is realistic to expect that the experiments will be able to extract a consistent, data-driven signal model to be used in all Dalitz-plot analysis, yielding systematic uncertainties significantly decreased with respect to the results from $B$-factories.
Finally for LHCb, since this experiment cannot perform $B$-meson counting as in a $B$-factory environment, the branching fractions need to be normalised with respect to measurements performed at and Belle, until the advent of Belle II. This prospective study therefore is split into two periods: a first one based on the assumption of new results from LHCb Run1+Run2 only, and a second one using the complete set of LHCb and Belle II results. The corresponding inputs are gathered in App. \[app:refprosp\]. We use the reference scenario described in Tab. \[tab:IdealCase\] for the central values, so that we can guarantee the self-consistency of the inputs and we avoid reducing the uncertainties artificially because of barely compatible measurements (which would occur if we used the central values of the current data and rescaled the uncertainties). The expected uncertainties, obtained from the extrapolations discussed previously, are described in Tab. \[tab:LHCbAndBelleII\].
The blue area in Fig. \[fig:LHCbAndBelleII2023\] illustrates the potential for the first step of our prospective study ($B$-factories and LHCb Run1+Run2). For the input values used in the prospective, the modulus of the $P^{+-}/T^{+-}$ ratio will be constrained with a relative $10\%$ accuracy, and its complex phase will be constrained within $3$ degrees (we discuss 68% C.L. ranges in the following, whereas Fig. \[fig:LHCbAndBelleII2023\] shows 95% C.L. regions). Slightly tighter upper bounds on the $|T^{00}_{\rm C}/T^{+-}|$ and $|N^{0+}/T^{+-}|$ ratios may be set, albeit the relative phases of these rations will remain very poorly constrained. Assuming that the electroweak penguin is in agreement with the CPS/GPSZ prediction, its modulus will be constrained within $45\%$ and its phase within $14$ degrees.
The addition of results from the Belle II experiment corresponds to the second step of this prospective study. As illustrated by the green area in Fig. \[fig:LHCbAndBelleII2023\], the uncertainties on the modulus and phase of the $P^{+-}/T^{+-}$ ratio will decrease by factors of $1.4$ and $2.5$, respectively. Owing to the addition of precision measurements by Belle II of the $B^0\to K^{*0}\pi^0$ Dalitz-plot parameters from the amplitude analysis of the $B^0\to K^+\pi^-\pi^0$ modes, the $T^{00}_{\rm C}/T^{+-}$ ratio can be constrained within a $22\%$ uncertainty for its modulus, and within $10$ degrees for its phase. Similarly, the uncertainties on the modulus and phase of the $P_{\rm EW}/T_{3/2}$ ratio will decrease by factors $2.7$ and $2.9$, respectively. Concerning the colour-suppressed electroweak penguin, for which only a mild upper bound on its modulus was achievable within the first step of the prospective, can now be measured within a $22\%$ uncertainty for its modulus, and within $8$ degrees for its phase. Finally, the less stringent constraint will be achieved for the annihilation parameter. While its modulus can nevertheless be constrained between 0.3 and 1.5, the phase of this ratio may remain unconstrained in value, with just the sign of the phase being resolved. We add that one can also expect Belle II measurements for $B^+\to K^+\pi^0\pi^0$ and $B^0\to K_S\pi^0\pi^0$, however with larger uncertainties, so that we have not taken into account these decays.
In total, precise constraints on almost all hadronic parameters in the $B\rightarrow K^\star\pi$ system will be achieved using the Dalitz-plot results from the LHCb and Belle II experiments, with a resolution of the current phase ambiguities. These constraints can be compared with various theoretical predictions, proving an important tool for testing models of hadronic contributions to charmless $B$ decays.
Conclusion
==========
Non-leptonic B meson decays are very interesting processes both as probes of weak interaction and as tests of our understanding of QCD dynamics. They have been measured extensively at $B$-factories as well as at the LHCb experiment, but this wealth of data has not been fully exploited yet, especially for the pseudoscalar-vector modes which are accessible through Dalitz-Plot analyses of $B\to K\pi\pi$ modes. We have focused on the $B\to K^*\pi$ system which exhibits a large set of observables already measured. Isospin analysis allows us to express this decay in terms of CKM parameters and 6 complex hadronic amplitudes, but reparametrisation invariance prevents us from extracting simultaneously information on the weak phases and the hadronic amplitudes needed to describe these decays. We have followed two different approaches to exploit this data: either we extracted information on the CKM phase (after setting a condition on some of the hadronic amplitudes), or we determined of hadronic amplitudes (once we set the CKM parameters to their value from the CKM global fit [@Charles:2004jd; @Charles:2015gya; @CKMfitterwebsite]).
In the first case, we considered two different strategies. We first reconsidered the CPS/GPSZ strategy proposed in Ref. [@Ciuchini:2006kv; @Gronau:2006qn], amounting to setting a bound on the electroweak penguin in order to extract an $\alpha$-like constraint. We used a reference scenario inspired by the current data but with consistent central values and much smaller uncertainties in order to probe the robustness of the CPS/GPSZ method: it turns out that the method is easily biased if the bound on the electroweak penguin is not correct, even by a small amount. Unfortunately, this bound is not very precise from the theoretical point of view, which casts some doubt on the potential of this method to constrain $\alpha$. We have then considered a more promising alternative, consisting in setting a bound on the annihilation contribution. We observed that we could obtain an interesting stable $\beta$-like constraint and we discussed its potential to extract confidence intervals according to the accuracy of the bound used for the annihilation contribution.
In a second stage, we discussed how the data constrain the hadronic amplitudes, assuming the values of the CKM parameters. We performed an average of and Belle data in order to extract constraints on various ratios of hadronic amplitudes, with the issue that some of these data contain several solutions to be combined in order to obtain a single set of inputs for the Dalitz-plot observables. The ratio $P^{+-}/T^{+-}$ is not very well constrained and exhibits two distinct preferred solutions, but it is not large and supports the expect penguin suppression. On the other hand, colour or electroweak suppression does not seem to hold, as illustrated by $|P_{\rm EW}/P^{+-}|$ (around 2), $|P_{\rm EW}^{\rm C}/P_{\rm EW}|$ (around 1) or $|T^{00}_{\rm C}/T^{+-}|$ (mildly favouring values around 1). We however recall that some of these conclusions are very dependent on the measurement on $\varphi^{00,+-}$ phase differences measured in $B^0\to K^+\pi^+\pi^0$: removing this input turns the ranges into mere upper bounds on these ratios of hadronic amplitudes.
For illustration purposes, we compared these results with typical theoretical expectations. We determined the hadronic amplitudes using an updated implementation of QCD factorisation. A good overall agreement between theory and experiment is found for most of the ratios of hadronic amplitudes, even though the experimental determinations remain often less accurate than the theoretical determinations in most instances. Nevertheless, two quantities still feature interesting properties. The ratio $P^{+-}/P_{\rm EW}$ could provide interesting constraints on the models used to describe power-suppressed contributions in QCD factorisation, keeping in mind the (precise) experimental determination of this ratio relies strongly on the $\varphi^{00,+-}$ phases measured by , as discussed in the previous paragraph. The ratio $P_{\rm EW}^C/P_{\rm EW}$ is determined with similar accuracies theoretically and experimentally, but the two determinations are not in good agreement, suggesting that this quantity could also be used to constrain QCD factorisation parameters.
Finally, we performed prospective studies, considering two successive stages based first on LHCb data from Run1 and Run2, then on the additional input from Belle II. Using our reference scenario and extrapolating the uncertainties of the measurements at both stages, we determined the confidence regions for the moduli and phases of the ratios of hadronic amplitudes. The first stage (LHCb only) would correspond to a significant improvement for $P^{+-}/T^{+-}$ and $P_{\rm EW}/T_{3/2}$, whereas the second stage (LHCb+Belle II) would yield tight constraints on $N^{0+}/T^{+-}$, $P_{\rm EW}^C/T^{+-}$ and $T^{00}_{\rm C}/T^{+-}$.
Non-leptonic $B$-meson decays remain an important theoretical challenge, and any contender should be able to explain not only the pseudoscalar-pseudoscalar modes but also the pseudoscalar-vector modes. Unfortunately, the current data do not permit such extensive tests, even though they hint at potential discrepancies with theoretical expectations concerning the hierarchies of hadronic amplitudes. However, our study suggests that a more thorough analysis of $B\to K\pi\pi$ Dalitz plots from LHCb and Belle II could allow for a precise determination of the hadronic amplitudes involved in $B\to K^*\pi$ decays thanks to the isobar approximation for three-body amplitudes. This will definitely shed some light on the complicated dynamics of weak and strong interaction at work in pseudo-scalar-vector modes, and it will provide important tests of our understanding of non-leptonic $B$-meson decays.
Acknowledgments
===============
We would like to thank all our collaborators from the CKMfitter group for useful discussions, and Reina Camacho Toro for her collaboration on this project at an early stage. This project has received funding from the European Union Horizon 2020 research and innovation programme under the grant agreements No 690575. No 674896 and No. 692194. SDG acknowledges partial support from Contract FPA2014-61478-EXP.
Current experimental inputs {#App:Exp_inputs}
===========================
The full set real-valued physical observables, derived from the experimental inputs from and Belle, is described in the following sections. The errors and correlation matrices include both statistical and systematic uncertainties.
[l|c|ccc]{} $B^0\rightarrow K^0_S\pi^+\pi^-$ & $~~~~~~~~~~$Global min$~~~~~~~~~~$ & ${\mathcal Re}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right]$ & ${\mathcal Im}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right]$ & ${\mathcal B}(K^{*+}\pi^-)$\
${\mathcal Re}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right]$ & $0.428 \pm 0.473$ & 1.00 & 0.90 & 0.02\
${\mathcal Im}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right]$ & $-0.690 \pm 0.302$ & & 1.00 & -0.06\
${\mathcal B}(K^{*+}\pi^-) (\times 10^{-6})$ & $8.290 \pm 1.189$ & & & 1.00\
[l|c|ccc]{} $B^0\rightarrow K^0_S\pi^+\pi^-$ & Local min ($\Delta {\rm NLL} = 0.16$) & ${\mathcal Re}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right]$ & ${\mathcal Im}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right]$ & ${\mathcal B}(K^{*+}\pi^-)$\
${\mathcal Re}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right]$ & $-0.819 \pm 0.116$ & 1.00 & -0.19 & -0.15\
${\mathcal Im}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right]$ & $-0.049 \pm 0.494$ & & 1.00 & -0.01\
${\mathcal B}(K^{*+}\pi^-) (\times 10^{-6})$ & $8.290 \pm 1.189$ & & & 1.00\
[l|c]{} $B^+\rightarrow K^+\pi^-\pi^+$ & Value\
$\left| \frac{\overline{A}(\overline{K}^{*0}\pi^-)}{A(K^{*0}\pi^+)} \right|$ & $1.033 \pm 0.047$\
${\mathcal B}(K^{*0}\pi^+) (\times 10^{-6})$ & $10.800 \pm 1.389$\
[l|c|cccccc]{} $B^0\rightarrow K^+\pi^-\pi^0$ & Value & $\left| \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right|$ & ${\mathcal Re}\left[ \frac{A(K^{*0}\pi^0)}{A(K^{*+}\pi^-)} \right]$ & ${\mathcal Im}\left[ \frac{A(K^{*0}\pi^0)}{A(K^{*+}\pi^-)} \right]$ & ${\mathcal Re}\left[ \frac{\overline{A}(\overline{K}^{*0}\pi^0)}{\overline{A}(K^{*-}\pi^+)} \right]$ & ${\mathcal Re}\left[ \frac{\overline{A}(\overline{K}^{*0}\pi^0)}{\overline{A}(K^{*-}\pi^+)} \right]$ & ${\mathcal B}(K^{*0}\pi^0)$\
$\left| \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right|$ & $0.742 \pm 0.091$ & 1.00 & 0.00 & 0.03 & -0.22 & -0.11 & -0.06\
${\mathcal Re}\left[ \frac{A(K^{*0}\pi^0)}{A(K^{*+}\pi^-)} \right]$ & $0.562 \pm 0.148$ & & 1.00 & 0.68 & 0.33 & -0.01 & 0.44\
${\mathcal Im}\left[ \frac{A(K^{*0}\pi^0)}{A(K^{*+}\pi^-)} \right]$ & $-0.227 \pm 0.296$ & & & 1.00 & -0.07 & 0.00 & -0.13\
${\mathcal Re}\left[ \frac{\overline{A}(\overline{K}^{*0}\pi^0)}{\overline{A}(K^{*-}\pi^+)} \right]$ & $0.701 \pm 0.126$ & & & & 1.00 & 0.25 & 0.55\
${\mathcal Im}\left[ \frac{\overline{A}(\overline{K}^{*0}\pi^0)}{\overline{A}(K^{*-}\pi^+)} \right]$ & $-0.049 \pm 0.376$ & & & & & 1.00 & -0.02\
${\mathcal B}(K^{*0}\pi^0) (\times 10^{-6})$ & $3.300 \pm 0.640$ & & & & & & 1.00\
[l|c|cccccc]{} $B^+\rightarrow K^0_S\pi^+\pi^0$ & Value & $\left| \frac{\overline{A}(K^{*-}\pi^0)}{A(K^{*+}\pi^0)} \right|$ & ${\mathcal Re}\left[ \frac{A(K^{*+}\pi^0)}{A(K^{*0}\pi^+)} \right]$ & ${\mathcal Im}\left[ \frac{A(K^{*+}\pi^0)}{A(K^{*0}\pi^+)} \right]$ & ${\mathcal Re}\left[ \frac{\overline{A}(K^{*-}\pi^0)}{\overline{A}(\overline{K}^{*0}\pi^-)} \right]$ & ${\mathcal Im}\left[ \frac{\overline{A}(K^{*-}\pi^0)}{\overline{A}(\overline{K}^{*0}\pi^-)} \right]$ & ${\mathcal B}(K^{*+}\pi^0)$\
$\left| \frac{\overline{A}(K^{*-}\pi^0)}{A(K^{*+}\pi^0)} \right|$ & $0.533 \pm 1.403$ & 1.00 & -0.26 & 0.01 & -0.70 & -0.22 & -0.16\
${\mathcal Re}\left[ \frac{A(K^{*+}\pi^0)}{A(K^{*0}\pi^+)} \right]$ & $1.415 \pm 6.952$ & & 1.00 & -0.23 & 0.12 & -0.51 & 0.90\
${\mathcal Im}\left[ \frac{A(K^{*+}\pi^0)}{A(K^{*0}\pi^+)} \right]$ & $-0.189 \pm 3.646$ & & & 1.00 & -0.39 & 0.23 & -0.28\
${\mathcal Re}\left[ \frac{\overline{A}(K^{*-}\pi^0)}{\overline{A}(\overline{K}^{*0}\pi^-)} \right]$& $-0.106 \pm 2.687$ & & & & 1.00 & 0.23 & 0.03\
${\mathcal Im}\left[ \frac{\overline{A}(K^{*-}\pi^0)}{\overline{A}(\overline{K}^{*0}\pi^-)} \right]$& $-0.851 \pm 4.278$ & & & & & 1.00 & -0.82\
${\mathcal B}(K^{*+}\pi^0) (\times 10^{-6})$ & $9.200 \pm 1.480$ & & & & & & 1.00\
[l|c]{} $B^+\to K^{*+}\pi^0$ in $B^+\to K^+\pi^0\pi^0$ & value\
${\mathcal B}(K^{*+}\pi^0)$ & $(8.2 \pm 1.5 \pm 1.1)\times10^{-6}$\
$A_{CP}(K^{*+}\pi^0)$ & $ -0.06 \pm 0.24 \pm 0.04$\
results {#App:babar_inputs}
--------
In this section, we describe the set of experimental inputs from the experiment.
- $B^0\rightarrow K^0_S\pi^+\pi^-$ [@Aubert:2009me]. Two almost degenerate solutions were found differing only by $0.16$ negative-log-likelihood ($\Delta {\rm NLL}$) units. The central values and correlation matrix of the measured observables for both solutions are shown in Tab. \[tab:KSPiPi\_babar\].
- $B^+\rightarrow K^+\pi^-\pi^+$ [@Aubert:2008bj]. The central values of the observables for this analysis are shown in Tab. \[tab:KPiPi\_babar\]. A linear correlation of $2\%$ was found between $\left| \frac{\overline{A}(\overline{K}^{*0}\pi^-)}{A(K^{*0}\pi^+)} \right|$ and ${\mathcal B}(K^{*0}\pi^+)$.
- $B^0\rightarrow K^+\pi^-\pi^0$ [@BABAR:2011ae]. The central values and correlation matrix of the measured observables for this analysis are shown in Tab. \[tab:KPiPi0\_babar\].
- $B^+\rightarrow K^0_S\pi^+\pi^0$ [@Lees:2015uun]. The central values and correlation matrix of the measured observables for this analysis are shown in Tab. \[tab:K0PiPi0\_babar\].
- $B^+\to K^{*+}(892)\pi^0$ quasi-two-body contribution to the $B^+\to K^+\pi^0\pi^0$ final state [@Lees:2011aaa]. The measured branching ratio and $CP$ asymmetry are shown in Tab. \[tab:KstPi0\] and they are used as uncorrelated inputs.
[l|c|ccc]{} $B^0\rightarrow K^0_S\pi^+\pi^-$ & $~~~~~~~~~$Global min$~~~~~~~~~$ & ${\mathcal Re}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right]$ & ${\mathcal Im}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right]$ & ${\mathcal B}(K^{*+}\pi^-)$\
${\mathcal Re}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right]$ & $0.790 \pm 0.145$ & 1.00 & 0.62 & -0.04\
${\mathcal Im}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right]$ & $-0.206 \pm 0.398$ & & 1.00 & 0.00\
${\mathcal B}(K^{*+}\pi^-) (\times 10^{-6})$ & $8.400 \pm 1.449$ & & & 1.00\
[l|c|ccc]{} $B^0\rightarrow K^0_S\pi^+\pi^-$ & Local min ($\Delta {\rm NLL} = 7.5$) & ${\mathcal Re}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right]$ & ${\mathcal Im}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right]$ & ${\mathcal B}(K^{*+}\pi^-)$\
${\mathcal Re}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right]$ & $0.808 \pm 0.110$ & 1.00 & 0.01 & -0.06\
${\mathcal Im}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right]$ & $0.010 \pm 0.439$ & & 1.00 & 0.00\
${\mathcal B}(K^{*+}\pi^-) (\times 10^{-6})$ & $8.400 \pm 1.449$ & & & 1.00\
$B^+\rightarrow K^+\pi^-\pi^+$ value
------------------------------------------------------------------------------ ---------------------
$\left| \frac{\overline{A}(\overline{K}^{*0}\pi^-)}{A(K^{*0}\pi^+)} \right|$ $0.861 \pm 0.059$
${\mathcal B}(K^{*0}\pi^+) (\times 10^{-6})$ $9.670 \pm 1.061$
: Central values of the observables from the Belle $B^+\rightarrow K^+\pi^-\pi^+$ analysis.[]{data-label="tab:KPiPi_belle"}
Belle results {#App:belle_inputs}
-------------
In this section, we describe the set of experimental inputs from the Belle experiment.
- $B^0\rightarrow K^0_S\pi^+\pi^-$ [@Dalseno:2008wwa]. Two solutions were found differing by $7.5$ $\Delta {\rm NLL}$. The central values and correlation matrix of the measured observables for both solutions are shown in Tab. \[tab:KSPiPi\_belle\].
- $B^+\rightarrow K^+\pi^-\pi^+$ [@Garmash:2006bj]. The central values of the observables for this analysis are shown in Tab. \[tab:KPiPi\_belle\]. A nearly vanishing correlation was found between $\left| \frac{\overline{A}(\overline{K}^{*0}\pi^-)}{A(K^{*0}\pi^+)} \right|$ and ${\mathcal B}(K^{*0}\pi^+)$.
Combined and Belle results {#App:comb_inputs}
--------------------------
The and Belle results for the $B^0\rightarrow K^0_S\pi^+\pi^-$ and $B^+\rightarrow K^+\pi^-\pi^+$ analyses shown previously have been combined in the usual way for sets of independent measurements. The combination for the $B^+\rightarrow K^+\pi^-\pi^+$ mode is straightforward as the results exhibit only one solution, as shown in Fig. \[fig:Combination\_babarbelle\_KPiPi\]. The resulting central values are shown in Tab. \[tab:KPiPiKSPiPi\_babarbelle\]. A vanishing linear correlation is found between $\left| \frac{\overline{A}(\overline{K}^{*0}\pi^-)}{A(K^{*0}\pi^+)} \right|$ and ${\mathcal B}(K^{*0}\pi^+)$.
The combination of the and Belle measurements for the $B^0\rightarrow K^0_S\pi^+\pi^-$ mode is more complicated as the results feature several solutions which are relatively close in units of $\Delta {\rm NLL}$. In order to combine this measurements we proceed as follows:
- We combine each solution of the analysis with each one of the Belle results.
- In the goodness of fit of the combination ($\chi^2_{\rm min}$), we add the $\Delta {\rm NLL}$ of each and Belle solution. In the case of the global minimum the corresponding $\Delta {\rm NLL}$ is zero.
- Finally, we take the envelope of the four combinations as the final result.
We find the following $\chi^2_{\rm min}$ for the four combinations: 1.1, 8.7, 9.5 and 98.3. As the closest combination from the global minimum differs by 7.6 units in $\chi^2_{\rm min}$, we have decided to focus on the global minimum for the phenomenological analysis. The combination for this global minimum is shown in Fig. \[fig:Combination\_babarbelle\_KSPiPi\]. The resulting central values and covariance matrix are shown in Tab. \[tab:KPiPiKSPiPi\_babarbelle\].
![Contours at 1 (solid) and 2 (dotted) $\sigma$ in the $\left| \frac{\overline{A}(\overline{K}^{*0}\pi^-)}{A(K^{*0}\pi^+)} \right|$ vs ${\mathcal B}(K^{*0}\pi^+)$ plane for the (black) and Belle (red) results, as well as the combination (blue).[]{data-label="fig:Combination_babarbelle_KPiPi"}](Inputs_BpToKpPimPip_Comb_BaBarAbdBelle.eps){width="8.5cm"}
{width="5.9cm"} {width="5.9cm"} {width="5.9cm"}
[l|c]{} $B^+\rightarrow K^+\pi^-\pi^+$ & Value\
$\left| \frac{\overline{A}(\overline{K}^{*0}\pi^-)}{A(K^{*0}\pi^+)} \right|$ & $0.965 \pm 0.037$\
${\mathcal B}(K^{*0}\pi^+) (\times 10^{-6})$ & $10.062 \pm 0.835$\
[l|c|ccc]{} $B^0\rightarrow K^0_S\pi^+\pi^-$ & Value & ${\mathcal Re}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right]$ & ${\mathcal Im}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right]$ & ${\mathcal B}(K^{*+}\pi^-)$\
${\mathcal Re}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right]$ & $0.698 \pm 0.120$ & 1.00 & 0.58 & -0.01\
${\mathcal Im}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right]$ & $-0.506 \pm 0.146$ & & 1.00 & -0.09\
${\mathcal B}(K^{*+}\pi^-) (\times 10^{-6})$ & $8.340 \pm 0.910$ & & & 1.00\
These combined results for the $B^0\rightarrow K^0_S\pi^+\pi^-$ and $B^+\rightarrow K^+\pi^-\pi^-$ modes are used with the results for the $B^0\rightarrow K^+\pi^-\pi^0$ and $B^+\rightarrow K^0_S \pi^+\pi^0$ as inputs for the phenomenological analysis using the current experimental measurements.
Two-body non leptonic amplitudes in QCD factorisation {#app:QCDFinputs}
=====================================================
We compute the $B\to K^*\pi$ amplitudes in the framework of QCD factorisation, using the results of Ref. [@Beneke:2003zv]. We take the semileptonic $B\to\pi$ and $B\to K\pi$ form factors from computations based on Light-Cone Sum Rules [@Ball:2004rg; @Straub:2015ica]. The parameters for the light-meson distribution amplitudes that enter hard-scattering contributions are consistently taken from the last two references. On the other hand the first inverse moment of the $B$-meson distribution amplitude $\lambda_B$ is taken from Ref. [@Braun:2003wx]. Quark masses are taken from review by the FLAG group [@Aoki:2016frl]. Our updated inputs are summarised in Table \[tab:QCDFinputs\].
Input Value Input Value
----------------- ------------------------- ----------------------- -----------------------
$\alpha_1(K^*)$ $0.06\pm 0\pm 0.04$ $\alpha_1(K^*,\perp)$ $0.04\pm 0\pm 0.03$
$\alpha_2(K^*)$ $0.16\pm 0\pm 0.09$ $\alpha_2(K^*,\perp)$ $0.10\pm 0\pm 0.08$
$f_\perp(K^*)$ $0.159\pm 0\pm 0.006$ $A_0[B\to K^*](0)$ $0.356\pm 0\pm 0.046$
$\alpha_2(\pi)$ $0.062\pm 0\pm 0.054$ $F_0[B\to\pi](0)$ $0.258\pm 0\pm 0.031$
$\lambda_B$ $0.460\pm 0\pm 0.110$ $\bar m_b$ $4.17$
$\bar m_s$ $0.0939\pm 0\pm 0.0011$ $m_q/m_s$ $\sim 0$
: Input values for the hadronic parameters that enter QCD factorisation predictions: moments of the distribution amplitudes for mesons, decay constants, form factors and quark masses. Dimensionful quantities are in GeV. The $\pm 0$ in second position means that all uncertainties are considered as coming from a theoretical origin and they are treated according to the Rfit approach. See the text for references.[]{data-label="tab:QCDFinputs"}
We stress that the calculations of Ref. [@Beneke:2003zv] correspond to Next-to-Leading Order (NLO). Since then, some NNLO contributions have been computed [@BHWS; @Bell:2007tv; @Bell:2009nk; @Beneke:2009ek; @Bell:2015koa], that we neglect in view of the sizeable uncertainties on the input parameters: this is sufficient for our illustrative purposes (see Section \[QCDFcomparison\]).
Reference scenario and prospective studies {#app:refprosp}
==========================================
Some of the experimental results collected in App. \[App:Exp\_inputs\] are affected by large uncertainties, and the central values are not always fully consistent with SM expectations. This is not a problem when we want to extract values of the hadronic parameters from the data, but it makes rather unclear the discussion of the accuracy of specific models (say, for the extraction of weak angles) or the prospective studies assuming improved experimental measurements, see Secs. \[sec:CKM\] and \[sec:prospect\].
For this reason, we design a reference scenario described in Tab. \[tab:IdealCase\]. The values on hadronic parameters are chosen to reproduce the current best averages of branching fractions and $CP$ asymmetries in $B\rightarrow K^*\pi$ roughly. As most observable phase differences among these modes are poorly constrained by the results currently available, we do no attempt at reproducing their central values and we use the values resulting from the hadronic parameters. The hadronic amplitudes are constrained to respect the naive assumptions: $|P_{\rm EW}/T_{3/2}| \simeq 1.35\%$, $|P^C_{\rm EW}| < |P_{\rm EW}|$ and $|T^{00}_{\rm C}| < |T^{+-}|$. The best values of the hadronic parameters yield the values of branching ratios and $CP$ asymmetries gathered in Tab. \[tab:IdealCase\]. As can be seen, the overall agreement is fair, but it is not good for all observables. Indeed, as discussed in Sec. \[sec:Hadronic\], the current data do not favour all the hadronic hierarchies that we have imposed to obtain our reference scenario in Tab. \[tab:IdealCase\].
For the studies of different methods to extract CKM parameters described in Sec. \[sec:CKM\], we fit the values of hadronic parameters by assigning small, arbitrary, uncertainties to the physical observables: $\pm 5\%$ for branching ratios, $\pm 0.5\%$ for $CP$ asymmetries, and $\pm 5^\circ$ for interference phases.
For the prospective studies described in Sec. \[sec:prospect\], we estimate future experimental uncertainties at two different stages. We first consider a list of expected measurements from LHCb, using the combined Run1 and Run2 data. We then reassess the expected results including Belle II measurements. Our method to project uncertainties in the two stages is based on the statistical scaling of data samples ($1/\sqrt{N_{\rm evts}}$), corrected for additional factors due to particular detector performances and analysis technique features, as described below.
LHCb Run1 and Run2 data will significantly increase the statistics mainly for the fully charged final states $B^0\rightarrow K^0_S(\rightarrow \pi^+\pi^-)\pi^+\pi^-$ and $B^+\rightarrow K^+\pi^-\pi^+$, with an expected increase of about $3$ and $40$, respectively [@B0toKspipi_LHCb:2012iva; @BtoKpipi_LHCb:2013iva]. For these modes, we assume a signal-to-background ratio similar to the ones measured at $B$ factories (this may represent an underestimation of the potential sensitivity of LHCb data, but this assumption has a very minor impact on the results of our prospective study). The statistical scaling factor thus defined can be applied as such to direct $CP$ asymmetries, but some additional aspects must be considered in the scaling of uncertainties for other observables. For time-dependent $CP$ asymmetries, the difference in flavour-tagging performances (the effective tagging efficiency $Q$) should be taken into account. In the $B$-factory environment, a quality factor $Q_{\rm B-factories} \sim 30$ [@FavourTagging_BaBar:2009iva; @FavourTagging_Belle:2012iva] was achieved, while for LHCb a smaller value is used ($Q_{\rm LHCb} \sim 3$ [@FavourTagging_LHCb:2012iva]), which entails an additional factor $(Q_{\rm B-factories}/Q_{\rm LHCb})^{1/2} \sim 3.2$ in the scaling of uncertainties. For branching ratios, LHCb is not able to directly count the number of $B$ mesons produced, and it is necessary to resort to a normalisation using final states for which the branching ratio has been measured elsewhere (mainly at $B$-factories). This additional source of uncertainty is taken into account in the projection of the error. Finally, in our prospective studies, we adopt the pessimistic view of neglecting potential measurements from LHCb for modes with $\pi^0$ mesons in the final state (e.g., $B^0\rightarrow K^+\pi^-\pi^0$ and $B^+\rightarrow K^0_S\pi^+\pi^0$), as it is difficult to anticipate the evolution in the performances for $\pi^0$ reconstruction and phase space resolution.
Belle II [@Urquijo:2015qsa] expects to surpass by a factor of $\sim 50$ the total statistics collected by the $B$-factories. As the experimental environments will be very similar, we just scale the current uncertainties by this statistical factor.
Starting from the statistical uncertainties from Babar and scaling them according to the above procedure, we obtain our projections of uncertainties on physical observables, shown in Tab. \[tab:LHCbAndBelleII\], where the current uncertainties are compared with the projected ones for the first ($B$-factories combined with LHCb Run1 and Run2) and second (adding Belle II) stages described previously.
[lcc|lcc]{} Hadronic amplitudes & Magnitude & Phase ($^\circ$) & Observable & Measurement & Value\
$T^{+-}$ & 2.540 & 0.0 & ${\mathcal B}(B^0\rightarrow K^{*+}\pi^-)$ & $8.4 \pm 0.8$ & $7.1$\
$T^{00}_{\rm C}$ & 0.762 & 75.8 & ${\mathcal B}(B^0\rightarrow K^{*0}\pi^0)$ & $3.3 \pm 0.6$ & $1.6$\
$N^{0+}$ & 0.143 & 108.4 & ${\mathcal B}(B^+\rightarrow K^{*+}\pi^0)$ & $8.2 \pm 1.8$ & $8.5$\
$P^{+-}$ & 0.091 & -6.5 & ${\mathcal B}(B^+\rightarrow K^{*0}\pi^+)$ & $10.1^{+0.8}_{-0.9}$ & $10.9$\
$P_{\rm EW}$ & 0.038 & 15.2 & $A_{CP}(B^0\rightarrow K^{*+}\pi^-)$ & $-0.23 \pm 0.06$ & $-0.129$\
$P_{\rm EW}^{\rm C}$ & 0.029 & 101.9 & $A_{CP}(B^0\rightarrow K^{*0}\pi^0)$ & $-0.15 \pm 0.13$ & $+0.465$\
$\left|\frac{V_{ts}V_{tb}^*P^{+-}}{V_{us}V_{ub}^*T^{+-}}\right|$ & 1.809 & & $A_{CP}(B^+\rightarrow K^{*+}\pi^0)$ & $-0.39 \pm 0.12$ & $-0.355$\
$\left|T^{00}_{\rm C}/T^{+-}\right|$ & 0.300 & & $A_{CP}(B^+\rightarrow K^{*0}\pi^+)$ & $+0.038 \pm 0.042$ & $+0.039$\
$\left|N^{0+}/T^{00}_{\rm C}\right|$ & 0.187 & & & &\
$\left|P_{\rm EW}/P^{+-}\right|$ & 0.421 & & & &\
$\left|P_{\rm EW}/(T^{+-} + T^{00}_{\rm C})\right|/R$ & 1.009 & & & &\
$\left|P_{\rm EW}^{\rm C}/P_{\rm EW}\right|$ & 0.762 & & & &\
Observable Analysis Current uncertainty LHCb (Run1+Run2) LHCb+Belle II
------------------------------------------------------------------------------------------------------ ---------------------------------- --------------------- ------------------ ---------------
${\mathcal Re}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right]$ $B^0\rightarrow K^0_S\pi^+\pi^-$ $0.11$ $0.04$ $0.01$
${\mathcal Im}\left[ \frac{q}{p} \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right]$ $B^0\rightarrow K^0_S\pi^+\pi^-$ $0.16$ $0.11$ $0.02$
${\mathcal B}(K^{*+}\pi^-)$ $B^0\rightarrow K^0_S\pi^+\pi^-$ $0.69$ $0.32$ $0.09$
$\left| \frac{\overline{A}(K^{*-}\pi^+)}{A(K^{*+}\pi^-)} \right|$ $B^0\rightarrow K^+\pi^-\pi^0$ $0.06$ $0.06$ $0.01$
${\mathcal Re}\left[ \frac{A(K^{*0}\pi^0)}{A(K^{*+}\pi^-)} \right]$ $B^0\rightarrow K^+\pi^-\pi^0$ $0.11$ $0.11$ $0.02$
${\mathcal Im}\left[ \frac{A(K^{*0}\pi^0)}{A(K^{*+}\pi^-)} \right]$ $B^0\rightarrow K^+\pi^-\pi^0$ $0.23$ $0.23$ $0.03$
${\mathcal Re}\left[ \frac{\overline{A}(\overline{K}^{*0}\pi^0)}{\overline{A}(K^{*-}\pi^+)} \right]$ $B^0\rightarrow K^+\pi^-\pi^0$ $0.10$ $0.10$ $0.01$
${\mathcal Im}\left[ \frac{\overline{A}(\overline{K}^{*0}\pi^0)}{\overline{A}(K^{*-}\pi^+)} \right]$ $B^0\rightarrow K^+\pi^-\pi^0$ $0.30$ $0.30$ $0.04$
${\mathcal B}(K^{*0}\pi^0) $ $B^0\rightarrow K^+\pi^-\pi^0$ $0.35$ $0.35$ $0.05$
$\left| \frac{\overline{A}(\overline{K}^{*0}\pi^-)}{A(K^{*0}\pi^+)} \right|$ $B^+\rightarrow K^+\pi^-\pi^+$ $0.04$ $0.005$ $0.004$
${\mathcal B}(K^{*0}\pi^+)$ $B^+\rightarrow K^+\pi^-\pi^+$ $0.81$ $0.50$ $0.11$
$\left| \frac{\overline{A}(K^{*-}\pi^0)}{A(K^{*+}\pi^0)} \right|$ $B^+\rightarrow K^0_S\pi^+\pi^0$ $0.15$ $0.15$ $0.02$
${\mathcal Re}\left[ \frac{A(K^{*+}\pi^0)}{A(K^{*0}\pi^+)} \right]$ $B^+\rightarrow K^0_S\pi^+\pi^0$ $0.16$ $0.16$ $0.02$
${\mathcal Im}\left[ \frac{A(K^{*+}\pi^0)}{A(K^{*0}\pi^+)} \right]$ $B^+\rightarrow K^0_S\pi^+\pi^0$ $0.30$ $0.30$ $0.04$
${\mathcal Re}\left[ \frac{\overline{A}(K^{*-}\pi^0)}{\overline{A}(\overline{K}^{*0}\pi^-)} \right]$ $B^+\rightarrow K^0_S\pi^+\pi^0$ $0.21$ $0.21$ $0.03$
${\mathcal Im}\left[ \frac{\overline{A}(K^{*-}\pi^0)}{\overline{A}(\overline{K}^{*0}\pi^-)} \right]$ $B^+\rightarrow K^0_S\pi^+\pi^0$ $0.13$ $0.13$ $0.02$
${\mathcal B}(K^{*+}\pi^0)$ $B^+\rightarrow K^0_S\pi^+\pi^0$ $0.92$ $0.92$ $0.13$
[99]{} A. J. Bevan [*et al.*]{} \[ and Belle Collaborations\], Eur. Phys. J. C [**74**]{} (2014) 3026 doi:10.1140/epjc/s10052-014-3026-9 \[arXiv:1406.6311 \[hep-ex\]\]. R. Aaij [*et al.*]{} \[LHCb Collaboration\], Eur. Phys. J. C [**73**]{} (2013) no.4, 2373 doi:10.1140/epjc/s10052-013-2373-2 \[arXiv:1208.3355 \[hep-ex\]\]. N. Cabibbo, Phys. Rev. Lett. [**10**]{} (1963) 531. doi:10.1103/PhysRevLett.10.531 M. Kobayashi and T. Maskawa, Prog. Theor. Phys. [**49**]{} (1973) 652. doi:10.1143/PTP.49.652 O. Deschamps [*et al.*]{}, work in progress.
J. Charles [*et al.*]{} \[CKMfitter Group\], Eur. Phys. J. C [**41**]{} (2005) 1 \[arXiv:hep-ph/0406184\]. Updates and numerical results on the CKMfitter group web site: `http://ckmfitter.in2p3.fr/`
J. Charles [*et al.*]{}, Phys. Rev. D [**91**]{} (2015) no.7, 073007 doi:10.1103/PhysRevD.91.073007 \[arXiv:1501.05013 \[hep-ph\]\]. P. Koppenburg and S. Descotes-Genon, arXiv:1702.08834 \[hep-ex\]. J. Charles, S. Descotes-Genon, Z. Ligeti, S. Monteil, M. Papucci and K. Trabelsi, Phys. Rev. D [**89**]{} (2014) no.3, 033016 doi:10.1103/PhysRevD.89.033016 \[arXiv:1309.2293 \[hep-ph\]\]. A. Lenz, U. Nierste, J. Charles, S. Descotes-Genon, H. Lacker, S. Monteil, V. Niess and S. T’Jampens, Phys. Rev. D [**86**]{} (2012) 033008 doi:10.1103/PhysRevD.86.033008 \[arXiv:1203.0238 \[hep-ph\]\]. A. Lenz [*et al.*]{}, Phys. Rev. D [**83**]{} (2011) 036004 doi:10.1103/PhysRevD.83.036004 \[arXiv:1008.1593 \[hep-ph\]\]. O. Deschamps, S. Descotes-Genon, S. Monteil, V. Niess, S. T’Jampens and V. Tisserand, Phys. Rev. D [**82**]{} (2010) 073012 doi:10.1103/PhysRevD.82.073012 \[arXiv:0907.5135 \[hep-ph\]\].
M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Phys. Rev. Lett. [**83**]{} (1999) 1914 doi:10.1103/PhysRevLett.83.1914 \[hep-ph/9905312\]. M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Nucl. Phys. B [**591**]{} (2000) 313 doi:10.1016/S0550-3213(00)00559-9 \[hep-ph/0006124\]. M. Beneke and M. Neubert, Nucl. Phys. B [**675**]{} (2003) 333 doi:10.1016/j.nuclphysb.2003.09.026 \[hep-ph/0308039\]. M. Beneke, J. Rohrer and D. Yang, Nucl. Phys. B [**774**]{} (2007) 64 doi:10.1016/j.nuclphysb.2007.03.020 \[hep-ph/0612290\]. H. n. Li, Phys. Rev. D [**66**]{} (2002) 094010 doi:10.1103/PhysRevD.66.094010 \[hep-ph/0102013\]. H. n. Li and K. Ukai, Phys. Lett. B [**555**]{} (2003) 197 doi:10.1016/S0370-2693(03)00049-2 \[hep-ph/0211272\]. H. n. Li, Prog. Part. Nucl. Phys. [**51**]{} (2003) 85 doi:10.1016/S0146-6410(03)90013-5 \[hep-ph/0303116\]. A. Ali, G. Kramer, Y. Li, C. D. Lu, Y. L. Shen, W. Wang and Y. M. Wang, Phys. Rev. D [**76**]{} (2007) 074018 doi:10.1103/PhysRevD.76.074018 \[hep-ph/0703162 \[HEP-PH\]\]. H. n. Li, CERN Yellow Report CERN-2014-001, pp.95-135 doi:10.5170/CERN-2014-001.95 \[arXiv:1406.7689 \[hep-ph\]\]. W. F. Wang and H. n. Li, Phys. Lett. B [**763**]{} (2016) 29 doi:10.1016/j.physletb.2016.10.026 \[arXiv:1609.04614 \[hep-ph\]\]. C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. D [**67**]{} (2003) 071502 doi:10.1103/PhysRevD.67.071502 \[hep-ph/0211069\]. M. Beneke and T. Feldmann, Nucl. Phys. B [**685**]{} (2004) 249 doi:10.1016/j.nuclphysb.2004.02.033 \[hep-ph/0311335\]. C. W. Bauer, D. Pirjol, I. Z. Rothstein and I. W. Stewart, Phys. Rev. D [**70**]{} (2004) 054015 doi:10.1103/PhysRevD.70.054015 \[hep-ph/0401188\]. C. W. Bauer, I. Z. Rothstein and I. W. Stewart, Phys. Rev. D [**74**]{} (2006) 034010 doi:10.1103/PhysRevD.74.034010 \[hep-ph/0510241\]. T. Becher, A. Broggio and A. Ferroglia, Lect. Notes Phys. [**896**]{} (2015) doi:10.1007/978-3-319-14848-9 \[arXiv:1410.1892 \[hep-ph\]\]. S. Descotes-Genon and C. T. Sachrajda, Nucl. Phys. B [**625**]{} (2002) 239 doi:10.1016/S0550-3213(02)00017-2 \[hep-ph/0109260\]. M. Ciuchini, E. Franco, G. Martinelli, M. Pierini and L. Silvestrini, Phys. Lett. B [**515**]{} (2001) 33 doi:10.1016/S0370-2693(01)00700-6 \[hep-ph/0104126\]. M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Phys. Rev. D [**72**]{} (2005) 098501 doi:10.1103/PhysRevD.72.098501 \[hep-ph/0411171\]. A. V. Manohar and I. W. Stewart, Phys. Rev. D [**76**]{} (2007) 074002 doi:10.1103/PhysRevD.76.074002 \[hep-ph/0605001\]. H. n. Li and S. Mishima, Phys. Rev. D [**83**]{} (2011) 034023 doi:10.1103/PhysRevD.83.034023 \[arXiv:0901.1272 \[hep-ph\]\]. F. Feng, J. P. Ma and Q. Wang, arXiv:0901.2965 \[hep-ph\]. M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Eur. Phys. J. C [**61**]{} (2009) 439 doi:10.1140/epjc/s10052-009-1028-9 \[arXiv:0902.4446 \[hep-ph\]\]. T. Becher and G. Bell, Phys. Lett. B [**713**]{} (2012) 41 doi:10.1016/j.physletb.2012.05.016 \[arXiv:1112.3907 \[hep-ph\]\]. M. Beneke, Nucl. Part. Phys. Proc. [**261-262**]{} (2015) 311 doi:10.1016/j.nuclphysbps.2015.03.021 \[arXiv:1501.07374 \[hep-ph\]\]. B. Aubert [*et al.*]{} \[ Collaboration\], Phys. Rev. D [**80**]{} (2009) 112001 doi:10.1103/PhysRevD.80.112001 \[arXiv:0905.3615 \[hep-ex\]\]. B. Aubert [*et al.*]{} \[ Collaboration\], Phys. Rev. D [**78**]{}, 012004 (2008) doi:10.1103/PhysRevD.78.012004 \[arXiv:0803.4451 \[hep-ex\]\]. J. P. Lees [*et al.*]{} \[ Collaboration\], Phys. Rev. D [**83**]{}, 112010 (2011) doi:10.1103/PhysRevD.83.112010 \[arXiv:1105.0125 \[hep-ex\]\]. J. P. Lees [*et al.*]{} \[ Collaboration\], arXiv:1501.00705 \[hep-ex\]. J. P. Lees [*et al.*]{} \[ Collaboration\], Phys. Rev. D [**84**]{}, 092007 (2011) doi:10.1103/PhysRevD.84.092007 arXiv:1109.0143 \[hep-ex\].
A. Garmash [*et al.*]{} \[Belle Collaboration\], Phys. Rev. Letters [**96**]{} (2006) 251803 doi:10.1103/PhysRevD.79.072004 \[arXiv:0811.3665 \[hep-ex\]\]. J. Dalseno [*et al.*]{} \[Belle Collaboration\], Phys. Rev. D [**79**]{} (2009) 072004 doi:10.1103/PhysRevD.79.072004 \[arXiv:0811.3665 \[hep-ex\]\].
B. Bhattacharya, M. Gronau and J. L. Rosner, Phys. Lett. B [**726**]{} (2013) 337 doi:10.1016/j.physletb.2013.08.062 \[arXiv:1306.2625 \[hep-ph\]\]. B. Bhattacharya and D. London, JHEP [**1504**]{} (2015) 154 doi:10.1007/JHEP04(2015)154 \[arXiv:1503.00737 \[hep-ph\]\].
B. Bhattacharya, M. Gronau, M. Imbeault, D. London and J. L. Rosner, Phys. Rev. D [**89**]{} (2014) no.7, 074043 doi:10.1103/PhysRevD.89.074043 \[arXiv:1402.2909 \[hep-ph\]\], and references therein J. H. Alvarenga Nogueira [*et al.*]{}, arXiv:1605.03889 \[hep-ex\]. K. Abe [*et al.*]{} \[Belle Collaboration\], arXiv:0708.1845 \[hep-ex\]. B. Aubert [*et al.*]{} \[BaBar Collaboration\], Phys. Rev. D [**76**]{} (2007) 071101 doi:10.1103/PhysRevD.76.071101 \[hep-ex/0702010\]. L. A. Pérez Pérez, “Time-Dependent Amplitude Analysis of $B^0 \to K_s \pi^+ \pi^-$ decays with the Experiment and constraints on the [CKM]{} matrix using the $B \to K^*\pi$ and $B \to \rho K$ modes,” TEL-00379188. Y. Nir and H. R. Quinn, Phys. Rev. Lett. [**67**]{} (1991) 541. doi:10.1103/PhysRevLett.67.541 M. Gronau and J. Zupan, Phys. Rev. D [**71**]{} (2005) 074017 doi:10.1103/PhysRevD.71.074017 \[hep-ph/0502139\]. F. J. Botella and J. P. Silva, Phys. Rev. D [**71**]{} (2005) 094008 doi:10.1103/PhysRevD.71.094008 \[hep-ph/0503136\]. M. Beneke and S. Jäger, Nucl. Phys. B [**768**]{} (2007) 51 doi:10.1016/j.nuclphysb.2007.01.016 \[hep-ph/0610322\].
G. Bell and V. Pilipp, Phys. Rev. D [**80**]{} (2009) 054024 doi:10.1103/PhysRevD.80.054024 \[arXiv:0907.1016 \[hep-ph\]\]. G. Bell, M. Beneke, T. Huber and X. Q. Li, Phys. Lett. B [**750**]{} (2015) 348 doi:10.1016/j.physletb.2015.09.037 \[arXiv:1507.03700 \[hep-ph\]\].
F. J. Botella, D. London and J. P. Silva, Phys. Rev. D [**73**]{} (2006) 071501 doi:10.1103/PhysRevD.73.071501 \[hep-ph/0602060\].
M. Gronau and D. London, Phys. Rev. Lett. [**65**]{} (1990) 3381. doi:10.1103/PhysRevLett.65.3381 A. J. Buras and R. Fleischer, Eur. Phys. J. C [**11**]{} (1999) 93 doi:10.1007/s100529900201, 10.1007/s100520050617 \[hep-ph/9810260\]. M. Neubert and J. L. Rosner, Phys. Lett. B [**441**]{} (1998) 403 doi:10.1016/S0370-2693(98)01194-0 \[hep-ph/9808493\]. M. Neubert and J. L. Rosner, Phys. Rev. Lett. [**81**]{} (1998) 5076 doi:10.1103/PhysRevLett.81.5076 \[hep-ph/9809311\]. M. Gronau, Phys. Rev. Lett. [**91**]{} (2003) 139101 doi:10.1103/PhysRevLett.91.139101 \[hep-ph/0305144\]. M. Ciuchini, M. Pierini and L. Silvestrini, Phys. Rev. D [**74**]{} (2006) 051301 doi:10.1103/PhysRevD.74.051301 \[hep-ph/0601233\]. M. Gronau, D. Pirjol, A. Soni and J. Zupan, Phys. Rev. D [**75**]{} (2007) 014002 doi:10.1103/PhysRevD.75.014002 \[hep-ph/0608243\]. Y. Amhis [*et al.*]{}, arXiv:1612.07233 \[hep-ex\]. J. Charles, S. Descotes-Genon, V. Niess and L. Vale Silva, arXiv:1611.04768 \[hep-ph\]. R. Aaij [*et al.*]{} \[LHCb Collaboration\], Phys. Rev. D [**90**]{}, no. 11, 112004 (2014) doi:10.1103/PhysRevD.90.112004 \[arXiv:1408.5373 \[hep-ex\]\].
P. Urquijo, Nucl. Part. Phys. Proc. [**263-264**]{}, 15 (2015). doi:10.1016/j.nuclphysbps.2015.04.004. Belle II web page: https://www.belle2.org/
R. Aaij [*et al.*]{} \[LHCb Collaboration\], Phys. Rev. Lett. [**111**]{} (2013) 101801 \[arXiv:1306.1246 \[hep-ex\]\].
LHCb-CONF-2012-023, July 6, 2012 LHCb-CONF-2012-026, July 11, 2012 B. Aubert [*et al.*]{} \[Collaboration\], Phys. Rev. D [**79**]{}, (2009) 072009 doi:10.1103/PhysRevD.79.072009 \[arXiv:0902.1708 \[hep-ex\]\]. I. Adachi [*et al.*]{} \[Belle Collaboration\], Phys. Rev. Lett. [**108**]{}, 171802 (2012) doi:10.1103/PhysRevLett.108.171802 \[arXiv:1201.4643 \[hep-ex\]\]. P. Ball and R. Zwicky, Phys. Rev. D [**71**]{} (2005) 014029 doi:10.1103/PhysRevD.71.014029 \[hep-ph/0412079\]. A. Bharucha, D. M. Straub and R. Zwicky, JHEP [**1608**]{} (2016) 098 doi:10.1007/JHEP08(2016)098 \[arXiv:1503.05534 \[hep-ph\]\].
V. M. Braun, D. Y. Ivanov and G. P. Korchemsky, Phys. Rev. D [**69**]{} (2004) 034014 doi:10.1103/PhysRevD.69.034014 \[hep-ph/0309330\].
S. Aoki [*et al.*]{}, Eur. Phys. J. C [**77**]{} (2017) no.2, 112 doi:10.1140/epjc/s10052-016-4509-7 \[arXiv:1607.00299 \[hep-lat\]\].
G. Bell, talk given at the workshop *Future Challenges in Non-Leptonic B Decays: Theory and Experiment*, Bad Honnef (Germany), 10-12 February 2016 \[\].
G. Bell, Nucl. Phys. B [**795**]{} (2008) 1 doi:10.1016/j.nuclphysb.2007.09.006 \[arXiv:0705.3127 \[hep-ph\]\]. G. Bell, Nucl. Phys. B [**822**]{} (2009) 172 doi:10.1016/j.nuclphysb.2009.07.012 \[arXiv:0902.1915 \[hep-ph\]\]. M. Beneke, T. Huber and X. Q. Li, Nucl. Phys. B [**832**]{} (2010) 109 doi:10.1016/j.nuclphysb.2010.02.002 \[arXiv:0911.3655 \[hep-ph\]\].
| ArXiv |
---
abstract: 'We report a femtosecond response in photoinduced magnetization rotation in the ferromagnetic semiconductor GaMnAs, which allows for detection of a four-state magnetic memory at the femtosecond time scale. The temporal profile of this cooperative magnetization rotation exhibits a discontinuity that reveals two distinct temporal regimes, marked by the transition from a highly non-equilibrium, carrier-mediated regime within the first 200 fs, to a thermal, lattice-heating picosecond regime.'
author:
- 'J. Wang'
- 'I. Cotoros'
- 'X. Liu'
- 'J. Chovan'
- 'J. K. Furdyna'
- 'I. E. Perakis'
- 'D. S. Chemla'
title: |
Memory Effect in the Photoinduced Femtosecond Rotation of Magnetization\
in the Ferromagnetic Semiconductor GaMnAs
---
![(Color online) Static magnetic memory. (a)-(b): Sweeping a slightly tilted [*B*]{} field (5$^o$ from the [*Z*]{}-axis and 33$^o$ from the [*X*]{}-axis) up (dashed line) and down (solid line) leads to consecutive 90$^o$ magnetization switchings between the [*XZ*]{} and [*YZ*]{} planes, manifesting as a “major" hysteresis loop in the Hall magneto-resistivity. (c)-(d): “Minor" hysteresis loop with [*B*]{} field sweeping in the vicinity of 0T. The magnetic memory state X$-$(0) or Y$+$(0) is parallel to one of the easy axis directions in the [*XY*]{} plane. ](Fig1f.eps)
\[dep\]
Magnetic materials displaying [*carrier-mediated*]{} ferromagnetic order offer fascinating opportunities for non-thermal, potentially [*femtosecond*]{} manipulation of magnetism. A model system of such materials is Mn-doped III-V ferromagnetic semiconductors that have received a lot of attention lately [@ohno1998]. On the one hand, their magnetic properties display a strong response to excitation with light or electrical gate and current via carrier density tuning [@Koshiharaetal97PRL; @ohnoetal2000; @wangetalPRL2007]. On the other hand, the strong coupling ($\sim$1 eV in GaMnAs) between carriers (holes) and Mn ions inherent in carrier-mediated ferromagnetism could enable a [*femtosecond*]{} cooperative magnetic response induced by photoexcited carriers. Indeed, the existence of a very early non-equilibrium, non-thermal femtosecond regime of collective spin rotation in (III,Mn)Vs has been predicted theoretically [@ChovanetalPRL2006]. In addition, a coherent driving mechanism for femtosecond spin rotation via [*virtual*]{} excitations has also been recently demonstrated in antiferro- and ferri-magnets [@coherent]. Nevertheless, all prior studies of photoexcited magnetization rotation in ferromagnetic (III,Mn)Vs showed dynamics on the few picosecond timescale, which accesses the quasi-equilibrium, quasi-thermal, lattice-heating regime [@psRotationGaMnAs]. Up to now in these materials, the main observation on the femtosecond time scale has been photoinduced demagnetization [@wangetalPRL2005; @wangetalreview2006; @wang2008; @Cywinski_PRB07].
Custom-designed (III,Mn)V hetero- and nano-structures show rich magnetic memory effects. One prominent example is GaMnAs-based four-state magnetic memory, where “giant” magneto-optical and magneto-transport effects allow for ultrasensitive magnetic memory readout [@fourstate]. However, all detection schemes demonstrated so far have been static measurements. Achieving an understanding of collective magnetic phenomena on the femtosecond time scale is critical for terahertz detection of magnetic memory and therefore essential for developing realistic “spintronic” devices and large-scale functional systems.
In this Letter, we report on photoinduced [*femtosecond*]{} collective magnetization rotation that allows for femtosecond detection of magnetic memory in GaMnAs. Our time-resolved magneto-optical Kerr effect (MOKE) technique directly reveals a photoinduced four-state magnetic hysteresis via a quasi-instantaneous magnetization rotation. We observe for the first time a distinct initial temporal regime of the magnetization rotation within the first $\sim$200 fs, during the photoexcitation and highly non-equilibrium, non-thermal carrier redistribution times. We attribute the existence of such a regime to a [*carrier-mediated*]{} effective magnetic field pulse, arising without assistance from either lattice heating or demagnetization.
The main sample studied was grown by low-temperature molecular beam epitaxy (MBE), and consisted of a 73-nm Ga$_{0.925}$Mn$_{0.075}$As layer on a 10 nm GaAs buffer layer and a semi-insulating GaAs \[100\] substrate. The Curie temperature and hole density were 77 K and $3 \times 10^{20}$ cm$^{-3}$, respectively. As shown in Fig. 1, our structure exhibits a four-state magnetic memory functionality. By sweeping an external magnetic field B aligned nearly perpendicularly to the sample normal, with small components in both the [*X*]{} and [*Y*]{} directions in the sample plane, one can sequentially access four magnetic states, X$+\rightarrow$Y$-\rightarrow$X$-\rightarrow$Y$+$, via abrupt 90$^o$ magnetization ($\mathbf{M}$) switchings between the [*XZ*]{} and [*YZ*]{} planes \[Fig. 1(a)\]. In these magnetic states, $\mathbf{M}$ aligns along a direction arising as a combination of the external B field and the anisotropy fields, which point along the in-plane easy axes \[100\] and \[010\]. The multistep magnetic switchings manifest themselves as abrupt jumps in the four-state hysteresis in the Hall magneto-resistivity $\rho _{Hall}$ \[Fig. 1(b)\] (planar Hall effect [@fourstate]). The continuous slopes of $\rho _{Hall}$ indicate a coherent out-of-plane $\mathbf{M}$ rotation during the perpendicular magnetization reversal (anomalous Hall effect [@ohno1998]). Fig. 1(c)-(d) show the B scans in the vicinity of 0T, with the field turning points between the coercivity fields, i.e., $B_{c1}<\left|B\right|<B_{c2}$. This leads to a “minor” hysteresis loop, accessesing two magnetic memory states at $B=$0T: X$-$(0) and Y$+$(0).
We now turn to the transient magnetic phenonmena. We performed time-resolved MOKE spectroscopy [@wangetalreview2006] using 100 fs laser pulses. The linearly polarized ($\sim$12 degree from the crystal axis \[100\]) UV pump beam was chosen at 3.1 eV, with peak fluence $\sim$ 10$\mu$J/cm$^2$. A NIR beam at 1.55 eV, kept nearly perpendicular to the sample ($\sim$ 0.65 degree from the normal), was used as probe. The signal measured in this polar geometry reflects the out-of-plane magnetization component, M$_z$.
![(Color) Photoinduced femtosecond four-state magnetic hysteresis. (a) B field scans of $\triangle \theta _{k}$ at 5K for time delays $\triangle t=$ -1 ps, 600 fs, and 3.3 ps. The traces are vertically offset for clarity. Inset (left): temporal profiles of normalized Kerr ($\theta _{k}$) and ellipticity ($\eta _{k}$) angle changes at 1.0T; Inset (right): static magnetization curve at 5K ($\sim$4 mrad), measured in the same experimental condition (but without the pump pulse). (b) Temporal profiles of photoinduced $\triangle \theta _{k}$ for the four magnetic states. Shaded area: pump–probe cross–correlation.[]{data-label="mag-dep"}](Fig2f.eps)
Fig. 2(a) shows the B field scan traces of the photoinduced change, $\Delta\theta_K$, in the Kerr rotation angle at three time delays, $\triangle t=$ -1 ps, 600 fs, and 3.3 ps. The magnetic origin of this femtosecond MOKE response [@KoopmansetAl00PRL] was confirmed by control measurements showing a complete overlap of the pump–induced rotation ($\theta _{k}$) and ellipticity ($\eta _{k}$) changes \[left inset, Fig. 2(a)\]. $\Delta\theta_K$ is negligible at $\triangle t=$-1 ps. However, a mere $\triangle t=$600 fs after photoexcitation, a clear photoinduced four-state magnetic hysteresis is observed in the magnetic field dependence of $\Delta\theta_K$ (and therefore $\Delta M_z$), with four abrupt switchings at $\left|B_{c1}\right|=$0.074T and $\left|B_{c1}\right|=$0.33T due to the magnetic memory effects. As marked by the arrows in Fig. 2(a), the four magnetic states X$+$, X$-$, Y$-$, Y$+$ for $\left|B\right|=$0.2T give different photoinduced $\Delta\theta_K$. It is critical to note that the steady-state MOKE curve, i.e. $\theta_K$ without pump field, doesn’t show any sign of magnetic switching or memory behavior \[right inset, Fig. 2(a)\]; these arise from the pump photoexcitation. The B field scans also show a saturation behavior at $\left|B\right|>$0.6T, to be discussed later. We note that the photo–induced hysteresis loops at $\triangle t=$3.3 ps and 600 fs sustain similar shapes, with only slightly larger amplitudes at 3.3 ps. This observation confirms that the dynamic magnetic processes responsible for the abrupt switchings occur on a femtosecond time scale. Fig. 2(b) shows the photoinduced $\Delta\theta_K$ dynamics for the four initial states X$+$, X$-$, Y$-$, and Y$+$. An extremely fast $\Delta\theta_K$ develops within 200 fs, with magnitude and sign that distinctly differ, depending on the initially prepared state, consistent with Fig.2 (a). The substantial difference in $\Delta\theta_K$ under the same B field - for instance between the X+ and Y+ states - shows that the magnetic dynamics is not due to simple demagnetization [@BeaurepaireetalPRL96; @wangetalPRL2005; @wangetalreview2006].
The photoinduced dynamics of the zero–B field memory states \[Fig. 1(c)\] elucidates the salient features of the femtosecond magnetic processes. Fig. 3(a) shows the temporal profiles of the photo-induced $\Delta\theta_K$ for X$-$(0) and Y$+$(0) initial states. Since the initial magnetization vector lies within the sample plane, $\Delta \theta _{k}$ in the first 200 fs reveals an out–of–plane spin rotation, with negligible contribution from demagnetization. More intriguingly, the $\mathbf{M}$ in X$-$ and Y$+$ initial states rotates to different [*Z*]{}-axis directions, as illustrated in Fig. 3(b). This leads to opposite signs of the photoinduced signals and is responsible for the four-state magnetic switchings. Furthermore, the observation of an initial discontinuity in the temporal profiles of the $\mathbf{M}$ tilt reveals [*two distinct temporal regimes*]{}, marked in Fig. 3(a): a substantial magnetization rotation concludes after the first 200 fs and is followed by a [*much slower*]{} rotation change afterwards (over 100’s of ps).
![(Color online) (a) Photo-induced $\Delta \theta _{k}$ for two in-plane magnetic memory states, shown together with the pump-probe cross-correlation (shaded). The opposite, out–of–plane $\mathbf{M}$ rotations for the X$-$(0) and Y$+$(0) are illustrated in (b).[]{data-label="power"}](Fig3f.eps)
We now discuss the origin of the observed femtosecond magnetization rotation. In the previously held picture of light–induced magnetization rotation in ferromagnets, the photoexcitation alters the anisotropy fields via quasi-equilibrium mechanisms, such as heating of the lattice (magneto-crystalline anisotropy) or heating of the spins (shape anisotropy) [@heating]. Since the in-plane magnetic memory states of Fig. 1(c) have negligible shape anisotropy, a significant B field within the standard picture can only occur on a time scale of several picoseconds via the lattice heating mechanism. However, it has been shown theoretically [@ChovanetalPRL2006; @ChovanetalPRB2008] that the Mn spin in GaMnAs can respond quasi-instantaneously to a femtosecond effective magnetic field pulse generated by hole spins via nonlinear optical processes assisted by interactions. This light–induced B field pulse may be thought of as a femtosecond modification of the magnetic anisotropy fields. In the realistic system, one needs to also treat microscopically the transient magnetic anisotropy changes, due to the complex valence bands and highly non–thermal hole populations in the femtosecond regime, which drastically affect the photoexcited carrier spin. Due to the [*hole-mediated*]{} effective exchange interaction between Mn spins, the anisotropy fields in GaMnAs result from the coupling of several valence bands by the [*spin-orbit interaction*]{} and depend on the transient hole distribution and coherences between different bands [@ChovanetalPRB2008]. In the static case, recent experimental [@anisotropy-exp] and theoretical [@anisotropy-the] investigations have shown that increasing the hole density significantly reduces the cubic anisotropy (K$_c$) along the \[100\] direction, while enhancing the uniaxial anisotropy (K$_u$) along \[1-10\]. One therefore expects that the photoexcited hole population turns on an effective magnetic field pulse ($\Delta B_{c}$) along the \[1-10\] direction \[Fig. 4(a)\]. This photo-triggered $\Delta B_{c}$ then exerts a spin torque on the $\mathbf M$ vector, $\Delta \overrightarrow B_{c}\times
\overrightarrow { M}$, and pulls it away from the sample plane. The directions of these spin torques for the X$-$(0) and Y$+$(0) states are opposite, leading to different $\mathbf M$ rotation paths \[Fig. 3\]. Since this mechanism is mediated by the the non–thermal holes, the appearance of $\Delta B_{c}$ is quasi-instantaneous, limited only by the pulse duration of $\sim$100 fs [@ChovanetalPRB2008]. This femtosecond magnetic anisotropy contribution from the non–thermal photoexcited carriers should be contrasted to the quasi–thermal contribution, arising from, e.g., the transient lattice temperature elevation on the picosecond time scale [@psRotationGaMnAs].
Next we turn to the origin of the discontinuity that reveals the [*two temporal regimes*]{} in the collective magnetization rotation \[Fig. 3\]. The quick termination of the initial magnetization tilt implies that the effective $\Delta B_{c}$ pulse induced by the photoexcitation decays within the first hundreds of femtoseconds. The photoexcitation of a large (as compared to the ground state anisotropy field) $\Delta B_{c}$ requires an extensive [*non-thermal*]{} distribution of transient holes in [*the high momentum states*]{} of the valence band [@anisotropy-the]. This is due to the large spin anisotropy of these hole states, empty in the unexcited sample, via their strong spin-orbit interaction. In our experiment, immediately following photoexcitation at 3.1 eV, a large density of transient holes distribute themselves over almost half of the Brillouin zone along the L\[111\] direction. The Mn–hole spin exchange interaction is also believed to be enhanced along \[111\] due to strong p-d orbital hybridization [@BurchPRB2004]. Consequently, these photoexcited holes contribute strongly to the magnetic anisotropy fields. The following rapid relaxation and thermalization of the high momentum holes, due to carrier-carrier and carrier-phonon scattering, reduce $\Delta B_{c}$ within a few hundred femtoseconds. The subsequent picosecond magnetization rotation process arises from the change in magnetic anisotropy induced by the lattice temperature elevation. Our results reveal a complex scenario of collective spin rotation, marked by the transition from a non-equilibrium, carrier-mediated regime ($<$200 fs) to a thermal, lattice-heating regime on the ps time scale.
![(Color online) (a) Schematics of the photoexcited carrier–induced anisotropy field $\Delta B_{c}$. (b) Simulations of $\Delta M_{z}/M_0$ for the two magnetic memory states. Parameters used in the calculation are $K_{c}=1.198\cdot 10^{-2}meV$, $K_{u}=0.373\cdot 10^{-2}meV$, $K_{3}=0.746\cdot 10^{-2}meV$, $T_{1}=330$fs and $3\% $ of photoexcited carriers. (c) Schematics of the photoinduced M$_{z}$ for the X$-$, X$+$, Y$-$ and Y$+$ states at $\left|B\right|=$0.2T. []{data-label="temperature"}](Fig4f.eps)
We modelled the transient anisotropy phenomenologically by deriving $\Delta B_{c}$ from the magnetic free energy, $$E_{anis}=-{K_{c}\over S^{4}}
S_{x}^{2}S_{y}^{2}+{K_{u}\over S^{2}}
S_{x}^{2}+{K_{3}\over S^{2}}
S_{z}^{2},$$ describing cubic (K$_c$) and uniaxial (K$_u$) contributions, and added a time–dependent modification of $K_{c}/K_{u}$ due to the strongly anisotropic photoexcited hole states [@ChovanetalPRB2008]. The corresponding contribution to the Mn spin equation of motion is $\partial
_{t}{\bf S}={\bf S}\times {\bf H}_{anis}$, where ${\bf H}_{anis}=-
{\partial E_{anis}\over
\partial {\bf S}}$. The light–induced change in the magnitude of $K_{c}/K_{u}$ increases during the pulse and then decreases with the energy relaxation time (T$_1$) of the high–momentum photoexcited holes. The results of our calculation are shown in Fig. 4(b), which gives a similar time dependence of the normalized $\Delta M_{z}$, with magnitude $\sim$ 0.4$\%$ of the total magnetization M$_0$ ($\sim$4 mrad at 5K), which compares well with the experiment.
Finally, Fig. 4(c) illustrates the femtosecond detection of the four-state magnetic memory shown in Fig. 2. By incorporating both the photo-induced rotation (red arrows) and the demagnetization (green arrows) effects, we can visualize the different M$_z$ changes for the four magnetic states, consistent with our observation. Demagnetization results in the high field saturation behaviour observed in Fig. 2(a). For $\left|B\right|>0.60T$, $\mathbf M$ is aligned mostly along the sample normal. Then the photo-induced signals arise from the decrease in the $\mathbf M$ amplitude, which is more or less constant with respect to the field.
In conclusion, we report on the femtosecond magnetic response of photoinduced magnetization rotation in GaMnAs, which allows for femtosecond detection of four-state magnetic memory. Our observations unequivocally identify a [*non-thermal, carrier-mediated*]{} mechanism of magnetization rotation, relevant only in the [*femtosecond*]{} regime, without assistance of either lattice heating or demagnetization. This femtosecond cooperative magnetic phenomenon may represent an as-yet-undiscovered universal principle in all carrier-mediated ferromagnetic materials - a class of rapidly emerging “multi-functional" materials with significant potential for future applications, e.g., the oxides with promise of far above room temperature Curie temperature.
This work was supported by the Office of Basic Energy Sciences of the US Department of Energy under Contract No. DE-AC02-05CH11231, by the National Science Foundation DMR-0603752, and by the EU STREP program HYSWITCH.
[10]{} bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
H. Ohno, Science [**281**]{}, 951 (1998).
., ****, ().
H. Ohno [*et al*]{}., Nature [**408**]{}, 944 (2000). ., ****, ().
., ****, (); ****, ().
A. V. Kimel [*et al*]{}., Nature [**435**]{}, 655 (2005); F. Hansteen [*et al*]{}, Phys. Rev. Lett. 95, 047402 (2005).
J. Qi, [*et al*]{}., App. Phys. Lett. [**91**]{}, 112506 (2007); D. M. Wang [*et al*]{}., Phys. Rev. B [**75**]{}, 233308 (2007); Y. Hashimoto [*et al*]{}., Phys. Rev. Lett. [**100**]{}, 067202 (2008)
., ****, ().
., ****, ().
J. Wang [*et al*]{}., arXiv:0804.0451 (2008).
L. Cywi[ń]{}ski [*et al*]{}., ****, ().
H. Tang [*et al*]{}., Phys. Rev. Lett. [**90**]{}, 107201 (2003). ., ****, ().
E. Beaurepaire [*et al*]{}., Phys. Rev. Lett. [**76**]{}, 4250 (1996).
M. V. Kampen [*et al*]{}, Phys. Rev. Lett. [**88**]{}, 227201 (2002); M. Vomir [*et al*]{}, Phys. Rev. Lett. 94, 237601 (2005).
J. Chovan and I. E. Perakis, Phys. Rev. B. 77, 085321 (2008); and unpublished.
X. Liu [*et al*]{}., Phys. Rev. B. 71, 035307 (2005); K. Hamaya [*et al*]{}., Phys. Rev. B 74, 045201 (2006)
T. Dietl [*et al*]{}., Phys. Rev. B. 63, 195205 (2001); M. Abolfath [*et al*]{}., Phys. Rev. B 63, 055418 (2001)
K. S. Burch [*et al*]{}., Phys. Rev. B 70, 205208 (2004)
| ArXiv |
---
abstract: 'In this paper, a Sturm-Liouville boundary value problem equiped with conformable fractional derivates is considered. We give some uniqueness theorems for the solutions of inverse problems according to the Weyl function, two given spectra and classical spectral data. We also study on half-inverse problem and prove a Hochstadt and Lieberman-type theorem.'
author:
- 'A. Sinan Ozkan'
- İbrahim Adalar
title: 'Inverse problems for a conformable fractional Sturm-Liouville operator'
---
**Introduction**
=================
Inverse spectral problems consist in recovering the coefficients of an operator from some given data; for example Weyl function, spectral function, nodal points and some special sequences which consist of some spectral values. Various inverse problems for the classical Sturm-Liouville operator have been studied for about ninety years (see [@Ambar], [@Borg]-[horv]{}, [@lev], [@levitan], [@marc], [@marc2], [@troos], [@ozk] and the references therein). Since these kinds of problems appear in mathematical physics, mechanics, electronics, geophysics and other branches of natural sciences the literature on this area is vast.
Fractional derivative which is as old as calculus appears by a question of L’Hospital to Leibniz in 1695. He asked what does it mean $\frac{d^{n}f}{%
dx^{n}}$ if $n=1/2$. Later on, many researchers tried to give a definition of a fractional derivative. Most of them used an integral form for the fractional derivative (see [@mil], [@old]). However, almost all of them fail to satisfy some of the basic properties owned by usual derivatives, for example chain rule, the product rule, mean value theorem and etc. In 2014, the authors Khalil et al. introduced a new simple well-behaved definition of the fractional derivative called conformable fractional derivative [@hal]. One year later, Abdeljawad gave the fractional versions of some important concepts e.g. chain rule, exponential functions, Gronwall’s inequality, integration by parts, Taylor power series expansions and etc [@abd]. Also, other basic properties on conformable derivative can be found in [@atan]. It seems to satisfy all the requirements of the standard derivative. Because of its effectiveness and applicability, conformable derivative has received a lot of attention and has applied quickly to various areas.
In recent years, some new fractional Sturm-Liouville problems have been studied (see [@Bp], [@al], [@kro], [@kli], [@riv]). These problems appear in various branches of natural sciences (see [@bal], [@main], [@mon], [@pal], [@sil]). Although the inverse Sturm-Liouville problems with classical derivation are studied extensively, there is only one study about this subject with conformable fractional derivation. Mortazaasl and Akbarfam gave a solution of inverse nodal problem for conformable fractional Sturm-Liouville operator in [@Oz].
In the present paper, we consider a conformable fractional Sturm-Liouville boundary value problem and give uniqueness theorems for the solution of inverse problem according to the Weyl function, two eigenvalues-sets and the sequences which consist of eigenvalues and norming constants. We also study on half-inverse problem and prove a Hochstadt and Lieberman-type theorem.
**Preliminaries**
=================
Before presenting our main results, we recall the some important concepts of the conformable fractional calculus theory.
Let $f:[0,\infty )\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ be a given function. Then, the conformable fractional derivative of order $%
0<\alpha \leq 1$ of $f$ at $x>0$ is defined by:$$D^{\alpha }f(x)=\underset{h\rightarrow 0}{\lim }\frac{f(x+hx^{1-\alpha
})-f(x)}{h},$$and the fractional derivative at $0$ is defined as $D^{\alpha }f(0)=\underset%
{x\rightarrow 0^{+}}{\lim }D^{\alpha }f(x).$
Let $f:[0,\infty )\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ be a given function. The conformable fractional integral of $f$ of order $%
\alpha $ is defined by:$$I_{\alpha }f(x)=\int\limits_{0}^{x}f(t)d_{\alpha
}t=\int\limits_{0}^{x}t^{\alpha -1}f(t)dt,$$for all $x>0.$
We collect some necessary relations in the following lemma.
Let $f,g$ be $\alpha $-differentiable at $x,$ $x>0.$
i\) $D_{x}^{\alpha }(af+bg)=aD_{x}^{\alpha }f+bD_{x}^{\alpha }g,$ $\forall
a,b\in
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
,$
ii\) $D_{x}^{\alpha }(x^{a})=ax^{a-\alpha },$ $\forall a\in
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
,$
iii\) $D_{x}^{\alpha }(c)=0,$ ($c$ is a constant)
iv\) $D_{x}^{\alpha }(fg)=D_{x}^{\alpha }(f)g+fD_{x}^{\alpha }(g)$,
v\) $D_{x}^{\alpha }(f/g)=\frac{D_{x}^{\alpha }(f)g-fD_{x}^{\alpha }(g)}{g^{2}%
},$
vi\) if $f$ is a continuous function, then for all $x>0,$ we have $\
D_{x}^{\alpha }I_{\alpha }f(x)=f(x),$
vii\) if $f$ is a differentiable function, then we have$\ D_{x}^{\alpha
}f(x)=x^{1-\alpha }f^{^{\prime }}(x),$
Let $f,g:(0,\infty )\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ be $\alpha $-differentiable functions and $h(x)=f(g(x)).$ Then, $h(x)$ is $%
\alpha $-differentiable and for all $x\neq 0$ and $g(x)\neq 0,$$$(D_{x}^{\alpha }h)(x)=(D_{x}^{\alpha }f)(g(x))(D_{x}^{\alpha }g)(x)g^{\alpha
-1}(x),$$if $x=0,$ then $\ \ (D_{x}^{\alpha }h)(0)=\underset{x\rightarrow 0^{+}}{\lim
}(D_{x}^{\alpha }f)(g(x))(D_{x}^{\alpha }g)(x)g^{\alpha -1}(x).$
For further knowledge about the conformable fractional derivative, the reader is referred to [@abd] and [@atan], [@hal].
Let us consider the following boundary value problem $L_{\alpha }(q(x),h,H)$ $$\begin{aligned}
&&\text{\ }\left. \ell y:=-D_{x}^{\alpha }D_{x}^{\alpha }y+q(x)y=\lambda y%
\text{, \ \ }0<x<\pi \right. \medskip \\
&&\text{ }\left. U(y):=D_{x}^{\alpha }y(0)-hy(0)=0\right. \medskip \\
&&\text{ }\left. V(y):=D_{x}^{\alpha }y(\pi )+Hy(\pi )=0\right. \medskip\end{aligned}$$where $D_{x}^{\alpha }$ is the conformable fractional (CF) derivative of order $\alpha ,$ $0<\alpha \leq 1,$ $q(t)$ is real valued continuous function on $\left[ 0,\pi \right] $, $h,H\in
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ and $\lambda $ is the spectral parameter.
Let the functions $\varphi (x,\lambda )$ and $\psi (x,\lambda )$ be the solutions of (1) under the initial conditions $$\varphi (0,\lambda )=1\text{, }D_{x}^{\alpha }\varphi (0,\lambda )=h\text{
and }\psi (\pi ,\lambda )=1,D_{x}^{\alpha }\psi (\pi ,\lambda )=-H$$respectively. These solutions are entire according to $\lambda $ for each fixed $x$ in $\left[ 0,\pi \right] $ and they satisfy the following asymptotic formulas [@Oz]: $$\begin{aligned}
\varphi (x,\lambda ) &=&\cos (\frac{\sqrt{\lambda }}{\alpha }x^{\alpha
})+O\left( \dfrac{1}{\sqrt{\lambda }}\exp (\frac{\left\vert \tau \right\vert
}{\alpha }x^{\alpha })\right) \\
D_{x}^{\alpha }\varphi (x,\lambda ) &=&-\sqrt{\lambda }\sin (\frac{\sqrt{%
\lambda }}{\alpha }x^{\alpha })+O\left( \exp (\frac{\left\vert \tau
\right\vert }{\alpha }x^{\alpha })\right) \\
\psi (x,\lambda ) &=&\cos (\frac{\sqrt{\lambda }}{\alpha }(\pi ^{\alpha
}-x^{\alpha }))+O\left( \dfrac{1}{\sqrt{\lambda }}\exp (\frac{\left\vert
\tau \right\vert }{\alpha }(\pi ^{\alpha }-x^{\alpha }))\right) \\
D_{x}^{\alpha }\psi (x,\lambda ) &=&\sqrt{\lambda }\sin (\frac{\sqrt{\lambda
}}{\alpha }(\pi ^{\alpha }-x^{\alpha }))+O\left( \exp (\frac{\left\vert \tau
\right\vert }{\alpha }(\pi ^{\alpha }-x^{\alpha }))\right)\end{aligned}$$where $\tau :=\func{Im}\sqrt{\lambda }.$ The function
$$W_{\alpha }[\psi (x,\lambda ),\varphi (x,\lambda )]=\psi (x,\lambda
)D_{x}^{\alpha }\varphi (x,\lambda )-\varphi (x,\lambda )D_{x}^{\alpha }\psi
(x,\lambda )$$
is called as the fractional Wronskian of $\psi $ and $\varphi .$ It is proven in [@Oz] that $W_{\alpha }$ does not depend on $x$ and it can be written as $W_{\alpha }[\psi (x,\lambda ),\varphi (x,\lambda )]=\Delta
(\lambda )=V(\varphi )=-U(\psi ).$ Put $G_{\delta }:=\left\{ \sqrt{\lambda }%
:\left\vert \sqrt{\lambda }-\frac{\alpha }{\pi ^{\alpha -1}}k\right\vert
\geq \delta ,\text{ }k=0,1,2,\ldots \right\} ,$ where $\delta $ is a sufficiently small positive number. The function $\Delta (\lambda )$ satisfies the inequality $$\left\vert \Delta (\lambda )\right\vert \geq C_{\delta }\left\vert \sqrt{%
\lambda }\right\vert \exp (\frac{\left\vert \tau \right\vert }{\alpha }\pi
^{\alpha }),\text{ \ }\sqrt{\lambda }\in G_{\delta },\left\vert \sqrt{%
\lambda }\right\vert \geq \rho ^{\ast },$$for sufficiently large $\rho ^{\ast }=\rho ^{\ast }(\delta ).$
Let $\left\{ \lambda _{n}\right\} _{n\geq 0}$ be the eigenvalues sets of $%
L_{\alpha }(q(x),h,H)$. The numbers $\lambda _{n}$ are real, simple and satisfy the following asymptotic estimate:$$\sqrt{\lambda _{n}}=\left( \frac{\alpha }{\pi ^{\alpha -1}}\right) n+\frac{%
\omega _{\alpha }}{n\pi }+\frac{\kappa _{\alpha _{n}}}{n},\text{ }\kappa
_{\alpha _{n}}\in l_{\alpha }^{2}$$where $\omega _{\alpha }=h+H+\frac{1}{2}\int\limits_{0}^{\pi }q(t)d_{\alpha
}t$ [@Oz].
**Uniqueness Theorems**
=======================
Together with $L_{\alpha }$, we consider a boundary value problem $%
\widetilde{L}_{\alpha }=L(\widetilde{q}(x),\widetilde{h},\widetilde{H})$ of the same form but with different coefficients. We assume that if a certain symbol $s$ denotes an object related to $L_{\alpha }$ , then $\widetilde{s}$ will denote an analogous object related to $\widetilde{L}_{\alpha }.$
According to the Weyl function
------------------------------
Let $S(x,\lambda )$ be a solution of (1) that satisfies the conditions $%
S(0,\lambda )=1$, $D_{x}^{\alpha }S(0,\lambda )=1.$ It is clear that $%
W_{\alpha }[\varphi (x,\lambda ),S(x,\lambda )]=1$ and the function $\psi
(x,\lambda )$ can be represented by
$$\frac{\psi (x,\lambda )}{\Delta (\lambda )}=S(x,\lambda )-M(\lambda )\varphi
(x,\lambda )$$
where $M(\lambda )=\dfrac{-\psi (0,\lambda )}{\Delta (\lambda )}$ is called as Weyl function.
If $M(\lambda )=\widetilde{M}(\lambda )$ then $q(x)=\widetilde{q}(x),$ a.e. in $\left[ 0,\pi \right] $, $h=\widetilde{h}$ and $H=\widetilde{H}.$
Let us consider the functions $P_{1}(x,\lambda )$ and $P_{2}(x,\lambda )$ which are defined by the following formulas$$\begin{aligned}
P_{1}(x,\lambda ) &=&\varphi (x,\lambda )D_{x}^{\alpha }\widetilde{\phi }%
(x,\lambda )-\phi (x,\lambda )D_{x}^{\alpha }\widetilde{\varphi }(x,\lambda
), \\
P_{2}(x,\lambda ) &=&\phi (x,\lambda )\widetilde{\varphi }(x,\lambda
)-\varphi (x,\lambda )\widetilde{\phi }(x,\lambda ),\end{aligned}$$where $\phi (x,\lambda )=\frac{\psi (x,\lambda )}{\Delta (\lambda )}.$ It is easy to see that the functions $P_{1}(x,\lambda )$ and $P_{2}(x,\lambda )$ are meromorphic with respect to $\lambda .$ Moreover, $M(\lambda )=%
\widetilde{M}(\lambda )$ and (11) yield that $P_{1}$ and $P_{2}$ are entire in $\lambda .$ It follows from asymptotic formulas (5)-(9) that $%
P_{1}(x,\lambda )=O(1)$ and $P_{2}(x,\lambda )=O\left( \dfrac{1}{\sqrt{%
\lambda }}\right) $. Therefore, we obtain $P_{1}(x,\lambda )=A(x)$ and $%
P_{2}(x,\lambda )=0$ by well-known Liouville’s theorem. From (12) and (13) we get $$\varphi (x,\lambda )=A(x)\widetilde{\varphi }(x,\lambda )\text{ and }\phi
(x,\lambda )=A(x)\widetilde{\phi }(x,\lambda ).$$On the other hand, since$$W_{\alpha }[\varphi (x,\lambda ),\phi (x,\lambda )]=\frac{W_{\alpha
}[\varphi (x,\lambda ),\psi (x,\lambda )]}{\Delta (\lambda )}=1$$and similarly $W_{\alpha }[\widetilde{\varphi }(x,\lambda ),\widetilde{\phi }%
(x,\lambda )]=1,$ then $A^{2}(x)=1$ for all $x\in \left[ 0,\pi \right] .$ Taking into account the asymptotic expressions of $\varphi (x,\lambda )$ and $\widetilde{\varphi }(x,\lambda ),$ we get $A(x)\equiv 1.$ Hence $\varphi
(x,\lambda )=\widetilde{\varphi }(x,\lambda ),$ $\phi (x,\lambda )=%
\widetilde{\phi }(x,\lambda )$ and so $q(x)=\widetilde{q}(x),$ a.e. in $%
\left[ 0,\pi \right] $, $h=\widetilde{h}$ and $H=\widetilde{H}.$
According to two given spectra or a spectrum and norming cons-tants
-------------------------------------------------------------------
We consider the boundary value problem $L_{\alpha ,1}$ with the condition $%
y(0,\lambda )=0$ instead of (2) in $L_{\alpha }$. Let $\{\xi _{n}\}_{n\geq
0} $ be the eigenvalues of the problem $L_{\alpha ,1}$. It is obvious that $%
\xi _{n}$ are zeros of $\Delta _{1}(\xi ):=\psi (0,\xi ).$
If$\ \left\{ \lambda _{n},\xi _{n}\right\} _{n\geq 0}=\left\{ \widetilde{%
\lambda }_{n},\widetilde{\xi }_{n}\right\} _{n\geq 0}$ then $q(x)=\widetilde{%
q}(x),$ a.e. in $\left[ 0,\pi \right] $, $h=\widetilde{h}$ and $H=\widetilde{%
H}.$
According to the Theorem 3.11 in [@Oz], the function $\Delta (\lambda )$ can be represented as follows$$\Delta (\lambda )=\frac{\pi ^{3\alpha -2}}{\alpha ^{3}}\left( \lambda
_{0}-\lambda \right) \prod\limits_{n=0}^{\infty }\left( \dfrac{\lambda
_{n}-\lambda }{n^{2}}\right) .$$Therefore, $\Delta (\lambda )\equiv \widetilde{\Delta }(\lambda )$ (similarly $\Delta _{1}(\xi )\equiv \widetilde{\Delta }_{1}(\xi )$)$,$ when $%
\lambda _{n}=\widetilde{\lambda }_{n}$ ($\xi _{n}=\widetilde{\xi }_{n}$) for all $n.$ Consequently, $M(\lambda )\equiv \widetilde{M}(\lambda )\ $and so the proof is completed by Theorem 1.
Denote $\alpha _{n}=\left\Vert \varphi (x,\lambda _{n})\right\Vert
_{2,\alpha }^{2}=\int\limits_{0}^{\pi }\varphi ^{2}(x,\lambda _{n})d_{\alpha
}x.$ Then, we have $\beta _{n}\alpha _{n}=-\Delta ^{\prime }(\lambda _{n})$, where $\beta _{n}=\varphi (\pi ,\lambda _{n})=\dfrac{1}{\psi (0,\lambda _{n})%
}$.
The numbers $\alpha _{n}$ in Lemma 1 are called norming constants.
If $\left\{ \lambda _{n},\alpha _{n}\right\} _{n\geq 0}=\left\{ \widetilde{%
\lambda }_{n},\widetilde{\alpha }_{n}\right\} _{n\geq 0}$ then $q(x)=%
\widetilde{q}(x),$ a.e. in $\left[ 0,\pi \right] $, $h=\widetilde{h}$ and $H=%
\widetilde{H}$.
Since $\lambda _{n}=\widetilde{\lambda }_{n},$ $\Delta (\lambda )\equiv
\widetilde{\Delta }(\lambda ).$ Therefore, it is obtained by using Lemma 3 that $\beta _{n}=\widetilde{\beta }_{n}$ and so $\psi (0,\lambda _{n})=%
\widetilde{\psi }(0,\lambda _{n}).$ Hence the function $$G(\lambda ):=\dfrac{\psi (0,\lambda )-\widetilde{\psi }(0,\lambda )}{\Delta
(\lambda )}$$is entire on $\lambda .$ Moreover, one can obtained from (7) and (9) that $%
G(\lambda )=O(\dfrac{1}{\lambda })$ for $\left\vert \lambda \right\vert
\rightarrow \infty .$ Thus $G(\lambda )\equiv 0$ and $\psi (0,\lambda
)\equiv \widetilde{\psi }(0,\lambda ).$ Finally, we get $M(\lambda )=%
\widetilde{M}(\lambda )$ and so obtain our desired result by Theorem 1.
According to the mixed data
---------------------------
The next theorem is a generalized version of well-known Hochstadt and Lieberman theorem in the classical Sturm-Liouville theory.
If $\left\{ \lambda _{n}\right\} _{n\geq 0}=\left\{ \widetilde{\lambda }%
_{n}\right\} _{n\geq 0}$, $H=\widetilde{H}$ and $q(x)=\widetilde{q}(x)$ on $%
\left( \pi /2,\pi \right) $ then $q(x)=\widetilde{q}(x),$ a.e. in $[0,\pi ]$ and $h=\widetilde{h}$.
It is clear that the following equality holds$$D_{x}^{\alpha }\left[ \widetilde{\varphi }(x,\lambda )D_{x}^{\alpha }\varphi
(x,\lambda )-\varphi (x,\lambda )D_{x}^{\alpha }\widetilde{\varphi }%
(x,\lambda )\right] =\left[ q(x)-\widetilde{q}(x)\right] \varphi (x,\lambda )%
\widetilde{\varphi }(x,\lambda )$$By integrating (in the conformable fractional integral) both sides of this equality on $\left[ 0,\pi \right] ,$ we obtain$$\left[ \widetilde{\varphi }(x,\lambda )D_{x}^{\alpha }\varphi (x,\lambda
)-\varphi (x,\lambda )D_{x}^{\alpha }\widetilde{\varphi }(x,\lambda )\right]
_{0}^{\pi }=\int\limits_{0}^{\pi }\left[ q(t)-\widetilde{q}(t)\right]
\varphi (t,\lambda )\widetilde{\varphi }(t,\lambda )d_{\alpha }t.$$Since $q(x)=\widetilde{q}(x)$ on $\left( \pi /2,\pi \right) $ and from (4), it is obvious that $$\left. \widetilde{\varphi }(\pi ,\lambda )D_{x}^{\alpha }\varphi (\pi
,\lambda )-\varphi (\pi ,\lambda )D_{x}^{\alpha }\widetilde{\varphi }(\pi
,\lambda )=h-\widetilde{h}+\int\limits_{0}^{\pi /2}\left[ q(t)-\widetilde{q}%
(t)\right] \varphi (t,\lambda )\widetilde{\varphi }(t,\lambda )d_{\alpha
}t\right.$$Let$$\left. H(\lambda ):=h-\widetilde{h}+\int\limits_{0}^{\pi /2}\left[ q(t)-%
\widetilde{q}(t)\right] \varphi (t,\lambda )\widetilde{\varphi }(t,\lambda
)d_{\alpha }t\right.$$Since $\widetilde{\varphi }(\pi ,\lambda _{n})D_{x}^{\alpha }\varphi (\pi
,\lambda _{n})-\varphi (\pi ,\lambda _{n})D_{x}^{\alpha }\widetilde{\varphi }%
(\pi ,\lambda _{n})=0,$ $H(\lambda _{n})=0$ for all $n$ and so $\chi
(\lambda ):=\dfrac{H(\lambda )}{\Delta (\lambda )}$ is entire on $\lambda .$ On the other hand, from the asymptotic expressions of $\varphi (x,\lambda )$ and $\widetilde{\varphi }(x,\lambda ),$ it can be calculated that $%
\left\vert \chi (\lambda )\right\vert \leq \frac{C}{\left\vert \sqrt{\lambda
}\right\vert }$ for sufficiently large $\left\vert \lambda \right\vert $. By Liouville’s Theorem, we get $\chi (\lambda )=0$ for all $\lambda $. Hence $%
H(\lambda )\equiv 0$.
By integrating again both sides of the equality (15) on $\left( 0,\pi
/2\right) $, we get$$\widetilde{\varphi }(\pi /2,\lambda )D_{x}^{\alpha }\varphi (\pi /2,\lambda
)=\varphi (\pi /2,\lambda )D_{x}^{\alpha }\widetilde{\varphi }(\pi
/2,\lambda )$$Put $\psi (x,\lambda ):=\varphi (\left( \left( \pi /2\right) ^{\alpha
}-x^{\alpha }\right) ^{1/\alpha },\lambda ).$ From Lemma 2, it is clear that $\psi (x,\lambda )$ is the solution of the following initial value problem $$\begin{aligned}
&&\left. -D_{x}^{\alpha }D_{x}^{\alpha }y+q(\left( \left( \pi /2\right)
^{\alpha }-x^{\alpha }\right) ^{1/\alpha })y=\lambda y,\text{ }x\in (0,\pi
/2)\right. \\
&&\left. y(\pi /2,\lambda )=1,\text{ }D_{x}^{\alpha }y(\pi /2,\lambda
)=-h\right.\end{aligned}$$It follows from (16) that$$\widetilde{\psi }(0,\lambda )D_{x}^{\alpha }\psi (0,\lambda )=\psi
(0,\lambda )D_{x}^{\alpha }\widetilde{\psi }(0,\lambda ).$$Taking into account Theorem 1, it is concluded that $q(x)=$ $\widetilde{q}%
(x) $ on $\left[ 0,\pi /2\right] $ and $h=\widetilde{h}.$ This completes the proof.
[99]{} Abdeljawad, T.: On conformable fractional calculus. J Comput Appl Math. 279, 57–66 (2015)
Ambarzumyan, V.A.: Über eine Frage der Eigenwerttheorie. Z. Phys. 53, 690–695 (1929)
Allahverdiev, Bilender P., Tuna H., Yalçinkaya Y.: Conformable fractional Sturm-Liouville equation. Mathematical Methods in the Applied Sciences. 42(10), 3508-3526 (2019)
Al-Towailb, M.A: A q-fractional approach to the regular Sturm-Liouville problems. Electron J Differ Equ. 88, 1–13 2017
Atangana A, Baleanu D, Alsaedi A.: New properties of conformable derivative. Open Math. 13, 889–898 (2015)
Baleanu, D., Guvenc, Z.B., Machado, J.T.: New trends in nanotechnology and fractional calculus applications. Springer, New York (2010)
Borg, G.: Eine Umkehrung der Sturm–Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte. Acta Math. 78, 1–96 (1946)
Chadan, K., Colton, D., Päivärinta, L., Rundell, W.: An introduction to inverse scattering and inverse spectral problems. Society for Industrial and Applied Mathematics, (1997)
Freiling, G., Yurko V.A.: Inverse Sturm–Liouville Problems and their Applications, Nova Science, New York (2001)
Gelfand, I. M., Levitan, B. M.: On the determination of a differential equation from its spectral function, Trans. Amer. Math. Soc. 1, 253-304 (1951)
Gesztesy, F., Simon, B.: Inverse spectral analysis with partial information on the potential. II. The case of discrete spectrum. Trans. Am. Math. Soc. 352(6), 2765-2787 (2000)
Hald, O.H.: McLaughlin, J.R.: Solution of inverse nodal problems. Inverse Problems. 5, 307–347 (1989)
Hald, O. H.: Discontinuous inverse eigenvalue problems. Comm. Pure Appl. Math., 37, 539–577 (1984)
Hald, O. H.: The inverse Sturm-Liouville problem with symmetric, potentials, Acta Math. 141, 263-291 (1978)
Hochstadt, H.: Lieberman, B.: An inverse Sturm–Liouville problem with mixed given data. SIAM Journal on Applied Mathematics. 34(4), 676-680 (1978)
Horvath, M.: Inverse spectral problems and closed exponential systems, Ann. of Math. 162 885-918 (2005)
Khalil, R., Al Horani, M., Yousef, A., et al.: A newdefinition of fractional derivative. J Comput Appl Math. 264, 65–70 (2014)
Khosravian-Arab, H., Dehghan, M., Eslahchi, M.R.: Fractional Sturm-Liouville boundary value problems in unbounded domains, theory and applications. J Comput Phys. 299, 526–560 (2015)
Klimek, M, Agrawaln, OP.: Fractional Sturm-Liouville problem. Comput Math Appl. 6, 795–812 (2013)
Levinson, N.: The inverse Sturm-Liouville problem. Math. Tidsskr. B, 25, 25-30 (1949)
Levitan, B.M., Sargsjan, I.S.: Sturm-Liouville and Dirac Operators. Kluwer, Dordrecht (1991)
Mainardi, F.: Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models. London, UK: Imperial College Press, (2010)
Marchenko, V. A.: Concerning the theory of a differential operator of second order, Dokl. Akad. Nauk. SSR, 72, 457-470 (1950)
Marchenko, V.A.: Some questions in the theory of one-dimensional linear differential operators of the second order. I. Tr.Mosk. Mat. Obs. 1, 327-420 (1952) (Russian). English transl. in Am.Math. Soc. Trans. 2,1-104 (1973)
Miller, K.S.: An Introduction to Fractional Calculus and Fractional Differential Equations, J. Wiley and Sons, New York, (1993)
Mochizuki, K., Trooshin, I.: Inverse problem for interior spectral data of Sturm-Liouville operator, J. Inverse Ill-posed Probl. 9, 425-433 (2001)
Monje, C.A., Chen, Y., Vinagre, B.M., et al.: Fractional-order systems and controls: fundamentals and applications. Springer, London (2010)
Mortazaasl, H., Akbarfam, A.J.: Trace formula and inverse nodal problem for a conformable fractional Sturm-Liouville problem. Inverse Problems in Science and Engineering, 1-32 (2009)
Oldham, K., Spanier, J.: The Fractional Calculus, Theory and Applications of Differentiation and Integration of Arbitrary Order. Academic Press, USA (1974)
Ozkan, A. S.: Inverse Sturm–Liouville problems with eigenvalue dependent boundary and discontinuity conditions. Inverse Probl. Sci. Eng., 20(6), 857–868 (2012)
Pálfalvi, A.: Efficient solution of a vibration equation involving fractional derivatives. Int J Nonlin Mech. 45, 169–175 (2010)
Rivero, M, Trujillo, J.J., Velasco M.P.: A fractional approach to the Sturm-Liouville problem. Centr Eur J Phys. 11, 1246–1254 (2013)
Silva, M.F., Machado, J.A.T.: Fractional order PD $\alpha $ joint control of legged robots. J Vib Control. 12, 1483–1501 (2006)
| ArXiv |
---
author:
- |
Meelis Kull\
Department of Computer Science\
University of Tartu\
`[email protected]`\
Miquel Perello-Nieto\
Department of Computer Science\
University of Bristol\
`[email protected]`\
Markus Kängsepp\
Department of Computer Science\
University of Tartu\
`[email protected]`\
Telmo Silva Filho\
Department of Statistics\
Universidade Federal da Paraíba\
`[email protected]`\
Hao Song\
Department of Computer Science\
University of Bristol\
`[email protected]`\
Peter Flach\
Department of Computer Science\
University of Bristol and\
The Alan Turing Institute\
`[email protected]`\
title: |
Beyond temperature scaling:\
Obtaining well-calibrated multi-class probabilities with Dirichlet calibration\
Supplementary material
---
Source code
===========
The instructions and code for the experiments can be found on .
Proofs
======
\[thm:equiv\] The parametric families $\vmuh_{DirGen}(\vq; \valpha,\vpi)$, $\vmuh_{DirLin}(\vq; \MW,\vb)$ and $\vmuh_{Dir}(\vq; \MA,\vc)$ are equal, i.e. they contain exactly the same calibration maps.
We will prove that:
1. every function in $\vmuh_{DirGen}(\vq; \valpha,\vpi)$ belongs to $\vmuh_{DirLin}(\vq; \MW,\vb)$;
2. every function in $\vmuh_{DirLin}(\vq; \MW,\vb)$ belongs to $\vmuh_{Dir}(\vq; \MA,\vc)$;
3. every function in $\vmuh_{Dir}(\vq; \MA,\vc)$ belongs to $\vmuh_{DirGen}(\vq; \valpha,\vpi)$.
#### 1.
Consider a function $\muh(\vq)=\vmuh_{DirGen}(\vq; \valpha,\vpi)$. Let us start with an observation that any vector $\vx=(x_1,\dots,x_k)\in(0,\infty)^k$ with only positive elements can be renormalised to add up to $1$ using the expression $\softmax(\vln(\vx))$, since: $$\begin{aligned}
\softmax(\vln(\vx))=\vexp(\vln(\vx))/(\sum_i \exp(\ln(x_i)))=\vx/(\sum_i x_i)\end{aligned}$$ where $\vexp$ is an operator applying exponentiation element-wise. Therefore, $$\begin{aligned}
\muh(\vq)=\softmax(\vln(\pi_1 f_1(\vq),\dots,\pi_k f_k(\vq)))
% =\softmax(\vln\pi+\vln(f_1(\vq),\dots,f_k(\vq)))\end{aligned}$$ where $f_i(\vq)$ is the probability density function of the distribution $Dir(\valpha^{(i)})$ where $\valpha^{(i)}$ is the $i$-th row of matrix $\valpha$. Hence, $f_i(\vq)=\frac{1}{B(\valpha^{(i)})}\prod_{j=1}^k q_j^{\alpha_{ij}-1}$, where $B(\cdot)$ denotes the multivariate beta function. Let us define a matrix $\MW$ and vector $\vb$ as follows: $$\begin{aligned}
w_{ij}=\alpha_{ij}-1,\qquad b_i=\ln(\pi_i)-\ln(B(\valpha^{(i)}))\end{aligned}$$ with $w_{ij}$ and $\alpha_{ij}$ denoting elements of matrices $\MW$ and $\valpha$, respectively, and $b_i,\pi_i$ denoting elements of vectors $\vb$ and $\vpi$. Now we can write $$\begin{aligned}
\ln(\pi_i f_i(\vq))
&=\ln(\pi_i)-\ln(B(\valpha^{(i)}))+\ln\prod_{j=1}^k q_j^{\alpha_{ij}-1} \\
&=\ln(\pi_i)-\ln(B(\valpha^{(i)}))+\sum_{j=1}^k (\alpha_{ij}-1)\ln(q_j) \\
&=b_i+\sum_{j=1}^k w_{ij}\ln(q_j)\end{aligned}$$ and substituting this back into $\muh(\vq)$ we get: $$\begin{aligned}
\muh(\vq)&=\softmax(\vln(\pi_1 f_1(\vq),\dots,\pi_k f_k(\vq))) \\
&=\softmax(\vb+\MW\vln(\vq))=\muh_{DirLin}(\vq; \MW,\vb)\end{aligned}$$
#### 2.
Consider a function $\muh(\vq)=\vmuh_{DirLin}(\vq; \MW,\vb)$. Let us define a matrix $\MA$ and vector $\vc$ as follows: $$\begin{aligned}
a_{ij}=w_{ij}-\min_{i}w_{ij},\qquad \vc=\softmax(\MW\,\vln\,\vu+\vb)\end{aligned}$$ with $a_{ij}$ and $w_{ij}$ denoting elements of matrices $\MA$ and $\MW$, respectively, and $\vu=(1/k,\dots,1/k)$ is a column vector of length $k$. Note that $\MA\,\vx=\MW\,\vx+const_1$ and $\vln\,\softmax(\vx)=\vx+const_2$ for any $x$ where $const_1$ and $const_2$ are constant vectors (all elements are equal), but the constant depends on $\vx$. Taking into account that $\softmax(\vv+const)=\softmax(\vv)$ for any vector $\vv$ and constant vector $const$, we obtain: $$\begin{aligned}
\muh_{Dir}(\vq; \MA,\vc)
&=\softmax(\MA\,\vln\,\frac{\vq}{1/k}+\vln\,\vc)
=\softmax(\MW\,\vln\,\frac{\vq}{1/k}+const_1+\vln\,\vc) \\
&=\softmax(\MW\,\vln\,\vq-\MW\,\vln\,\vu+const_1+\vln\,\softmax(\MW\,\vln\,\vu+\vb)) \\
&=\softmax(\MW\,\vln\,\vq-\MW\,\vln\,\vu+const_1+\MW\,\vln\,\vu+\vb+const_2) \\
&=\softmax(\MW\,\vln\,\vq+\vb+const_1+const_2)
=\softmax(\MW\,\vln\,\vq+\vb)=\muh_{DirLin}(\vq; \MW,\vb)\\
&=\muh(\vq)\end{aligned}$$
#### 3.
Consider a function $\muh(\vq)=\vmuh_{Dir}(\vq; \MA,\vc)$. Let us define a matrix $\valpha$, vector $\vb$ and vector $\pi$ as follows: $$\begin{aligned}
\alpha_{ij}=a_{ij}+1,\qquad \vb=\vln\,\vc-\MA\,\vln\,\vu,\qquad \pi_i=\exp(b_i)\cdot B(\valpha^{(i)})\end{aligned}$$ with $\alpha_{ij}$ and $a_{ij}$ denoting elements of matrices $\valpha$ and $\MA$, respectively, and $\vu=(1/k,\dots,1/k)$ is a column vector of length $k$. We can now write: $$\begin{aligned}
\muh(\vq)&=\muh_{Dir}(\vq; \MA,\vc)
=\softmax(\MA\,\vln\,\frac{\vq}{1/k}+\vln\,\vc)
=\softmax(\MA\,\vln\,\vq-\MA\,\vln\,\vu+\vln\,\vc) \\
&=\softmax((\valpha-1)\vln\,\vq+\vb)\end{aligned}$$ Element $i$ in the vector within the softmax is equal to: $$\begin{aligned}
\sum_{j=1}^k (\valpha_{ij}-1)\ln(q_j)+b_j
&= \sum_{j=1}^k (\valpha_{ij}-1)\ln(q_j) +\ln(\pi_i\cdot\frac{1}{B(\valpha^{(i)})}) \\
&= \ln(\pi_i\cdot \frac{1}{B(\valpha^{(i)})} \prod_{j=1}^k q_j^{\valpha_{ij}-1}) \\
&= \ln(\pi_i\cdot f_i(\valpha^{(i)}))\end{aligned}$$ and therefore: $$\begin{aligned}
\muh(\vq)=\softmax((\valpha-1)\vln(\vq)+\vb)=\softmax(\ln(\pi_i\cdot f_i(\valpha^{(i)})))=\vmuh_{DirGen}(\vq; \valpha,\vpi)\end{aligned}$$
The following proposition proves that temperature scaling can be viewed as a general-purpose calibration method, being a special case within the Dirichlet calibration map family.
Let us denote the temperature scaling family by $\muh'_{TempS}(\vz; t)=\softmax(\vz/t)$ where $\vz$ are the logits. Then for any $t$, temperature scaling can be expressed as $$\begin{aligned}
\muh'_{TempS}(\vz; t)=\muh_{DirLin}(\softmax(\vz); \frac{1}{t}\MI, \vzero)\end{aligned}$$ where $\MI$ is the identity matrix and $\vzero$ is the vector of zeros.
Let us first observe that for any $\vx\in\sR^k$ there exists a constant vector $const$ (all elements are equal) such that $\vln\,\softmax(\vx)=\vx+const$. Furthermore, $\softmax(\vv+const)=\softmax(\vv)$ for any vector $\vv$ and any constant vector $const$. Therefore, $$\begin{aligned}
\muh_{DirLin}(\softmax(\vz); \frac{1}{t}\MI, \vzero)
&=\softmax(\frac{1}{t}\,\MI\,\vln\,\softmax(\vz))) \\
&=\softmax(\frac{1}{t}\,\MI\,(\vz+const)) \\
&=\softmax(\frac{1}{t}\,\MI\,\vz+\frac{1}{t}\,\MI\, const) \\
&=\softmax(\vz/t+const') \\
&=\softmax(\vz/t) \\
&=\muh'_{TempS}(\vz; t)\end{aligned}$$ where $const'=\frac{1}{t}\,\MI\, const$ is a constant vector as a product of a diagonal matrix with a constant vector.
Dirichlet calibration
=====================
In this section we show some examples of reliability diagrams and other plots that can help to understand the representational power of Dirichlet calibration compared with other calibration methods.
Reliability diagram examples
----------------------------
We will look at two examples of reliability diagrams on the original classifier and after applying $6$ calibration methods. Figure \[fig:mlp:bs:reldiag\] shows the first example for the 3 class classification dataset *balance-scale* and the classifier MLP. This figure shows the confidence-reliability diagram in the first column and the classwise-reliability diagrams in the other columns. Figure \[fig:nb:reldiag:mlp:bal:uncal\] shows how posterior probabilities from the MLP have small gaps between the true class proportions and the predicted means. This visualisation may indicate that the original classifier is already well calibrated. However, when we separate the reliability diagram per class, we notice that the predictions for the first class are underconfident; as indicated by low mean predictions containing high proportions of the true class. On the other hand, classes 2 and 3 are overconfident in the regions of posterior probabilities compressed between $[0.2, 0.5]$ while being underconfident in higher regions. The discrepancy shown by analysing the individual reliability diagrams seems to compensate in the general picture of the aggregated one.
The following subfigures show how the different calibration methods try to reduce ECE, occasionally increasing the error. As can be seen in Table \[table:mlp:balance:ece\], Dirichlet L2 and One-vs.Rest isotonic regression obtain the lowest ECE while One-vs.Rest frequency binning makes the original calibration worse. Looking at Figure \[fig:nb:reldiag:mlp:bal:temp\] it is possible to see how temperature scaling manages to reduce the overall overconfidence in the higher range of probabilities for classes 2 and 3, but makes the situation worse in the interval $[0.2, 0.6]$. However, it manages to reduce the overall ECE.
[.26]{} ![Confidence-reliability diagrams in the first column and classwise-reliability diagrams in the remaining columns, for a real experiment with the multilayer perceptron classifier on the balance-scale dataset and a subset of the calibrators. All the test partitions from the 5 times 5-fold-cross-validation have been aggregated to draw every plot.[]{data-label="fig:mlp:bs:reldiag"}](figures/results/mlp_balance-scale_uncalibrated_conf_rel_diagr.pdf "fig:"){width="\linewidth"}
[.71]{} ![Confidence-reliability diagrams in the first column and classwise-reliability diagrams in the remaining columns, for a real experiment with the multilayer perceptron classifier on the balance-scale dataset and a subset of the calibrators. All the test partitions from the 5 times 5-fold-cross-validation have been aggregated to draw every plot.[]{data-label="fig:mlp:bs:reldiag"}](figures/results/mlp_balance-scale_uncalibrated_rel_diagr_perclass.pdf "fig:"){width="\linewidth"}
[.26]{} ![Confidence-reliability diagrams in the first column and classwise-reliability diagrams in the remaining columns, for a real experiment with the multilayer perceptron classifier on the balance-scale dataset and a subset of the calibrators. All the test partitions from the 5 times 5-fold-cross-validation have been aggregated to draw every plot.[]{data-label="fig:mlp:bs:reldiag"}](figures/results/mlp_balance-scale_binning_freq_conf_rel_diagr.pdf "fig:"){width="\linewidth"}
[.71]{} ![Confidence-reliability diagrams in the first column and classwise-reliability diagrams in the remaining columns, for a real experiment with the multilayer perceptron classifier on the balance-scale dataset and a subset of the calibrators. All the test partitions from the 5 times 5-fold-cross-validation have been aggregated to draw every plot.[]{data-label="fig:mlp:bs:reldiag"}](figures/results/mlp_balance-scale_binning_freq_rel_diagr_perclass.pdf "fig:"){width="\linewidth"}
[.26]{} ![Confidence-reliability diagrams in the first column and classwise-reliability diagrams in the remaining columns, for a real experiment with the multilayer perceptron classifier on the balance-scale dataset and a subset of the calibrators. All the test partitions from the 5 times 5-fold-cross-validation have been aggregated to draw every plot.[]{data-label="fig:mlp:bs:reldiag"}](figures/results/mlp_balance-scale_binning_width_conf_rel_diagr.pdf "fig:"){width="\linewidth"}
[.71]{} ![Confidence-reliability diagrams in the first column and classwise-reliability diagrams in the remaining columns, for a real experiment with the multilayer perceptron classifier on the balance-scale dataset and a subset of the calibrators. All the test partitions from the 5 times 5-fold-cross-validation have been aggregated to draw every plot.[]{data-label="fig:mlp:bs:reldiag"}](figures/results/mlp_balance-scale_binning_width_rel_diagr_perclass.pdf "fig:"){width="\linewidth"}
[.26]{} ![Confidence-reliability diagrams in the first column and classwise-reliability diagrams in the remaining columns, for a real experiment with the multilayer perceptron classifier on the balance-scale dataset and a subset of the calibrators. All the test partitions from the 5 times 5-fold-cross-validation have been aggregated to draw every plot.[]{data-label="fig:mlp:bs:reldiag"}](figures/results/mlp_balance-scale_isotonic_conf_rel_diagr.pdf "fig:"){width="\linewidth"}
[.71]{} ![Confidence-reliability diagrams in the first column and classwise-reliability diagrams in the remaining columns, for a real experiment with the multilayer perceptron classifier on the balance-scale dataset and a subset of the calibrators. All the test partitions from the 5 times 5-fold-cross-validation have been aggregated to draw every plot.[]{data-label="fig:mlp:bs:reldiag"}](figures/results/mlp_balance-scale_isotonic_rel_diagr_perclass.pdf "fig:"){width="\linewidth"}
[.26]{} ![Confidence-reliability diagrams in the first column and classwise-reliability diagrams in the remaining columns, for a real experiment with the multilayer perceptron classifier on the balance-scale dataset and a subset of the calibrators. All the test partitions from the 5 times 5-fold-cross-validation have been aggregated to draw every plot.[]{data-label="fig:mlp:bs:reldiag"}](figures/results/mlp_balance-scale_dirichlet_fix_diag_conf_rel_diagr.pdf "fig:"){width="\linewidth"}
[.71]{} ![Confidence-reliability diagrams in the first column and classwise-reliability diagrams in the remaining columns, for a real experiment with the multilayer perceptron classifier on the balance-scale dataset and a subset of the calibrators. All the test partitions from the 5 times 5-fold-cross-validation have been aggregated to draw every plot.[]{data-label="fig:mlp:bs:reldiag"}](figures/results/mlp_balance-scale_dirichlet_fix_diag_rel_diagr_perclass.pdf "fig:"){width="\linewidth"}
[.26]{} ![Confidence-reliability diagrams in the first column and classwise-reliability diagrams in the remaining columns, for a real experiment with the multilayer perceptron classifier on the balance-scale dataset and a subset of the calibrators. All the test partitions from the 5 times 5-fold-cross-validation have been aggregated to draw every plot.[]{data-label="fig:mlp:bs:reldiag"}](figures/results/mlp_balance-scale_ovr_dir_full_conf_rel_diagr.pdf "fig:"){width="\linewidth"}
[.71]{} ![Confidence-reliability diagrams in the first column and classwise-reliability diagrams in the remaining columns, for a real experiment with the multilayer perceptron classifier on the balance-scale dataset and a subset of the calibrators. All the test partitions from the 5 times 5-fold-cross-validation have been aggregated to draw every plot.[]{data-label="fig:mlp:bs:reldiag"}](figures/results/mlp_balance-scale_ovr_dir_full_rel_diagr_perclass.pdf "fig:"){width="\linewidth"}
[.26]{} ![Confidence-reliability diagrams in the first column and classwise-reliability diagrams in the remaining columns, for a real experiment with the multilayer perceptron classifier on the balance-scale dataset and a subset of the calibrators. All the test partitions from the 5 times 5-fold-cross-validation have been aggregated to draw every plot.[]{data-label="fig:mlp:bs:reldiag"}](figures/results/mlp_balance-scale_dirichlet_full_l2_conf_rel_diagr.pdf "fig:"){width="\linewidth"}
[.71]{} ![Confidence-reliability diagrams in the first column and classwise-reliability diagrams in the remaining columns, for a real experiment with the multilayer perceptron classifier on the balance-scale dataset and a subset of the calibrators. All the test partitions from the 5 times 5-fold-cross-validation have been aggregated to draw every plot.[]{data-label="fig:mlp:bs:reldiag"}](figures/results/mlp_balance-scale_dirichlet_full_l2_rel_diagr_perclass.pdf "fig:"){width="\linewidth"}
In the second example we show $3$ calibration methods for a 4 class classification problem (car dataset) applied on the scores of an Adaboost SAMME classifier. Figure \[fig:nb:reldiag:adas:car\] shows one reliability diagram per class ($C_1$ *acceptable*, $C_2$ *good*, $C_3$ *unacceptable*, and $C_4$ *very good*).
From this Figure we can see that the uncalibrated model is underconfident for classes 1, 2 and 3, showing posterior probabilities never higher than $0.7$, while having true class proportions higher than $0.7$ in the mentioned interval. We can see that after applying some of the calibration models the posterior probabilities reach higher probability values.
As can be seen in Table \[table:adas:car:ece\], Dirichlet L2 and One-vs.Rest Isotonic Regression obtain the lowest ECE while Temperature Scaling makes the original calibration worse. Figure \[fig:nb:reldiag:class:adas:car:dirl2\] shows how Dirichlet calibration with L2 regularisation achieved the largest spread of probabilities, also reducing the error mean gap with the predictions and the true class proportions. On the other hand, temperature scaling reduced ECE for class 1, but hurt the overall performance for the other classes.
[.9]{} ![Reliability diagrams per class for a real experiment with the classifier Ada boost SAMME on the car dataset and $3$ calibrators. The test partitions from the 5 times 5-fold-cross-validation have been aggregated to draw every plot.[]{data-label="fig:nb:reldiag:adas:car"}](figures/results/adas_car_uncalibrated_rel_diagr_perclass.pdf "fig:"){width="\linewidth"}
[.9]{} ![Reliability diagrams per class for a real experiment with the classifier Ada boost SAMME on the car dataset and $3$ calibrators. The test partitions from the 5 times 5-fold-cross-validation have been aggregated to draw every plot.[]{data-label="fig:nb:reldiag:adas:car"}](figures/results/adas_car_isotonic_rel_diagr_perclass.pdf "fig:"){width="\linewidth"}
[.9]{} ![Reliability diagrams per class for a real experiment with the classifier Ada boost SAMME on the car dataset and $3$ calibrators. The test partitions from the 5 times 5-fold-cross-validation have been aggregated to draw every plot.[]{data-label="fig:nb:reldiag:adas:car"}](figures/results/adas_car_dirichlet_fix_diag_rel_diagr_perclass.pdf "fig:"){width="\linewidth"}
[.9]{} ![Reliability diagrams per class for a real experiment with the classifier Ada boost SAMME on the car dataset and $3$ calibrators. The test partitions from the 5 times 5-fold-cross-validation have been aggregated to draw every plot.[]{data-label="fig:nb:reldiag:adas:car"}](figures/results/adas_car_dirichlet_full_l2_rel_diagr_perclass.pdf "fig:"){width="\linewidth"}
A more detailed depiction of the previous reliability diagrams can be seen in Figure \[fig:nb:pos:scores:class\]. In this case, the posterior probabilities are not introduced in bins, but a boxplot summarises their full distribution. The first observation here is, for the *good* and *very good* classes, the uncalibrated model tends to predict probability vectors with small variance, i.e. the outputs do not change much among different instances. Among the calibration approaches, temperature scaling still maintains this low level of variance, while both isotonic and Dirichlet L2 manage to show a higher variance on the outputs. While this observation cannot be justified here without quantitative analysis, another observation clearly shows an advantage of using Dirichlet L2. For the *acceptable* class, only Dirichlet L2 is capable of providing the highest mean probability for the correct class, while the other three methods tend to put higher probability mass on the *unacceptable* class on average.
[.24]{} ![Effect of Dirichlet Calibration on the scores of Ada boost SAMME on the *car* dataset which is composed of $4$ classes (*acceptable*, *good*, *unacceptable*, and *very good*). The whiskers of each box indicate the 5th and 95th percentile, the notch around the median indicates the confidence interval. The [green]{} error bar to the right of each box indicates one standard deviation on each side of the mean. In each subfigure, the first boxplot corresponds to the posterior probabilities for the samples of class 1, divided in 4 boxes representing the posterior probabilities for each class. A good classifier should have the highest posterior probabilities in the box corresponding to the true class. In Figure \[fig:nb:pos:scores:class:adas:car:uncal\] it is possible to see that the first class (*acceptable*) is missclassified as belonging to the third class (*unacceptable*) with high probability values, while Dirichlet Calibration is able to alleviate that problem. Also, for the second and fourth true classes (*good*, and *very good*) the original classifier uses a reduced domain of probabilities (indicative of underconfidence), while Dirichlet calibration is able to spread these probabilities with more meaningful values (as indicated by a reduction of the calibration losses; See Figure \[fig:nb:reldiag:adas:car\]). []{data-label="fig:nb:pos:scores:class"}](figures/results/adas_car_uncalibrated_positive_scores_per_class.pdf "fig:"){width="\linewidth"}
[.24]{} ![Effect of Dirichlet Calibration on the scores of Ada boost SAMME on the *car* dataset which is composed of $4$ classes (*acceptable*, *good*, *unacceptable*, and *very good*). The whiskers of each box indicate the 5th and 95th percentile, the notch around the median indicates the confidence interval. The [green]{} error bar to the right of each box indicates one standard deviation on each side of the mean. In each subfigure, the first boxplot corresponds to the posterior probabilities for the samples of class 1, divided in 4 boxes representing the posterior probabilities for each class. A good classifier should have the highest posterior probabilities in the box corresponding to the true class. In Figure \[fig:nb:pos:scores:class:adas:car:uncal\] it is possible to see that the first class (*acceptable*) is missclassified as belonging to the third class (*unacceptable*) with high probability values, while Dirichlet Calibration is able to alleviate that problem. Also, for the second and fourth true classes (*good*, and *very good*) the original classifier uses a reduced domain of probabilities (indicative of underconfidence), while Dirichlet calibration is able to spread these probabilities with more meaningful values (as indicated by a reduction of the calibration losses; See Figure \[fig:nb:reldiag:adas:car\]). []{data-label="fig:nb:pos:scores:class"}](figures/results/adas_car_isotonic_positive_scores_per_class.pdf "fig:"){width="\linewidth"}
[.24]{} ![Effect of Dirichlet Calibration on the scores of Ada boost SAMME on the *car* dataset which is composed of $4$ classes (*acceptable*, *good*, *unacceptable*, and *very good*). The whiskers of each box indicate the 5th and 95th percentile, the notch around the median indicates the confidence interval. The [green]{} error bar to the right of each box indicates one standard deviation on each side of the mean. In each subfigure, the first boxplot corresponds to the posterior probabilities for the samples of class 1, divided in 4 boxes representing the posterior probabilities for each class. A good classifier should have the highest posterior probabilities in the box corresponding to the true class. In Figure \[fig:nb:pos:scores:class:adas:car:uncal\] it is possible to see that the first class (*acceptable*) is missclassified as belonging to the third class (*unacceptable*) with high probability values, while Dirichlet Calibration is able to alleviate that problem. Also, for the second and fourth true classes (*good*, and *very good*) the original classifier uses a reduced domain of probabilities (indicative of underconfidence), while Dirichlet calibration is able to spread these probabilities with more meaningful values (as indicated by a reduction of the calibration losses; See Figure \[fig:nb:reldiag:adas:car\]). []{data-label="fig:nb:pos:scores:class"}](figures/results/adas_car_dirichlet_fix_diag_positive_scores_per_class.pdf "fig:"){width="\linewidth"}
[.24]{} ![Effect of Dirichlet Calibration on the scores of Ada boost SAMME on the *car* dataset which is composed of $4$ classes (*acceptable*, *good*, *unacceptable*, and *very good*). The whiskers of each box indicate the 5th and 95th percentile, the notch around the median indicates the confidence interval. The [green]{} error bar to the right of each box indicates one standard deviation on each side of the mean. In each subfigure, the first boxplot corresponds to the posterior probabilities for the samples of class 1, divided in 4 boxes representing the posterior probabilities for each class. A good classifier should have the highest posterior probabilities in the box corresponding to the true class. In Figure \[fig:nb:pos:scores:class:adas:car:uncal\] it is possible to see that the first class (*acceptable*) is missclassified as belonging to the third class (*unacceptable*) with high probability values, while Dirichlet Calibration is able to alleviate that problem. Also, for the second and fourth true classes (*good*, and *very good*) the original classifier uses a reduced domain of probabilities (indicative of underconfidence), while Dirichlet calibration is able to spread these probabilities with more meaningful values (as indicated by a reduction of the calibration losses; See Figure \[fig:nb:reldiag:adas:car\]). []{data-label="fig:nb:pos:scores:class"}](figures/results/adas_car_dirichlet_full_l2_positive_scores_per_class.pdf "fig:"){width="\linewidth"}
Experimental setup {#sec:exp}
==================
In this section we provide the detailed description of the experimental setup on a variety of non-neural classifiers and datasets. While our implementation of Dirichlet calibration is based on standard Newton-Raphson with multinomial logistic loss and L2 regularisation, as mentioned at the end of Section 3, existing implementations of logistic regression (e.g. scikit-learn) with the log transformed predicted probabilities can also be easily applied.
Datasets and performance estimation
-----------------------------------
The full list of datasets, and a brief description of each one including the number of samples, features and classes is presented in Table \[tab:data\].
Figure \[fig:ds:partition\] shows how every dataset was divided in order to get an estimated performance for every combination of dataset, classifier and calibrator. Each dataset was divided using 5 times 5-fold-cross-validation to create 25 test partitions. For each of the 25 partitions the corresponding training set was divided further with a 3-fold-cross-validation for wich the bigger portions were used to train the classifiers (and validate the calibratiors if they had hyperparameters), and the small portion was used to train the calibrators. The 3 calibrators trained in the inner 3-folds were used to predict the corresponding test partition, and their predictions were averaged in order to obtain better estimates of their performance with the 7 different metrics (accuracy, Brier score, log-loss, maximum calibration error, confidence-ECE, classwise-ECE and the p test statistic of the ECE metrics). Finally, the 25 resulting measures were averaged.
\[tab:data\]
{width="0.9\linewidth"} \[fig:ds:partition\]
Full example of statistical analysis {#sec:exp:example}
------------------------------------
The following is a full example of how the final rankings and statistical tests are computed. For this example, we will focus on the metric log-loss, and we will start with the naive Bayes classifier. Table \[table:nbayes:loss\] shows the estimated log-loss by averaging the 5-times 5-fold cross-validation log-losses of the inner 3-fold aggregated predictions. The sub-indices are the ranking of every calibrator for each dataset (ties in the ranking share the averaged rank). The resulting table of sub-indices is used to compute the Friedman test statistic, resulting in a value of $73.8$ and a p-value of $6.71e^{-14}$ indicating statistical difference between the calibration methods. The last row contains the average ranks of the full table, which is shown in the corresponding critical difference diagram in Figure \[fig:cd:nbayes:loss\]. The critical difference uses the Bonferroni-Dunn one-tailed statistical test to compute the minimum ranking distance that is shown in the Figure, indicating that for this particular classifier and metric the Dirichlet calibrator with L2 regularisation is significantly better than the other methods.
[.49]{} {width="\linewidth"}
[.49]{} {width="\linewidth"}
The same process is applied to each of the $11$ classifiers for every metric. Table \[table:loss\] shows the final average results of all classifiers. Notice that the row corresponding to naive Bayes has the rounded average rankings from Figure \[fig:cd:nbayes:loss\].
Results {#sec:res}
=======
In this Section we present all the final results, including ranking tables for every metric, critical difference diagrams, the best hyperparameters selected for Dirichlet calibration with L2 regularisation, Frequency binning and Width binning; a comparison of how calibrated the $11$ classifiers are, and additional results on deep neural networks.
Final ranking tables for all metrics {#sec:res:rank}
------------------------------------
We present here all the final ranking tables for all metrics (Tables \[table:acc\], \[table:loss\], \[table:brier\], \[table:mce\], \[table:conf-ece\], \[table:cw-ece\], \[table:p-conf-ece\], and \[table:p-cw-ece\]). For each ranking, a lower value is indicative of a better metric value (eg. a higher accuracy corresponds to a lower ranking, while a lower log-loss corresponds to a lower ranking as well). Additional details on how to interpret the tables can be found in Section \[sec:exp:example\].
Final critical difference diagrams for every metric
---------------------------------------------------
In order to perform a final comparison between calibration methods, we considered every combination of dataset and classifier as a group $n = \#datasets \times \#classifiers$, and ranked the results of the $k$ calibration methods. With this setting, we have performed the Friedman statistical test followed by the one-tailed Bonferroni-Dunn test to obtain critical differences (CDs) for every metric (See Figure \[fig:multi:cd:all\]). The results showed Dirichlet L2 as the best calibration method for the measures accuracy, log-loss and p-cw-ece with statistical significance (See Figures \[fig:multi:cd:acc\] \[fig:multi:cd:logloss\], and \[fig:multi:cd:p-cw-ece\]), and in the group of the best calibration methods in the rest of the metrics with statistical significance, but no difference within the group. It is worth mentioning that Figure \[fig:multi:cd:logloss\] showed statistical difference between Dirichlet L2, OvR Beta, OvR width binning, and the rest of the calibrators in one group; in the mentioned order.
[.49]{} ![Critical difference of the average of multiclass classifiers.[]{data-label="fig:multi:cd:all"}](figures/results/crit_diff_acc_v2 "fig:"){width="\linewidth"}
[.49]{} ![Critical difference of the average of multiclass classifiers.[]{data-label="fig:multi:cd:all"}](figures/results/crit_diff_brier_v2 "fig:"){width="\linewidth"}
[.49]{} ![Critical difference of the average of multiclass classifiers.[]{data-label="fig:multi:cd:all"}](figures/results/crit_diff_loss_v2 "fig:"){width="\linewidth"}
[.49]{} ![Critical difference of the average of multiclass classifiers.[]{data-label="fig:multi:cd:all"}](figures/results/crit_diff_mce_v2 "fig:"){width="\linewidth"}
[.49]{} ![Critical difference of the average of multiclass classifiers.[]{data-label="fig:multi:cd:all"}](figures/results/crit_diff_conf-ece_v2.pdf "fig:"){width="\linewidth"}
[.49]{} ![Critical difference of the average of multiclass classifiers.[]{data-label="fig:multi:cd:all"}](figures/results/crit_diff_cw-ece_v2.pdf "fig:"){width="\linewidth"}
[.49]{} ![Critical difference of the average of multiclass classifiers.[]{data-label="fig:multi:cd:all"}](figures/results/crit_diff_p-conf-ece_v2.pdf "fig:"){width="\linewidth"}
[.49]{} ![Critical difference of the average of multiclass classifiers.[]{data-label="fig:multi:cd:all"}](figures/results/crit_diff_p-cw-ece_v2.pdf "fig:"){width="\linewidth"}
[.4]{} ![Proportion of times each calibrator passes a calibration p-test with a p-value higher than 0.05.[]{data-label="fig:cal:p:ece"}](figures/results/p_table_calibrators_p-conf-ece.pdf "fig:"){width="\linewidth"}
[.4]{} ![Proportion of times each calibrator passes a calibration p-test with a p-value higher than 0.05.[]{data-label="fig:cal:p:ece"}](figures/results/p_table_calibrators_p-cw-ece.pdf "fig:"){width="\linewidth"}
Best calibrator hyperparameters
-------------------------------
[.30]{} ![Histogram of the selected hyperparameters during the inner 3-fold-cross-validation[]{data-label="fig:hyper"}](figures/results/bars_hyperparameters_all_Dirichlet_L2.pdf "fig:"){width="\linewidth"}
[.30]{} ![Histogram of the selected hyperparameters during the inner 3-fold-cross-validation[]{data-label="fig:hyper"}](figures/results/bars_hyperparameters_all_OvR_Freq_Bin.pdf "fig:"){width="\linewidth"}
[.30]{} ![Histogram of the selected hyperparameters during the inner 3-fold-cross-validation[]{data-label="fig:hyper"}](figures/results/bars_hyperparameters_all_OvR_Width_Bin.pdf "fig:"){width="\linewidth"}
Figure \[fig:hyper\] shows the best hyperparameters for every inner 3-fold-cross-validation. Dirichlet L2 (Figure \[fig:hyper:dir:l2\]) shows a preference for regularisation hyperparameter $\lambda = 1e^{-3}$ and lower values. Our current minimum regularisation value of $1e^{-7}$ is also being selected multiple times, indicating that lower values may be optimal in several occasions. However, this fact did not seem to hurt the overall good results in our experiments. One-vs.-Rest frequency binning tends to prefer $10$ bins of equal number of samples, while One-vs.Rest width binning prefers $5$ equal sized bins (See Figures \[fig:hyper:freq:bin\] and \[fig:hyper:width:bin\] respectively).
Comparison of classifiers
-------------------------
In this Section we compare all the classifiers without post-hoc calibration on $17$ of the datasets; from the total of $21$ datasets *shuttle*, *yeast*, *mfeat-karhunen* and *libras-movement* were removed from this analysis as at least one classifier was not able to complete the experiment.
[.49]{} ![Critical difference of uncalibrated classifiers.[]{data-label="fig:uncal:cd:all"}](figures/results/crit_diff_uncal_classifiers_acc "fig:"){width="\linewidth"}
[.49]{} ![Critical difference of uncalibrated classifiers.[]{data-label="fig:uncal:cd:all"}](figures/results/crit_diff_uncal_classifiers_loss "fig:"){width="\linewidth"}
[.49]{} ![Critical difference of uncalibrated classifiers.[]{data-label="fig:uncal:cd:all"}](figures/results/crit_diff_uncal_classifiers_brier "fig:"){width="\linewidth"}
[.49]{} ![Critical difference of uncalibrated classifiers.[]{data-label="fig:uncal:cd:all"}](figures/results/crit_diff_uncal_classifiers_mce "fig:"){width="\linewidth"}
[.49]{} ![Critical difference of uncalibrated classifiers.[]{data-label="fig:uncal:cd:all"}](figures/results/crit_diff_uncal_classifiers_conf-ece "fig:"){width="\linewidth"}
[.49]{} ![Critical difference of uncalibrated classifiers.[]{data-label="fig:uncal:cd:all"}](figures/results/crit_diff_uncal_classifiers_cw-ece "fig:"){width="\linewidth"}
[.49]{} ![Critical difference of uncalibrated classifiers.[]{data-label="fig:uncal:cd:all"}](figures/results/crit_diff_uncal_classifiers_p-conf-ece "fig:"){width="\linewidth"}
[.49]{} ![Critical difference of uncalibrated classifiers.[]{data-label="fig:uncal:cd:all"}](figures/results/crit_diff_uncal_classifiers_p-cw-ece "fig:"){width="\linewidth"}
Figure \[fig:uncal:cd:all\] shows the Critical Difference diagram for all the $8$ metrics. In particular, the MLP and the SVC with linear kernel are always in the group with the higher rankings and never in the lowest. Similarly, random forest is consistently in the best group, but in the worst group as well in $4$ of the measures. SVC with radial basis kernel is in the best group $6$ times, but $3$ times in the worst. On the other hand, naive Bayes and Adaboost SAMME are consistently in the worst group and never in the best one. The rest of the classifiers did not show a clear ranking position.
[.4]{} ![Proportion of times each classifier is already calibrated with different p-tests.[]{data-label="fig:uncal:p:ece"}](figures/results/p_table_classifiers_p-conf-ece.pdf "fig:"){width="\linewidth"}
[.4]{} ![Proportion of times each classifier is already calibrated with different p-tests.[]{data-label="fig:uncal:p:ece"}](figures/results/p_table_classifiers_p-cw-ece.pdf "fig:"){width="\linewidth"}
Figures \[fig:uncal:p:cw:ece\] and \[fig:uncal:p:conf:ece\] show the proportion of times each classifier passed the p-conf-ECE and p-cw-ECE statistical test for all datasets and cross-validation folds.
Deep neural networks
--------------------
In this section, we provide further discussion about results from the deep networks experiments. These are given in the form of critical difference diagrams (Figure \[fig:dnn:cd:all\]) and tables (Tables \[table:res:dnn:loss\]-\[table:res:dnn:pece\_cw\]) both including the following measures: error rate, log-loss, Brier score, maximum calibration error (MCE), confidence-ECE (conf-ECE), classwise-ECE (cw-ECE), as well as significance measures p-conf-ECE and p-cw-ECE.
In addition, Table \[table:res:dnn:ms\_vs\_vecs\] compares MS-ODIR and vector scaling on log-loss. On the table, we also added MS-ODIR-zero which was obtained from the respective MS-ODIR model by replacing the off-diagonal entries with zeroes. Each experiment is replicated three times with different splits on datasets. This is done to compare the stability of the methods. In each replication, the best scoring model is written in bold.
Finally, Figure \[fig:res:rd\_ece:class4\] shows that temperature scaling systematically under-estimates class 4 probabilities on the model [c10\_resnet\_wide32]{} on CIFAR-10.
[.49]{} ![Critical difference of the deep neural networks.[]{data-label="fig:dnn:cd:all"}](figures/cd_diag_dnn/crit_diff_Error_v2.pdf "fig:"){width="\linewidth"}
[.49]{} ![Critical difference of the deep neural networks.[]{data-label="fig:dnn:cd:all"}](figures/cd_diag_dnn/crit_diff_Brier_v2.pdf "fig:"){width="\linewidth"}
[.49]{} ![Critical difference of the deep neural networks.[]{data-label="fig:dnn:cd:all"}](figures/cd_diag_dnn/crit_diff_Loss_v2.pdf "fig:"){width="\linewidth"}
[.49]{} ![Critical difference of the deep neural networks.[]{data-label="fig:dnn:cd:all"}](figures/cd_diag_dnn/crit_diff_MCE_v2.pdf "fig:"){width="\linewidth"}
[.49]{} ![Critical difference of the deep neural networks.[]{data-label="fig:dnn:cd:all"}](figures/cd_diag_dnn/crit_diff_ECE_v2.pdf "fig:"){width="\linewidth"}
[.49]{} ![Critical difference of the deep neural networks.[]{data-label="fig:dnn:cd:all"}](figures/cd_diag_dnn/crit_diff_ECE_CW_v2.pdf "fig:"){width="\linewidth"}
[.49]{} ![Critical difference of the deep neural networks.[]{data-label="fig:dnn:cd:all"}](figures/cd_diag_dnn/crit_diff_pECE_v2.pdf "fig:"){width="\linewidth"}
[.49]{} ![Critical difference of the deep neural networks.[]{data-label="fig:dnn:cd:all"}](figures/cd_diag_dnn/crit_diff_pECE_cw_v2.pdf "fig:"){width="\linewidth"}
![Reliability diagrams of c10\_resnet\_wide32 on CIFAR-10: (a) classwise-reliability for class 4 after temperature scaling; (b) classwise-reliability for class 4 after Dirichlet calibration.[]{data-label="fig:res:rd_ece:class4"}](figures/results/figure_RD_ECE_class4.pdf){width="\linewidth"}
| ArXiv |
---
abstract: 'The equilibrium configuration of an engineering structure, able to withstand a certain loading condition, is usually associated with a local minimum of the underlying potential energy. However, in the nonlinear context, there may be other equilibria present, and this brings with it the possibility of a transition to an alternative (remote) minimum. That is, given a sufficient disturbance, the structure might buckle, perhaps suddenly, to another shape. This paper considers the dynamic mechanisms under which such transitions (typically via saddle points) occur. A two-mode Hamiltonian is developed for a shallow arch/buckled beam. The resulting form of the potential energy—two stable wells connected by rank-1 saddle points—shows an analogy with resonance transitions in celestial mechanics or molecular reconfigurations in chemistry, whereas here the transition corresponds to switching between two stable structural configurations. Then, from Hamilton’s equations, the analytical equilibria are determined and linearization of the equations of motion about the saddle is obtained. After computing the eigenvalues and eigenvectors of the coefficient matrix associated with the linearization, a symplectic transformation is given which puts the Hamiltonian into normal form and simplifies the equations, allowing us to use the conceptual framework known as tube dynamics. The flow in the equilibrium region of phase space as well as the invariant manifold tubes in position space are discussed. Also, we account for the addition of damping in the tube dynamics framework, which leads to a richer set of behaviors in transition dynamics than previously explored.'
author:
- Jun Zhong
- 'Lawrence N. Virgin'
- 'Shane D. Ross'
title: 'A Tube Dynamics Perspective Governing Stability Transitions: An Example Based on Snap-through Buckling'
---
potential energy ,transients ,tube dynamics ,dynamic buckling ,invariant manifolds ,Hamiltonian
Introduction
============
The nonlinear behavior of slender structures under loading is often dominated by a potential energy function that possesses a number of stationary points corresponding to various equilibrium configurations [@WiVi2016; @collins2012isomerization]. Some are stable (local minima, or ‘well’), some are unstable (local maxima or ‘hill-top’), and some correspond to saddle points, i.e., a shape with opposite curvature in different directions, but still unstable, having both stable and unstable directions. Interestingly, although difficult to observe experimentally, it is these saddle points that can have a profound organizing effect on global trajectories in a dynamics context. Thus, under a nominally fixed set of loads or a given configuration we may have the situation in which a system is at rest in a position of stable equilibrium, but, given sufficiently large perturbation (input of energy) may transition to a remote stable equilibrium [@virgin2017geometric], or even collapse completely [@das2009symmetry; @das2009pull]. The path taken during this transition is associated with the least energetic route, and this will typically correspond to a passage close to a saddle point: it is easier to take a path around a mountain than going directly over its peak.
For a single mechanical degree of freedom the transition from one potential energy minimum to another is relatively unambiguous [@mann2009energy; @Thompson1984]. We can think of a twin-well oscillator and how it has no choice but to pass over an intermediate hilltop in transitioning to an adjacent minimum. For high-order systems trajectories have many more possible paths. But a system with two mechanical degrees of freedom (configuration space), and thus a 4 dimensional phase space, offers an intermediate situation: compelling conceptual clarity (i.e., the potential energy can be thought of as a surface or landscape), but still retaining a wider range of potential behavior over and above the aforementioned single oscillator (i.e., multiple ways of traversing and perhaps escaping from one potential well to another).
For the two degree of freedom system, the analog of the hilltop is the saddle point of the potential energy surface. The linearized dynamics near such a point yield an oscillatory mode and an exponential mode, with both asymptotically stable and unstable directions. For energies slightly above the saddle point, there is a bottleneck to the energy surface [@NaRo2017; @KoLoMaRo2000]. Transitions from one side of the bottleneck can be understood in terms of sets of trajectories which are bounded by topological cylinders. The transition dynamics, which has in some contexts been known as tube dynamics [@Conley1968; @LlMaSi1985; @OzDeMeMa1990; @DeMeTo1991; @DeLeon1992; @Topper1997; @KoLoMaRo2000; @GaKoMaRo2005; @GaKoMaRoYa2006; @MaRo2006; @KoLoMaRo2011], has been developed for studying transitions between stable states (the potential wells) in a number of disparate contexts, and here it is applied to a structural mechanics situation in which snap-through buckling [@collins2012isomerization] is the key phenomenological transition. Conditions are determined whereby the initial energy imparted to the system is characterized in terms of subsequent escape from the initial potential well.
The Paradigm: Snap-through of an Arch/Buckled Beam
==================================================
A classic example of a saddle-node bifurcation in structural mechanics is the symmetric snap-through buckling of a shallow arch, in an essentially co-dimension 1 bifurcation [@Thompson1984]. However, if the arch (or equivalently a buckled beam) is [*not*]{} shallow then the typical mechanism of instability is an asymmetric snap-through, requiring two modes (symmetric and asymmetric) for characterization, and the instability corresponds to a subcritical pitchfork bifurcation. In both of these cases the transition is sudden and associated with a fast dynamic jump, since there is no longer any locally available stable equilibrium. This behavior is generic regardless of boundary conditions and is also exhibited by the laterally-loaded buckled beam [@Murphy1996; @Wiebe2013]. We shall focus on this latter example, for relative simplicity of introduction. The essential focus here is that the underlying potential energy of this system consists of two potential energy wells (the original unloaded equilibrium and the snapped-through equilibrium), an unstable hilltop (the intermediate, straight, unstable equilibrium) and two saddle-points. The symmetry of this system is broken by small geometric imperfections. The question is: [*how does the system escape its local potential energy well*]{} in a dynamical systems sense?
Suppose we have a moderately buckled beam. If a central point load is applied then the beam deflects initially in a purely symmetric mode, as shown by the red line in Fig. \[fig:arch\](a), following the $\alpha$ loading path.
![ (a) A schematic load-deflection characteristic, (b) the two dominant degrees of freedom.[]{data-label="fig:arch"}](arch1_mod.png){width="100.00000%"}
Upon a quasi-static increase in the load $P$, point $C$ is reached (a subcritical pitchfork bifurcation) and the arch quickly snaps-through (a thoroughly dynamic event) with a significant asymmetric component in the deflection and the system settles into its inverted position $D$ [@virgin2017geometric]. This behavior is captured by considering a two-mode analysis: sag $S$ (symmetric) and angle $A$ (asymmetric), or alternatively we can use the harmonic coordinates $X$ and $Y$, respectively, corresponding to the modes in part (b). In an approximate analysis they might be the lowest two buckling modes or free vibration modes from a standard eigen-analysis. Suppose we load the beam to a value slightly below the snap value at $P_C$, and fix it at that value. In this case there will be the five equilibria mentioned earlier: three equilibria purely in sag (two stable and an unstable one between them), and two saddles, with significant angular components but geometrically opposed [@WiVi2016]. Small geometric imperfections (in $A$ and/or $S$) will break the symmetry and influence which path is more likely to be followed. In this fixed configuration we can then think of the system in dynamic terms, and consider the range of initial conditions (including velocity, perhaps caused by an impact force) that might push the system from a point on path $\alpha$ to a point on path $\phi$.
#### Governing equations
In this analysis a slender buckled beam with thickness $d$, width $b$ and length $L$ is considered. A Cartesian coordinate system $o \textendash xyz$ is established on the mid-plane of the beam in which $o$ is the origin, $x,y$ the directions along the length and width directions and $z$ the downward direction normal to the mid-plane. Based on Euler-Bernoulli beam theory [@zhong2016analysis; @WiVi2016], the displacement field $(u_1,u_3)$ of the beam along $(x,z)$ directions can be written as $$\begin{split}
u_1(x,z,t)&= u(x,t)-z \frac{\partial w(x,t)}{\partial x}\\
u_3(x,z,t)&= w(x,t)
\label{disp_field}
\end{split}$$ where $u(x,t)$ and $w(x,t)$ are the axial and transverse displacements of an arbitrary point on the mid-plane of the beam. Considering the von Kámán-type geometrical nonlinearity, the total axial strain can be obtained as $$\begin{split}
\varepsilon^*_x= \frac{\partial u}{\partial x} - z \frac{\partial^2 w}{ \partial x^2}+ \frac{1}{2} \left(\frac{\partial w}{\partial x}\right)^2
\label{total_strain}
\end{split}$$ For a moderately buckled-beam, we need to consider the initial strain $\varepsilon_0$ produced by initial deflection $w_0$ which is $$\begin{split}
\varepsilon_0 = -z \frac{\partial^2 w_0}{ \partial x^2} + \frac{1}{2} \left(\frac{\partial w_0}{\partial x}\right)^2
\label{init_strain}
\end{split}$$ Then the change in strain $\varepsilon_x$ can be expressed as $$\begin{split}
\varepsilon_x = \varepsilon^*_x - \varepsilon_0 = \frac{\partial u}{\partial x} - z \left(\frac{\partial^2 w}{\partial x^2} - \frac{\partial^2 w_0}{\partial x^2}\right) + \frac{1}{2} \left[ \left(\frac{\partial w}{\partial x}\right)^2 - \left(\frac{\partial w_0}{\partial x}\right)^2\right]
\label{strain_change}
\end{split}$$
Here we just consider homogeneous isotropic materials with Young’s modulus $E$, and allow for the possibility of thermal loading. The axial stress $\sigma_x$ can be obtained according to the one dimensional constitutive equation, as $$\begin{split}
\sigma_x= E \varepsilon_x - E \alpha_x \Delta T
\label{constitutive}
\end{split}$$ where $\alpha_x$ is the thermal expansion coefficient and $\Delta T $ is the temperature increment from the reference temperature at which the beam is in a stress free state. Thermal loading is introduced as a convenient way of controlling the initial equilibrium shapes (and hence the potential energy landscape) via axial loading.
The strain energy $\mathcal{V}(x,z,t)$ is $$\begin{split}
\mathcal{V}(x,z,t) &= \frac{b}{2} \int_0^L \int_{- \frac{d}{2}}^{ \frac{d}{2}} \sigma_x \varepsilon_x \mathrm{d}z \mathrm{d}x
\label{strain_energy}
\end{split}$$ Ignoring the axial inertia term, the kinetic energy $\mathcal{T}(x,z,t)$ of the buckled beam is $$\begin{split}
\mathcal{T}(x,z,t)= \frac{b}{2} \int_0^L \int_{- \frac{d}{2}}^{ \frac{d}{2}} \rho \dot w^2 \mathrm{d}z \mathrm{d}x
\label{kinetic_energy}
\end{split}$$ where $\rho$ is the mass density. In addition, the dot over the quantity is the derivative with respective to time.
The governing equations can be obtained by Hamilton’s principle which requires that $$\begin{split}
\delta \int_{t_0}^t \left[ \mathcal{T}(x,z,t) - \mathcal{V}(x,z,t) \right] \mathrm{d} t
\label{Hamilton_prin} + \int_{t_0}^{t} \delta W_{nc} \mathrm{d}t =0
\end{split}$$ where $\delta$ denotes the variational operator, $t_0$ and $t$ the initial and current time. $\delta W_{nc}$ is the variation of the virtual work done by non-conservative force (damping) which is expressed as $$\begin{split}
\delta W_{nc} = - c_d \dot w \delta w
\end{split}$$ where $c_d$ is the coefficient of (linear viscous) damping. In subsequent analysis, and related to typical practical situations, the damping will be small.
After some manipulation, the governing equations in terms of axial force $N_x$ and bending moment $M_x$ can be obtained as [@zhong2016analysis] $$\begin{split}
&\frac{\partial N_x}{\partial x}=0\\
&\frac{\partial^2 M_x}{\partial x^2}+ N_x \frac{\partial^2 w}{\partial x^2} = \rho A \ddot w + c_d \dot w
\label{govern_eq_force}
\end{split}$$ where $N_x$ and $M_x$ are defined as $$\begin{split}
\left(N_x, M_x \right)=b \int_{- \frac{d}{2}}^{ \frac{d}{2}}
\end{split} \sigma_x \left(1,z \right) \mathrm{d} z
\label{force_definition}$$ By using , and , the force $N_x$ and moment $M_x$ in can be rewritten as $$\begin{split}
N_x&=E A \left[\frac{\partial u}{\partial x} + \frac{1}{2} \left( \left(\frac{\partial w}{\partial x}\right)^2 - \left(\frac{\partial w_0}{\partial x}\right)^2\right) \right] -N_T\\
M_x&=- E I \left(\frac{\partial^2 w}{\partial x^2} - \frac{\partial^2 w_0}{\partial x^2}\right)
\label{force_moment}
\end{split}$$ where $A$ and $I$ denote the cross-sectional area and moment of inertia; $N_T=EA \alpha_x \Delta T$, the axial thermal loads. Thus, $EA$ and $EI$ are the axial stiffness and bending stiffness, respectively.
Here we just consider a clamped-clamped beam with in-plane immovable ends. The boundary conditions are $$\begin{split}
x=0,L: u=w=\frac{\partial w}{\partial x}=0
\label{boundary_condi}
\end{split}$$ Note that from the first equation in , it is clear that the axial force $N_x$ is constant along the axial direction. In this case, integrating the axial force along the $x$ axis and using the boundary conditions $u(0,t)=u(L,t)=0$, one can obtain $$\begin{split}
N_x= \frac{E A}{2L} \int_0^L \left[ \left(\frac{\partial w}{\partial x} \right)^2 - \left(\frac{\partial w_0}{\partial x} \right)^2 \right] \mathrm{d} x -N_T
\label{axial_force}
\end{split}$$
Using $M_x$ in and $N_x$ in , the second equation in can be described in terms of the transverse displacement $w$ as [@WiVi2016] $$\begin{split}
\rho A \ddot{w} + c_d \dot w + EI \left(\frac{\partial^4 w}{\partial x^4} - \frac{\partial^4 w_0}{\partial x^4}\right) + \left[N_T - \frac{EA}{2L} \int_0^L \left( \left(\frac{\partial w}{\partial x} \right)^2 - \left(\frac{\partial w_0}{\partial x} \right)^2 \right) \mathrm{d} x \right] \frac{\partial^2 w}{\partial x^2} =0
\label{eq:PDE}
\end{split}$$ where $w$ and $w_0$ are the current deflection and initial geometrical imperfection, respectively; $\rho$ is the mass density; $c_d$ is the damping coefficient; $A$ and $I$ are the area and the moment of inertia of the cross-section, respectively; $E$ is the Young’s modulus. Given the immovable ends it is natural to consider the effective externally applied axial force to be replaced by a thermal loading term: this is the primary destabilizing nonlinearity in the system.
As mentioned earlier, clamped-clamped boundary conditions are considered. Thus we make use of the mode shapes $$\begin{split}
&\phi_n = \alpha_n \left[\sinh \frac{\beta_n x}{L} - \sin \frac{\beta_n x}{L} + \delta_n \left(\cosh \frac{\beta_n x}{L} - \cos \frac{\beta_n x}{L} \right) \right],\\
&\delta_n = \frac{\sinh \beta_n - \sin \beta_n}{\cos \beta_n - \cosh \beta_n},\\
&\cos \beta_n \cosh \beta_n =1,\\
& \alpha_1 = -0.6186, \ \ \ \alpha_2 = -0.6631
\label{mode:Virgin}
\end{split}$$ and describe the deflected shape in terms of a two-degree-of-freedom approximation $$\begin{split}
w(x,t)&= X(t) \phi_1(x) +Y(t) \phi_2(x),\\
w_0(x)&= \gamma_1 \phi_1(x) + \gamma_2 \phi_2(x)
\end{split}$$ where the initial imperfections are given by $w_0$. Substituting the assumed solution into the equation of motion \[eq:PDE\] yields $$\begin{split}
&\rho A \int_0^L \phi_i \ddot w \mathrm{d}x + c_d \int_0^L \phi_i \dot w \mathrm{d}x+ EI \int_0^L \frac{\partial^2 \phi_i}{\partial x^2} \left(\frac{\partial^2 w}{\partial x^2} - \frac{\partial^2 w_0}{\partial x^2} \right) \mathrm{d} x\\
& - \left[N_T - \frac{EA}{2L} \int_0^L \left(\left(\frac{\partial w}{\partial x} \right)^2 - \left(\frac{\partial w_0}{\partial x} \right)^2 \right) \mathrm{d}x \right] \int_0^L \frac{\partial \phi_i}{\partial x} \frac{\partial w}{\partial x} \mathrm{d}x =0
\label{virtual}
\end{split}$$ Using the specific forms of $\phi_i$ in and noticing each mode shape is orthogonal, the nonlinear equations can be obtained $$\begin{split}
& M_1 \ddot X + C_1 \dot X + K_1 \left(X - \gamma_1 \right) - N_T G_1 X - \frac{EA}{2L}G_1^2 \left(\gamma_1^2 X -X^3 \right) - \frac{EA}{2L} G_1 G_2 \left(\gamma_2^2 X -X Y^2 \right)=0\\
& M_2 \ddot Y + C_2 \dot Y + K_2 \left(Y - \gamma_2 \right) - N_T G_2 Y - \frac{EA}{2L}G_2^2 \left(\gamma_2^2 Y -Y^3 \right) - \frac{EA}{2L} G_1 G_2 \left(\gamma_1^2 Y -X^2 Y \right)=0
\label{odes}
\end{split}$$ where $$\begin{split}
\left(M_i, C_i \right) = \left(\rho A, c_d \right) \int_0^L \phi_i^2 \mathrm{d}x, \ \ K_i = EI \int_0^L \left(\frac{\partial^2 \phi_i}{\partial x^2} \right)^2 \mathrm{d}x, \ \ G_i = \int_0^L \left(\frac{\partial \phi_i}{\partial x} \right)^2 \mathrm{d}x
\label{Galerkin-coefficient}
\end{split}$$ The kinetic energy and potential energy, respectively, can be represented as $$\begin{split}
\mathcal{T}(\dot X, \dot Y)=& \frac{1}{2} M_1 \dot X^2 + \frac{1}{2} M_2 \dot Y^2,\\
\mathcal{V}(X, Y)=& - K_1 \gamma_1 X - K_2 \gamma_2 Y + \frac{1}{2} K_1 X^2 + \frac{1}{2} K_2 Y^2 - \frac{1}{2} N_T\left( G_1 X^2 + G_2 Y^2 \right) \\
& - \frac{EA}{2L}G_1^2 \left(\frac{1}{2}\gamma_1^2 X^2 - \frac{1}{4}X^4 \right) - \frac{EA}{2L}G_2^2 \left(\frac{1}{2} \gamma_2^2 Y^2 - \frac{1}{4}Y^4 \right) \\
& - \frac{EA}{2L} \frac{G_1 G_2}{2} \left(\gamma_2^2 X^2 + \gamma_1^2 Y^2 -X^2 Y^2 \right).\\
% & + \frac{EA}{8L} \left(\gamma_1^2 G_1 + \gamma_2^2 G_2 \right)^2 + \frac{1}{2} K_1 \gamma_1^2 + \frac{1}{2} K_2 \gamma_2^2
\end{split}$$ For physically reasonable coefficients we have a number of equilibrium possibilities. For small values of $N_T$ we have an essentially linear system, dominated by the trivial (straight) equilibrium configuration, and thus an isolated center (minimum of the potential energy). This relates back to the situation in Figure \[fig:arch\] for a small value of $P$. But for larger values of $P$, for example a little below $P_c$, the system typically possesses a number of equilibria, some of which are stable and some of which are not. Some typical forms are shown in Figure \[fig:shape\](a) in which the five dots are the equilibrium points where W$_1$ and W$_2$ are within the two stable wells; S$_1$ and S$_2$ two unstable saddle points; H the unstable hilltop. Thus, we might have the system sitting (in equilibrium) at point W$_1$. If it is then subject to a disturbance [*with the right size and direction*]{} (in the dynamical context), we might expect the system to transition to the remote equilibrium at W$_2$. This might occur when the system is subject to a large impact force, for example [@Wiebe2013]. It is anticipated (and will later be shown) that the typically easiest transition will be associated with (an asymmetric) passage close to S$_1$ or S$_2$, and generally avoiding H. In Figure \[fig:shape\](b) is shown the same system but now with a small geometric imperfection in both modes (i.e., $\gamma_1 \ne 0$ and $\gamma_2 \ne 0$). In this case the symmetry of the system is broken, and given the relative energy associated with the saddle points it is anticipated (and will also be shown later) that optimal escape will tend to occur via S$_1$.
![ Contours of potential energy: (a) the symmetric system, $\gamma_1 = \gamma_2 = 0$, (b) with small initial imperfections in both modes, i.e., $\gamma_1$ and $\gamma_2 $ are nonzero.[]{data-label="fig:shape"}](shape_mod.png){width="100.00000%"}
Note that eqs. can also be obtained from Lagrange’s equations, $$\frac{d}{dt}\left( \frac{\partial \mathcal{L}}{ \partial \dot q_i}\right) -\frac{\partial \mathcal{L}}{ \partial q_i} = - C_i \dot q_i, \quad i =1, 2$$ when $q_1 = X$ and $q_2=Y$, and the Lagrangian is $$\mathcal{L}(X,Y,\dot X,\dot Y) = \mathcal{T}(\dot X, \dot Y) - \mathcal{V}(X, Y)$$
To transform this to a Hamiltonian system, one defines the generalized momenta, $$\begin{split}
p_i = \frac{\partial \mathcal{L}}{ \partial \dot q_i} = M_i \dot q_i
\end{split}$$ so $p_X = M_1 \dot X$ and $p_Y = M_2 \dot Y$, in which case, the kinetic energy is $$\mathcal{T}(X,Y,p_X,p_Y) = \frac{1}{2 M_1} p_X^2 + \frac{1}{2 M_2} p_Y^2$$ and the Hamiltonian is $$\mathcal{H}(X,Y,p_X,p_Y) = \mathcal{T} + \mathcal{V}$$ and Hamilton’s equations (with damping) [@Greenwood2003] are $$\begin{split}
\dot X &= \frac{\partial \mathcal{H}}{\partial p_X}=\frac{p_X}{M_1} \\
\dot Y &= \frac{\partial \mathcal{H}}{\partial p_Y}=\frac{p_y}{M_2} \\
\dot p_X &= - \frac{\partial \mathcal{H}}{\partial X} - C_H p_X=- \frac{\partial \mathcal{V}}{\partial X} - C_H p_X \\
\dot p_Y &= - \frac{\partial \mathcal{H}}{\partial Y} - C_H p_Y=- \frac{\partial \mathcal{V}}{\partial Y} - C_H p_Y \\
\label{eq:eomHam}
\end{split}$$ where $$\begin{split}
\frac{\partial \mathcal{V}}{\partial X}=& K_1 \left(X - \gamma_1 \right) - N_T G_1 X - \frac{EA}{2L}G_1^2 \left(\gamma_1^2 X -X^3 \right) - \frac{EA}{2L} G_1 G_2 \left(\gamma_2^2 X -X Y^2 \right),\\
\frac{\partial \mathcal{V}}{\partial Y}=& K_2 \left(Y - \gamma_2 \right) - N_T G_2 Y - \frac{EA}{2L}G_2^2 \left(\gamma_2^2 Y -Y^3 \right) - \frac{EA}{2L} G_1 G_2 \left(\gamma_1^2 Y -X^2 Y \right)
\end{split}$$ and $C_H=C_1/M_1=C_2/ M_2$ is the damping coefficient in the Hamiltonian system which can be easily found by comparing and , and using the relations of $M_i$ and $C_i$ in .
We assume the lower saddle point S$_1$ has the smaller potential energy compared to S$_2$, thus the energy of S$_1$ is the critical energy for snap-though, and we initially focus attention on the dynamic behavior around the region of S$_1$. The linearized equations of about S$_1$ with position $(X_e,Y_e)$ can be written as $$\begin{split}
\dot x&= \frac{p_x}{M_1}\\
\dot y&= \frac{p_y}{M_2}\\
\dot p_x&= A_{31} x + A_{32} y - C_H p_x\\
\dot p_y&= A_{32} x + A_{42} y - C_H p_y
\label{linearization}
\end{split}$$ where $(x,y,p_x,p_y)= (X,Y,p_X,p_Y) - (X_e,Y_e,0,0)$ and $$\begin{split}
& A_{31}= -K_1 + N_T G_1 + \frac{E A G_1^2 \left(\gamma_1^2 - 3 X_e^2 \right)}{2L} + \frac{E A G_1 G_2 \left(\gamma_2^2 -Y_e^2 \right)}{2L},\\
& A_{32}= - \frac{E A G_1 G_2 X_e Y_e}{L},\\
& A_{42}= -K_2 + N_T G_2 + \frac{E A G_2^2 \left(\gamma_2^2 - 3 Y_e^2 \right)}{2L} + \frac{E A G_1 G_2 \left(\gamma_1^2 -X_e^2 \right)}{2L}
\label{lin paras}
\end{split}$$
If we replace the position of S$_1$ by the position of W$_1$, we can still use the linearized equations in to obtain the natural frequencies of the shallow arch near W$_1$ as $$\omega_{1,2}^{(d)} = w_{1,2}^{(c)} \sqrt{1-\xi_{1,2}^2}$$ where $\omega_{1,2}^{(c)}$ are the first two natural frequencies for the conservative system and $\xi_{1,2}$ are the viscous damping factors with the forms $$\omega_{1,2}^{(c)}=\frac{(b_{\omega} \mp \sqrt{b_{\omega}^2 - 4 c_{\omega}} )}{2}, \hspace{0.5in} \xi_{1,2} = \frac{C_H}{2 \omega_{1,2}^{(c)}}$$ and $$b_{\omega}=- \frac{A_{31}}{M_1} - \frac{A_{42}}{M_2}, \hspace{0.5in} c_{\omega}=\frac{A_{31} A_{42} - A_{32}^2}{M_1 M_2}$$
#### Non-dimensional equations of motion
In order to reduce the parameters, some non-dimensional quantities are introduced, $$\begin{split}
&\left( L_x, L_y \right)= L \left(1,\sqrt{ \frac{M_1}{M_2}} \right), \omega_0= \frac{ \sqrt{- A_{32} }}{ \left( M_1 M_2\right)^ \frac{1}{4}}, \tau= \omega_0 t, \left(\bar q_1 , \bar q_2 \right)= \left( \frac{x}{L_x}, \frac{y}{L_y} \right),\\
&\left(\bar p_1 , \bar p_2 \right)= \frac{1}{\omega_0} \left( \frac{p_x}{ L_x M_1},\frac{p_y}{ L_y M_2}\right), \left( c_x , c_y \right)= \frac{1}{ \omega_{0}^2} \left( \frac{A_{31}}{M_1}, \frac{A_{42}}{M_2} \right), c_1= \frac{C_H}{\omega_0}
\label{dimless quan}
\end{split}$$
Using the non-dimensional parameters in , the non-dimensional linearized equations are written as $$\begin{split}
\dot {\bar q}_1 &= \bar p_1,\\
\dot {\bar q}_2 &= \bar p_2,\\
\dot {\bar p}_1 &= c_x \bar q_1 - \bar q_2 - c_1 \bar p_1,\\
\dot {\bar p}_2 &= - \bar q_1 + c_y \bar q_2 - c_1 \bar p_2
\label{nond eq}
\end{split}$$ Written in matrix form, with column vector $\bar z=(\bar q_1 , \bar q_2 , \bar p_1 , \bar p_2)$, we have $$\dot {\bar z} = A \bar z + D \bar z$$ where $$A = \begin{pmatrix}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
c_x & -1 & 0 & 0 \\
-1 & c_y & 0 & 0
\end{pmatrix},
\hspace{0.5in}
D = \begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & -c_1 & 0 \\
0 & 0 & 0 & -c_1
\end{pmatrix}
\label{A_and_D_matrix}$$ are the Hamiltonian part and damping part of the linear equations, respectively.
Linearized Conservative Hamiltonian System {#linearization of conserve}
==========================================
Solutions near the equilibria
-----------------------------
#### Eigenvalues and eigenvectors
In this section, we will discuss the linear dynamical behaviors of a buckled beam in the Hamiltonian system without taking account of any energy dissipation which makes $c_1=0$ (i.e., $C_H = 0$). Thus, the equations of motion are given as $$\dot {\bar z} = A \bar z
\label{conservative eqns}$$ The system can be viewed as resulting from a quadratic Hamiltonian, $$\mathcal{H}_2= \tfrac{1}{2}\bar p_1 ^2 + \tfrac{1}{2}\bar p_2 ^2
- \tfrac{1}{2}c_x \bar q_1 ^2 - \tfrac{1}{2}c_y \bar q_2 ^2 + \bar q_1 \bar q_2
\label{H_2_bar}$$ which can be written in matrix form $$\mathcal{H}_2 = \frac{1}{2} \bar z ^T B \bar z$$ where $$\begin{split}
B = J^{T}A =\begin{pmatrix}
-c_x & 1 & 0 & 0 \\
1 & -c_y & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
\end{split}$$ and ${J}$ is the $4 \times 4$ canonical symplectic matrix $$\begin{aligned}
\begin{split}
{J} = \begin{pmatrix}
{\bf 0} & {I}_2\\
-{I}_2 & {\bf 0}\\
\end{pmatrix}
\end{split}\end{aligned}$$ where ${I}_2$ is the $2 \times 2$ identity matrix.
The characteristic polynomial of is $$\begin{aligned}
p( \beta ) = \beta^4 - ( c_x + c_y ) \beta^2 + c_x c_y - 1\end{aligned}$$ Let $\alpha= \beta^2$, then the roots of $ p(\alpha)=0 $ are as follows $$\begin{split}
\alpha_1= \frac{c_x + c_y + \sqrt{ \left( c_x -c_y \right)^2 + 4}}{2},\\ \alpha_2= \frac{c_x + c_y - \sqrt{ \left( c_x - c_y \right)^2 + 4}}{2}
\label{charoots}
\end{split}$$
Generally, in $c_x > 0$ and $c_y <0$. In this case, $\alpha_1 > 0$ and $\alpha_2 < 0$. It follows that this equilibrium point is of the type saddle $\times$ center. Here we define $\lambda=\sqrt{\alpha_1}$ and $\omega_p=\sqrt{-\alpha_2}$. Thus, the eigenvectors are given by $$\begin{split}
\left(1, c_x - \beta^2 , \beta , c_x \beta -\beta^3 \right),
\label{gener egvec}
\end{split}$$ where $\beta$ denotes one of the eigenvalues.
After substituting $\beta = i \omega_p $ into and separating real and imaginary parts as $u_{\omega_p} + i v_{\omega_p}$, we obtain two corresponding eigenvectors $$\begin{split}
u_ {\omega_p}&= \left( 1, c_x+ \omega_p^2 , 0 , 0 \right),\\
v_ {\omega_p}&= \left( 0, 0 , \omega_p ,c_x \omega_p+ \omega_p^3 \right),
\label{eivect I}
\end{split}$$
Moreover, the other two eigenvectors associated with the pair of real eigenvalues $\pm \lambda $ can be taken as $$\begin{split}
u_{+ \lambda}&= \left( 1, c_x - \lambda^2 , \lambda , c_x \lambda - \lambda^3 \right),\\
u_{- \lambda}&=- \left( 1, c_x - \lambda^2 , - \lambda , \lambda^3 - c_x \lambda \right)
\label{eivect R}
\end{split}$$
#### Symplectic change of variables
We consider the linear symplectic change of variables from $(\bar q_1, \bar q_2, \bar p_1, \bar p_2)$ to $(q_1 , q_2 , p_1 , p_2)$, $$\left( \begin{array}{c c c c} \bar q_1 \\ \bar q_2 \\ \bar p_1 \\ \bar p_2
\end{array}
\right) =
C
\left(
\begin{array}{c c c c} q_1 \\ q_2 \\ p_1 \\ p_2
\end{array}
\right)$$ \[CT\] where the columns of the matrix $C$ are given by the eigenvectors, $$\begin{split}
C=\left(u_{ + \lambda} , u_{ \omega_p} , u_{ - \lambda} , v_{\omega_p} \right)
\label{change matrix}
\end{split}$$ and where the vectors are written as column vectors.
Then we find $$\begin{split}
C^T J C=\begin{pmatrix}
0 & \bar D\\
-\bar D & 0
\end{pmatrix}, \hspace{0.5in}
\bar D= \begin{pmatrix}
d_\lambda & 0\\
0 & d_{\omega_p}
\end{pmatrix}
\end{split}$$ where $$\begin{split}
d_\lambda &= \lambda [4 - 2 (c_x -c_y) ( \lambda^2 -c_x)]\\
d_{\omega_p}&= \frac{\omega_p}{2} [4 +2 (c_x - c_y) ( \omega_p^2 + c_x)]
\end{split}$$
In order to obtain a symplectic form which satisfies $C^T J C =J$, we need to rescale the columns of $C$. The scaling is given by factors $s_1 = \sqrt{ d_\lambda}$ and $s_2 = \sqrt{d_{\omega_p}}$. In this case, the final form of the symplectic matrix $C$ is given by $$\begin{split}
C= \begin{pmatrix}
\frac{1}{s_1} & \frac{1}{s_2} & - \frac{1}{s_1} & 0\\
\frac{c_x - \lambda^2}{s_1} & \frac{ \omega_p^2 + c_x}{s_2} & \frac{\lambda^2 - c_x}{s_1} & 0\\
\frac{\lambda}{s_1} & 0 & \frac{ \lambda}{s_1} & \frac{\omega_p}{s_2} \\
\frac{c_x \lambda - \lambda^3}{s_1} & 0 & \frac{c_x \lambda - \lambda^3}{s_1} & \frac{c_x \omega_p + \omega_p^3 }{s_2}
\end{pmatrix}
\label{sym matrix}
\end{split}$$
The Hamiltonian can be rewritten in the simplified, normal form, $$\label{Enlin}
\mathcal{H}_2= \lambda q_1 p_1 + \tfrac{1}{2} \omega_p(q_2 ^2 +p_2^2)$$ with corresponding linearized equations, $$\begin{split}
\dot q_1&= ~~\lambda q_1, \\
\dot p_1&= - \lambda p_1,\\
\dot q_2&= ~~\omega_p p_2, \\
\dot p_2&= - \omega_p q_2
\label{new equa}
\end{split}$$ Written in matrix form, with column vector $z=(q_1 , q_2 , p_1 , p_2)$, we have $$\dot z = \Lambda z$$ where $$\begin{split}
\Lambda = C^{-1}AC = \left( \begin{array}{rrrr}
\lambda & 0 & 0 & 0 \\
0 & 0 & 0 & \omega_p \\
0 & 0 & -\lambda & 0 \\
0 & -\omega_p & 0 & 0
\end{array} \right)
\label{Lambda_matrix}
\end{split}$$
The solution of can be written as $$\begin{split}
& q_1= q_1^0 e^{ \lambda t}, \ \ \ p_1= p_1^0 e^{ - \lambda t}\\
& q_2 + i p_2 = \left(q_2^0 + i p_2^0 \right) e^{-i \omega_p t}
\end{split}$$ Note that the three functions $$f_1 = q_1 p_1, \quad
f_2 = q_2^2 + p_2^2, \quad
f_3 = \mathcal{H}_2$$ are constants of motion under the Hamiltonian system .
Boundary of transit and non-transit orbits {#sec:separatrix}
------------------------------------------
#### The Linearized Phase Space
For positive $h$ and $c$, the equilibrium or bottleneck region $\mathcal{R}$ (sometimes just called the neck region), which is determined by $$\mathcal{H}_2=h, \quad \mbox{and} \quad |p_1-q_1|\leq c,$$ is homeomorphic to the product of a 2-sphere and an interval $I$, $S^2\times I$; namely, for each fixed value of $p_1 -q_1 $ in the interval $I=[-c,c]$, we see that the equation $\mathcal{H}_2=h$ determines a 2-sphere $$\label{2-sphere}
\tfrac{\lambda }{4}(q_1 +p_1 )^2
+ \tfrac{1}{2}\omega_p (q_2^2+p_2^2)
=h+\tfrac{\lambda }{4}(p_1 -q_1 )^2.$$ Suppose $a \in I$, then can be re-written as $$\label{2-sphere2}
x_1^2 + q_2^2+p_2^2
= r^2,$$ where $x_1 = \sqrt{\tfrac{1 }{2}\tfrac{\lambda}{\omega_p}}(q_1 +p_1 )$ and $r^2=\tfrac{2}{\omega_p}(h+\tfrac{\lambda }{4}a^2)$, which defines a 2-sphere of radius $r$ in the three variables $x_1$, $q_2$, and $p_2$.
The bounding 2-sphere of $\mathcal{R}$ for which $p_1 -q_1 = c$ will be called $n_1$ (the “left” bounding 2-sphere), and that where $p_1 -q_1 = -c$, $n_2$ (the “right” bounding 2-sphere). See Figure \[fig5\].
![\[fig5\][ The flow in the equilibrium region has the form saddle $\times$ center. On the left is shown the projection onto the $(p_1,q_1)$ plane, the saddle projection. For the conservative dynamics, the Hamiltonian function $\mathcal{H}_2$ remains constant at $h>0$. Shown are the periodic orbit (black dot at the center), the asymptotic orbits (labeled A), two transit orbits (T) and two non-transit orbits (NT). ]{}](linear_projections_n_mod.png){width="\textwidth"}
We call the set of points on each bounding 2-sphere where $q_1 + p_1 = 0$ the equator, and the sets where $q_1 + p_1 > 0$ or $q_1 + p_1 < 0$ will be called the northern and southern hemispheres, respectively.
#### The Linear Flow in $\mathcal{R}$
To analyze the flow in $\mathcal{R}$, consider the projections on the ($q_1, p_1$)-plane and the $(q_2,p_2)$-plane, respectively. In the first case we see the standard picture of a saddle point in two dimensions, and in the second, of a center consisting of harmonic oscillator motion. Figure \[fig5\] schematically illustrates the flow. With regard to the first projection we see that $\mathcal{R}$ itself projects to a set bounded on two sides by the hyperbola $q_1p_1 = h/\lambda $ (corresponding to $q_2^2+p_2^2=0$, see ) and on two other sides by the line segments $p_1-q_1= \pm c$, which correspond to the bounding 2-spheres, $n_1$ and $n_2$, respectively.
Since $q_1p_1$ is an integral of the equations in $\mathcal{R}$, the projections of orbits in the $(q_1,p_1)$-plane move on the branches of the corresponding hyperbolas $q_1p_1 =$ constant, except in the case $q_1p_1=0$, where $q_1 =0$ or $p_1 =0$. If $q_1p_1 >0$, the branches connect the bounding line segments $p_1 -q_1 =\pm c$ and if $q_1p_1 <0$, they have both end points on the same segment. A check of equation shows that the orbits move as indicated by the arrows in Figure \[fig5\].
To interpret Figure \[fig5\] as a flow in $\mathcal{R}$, notice that each point in the $(q_1,p_1)$-plane projection corresponds to a 1-sphere $S^1$ in $\mathcal{R}$ given by $$q_2^2+p_2^2
=\tfrac{2 }{\omega_p}(h-\lambda q_1p_1) .$$ Of course, for points on the bounding hyperbolic segments ($q_1p_1 =h/\lambda $), the 1-sphere collapses to a point. Thus, the segments of the lines $p_1-q_1 =\pm c$ in the projection correspond to the 2-spheres bounding $\mathcal{R}$. This is because each corresponds to a 1-sphere crossed with an interval where the two end 1-spheres are pinched to a point.
We distinguish nine classes of orbits grouped into the following four categories:
1. The point $q_1 =p_1 =0$ corresponds to an invariant 1-sphere $S^1_h$, an unstable [**period orbit**]{} in $\mathcal{R}$. This 1-sphere is given by $$\label{3-sphere}
q_2^2+p_2^2=\tfrac{2 }{\omega_p}h,
\hspace{0.3in} q_1 =p_1 =0.$$ It is an example of a normally hyperbolic invariant manifold (NHIM) (see [@Wiggins1994]). Roughly, this means that the stretching and contraction rates under the linearized dynamics transverse to the 1-sphere dominate those tangent to the 1-sphere. This is clear for this example since the dynamics normal to the 1-sphere are described by the exponential contraction and expansion of the saddle point dynamics. Here the 1-sphere acts as a “big saddle point”. See the black dot at the center of the $(q_1,p_1)$-plane on the left side of Figure \[fig5\].
2. The four half open segments on the axes, $q_1p_1 =0$, correspond to four cylinders of orbits asymptotic to this invariant 1-sphere $S^1_h$ either as time increases ($p_1 =0$) or as time decreases ($q_1 =0$). These are called [**asymptotic**]{} orbits and they form the stable and the unstable manifolds of $S^1_h$. The stable manifolds, $W^s_{\pm}(S^1_h)$, are given by $$\label{stable_manifold}
q_2^2+p_2^2=\tfrac{2 }{\omega_p}h,
\hspace{0.3in} q_1 =0,
\hspace{0.3in} p_1 ~{\rm arbitrary}.$$ $W^s_+(S^1_h)$ (with $p_1>0$) is the branch going entering from $n_1$ and $W^s_-(S^1_h)$ (with $p_1<0$) is the branch going entering from $n_2$. The unstable manifolds, $W^u_{\pm}(S^1_h)$, are given by $$\label{unstable_manifold}
q_2^2+p_2^2=\tfrac{2 }{\omega_p}h,
\hspace{0.3in} p_1 =0,
\hspace{0.3in} q_1 ~{\rm arbitrary}$$ $W^u_+(S^1_h)$ (with $q_1>0$) is the branch exiting from $n_2$ and $W^u_-(S^1_h)$ (with $q_1<0$) is the branch exiting from $n_1$. See the four orbits labeled A of Figure \[fig5\].
3. The hyperbolic segments determined by $q_1p_1 ={\rm constant}>0$ correspond to two cylinders of orbits which cross $\mathcal{R}$ from one bounding 2-sphere to the other, meeting both in the same hemisphere; the northern hemisphere if they go from $p_1-q_1 =+c$ to $p_1-q_1 =-c$, and the southern hemisphere in the other case. Since these orbits transit from one realm to another, we call them [**transit**]{} orbits. See the two orbits labeled T of Figure \[fig5\].
4. Finally the hyperbolic segments determined by $q_1p_1 = {\rm
constant}<0$ correspond to two cylinders of orbits in $\mathcal{R}$ each of which runs from one hemisphere to the other hemisphere on the same bounding 2-sphere. Thus if $q_1 >0$, the 2-sphere is $n_1$ ($p_1
-q_1 =-c$) and orbits run from the southern hemisphere ($q_1 +p_1 <0$) to the northern hemisphere ($q_1
+p_1 >0$) while the converse holds if $q_1 <0$, where the 2-sphere is $n_2$. Since these orbits return to the same realm, we call them [**non-transit**]{} orbits. See the two orbits labeled NT of Figure \[fig5\].
Trajectories in the neck region
-------------------------------
We now examine the appearance of the orbits in configuration space, that is, in $(\bar q_1,\bar q_2)$-plane. In configuration space, $\mathcal{R}$ appears as the neck region connecting two realms, so trajectories in $\mathcal{R}$ will be transformed back to the neck region. It should pointed out that at each moment in time, all trajectories must evolve within the energy boundaries which are zero velocity curves (corresponding to $ \bar p_1 = \bar p_2 = 0$) given by solving for $\bar q_2$ as a function of $\bar q_1$, $$\bar q_2( \bar q_1) = \frac{\bar q_1 \pm \sqrt{\bar q_1^2 - 2 c_y (h + \tfrac{c_x}{2} \bar q_1^2)}}{c_y}$$
Recall that in order to obtain the analytical solutions for $\bar z = (\bar q_1, \bar q_2, \bar p_1, \bar p_2)$, system $\bar z$ has been transformed into system $z = (q_1, q_2, p_1, p_2)$ by using the symplectic matrix $C$ consisting of generalized (re-scaled) eigenvectors $u_{+ \lambda}, u_{- \lambda}, u_{\omega_p}, v_{\omega_p}$ with corresponding eigenvalues $ \pm \lambda$ and $\pm i \omega_p$. Thus, the system $z$ should be transformed back to system $\bar z$ which generates the following general (real) solution with the form $$\begin{split}
\bar z(t) = \left(\bar q_1, \bar q_2, \bar p_1, \bar p_2 \right)^T = q_1^0 e^{\lambda t} u_{+ \lambda} + p_1^0 e^{- \lambda t} u_{- \lambda} + \mathrm{Re} \left[\beta_0 e^{- i \omega_p t} \left(u_{\omega_p} - i v_{\omega_p} \right) \right]
\label{whole-gener-sol}
\end{split}$$ where $q_1^0, p_1^0$ are real and $\beta_0 = q_2^0 + i p_2^0$ is complex.
Upon inspecting this general solution, we see that the solutions on the energy surface fall into different classes depending upon the limiting behaviors of $ \bar q_1, \bar q_2$ as $t$ tends to plus or minus infinity. Notice that $$\begin{split}
\bar q_1(t) &= \frac{q_1^0}{s_1} e^{\lambda t} - \frac{p_1^0}{s_1} e^{-\lambda t} + \frac{1}{s_2} \left( q_2^0 \cos \omega_p t + p_2^0 \sin \omega_p t \right)\\
\bar q_2(t) &= \frac{c_x-\lambda^2}{s_1} q_1^0 e^{\lambda t} + \frac{\lambda^2 - c_x}{s_1} p_1^0 e^{-\lambda t} + \frac{\omega_p^2 +c_x}{s_2} \left( q_2^0 \cos \omega_p t + p_2^0 \sin \omega_p t \right)\\
\label{conser-sol}
\end{split}$$ Thus, if $t\rightarrow + \infty$, then $\bar q_1 (t)$ is dominated by its $q_1^0$ term. Hence, $\bar q_1 (t)$ tends to minus infinity (staying on the left-hand side), is bounded (staying around the equilibrium point), or tends to plus infinity (staying on the right-hand side) according to $q_1^0 < 0 $, $q_1^0=0$ and $q_1^0>0$. See Figure \[fig:Conley\]. The same statement holds if $t\rightarrow - \infty$ and $-p_1^0$ replaces $q_1^0$. Different combinations of the signs of $q_1^0$ and $p_1^0$ will give us again the same nine classes of orbits which can be grouped into the same four categories.
1. If $q_1^0 = p_1^0 = 0$, we obtain a periodic solution. The periodic orbit projects onto the $(\bar q_1, \bar q_2)$ plane as a line with the following expression $$\begin{split}
\bar q_1 & = \frac{1}{s_2} \left( q_2^0 \cos \omega_p t + p_2^0 \sin \omega_p t \right)\\
\bar q_2 & = \frac{ \omega_p^2 + c_x}{s_2} \left(q_2^0 \cos \omega_p t + p_2^0 \sin \omega_p t \right)\\
& =\left( \omega_p^2 + c_x \right) \bar q_1
\end{split}$$ Notice and $\mathcal{H}_2$ now can be rewritten as $\mathcal{H}_2 = \omega_p |\beta_0|^2 / 2$. Thus, since $\mathcal{H}_2=h$, the length of the periodic orbit is $\sqrt{ 2 h \left[(\omega_p^2 + c_x)^2+1 \right] / \left(\omega_p s_2^2 \right)}$. Note that the length of the line goes to zero with $h$.
2. Orbits with $q_1^0 p_1^0=0$ are asymptotic orbits. They are asymptotic to the periodic orbit.
1. When $q_1^0=0$ , the general solutions for $\bar q_1, \bar q_2$ are $$\begin{split}
\bar q_1 &= - \frac{p_1}{s_1}+\frac{q_2}{s_2} \\
\bar q_2 &= \frac{ \lambda^2 - c_x}{s_1} p_1 + \frac{\omega_p^2 + c_x}{s_2} q_2 \\
& = \left(c_x - \lambda^2 \right) \bar q_1 + \frac{\lambda^2 + \omega_p^2}{s_2} \left(q_2^0 \cos \omega_p t + p_2^0 \sin \omega_p t \right)
\end{split}$$ Thus, the orbits with $q_1^0=0$ project into a strip $S$ in the $(\bar q_1, \bar q_2)$-plane bounded by $$\begin{split}
\bar q_2 =\left(c_x - \lambda^2 \right) \bar q_1 \pm \frac{\lambda^2 + \omega_p^2}{s_2} \sqrt{\frac{2 h}{\omega_p}}
\label{strip boundary}
\end{split}$$
2. For $p_1^0=0$, following the same procedure as $q_1^0=0$, we have $$\begin{split}
\bar q_1 &= \frac{q_1}{s_1} + \frac{q_2}{s_2}\\
\bar q_2 &= \left(c_x - \lambda^2\right) \bar q_1 + \frac{\lambda^2 + \omega_p^2}{s_2} \left(q_2^0 \cos \omega_p t + p_2^0 \sin \omega_p t \right)
\end{split}$$ Notice that these two asymptotic orbits with $q_1^0=0$ and $p_1^0=0$ share the same strip $S$ and the same boundaries governed by . Also, since the slopes of the periodic orbit and the strip satisfies $\left(c_x - \lambda^2 \right) \left(c_x + \omega_p^2 \right)=-1$, the periodic orbit is perpendicular to the strip. In other words, the length of the periodic orbit is exactly the same as the width of the strip.
3. Orbits with $q_1^0 p_1^0 >0$ are transit orbits because they cross the equilibrium region $R$ from $- \infty$ (the left-hand side) to $+ \infty$ (the right-hand side) or vice versa.
4. Orbits with $q_1^0 p_1^0 <0$ are non-transit orbits
![ The flow of the conservative system in $\mathcal{R}$, the equilibrium region projected onto the $xy$ configuration space, for a fixed value of energy, $\mathcal{H}_2=h>0$. For any point on the bounding vertical lines $n_1$ or $n_2$ (dashed), there is a wedge of velocity directions inside of which the trajectories are transit orbits, and outside of which are non-transit orbits. The boundary of the wedge gives the orbits asymptotic to the single unstable periodic orbit in the neck for this energy. Shown are a typical asymptotic orbit; two transit orbits (dashed); and two non-transit orbits (dotted). []{data-label="fig:Conley"}](lin_conser_position_local_paper.png){width="67.00000%"}
To study the flow in position space, Figure \[fig:Conley\] gives the four categories of orbits. From , we can see that for transit orbits and non-transit orbits, the signs of $q_1^0 p_1^0$ must satisfy $q_1^0 p_1^0>0$ and $q_1^0 p_1^0<0$,respectively.
In Figure \[fig:Conley\], $S$ is the strip mentioned above. Outside of the strip, the signs of $q_1^0$ and $p_1^0$ are independent of the direction of the velocity. These signs can be determined in each of the components of the equilibrium region $\mathcal{R}$ complementary to the strip. For example, in the left two components, $q_1^0<0$ and $p_1^0>0$, while in the right two components $q_1^0>0$ and $p_1^0<0$. Therefore, $q_1^0 p_1^0<0$ in all components and only non-transit orbits project on to these four components.
Inside the strip the situation is more complicated since in $S$ the signs of $q_1^0$ and $p_1^0$ depend on the direction of the velocity. At each position $(\bar q_1, \bar q_2)$ inside the strip there exists the so-called ‘wedge’ of velocities in which $q_1^0 p_1^0>0$ which was first found by Conley (1968) [@Conley1968] in the restricted three-body problem. See the shaded wedges in Figure \[fig:Conley\]. The existence and the angle of the wedge of velocity will be given in the next part. For simplicity we have indicated this dependence only on the two vertical bounding line segments in Figure \[fig:Conley\]. For example, consider the intersection of strip $S$ with left-most vertical line. On the subsegment so obtained there is at each point a wedge of velocity in which both $q_1^0$ and $p_1^0$ are positive, so that orbits with velocity interior to the wedge are transit orbits $(q_1^0 p_1^0>0)$. Of course, orbits with velocity on the boundary ot the wedge are asymptotic $(q_1^0 p_1^0 = 0)$, while orbits with velocity outside of the wedge are non-transit. The situation on the other subsegment is similar.
#### The wedge of velocities
To establish the wedge of velocity and obtain its angle, we need to use the following fact that all the inner products of one generalized eigenvector and another generalized eigenvector associated with $B$ are zero except for $$\begin{split}
u_{+ \lambda}^T B u_{- \lambda} &=u_{- \lambda}^T B u_{+ \lambda} = \lambda \\
u_{\omega_p}^T B u_{\omega_p} &= v_{\omega_p}^T B v_{\omega_p} = \omega_p \\
\end{split}$$
Using this condition, we have the following relations, as $$\begin{split}
\lambda &= u_{+ \lambda}^T B u_{- \lambda}\\
\Rightarrow \lambda q_1^0 &= q_1^0 u_{+ \lambda}^T B u_{- \lambda}\\
\Rightarrow \lambda q_1^0 &= e^{- \lambda t} \left(q_1^0 e^{\lambda t} u_{+ \lambda} \right)^T B u_{- \lambda}\\
\Rightarrow \lambda q_1^0 &= e^{- \lambda t} \bar z^T B u_{- \lambda}\\
\Rightarrow \lambda q_1^0 &= e^{- \lambda t} \left(\frac{\lambda^2}{s_1} \bar q_1 - \frac{ 1-c_xc_y + c_y \lambda^2}{s_1} \bar q_2 + \frac{\lambda}{s_1} \bar p_1 + \frac{c_x \lambda - \lambda^3}{s_1} \bar p_2 \right)
\end{split}$$ Using similar arguments, we can also obtain $$\begin{split}
\lambda p_1^0 &= e^{\lambda t} \left(- \frac{\lambda^2}{s_1} \bar q_1 + \frac{1- c_x c_y + c_y \lambda^2}{s_1} \bar q_2 + \frac{\lambda}{s_1} \bar p_1 + \frac{c_x \lambda - \lambda^3}{s_1} \bar p_2 \right)
\end{split}$$ Thus, we obtain the following relations $$\begin{split}
\lambda q_1^0 e^{\lambda t}&= \frac{\lambda^2}{s_1} \bar q_1 - \frac{1- c_x c_y + c_y \lambda^2}{s_1} \bar q_2 + \frac{\lambda}{s_1} \bar p_1 + \frac{c_x \lambda - \lambda^3}{s_1} \bar p_2\\
\lambda p_1^0 e^{- \lambda t} &=- \frac{\lambda^2}{s_1} \bar q_1 + \frac{1- c_x c_y + c_y \lambda^2}{s_1} \bar q_2 + \frac{\lambda}{s_1} \bar p_1 + \frac{c_x \lambda - \lambda^3}{s_1} \bar p_2
\label{chi prepare}
\end{split}$$
Let $\chi$ be the angles determined by $$\begin{split}
\cos \chi &= \frac{1}{\sqrt{ \left(\lambda^2 -c_x \right)^2 + 1}}, \ \ \ \ \ \ \sin \chi = \frac{\lambda^2 -c_x}{\sqrt{ \left(\lambda^2 -c_x \right)^2 + 1}}.\\
% \cos \chi_2 &= - \frac{1}{\sqrt{ \left(\lambda^2 -c_x \right)^2 + 1}}, \ \ \ \ \sin \chi_2 = - \frac{\lambda^2 -c_x}{\sqrt{ \left(\lambda^2 -c_x \right)^2 + 1}}
\end{split}$$ Furthermore, let $$\begin{split}
\bar p_1 = \rho \cos \theta, \ \ \ \bar p_2= \rho \sin \theta
\end{split}$$ and $$\begin{split}
\gamma &= \left( \frac{\lambda^2}{s_1} \bar q_1 - \frac{1- c_x c_y + c_y \lambda^2}{s_1} \bar q_2 \right) \left[\frac{\lambda^2}{s_1^2} \left(\bar p_1^2 + \bar p_2 ^2\right) \left(\left(\lambda^2 -c_x \right)^2 + 1 \right) \right]^{- \frac{1}{2}}\\
\label{gamma}
\end{split}$$ Using , can be rewritten as $$\begin{split}
\lambda q_1^0 e^{\lambda t} \left[\frac{\lambda^2}{s_1^2} \left(\bar p_1^2 + \bar p_2 ^2\right) \left(\left(\lambda^2 -c_x \right)^2 + 1 \right) \right]^{- \frac{1}{2}} &= \gamma + \cos(\theta - \chi)\\
\lambda p_1^0 e^{- \lambda t} \left[\frac{\lambda^2}{s_1^2} \left(\bar p_1^2 + \bar p_2 ^2\right) \left(\left(\lambda^2 -c_x \right)^2 + 1 \right) \right]^{- \frac{1}{2}} &= - \gamma + \cos(\theta - \chi)
\label{angle of wedge}
\end{split}$$
So far, the signs of $q_1^0$ and $p_1^0$ can be determined using Eq. . From Eq. , it can be concluded that $\gamma$ is only dependent on the position $(\bar q_1, \bar q_2)$, because $\bar p_1^2 + \bar p_2 ^2$ can be obtained from Eq. once the position is given. Outside the strip, we have $\mid \gamma \mid >1 $. In this case, the signs of $q_1^0$ and $p_1^0$ are independent of the direction of velocity and are always opposite, which makes $q_1^0 p_1^0<0$. Thus, only non-transit orbit exist in these regions. Inside the strip, we have $\mid \gamma \mid <1 $. This situation is quite different since the signs of $q_1^0$ and $p_1^0$ are dependent on the angle of velocity. Because for transit orbits, the sign of $q_1^0 p_1^0$ must be positive. Thus, we can vary $\theta$ (the direction of velocity) to satisfy this condition, and the wedge of velocity can be determined. It should be noted that the wedge of velocity can only exist inside the strip $S$: outside of $S$, no transit orbit exists.
Linearized Dissipative Hamiltonian System {#linearization of dissipative}
=========================================
Solutions near the equilibria
-----------------------------
For the dissipative system, we still use the symplectic matrix $C$ as in to transform to the eigenbasis, i.e., transform $\bar z=(\bar q_1 , \bar q_2 , \bar p_1 , \bar p_2)$ to $z=(q_1 , q_2 , p_1 , p_2)$. The equations of motion now become $$\dot z = \Lambda z + \Delta z$$ where $\Lambda = {C}^{-1}{ {A}}{C}$ from before and the transformed damping matrix is, $$\begin{split}
\Delta = { C}^{-1}{{D}}{C} = -c_1 \left( \begin{array}{rrrr}
\tfrac{1}{2} & 0 & \tfrac{1}{2} & 0 \\
0 & 0 & 0 & 0 \\
\tfrac{1}{2} & 0 & \tfrac{1}{2} & 0 \\
0 & 0 & 0 & 1
\end{array} \right)
\label{zeta_matrix}
\end{split}$$ which results in
$$\begin{aligned}
&\begin{cases}
\dot q_1 = \left(\lambda - \frac{ c_1}{2} \right) q_1 - \frac{ c_1}{2} p_1\\
\dot p_1 = - \frac{ c_1}{2} q_1 + \left(- \lambda -\frac{ c_1 }{2} \right) p_1
\label{Hd q1}
\end{cases}\\
&\begin{cases}
\dot q_2 = \omega_p p_2\\
\dot p_2 = - \omega_p q_2 - c_1 p_2
\label{Hd q2}
\end{cases}
\end{aligned}$$
Notice that the dynamics on the $(q_1,p_1)$ plane and $(q_2,p_2)$ plane are uncoupled.
The fourth-order characteristic polynomial is thus decomposable into $p(\beta)= p_1(\beta) p_2(\beta)$, where the second-order characteristic polynomials for and are
p\_1()= \^2 + c\_1 - \^2\
p\_2()= \^2 + c\_1 + \_p\^2
\[Hd poly\]
Considering $c_1$ is positive and $c_1^2$ is smaller compared with $4 \omega_p^2$, the determinants for are
\_1 = c\_1\^2 + 4 \^2 > 0\
\_2 = c\_1\^2 - 4 \_p\^2 <0
The corresponding eigenvalues are
$$\begin{aligned}
&\begin{cases}
\beta_1= \frac{ - c_1 + \sqrt{ c_1^2 + 4 \lambda^2}}{2}\\
\beta_2= \frac{ - c_1 - \sqrt{ c_1^2 + 4 \lambda^2}}{2}
\end{cases}\\
&\begin{cases}
\beta_3= -\delta + i \omega_d \\
\beta_4= -\delta - i \omega_d \\
\end{cases}
\end{aligned}$$
where $\delta = \frac{c_1}{2}, \omega_d =\omega_p \sqrt{ 1 - \xi_d^2}$ and $\xi_d=\frac{\delta}{\omega_p}$, with the corresponding eigenvectors $$\begin{split}
u_{\beta_1}&= \left( \frac{c_1}{2}, \lambda - \frac{1}{2} \sqrt{c_1^2 + 4 \lambda^2} \right)\\
u_{\beta_2}&= \left( \frac{c_1}{2}, \lambda + \frac{1}{2} \sqrt{c_1^2 + 4 \lambda^2} \right)\\
u_{\beta_3}&= \left( \omega_p, - \delta + i \omega_d \right)\\
u_{\beta_4}&= \left( \omega_p, - \delta - i \omega_d \right)
\label{Hd imagin}
\end{split}$$
Thus, the general solutions for the $\left(q_1,p_1\right)$ and $\left(q_2,p_2\right)$ systems are
$$\begin{aligned}
&\begin{cases}
q_1= k_1 e^{\beta_1 t} + k_2 e^{\beta_2 t}\\
p_1= k_3 e^{\beta_1 t} + k_4 e^{\beta_2 t}
\end{cases}\\
&\begin{cases}
q_2= k_5 e^{- \delta t} \cos{\omega_d t} + k_6 e^{- \delta t} \sin{\omega_d t}\\
p_2= \frac{k_5 }{\omega_p} e^{- \delta t} \left(-\delta \cos{\omega_d t} - \omega_d \sin{\omega_d t} \right) +\frac{k_6 }{\omega_p} e^{- \delta t} \left(\omega_d \cos{\omega_d t - \delta \sin{\omega_d t}} \right)\\
\end{cases}
\end{aligned}$$
\[Hd gener solut\]
where $$\begin{split}
k_1 &= \frac{q_1^0 \left(2 \lambda + \sqrt{c_1^2 + 4 \lambda^2} \right)-c_1 p_1^0 }{2\sqrt{c_1^2 + 4 \lambda^2}}, \hspace{0.5in} k_2 =\frac{q_1^0 \left(-2 \lambda + \sqrt{c_1^2 + 4 \lambda^2} \right)+c_1 p_1^0 }{2\sqrt{c_1^2 + 4 \lambda^2}},\\
k_3 &= \frac{p_1^0 \left(-2 \lambda + \sqrt{c_1^2 + 4 \lambda^2} \right)-c_1 q_1^0 }{2\sqrt{c_1^2 + 4 \lambda^2}}, \hspace{0.5in} k_4 = \frac{p_1^0 \left(2 \lambda + \sqrt{c_1^2 + 4 \lambda^2} \right)+c_1 q_1^0 }{2\sqrt{c_1^2 + 4 \lambda^2}},\\
k_5&=q^0_2 , \hspace{0.5in} k_6=\frac{p^0_2 \omega_p + q^0_2 \delta}{\omega_d}
\end{split}$$ Note that $k_1=q_1^0$, $k_4=p_1^0$, $k_2=k_3=0$, $k_5=q_2^0$ and $k_6=p_2^0 $ if $c_1=c_2=0$.
Taking total derivative with respect to $t$ of the Hamiltonian along trajectories gives us $$\begin{split}
\frac{d \mathcal{H}_2}{d t} = - \tfrac{1}{2} c_1 \lambda \left(q_1 + p_1 \right)^2 - c_1 \omega_p p_2^2 \le 0
\label{H2-rate of change-equal}
\end{split}$$ which means the Hamiltonian is non-increasing, and will generally decrease due to damping.
Boundary of transit and non-transit orbits {#boundary-of-transit-and-non-transit-orbits}
------------------------------------------
#### The Linear Flow in $\mathcal{R}$
Similar to the discussions for the conservative system, we still choose an equilibrium region $\mathcal{R}$ bounded by regions which project to the lines $n_1$ and $n_2$ in the $(q_1,p_1)$-plane (see Figure \[flow-damped\]). To analyze the flow in $\mathcal{R}$, we consider the projections on the $(q_1,p_1)$-plane and the $(q_2,p_2)$-plane, respectively. In the first case we see the standard picture of saddle point, now rotated compared to the conservative case, and in the second, of a stable focus which is a damped oscillator with frequency $\omega_d=\omega_p \sqrt{1- \xi_d^2}$, where $\xi_d=\frac{c_1}{2 \omega_p}$ - the viscous damping factor (damping ralative to critical damping). Notice that the frequency $\omega_d$ for the damped system decreases with increased damping, but only very slightly for lightly damped systems.
![\[flow-damped\][ The flow in the equilibrium region around S$_1$ for the dissipative system has the form saddle $\times$ focus. On the left is shown the projection onto the $(p_1,q_1)$ plane, the saddle projection. The asymptotic orbits (labeled A) on this projection are the saddle-type asymptotic orbits, and are rotated clockwise compared to the conservative system. They still form the separatrix between transit orbits (T) and two non-transit orbits (NT). The black dot at the center represents trajectories with only a focus projection, thus oscillatory dynamics decaying onto the point S$_1$. As the energy, the Hamiltonian function $\mathcal{H}_2$, is decreasing, the boundary is no longer equal to $q_1 p_1 = h/\lambda$, as it is for the conservative case, where $\mathcal{H}_2=h$ is the initial value of the energy for those trajectories entering through the left or right side bounding sphere (i.e., $n_1$ or $n_2$, respectively). These boundaries (the boundary of the shaded region) still correspond to the fastest trajectories through the neck region for a given $h$. ]{}](eigenspace_damp_both_paper.png){width="\textwidth"}
We distinguish nine classes of orbits grouped into the following four categories:
1. The point $q_1=p_1=0$ corresponds to a [**focus-type asymptotic**]{} orbit with motion purely in the $(q_2,p_2)$-plane (see black dot at the origin of the $(q_1,p_1)$-plane in Figure \[flow-damped\]). Such orbits are asymptotic to the equilibrium point S$_1$ itself. Due to the effect of damping, the periodic orbit in the conservative system, which is an invariant 1-sphere $S_h^1$ mentioned in , does not exist.
2. The four half open segments on the lines governed by $q_1= c_1 p_1/(2 \lambda \pm \sqrt{c_1^2 + 4 \lambda^2}) $ correspond to [**saddle-type asymptotic**]{} orbits. See the four orbits labeled A in Figure \[flow-damped\]. These orbits have motion in both the $(q_1,p_1)$- and $(q_2,p_2)$-planes.
3. The segments which cross $\mathcal{R}$ from one boundary to the other, i.e., from $p_1 - q_1=+c$ to $p_1 - q_1=-c$ in the northern hemisphere, and vice versa in the southern hemisphere, correspond to [**transit**]{} orbits. See the two orbits labeled $T$ of Figure \[flow-damped\].
4. Finally the segments which run from one hemisphere to the other hemisphere on the same boundary, namely which start from $p_1 - q_1 = \pm c$ and return to the same boundary, correspond to [**non-transit**]{} orbits. See the two orbits labeled NT of Figure \[flow-damped\].
Trajectories in the neck region
-------------------------------
Following the same procedure of analysis as for conservative system, the general solution to the dissipative system can be obtained by $\bar z=C z$ which gives $$\begin{split}
\bar q_1 &= \frac{k_1 - k_3}{s_1} e^{\beta_1 t} - \frac{k_4 - k_2}{s_1} e^{\beta_2 t} + \frac{q_2}{s_2}\\
\bar q_2 &= \frac{k_1 - k_3}{s_1} (c_x-\lambda^2) e^{\beta_1 t} - \frac{k_4 - k_2}{s_1} (c_x-\lambda^2) e^{\beta_2 t} + \frac{\omega_p^2 + c_x}{s_2} q_2
\label{diss sol}
\end{split}$$
Similar to the situation in the conservative system, the solutions for the dissipative system on the energy surface fall into different classes depending upon the limiting behaviors. See Figure \[lin\_damp\_position\_paper\].
![\[lin\_damp\_position\_paper\][ The flow of the dissipative system in $\mathcal{R}$, the equilibrium region projected onto the $xy$ configuration space, for trajectories starting at a fixed value of energy, $\mathcal{H}_2=h$, on either the right or left side vertical boundaries. As before, for any point on a bounding vertical line (dashed), there is a wedge of velocities inside of which the trajectories are transit orbits, and outside of which are non-transit orbits. For a given fixed energy, the wedge for the dissipative system is a subset of the wedge for the conservative system. The boundary of the wedge gives the orbits asymptotic (saddle-type) to the equilibrium point S$_1$. ]{}](lin_damp_position_paper.png){width="\textwidth"}
From we know that the conditions $k_1-k_3>0$, $k_1 - k_3=0$ and $k_1-k_3<0$ make $\bar q_1$ tend to minus infinity, are bounded or tend to plus infinity if $t \rightarrow \infty$. See Figure \[flow-damped\]. The same statement holds if $t \rightarrow- \infty$ and $k_2 -k_4$ replaces $k_1- k_3$. Nine classes of orbits can be given according to different combinations of the sign of $k_1-k_3$ and $k_2 - k_4$ which can be classified into the following four categories:
1. Orbits with $k_1 - k_3 = k_4-k_2 = 0$ are [**focus-type asymptotic**]{} orbits $$\begin{split}
\bar q_1 = q_2 / s_2, \hspace{0.5in} \bar q_2 = \left(\omega_p^2 + c_x \right) \bar q_1
\end{split}$$ Notice the presence of $q_2$ in reveals that the amplitude of the periodic orbit will gradually decease at the rate of $e^{-\delta t}$ with time. The larger the damping, the faster the rate will be.
2. Orbits with $ \left(k_1 - k_3\right) \left(k_4 - k_2 \right)= 0$ are [**saddle-type asymptotic**]{} orbits $$\bar q_2 =\left(c_x-\lambda^2 \right) \bar q_1 + \frac{\lambda^2+\omega_p^2}{s_2} q_2
\label{damped-sympt}$$ In similarity with the shrinking of the length of the periodic orbit, the amplitude of asymptotic orbits are also shrinking.
3. Orbits with $\left(k_1 - k_3\right) \left(k_4 - k_2 \right)>0$ are [**transit**]{} orbits
4. Orbits with $\left(k_1 - k_3\right) \left(k_4 - k_2 \right)<0$ are [**non-transit**]{} orbits
#### Wedge of velocities
We previously obtained the wedge of velocities for the conservative system. However, this method is no longer effective for the dissipative system. Thus, another approach will be pursued here.
Based on the eigenvectors in , we can conclude that the directions of stable asymptotic orbits are along $u_{\beta_2}=\left(\frac{c_1}{2}, \lambda + \frac{1}{2} \sqrt{c_1^2 + 4 \lambda^2} \right)$. In this case, all asymptotic orbits in the transformed system must start on the line $$q_1=k_p p_1
\label{R:asymp-formula}$$ where $k_p=c_1 / (2 \lambda + \sqrt{c_1^2 + 4 \lambda^2})$. For a specific point $\left(\bar q_{10}, \bar q_{20} \right)$, the initial conditions in position space and transformed space are defined as $\left(\bar q_{10}, \bar q_{20}, \bar p_{10}, \bar p_{20} \right)$ and $\left(q_{10}, q_{20}, p_{10}, p_{20} \right)$, respectively. Using Eq. and the change of variables in , we can obtain $p_{10}$, $q_{20}$, $p_{20}$ and $\bar p_{20}$ in terms of $\bar q_{10}$, $\bar q_{20}$ and $\bar p_{10}$. With $p_{10}$, $q_{20}$, $p_{20}$ and $\bar p_{20}$ in hand, the normal form of the Hamiltonian can be rewritten as $$a_p \bar p_{10}^2 + b_p \bar p_{10}+c_p=0
\label{quadratic}$$ where $$\begin{split}
& a_p=\frac{s_2^2}{2 \omega_p}, \hspace{0.5in} b_p=\frac{\lambda s_2^2 (1+k_p) \left[\bar q_2- \bar q_1 \left(c_x + \omega_p^2 \right) \right]}{\omega_p \left(k_p-1 \right) \left(\lambda^2 + \omega_p^2 \right)},\\
& c_p = \left(\sum\limits_{i=1}^{4} c_p^{(i)}\right)/ \left[2 \omega_p \left(k_p-1 \right)^2 \left(\lambda^2 + \omega_p^2 \right)^2 \right]- h,\\
& c_p^{(1)}=2 k_p s_1^2 \lambda \omega_p \left[\bar q_2- \bar q_1 \left(c_x + \omega_p^2 \right) \right]^2, \\
& c_p^{(2)}= 8 k_p s_2^2 \lambda^2 \omega_p^2 \bar q_1 \left(c_x \bar q_1 - \bar q_2 \right),\\
& c_p^{(3)}=s_2^2 \lambda^2 \left(1+k_p \right)^2 \left[\left(c_x \bar q_1- \bar q_2 \right)^2+ \bar q_1^2\omega_p^4 \right],\\
& c_p^{(4)}=s_2^2 \omega_p^2 \left(k_p -1 \right)^2 \left[\left(c_x \bar q_1 - \bar q_2) \right)^2+ \bar q_1^2 \lambda^4 \right]
\end{split}$$
For the existence of real solutions, the determinant of quadratic equation should satisfy the condition $\vartriangle = b_p^2 - 4 a_p c_p \geq 0$: $\vartriangle=0$ is the critical condition for $p_{10}$ to have real solutions. Noticing $\left(c_x - \lambda^2 \right) \left(c_x + \omega_p^2 \right)=-1$, the critical condition gives an ellipse of the form
$$\frac{\left(\bar q_{10} \cos \vartheta + \bar q_{20} \sin \vartheta \right)^2}{a_e^2} +\frac{\left(- \bar q_{10} \sin \vartheta + \bar q_{20} \cos \vartheta \right)^2}{b_e^2} = 1,
\label{R:standard-ellipse}$$
where $$\begin{split}
& a_e= \sqrt{\frac{2 h \left(\lambda^2 + \omega_p^2\right)^2 \left(c_x + \omega_p^2 \right)^2}{\omega_p s_2^2 \left[ \left(c_x + \omega_p^2 \right)^2 + 1\right]}}, \hspace{0.7in} b_e= \sqrt{\frac{h \left(k_p - 1\right)^2 \left(\lambda^2 + \omega_p^2 \right)^2}{\lambda k_p s_1^2 \left[ \left(c_x + \omega_p^2 \right)^2 + 1\right]}},\\
& \cos \vartheta = \frac{1}{\sqrt{\left(c_x + \omega_p^2 \right)^2 + 1 }}, \hspace{1.3in} \sin \vartheta= \frac{\left(c_x + \omega_p^2 \right)}{\sqrt{\left(c_x + \omega_p^2 \right)^2 + 1 } }\\
\end{split}$$ The ellipse is counterclockwise tilted by $\vartheta$ from a standard ellipse $\bar q_{10}^2/a_e^2 + \bar q_{20}^2 /b_e^2=1$. The ellipse governed by is the critical condition that $\bar p_{10}$ exists, so it is the boundary for asymptotic orbits. In other words, inside the ellipse, transit orbits exist, while outside the ellipse, transit orbits do not exist. As a result, we refer to the ellipse as the [**ellipse of transition**]{} (see Figure \[lin\_damp\_position\_paper\](b)). Note that on the boundary of the ellipse, there is only one asymptotic orbit (i.e., the wedge has collapsed into a single direction). The solutions to are given by $$\bar p_{10}=\frac{-b_p \pm \sqrt{b_p^2 - 4 a_p c_p}}{2 a_p}$$ and the expression for $\bar p_{20}$ is $$\begin{aligned}
\bar p_{20}=\bar p_{10} \left(c_x + \omega_p^2 \right) + \frac{\lambda \left(1+ k_p \right) \left[\bar q_{20} -\bar q_{10} \left(c_x + \omega_p^2 \right) \right]}{k_p -1}\end{aligned}$$ Up to now, the initial conditions $(\bar q_{10}, \bar q_{20}, \bar p_{10}, \bar p_{20})$ for the asymptotic orbits at a specific position have been obtained. The interior angle determined by these two initial velocities defines the wedge of velocites: $\theta =\arctan \left(\bar p_{20}/\bar p_{10} \right)$. The boundary of this wedge correspond to the asymptotic orbits. In fact, the wedge for the conservative system can be obtained by this method by taking $c_1$ as zero.
Figure \[lin\_damp\_position\_paper\] illustrates the projection on the configuration space in the equilibrium region. In the dissipative system, one important finding is the existence of the ellipse of transition given by . The length of the major and minor axes of the ellipse are $a_e$ and $b_e$, respectively. For small damping, the major axis is much larger than the minor axis so that it reaches far beyond the neck region. Thus, here we give the local flow near the neck region as shown in Figure \[lin\_damp\_position\_paper\](a). We show a zoomed-out view revealing the entire ellipse in Figure \[lin\_damp\_position\_paper\](b). The asymptotic orbits in the dissipative system are bounded by the ellipse (which is different from the asymptotic orbits in the conservative system, which are bounded by the strip). Moreover, in the conservative system, all asymptotic orbits can reach the boundary of the strip with the period of $ 2 \pi / \omega_p$, while the asymptotic orbits in the dissipative system can never reach the boundary of the ellipse after they start due to damping. Notice that $a_e$ goes to zero when $c_1$ is large enough.
Outside the ellipse, $\vartriangle= b_p^2 - 4 a_p c_p<0$, only non-transit orbits project onto this region. Thus we can conclude that the signs of $k_1 - k_3$ and $k_4 - k_2$ are independent of the direction of the velocity and can be determined in each of the components of the equilibrium region $R$ complementary to the ellipse. For example, in the left-most component, $k_1 - k_3$ is negative and $k_4 - k_2$ is positive, while in the right-most components, $k_1 - k_3$ is positive and $k_4 - k_2$ is negative.
Inside the ellipse the situation is more complex due to the existence of the wedge of velocity. For simplicity we still just show the wedges on the two vertical bounding line segments in Figure \[lin\_damp\_position\_paper\]. For example, consider the intersection of the strip with the left-most vertical line. At each position on the subsegment, one wedge of velocity exists in which $k_1 - k_2 $ is positive. The orbits with velocity interior to the wedge are transit orbits, and $k_4 - k_2$ is always positive. Orbits with velocity on the boundary of the wedge are asymptotic ($(k_1 - k_3) (k_4 - k_2)=0$), while orbits with velocity outside of the wedge are non-transit ($(k_1 - k_3) (k_4 - k_2)<0$). Notice that in Figure \[lin\_damp\_position\_paper\], the grey shaded wedges are the wedges for the dissipative system, while the blacked shaded wedges partially covered by the grey ones are for conservative system (hardly visible for the parameters shown in the figure). The shrinking of the wedges from the conservative system to the dissipative system is caused by damping. This confirms the expectation that an increase in damping decreases the proportion of the transit orbits.
Transition Tubes
================
In this section, we go step by step through the numerical construction of the boundary between transit and non-transit orbits in the nonlinear system . We combine the geometric insight of the previous sections with numerical methods to demonstrate the existence of ‘transition tubes’ for both the conservative and damped systems. Particular attention is paid to the modification of phase space transport as damping is increased, as this has not been considered previously.
#### Tube dynamics
The dynamic snap-through of the shallow arch can be understood as trajectories escaping from a potential well with energy above a critical level: the energy of the saddle point S$_1$. However, even if the energy of the system is higher than critical, the snap-through may not occur. The dynamical boundary between snap-through and non-snap-through behavior can be systematically understood by [**tube dynamics**]{}. Tube dynamics [@Conley1968; @LlMaSi1985; @OzDeMeMa1990; @DeMeTo1991; @DeLeon1992; @Topper1997; @KoLoMaRo2000; @GaKoMaRo2005; @GaKoMaRoYa2006; @MaRo2006; @KoLoMaRo2011] supplies a large-scale picture of transport; transport between the largest features of the phase space—the potential wells. In the conservative system, the stable and unstable manifolds with a $S^1 \times \mathbb{R}$ geometry act as [**tubes**]{} emanating from the periodic orbits. While found above for the linearized system near S$_1$, these structures persist in the full nonlinear system The manifold tubes (usually called [**transition tubes**]{} in tube dynamics), formed by pieces of asymptotic orbits, separate two distinct types of orbits: transit orbits and non-transit orbits, corresponding to snap-through and non-snap-through in the present problem. The transit orbits, passing from one region to another through the bottleneck, are those inside the transition tubes. The non-transit orbits, bouncing back to their region of origin, are those outside the transition tubes. Thus, the transition tubes can mediate the global transport of states between snap-through and non-snap-through. In the dissipative system, similar transition tubes also exist. Even in systems where stochastic effects are present, the influence of these structures remains [@NaRo2017].
Algorithm for computing transition tubes
----------------------------------------
For the conservative system, Ref. [@KoLoMaRo2011] gives a general numerical method to obtain the transition tubes. The key steps are (1) to find the periodic orbits restricted to a specified energy using differential correction and numerical continuation based on the initial conditions obtained from the linearized system at first, then (2) to compute the manifold tubes of the periodic orbits in the nonlinear system (i.e., ‘globalizing’ the manifolds), and finally (3) to obtain the intersection of the Poincaré surface of section and global manifolds. See details in Ref. [@KoLoMaRo2011]. The method is effective in the conservative system, but not applicable to the dissipative system, since due to loss of conservation of energy, no periodic orbit exists. Thus, we provide another method as follows.
[**Step 1: Select an appropriate energy.**]{} We first need to set the energy to an appropriate value such that the snap-though behavior exists. Once the energy is given, it remains constant in the conservative system. In our example, the critical energy for snap-through is the energy of S$_1$. Thus, we can choose an energy which is between that of S$_1$ and S$_2$. In this case, all transit orbits can just escape from W$_1$ to W$_2$ through S$_1$. Notice that the potential energy determines the width of the bottleneck and the size of the transition tubes which hence determines the relative fraction of transit orbits in the phase space. A representative energy case is shown in Figure \[non\_sections\_paper\], which also establishes our location for Poincaré sections $\Sigma_1$ and $\Sigma_2$ which are at $X=$constant lines passing through W$_1$ and W$_2$ respectively, and with $p_X>0$.
![\[non\_sections\_paper\][ For a representative energy above the saddle point S$_1$, we show the unstable periodic orbit in the neck region around S$_1$. It projects to a single line going between the upper and lower energy boundary curves, and arrows are shown for convenience. We show the Poincaré sections $\Sigma_1$ and $\Sigma_2$ which are defined by $X$ values equal to that of the two stable equilibria in the center of the left and right side wells, W$_1$ and W$_2$, respectively. The arrows on the vertical lines indicate that these Poincaré sections are also defined by positive $X$ momentum. ]{}](non_sections_paper.png){width="70.00000%"}
[**Step 2: Compute the approximate transition tube and its intersection on a Poincaré section.**]{} We have analyzed the flow of linearized system in both phase space and position space which classifies orbits into four classes. It shows that in the conservative system the stable manifolds correspond to the boundary between transit orbits and non-transit orbits. Thus, we can choose this manifold as the starting point. We start by considering the approximation of transition tubes for the conservative system.
[*Determine the initial condition.*]{} The stable manifold divides the transit orbits and non-transit orbits for all trajectories headed toward a bottleneck. Thus, we can use the stable manifold to obtain the initial condition. Considering the general solutions of the linearized equations , we can let $p_1^0=c$, $q_1^0=0$, $q_2^0=A_q$ and $p_2^0=A_p$. Notice that $$A_q^2+A_p^2=2 h/\omega_p
\label{circle}$$ which forms a circle in the center projection, so in the next computational procedure we should pick up $N$ points on the circle with a constant arc length interval. Each $A_q$ and $A_p$ determined by these sampling points along with $p_1^0=c$ and $q_1^0=0$ can be used as initial conditions. When first transformed back to the position space and then transformed to dimensional quantities, this yields an initial condition $$\begin{pmatrix}
X_0\\ Y_0\\p_{X0} \\ p_{Y0}
\end{pmatrix}
=
\begin{pmatrix}
x_e\\ y_e\\0\\ 0
\end{pmatrix} + \begin{pmatrix}
L_x\\ L_y\\ \omega_0 L_x M_1\\ \omega_0 L_y M_2
\end{pmatrix}^T
C
\begin{pmatrix}
c\\0\\A_q\\A_p
\end{pmatrix}
\label{Initial}$$
[*Integrate backward and obtaining Poincaré section.*]{} Using the $N$ initial conditions yielded by varying $A_q$ and $A_p$ governed by and integrating the nonlinear equations of motions in in the backward direction, we obtain a tube, formed by the $N$ trajectories, which is a linear approximation for the transition tube. Choosing the Poincaré surface-of-section $\Sigma_1$ is shown in Figure \[non\_sections\_paper\], corresponding to $X=X_{{\rm W}_1}$ and $p_X>0$.
[**Step 3: Compute the real transition tube by the bisection method.**]{} We have obtained a Poincaré section which is the intersection of the approximate transition tube and the surface $\Sigma_1$. First pick a point (noted as $p_i$) which is almost the center of the closed curve. The line from $p_i$ to each of the $N$ points on the Poincaré map will form a ray. The point $p_i$ inside the curve in general is a transit orbit. Then choose another point on each radius which is a non-transit orbit, noted as $p_o$. With the approach described above, we can use the bisection method to obtain the boundary of the transition tube on a specific radius (cf. [@AnEaLo2017]). Picking the midpoint (marked by $p_m$) as the initial condition and carrying out forward integration for the nonlinear equation of motion in , we can estimate if this trajectory can transit or not. If it is a transit orbit, note it as $p_i$, or note it as $p_o$. Continuing this procedure again until the distance between $p_i$ and $p_o$ reaches a specified tolerance, the boundary of the tube on this ray is estimated. Thus, the real transition tube for the conservative system can be obtained if the same procedure is carried out for all angles. A related method is described in [@OnYoRo2017], adapting an approach of [@GaMaDuCa2009].
For the dissipative system, the size of the transition tubes for a given energy on $\Sigma_1$ will shrink. Using the bisection method and following the same procedure as for conservative system, the transition tube for the dissipative system will be obtained.
Numerical results and discussion
--------------------------------
To visualize the tube dynamics for the arch, several examples will be given. According to the steps mentioned above, we can obtain the transition tubes for both the conservative system and dissipative systems. For all results, the geometries of the arch are selected as $b=12.7$ mm $d=0.787$ mm, $L=228.6$ mm. The Young’s modulus and the mass density are $E=153.4$ GPa and $\rho=7567 \ \mathrm{kg \ m^{-3}}$. The selected thermal load corresponds to $184.1$ N, while the initial imperfections are $\gamma_1 = 0.082$ mm and $\gamma_2 = -0.077$ mm. These values match the parameters given in the experimental study [@WiVi2016]. For all the numerical results given in this section, the initial energy of the system is set at 3.68$\times 10^{-4}$ J - above the energy of saddle point S$_1$, so that the equilibrium point $W_1$ is inside the configuration space projection. This choice of initial energy will make it possible to compare the numerical results with the experimental results which are planned for future work.
#### Transition tubes for conservative system
For conservative system, the Hamiltonian is a constant of motion. In Figure \[non\_conservative\_tube\_all\_paper\], we show the configuration space projection of the transition tube and the Poincaré sections on $\Sigma_1$ and $\Sigma_2$ which are closed curves. In Figure \[non\_conservative\_tube\_all\_paper\] are shown all the trajectories which form the transition tube boundary starting from $\Sigma_1$ and ending up at $\Sigma_2$, flowing from left to right through the neck region.
![\[non\_conservative\_tube\_all\_paper\][ A transition tube from the left well to the right well, obtained using the method described in the text. The upper figure shows the configuration space projection. The lower left shows the tube boundary (closed curve) on Poincaré section $\Sigma_1$, which separates transit and non-transit trajectories. The lower right shows the corresponding tube boundary (closed curve) on Poincaré section $\Sigma_2$. ]{}](non_conservative_tube_all_paper.png){width="\textwidth"}
Due to the the conservation of energy, the size of the transition tube is constant during evolution, which corresponds to the cross-sectional area of the transition tube. It should be noted that the areas of the tube Poincaré sections on $\Sigma_1$ and $\Sigma_2$ in Figure \[non\_conservative\_tube\_all\_paper\] are equal, due to the integral invariants of Poincaré for a system obeying Hamilton’s canonical equations (with no damping). Moreover, note that the size of the transition tube, the boundary of the transit orbits, is determined by the energy. For a lower energy, the size of the transition tube is smaller or vice versa. In other words, the area of the Poincaré sections on $\Sigma_1$ and $\Sigma_2$ is determined by the energy. In fact, the cross-sectional area of the transition tube is proportional to the energy above the saddle point S$_1$ [@MacKay1990]. As mentioned before, the transition tube separates the transit orbits and non-transit orbits, which correspond to snap-through and non-snap-through. The orbit inside the transition tube can transit, while the orbit outside the transition tube cannot transit.
#### Transition tubes for dissipative system
Unlike the conservation of energy in conservative system, the energy in the dissipative system is decreasing with time.
![\[non\_damp\_tube\_all\_paper\][ A transition tube from the left well to the right well, obtained using the method described in the text, for the case of damping. The upper figure shows the configuration space projection. The lower left shows the tube boundary (closed curve) on Poincaré section $\Sigma_1$ which separates transit and non-transit trajectories for initial conditions all with a given fixed initial energy. The lower right shows the corresponding image under the flow on Poincaré section $\Sigma_2$. Due to the damping, and a range of times spent in the neck region, spiraling is visible in this 2D projection since trajectories which spend longer in the neck will be at lower total energies. Compare with Figure \[non\_conservative\_tube\_all\_paper\]. ]{}](non_damp_tube_all_paper.png){width="\textwidth"}
Figure \[non\_damp\_tube\_all\_paper\] shows the configuration space projection of the transition tube and the Poincaré sections on $\Sigma_1$ and $\Sigma_2$. In Figure \[non\_damp\_tube\_all\_paper\] the transition tube starts from $\Sigma_1$ and ends up with $\Sigma_2$ flowing from left to right through the neck region, as shown previously for the conservative system. From the figure, we can observe the distinct reduction in the size of the transition tube, especially near the neck region. To show this, the scale of the Poincaré section projections is the same as in Figure \[non\_conservative\_tube\_all\_paper\]. During the evolution, the energy of the system is decreasing due to damping. The trajectories spend a great amount of time crossing the neck region, resulting in the total energy decreasing dramatically (and influencing the size of the transition tubes to the right of the neck region). Thus, the transition tube is spiraling in the neck region so that Poincaré $\Sigma_2$ is not a closed curve, nor are the trajectories at a constant energy. The $\Sigma_2$ plot is merely a projection onto the $(Y,p_Y)$-plane to give an idea of the actual co-dimension 1 tube boundary in the 4-dimensional phase space. Note the clear differences between Figure \[non\_conservative\_tube\_all\_paper\] and Figure \[non\_damp\_tube\_all\_paper\]. The dramatic shrinking of tubes near the neck region is due almost entirely to the linearized dynamics near the saddle point. To confirm this, we present the linear transition tube obtained by the analytical solutions for the linearized dissipative system in Figure \[lin\_damp\_tube\_paper\].
![\[lin\_damp\_tube\_paper\][ A transition tube from the left side boundary ($n_1$) to the right side boundary ($n_2$) of the equilibrium region around saddle point S$_1$, obtained for the linear damped system. Notice that the shrinking of the tube is observed as in the nonlinear system, Figure \[non\_damp\_tube\_all\_paper\], here seen in terms of the width of the projected strip onto configuration space. ]{}](lin_damp_tube_paper.png){width="70.00000%"}
#### Effect of damping on the transition tubes
In order to further quantify how damping affects the size of transition tubes, we present the tube Poincaré section on $\Sigma_1$ with different damping in Figure \[multi\_damping\_Poincare\_both\_paper\]. In Figure \[multi\_damping\_Poincare\_both\_paper\](a), we can see the canonical area ($\int_\mathcal{A} p_Y dY$) decreases with increasing damping. Thus, the proportion of transition trajectories will be fewer if the damping increases. Note that when the damping changes, different transition tubes almost share the same center which corresponds to the fastest trajectories. Figure \[multi\_damping\_Poincare\_both\_paper\](b) shows the relation between the damping and the projected canonical area ($\int_\mathcal{A} p_Y dY$), which is related to the relative number of transit compared to non-transit orbits. It shows that an increase in damping decreases the projected area. When the damping is small, the relation between the damping and the area is linear, while when the damping is large, the relation becomes slightly nonlinear. Note that generally in mechanical/structural experiments the non-dimensional damping factor $\xi_d$ is less than $5\%$ which corresponds to a damping coefficient $C_H$ less than $107.3 \ \mathrm{s^{-1}}$ (see the shaded region in Figure \[multi\_damping\_Poincare\_both\_paper\](b)). Furthermore, note that for the initial energy depicted in Figure \[multi\_damping\_Poincare\_both\_paper\], there are are no transit orbits starting on $\Sigma_1$ for $C_H$ greater than about $185 \ \mathrm{s^{-1}}$.
![\[multi\_damping\_Poincare\_both\_paper\][ The effect of the damping coefficient $C_H$ on the area of the transition tube on Poincaré section $\Sigma_1$ is shown. For a fixed initial energy above the saddle, the projection on the canonical plane $(Y,p_Y)$ is shown in (a) and the area is plotted in (b). In (b), the shaded region indicates the experimentally observed range of damping coefficients, which correspond to non-dimensional damping factor $\xi_d$ less than 5%. ]{}](multi_damping_Poincare_both_paper.png){width="\textwidth"}
#### Demonstration of trajectories inside and outside the transition tube
To illustrate the effectiveness of the transition tubes, we choose three points on $\Sigma_1$ (see A, B and C in Figure \[time-history-Poincare-all-mod-paper\](a)) as the initial conditions and integrate forward to see their evolution.
![\[time-history-Poincare-all-mod-paper\][ Several example trajectories are shown, starting from the stable well point W$_1$. The initial conditions from Poincaré section $\Sigma_1$ are shown in (a) for a fixed initial energy, along with the transition tube boundaries for the conservative case and a damped case. In (b), we show the trajectories for points A and B, for the conservative case where A is just outside the tube boundary and B is just inside. In (c), we show the trajectories for points B and C, for the damped case where B is just outside the tube boundary and C is just inside. In (d), we illustrate the effect of damping by starting the same initial condition, B, but showing the trajectory in the conservative case as trajectory B and the damped case as trajectory B$^{\prime}$. ]{}](time-history-Poincare-all-mod-paper.png){width="\textwidth"}
Note that all the trajectories corresponding to these three points have the same initial energy and start from a configuration identical to the equilibrium point $W_1$, but with different initial velocity directions. Figure \[time-history-Poincare-all-mod-paper\](b) shows the trajectories A and B in the conservative system where A is outside the tube boundary and B is inside the tube boundary. In the figure, trajectory B transits through the neck region and trajectory A bounces back. Figure \[time-history-Poincare-all-mod-paper\](c) shows trajectories B and C in the dissipative system. Like the situation in the conservative system, trajectory C which is inside the tube can transit, while trajectory B which is outside the tube cannot. Figure \[time-history-Poincare-all-mod-paper\](d) shows the effect of damping on the transit condition for the trajectories B and B$^{\prime}$ with the same initial condition. Trajectory B is simulated using the conservative system and trajectory B$^{\prime}$ is simulated using the dissipative system. It shows that the damping changes the transit condition that a transit orbit B in the conservative system becomes non-transit orbit B$^{\prime}$ in the dissipative system, both starting from the same initial condition. From Figure \[time-history-Poincare-all-mod-paper\], we can conclude the transition tube can effectively estimate the snap-through transitions both in conservative systems and dissipative systems.
Finally, we point out that the transition tubes are the boundary for transit orbits that transition [*the first time*]{}. For example, trajectory A in Figure \[time-history-Poincare-all-mod-paper\](b) stays outside of the transition tube so that it returns near the neck region at first, but, unless it happens to be on a KAM torus or a stable manifold of such a torus, it will ultimately transit as long as it does not form a periodic orbit near the potential well W$_1$, since the energy is above the critical energy for transition and is conservative.
Conclusions
===========
Tube dynamics is a conceptual dynamical systems framework initially used to study the isomerization reactions in chemistry [@OzDeMeMa1990; @DeMeTo1991; @DeLeon1992; @Topper1997; @JaFaUz1999] as well as other fields, like resonance transitions in celestial mechanics [@LlMaSi1985; @KoLoMaRo2000; @JaRoLoMaFaUz2002; @GaKoMaRoYa2006; @MaRo2006] and capsize in ship dynamics [@NaRo2017]. Here we extend the application of tube dynamics to structural mechanics: the snap-through of a shallow arch, or buckled-beam. In general, slender elastic structures are capable of exhibiting a variety of (co-existing) equilibrium shapes, and thus, given a disturbance, tube dynamics sheds light on how such a system might be caused to transition between available, stable equilibrium configurations. Moreover, it is the first time, to the best of our knowledge, that tube dynamics has been worked out for a dissipative system, which increases the generality of the approach.
The snap-through transition of an arch was studied via a two-mode truncation of the governing partial differential equations based on Euler-Bernoulli beam theory. Via analysis of the linearized Hamiltonian equations around the saddle, the analytical solutions for both the conservative and dissipative systems were determined and the corresponding flows in the equilibrium region of eigenspace and configuration space were discussed. The results show that all transit orbits, corresponding to snap-through, must evolve from a wedge of velocities which are restricted to a strip in configuration space in the conservative system, and by an ellipse in the corresponding dissipative system when damping is included. Using the results from the linearization as an approximation, the transition tubes based on the full nonlinear equations for both the conservative and dissipative system were obtained by the bisection method. The orbits inside the transition tubes can transit, while the orbits outside the tubes cannot. Results also show that the damping makes the size of the transition tubes smaller, which corresponds to the degree, or amount, of orbits that transit. When the damping is small, it has a nearly linear effect on the size of the transition tubes. Further study of the dynamic behaviors of the arch can lead to more immediate application structural mechanics. For example, many structural systems possess multiple equilibria, and the manner in which the governing potential energy changes with a control parameter is, of course, the essence of bifurcation theory. However, under nominally fixed conditions, the present paper directly assesses the energy required to (dynamically) perturb a structural system beyond the confines of its immediate potential energy well. In future work, a three-mode truncation may be introduced to study such systems. High order approximations will present higher index saddles which will modify the tube dynamics framework presented here (cf. [@collins2011index] [@HallerUzer2011] [@Nagahata2013]). Furthermore, experiments will be carried out to show the effectiveness of the present approach to prescribe initial conditions which lead to dynamic buckling.
Acknowledgements
================
This work was supported in part by the National Science Foundation under awards 1150456 (to SDR) and 1537349 (to SDR and LNV). One of the authors (SDR) acknowledges enjoyable interactions during the past decade with Professor Romesh Batra, who is being honored by this issue.
[10]{} \[1\][[\#1]{}]{}
Wiebe, R. and Virgin, L. N. \[2016\] On the experimental identification of unstable static equilibria. (2190):20160172.
Collins, P., Ezra, G. S. and Wiggins, S. \[2012\] Isomerization dynamics of a buckled nanobeam. (5):056218.
Virgin, L., Guan, Y. and Plaut, R. \[2017\] On the geometric conditions for multiple stable equilibria in clamped arches. :8–14.
Das, K. and Batra, R. \[2009\] Symmetry breaking, snap-through and pull-in instabilities under dynamic loading of microelectromechanical shallow arches. (11):115008.
Das, K. and Batra, R. \[2009\] Pull-in and snap-through instabilities in transient deformations of microelectromechanical systems. (3):035008.
Mann, B. \[2009\] Energy criterion for potential well escapes in a bistable magnetic pendulum. (3):864–876.
Thompson, J. M. T. and Hunt, G. W. \[1984\] [*Elastic Instability Phenomena*]{}. Wiley.
Naik, S. and Ross, S. D. \[2017\] Geometry of escaping dynamics in nonlinear ship motion. :48 – 70.
Koon, W. S., Lo, M. W., Marsden, J. E. and Ross, S. D. \[2000\] Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics. :427–469.
Conley, C. C. \[1968\] Low energy transit orbits in the restricted three-body problem. :732–746.
Llibre, J., Martinez, R. and Simó, C. \[1985\] Transversality of the invariant manifolds associated to the [L]{}yapunov family of periodic orbits near [L]{}2 in the restricted three-body problem. :104–156.
, A. M., [De Leon]{}, N., Mehta, M. A. and Marston, C. C. \[1990\] Geometry and dynamics of stable and unstable cylinders in [H]{}amiltonian systems. :265–285.
, N., Mehta, M. A. and Topper, R. Q. \[1991\] Cylindrical manifolds in phase space as mediators of chemical reaction dynamics and kinetics. [I]{}. [T]{}heory. :8310–8328.
, N. \[1992\] Cylindrical manifolds and reactive island kinetic theory in the time domain. :285–297.
Topper, R. Q. \[1997\] Visualizing molecular phase space: nonstatistical effects in reaction dynamics. In [*Reviews in Computational Chemistry*]{} (edited by K. B. Lipkowitz and D. B. Boyd), vol. 10, chap. 3, 101–176. VCH Publishers, New York.
Gabern, F., Koon, W. S., Marsden, J. E. and Ross, S. D. \[2005\] Theory and Computation of Non-[RRKM]{} Lifetime Distributions and Rates in Chemical Systems with Three or More Degrees of Freedom. :391–406.
Gabern, F., Koon, W. S., Marsden, J. E., Ross, S. D. and Yanao, T. \[2006\] Application of tube dynamics to non-statistical reaction processes. :167–172.
Marsden, J. E. and Ross, S. D. \[2006\] New methods in celestial mechanics and mission design. :43–73.
Koon, W. S., Lo, M. W., Marsden, J. E. and Ross, S. D. \[2011\] [*Dynamical Systems, the Three-Body Problem and Space Mission Design*]{}. Marsden Books, ISBN 978-0-615-24095-4.
Murphy, K. D., Virgin, L. N. and Rizzi, S. A. \[1996\] Experimental snap-through boundaries for acoustically excited, thermally buckled plates. :312–317.
Wiebe, R., Virgin, L. N., Stanciulescu, I., Spottswood, S. M. and Eason, T. G. Characterizing Dynamic Transitions Associated with Snap-Through: A Discrete System. .
Zhong, J., Fu, Y., Chen, Y. and Li, Y. \[2016\] Analysis of nonlinear dynamic responses for functionally graded beams resting on tensionless elastic foundation under thermal shock. :272–277.
Greenwood, D. T. \[2003\] [*Advanced Dynamics*]{}. Cambridge University Press.
Wiggins, S. \[1994\] [*Normally Hyperbolic Invariant Manifolds in Dynamical Systems*]{}. Springer-Verlag, New York.
Anderson, R. L., Easton, R. W. and Lo, M. W. \[2017\] Isolating blocks as computational tools in the circular restricted three-body problem. :38 – 50.
Onozaki, K., Yoshimura, H. and Ross, S. D. \[2017\] Tube dynamics and low energy [E]{}arth-[M]{}oon transfers in the 4-body system. ( ):to appear.
Gawlik, E. S., Marsden, J. E., [Du [T]{}oit]{}, P. C. and Campagnola, S. \[2009\] Lagrangian coherent structures in the planar elliptic restricted three-body problem. :227–249. .
MacKay, R. S. \[1990\] Flux over a saddle. :425–427.
Jaffé, C., Farrelly, D. and Uzer, T. \[1999\] Transition state in atomic physics. :3833–3850.
Jaffé, C., Ross, S. D., Lo, M. W., Marsden, J. E., Farrelly, D. and Uzer, T. \[2002\] Theory of asteroid escape rates. :011101.
Collins, P., Ezra, G. S. and Wiggins, S. \[2011\] Index k saddles and dividing surfaces in phase space with applications to isomerization dynamics. (24):244105.
Haller, G., Uzer, T., Palacian, J., Yanguas, P. and Jaffe, C. \[2011\] Transition state geometry near higher-rank saddles in phase space. (2):527.
Nagahata, Y., Teramoto, H., Li, C.-B., Kawai, S. and Komatsuzaki, T. \[2013\] Reactivity boundaries for chemical reactions associated with higher-index and multiple saddles. :042923.
| ArXiv |
David G. CHARLTON$^{a}$, Guido MONTAGNA$^{b,c}$,\
Oreste NICROSINI$^{c,b}$ and Fulvio PICCININI$^{c}$
[*$^a$Royal Society University Research Fellow, School of Physics and Space Research, University of Birmingham, Birmingham B15 2TT, UK*]{}\
[*$^b$Dipartimento di Fisica Nucleare e Teorica, Università di Pavia, via A. Bassi n. 6 - 27100 PAVIA - ITALY*]{}\
[*$^c$ INFN, Sezione di Pavia, via A. Bassi n. 6 - 27100 PAVIA - ITALY*]{}
Program classification: 11.1\
[The Monte Carlo program [WWGENPV]{}, designed for computing distributions and generating events for four-fermion production in $e^+ e^- $ collisions, is described. The new version, 2.0, includes the full set of the electroweak (EW) tree-level matrix elements for double- and single-$W$ production, initial- and final-state photonic radiation including $p_T / p_L$ effects in the Structure Function formalism, all the relevant non-QED corrections (Coulomb correction, naive QCD, leading EW corrections). An hadronisation interface to [JETSET]{} is also provided. The program can be used in a three-fold way: as a Monte Carlo integrator for weighted events, providing predictions for several observables relevant for $W$ physics; as an adaptive integrator, giving predictions for cross sections, energy and invariant mass losses with high numerical precision; as an event generator for unweighted events, both at partonic and hadronic level. In all the branches, the code can provide accurate and fast results. ]{}
[*Program obtainable from:*]{} CPC Program Library, Queen’s University of Belfast, N. Ireland (see application form in this issue)
[*Reference to original program:*]{} [WWGENPV]{}; [*Cat. no.:*]{} ACNT; [*Ref. in CPC:* ]{} [**90**]{} (1995) 141
[*Authors of original program:*]{} Guido Montagna, Oreste Nicrosini and Fulvio Piccinini
[*The new version supersedes the original program*]{}
[*Computer for which the new version is designed:*]{} DEC ALPHA 3000, HP 9000/700 series; [*Installation:*]{} INFN, Sezione di Pavia, via A. Bassi 6, 27100 Pavia, Italy
[*Keywords:*]{} $e^+ e^-$ collisions, LEP, $W$-mass measurement, radiative corrections, QED corrections, QCD corrections, Minimal Standard Model, four-fermion final states, electron structure functions, Monte Carlo integration/simulation, hadronisation.
The precise measurement of the $W$-boson mass $M_W$ constitutes a primary task of the forthcoming experiments at the high energy electron–positron collider LEP2 ($2 M_W \leq \sqrt{s} \leq 210$ GeV). A meaningful comparison between theory and experiment requires an accurate description of the fully exclusive processes $e^+ e^- \to 4f$, including the main effects of radiative corrections, with the final goal of providing predictions for the distributions measured by the experiments.
Same as in the original program, as far as weighted event integration and unweighted event generation are concerned. Adaptive Monte Carlo integration for high numerical precision purposes is added. Optional hadronic interface in the generation branch is supplied.
The most promising methods for measuring the $W$-boson mass at LEP2 are the so called “threshold” and “direct reconstruction” methods \[5\]. For the first one, a precise evaluation of the threshold cross section is required. For the second one, a precise description of the invariant-mass shape of the hadronic system in semileptonic decays is mandatory. In order to meet these requirements, the previous version of the program has been improved by extending the class of the tree-level EW diagrams taken into account, by including $p_T / p_L $ effects both in initial- and final-state QED radiation, by supplying an hadronic interface in the generation branch.
While the semileptonic decay channels are complete at the level of the Born approximation EW diagrams ([CC11/CC20]{} diagrams), neutral current backgrounds are neglected in the fully hadronic and leptonic decay channels. QED radiation is treated at the leading logarithmic level. Due to the absence of a complete ${\cal O}
(\alpha)$/${\cal O} (\alpha_s)$ diagrammatic calculation, the most relevant EW and QCD corrections are effectively incorporated according to the recipe given in \[6\]. No anomalous coupling effects are at present taken into account.
As adaptive integrator, the code provides cross section and energy and invariant-mass losses with a relative accuracy of about 1% in 8 min on HP 9000/735. As integrator of weighted events, the code produces about $10^5$ events/min on the same system. The generation of a sample of $10^3$ hadronised unweighted events requires about 8 min on the same system.
Subroutines from the library of mathematical subprograms [NAGLIB]{} \[3\] for the numerical integrations are used in the program, when the adaptive integration branch is selected.
\[1\] CERN Program Library, CN Division, CERN, Geneva. \[2\] NAG Fortran Library Manual Mark 16 (Numerical Algorithms Group, Oxford, 1991). \[3\] T. Sjöstrand, Comp. Phys. Commun. [**82**]{} (1994) 74; Lund University Report LU TP 95-20 (1995). \[4\] F. James, Comput. Phys. Commun. 79 (1994) 111. \[5\] [*Physics at LEP2*]{}, CERN Report 96-01, Theoretical Physics and Particle Physics Experiments Divisions, G. Altarelli, T. Sjöstrand and F. Zwirner, eds., Vols. 1 and 2, Geneva, 19 February 1996. \[6\] W. Beenakker, F. Berends et al., “WW cross-sections and distributions”, in \[5\], Vol. 1, pag. 79.
Introduction
============
The precise measurement of the $W$-boson mass $M_W$ constitutes a primary task of the forthcoming experiments at the high energy electron–positron collider LEP2 ($2 M_W \leq \sqrt{s} \leq 210$ GeV). A meaningful comparison between theory and experiment requires an accurate description of the fully exclusive processes $e^+ e^- \to 4f$, including the main effects of radiative corrections, with the final goal of providing predictions for the distributions measured by the experiments. A large effort in the direction of developing tools dedicated to the investigation of this item has been spent within the Workshop “Physics at LEP2”, held at CERN during 1995. Such an effort has led to the development of several independent four-fermion codes, both semianalytical and Monte Carlo, extensively documented in [@wweg]. [WWGENPV]{} is one of these codes, and the aim of the present paper is to describe in some detail the developments performed with respect to the original version [@cpcww], where a description of the formalism adopted and the physical ideas behind it can be found.
As discussed in [@wmass], the most promising methods for measuring the $W$-boson mass at LEP2 are the so called “threshold” and “direct reconstruction” methods. For the first one, a precise evaluation of the threshold cross section is required. For the second one, a precise description of the invariant-mass shape of the hadronic system in semileptonic and hadronic decays is mandatory. In order to meet these requirements, the previous version of the program has been improved, both from the technical and physical point of view.
On the technical side, in addition to the “weighted event integration” and “unweighted event generation” branches, the present version can also be run as an “adaptive Monte Carlo” integrator, in order to obtain high numerical precision results for cross sections and other relevant observables. In the “weighted event integration” branch, a “canonical” output can be selected, in which several observables are processed in parallel together with their most relevant moments [@wweg]. Moreover, the program offers the possibility of generating events according to a specific flavour quantum number assignment for the final-state fermions, or of generating “mixed samples”, namely a fully leptonic, fully hadronic or semileptonic sample.
On the physical side, the class of tree-level EW diagrams taken into account has been extended to include all the single resonant diagrams ([CC11/CC20]{}), in such a way that all the charged current processes are covered. Motivated by the physical relevance of keeping under control the effects of the transverse degrees of freedom of photonic radiation, both for the $W$ mass measurement and for the detection of anomalous couplings, the contribution of QED radiation has been fully developed in the leading logarithmic approximation, going beyond the initial-state, strictly collinear approximation, to include $p_T / p_L $ effects both for initial- and final-state photons. Last, an hadronic interface to [JETSET]{} in the generation branch has been added.
In the present version the neutral current backgrounds are neglected in the fully hadronic and leptonic decay channels, but this is not a severe limitation of the program since, at least in the LEP2 energy range, these backgrounds can be suppressed by means of proper invariant mass cuts. On the other hand, the semileptonic decay channels are complete at the level of the Born approximation EW diagrams ([CC11/CC20]{} diagrams), and this feature allows to treat at best those channels that are expected to be the most promising for the direct reconstruction of the $W$ mass, because free of systematics such as the “color reconnection” and the “Bose-Einstein correlation” problems.
In the development of the code, particular attention has been paid to the possibility of obtaining precise results in relatively short CPU time. As shown in [@wweg], [WWGENPV]{} is one of the most precise four fermion Monte Carlo’s from the numerical point of view. This feature allows the use of the code also for fitting purposes.
The most important new features
===============================
In the following we list and briefly describe the most important technical and physical developments implemented in the new version of [WWGENPV]{}.
TECHNICAL IMPROVEMENTS
The present version of the program consists of three branches, two of them already present in the original version but upgraded in some respect, the third one completely new.
- Unweighted event generation branch. This branch, meant for simulation purposes, has been improved by supplying an option for an hadronisation interface (see more details later on).
- Weighted event integration branch. This branch, intended for computation only, includes as a new feature an option for selecting a “canonical” output containing predictions for several observables and their most relevant moments together with a Monte Carlo estimate of the errors. According to the strategy adopted in [@wweg], the first four Chebyshev/power moments of the following quantities are computed: the production angle of the $W^+$ with respect to the positron beam ([TNCTHW, N=1,2,3,4]{}), the production angle $\vartheta_{d}$ of the down fermion with respect to the positron beam ([TNCTHD]{}), the decay angle $\vartheta_{d}^*$ of the down fermion with respect to the direction of the decaying $W^-$ measured in its rest frame ([TNCTHSR]{}), the energy $E_{d}$ of the down fermion normalized to the beam energy ([XDN]{}), the sum of the energies of all radiated photons ([XGN]{}) normalized to the beam energy, the lost and visible photon energies normalized to the beam energy ([XGNL]{} and [XGNV]{}), respectively, and, finally $$\begin{aligned}
<x_m> \, = \, {1 \over \sigma} \, \int \left( { { \sqrt{s_+} +
\sqrt{s_-} -2 M_W} \over {2 E_b} } \right) \, d \sigma\end{aligned}$$ where $s_+$ and $s_-$ are the invariant masses of the $W^+$ and $W^-$ decay products, respectively ([XMN]{}).
- Adaptive integration branch. This new branch is intended for computation only, but offers high precision performances. On top of the importance sampling, an adaptive Monte Carlo integration algorithm is used. The code returns the value of the cross section together with a Monte Carlo estimate of the error. Moreover, if QED corrections are taken into account, also the average energy and invariant mass losses are printed. The program must be linked to NAG library for the Monte Carlo adaptive routine. Full consistency between non-adaptive and adaptive integrations has been explicitly proven.
In each of these three branches the user is asked to specify the four-fermion final state which is required. The final states at present available are those containing [CC03]{} diagrams as a subset. Their list, as appears when running the code, is the following:
PURELY LEPTONIC PROCESSES
[0] ---> E+ NU_E E- BAR NU_E
[1] ---> E+ NU_E MU- BAR NU_MU
[2] ---> E- BAR NU_E MU+ NU_MU
[3] ---> E+ NU_E TAU- BAR NU_TAU
[4] ---> E- BAR NU_E TAU+ NU_TAU
[5] ---> MU+ NU_MU MU- BAR NU_MU
[6] ---> MU+ NU_MU TAU- BAR NU_TAU
[7] ---> MU- BAR NU_MU TAU+ NU_TAU
[8] ---> TAU+ NU_TAU TAU- BAR NU_TAU
SEMILEPTONIC PROCESSES
[9] ---> E+ NU_E D BAR U
[10] ---> E- BAR NU_E BAR D U
[11] ---> E+ NU_E S BAR C
[12] ---> E- BAR NU_E BAR S C
[13] ---> MU+ NU_MU D BAR U
[14] ---> MU- BAR NU_MU BAR D U
[15] ---> MU+ NU_MU S BAR C
[16] ---> MU- BAR NU_MU BAR S C
[17] ---> TAU+ NU_TAU D BAR U
[18] ---> TAU- BAR NU_TAU BAR D U
[19] ---> TAU+ NU_TAU S BAR C
[20] ---> TAU- BAR NU_TAU BAR S C
HADRONIC PROCESSES
[21] ---> D BAR U BAR D U
[22] ---> D BAR U BAR S C
[23] ---> S BAR C BAR D U
[24] ---> S BAR C BAR S C
MIXED SAMPLES
[25] ---> LEPTONIC SAMPLE
[26] ---> SEMILEPTONIC SAMPLE
[27] ---> HADRONIC SAMPLE
It is worth noting that in the weighted event integration and unweighted event generation branches, besides the possibility of selecting a specific four-fermion final state, an option is present for considering three realistic mixed samples corresponding to the fully leptonic, fully hadronic or semileptonic decay channel, respectively. When the generation of a mixed sample is required, as a a first step the cross section for each contributing channel is calculated; as a second step, the unweighted events are generated for each contributing channel with a frequency given by the weight of that particular channel with respect to the total.
In the generation branch, if the hadronisation interface is not enabled, an $n$-tuple is created with the following structure:
'X_1','X_2','EB' ! x_{1,2} represent the
! energy fractions of incoming e^- and
! e^+ after ISR; EB is the beam energy;
'Q1X','Q1Y','Q1Z','Q1LB' ! x,y,z components of the momentum
! of particle 1 and the particle label
! according to PDG; the final-state
! fermions are assumed to be massless;
'Q2X','Q2Y','Q2Z','Q2LB' ! as above, particle 2
'Q3X','Q3Y','Q3Z','Q3LB' ! " " " 3
'Q4X','Q4Y','Q4Z','Q4LB' ! " " " 4
'AK1X','AK1Y','AK1Z' ! x,y,z components of the momentum of
! the photon from particle 1;
! they are 0 if no FSR has been chosen
! and/or if particle 1 is a neutrino;
'AK2X','AK2Y','AK2Z' ! as above, particle 2
'AK3X','AK3Y','AK3Z' ! " " " 3
'AK4X','AK4Y','AK4Z' ! " " " 4
'AKEX','AKEY','AKEZ' ! x,y,z components of the momentum of
! the photon from the initial-state
! electron; they are 0 if no ISR has
! been chosen;
'AKPX','AKPY','AKPZ' ! as above, initial-state positron;
If the hadronisation interface is enabled, fully hadronised events are instead made available to a user routine in the [/HEPEVT/]{} format (see below).
PHYSICAL IMPROVEMENTS\
The main theoretical developments with respect to the original version concern the inclusion of additional matrix elements to the tree-level kernel and a more sophisticated treatment of the photonic radiation, beyond the initial-state, strictly collinear approximation. Moreover, an hadronisation interface to [ JETSET]{} has been also provided.
[*Tree-level EW four-fermion diagrams*]{} – In addition to double-resonant charged-current diagrams [CC03]{} already present in the previous version, the matrix element includes also the single-resonant charged-current diagrams [CC11]{} for $\mu $ and $\tau$’s in the final state, and [CC20]{} for final states containing electrons. This allows a complete treatment at the level of four-fermion EW diagrams of the semileptonic sample, which appears the most promising and cleanest for the direct mass reconstruction method due to the absence of potentially large “interconnection” effects [@wmass]. Concerning [CC20]{} diagrams, the importance sampling technique has been extended to take care of the peaking behaviour of the matrix element when small momenta of the virtual photon are involved. As a consequence of the fact that the tree-level matrix-element is computed in the massless limit, a cut on the minimum electron (positron) scattering angle must be imposed. The inclusion of such a cut eliminates the problems connected with gauge-invariance in the case of [CC20]{} processes, for which the present version does not include any so-called “reparation” scheme [@bhf; @wwcd]. Anyway, when for instance a set of “canonical cuts” [@wweg] is used, the numerical relevance of such gauge-invariance restoring schemes has been shown [@bhf; @wwcd] to be negligible compared with the expected experimental accuracy.
[*Photonic corrections*]{} – As far as photonic effects are concerned, the original version, as stated above, included only leading logarithmic initial-state corrections in the collinear approximation within the SF formalism. The treatment has been extended in a two-fold way: the contribution of final-state radiation has been included and the $p_T / p_L$ effects have been implemented both for initial- and final-state radiation. The inclusion of the transverse degrees of freedom has been achieved by generating the fractional energy $x_{\gamma}$ of the radiated photons by means of resummed electron structure functions $D(x; s)$ [@sf] ($x_{\gamma} = 1 - x$) and the angles using an angular factor inspired by the pole behaviour $1 / (p \cdot k)$ for each charged emitting fermion. This allows to incorporate leading QED radiative corrections originating from infrared and collinear singularities, taking into account at the same time the dominant kinematic effects due to non-strictly collinear photon emission, in such a way that the universal factorized photonic spectrum is recovered. According to this procedure, the leading logarithmic corrections from initial- and final-state radiation are isolated as a gauge-invariant subset of the full calculation (not yet available) of the electromagnetic corrections to $e^+ e^- \to 4f$. Due to the inclusion of $p_T$-carrying photons at the level of initial-state radiation, the Lorentz boost allowing the reconstruction of the hard-scattering event from the c.m. system to the laboratory one has been generalized to keep under control the $p_T$ effects on the beam particles. Final state radiation and $p_T / p_L$ effects are not taken into account in the adaptive integration branch.
[*Non-QED corrections*]{} – Coulomb correction is treated as in the original version on double-resonant [CC03]{} diagrams. QCD corrections are implemented in the present version in the naive form according to the recipe described in [@wweg; @wwcd]. The treatment of the leading EW contributions is unchanged with respect to the original version.
[*Hadronisation*]{} – Final-state quarks issuing from the electroweak 4-fermion scattering are not experimentally observable. An hadronisation interface is provided to the [JETSET]{} package [@jetset] to allow events to be extrapolated to the hadron level, for example for input to a detector simulation program. Specifically, the 4-fermion event structure is converted to the [/HEPEVT/]{} convention, then [JETSET]{} is called to simulate QCD partonic evolution (via routines [LUJOIN]{} and [LUSHOW]{}) and hadronisation (routine [LUEXEC]{}). In making this conversion, masses must be added to the outgoing fermions, considered massless in the hard scattering process. This is done by rescaling the fermion momenta by a single scale factor, keeping the flight directions fixed in the rest frame of the four fermion system. In the QCD evolution phase, strings join quarks coming from the same W decay. The virtuality scale of the QCD evolution is taken to be the invariant mass-squared of each evolving fermion pair. No colour reconnection is included by default, although it could be implemented by appropriate modification of routine [WWGJIF]{} if required. Bose - Einstein correlations are neglected in the present version. The resulting event structure is then made available to the user in the [/HEPEVT/]{} common block via a routine [WWUSER]{} for further analysis, such as writing out for later input to a detector simulation program. The [WWUSER]{} routine is also called at program initialisation time to allow the user to set any non-standard [JETSET]{} program options, for example, and at termination time to allow any necessary clean-up. A dummy [WWUSER]{} routine is supplied with the program. The only [JETSET]{} option which is changed by [WWGENPV]{} from its default value controls emission of gluons and photons by final-state partons[^1], turning off final-state photon emission simulation from [JETSET]{} if activated in [WWGENPV]{}, to avoid double counting.\
All the new features of the program can be switched on/off by means of separate flags, as described in the following.
Input
=====
Here we give a short explanation of the input parameters and flags required when running the program.
****
OGEN(CHARACTER*1)
It controls the use of the program as a Monte Carlo event generator of unweighted events ([OGEN = G]{}) or as a Monte Carlo/adaptive integrator for weighted events ([OGEN = I]{}).
****
RS(REAL*8)
The centre-of-mass energy (in GeV).
****
OFAST(CHARACTER*1)
It selects ([OFAST = Y]{}) the adaptive integration branch, when [OGEN = I]{}. When this choice is done, the required relative accuracy of the numerical integration has to be supplied by means of the [REAL\*8]{} variable [EPS]{}.
****
NHITWMAX(INTEGER)
Required by the Monte Carlo integration branch. It is the maximum number of calls for the Monte Carlo loop.
****
NHITMAX(INTEGER)
Required by the event-generation branch. It is the maximum number of hits for the hit-or-miss procedure.
****
IQED(INTEGER)
This flag allows the user to switch on/off the contribution of the initial-state radiation. If [IQED = 0]{} the distributions are computed in lowest-order approximation, while for [IQED = 1]{} the QED corrections are included in the calculation.
****
OPT(CHARACTER*1)
This flag controls the inclusion of $p_T / p_L$ effects for the initial-state radiation. It is ignored in the adaptive integration branch where only initial-state strictly collinear radiation is allowed.
****
OFS(CHARACTER*1)
It is the option for including final-state radiation. It is assumed that final-state radiation can be switched on only if initial-state radiation including $p_T / p_L$ effects is on, in which case final-state radiation includes $p_T / p_L$ effects as well. Ignored in the adaptive integration branch.
****
ODIS(CHARACTER*1)
Required by the integration branch. It selects the kind of experimental distribution. For [ODIS = T]{} the program computes the total cross section (in pb) of the process; for [ODIS = W]{} the value of the invariant-mass distribution $d \sigma / d M$ of the system $d \bar u$ ([IWCH = 1]{}) or of the system ${\bar d} u$ ([IWCH = 2]{}) is returned (in pb/GeV).
****
OWIDTH(CHARACTER*1)
It allows a different choice of the value of the $W$-width. [OWIDTH = Y]{} means that the tree-level Standard Model formula for the $W$-width is used; [OWIDTH = N]{} requires that the $W$-width is supplied by the user in GeV.
****
NSCH(INTEGER)
The value of [NSCH]{} allows the user to choose the calculational scheme for the weak mixing angle and the gauge coupling. Three choices are available. If [NSCH=1]{}, the input parameters used are $G_F, M_W, M_Z$ and the calculation is performed at tree level. If [NSCH = 2]{} or [3]{}, the input parameters used are $\alpha(Q^2), G_F, M_W$ or $\alpha(Q^2), G_F, M_Z$, respectively, and the calculation is performed using the QED coupling constant at a proper scale $Q^2$, which is requested as further input. The recommended choice is [NSCH = 2]{}, consistently with [@wweg].
****
OCOUL(CHARACTER*1)
This flag allows the user to switch on/off the contribution of the Coulomb correction. Unchanged with respect to the old version of the program.
****
OQCD(CHARACTER*1)
This flag allows the user to switch on/off the contribution of the naive QCD correction.
****
ICHANNEL(INTEGER)
A channel corresponding to a specific flavour quantum number assignment can be chosen.
****
ANGLMIN(REAL*8)
The minimum electron (positron) scattering angle (deg.) in the laboratory frame. It is ignored when [CC20]{} graphs are not selected.
****
SRES(CHARACTER*1)
Option for switching on/off single-resonant diagrams ([CC11]{}).
****
OCC20(CHARACTER*1)
Option for switching on/off single-resonant diagrams when electrons (positrons) occur in the final state ([CC20]{}).
****
OOUT(CHARACTER*1)
Option for “canonical” output containing results for several observables and their most important moments. It is active only in the Monte Carlo integration branch.
****
OHAD(CHARACTER*1)
Option for switching on/off hadronisation interface in the unweighted event generation branch.
Test run output
===============
The typical new calculations that can be performed with the updated version of the program are illustrated in the following examples.
An example of adaptive integration is provided. The process considered is $e^+ e^- \to e^+ \nu_e d \bar u$ ([CC20]{}). The output gives the cross section, together with the energy and invariant-mass losses from initial-state radiation. “Canonical” cuts are imposed as in [@wweg]. The input card is as follows:
OGEN = I
RS = 190.D0
OFAST = Y
EPS = 1.D-2
IQED = 1
OPT = N
OFS = N
ODIS = T
OWIDTH = Y
NSCH = 2
ALPHM1 = 128.07D0
OCOUL = N
OQCD = N
ICHANNEL = 9
ANGLMIN = 10.D0
SRES = Y
OCC20 = Y
OOUT = N
An example of weighted event integration is provided. Here the process considered is $e^+ e^- \to \mu^+ \nu_{\mu}
d \bar u$ ([CC11]{}). “Canonical” cuts are imposed as before. The “canonical” output is provided. The input card differs from the previous one as follows:
RS = 175.D0
OFAST = N
NHITWMAX = 100000
OPT = Y
OFS = Y
OCOUL = Y
OQCD = Y
ICHANNEL = 13
OCC20 = N
OOUT = Y
An example of unweighted event generation including hadronisation is provided. A sample of 100 events corresponding to the full semileptonic channel is generated. The detailed list of an hadronised event is given. The input card differs from the first one as follows:
OGEN = G
RS = 175.D0
NHITMAX = 100
OPT = Y
OFS = Y
OCOUL = Y
OQCD = Y
ICHANNEL = 26
ANGLMIN = 5.D0
OHAD = Y
Conclusions
===========
The program [WWGENPV 2.0]{} has been described. In its present version it allows the treatment of all the four-fermion reactions including the [CC03]{} class of diagrams as a subset. This means that all the semileptonic channels are complete from the tree-level diagrams point of view, whereas fully leptonic and fully hadronic channels are treated in the CC approximation. Since the most promising channels for the $W$ mass reconstruction are the semileptonic ones, the present version of the code allows a precise analysis of such data. Moreover, NC backgrounds can be suppressed by proper invariant mass cuts, so that the code is also usable for fully hadronic and leptonic events analysis, with no substantial loss of reliability. Initial- and final-state QED radiation is taken into account within the SF formalism, including finite $p_T / p_L$ effects in the leading logarithmic approximation. The Coulomb correction is included for the [CC03 ]{} graphs. Naive QCD and leading EW corrections are implemented as well. An hadronic interface to [JETSET]{} is also provided.
The code as it stands is a valuable tool for the analysis of LEP2 data, with particular emphasis to the threshold and direct reconstruction methods for the measurement of the $W$-boson mass. Speed and high numerical accuracy allow the use of the program also for fitting purposes.
The code is supported. Future releases of [WWGENPV]{} will include:
- an interface to the code [HIGGSPV]{} [@wweg; @egdp] in order to treat all the possible four-fermion processes in the massless limit, including Higgs-boson signals;
- implementation of anomalous couplings;
- implementation of CKM effects;
- the extension of the hadronic interface to [HERWIG]{} [@herwig].
[99]{}
D. Bardin, R. Kleiss et al., “Event generators for WW physics”, in [*Physics at LEP2*]{}, CERN [**96-01**]{}, Theoretical Physics and Particle Physics Experiments Divisions, G. Altarelli, T. Sjöstrand and F. Zwirner, eds., Vol. 2, pag. 3, Geneva, 19 February 1996.
G. Montagna, O. Nicrosini and F. Piccinini, Comput. Phys. Commun. [**90**]{} (1995) 141.
Z. Kunszt, W.J. Stirling et al., “Determination of the mass of the $W$ boson”, in [*Physics at LEP2*]{}, CERN [**96-01**]{}, Theoretical Physics and Particle Physics Experiments Divisions, G. Altarelli, T. Sjöstrand and F. Zwirner, eds., Vol. 1, pag. 141, Geneva, 19 February 1996.
E. N. Argyres et al., Phys. Lett. [**B358**]{} (1995) 339.
W. Beenakker, F. Berends et al., “WW cross-sections and distributions”, in [*Physics at LEP2*]{}, CERN [**96-01**]{}, Theoretical Physics and Particle Physics Experiments Divisions, G. Altarelli, T. Sjöstrand and F. Zwirner, eds., Vol. 1, pag. 79.
T. Sjöstrand, Comp. Phys. Commun. [**82**]{} (1994) 74; Lund University Report LU TP 95-20 (1995).
M.L. Mangano, G. Ridolfi et al., “Event Generators for Discovery Physics”, in [*Physics at LEP2*]{}, CERN [**96-01**]{}, Theoretical Physics and Particle Physics Experiments Divisions, G. Altarelli, T. Sjöstrand and F. Zwirner, eds., Vol.2, pag. 299.
G. Marchesini, B. R. Webber et al., Comput. Phys. Commun. [**67**]{} (1992) 465.
HADRONISATION VIA JETSET
WWGJIF: Initialising WWGENPV to JETSET interface
PHOTON FSR WILL BE HANDLED BY WWGENPV:
TURNING OFF PHOTON FSR FROM JETSET
1
SQRT(S) = 175.0 GEV
M_Z = 91.1888 GEV
G_Z = 2.4974 GEV
M_W = 80.23 GEV
G_W = 2.08823657515165 GEV
STH2 = .2310309124510679
GVE = -1.409737267299689E-02
GAE = -.185794027211796
GWF = .230409927395451
GWWZ = -.571479454308384
Event listing (summary)
I particle/jet KS KF orig p_x p_y p_z E m
1 !e+! 21 -11 0 .000 .000 -87.500 87.500 .001
2 !e-! 21 11 0 .000 .000 87.500 87.500 .001
3 !e+! 21 -11 1 .000 .000 -87.500 87.500 .001
4 !e-! 21 11 2 .000 .000 87.500 87.500 .001
5 gamma 1 22 1 .000 .000 .000 .000 .000
6 nu_e 1 12 0 45.589 -8.504 -7.296 46.945 .000
7 e+ 1 -11 0 -32.518 1.009 -24.213 40.555 .001
8 gamma 1 22 7 -.063 -.003 -.054 .083 .000
9 (c~) A 11 -4 0 -6.482 5.087 -14.629 16.844 1.350
10 (g) V 11 21 0 -.912 -.166 -1.330 1.621 .000
11 (g) A 11 21 0 -1.717 2.148 .149 2.754 .000
12 (g) V 11 21 0 -.404 -.150 -.153 .457 .000
13 (g) A 11 21 0 .524 .257 -.485 .759 .000
14 (g) V 11 21 0 .568 5.813 -3.909 7.028 .000
15 (g) A 11 21 0 -.833 .123 -.634 1.054 .000
16 (g) V 11 21 0 -1.616 -.218 .984 1.904 .000
17 (g) A 11 21 0 -.893 -.594 -.345 1.127 .000
18 (g) V 11 21 0 -.064 -1.133 .124 1.141 .000
19 (u) A 11 2 0 -1.275 -.657 4.823 5.032 .006
20 (u~) V 11 -2 0 -1.276 .758 3.201 3.528 .006
21 (g) A 11 21 0 .733 -.103 2.158 2.281 .000
22 (g) V 11 21 0 .691 -.234 2.326 2.438 .000
23 (g) A 11 21 0 .410 -1.788 16.927 17.027 .000
24 (s) V 11 3 0 -.461 -1.646 22.354 22.420 .199
25 (string) 11 92 9 -13.104 10.511 -15.404 39.722 32.532
26 (D~0) 11 -421 25 -6.057 4.073 -13.996 15.894 1.865
27 (pi0) 11 111 25 -1.112 1.280 -.302 1.728 .135
28 (rho-) 11 -213 25 -1.158 1.158 -2.172 2.803 .676
29 (omega) 11 223 25 -.792 .558 .401 1.303 .774
30 (Sigma*~+) 11 -3114 25 .123 2.038 -1.332 2.787 1.351
31 (K*-) 11 -323 25 .319 1.963 -1.639 2.729 .896
32 (Delta+) 11 2214 25 -.155 1.330 -1.314 2.263 1.266
33 (rho-) 11 -213 25 -1.026 .221 .345 1.292 .670
34 (pi0) 11 111 25 -.256 .036 -.546 .619 .135
35 (rho0) 11 113 25 -.488 -.118 .538 1.103 .822
36 pi+ 1 211 25 -.420 .013 -.275 .521 .140
37 pi- 1 -211 25 -.612 -.620 .002 .882 .140
38 (rho+) 11 213 25 -.474 -1.082 1.048 1.662 .520
39 (rho0) 11 113 25 -.997 -.339 3.839 4.136 1.122
40 (string) 11 92 20 .096 -3.014 46.967 47.695 7.733
41 p~- 1 -2212 40 -.233 .669 3.532 3.723 .938
42 p+ 1 2212 40 .259 -.320 2.633 2.826 .938
43 (Delta~--) 11 -2224 40 -.208 -.282 7.298 7.431 1.357
44 pi+ 1 211 40 .568 -.518 3.830 3.909 .140
45 (Lambda0) 11 3122 40 -.290 -2.562 29.673 29.806 1.116
46 (K*+) 11 323 26 -3.673 2.525 -9.054 10.129 .875
47 (rho-) 11 -213 26 -2.384 1.548 -4.942 5.765 .856
48 gamma 1 22 27 -1.055 1.186 -.310 1.617 .000
49 gamma 1 22 27 -.058 .094 .008 .111 .000
50 pi- 1 -211 28 -.837 1.095 -1.923 2.370 .140
51 (pi0) 11 111 28 -.321 .063 -.249 .433 .135
52 pi+ 1 211 29 -.610 .314 .197 .727 .140
53 pi- 1 -211 29 -.236 .248 .217 .429 .140
54 (pi0) 11 111 29 .055 -.005 -.014 .146 .135
55 (Lambda~0) 11 -3122 30 .085 1.705 -.918 2.237 1.116
56 pi+ 1 211 30 .038 .333 -.414 .551 .140
57 (K~0) 11 -311 31 .462 1.760 -1.507 2.414 .498
58 pi- 1 -211 31 -.143 .203 -.133 .315 .140
59 n0 1 2112 32 .123 .910 -1.024 1.665 .940
60 pi+ 1 211 32 -.277 .420 -.290 .598 .140
61 pi- 1 -211 33 .005 .082 .159 .227 .140
62 (pi0) 11 111 33 -1.031 .138 .186 1.065 .135
63 gamma 1 22 34 -.201 -.017 -.301 .362 .000
64 gamma 1 22 34 -.055 .053 -.245 .257 .000
65 pi- 1 -211 35 -.071 -.134 .622 .655 .140
66 pi+ 1 211 35 -.417 .016 -.084 .448 .140
67 pi+ 1 211 38 .032 -.277 .386 .497 .140
68 (pi0) 11 111 38 -.506 -.804 .661 1.166 .135
69 pi- 1 -211 39 -.186 .113 .094 .275 .140
70 pi+ 1 211 39 -.811 -.452 3.745 3.861 .140
71 p~- 1 -2212 43 -.135 -.544 5.493 5.600 .938
72 pi- 1 -211 43 -.073 .262 1.805 1.831 .140
73 n0 1 2112 45 -.262 -2.385 26.596 26.721 .940
74 (pi0) 11 111 45 -.028 -.177 3.077 3.085 .135
75 (K0) 11 311 46 -2.419 1.634 -6.536 7.176 .498
76 pi+ 1 211 46 -1.253 .891 -2.517 2.953 .140
77 pi- 1 -211 47 -.404 .604 -1.633 1.792 .140
78 (pi0) 11 111 47 -1.980 .945 -3.310 3.973 .135
79 gamma 1 22 51 -.223 .051 -.091 .246 .000
80 gamma 1 22 51 -.098 .012 -.158 .187 .000
81 gamma 1 22 54 .048 -.066 .004 .082 .000
82 gamma 1 22 54 .006 .062 -.017 .064 .000
83 p~- 1 -2212 55 .064 1.274 -.756 1.755 .938
84 pi+ 1 211 55 .021 .431 -.161 .482 .140
85 (K_S0) 11 310 57 .462 1.760 -1.507 2.414 .498
86 gamma 1 22 62 -.111 .044 -.011 .120 .000
87 gamma 1 22 62 -.920 .094 .197 .945 .000
88 gamma 1 22 68 -.034 -.065 .012 .074 .000
89 gamma 1 22 68 -.472 -.740 .649 1.091 .000
90 gamma 1 22 74 -.035 -.088 2.449 2.451 .000
91 gamma 1 22 74 .007 -.089 .628 .634 .000
92 (K_S0) 11 310 75 -2.419 1.634 -6.536 7.176 .498
93 gamma 1 22 78 -1.877 .868 -3.149 3.768 .000
94 gamma 1 22 78 -.103 .077 -.160 .205 .000
95 pi- 1 -211 85 .270 .383 -.379 .618 .140
96 pi+ 1 211 85 .192 1.377 -1.128 1.796 .140
97 pi+ 1 211 92 -.563 .463 -2.034 2.165 .140
98 pi- 1 -211 92 -1.857 1.171 -4.502 5.011 .140
sum: .00 .000 .000 .000 175.000 175.000
NHIT = 100
XSECT FOR UNWEIGHTED EVENTS
EFF = 1.887112905965163E-03
XSECT = 6.79360646147459 +- .6787193283232224 (PB)
NBIAS/NMAX = .0
USER TERMINATION
A TOTAL OF 100 EVENTS WERE GENERATED
THE TOTAL CROSS-SECTION WAS: 6.793606281280517E-09 MB
[^1]: parameter [MSTJ(41)]{}
| ArXiv |
---
abstract: 'In this paper, we give a family of rational maps whose Julia sets are Cantor circles and show that every rational map whose Julia set is a Cantor set of circles must be topologically conjugate to one map in this family on their corresponding Julia sets. In particular, we give the specific expressions of some rational maps whose Julia sets are Cantor circles, but they are not topologically conjugate to any McMullen maps on their Julia sets. Moreover, some non-hyperbolic rational maps whose Julia sets are Cantor circles are also constructed.'
address:
- 'Weiyuan Qiu, School of Mathematical Sciences, Fudan University, Shanghai, 200433, P. R. China'
- 'Fei Yang, Department of Mathematics, Nanjing University, Nanjing, 210093, P. R. China'
- 'Yongcheng Yin, Department of Mathematics, Zhejiang University, Hangzhou, 310027, P. R. China'
author:
- WEIYUAN QIU
- FEI YANG
- YONGCHENG YIN
title: RATIONAL MAPS WHOSE JULIA SETS ARE CANTOR CIRCLES
---
Introduction
============
The study of the topological properties of the Julia sets of rational maps is a central problem in complex dynamics. For each degree at least two polynomial with a disconnected Julia set, it was proved that all but countably many components of the Julia set are single points in [@QY]. For rational maps, the Julia sets may exhibit more complex topological structures. Pilgrim and Tan proved that if the Julia set of a hyperbolic (more generally, geometrically finite) rational map is disconnected, then, with the possible exception of finitely many periodic components and their countable collection of preimages, every Julia component is either a point or a Jordan curve [@PT Theorem 1.2]. In this paper, we will consider one class of rational maps whose Julia sets possess simple topological structure: each Julia component is a Jordan curve.
A subset of the Riemann sphere $\overline{\mathbb{C}}$ is called a *Cantor set of circles* (sometimes *Cantor circles* in short) if it consists of uncountably many closed Jordan curves which is homeomorphic to $\mathcal{C}\times \mathbb{S}^1$, where $\mathcal {C}$ is the middle third Cantor set and $\mathbb{S}^1$ is the unit circle. The first example of rational map whose Julia set is a Cantor set of circles was discovered by McMullen (see [@Mc $\S$7]). He showed that if $f(z)=z^2+\lambda/z^3$ and $\lambda$ is small enough, then the Julia set of $f$ is a Cantor set of circles. Later, many authors focus on the following family, which is commonly referred as the *McMullen maps*: $$\label{McMullen}
g_{\eta}(z)=z^k+\eta/z^l,$$ where $k,l\geq 2$ and $\eta\in\mathbb{C}\setminus\{0\}$ (see [@DLU; @St; @QWY] and the references therein). These special rational maps can be viewed as a perturbation of the simple polynomial $g_0(z)=z^k$ if $\eta$ is small. It is known that when $1/k+1/l<1$, there exists a punched neighborhood $\mathcal{M}$ centered at origin in the parameter space, which is called the *McMullen domain*, such that when $\eta\in\mathcal{M}$, then the Julia set of $g_\eta$ is a Cantor set of circles (see [@Mc $\S$7] for $k=2,l=3$ and [@DLU $\S$3] for the general cases).
The following three questions arise naturally: (1) Besides McMullen maps, do there exist any other rational maps whose Julia sets are Cantor circles? (2) If the answer to the first question is yes, what do they look like? Or in other words, can we find specific expressions for them? (3) Can we find out all rational maps whose Julia sets are Cantor circles in some sense? This paper will give affirmative answers to these questions.
By quasiconformal surgery, we can obtain many new rational maps after perturbing the immediate super-attracting basin centered at $\infty$ of $g_\eta$ into a geometric one. Fix one of them, then this map is not topologically conjugate to $g_\eta$ on the whole $\overline{\mathbb{C}}$. But they are topologically conjugate to each other on their corresponding Julia sets. In particular, $h_{c,\eta}(z)=\frac{1}{z}\circ (z^k+c)\circ\frac{1}{z}+\eta/z^l$ is an example, where $1/k+1/l<1$ and $c,\eta\in\mathbb{C}\setminus\{0\}$ are both small enough. However, these types of rational maps can be also regarded as the McMullen maps essentially, which are not what we want to find since they can be obtained by doing a surgery only on the Fatou sets of the genuine McMullen maps. So it will be very interesting to find other types of rational maps with Cantor circles Julia sets which are not topologically conjugate to any McMullen maps on their corresponding Julia sets.
The existence of of types of rational maps ‘essentially’ different from McMullen maps was known previously (see [@HP $\S\S$1,2]). Here, ‘essentially’ means there exists no topological conjugacy between the Julia sets of McMullen maps and the rational maps whose Julia sets are Cantor circles. In this paper, we will give the specific expressions for these types of rational maps, not only including the cases discussed in [@HP], but also covering all the rational maps whose Julia sets are Cantor circles ‘essentially’ (see Theorem \[this-is-all\]).
Let $p\in\{0,1\}$, $n\geq 2$ be an integer and $d_1,\cdots,d_n$ be $n$ positive integers such that $\sum_{i=1}^{n}(1/d_i)<1$. We define $$\label{family}
f_{p,d_1,\cdots,d_n}(z)=z^{(-1)^{n-p} d_1}\prod_{i=1}^{n-1}(z^{d_i+d_{i+1}}-a_i^{d_i+d_{i+1}})^{(-1)^{n-i-p}},$$ where $a_1,\cdots,a_{n-1}$ are $n-1$ small complex numbers satisfying $0<|a_1|<\cdots<|a_{n-1}|<1$. In particular, if $n=2$, then $f_{1,d_1,d_2}(z)=z^{d_2}-a_1^{d_1+d_2}/z^{d_1}$ is the McMullen map that has been well studied by many authors. Moreover, $f_{0,d_1,d_2}(z)=z^{d_1}/(z^{d_1+d_2}-a_1^{d_1+d_2})$ is conformally conjugate to the McMullen map $z\mapsto z^{d_1}+\eta/z^{d_2}$ for some $\eta\neq 0$. The degrees of $f_{p,d_1,\cdots,d_n}$ at $0$ and $\infty$ are $d_1$ and $d_n$ respectively and $\text{deg} (f_{p,d_1,\cdots,d_n})=\sum_{i=1}^{n}d_i$. For each element in the family , it is easy to check that $0$ and $\infty$ belong to the Fatou set of $f_{p,d_1,\cdots,d_n}$. Let $D_0$ and $D_\infty$ be the Fatou components containing $0$ and $\infty$ respectively. There are four cases (we use $f$ to replace $f_{p,d_1,\cdots,d_n}$ temporarily):
\(1) If $p=1$ and $n$ is odd, then $f(D_0)=D_0$ and $f(D_\infty)=D_\infty$;
\(2) If $p=1$ and $n$ is even, then $f(D_0)=D_\infty$ and $f(D_\infty)=D_\infty$;
\(3) If $p=0$ and $n$ is odd, then $f(D_0)=D_\infty$ and $f(D_\infty)=D_0$;
\(4) If $p=0$ and $n$ is even, then $f(D_0)=D_0$ and $f(D_\infty)=D_0$.
Firstly we will find suitable parameters $a_i$ in , where $1\leq i\leq n-1$, such the Julia set of each $f_{p,d_1,\cdots,d_n}$ in the four cases stated above is a Cantor set of circles.
\[parameter\] For each given $p\in\{0,1\}$, $n\geq 2$ and $d_1,\cdots,d_n$ satisfying $\sum_{i=1}^{n}(1/d_i)<1$, there exist suitable parameters $a_i$, where $1\leq i\leq n-1$ such that the Julia set of $f_{p,d_1,\cdots,d_n}$ is a Cantor set of circles.
The specific value ranges of $a_i$ are given in §\[sec-loc-crit\], where $1\leq i\leq n-1$ (see , and Theorem \[parameter-restate\]). These rational maps can be seen as the perturbations of $z^{d_n}$ or $z^{-d_n}$ (according to whether $p=1$ or 0) since each $a_i$ can be arbitrarily small (see Theorem \[parameter-restate\]). Moreover, it will be shown that if $n\geq 3$, then each $f_{p,d_1,\cdots,d_n}$ is not topologically conjugate to any McMullen maps on their corresponding Julia sets (see Theorem \[no-topo-equiv\]). This means that we have found the specific expressions of rational maps whose Julia sets are Cantor circles which are ‘essentially’ different from McMullen maps.
For example, let $p=1$, $n=4$, $d_1=d_2=d_3=d_4=5$ and define $$\label{an-example}
f_{1,5,5,5,5}(z)=\frac{(z^{10}-a_1^{10})(z^{10}-a_3^{10})}{z^5(z^{10}-a_2^{10})},$$ where $a_1=0.00025,a_2=0.005$ and $a_3=0.1$. By a straightforward calculation or using Theorem \[parameter-restate\] and Remark \[range-unif\], one can show that the Julia set of $f_{1,5,5,5,5}$ is a Cantor set of circles (see Figure \[Fig\_Cantor-cicle\]). The dynamics on the set of Julia components of $f_{1,5,5,5,5}$ is conjugate to the one-sided shift on four symbols $\Sigma_4:=\{0,1,2,3\}^{\mathbb{N}}$ while the set of Julia components of $g_\eta$ is conjugate to the one-sided shift on only two symbols $\Sigma_{2}:=\{0,1\}^{\mathbb{N}}$. This means that $f_{1,5,5,5,5}$ cannot be topologically conjugate to $g_\eta$ on their corresponding Julia sets.
![The Julia set of $f_{1,5,5,5,5}$ (left picture), which is not topologically conjugate to that of McMullen map $g_\eta(z)=z^3+0.001/z^3$ (right picture). The two Julia sets are both Cantor circles.[]{data-label="Fig_Cantor-cicle"}](Cantor-circle-Julia.png "fig:"){width="65mm"} ![The Julia set of $f_{1,5,5,5,5}$ (left picture), which is not topologically conjugate to that of McMullen map $g_\eta(z)=z^3+0.001/z^3$ (right picture). The two Julia sets are both Cantor circles.[]{data-label="Fig_Cantor-cicle"}](McMullen-Julia-Cantor-circle.png "fig:"){width="65mm"}
Note that if the Julia set $J(f)$ of a rational map $f$ is a Cantor set of circles, then there exist no critical points in $J(f)$ since each Julia component is a Jordan closed curve (see Lemma \[no-crit-on-J\]). This means that every periodic Fatou component of $f$ must be attracting or parabolic. In fact, we have following theorem.
\[this-is-all\] Let $f$ be a rational map whose Julia set is a Cantor set of circles. Then there exist $p\in\{0,1\}$, positive integers $n\geq 2$, and $d_1,\cdots, d_n$ satisfying $\sum_{i=1}^{n}(1/d_i)<1$ such that $f$ is topologically conjugate to $f_{p,d_1,\cdots,d_n}$ on their corresponding Julia sets for suitable parameters $a_i$, where $1\leq i\leq n-1$.
Since the dynamics on the Fatou set can be perturbed freely, it follows from Theorem \[this-is-all\] that we have found ‘all’ the possible rational maps whose Julia sets are Cantor circles. A rational map is *hyperbolic* if all critical points are attracted by attracting periodic orbits. For the regularity of the Julia components of $f_{p,d_1,\cdots, d_n}$, it can be shown that each Julia component of $f_{p,d_1,\cdots,d_n}$ is a quasicircle if $f_{p,d_1,\cdots,d_n}$ is hyperbolic (see Corollary \[Julia-comp\]).
If $\eta$ is small enough, then $g_\eta$ is hyperbolic (see [@DLU]). Now we construct some non-hyperbolic rational maps whose Julia sets are Cantor circles. Let $m,n\geq 2$ be two positive integers satisfying $1/m+1/n<1$ and $\lambda\in\mathbb{C}\setminus\{0\}$, we define $$P_\lambda(z)=\frac{\frac{1}{n}((1+z)^n-1)+\lambda^{m+n}z^{m+n}}{1-\lambda^{m+n}z^{m+n}}.$$ It is straightforward to verify that zero is a parabolic fixed point of $P_\lambda$ with multiplier one. We then have the following theorem.
\[non-hyper-cantor\] If $0<|\lambda|\leq 1/{(2^{10m}n^3)}$, then $P_\lambda$ is non-hyperbolic and its Julia set is a Cantor set of circles.
Inspired by Theorem \[parameter\], we can construct more non-hyperbolic rational maps whose Julia sets are Cantor circles. For simplicity, for each $n\geq 2$, we only consider the case $d_i=n+1$ for every $1\leq i\leq n$. For every $n\geq 2$, we define $$\label{family-para}
P_n(z)=A_n\,\frac{(n+1)z^{(-1)^{n+1} (n+1)}}{nz^{n+1}+1}\prod_{i=1}^{n-1}(z^{2n+2}-b_i^{2n+2})^{(-1)^{i-1}}+B_n,$$ where $b_1,\cdots,b_{n-1}$ are $n-1$ small complex numbers satisfying $1>|b_1|>\cdots>|b_{n-1}|>0$ and $$\label{A-B-n}
\begin{split}
A_n=\frac{1}{1+(2n+2)C_n}\prod_{i=1}^{n-1}(1-b_i^{2n+2})^{(-1)^i}, &~~B_n=\frac{(2n+2)C_n}{1+(2n+2)C_n} \\
\text{and}~C_n=\sum_{i=1}^{n-1}\frac{(-1)^{i-1}b_i^{2n+2}}{1-b_i^{2n+2}}. &
\end{split}$$
The terms $A_n$ and $B_n$ here can guarantee that $P_n(1)=1$ and $P_n'(1)=1$. Namely, $1$ is a parabolic fixed point of $P_n$ with multiplier one (see Lemma \[para-fixed\]).
\[parameter-parabolic\] For every $n\geq 2$ and $1\leq i\leq n-1$, if $|b_i|=s^i$ for $0<s\leq 1/(25n^2)$, then $P_n$ is non-hyperbolic and its Julia set is a Cantor set of circles.
It can be seen later the dynamics of $P_n$ on their Julia sets are conjugate to that of $f_{1,n+1,\cdots,n+1}$ for $n\geq 2$. One of the differences between their dynamics on the Fatou sets is the super-attracting basin of $f_{1,n+1,\cdots,n+1}$ at $\infty$ is replaced by a parabolic basin of $P_n$.
This paper is organized as follows: In §\[sec-loc-crit\], we do some estimates and prove Theorem \[parameter\]. In §\[sec-topo-conj\], we prove Theorem \[this-is-all\]. In §\[sec-para-mcm\], we show that the Julia set of $P_\lambda$ is a Cantor set of circles if $\lambda$ is small enough and prove Theorem \[non-hyper-cantor\]. We will prove Theorem \[parameter-parabolic\] in §\[sec-more-exam\] and leave a key lemma to the last section.
0.2cm *Notation*. We will use the following notations throughout the paper. Let $\mathbb{C}$ be the complex plane and $\overline{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$ the Riemann sphere. For $r>0$ and $a\in\mathbb{C}$, let $\mathbb{D}(a,r):=\{z\in\mathbb{C}:|z-a|<r\}$ be the Euclidean disk centered at $a$ with radius $r$. In particular, let $\mathbb{D}_r:=\mathbb{D}(0,r)$ be the disk centered at the origin with radius $r$ and $\mathbb{T}_r:=\partial\mathbb{D}_r$ be the boundary of $\mathbb{D}_r$. As usual, $\mathbb{D}:=\mathbb{D}_1$ and $\mathbb{S}^1:=\mathbb{T}_1$ denote the unit disk and the unit circle, respectively. For $0<r<R<+\infty$, let $\mathbb{A}_{r,R}:=\{z\in\mathbb{C}:r<|z|<R\}$ be the round annulus centered at the origin.
Location of the critical points and the hyperbolic case {#sec-loc-crit}
=======================================================
First we give some basic and useful estimations.
\[very-useful-est\] Let $n\geq 2$ be an integer, $a\in\mathbb{C}\setminus\{0\}$ and $0<\varepsilon<1/2$.
$(1)$ If $|z-a|\leq \varepsilon |a|$, then $|z^{n}-a^{n}|\leq ((1+\varepsilon)^{n}-1)\, |a|^{n}$;
$(2)$ If $|z^{n}-a^{n}|\leq \varepsilon |a|^{n}$, then $|a/z|^n<1+2\varepsilon$ and $|z-ae^{2\pi i{j}/{n}}|< \varepsilon |a|$ for some $1\leq j\leq n$;
$(3)$ If $0<\varepsilon<1/n$, then $n\varepsilon< (1+\varepsilon)^n-1< 3n\varepsilon$ and $n\varepsilon/3< 1-(1-\varepsilon)^n<n\varepsilon$.
Let $z=a(1+re^{i\theta})$ for $0\leq r\leq \varepsilon$ and $0\leq\theta<2\pi$, then $$|z^{n}-a^{n}|= |(1+re^{i\theta})^{n}-1|\cdot|a|^{n}\leq ((1+\varepsilon)^{n}-1)\, |a|^{n}.$$ This proves (1). The first statement in (2) follows from $|a/z|^n\leq 1/(1-\varepsilon)<1+2\varepsilon$ if $0<\varepsilon<1/2$. For the second statement, let $z^{n}=a^{n}(1+re^{i\theta})$ for $0\leq r\leq \varepsilon$ and $0\leq\theta<2\pi$, then $z=ae^{2\pi i{j}/{n}}(1+re^{i\theta})^{1/n}$ for some $1\leq j\leq n$ and we have $$|z-ae^{2\pi i{j}/{n}}|=|(1+re^{i\theta})^{1/n}-1|\cdot|a| \leq ((1+\varepsilon)^{1/n}-1)\cdot|a| < \varepsilon |a|$$ if $n\geq 2$. The claim (3) can be proved by using Lagrange’s mean value theorem to $x\mapsto x^n$ on the intervals $[1,1+\varepsilon]$ and $[1-\varepsilon,1]$ respectively. The proof is complete.
Fix $n\geq 2$ and let $d_1,\cdots,d_n\geq 2$ be $n$ positive numbers such that $\xi=\sum_{i=1}^{n}(1/d_i)<1$. We use $K\geq 3$ to denote the maximal number among $d_1,\cdots,d_n$. Let $u_1=s_1 K^{-5}$ and $v_1=s_1 K^{-2}$, where $$\label{range-s1}
0<s_1\leq \min\{K^{-5\xi/(1-\xi)},K^{5-2K}\}<1.$$ Let $u_0=s_0^{1+1/d_n+2(1-\xi)/3}$, $v_0=s_0^{1/d_n+(1-\xi)/3}$, where $$\label{range-s0}
0<s_0\leq \min\{2^{-(1-\xi)^{-1}(1+1/d_n-2\xi/3)^{-1}},(4K)^{-3/(1-\xi)},K^{-2K(1+1/d_n+2(1-\xi)/3)^{-1}}\}<1.$$ For $p\in\{0,1\}$, let $|a_{n-1,p}|=v_p^{1/d_{n}}$ and $|a_{i,p}|=u_p^{1/d_{i+1}}|a_{i+1,p}|$ be the $n-1$ parameters in the family $f_{p,d_1,\cdots,d_n}$, where $1\leq i\leq n-2$. Since the cases $p=0$ and $p=1$ can be discussed uniformly in general, we use $s$, $u$, $v$ and $a_i$, respectively, to denote $s_p$, $u_p$, $v_p$ and $a_{i,p}$ for simplicity when the situation is clear, where $1\leq i\leq n-1$.
\[esti-a1\] $(1)$ $u^{2/K}\leq K^{-4}$.
$(2)$ If $1\leq j\leq i\leq n-1$, then $|a_j/a_i|\leq u^{\frac{i-j}{K}}$.
$(3)$ If $p=1$, then
$(s/|a_1|)^{d_1}< su/(2v)=sK^{-3}/2$ and
$(|a_1|/s)^{d_1}v/2>K$.
$(4)$ If $p=0$, then
$2Ku/v<s$ and $1/(2Kv)>(2/s)^{1/d_n}$;
$(s/|a_1|)^{d_1}<sv/2<u^{1/2}/2$ and
$(|a_1|/s)^{d_1}u/(2v)>(2/s)^{1/d_n}$.
\(1) From and , we have $s_1\leq K^{5-2K}$ and $s_0\leq K^{-2K(1+1/d_n+2(1-\xi)/3)^{-1}}$. This means that $u_1^{2/K}=(s_1 K^{-5})^{2/K}\leq K^{-4}$ and $u_0^{2/K}\leq K^{-4}$.
\(2) If $j=i$, then (2) is trivial. Suppose that $1\leq j<i\leq n-1$, then $$|a_j/a_i|= u^{\frac{1}{d_{j+1}}+\cdots+\frac{1}{d_{i}}}\leq u^{\frac{i-j}{K}}$$ since $K\geq d_i$ for $1\leq i\leq n$. This proves (2).
\(3) If $p=1$, then $u=s K^{-5}$ and $v=s K^{-2}$. Since $s\leq K^{-5\xi/(1-\xi)}$, we have $s^{1-\xi}K^{5\xi}\leq 1$, so $$s^{1-\frac{1}{d_1}} s^{-(\frac{1}{d_2}+\cdots+\frac{1}{d_n})} K^{5(\frac{1}{d_2}+\cdots+\frac{1}{d_{n-1}})+\frac{2}{d_n}} 2^{\frac{1}{d_1}} K^{\frac{3}{d_1}}<1.$$ This is equivalent to $s^{1-\frac{1}{d_1}}2^{\frac{1}{d_1}} K^{\frac{3}{d_1}}/|a_1|<1$ since $$|a_1|=u^{\frac{1}{d_2}+\cdots+\frac{1}{d_{n-1}}}v^{\frac{1}{d_{n}}}=s^{\frac{1}{d_2}+\cdots+\frac{1}{d_n}}/K^{5(\frac{1}{d_2}+\cdots+\frac{1}{d_{n-1}})+\frac{2}{d_n}}.$$ So we have $(s/|a_1|)^{d_1}< su/(2v)=sK^{-3}/2$ and is proved. Moreover, (3b) can be derived from (3a) directly since $(|a_1|/s)^{d_1}>2K^3/s=2K/v$.
\(4) If $p=0$, then $u=s^{1+1/d_n+2(1-\xi)/3}$, $v=s^{1/d_n+(1-\xi)/3}$. From , we know $4Ks^{(1-\xi)/3}\leq 1$, which means $2Ku/v=2Ks^{1+(1-\xi)/3}<s$. Note that $2^{1+1/d_n}K s^{(1-\xi)/3}<1$, which is equivalent to $1/(2Kv)>(2/s)^{1/d_n}$. This ends the proof of (4a).
From , we know that $$\begin{split}
1\geq &~2s^{(1-\xi)(1+1/d_n-2\xi/3)}>2^{\frac{1}{d_1}}s^{(1-\xi)(1+1/d_n-2\xi/3)}\\
= &~ 2^{\frac{1}{d_1}} s^{1-\frac{1}{d_1}}/s^{(\frac{1}{d_2}+\cdots+\frac{1}{d_{n-1}})+\frac{1}{d_n}(\frac{1}{d_1}+\cdots+\frac{1}{d_{n}})+\frac{2\xi(1-\xi)}{3}}\\
> &~ 2^{\frac{1}{d_1}} s^{1-\frac{1}{d_1}}/s^{(\frac{1}{d_2}+\cdots+\frac{1}{d_{n-1}})+\frac{1}{d_n}(\frac{1}{d_1}+\cdots+\frac{1}{d_{n}})+\frac{1-\xi}{3}(\frac{1}{d_1}+2(\frac{1}{d_2}+\cdots+\frac{1}{d_{n-1}})+\frac{1}{d_n})}\\
= &~ s^{1-\frac{1}{d_1}} (2/v)^{\frac{1}{d_1}}/|a_1|.
\end{split}$$ This means that $(s/|a_1|)^{d_1}<sv/2=u^{1/2}s^{(1+1/d_n)/2}/2<u^{1/2}/2$. So (4b) holds.
The proof of (4c) is similar to (4b). We just need to note that $$1\geq 2s^{(1-\xi)(1+1/d_n-2\xi/3)}>2^{\frac{1}{d_1}(1+\frac{1}{d_n})}s^{(1-\xi)(1+1/d_n-2\xi/3)}
> (s/|a_1|)(2v/u)^{\frac{1}{d_1}}(2/s)^{\frac{1}{d_1 d_n}}.$$ This means that $(|a_1|/s)^{d_1}u/(2v)>(2/s)^{1/d_n}$.
In the following, we use $f$ to denote $f_{p,d_1,\cdots,d_n}$ for simplicity. Note that $0$ and $\infty$ are critical points of $f$ with multiplicity $d_1$ and $d_n$ respectively, and the degree of $f$ is $\sum_{i=1}^{n}d_i$. Denoting $D_i=d_i+d_{i+1}$, we have $5\leq D_i\leq 2K$, where $1\leq i\leq n-1$. Besides $0$ and $\infty$, the rest of the $\sum_{i=1}^{n-1}D_i$ critical points of $f$ are the solutions of $$\label{solu-crit}
(-1)^p \,z\,\frac{f'(z)}{f(z)}=\sum_{i=1}^{n-1}\frac{(-1)^{n-i}D_i z^{D_i}}{z^{D_i}-a_i^{D_i}}+(-1)^n d_1=0.$$
For $1\leq i\leq n-1$, let $\widetilde{CP}_i:=\{\widetilde{w}_{i,j}=r_i a_i \exp(\pi \textup{i}\frac{2j-1}{D_i}):1\leq j\leq D_i\}$ be the collection of $D_i$ points lying on the circle $\mathbb{T}_{r_i|a_i|}$ uniformly, where $r_i=\sqrt[D_i]{d_{i}/d_{i+1}}$. The following lemma shows that the $\sum_{i=1}^{n-1}D_i$ *free* critical points of $f$ are very ‘close’ to $\bigcup_{i=1}^{n-1} \widetilde{CP}_i$.
\[crit-close\] For every $\widetilde{w}_{i,j}\in\widetilde{CP}_i$, where $1\leq i\leq n-1$ and $1\leq j\leq D_i$, there exists $w_{i,j}$, which is a solution of , such that $|w_{i,j}-\widetilde{w}_{i,j}|<u^{\frac{2}{K}}|a_i|$. Moreover, $w_{i_1,j_1}= w_{i_2,j_2}$ if and only if $(i_1,j_1)=(i_2,j_2)$.
Note that the right side of equation is equivalent to $$\label{solu-crit-111}
(-1)^{n-i}\left(\frac{D_i z^{D_i}}{z^{D_i}-a_i^{D_i}}-d_{i}\right)+G_i(z)=0,$$ where $$\label{G_n}
G_{i}(z)=\sum_{1\leq j\leq n-1,\,j\neq i}\frac{(-1)^{n-j}D_j z^{D_j}}{z^{D_j}-a_j^{D_j}}+(-1)^n d_1+(-1)^{n-i}d_{i}.$$ After multiplying both sides of by $(z^{D_i}-a_i^{D_i})/d_{i+1}$, where $1\leq i\leq n-1$, we have $$\label{solu-crit-3}
(-1)^{n-i}(z^{D_i}+d_{i}a_i^{D_i}/d_{i+1})+(z^{D_i}-a_i^{D_i})\,G_{i}(z)/d_{i+1}=0.$$
Let $\Omega_{i}=\{z:|z^{D_i}+d_{i}a_i^{D_i}/d_{i+1}|\leq \varepsilon\,|a_i|^{D_i}\}$, where $\varepsilon=u^{\frac{2}{K}}$ and $1\leq i\leq n-1$. For every $z\in\Omega_{i}$, since $\varepsilon\leq K^{-4}$ by Lemma \[esti-a1\](1), we have $$\label{estim-0}
K^{-1}< d_{i}/d_{i+1}-\varepsilon\leq |z/a_i|^{D_i}\leq d_{i}/d_{i+1}+\varepsilon< K-1<K.$$ This means that $$\label{estim-00}
K^{-1}< |a_i/z|^{D_i}< K~~\text{and therefore}~~K^{-1}< |a_i/z|^{5}< K.$$
If $1\leq j<i$ and $z\in\Omega_{i}$, we have $$\label{less-than-1}
|{a_j}/{z}|^{D_i}\leq|{a_i}/{z}|^{D_i}|{a_{i-1}}/{a_i}|^{D_i}< K u^{1+d_{i+1}/d_{i}}<1.$$ Therefore, $|{a_j}/{z}|<1$. By the similar argument, it can be shown that $|z/a_j|<1$ if $i<j\leq n-1$ and $z\in\Omega_{i}$. If $1\leq j<i$, by Lemma \[esti-a1\](1) and (2) and , we have $$\label{estim-1}
|{a_j}/{z}|^{D_j}\leq|{a_i}/{z}|^{5}|{a_j}/{a_i}|^{5}< K\,\varepsilon^{5(i-j)/2}\leq K^{-9}.$$ Similarly, if $i<j\leq n-1$, we have $$\label{estim-2}
|{z}/{a_j}|^{D_j}\leq|{z}/{a_i}|^{5}|{a_i}/{a_j}|^{5}< K\,\varepsilon^{5(j-i)/2}\leq K^{-9}.$$ By definition, we have $$\label{estim-3}
\sum_{1\leq j<i}(-1)^{n-j}D_j+(-1)^n d_1+(-1)^{n-i}d_{i}=0.$$ From , (\[estim-1\]), (\[estim-2\]) and , we have $$\begin{split}
|G_{i}(z)|
= &~ \left|\sum_{1\leq j<i}\frac{(-1)^{n-j}D_j}{1-(a_j/z)^{D_j}}+
\sum_{i< j\leq n-1}\frac{(-1)^{n-j-1}D_j(z/a_j)^{D_j}}{1-(z/a_j)^{D_j}}+(-1)^n d_1+(-1)^{n-i}d_{i}\right|\\
\leq &~ 2\,K\,\left|\sum_{1\leq j<i}\frac{(-1)^{n-j}(a_j/z)^{D_j}}{1-(a_j/z)^{D_j}}+
\sum_{i< j\leq n-1}\frac{(-1)^{n-j-1}(z/a_j)^{D_j}}{1-(z/a_j)^{D_j}}\right|\\
< &~\frac{4 K^2}{1-K^{-9}}\,\sum_{k=1}^{n-1}\varepsilon^{5k/2}<
\frac{4 K^2}{1-K^{-9}}\,\frac{\varepsilon^{5/2}}{1-\varepsilon^{5/2}}<5\,K^2\,\varepsilon^{5/2}
\end{split}$$ since $\varepsilon^{5/2}\leq K^{-10}$. This means that if $z\in\Omega_{i}$, we have $$|z^{D_i}-a_i^{D_i}|\cdot|\,G_{i}(z)|/d_{i+1}< 3\,K^3\,\varepsilon^{5/2}|a_i|^{D_i} < \varepsilon|a_i|^{D_i}$$ by and Lemma \[esti-a1\](1).
From (\[solu-crit-3\]) and by Rouché’s Theorem, there exists a solution $w_{i,j}$ of such that $w_{i,j}\in\Omega_i$ for every $1\leq j\leq D_i$. In particular, $|w_{i,j}-\widetilde{w}_{i,j}|<\varepsilon|a_i|$ by the second statement of Lemma \[very-useful-est\](2). Note that for $1\leq i\leq n-2$, we have $$\label{differ-1}
|a_{i+1}|-|a_i|-2\varepsilon|a_i|-2\varepsilon|a_{i+1}|>|a_{i+1}|(1-2\varepsilon-(1+2\varepsilon)K^{-2})>0.$$ By Lemma \[esti-a1\](1) and $r_i=\sqrt[D_i]{d_{i}/d_{i+1}}\leq (K/2)^{1/5}$, we have, $$\label{differ-2}
\frac{r_i|a_i|\sin(\pi/D_i)}{\varepsilon|a_i|}\geq K^4(\frac{2}{K})^{1/5}\cdot\frac{2}{\pi}\cdot\frac{\pi}{2K}>K^2>1.$$ This means that $w_{i_1,j_1}= w_{i_2,j_2}$ if and only if $(i_1,j_1)=(i_2,j_2)$. The proof is complete.
For $1\leq i\leq n-1$, let $CP_i:=\{w_{i,j}: 1\leq j\leq D_i\}$ be the collection of $D_i$ free critical points of $f$ which lie close to the circle $\mathbb{T}_{r_i|a_i|}$ and denote $CV_i=f(CP_i)$.
\[nice-condition\] For every $1\leq i\leq n-1$, there exists an annular neighborhood $A_i$ containing $CP_i\cup\mathbb{T}_{r_i|a_i|}\cup\mathbb{T}_{|a_i|}$, such that
If $p=1$, then $f(\overline{A}_i)\subset\mathbb{D}_s$ for odd $n-i$ and $f(\overline{A}_i)\subset\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}_{K}$ for even $n-i$. In particular, the set of critical values of $f$ satisfies $\bigcup_{i=1}^{n-1}CV_i\subset\mathbb{D}_s \cup \overline{\mathbb{C}}\setminus\overline{\mathbb{D}}_{K}$. The disks $\overline{\mathbb{D}}_s$ and $\overline{\mathbb{C}}\setminus\mathbb{D}_{K}$ lie in the Fatou set of $f$ and $f^{-1}(\overline{\mathbb{A}}_{s,K})\subset \mathbb{A}_{s,K}$.
$(2)$ If $p=0$, then $f(\overline{A}_i)\subset\mathbb{D}_s$ for even $n-i$ and $f(\overline{A}_i)\subset\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}_{M}$ for odd $n-i$, where $M=(2/s)^{1/d_n}$. In particular, the set of critical values of $f$ satisfies $\bigcup_{i=1}^{n-1}CV_i\subset\mathbb{D}_s \cup \overline{\mathbb{C}}\setminus\overline{\mathbb{D}}_M$. The disks $\overline{\mathbb{D}}_s$ and $\overline{\mathbb{C}}\setminus\mathbb{D}_M$ lie in the Fatou set of $f$ and $f^{-1}(\overline{\mathbb{A}}_{s,M})\subset \mathbb{A}_{s,M}$.
Let $\varepsilon=u^{\frac{2}{K}}\leq K^{-4}$ be the number that appeared in Lemma \[crit-close\]. For every $1\leq i\leq n-1$, define the annulus $$A_i=\{z:(\min\{r_i,1\}-2\varepsilon)|a_i|<|z|<(\max\{r_i,1\}+2\varepsilon)|a_i|\}$$ where $r_i=\sqrt[D_i]{d_{i}/d_{i+1}}$. Obviously, $A_i\supset CP_i\cup\mathbb{T}_{r_i|a_i|}\cup\mathbb{T}_{|a_i|}$. By the definition, we have $$(2/K)^\frac{1}{D_i}\leq \min\{r_i,1\}\leq \max\{r_i,1\}\leq (K/2)^\frac{1}{D_i}.$$
If $z\in\overline{A}_i$, we have $$\label{esti-below}
|a_i/z|\leq\frac{1}{(2/K)^\frac{1}{D_i}-2\varepsilon}\leq \frac{(K/2)^\frac{1}{D_i}}{1-2K^{-4}(K/2)^{1/5}}<(K/2)^\frac{1}{D_i}(1+4/K^{19/5}).$$ and $$\label{esti-above}
|z/a_i|\leq(K/2)^\frac{1}{D_i}+2\varepsilon\leq (K/2)^\frac{1}{D_i}+2/K^4<(K/2)^\frac{1}{D_i}(1+1/K^3).$$ This means that $$\label{esti-below-new}
|a_i/z|^5<(K/2)^\frac{5}{D_i}(1+4/K^{19/5})^{5}<(K/2)\,e^{20/K^{19/5}}< (K/2)\,e^{20/3^{19/5}}<7K/10.$$ and also, $$\label{esti-above-new}
|z/a_i|^5<(K/2)^\frac{5}{D_i}(1+1/K^3)^{5}<(K/2)\,e^{5/K^3}< (K/2)\,e^{5/27}<7K/10.$$ Moreover, similar to the argument of and , we have $$\label{esti-other}
|a_i/z|^{d_{i}}+|z/a_i|^{d_{i+1}}<7K/5.$$
Recall that $|a_i/a_{i+1}|^{d_{i+1}}=u$ for every $1\leq i\leq n-2$ and $|a_{n-1}|^{d_n}=v$. Let $1\leq i_1\leq i_2\leq n-1$ and $p\in\{0,1\}$, we have $$\label{sequence}
\begin{split}
\prod_{j=i_1}^{i_2}|a_j|^{(-1)^{n-j-p}D_j}=
&~ |a_{i_1}|^{(-1)^{n-i_1-p}d_{i_1}}\,|a_{i_2}|^{(-1)^{n-i_2-p}d_{i_2+1}}
\,\prod_{j=i_1}^{i_2-1}\left|\frac{a_{j}}{a_{j+1}}\right|^{(-1)^{n-j-p}d_{j+1}} \\
= &~ |a_{i_1}|^{(-1)^{n-i_1-p}d_{i_1}}\,|a_{i_2}|^{(-1)^{n-i_2-p}d_{i_2+1}}\,u^{\frac{(-1)^{n-i_1-p}-(-1)^{n-i_2-p}}{2}}\\
= &~
\left\{ \begin{array}{ll} (|a_1|^{d_1}u/v)^{(-1)^p} &~~\text{if}~~i_1=1~~\text{and}~~i_2=n-1~~\text{is even} \\
(|a_1|^{-d_1}/v)^{(-1)^p} &~~\text{if}~~i_1=1~~\text{and}~~i_2=n-1~~\text{is odd}. \end{array} \right. \end{split}$$
By and the second equation of , we have $$\label{abs-f-n}
\begin{split}
&~|f(z)| \\
= &~ |z^{D_i}-a_i^{D_i}|^{(-1)^{n-i-p}}\,|z|^{(-1)^{n-p} d_1}\,\prod_{j=1}^{i-1}|z|^{(-1)^{n-j-p}D_j}
\,\prod_{j=i+1}^{n-1}|a_j|^{(-1)^{n-j-p}D_j}\cdot Q_i(z) \\
= &~ |1-(z/a_i)^{D_i}|^{(-1)^{n-i-p}}\,|{z}/{a_i}|^{(-1)^{n-i-p+1}d_{i}}\,|a_{n-1}|^{(-1)^{1-p}d_n}
\,u^{\frac{(-1)^{n-i-p}-(-1)^{1-p}}{2}}\cdot Q_i(z) \\
= &~ v^{(-1)^{1-p}}\,u^{\frac{(-1)^{n-i-p}-(-1)^{1-p}}{2}}\,|(a_i/z)^{d_{i}}-(z/a_i)^{d_{i+1}}|^{(-1)^{n-i-p}}\cdot Q_i(z)\\
&~
\left\{ \begin{array}{ll} \leq v^{(-1)^{1-p}}\,u^{\frac{1-(-1)^{1-p}}{2}}\,(|a_i/z|^{d_{i}}+|z/a_i|^{d_{i+1}})\,Q_i(z) &~~\text{if}~~n-i-p~~\text{is even} \\
\geq v^{(-1)^{1-p}}\,u^{\frac{-1-(-1)^{1-p}}{2}}\,(|a_i/z|^{d_{i}}+|z/a_i|^{d_{i+1}})^{-1}\,Q_i(z) &~~\text{if}~~n-i-p~~\text{is odd}, \end{array} \right. \end{split}$$ where $$\label{Q-i}
Q_i(z)=\prod_{j=1}^{i-1}\left|1-({a_j}/{z})^{D_j}\right|^{(-1)^{n-j-p}}
\prod_{j=i+1}^{n-1}\left|1-({z}/{a_j})^{D_j}\right|^{(-1)^{n-j-p}}.$$
For $1\leq i\leq n-1$, consider $z\in \overline{A}_i$. If $1\leq j<i$, by , we have $$\label{estim-1-2}
|{a_j}/{z}|^{D_j}\leq|{a_i}/{z}|^{5}|{a_j}/{a_i}|^{5}< 7K\,\varepsilon^{5(i-j)/2}/10<K^{-9}.$$ If $i<j\leq n-1$, then $$\label{estim-2-2}
|{z}/{a_j}|^{D_j}\leq|{z}/{a_i}|^{5}|{a_i}/{a_j}|^{5}< 7K\,\varepsilon^{5(i-j)/2}/10<K^{-9}.$$ by . Since $e^x<1+2x$ if $0<x\leq 1$ and $\varepsilon\leq K^{-4}$, by –, we have $$\label{Q-i-esti-1}
Q_i(z)< \prod_{k=1}^{\infty}\left(1+7K\,\varepsilon^{5k/2}/5\right)^2
\leq \exp\left(\frac{14\,K\,\varepsilon^{5/2}/5}{1-\varepsilon^{5/2}}\right)<1+K^{-5}<1.01.$$ and $$\label{Q-i-esti-2}
Q_i(z)> \prod_{k=1}^{\infty}\left(1+7K\,\varepsilon^{5k/2}/5\right)^{-2}
> 1/1.01 > 0.99.$$
For $p=1$, by Lemma \[esti-a1\](2) and (3a), for every $1\leq i\leq n-1$, if $|z|\leq s$, we have $$\label{last-estima-0}
|z^{D_i}/a_i^{D_i}| \leq |s/a_1|^{D_i}|a_1/a_i|^{D_i}\leq (sK^{-3}/2)^{\frac{5}{K}}u^{\frac{5(i-1)}{K}}.$$ If we notice Lemma \[esti-a1\](1), then $$\label{last-estima-00}
\sum_{i=1}^{n-1}|z^{D_i}/a_i^{D_i}| \leq \frac{(sK^{-3}/2)^{\frac{5}{K}}}{1-u^{\frac{5}{K}}}
\leq\frac{K^{\frac{10}{K}-10}}{1-K^{-10}}<1/200.$$
For $p=0$, by Lemma \[esti-a1\](2) and (4b), for every $1\leq i\leq n-1$, if $|z|\leq s$, we have $$\label{last-estima-0-lp}
|z^{D_i}/a_i^{D_i}| \leq |s/a_1|^{D_i}|a_1/a_i|^{D_i}\leq (u^{1/2}/2)^{\frac{5}{K}}u^{\frac{5(i-1)}{K}}.$$ By Lemma \[esti-a1\](1), then $$\label{last-estima-00-lp}
\sum_{i=1}^{n-1}|z^{D_i}/a_i^{D_i}| \leq \frac{(u^{1/2}/2)^{\frac{5}{K}}}{1-u^{\frac{5}{K}}}
\leq\frac{K^{-5}}{1-K^{-10}}<1/200.$$
Since $(1+2|a|)^{-1}\leq |1+a|^{\pm 1}\leq 1+2|a|$ if $0\leq |a|\leq 1/2$, by and , we know that $$\label{last-estima-1}
\prod_{i=1}^{n-1}\left|1-{z^{D_i}}/{a_i^{D_i}}\right|^{(-1)^{n-i-p}}
\leq \prod_{i=1}^{n-1}\left(1+2|z/a_i|^{D_i}\right)<e^{1/100}<K.$$ Therefore, $$\label{last-estima-2}
\prod_{i=1}^{n-1}\left|1-{z^{D_i}}/{a_i^{D_i}}\right|^{(-1)^{n-i-p}}
\geq \prod_{i=1}^{n-1}\left(1+2|z/a_i|^{D_i}\right)^{-1}>e^{-1/100}>1/K.$$
\(1) We first consider the case $p=1$. If $n-i$ is odd, by , and , if $z\in \overline{A}_i$ we have $$\label{bound-f-n-0}
|f(z)|\leq v\cdot (7K/5)\cdot 1.01<2Kv<s.$$ If $n-i$ is even, by , and , for $z\in \overline{A}_i$ we have $$\label{bound-f-n-1}
|f(z)|\geq (v/u)\cdot (7K/5)^{-1}\cdot 0.99>v/(2Ku)>K.$$
If $n$ is odd, by Lemma \[esti-a1\](3a), and , for every $z$ such that $|z|\leq s$, we have $$|f(z)| =|z|^{d_1} \prod_{i=1}^{n-1}|a_i|^{D_i(-1)^{n-i-1}} \prod_{i=1}^{n-1}\left|1-\frac{z^{D_i}}{a_i^{D_i}}\right|^{(-1)^{n-i-1}}
< |s/a_1|^{d_1}vu^{-1}\cdot1.02<s.$$ It follows that $f(\overline{\mathbb{D}}_{s})\subset\mathbb{D}_{s}$ for odd $n$. If $n$ is even and $|z|\leq s$, by Lemma \[esti-a1\](3b), and , we have $$|f(z)|=|a_1/z|^{d_1}v\,\prod_{i=1}^{n-1}\left|1-\frac{z^{D_i}}{a_i^{D_i}}\right|^{(-1)^{n-i-1}}
> |a_1/s|^{d_1}v/1.02>K.$$ Therefore $f(\overline{\mathbb{D}}_{s})\subset\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}_{K}$ for even $n$.
Note that $f$ is very ‘close’ to $z\mapsto z^{d_n}$ in the outside of $\mathbb{D}_{K}$ since $|a_i|^{D_i}$ is extremely small, where $1\leq i\leq n-1$. This means that $f$ may exhibit some dynamics of $z\mapsto z^{d_n}$ if $|z|\geq K$. More specifically, by arguments completely similar to those for –, if $|z|\geq K$, then $$\label{bound-lower-out-disk}
|f(z)|\geq |z|^{d_n} \prod_{i=1}^{n-1}\left(1+2\frac{|a_i|^{D_i}}{|z|^{D_i}}\right)^{-1}>K.$$ This means that $f(\overline{\mathbb{C}}\setminus\mathbb{D}_{K})\subset\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}_{K}$. Then we have $f^{-1}(\overline{\mathbb{A}}_{s,K})\subset \mathbb{A}_{s,K}$ for every $n\geq 2$ (see Figure \[Fig\_cantor-gene\]).
\(2) Now we consider the case $p=0$. If $n-i$ is even, by , , and Lemma \[esti-a1\](4a), if $z\in \overline{A}_i$ we have $$\label{bound-f-n-0-lp}
|f(z)|\leq v^{-1}u\cdot (7K/5)\cdot 1.01<2Ku/v<s.$$ If $n-i$ is odd, by , , and Lemma \[esti-a1\](4a), for $z\in \overline{A}_i$ we have $$\label{bound-f-n-1-lp}
|f(z)|\geq v^{-1}\cdot (7K/5)^{-1}\cdot 0.99>1/(2Kv)>M,$$ where $M=(2/s)^{1/d_n}$.
If $n$ is even, by Lemma \[esti-a1\](4b), and , for each $z$ such that $|z|\leq s$, we have $$|f(z)| =|z|^{d_1} \prod_{i=1}^{n-1}|a_i|^{D_i(-1)^{n-i}} \prod_{i=1}^{n-1}\left|1-\frac{z^{D_i}}{a_i^{D_i}}\right|^{(-1)^{n-i}}
< |s/a_1|^{d_1}v^{-1}\cdot e^{1/100}<s.$$ It follows that $f(\overline{\mathbb{D}}_{s})\subset\mathbb{D}_{s}$ for even $n$. If $n$ is odd and $|z|\leq s$, by Lemma \[esti-a1\](4c), and , we have $$|f(z)|=|a_1/z|^{d_1}uv^{-1}\,\prod_{i=1}^{n-1}\left|1-\frac{z^{D_i}}{a_i^{D_i}}\right|^{(-1)^{n-i}}\geq |a_1/s|^{d_1}uv^{-1}\cdot e^{-1/100}
> M.$$ Therefore $f(\overline{\mathbb{D}}_{s})\subset\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}_{M}$ for odd $n$.
If $|z|\geq M$, then $$\label{bound-lower-out-disk-lp}
|f(z)| =|z|^{-d_n} \prod_{i=1}^{n-1}\left|1-\frac{a_i^{D_i}}{z^{D_i}}\right|^{(-1)^{n-i}}
\leq M^{-d_n} \prod_{i=1}^{n-1}\left(1+\frac{2 |a_i|^{D_i}}{|z|^{D_i}}\right)<2M^{-d_n}=s.$$ This means that $f(\overline{\mathbb{C}}\setminus\mathbb{D}_M)\subset\mathbb{D}_{s}$. Then we have $f^{-1}(\overline{\mathbb{A}}_{s,M})\subset \mathbb{A}_{s,M}$ for every $n\geq 2$.
![Sketch illustrating of the mapping relation of $f_{1,d_1,\cdots,d_n}$, where $n$ is odd and even respectively (from left to right). The small stars denote the critical points and critical values, and the numbers shown at the bottom of the Figures denote the approximate coordinates.[]{data-label="Fig_cantor-gene"}](cantor-gene.pdf){width="130mm"}
\[parameter-restate\] If $|a_{n-1}|=(s_1 K^{-2})^{1/d_n}$ and $|a_i|=(s_1 K^{-5})^{1/d_{i+1}}|a_{i+1}|$ for $1\leq i\leq n-2$, where $s_1>0$ is small enough, then the Julia set of $f_{1,d_1,\cdots,d_n}$ is a Cantor set of circles. If $|a_{n-1}|=(s_0^{1/d_n+(1-\xi)/3})^{1/d_n}$ and $|a_i|=(s_0^{1+1/d_n+2(1-\xi)/3})^{1/d_{i+1}}|a_{i+1}|$ for $1\leq i\leq n-2$, where $s_0>0$ is small enough, then the Julia set of $f_{0,d_1,\cdots,d_n}$ is a Cantor set of circles.
We only focus on the case $p=1$ since the similar proof can be used to the case $p=0$ by using Lemma \[nice-condition\](2). We also use $f$ to denote $f_{1,d_1,\cdots,d_n}$ for simplicity. Let $U_i$ be the component of $f^{-1}(D)$ containing $a_i$, where $D=\mathbb{D}_s$ if $n-i$ is odd and $D=\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}_{K}$ if $n-i$ is even. By Lemma \[nice-condition\](1), it follows that the set of critical points $CP_i\subset U_i$ and $U_i$ is a connected domain containing the annulus $A_i$. Moreover, $U_i\cap U_{i+1}=\emptyset$ since $f(U_i)\cap f(U_{i+1})=\emptyset$ by Lemma \[nice-condition\](1), where $1\leq i<n-2$. This means that $U_i\cap U_j=\emptyset$ for different $i,j$. Suppose that $U_i$ has $m_i$ boundary components. Since there are exactly $D_i$ critical points in $U_i$ and $f:U_i\rightarrow D$ is a branched covering with degree $D_i$, then the Riemann-Hurwitz formula tells us $\chi_{U_i}=2-m_i=D_i\chi_{D}-D_i=0$, where $\chi$ denotes the Euler characteristic. This means that $m_i=2$ and therefore $U_i$ is an annulus surrounding the origin for every $1\leq i\leq n-1$.
For $1\leq i\leq n-2$, Let $V_{i+1}$ be the annular domain between $U_i$ and $U_{i+1}$. It is easy to see $f:V_{i+1}\rightarrow \mathbb{A}_{s,{K}}$ is a covering map with degree $d_{i+1}$. Note that every component of $f^{-1}(\mathbb{A}_{s,{K}})$ is an annulus since $\mathbb{A}_{s,{K}}$ is double connected and contains no critical values. It follows that there exist two annuli $V_1$ and $V_n$, which lie between $0$ and $U_1$, $U_{n-1}$ and $\infty$ respectively, such that $f:V_1,V_n\rightarrow \mathbb{A}_{s,{K}}$ are covering maps with degree $d_1$ and $d_n$ respectively. In fact, the restriction of $f$ on $\partial U_1$ and $\partial U_{n-1}$ has degree $d_1$ and $d_n$ respectively and there are no critical points in $V_1$ and $V_n$ (see Figure \[Fig\_cantor-gene\]).
The Julia set of $f$ is $J=\bigcap_{k\geq 0}f^{-k}(\mathbb{A}_{s,{K}})$. By the construction, the components of $J$ are compact sets nested between $0$ and $\infty$ since each inverse branch $f^{-1}:\mathbb{A}_{s,{K}}\rightarrow V_j$ is conformal for every $0\leq j\leq n$. Since the component of $J$ cannot be a point and $f$ is hyperbolic, every component of $J$ is a Jordan curve (actually quasicircle) by Theorem 1.2 in [@PT]. The dynamics on the set of Julia components of $f$ is isomorphic to the one-sided shift on $n$ symbols $\Sigma_{n}:=\{0,1,\cdots,n-1\}^{\mathbb{N}}$. In particular, $J$ is homeomorphic to $\Sigma_{n}\times\mathbb{S}^1$, which is a Cantor set of circles as desired. This ends the proof of Theorem \[parameter-restate\] and hence Theorem \[parameter\].
\[range-unif\] Since $f$ is hyperbolic, the Julia set of $f$ is also a Cantor set of circles if we perturb some $a_i$ gently, where $1\leq i\leq n-1$. In the first version of our manuscript of this paper, only $d_i=n+1$ for every $1\leq i\leq n$ was considered. In this case, it was shown that for every $n\geq 2$ and $1\leq i\leq n-1$, if $|a_{n-i}|=(\frac{n}{n+1})^{i-1}s^i$ for $0<s\leq 1/10$, then the Julia set of $f_{1,n+1,\cdots,n+1}$ is a Cantor set of circles.
\[no-topo-equiv\] Suppose that $a_i$ is chosen as in Theorem \[parameter\] such that the Julia set of $f_{p,d_1,\cdots,d_n}$ is a Cantor set of circles for $n\geq 3$, then $f_{p,d_1,\cdots,d_n}$ is not topologically conjugate to any McMullen maps on their corresponding Julia sets.
Since the dynamics on the set of Julia components of $f_{p,d_1,\cdots,d_n}$ is conjugate to the one-sided shift on $n$ symbols $\Sigma_n:=\{0,1,\cdots,n-1\}^{\mathbb{N}}$ and, in particular, the set of Julia components of $g_\eta$ is isomorphic to the one-sided shift on only two symbols $\Sigma_{2}:=\{0,1\}^{\mathbb{N}}$, this means that $f_{p,d_1,\cdots,d_n}$ cannot be topologically conjugate to $g_\eta$ on their corresponding Julia sets if $n\geq 3$.
Topological conjugacy between the Cantor circles Julia sets {#sec-topo-conj}
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In this section, we show that for any given rational map whose Julia set is a Cantor set of circles, there exists a map $f_{p,d_1,\cdots,d_n}$ in such that these two rational maps are topologically conjugate on their corresponding Julia sets.
\[no-crit-on-J\] If $f$ is a rational map whose Julia set is a Cantor set of circles. Then there exist no critical points in $J(f)$.
Suppose there exists a Julia component $J_0$ of $f$ containing a critical point $c_0$ of $f$ with multiplicity $d$. Then $f$ is not one to one in any small neighborhood of $c_0$. It is known $f(J_0)$ is a Julia component containing $f(c_0)$ [@Be Lemma 5.7.2]. Choose a small topological disk neighborhood $U$ of $f(c_0)$ such that $U\cap f(J_0)$ is a simple curve. The component of $f^{-1}(U)$ containing $c_0$ is mapped onto $U$ in the manner of $d+1$ to one. Note that the component $J'$ of $f^{-1}(U\cap f(J_0))$ containing $c_0$ is connected and contained in $J_0$. However, $J'$ possesses star-like structure and hence is not a simple curve. This contradicts to the assumption that $J_0$ is a Jordan closed curve since $J(f)$ is a Cantor set of circles.
We say that a compact set $X\subset \overline{\mathbb{C}}$ *separates* $0$ and $\infty$ if $0$ and $\infty$ lie in the two different components of $\overline{\mathbb{C}}\setminus X$ respectively. Let $X$ and $Y$ be two disjoint compact sets that both separate $0$ and $\infty$ respectively. We say $X\prec Y$ if $X$ is contained in the component of $\overline{\mathbb{C}}\setminus Y$ which contains $0$. Let $A$ be an annulus whose closure separates $0$ and $\infty$, we use $\partial_-A$ and $\partial_+A$ to denote the two components of the boundary of $A$ such that $\partial_-A\prec \partial_+A$.
\[this-is-all-resta\] Let $f$ be a rational map whose Julia set is a Cantor set of circles. Then there exist $p\in\{0,1\}$, positive integers $n\geq 2$ and $d_1,\cdots,d_n$ satisfying $\sum_{i=1}^{n}(1/d_i)<1$ such that $f$ is topologically conjugate to $f_{p,d_1,\cdots,d_n}$ on their corresponding Julia sets.
Let $J(f)$ be the Julia set of $f$ which is a Cantor set of circles, then every periodic Fatou component of $f$ must be attracting or parabolic by Lemma \[no-crit-on-J\]. We only prove the attracting (hyperbolic) case in detail and explain the parabolic case by using the work of Cui [@Cui].
In the following, we suppose that $f$ is hyperbolic. There exist exactly two simply connected Fatou components of $f$ and all other Fatou components are annuli. Let $\mathcal{D}$ and $\mathcal{A}$ be the collection of simply and doubly connected Fatou components of $f$ respectively. We claim that $f(\mathcal{D})\subset\mathcal{D}$ and there exists an integer $k\geq 1$ such that $f^{\circ k}(A)\in\mathcal{D}$ for every $A\in \mathcal{A}$. The assertion $f(\mathcal{D})\subset\mathcal{D}$ is obvious since the image of a simply connected Fatou component under a rational map is again simply connected. If $f(A_1)=A_2$, where $A_1,A_2\in\mathcal{A}$, then there exists no critical points in $A_1$ by Riemann-Hurwitz’s formula. This means that each $A\in\mathcal{A}$ cannot be periodic since the cycle of every periodic attracting Fatou component must contain at least one critical point. On the other hand, by Sullivan’s theorem, the Fatou components of a rational map cannot be wandering. This completes the proof of claim.
Up to a Mobius transformation, we can assume that $0$ and $\infty$, respectively, are belong to the two simply connected Fatou components of $f$, which are denoted by $D_0$ and $D_\infty$. Namely, $\mathcal{D}=\{D_0,D_\infty\}$. Since $f(\mathcal{D})\subset\mathcal{D}$, we first suppose that $f(D_0)=D_0$ and $f(D_\infty)=D_\infty$. Let $f^{-1}(D_0)=D_0\cup A_1\cup\cdots\cup A_m$, where $A_1,\cdots, A_m$ are $m$ annuli separating $0$ and $\infty$ such that $A_{i}\prec A_{i+1}$ for every $1\leq i\leq m-1$. It is easy to see $m\geq 1$. Otherwise, $D_0$ is completely invariant, then $J(f)=\partial D_0$ which contradicts to the assumption that $J(f)$ is a Cantor set of circles.
Suppose that $\deg (f|_{D_0}:D_0\rightarrow D_0)=d_1$ and $\deg (f|_{\partial_-A_i}:\partial_-A_i\rightarrow \partial D_0)=d_{2i}$ and $\deg (f|_{\partial_+A_i}:\partial_+A_i\rightarrow \partial D_0)=d_{2i+1}$ for $1\leq i\leq m$. It follows that $\deg(f)=\sum_{j=1}^{2m+1}d_j$. Let $W_1$ be the annular domain between $D_0$ and $A_1$ and $W_i$ be the annular domain between $A_{i-1}$ and $A_i$, where $2\leq i\leq m$. We have $f(W_i)=\overline{\mathbb{C}}\setminus \overline{D}_0$ and $\deg(f|_{W_i}:W_i\rightarrow\overline{\mathbb{C}}\setminus \overline{D}_0)=d_{2i-1}+d_{2i}$. This means that there exists at least one Fatou component $B_i\subsetneq W_i$ such that $f(B_i)=D_\infty$. If there exists $B_i'\neq B_i$ such that $B_i'\subsetneq W_i$ and $f(B_i')=D_\infty$, there must exist one component of $f^{-1}(D_0)$ in $W_i$, which contradicts the assumption that $A_1\cup\cdots\cup A_m$ is the collection of all annular components of $f^{-1}(D_0)$. So there exists exactly one Fatou component $B_i\subsetneq W_i$ such that $f(B_i)=D_\infty$ and $\deg(f|_{B_i}:B_i\rightarrow D_\infty)=d_{2i-1}+d_{2i}$. Similar argument can be used to show that $D_\infty$ is the only component of $f^{-1}(D_\infty)$ lying in the unbounded component of $\overline{\mathbb{C}}\setminus A_m$ which can be mapped onto $D_\infty$. Therefore, $f^{-1}(D_\infty)=B_1\cup\cdots\cup B_m\cup D_\infty$ and $\deg(f|_{D_\infty})=d_{2m+1}$ since $\deg(f)=\sum_{j=1}^{2m+1}d_j$. Denote $\overline{\mathbb{C}}\setminus ({D_0\cup D_\infty})$ by $E$. The preimage $f^{-1}(E)$ consists of $2m+1$ annuli components $E_1,\cdots,E_{2m+1}$ such that $E_i\prec E_{i+1}$ for $1\leq i\leq 2m$. The map $f:E_i\rightarrow E$ is a unramified covering map with degree $d_i$, where $1\leq i\leq 2m+1$ (see Figure \[Fig\_find-conj\]).
![Sketch illustrating of the mapping relation of $f$, where $d_i$, $1\leq i\leq 2m+1$ denote the degrees of the restriction of $f$ on the boundaries of Fatou components.[]{data-label="Fig_find-conj"}](find-conj.pdf){width="120mm"}
Let $n=2m+1$ and $p=1$. The assertion $\sum_{i=1}^{n}{1}/{d_i}<1$ follows from Grótzsch’s modulus inequality since each $E_i$ is essentially contained in $E$ and $\text{mod} (E_i)=\text{mod} (E)/d_i$. In the following, we will construct a quasiconformal map $\phi:\overline{\mathbb{C}}\rightarrow\overline{\mathbb{C}}$ which conjugates the dynamics on the Julia set of $f$ to that of $f_{1,d_1,\cdots,d_n}$.
For simplicity, we denote $f_{1,d_1,\cdots,d_n}$ by $F$. Note that $F(0)=0$ and $F(\infty)=\infty$. There exist two simply connected Fatou components $D_0'$ and $D_\infty'$, both are invariant under $F$ such that $0\in D_0'$ and $\infty\in D_\infty'$. From the proof of Theorem \[parameter\], we know that $F^{-1}(D_0')=D_0'\cup A_1'\cup\cdots\cup A_m'$, where $A_1',\cdots, A_m'$ are $m$ annuli separating $0$ and $\infty$ such that $A_{i}'\prec A_{i+1}'$ for every $1\leq i\leq m-1$. Moreover, $\deg (F|_{D_0'}:D_0'\rightarrow D_0')=d_1$ and $\deg (F|_{\partial_-A_i'}:\partial_-A_i'\rightarrow \partial D_0')=d_{2i}$ and $\deg (F|_{\partial_+A_i'}:\partial_+A_i'\rightarrow \partial D_0')=d_{2i+1}$ for $1\leq i\leq m$. Let $W_1'$ be the annular domain between $D_0'$ and $A_1'$ and $W_i'$ be the annular domain between $A_{i-1}'$ and $A_i'$, where $2\leq i\leq m$. There exists exactly one Fatou component $B_i'\subsetneq W_i'$ such that $F(B_i')=D_\infty'$ and $\deg(F|_{B_i'}:B_i'\rightarrow D_\infty')=d_{2i-1}+d_{2i}$. We have $F^{-1}(D_\infty')=B_1'\cup\cdots\cup B_m'\cup D_\infty'$ and $\deg(F|_{D_\infty'})=d_{2m+1}$. Similarly, let $E':=\overline{\mathbb{C}}\setminus ({D_0'\cup D_\infty'})$. There exist $2m+1$ annular components $E_1',\cdots,E_{2m+1}'$ of $F^{-1}(E')$ such that $E_i'\prec E_{i+1}'$ for $1\leq i\leq 2m$. The map $F:E_i'\rightarrow E'$ is a covering with degree $d_i$, where $1\leq i\leq 2m+1$.
By a quasiconformal surgery, it can be seen that $\partial D_0,\partial D_\infty,\partial D_0',\partial D_\infty'$ and their preimages are all quasicircles and the dilatation is bounded by a fixed constant. There exists a quasiconformal mapping $\phi_0:\overline{\mathbb{C}}\rightarrow\overline{\mathbb{C}}$ such that $\phi_0(D_0)=D_0'$ and $\phi_0(D_\infty)=D_\infty'$ hence $\phi_0(\partial D_0)=\partial D_0'$ and $\phi_0(\partial D_\infty)=\partial D_\infty'$. Moreover, $\phi_0$ can be chosen such that $\phi_0\circ f=F\circ\phi_0$ on $\partial D_0\cup \partial D_\infty$.
Now we construct a lift $\phi_{E_1}:E_1\rightarrow E_1'$ of $\phi_0:E\rightarrow E'$ as follows. For every $z\in E_1\setminus\partial_- E_1$, we choose a simple curve $\gamma:[0,1]\rightarrow E$ such that $\gamma(1)=f(z)$ and $\gamma(0)=w\in\partial_- E$. Since $f:E_1\rightarrow E$ is a covering map, there exists a unique lift $\widetilde{\gamma}:[0,1]\rightarrow E_1$ of $\gamma$ such that $\widetilde{\gamma}(1)=z$ and $\widetilde{w}:=\widetilde{\gamma}(0)\in\partial_- E_1$. Similarly, since $F:E_1'\rightarrow E'$ is a covering map, there exists a unique lift $\alpha:[0,1]\rightarrow E_1'$ of $\phi_0(\gamma):[0,1]\rightarrow E'$ such that $\alpha(0)=\phi_0(\widetilde{w})$ since $\phi_0\circ f=F\circ\phi_0$ on $\partial D_0=\partial_- E_1$. Define $\phi_{E_1}(z):=\alpha(1)$. We know that $\phi_0\circ f =F\circ \phi_{E_1}$ on $E_1$ and $\phi_{E_1}:E_1\rightarrow E_1'$ is quasiconformal since $f,F$ are both holomorphic covering maps with degree $d_1$ and $\phi_0:E\rightarrow E'$ is quasiconformal. Now some parts of $\phi_1:\overline{\mathbb{C}}\rightarrow\overline{\mathbb{C}}$ are defined as follows: $\phi_1|_{\overline{D}_0}=\phi_0|_{\overline{D}_0}$, $\phi_1|_{\overline{D}_\infty}=\phi_0|_{\overline{D}_\infty}$ and $\phi_1|_{E_1}=\phi_{E_1}$. Then, $\phi_1\circ f =F\circ \phi_1$ on $\partial E_1$. Similarly, there exists a unique quasiconformal mapping $\phi_{E_{2m+1}}:E_{2m+1}\rightarrow E_{2m+1}'$, which is the lift of $\phi_0:E\rightarrow E'$ such that $\phi_0\circ f =F\circ \phi_{E_{2m+1}}$ on $E_{2m+1}$. Define $\phi_1|_{E_{2m+1}}=\phi_{E_{2m+1}}$. Then, $\phi_1\circ f =F\circ \phi_1$ on $\partial E_{2m+1}$.
Unlike the cases of $E_1$ and $E_{2m+1}$, the lift $\phi_{E_i}:E_i\rightarrow E_i'$ of $\phi_0:E\rightarrow E'$ exists but is not unique for $2\leq i\leq 2m$. We first show the existence of $\phi_{E_i}$. Without loss of generality, suppose that $i$ is even. Since $f:\partial_- E_i\rightarrow\partial D_\infty$ and $F:\partial_- E_i'\rightarrow\partial D_\infty'$ are both covering mappings with degree $d_i$, there exists a lift (not unique) $\phi_{E_i}:\partial_- E_i\rightarrow \partial_- E_i'$ of $\phi_0:\partial D_\infty\rightarrow \partial D_\infty'$ such that $\phi_0\circ f =F\circ \phi_{E_i}$ on $\partial_-E_i$. By using the same method of defining $\phi_{E_1}$, there exists a unique lift of $\phi_0:E\rightarrow E'$ defined from $E_i$ to $E_i'$, which we denote also by $\phi_{E_i}$ such that $\phi_0\circ f =F\circ \phi_{E_i}$ on $E_i$. Note that $\phi_{E_i}:E_i\rightarrow E_i'$ is quasiconformal. Define $\phi_1|_{E_{i}}=\phi_{E_i}$. Then, $\phi_0\circ f =F\circ \phi_1$ on $\bigcup_{i=1}^{2m+1}E_i$ and $\phi_1\circ f =F\circ \phi_1$ on $\bigcup_{i=1}^{2m+1}\partial E_i$.
In order to unify the notations, let $D_{2i-1}:=B_i$ and $D_{2i}:=A_i$ for $1\leq i\leq m$. Then we have $D_i\prec D_{j}$ for $1\leq i<j\leq 2m$. We need to define $\phi_1$ on $\bigcup_{i=1}^{2m}D_i$. For every $D_i$, where $1\leq i\leq 2m$, its two boundary components $\partial_+ E_i$ and $\partial_- E_{i+1}$ are both quasicircles. Since $\phi_{E_{i}}$ and $\phi_{E_{i+1}}$ are both quasiconformal mappings, the map $\phi_1|_{\partial_+ E_{i}\cup\partial_- E_{i+1}}$ has a quasiconformal extension $\phi_{D_i}:\overline{D}_i\rightarrow \overline{D}_i'$ such that $\phi_{D_i}(D_i)=D_i'$. Now we obtain a quasiconformal mapping $\phi_1:\overline{\mathbb{C}}\rightarrow\overline{\mathbb{C}}$ defined as $\phi_1|_{E_i}:=\phi_{E_i}$, $\phi_1|_{D_j}=\phi_{D_j}$ and $\phi_1|_{D_0\cup D_\infty}=\phi_0$, where $1\leq i\leq 2m+1$ and $1\leq j\leq 2m$.
Next, we define $\phi_2$. First, let $\phi_2|_{D_j}=\phi_1$ for $j\in\{0,1,\cdots,2m,\infty\}$. Then we lift $\phi_1:E\rightarrow E'$ in an appropriate way to obtain $\phi_2:E_i\rightarrow E_i'$ for $1\leq i\leq 2m+1$. Finally, we check the continuity of the resulting map $\phi_2:\overline{\mathbb{C}}\rightarrow\overline{\mathbb{C}}$. Now let us make this precise. In order to guarantee the continuity of $\phi_2$ on $D_0\cup E_1$, we need to have $\phi_2|_{\partial_-E_1}=\phi_1$. Then there exists only one way to lift $\phi_1:E\rightarrow E'$ to obtain $\phi_2:E_1\rightarrow E_1'$. In order to guarantee the continuity of the lift $\phi_2$, we need to check the continuity of $\phi_2$ on the boundary $\partial_+E_1$ first. In fact, $\phi_0|_E$ and $\phi_1|_E$ are homotopic to each other and $\phi_1|_{\partial E}=\phi_0|_{\partial E}$, it follows that $\phi_2|_{\partial_+ E_1}=\phi_1|_{\partial_+ E_1}$ since $\phi_2|_{\partial_- E_1}=\phi_1|_{\partial_- E_1}$. This means that $\phi_2$ is continuous on $\partial_+ E_1$. Similarly, we can lift $\phi_1:E\rightarrow E'$ to obtain $\phi_2:E_i\rightarrow E_i'$ for $2\leq i\leq 2m+1$ and guarantee the continuity of $\phi_2$. Above all, the map $\phi_2:\overline{\mathbb{C}}\rightarrow\overline{\mathbb{C}}$ satisfies (1) $\phi_2$ is quasiconformal and the dilatation $K(\phi_2)=K(\phi_1)$; (2) $\phi_2|_{f^{-1}(D_0\cup D_\infty)}=\phi_1$; (3) $\phi_1\circ f=F\circ\phi_2$ on $\bigcup_{i=1}^{2m+1}E_i$ and hence $\phi_2\circ f=F\circ\phi_2$ on $f^{-2}(\partial D_0\cup \partial D_\infty)$.
Suppose we have obtained $\phi_k$ for some $k\geq 1$, then $\phi_{k+1}$ can be defined completely similarly to the process of the derivation of $\phi_2$ from $\phi_1$. Inductively, we can obtain a sequence of quasiconformal mappings $\{\phi_k\}_{k\geq 0}$ such that (1) $K(\phi_k)=K(\phi_1)\geq K(\phi_0)$ for $k\geq 1$; (2) $\phi_{k+1}(z)=\phi_{k}(z)$ for $z\in f^{-k}(D_0\cup D_\infty)$; (3) $\phi_k\circ f=F\circ\phi_k$ on $f^{-k}(\partial D_0\cup \partial D_\infty)$. This means that $\{\phi_k\}_{k\geq 0}$ forms a normal family. Take a convergent subsequence of $\{\phi_k\}_{k\geq 0}$ whose limit we denote by $\phi_\infty$, then $\phi_\infty$ is a quasiconformal mapping satisfying $\phi_\infty\circ f=F\circ\phi_\infty$ on $\bigcup_{k\geq 0} f^{-k}(\partial D_0\cup \partial D_\infty)$. Moreover, $K(\phi_\infty)\leq K(\phi_1)$. Since $\phi_\infty$ is continuous, $\phi_\infty\circ f=F\circ\phi_\infty$ holds on the closure of $\bigcup_{k\geq 0}f^{-k}(\partial D_0\cup \partial D_\infty)$, which is the Julia set of $f$. Therefore $\phi=\phi_\infty$ is the quasiconformal mapping we want to find which conjugates $f$ to $F$ on their corresponding Julia sets. This ends the proof of case $f(D_0)=D_0$ and $f(D_\infty)=D_\infty$.
The other three cases: (1) $f(D_0)=D_\infty$, $f(D_\infty)=D_\infty$; (2) $f(D_0)=D_\infty$, $f(D_\infty)=D_0$; and (3) $f(D_0)=D_0$, $f(D_\infty)=D_0$ can be proved completely similarly.
If one or both of the components $D_0$ and $D_\infty$ are parabolic, there exists a perturbation $f_\varepsilon$ of $f$ such that $f_\varepsilon$ is hyperbolic and the dynamics of $f_\varepsilon$ are topologically conjugate to that of $f$ on their corresponding Julia sets [@Cui]. Then $f$ has a ‘model’ in since $f_\varepsilon$ always does. This ends the proof of Theorem \[this-is-all-resta\] and hence Theorem \[this-is-all\].
From the proof of Theorem \[this-is-all-resta\] in the hyperbolic case, we have following immediate corollary.
\[Julia-comp\] If the parameters $a_i$ are chosen as in Theorem \[parameter\], where $1\leq i\leq n-1$, then each Julia component of $f_{p,d_1,\cdots,d_n}$ is a quasicircle.
Non-hyperbolic rational maps whose Julia sets are Cantor circles {#sec-para-mcm}
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The rational maps $$P_\lambda(z)=\frac{\frac{1}{n}((1+z)^n-1)+\lambda^{m+n}z^{m+n}}{1-\lambda^{m+n}z^{m+n}}$$ where $\lambda\in\mathbb{C}^*=\mathbb{C}\setminus\{0\}$ and $m,n\geq 2$ are both positive integers satisfying $1/m+1/n<1$ can be seen as a perturbation of the parabolic polynomial $$\widetilde{P}(z)=\frac{(1+z)^n-1}{n}.$$ Note that $\widetilde{P}$ has a parabolic fixed point at the origin with multiplier 1 and critical point $-1$ with multiplicity $n-1$. This means that there exists only one bounded and hence simply connected Fatou component of $\widetilde{P}$ in which all points are attracted to the origin. In particular, the Julia set of $\widetilde{P}$ is a Jordan curve with infinitely many cusps.
We hope that some properties of $\widetilde{P}$ stated above can be also hold for $P_\lambda$ when $\lambda$ is small. But obviously, there are lots of differences between $P_\lambda$ and $\widetilde{P}$. The degree of $P_\lambda$ is $m+n$ and $P_\lambda(\infty)=-1$. There are $2(m+n)-2$ critical points of $P_\lambda$: $m-1$ at $\infty$, $n-1$ are very close to $-1$ and the remaining $m+n$ critical points lie nearby the circle $\mathbb{T}_{r_0/|\lambda|}$, where $r_0=\sqrt[m+n]{n/m}$ (see Lemma \[crit-close-para\]). In fact, we will see that $P_\lambda$ can be viewed as a ‘parabolic’ McMullen map at the end of this section since $P_\lambda$ is conjugate to some $g_\eta$ on their corresponding Julia sets.
Firstly, we show that the fixed parabolic Fatou component of $\widetilde{P}$ contains the Euclidean disk $\mathbb{D}(-\frac{3}{4},\frac{3}{4})$ for every $n\geq 2$ and $P_\lambda$ maps $\mathbb{D}(-\frac{3}{4},\frac{3}{4})$ into itself if $\lambda$ is small enough.
\[key-lemma\] $(1)$ For every $n\geq 2$, $\widetilde{P}(\overline{\mathbb{D}}(-\frac{3}{4},\frac{3}{4}))\subset\mathbb{D}(-\frac{3}{4},\frac{3}{4})\cup\{0\}$.
$(2)$ If $0<|\lambda|<{1}/{(3n)}$, then $P_\lambda(\overline{\mathbb{D}}(-\frac{3}{4},\frac{3}{4}))\subset\mathbb{D}(-\frac{3}{4},\frac{3}{4})\cup\{0\}$. In particular, $\mathbb{D}(-\frac{3}{4},\frac{3}{4})$ lies in the parabolic Fatou component of $P_\lambda$ with parabolic fixed point $0$.
If $z\in\overline{\mathbb{D}}(-\frac{3}{4},\frac{3}{4})$, then $|\widetilde{P}(z)+1/n|=|1+z|^n/n\leq 1/n$. In particular, the inequality sign can be replaced by equality if and only if $z=0$. This ends the proof of (1).
The proof of (2) will be divided into two cases: $|z|$ is small and not too small. For every $z=-\frac{3}{4}+\frac{3}{4}e^{i\theta}\in\partial\mathbb{D}(-\frac{3}{4},\frac{3}{4})$, where $-\pi< \theta\leq\pi$, we have $|1+\widetilde{P}(z)|\leq 5/2$ by (1) and $|\lambda z|^{m+n}<1/2$ since $|\lambda|<1/(3n)$. This means that $$|P_\lambda(z)-\widetilde{P}(z)|=\left|\frac{\lambda^{m+n}z^{m+n}(1+\widetilde{P}(z))}{1-\lambda^{m+n}z^{m+n}}\right|\leq 5|\lambda z|^{m+n}.$$ Since $|z|=\frac{3}{4}|1-e^{i\theta}|=\frac{3}{4}|e^{-i\theta/2}-e^{i\theta/2}|=\frac{3}{2}|\sin\frac{\theta}{2}|\leq \frac{3}{4}|\theta|$ and $|\lambda|<1/(3n)$, we have $$\label{P-lambd-est}
|P_\lambda(z)-\widetilde{P}(z)|\leq 5\,({|\theta|}/{(4n)})^{m+n}.$$
On the other hand, since $|\sin\theta|\geq \frac{2}{\pi}|\theta|$ if $|\theta|\leq\frac{\pi}{2}$, we have $$\label{P-est-0}
\begin{split}
|\widetilde{P}(z)+{3}/{4}|
= &~ \left| \frac{(\frac{1}{4}+\frac{3}{4}e^{i\theta})^n-1}{n}+\frac{3}{4}\right|
\leq \frac{\left|\frac{1}{4}+\frac{3}{4}e^{i\theta}\right|^n-1}{n}+\frac{3}{4} \\
= &~ \frac{(1-\frac{3}{4}\sin^2\frac{\theta}{2})^{n/2}-1}{n}+\frac{3}{4}
\leq \frac{(1-\frac{3\theta^2}{4\pi^2})^{n/2}-1}{n}+\frac{3}{4}.
\end{split}$$ If $|\theta|<2\pi/n$, then $\frac{3\theta^2}{4\pi^2}<\frac{2}{n}$. By Lemma \[very-useful-est\](3), we have $$\label{P-est}
|\widetilde{P}(z)+{3}/{4}|\leq -\frac{\frac{n}{2}\cdot\frac{3\theta^2}{4\pi^2}}{3n}+\frac{3}{4}=\frac{3}{4}-\frac{\theta^2}{8\pi^2}.$$ Therefore, combining and , it follows that if $|\theta|<2\pi/n$, then $$|P_\lambda(z)+3/4| \leq |\widetilde{P}(z)+{3}/{4}|+|P_\lambda(z)-\widetilde{P}(z)|\leq \frac{3}{4}-\frac{\theta^2}{8\pi^2}+5\,(\frac{|\theta|}{4n})^{m+n}\leq 3/4.$$ If $2\pi/n\leq |\theta|\leq \pi$, from and , we know that $$\label{P-est-2}
|\widetilde{P}(z)+{3}/{4}|\leq \frac{3}{4}-\frac{1}{2 n^2}.$$ From and , it follows that if $2\pi/n\leq |\theta|\leq \pi$, then $$|P_\lambda(z)+3/4| \leq \frac{3}{4}-\frac{1}{2 n^2}+5\,(\frac{|\theta|}{4n})^{m+n}< 3/4.$$
Therefore, we have shown that $|P_\lambda(z)+\frac{3}{4}|\leq \frac{3}{4}$ for every $z\in\partial\mathbb{D}(-\frac{3}{4},\frac{3}{4})$ and $|P_\lambda(z)+\frac{3}{4}|= \frac{3}{4}$ if and only if $z=0$. The proof is complete.
As in the procedure in §2, now we locate the free critical points of $P_\lambda$. By a direct calculation, the bounded $m+2n-1$ critical points of $P_\lambda$ are the solutions of $$\label{crit-P-lamb}
(1+z)^{n-1}+\lambda^{m+n}z^{m+n-1}\{(1+m/n)[(1+z)^n+n-1]-z(1+z)^{n-1}\}=0.$$
\[nice-cond-para\] If $0<|\lambda|<{1}/{(3n)}$, then there are $n-1$ critical points of $P_\lambda$ in $\mathbb{D}(-1,|\lambda|)\subsetneq \mathbb{D}(-\frac{3}{4},\frac{3}{4})$.
If $|z+1|\leq |\lambda|<\frac{1}{3n}$, then $|z|\cdot|1+z|^{n-1}\leq (1+|\lambda|)|\lambda|^{n-1}<1$ and $$(1+m/n)\,|(1+z)^n+n-1|\leq (1+m/n)(|\lambda|^{n}+n-1)<m+n.$$ This means that if $|z+1|\leq |\lambda|$, then $$\begin{split}
&~ \left|\lambda^{m+n}z^{m+n-1}\{(1+m/n)[(1+z)^n+n-1]-z(1+z)^{n-1}\}\right|\\
< &~|\lambda|^{n-1}\cdot |\lambda z|^{m-1}|\lambda|^2|z|^n(m+n+1)
< |\lambda|^{n-1}\cdot (2n)^{1-m}(9n^2)^{-1}e^{1/3}(m+n+1)\\
< &~|\lambda|^{n-1}\cdot (m+n-1)/(2n)^{m+1}<|\lambda|^{n-1}.
\end{split}$$ By Rouché’s Theorem, the proof is completed.
Let $\widetilde{CP}:=\{\widetilde{w}_{j}=\frac{r_0}{\lambda} \exp(\pi i\frac{2j-1}{m+n}):1\leq j\leq m+n\}$ be the collection of the zeros of $m\lambda^{m+n}z^{m+n}+n=0$, where $r_0=\sqrt[m+n]{n/m}$. Since $h(x)=x^{1/x},x>0$ has maximal value $e^{1/e}<3/2$ at $x=e$, we have $$2/3<1/\sqrt[m]{m}<r_0<\sqrt[n]{n}<3/2.$$ The following lemma shows that the remaining $m+n$ critical points of $P_\lambda$ are very ‘close’ to $\widetilde{CP}$.
\[crit-close-para\] If $0<|\lambda|<{1}/{(2^m n^2)}$, then has a solution $w_j$ such that $|w_j-\widetilde{w}_j|<2(m+n)/m$, where $1\leq j\leq m+n$. Moreover, $w_i= w_j$ if and only if $i=j$.
Dividing $(1+z)^{n-1}$ on both sides of , we have $$1+\lambda^{m+n}z^{m+n-1}\left(\frac{m}{n}z+\frac{m+n}{n}\left(1+\frac{n-1}{(1+z)^{n-1}}\right)\right)=0.$$ Or, in more useful form $$\label{use-ful}
\frac{n}{m\lambda^{m+n}}+z^{m+n}+\frac{(m+n)z^{m+n-1}}{m}\left(1+\frac{n-1}{(1+z)^{n-1}}\right)=0.$$
Let $\Omega=\{z:|z^{m+n}+\frac{n}{m}\lambda^{-(m+n)}|\leq \beta|\lambda|\cdot\frac{n}{m}|\lambda|^{-(m+n)}\}$, where $\beta=\frac{2(m+n)}{mr_0}<\frac{3(m+n)}{m}$. If $z\in\Omega$, then $|\lambda^{m+n}z^{m+n}+\frac{n}{m}|<\beta|\lambda|\cdot\frac{n}{m}$ and $|z-\widetilde{w}_j|<\beta r_0$ for some $1\leq j\leq 2n$ by Lemma \[very-useful-est\](2). If $z\in\Omega$ and $0<|\lambda|<{1}/{(2^m n^2)}$, we have $$\label{P-estima}
\frac{n-1}{|1+z|^{n-1}}<\frac{n-1}{((|\lambda|^{-1}-\beta)r_0-1)^{n-1}}
<\frac{n-1}{(2^{m+1}n^2/3-3-2n/m)^{n-1}}<\frac{1}{15}$$ and $$\label{P-estima-1}
\beta|\lambda|\leq\frac{2(m+n)}{2^m n^2\cdot mr_0}<\frac{3}{2^m n}\left(\frac{1}{m}+\frac{1}{n}\right)<\frac{1}{4}, \text{~~therefore~~}
\frac{1+\beta|\lambda|}{2(1-\beta|\lambda|)}<\frac{5}{6}.$$ Therefore, if $z\in\Omega$ and $0<|\lambda|<{1}/{(2^m n^2)}$, from and , we have $$\begin{split}
&~ \left|\frac{(m+n)z^{m+n-1}}{m}\left(1+\frac{n-1}{(1+z)^{n-1}}\right)\right| =
\frac{m+n}{m|\lambda|^{m+n}}\left|\frac{\lambda^{m+n}z^{m+n}}{z}\left(1+\frac{n-1}{(1+z)^{n-1}}\right)\right|\\
< &~ \frac{m+n}{m|\lambda|^{m+n}}~\frac{(\beta|\lambda|+1)n/m}{r_0(1/|\lambda|-\beta)}\cdot \frac{16}{15}
= \frac{n\beta|\lambda|}{m|\lambda|^{m+n}}\,\frac{1+\beta|\lambda|}{2(1-\beta|\lambda|)}\cdot \frac{16}{15}
<\frac{n\beta|\lambda|}{m|\lambda|^{m+n}}.
\end{split}$$ Applying Rouché’s Theorem to and then using Lemma \[very-useful-est\](2), the proof of the first assertion is completed. By means of the same argument as , if $0<|\lambda|<{1}/{(2^m n^2)}$, we have $$\frac{(r_0/|\lambda|)\cdot\sin(\pi/(m+n))}{2(m+n)/m}\geq \frac{mr_0}{(m+n)^2|\lambda|}>\frac{2^{m+1}m}{3(m/n+1)^2}>1.$$ This means that $w_i= w_j$ if and only if $i=j$. The proof is complete.
Let $CP:=\{w_j:1\leq j\leq m+n\}$ be the $m+n$ critical points of $P_\lambda$ lying near the circle $\mathbb{T}_{r_0/|\lambda|}$ and $CV:=\{P_\lambda(w_j):1\leq j\leq m+n\}$. Let $CP_{-1}$ be the collection of $n-1$ critical points of $P_\lambda$ near $-1$ (see Lemma \[nice-cond-para\]) and $CV_{-1}=\{P_\lambda(z):z\in CP_{-1}\}$.
Let $T_0$ be the Fatou component of $P_\lambda$ containing the attracting petal at the origin and $U:=\mathbb{D}(-\frac{3}{4},\frac{3}{4})$. By Lemmas \[key-lemma\](2) and \[nice-cond-para\], we know that $CP_{-1}\cup CV_{-1}\subset U\subset T_0$. Since $P_\lambda(\infty)=-1$, it follows that there exists a neighborhood of $\infty$ such that $P_\lambda$ maps it to a neighborhood of $-1$. Let $T_\infty$ be the Fatou component such that $\infty\in T_\infty$ and $U_0,U_\infty$ be the component of $P_\lambda^{-1}(U)$ such that $0\in\overline{U}_0$ and $\infty\in U_\infty$. Obviously, we have $U\subset U_0\subset T_0$ and $U_\infty\subset T_\infty$.
\[Nice-cond-para-McM\] If $0<|\lambda|\leq {1}/{(2^{10m} n^3)}$, there exists an annular neighborhood $A_1$ of $CP$ containing $\mathbb{T}_{1/|\lambda|}\cup CP$ such that $P_\lambda(A_1)\subset \overline{U'}_\infty\subset U_\infty$, where $U'_\infty$ is a neighborhood of $\infty$.
It is known from Lemma \[crit-close-para\] that $CP$ is ‘almost’ lying uniformly on the circle $\mathbb{T}_{r_0/|\lambda|}$ and all the finite poles of $P_\lambda$ lie on the circle $\mathbb{T}_{1/|\lambda|}$. Define the annulus $$A_1=\{z:1/(2|\lambda|)<|z|<2/|\lambda|\}.$$ Note that $$\frac{r_0}{|\lambda|}+\frac{2(m+n)}{m}<\frac{3}{2|\lambda|}+2+\frac{2n}{m}<\frac{2}{|\lambda|}$$ and $$\frac{r_0}{|\lambda|}-\frac{2(m+n)}{m}>\frac{2}{3|\lambda|}-2-\frac{2n}{m}>\frac{1}{2|\lambda|}.$$ We have $\mathbb{T}_{1/|\lambda|}\cup CP\subset A_1$ by Lemma \[crit-close-para\]. If $z\in A_1$ and $|\lambda|\leq\frac{1}{2^{10m} n^3}$, then $$\label{import-1}
|P_\lambda(z)+1|\geq \frac{(|z|-1)^n}{n(|\lambda z|^{m+n}+1)}\geq\frac{(\frac{1}{2|\lambda|}-1)^n}{n(2^{m+n}+1)}=\frac{(1-2|\lambda|)^n}{2^n n|\lambda|^n(2^{m+n}+1)}>\frac{2}{|\lambda|^{1+\frac{n}{m}}}+1.$$ In fact, $$\frac{(1-2|\lambda|)^n}{2^{m+n}+1}>\frac{(1-\frac{2}{2^{10m}n^3})^n}{2^{m+n}+1}>\frac{0.9}{2^{m+n}+1}>\frac{1}{2^{m+n+1}}+2^n n|\lambda|^n.$$ This means that follows by $$2^{m+2n+2}\,n\,|\lambda|^n\leq |\lambda|^{1+n/m}.$$ This is true because $|\lambda|\leq\frac{1}{2^{10m} n^3}$. Now we have proved that if $z\in A_1$ and $|\lambda|\leq\frac{1}{2^{10m} n^3}$, then $|P_\lambda(z)|>\frac{2}{|\lambda|^{1+{n}/{m}}}$.
On the other hand, if $|z|\geq \frac{2}{|\lambda|^{1+{n}/{m}}}$, then $$|P_\lambda(z)+1|\leq \frac{(|z|+1)^n+1}{|\lambda z|^{m+n}-1}\leq
\frac{(1+|z|^{-1})^n+|z|^{-n}}{2^m-|z|^{-n}}<\frac{1}{2}.$$ This means that $P_\lambda(z)\in\mathbb{D}(-1,\frac{1}{2})\subset U$. Let $U'_\infty$ be the component of $P_\lambda^{-1}(\mathbb{D}(-1,\frac{1}{2}))$ containing $\{z:|z|\geq \frac{2}{|\lambda|^{1+{n}/{m}}}\}$, it follows that $P_\lambda(A_1)\subset \overline{U'}_\infty\subset U_\infty$ (see Figure \[Fig\_para-map\]).
![Sketch illustrating of the mapping relation of $P_\lambda$. The small pentagons denote the critical points.[]{data-label="Fig_para-map"}](parab-cantor.pdf){width="130mm"}
*Proof of Theorem \[non-hyper-cantor\]*. For every $\lambda$ such that $0<|\lambda|\leq {1}/{(2^{10m} n^3)}$, let $A:=\overline{\mathbb{C}}\setminus (U\cup U'_\infty)$. Since $P_\lambda:U'_\infty\rightarrow\mathbb{D}(-1,\frac{1}{2})$ is proper with degree $m$, it follows that $U'_\infty$ is simply connected and $A$ is an annulus. Note that $P_\lambda^{-1}(U'_\infty)$ is an annulus since there are $m+n$ critical points in $P_\lambda^{-1}(U'_\infty)$ and on which the degree of $P_\lambda$ is $m+n$. This means that $P_\lambda^{-1}(A)$ consists of two disjoint annuli $I_1$ and $I_2$ and $I_1\cup I_2\subset A$. The degree of the restriction of $P_\lambda$ on $I_1$ and $I_2$ are $m$ and $n$ respectively.
The following argument is very similar to that of Theorem \[parameter\]. The Julia set of $P_\lambda$ is $J_\lambda=\bigcap_{k\geq 0}P_\lambda^{-k}(A)$. By the construction, the components of $J_n$ are compact sets nested between $-1$ and $\infty$ since $P_\lambda^{-1}:A\rightarrow I_j$ is conformal for $j=1$ or $2$. Since the component of $J_n$ cannot be a point and the proof of Theorem 1.2 in [@PT] can also be applied to geometrically finite rational maps (see [@PT $\S$9] and [@TY]), we know that every component of $J_n$ is a Jordan curve. The dynamics of $P_\lambda$ on the set of Julia components is isomorphic to the one-sided shift on $2$ symbols $\Sigma_{2}:=\{0,1\}^{\mathbb{N}}$. In particular, $J_\lambda$ is homeomorphic to $\Sigma_{2}\times\mathbb{S}^1$, which is a Cantor set of circles as claimed. $\square$
From the proof of Theorem \[non-hyper-cantor\] and Theorem \[this-is-all-resta\], we know that the dynamics on the Julia set of $P_\lambda$ is conjugate to that of some $g_\eta$ with the form . Therefore, we can view $P_\lambda$ as a ‘parabolic’ McMullen map since the only difference is the super-attracting basin and its preimages of $g_\eta$ have been replaced by a fixed parabolic basin and its preimages of $P_\lambda$ (see Figure \[Fig\_C-C-C\]).
![The Julia set of $P_\lambda$, where $m=3,n=2$ and $\lambda$ is small enough such that $J_\lambda$ is a Cantor set of circles. All the Fatou components of $P_\lambda$ are iterated onto the fixed parabolic component (the ‘cauliflower’ in the center of this figure) with parabolic fixed point 1.[]{data-label="Fig_C-C-C"}](Cantor_Circle_Cauliflower.png){width="80mm"}
More Non-hyperbolic Examples {#sec-more-exam}
============================
In this section, we will construct more non-hyperbolic rational maps whose Julia sets are Cantor circles but they are not included by the previous section. Inspired by Theorem \[parameter\], for every $n\geq 2$, we define $$\label{family-para-restate}
P_n(z)=A_n\,\frac{(n+1)z^{(-1)^{n+1} (n+1)}}{nz^{n+1}+1}\prod_{i=1}^{n-1}(z^{2n+2}-b_i^{2n+2})^{(-1)^{i-1}}+B_n,$$ where $|b_i|=s^i$ for some $0<s\leq 1/(25n^2)$ and $$\label{A-B-n-restate}
A_n=\frac{1}{1+(2n+2)C_n}\prod_{i=1}^{n-1}(1-b_i^{2n+2})^{(-1)^i},~~B_n=\frac{(2n+2)C_n}{1+(2n+2)C_n}~~\text{and}~~
C_n=\sum_{i=1}^{n-1}\frac{(-1)^{i-1}b_i^{2n+2}}{1-b_i^{2n+2}}.$$
\[para-fixed\] $(1)$ $P_n(1)=1$ and $P_n'(1)=1$.
$(2)$ $1-s^{2n+1}/(n+1)<|A_n|<1+s^{2n+1}/(n+1)$ and $|B_n|<s^{2n+1}/(3n+3)$.
It is easy to see $P_n(1)=1$ by a straightforward calculation. Note that $$\label{solu-crit-parabolic}
F_n(z):=\frac{zP_n'(z)}{P_n(z)-B_n}=\sum_{i=1}^{n-1}\frac{(-1)^{i-1}(2n+2)z^{2n+2}}{z^{2n+2}-b_i^{2n+2}}+(-1)^{n+1} (n+1)-\frac{n(n+1)z^{n+1}}{nz^{n+1}+1}.$$ This means that $$\label{A-B-equation-1}
\begin{split}
&~ \frac{P_n'(1)}{P_n(1)-B_n} \\
= &~ (2n+2)\,\sum_{i=1}^{n-1}\frac{(-1)^{i-1}b_i^{2n+2}}{1-b_i^{2n+2}}+(2n+2)\,\sum_{i=1}^{n-1}(-1)^{i-1}+(-1)^{n+1} (n+1)-n\\
= &~ (2n+2)\,\sum_{i=1}^{n-1}\frac{(-1)^{i-1}b_i^{2n+2}}{1-b_i^{2n+2}}+1:=(2n+2)C_n+1.
\end{split}$$ Therefore, we have $$\label{A-B-equation-2}
P_n'(1)=(1-B_n)((2n+2)C_n+1)=1.$$ It follows that $1$ is a parabolic fixed point of $P_n$. This completes the proof of (1).
For (2), since $|1-b_i^{2n+2}|^{-1}\leq 1+2|b_1|^{2n+2}$ for $1\leq i\leq n-1$ and $0<s\leq 1/(25n^2)\leq 1/100$, then $$\label{C-n-estim}
\begin{split}
&~ (2n+2)|C_n|< (2n+2)\,(1+2|b_1|^{2n+2})\sum_{i=1}^{n-1}|b_i|^{2n+2}\\
\leq &~ \frac{(2n+2)(1+2s^{2n+2})s^{2n+2}}{1-s^{2n+2}}<\frac{s^{2n+1}}{4n+4}.
\end{split}$$ We have $$\label{B-n-estim}
|B_n|=\left|\frac{(2n+2)C_n}{1+(2n+2)C_n}\right|<(2n+2)|C_n|(1+(4n+4)|C_n|)<\frac{s^{2n+1}}{3n+3}$$ and $$\label{A-n-estim}
|A_n|<(1+(4n+4)|C_n|)\prod_{i=1}^{n-1}(1+2|b_i|^{2n+2})<(1+\frac{s^{2n+1}}{2n+2})(1+5s^{2n+2})<1+\frac{s^{2n+1}}{n+1}.$$ Moreover, we have $$\label{A-n-estim-lower}
|A_n|>(1-(2n+2)|C_n|)\prod_{i=1}^{n-1}(1-|b_i|^{2n+2})>(1-\frac{s^{2n+1}}{4n+4})(1-\frac{s^{2n+2}}{1-s^{2n+2}})>1-\frac{s^{2n+1}}{n+1}.$$ The proof is complete.
Let us first explain some ideas behind the construction. For $n\geq 2$, define $\widetilde{Q}(z)=(z^{n+1}+n)/(n+1)$ and $\varphi(z)=1/z$, then $Q(z):=\varphi\circ\widetilde{Q}\circ\varphi^{-1}(z)=(n+1)z^{n+1}/(nz^{n+1}+1)$ satisfies: $\infty$ is a critical point of $Q$ with multiplicity $n$ which is attracted to the parabolic fixed point $1$. Since $\{b_i\}_{1\leq i\leq n-1}$ are very small, the rational map $P_n$ can be viewed as a small perturbation of $Q$. The terms $A_n$ and $B_n$ here guarantee that $1$ is always a parabolic fixed point of $P_n$ (see Lemma \[para-fixed\]). It can be shown that $P_n$ maps an annular neighborhood of $\mathbb{T}_{|b_i|}$ into $T_0$ or $T_\infty$ according to whether $i$ is odd or even, where $T_0$ and $T_\infty$ denote the Fatou components containing $0$ and $\infty$ respectively (see Lemma \[lemma-want\]). The Fatou component $T_\infty$ is always parabolic while $T_0$ is attracting or mapped to $T_\infty$ according to whether $n$ is odd or even. The proof of Theorem \[parameter-parabolic\] will based on the mixed arguments as in the previous 2 sections.
If $|z|\leq 1$, then $|\widetilde{Q}(z)|\leq 1$. This means that the fixed parabolic Fatou component of $\widetilde{Q}$ contains the unit disk for every $n\geq 2$. Therefore, the parabolic Fatou component of $Q$ contains the exterior of the closed unit disk $\overline{\mathbb{C}}\setminus \overline{\mathbb{D}}$. Although the polynomial $Q$ has been perturbed into $P_n$, we still have following
\[key-lemma-complex\] $P_n(\overline{\mathbb{C}}\setminus \mathbb{D})\subset(\overline{\mathbb{C}}\setminus \overline{\mathbb{D}})\cup\{1\}$. In particular, the disk $\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}$ lies in the parabolic Fatou component of $P_n$ with parabolic fixed point $1$.
The proof of Lemma \[key-lemma-complex\] is very subtle, and will be delayed to next section.
\[sum-of\] For $n\geq 2$ and $1\leq i\leq n-1$, then $$\label{sum-equa}
\sum_{1\leq j<i}(-1)^j +\sum_{i<j\leq n-1}(-1)^{j-1}+\frac{1+(-1)^{n+1}}{2}=0.$$
The argument is based on several cases shown in Table \[Tab\_number\].
$\sum_{1\leqslant j<i}(-1)^j$ $\sum_{i<j\leqslant n-1}(-1)^{j-1}$ $(1+(-1)^{n+1})/{2}$
---------- ---------- ------------------------------- ------------------------------------- ----------------------
odd $n$ odd $i$ $0$ $-1$ $1$
even $i$ $-1$ $0$ $1$
even $n$ odd $i$ $0$ $0$ $0$
even $i$ $-1$ $1$ $0$
: The proof of Lemma \[sum-of\].[]{data-label="Tab_number"}
0.2cm
-0.5cm As before, we first locate the critical points of $P_n$. Note that $0$ and $\infty$ are both critical points of $P_n$ with multiplicity $n$ and the degree of $P_n$ is $n^2+n$. The remaining $2(n^2-1)$ critical points of $P_n$ are the solutions of $F_n(z)=0$ (see equation ).
For $1\leq i\leq n-1$, let $\widetilde{CP}_i:=\{\widetilde{w}_{i,j}=b_i \exp(\pi \textup{i}\frac{2j-1}{2n+2}):1\leq j\leq 2n+2\}$ be the collection of $2n+2$ points lying on $\mathbb{T}_{|b_i|}$ uniformly. The following lemma is similar to Lemmas \[crit-close\] and \[crit-close-para\].
\[crit-close-Parameter\] For every $\widetilde{w}_{i,j}\in\widetilde{CP}_i$, where $1\leq i\leq n-1$ and $1\leq j\leq 2n+2$, there exists $w_{i,j}$, which is a solution of $F_n(z)=0$, such that $|w_{i,j}-\widetilde{w}_{i,j}|<s^{n+1/2}|b_i|$. Moreover, $w_{i_1,j_1}= w_{i_2,j_2}$ if and only if $(i_1,j_1)=(i_2,j_2)$.
Note that $F_n(z)=0$ is equivalent to $$\label{solu-crit-2}
\sum_{i=1}^{n-1}(-1)^{i-1}\frac{z^{2n+2}+b_i^{2n+2}}{z^{2n+2}-b_i^{2n+2}}+\frac{1+(-1)^{n+1}}{2}-\frac{nz^{n+1}}{nz^{n+1}+1}=0.$$ Timing $z^{2n+2}-b_i^{2n+2}$ on both sides of (\[solu-crit-2\]), where $1\leq i\leq n-1$, we have $$\label{solu-crit-3-parabolic}
(-1)^{i-1}(z^{2n+2}+b_i^{2n+2})+(z^{2n+2}-b_i^{2n+2})\,G_{i}(z)=0,$$ where $$\label{G_n-parabolic}
G_{i}(z)=\sum_{1\leq j \leq n-1,\,j\neq i}(-1)^{j-1}\frac{z^{2n+2}+b_j^{2n+2}}{z^{2n+2}-b_j^{2n+2}}+\frac{1+(-1)^{n+1}}{2}-\frac{nz^{n+1}}{nz^{n+1}+1}.$$
Let $\Omega_{i}=\{z:|z^{2n+2}+b_i^{2n+2}|\leq s^{n+1/2}|b_i|^{2n+2}\}$, where $1\leq i\leq n-1$. If $z\in\Omega_i$, then $|z|^{n+1}\leq (1+s^{n+1/2})|b_i|^{n+1}\leq(1+s^{n+1/2})s^{n+1}$ by Lemma \[very-useful-est\](2). So $$\left|\frac{nz^{n+1}}{nz^{n+1}+1}\right|\leq \frac{n(1+s^{n+1/2})s^{n+1}}{1-n(1+s^{n+1/2})s^{n+1}}
\leq \frac{(1+100^{-5/2})s^{n+1/2}/5}{1-(1+100^{-5/2})100^{-5/2}/5}<0.3 \, s^{n+1/2}$$ since $s\leq 1/(25n^2)\leq 1/100$. For every $z\in\Omega_{i}$, if $1\leq j<i$, we have $$\label{estim-1-new}
|{z}/{b_j}|^{2n+2}=|{z}/{b_i}|^{2n+2}|{b_i}/{b_j}|^{2n+2}< (1+s^{n+1/2})\,s^{(2n+2)(i-j)}.$$ If $i<j\leq n-1$, by the first statement of Lemma \[very-useful-est\](2), we have $$\label{estim-2-new}
|{b_j}/{z}|^{2n+2}=|{b_i}/{z}|^{2n+2}|{b_j}/{b_i}|^{2n+2}\leq (1+2\cdot s^{n+1/2})\,s^{(2n+2)(j-i)}.$$ From (\[estim-1-new\]), (\[estim-2-new\]) and Lemma \[sum-of\], we have $$\label{bound-neww}
\begin{split}
&~ \left|G_{i}(z)+\frac{nz^{n+1}}{nz^{n+1}+1}\right|\\
= &~ \left|\sum_{1\leq j<i}(-1)^{j}\frac{1+(z/b_j)^{2n+2}}{1-(z/b_j)^{2n+2}}+
\sum_{i< j\leq n-1}(-1)^{j-1}\frac{1+(b_j/z)^{2n+2}}{1-(b_j/z)^{2n+2}}+\frac{1+(-1)^{n+1}}{2}\right|\\
< &~ 3\cdot(1+2\cdot s^{n+1/2})\,\left(\sum_{1\leq j<i}s^{(2n+2)(i-j)}+\sum_{i< j\leq n-1} s^{(2n+2)(j-i)}\right)\\
< &~6\cdot(1+2\cdot s^{n+1/2})^2\,s^{2n+2}.
\end{split}$$ The first inequality in (\[bound-neww\]) follows from the inequality $2x/(1-x)\leq 3x$ if $x<1/3$ (Here $x\leq (1+2\cdot s^{n+1/2})\,s^{2n+2}<10^{-10}$). So we have $$\label{bound-parabolic}
|G_{i}(z)|< ~6\cdot(1+2\cdot s^{n+1/2})^2\,s^{2n+2}+0.3 \, s^{n+1/2} < 0.4 \,s^{n+1/2}.$$ Therefore, if $z\in\Omega_{i}$, then $$|z^{2n+2}-b_i^{2n+2}|\cdot|\,G_{i}(z)|< (2+s^{n+1/2})|b_i|^{2n+2}\cdot 0.4 \,s^{n+1/2} < s^{n+1/2}|b_i|^{2n+2}.$$ From (\[solu-crit-3-parabolic\]) and by Rouché’s Theorem, there exists a solution $w_{i,j}$ of $F_n(z)=0$ such that $w_{i,j}\in\Omega_i$ for every $1\leq j\leq 2n+2$. In particular, $|w_{i,j}-\widetilde{w}_{i,j}|<s^{n+1/2}|b_i|$ by the second statement of Lemma \[very-useful-est\](2). The assertion $w_{i_1,j_1}= w_{i_2,j_2}$ if and only if $(i_1,j_1)=(i_2,j_2)$ can be verified similarly as and . The proof is complete.
For $1\leq i\leq n-1$, let $CP_i:=\{w_{i,j}: 1\leq j\leq 2n+2\}$ be the collection of critical points of $P_n$ which lie close to the circle $\mathbb{T}_{|b_i|}$.
\[lemma-want\] There exist $n-1$ annuli $\{A_i\}_{i=1}^{n-1}$ satisfying $A_{n-1}\prec \cdots\prec A_1$ and two simply connected domain $U_0$ and $U_\infty$ which contains $0$ and $\infty$ respectively, such that
$(1)$ $U_\infty\supset\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}$ and $P_n(\overline{U}_\infty)\subset U_\infty\cup\{1\}$;
$(2)$ $A_i\supset\mathbb{T}_{|b_i|}\cup CP_i$, $P_n(\overline{A}_i)\subset U_0$ for odd $i$ and $P_n(\overline{A}_i)\subset U_\infty$ for even $i$;
$(3)$ $P_n(\overline{U}_0)\subset U_\infty$ for even $n$ and $P_n(\overline{U}_0)\subset U_0$ for odd $n$.
Let $U_\infty:=\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}$ be the exterior of the closed unit disk. Then (1) is obvious if we apply Lemma \[key-lemma-complex\]. Let $\varepsilon=s^{n+1/2}$ and $A_i=\mathbb{A}_{|b_i|(1-2\varepsilon),|b_i|(1+2\varepsilon)}$. From , we know that $$\label{abs-f-n-yang}
|R_n(z)|: = \left|\frac{P_n(z)-B_n}{A_n}\cdot\frac{nz^{n+1}+1}{n+1}\right| = |z|^{(-1)^{n+1} (n+1)}\,|z^{2n+2}-b_i^{2n+2}|^{(-1)^{i-1}}H_i(z),$$ where $$\label{H-i-yang}
H_i(z)=\prod_{j=1}^{i-1}|b_j|^{(2n+2)(-1)^{j-1}}
\prod_{j=i+1}^{n-1}|z|^{(2n+2)(-1)^{j-1}}\cdot Q_i(z)$$ and $$\label{Q-i-yang}
Q_i(z)=\prod_{j=1}^{i-1}\left|1-({z}/{b_j})^{2n+2}\right|^{(-1)^{j-1}}
\prod_{j=i+1}^{n-1}\left|1-({b_j}/{z})^{2n+2}\right|^{(-1)^{j-1}}.$$
If $z\in \overline{A}_i$, where $1\leq i\leq n-1$, we have $$\label{Q-i-esti-1-yang}
Q_i(z)< \prod_{j=1}^{i-1}\left(1+3\,|{b_i}/{b_j}|^{2n+2}\right)
\prod_{j=i+1}^{n-1}\left(1+3\,|{b_j}/{b_i}|^{2n+2}\right)
< (1+6s^{2n+2})^2$$ and $$\label{Q-i-esti-2-yang}
Q_i(z)> \prod_{j=1}^{i-1}\left(1+3\,|{b_i}/{b_j}|^{2n+2}\right)^{-1}
\prod_{j=i+1}^{n-1}\left(1+3\,|{b_j}/{b_i}|^{2n+2}\right)^{-1}
> (1+6s^{2n+2})^{-2}.$$
Note that $\varepsilon=s^{n+1/2}\leq (5n)^{-2n-1}\leq 10^{-5}$. If $n$ is even and $1\leq i\leq n-1$ is odd, then for $z\in \overline{A}_i$, we have $$\begin{split}
|R_n(z)|
= &~ \frac{|z^{2n+2}-b_i^{2n+2}|}{|z|^{n+1}}\,\frac{1}{s^{(i-1)(n+1)}}\, Q_i(z)
< \frac{|b_i|^{n+1}(1+(1+2\varepsilon)^{2n+2})}{(1-2\varepsilon)^{n+1}}\,\frac{(1+6s^{2n+2})^2}{s^{(i-1)(n+1)}}\\
= &~\frac{1+(1+2\varepsilon)^{2n+2}}{(1-2\varepsilon)^{n+1}}(1+6s^{2n+2})^2 s^{n+1}<2.1\cdot s^{n+1}.
\end{split}$$ If $n$ and $1\leq i\leq n-1$ are both even, then for $z\in \overline{A}_i$, we have $$|R_n(z)|=\frac{|b_{i-1}|^{2n+2}|z|^{2n+2}}{|z|^{n+1}|z^{2n+2}-b_i^{2n+2}|}\,\frac{1}{s^{(i-2)(n+1)}}\, Q_i(z)
> \frac{(1-2\varepsilon)^{n+1}}{1+(1+2\varepsilon)^{2n+2}}\,(1-6s^{2n+2})^2 > 0.49.$$ This means that if $n$ is even and $1\leq i\leq n-1$ is odd, for $z\in \overline{A}_i$, we have $$\begin{split}
&~ |P_n(z)|<\left|\frac{2.1\cdot s^{n+1}\cdot (n+1)\,A_n}{nz^{n+1}+1}\right|+|B_n| \\
\leq &~ \frac{2.1\,(s^{n+1/2}/5)\cdot(1+s^{2n+1}/(n+1))}{1-n(1+2\varepsilon)s^{n+1}}+\frac{s^{2n+1}}{3n+3}<s^{n+1/2}
\end{split}$$ by Lemma \[para-fixed\](2). If $n$ and $1\leq i\leq n-1$ are both even, then for $z\in \overline{A}_i$, we have $$\begin{split}
|P_n(z)|
> &~ \left|\frac{0.49(n+1)A_n}{nz^{n+1}+1}\right|-|B_n| \\
\geq &~ \frac{0.49(n+1)(1-s^{2n+1}/(n+1))}{1+n(1+2\varepsilon)s^{n+1}}-\frac{s^{2n+1}}{3n+3}>\frac{n+1}{3}\geq 1.
\end{split}$$
By the completely similar arguments, one can show that if $n$ is odd, for $z\in \overline{A}_i$, we have $$\label{bound-f-n-3-parabolic}
|P_n(z)|<s^{n+1/2} \text{~for odd~} i \text{~and~} |P_n(z)|>1 \text{~for even~} i.$$ Let $U_0=\mathbb{D}_r$, where $r=s^{n+1/2}$. This proves (2).
If $n$ is odd, for every $z$ such that $|z|\leq s^{n+1/2}$, we have $$\begin{split}
|P_n(z)| \leq
& \left|\frac{(n+1)A_n}{nz^{n+1}+1}\right|\,|z|^{n+1} \prod_{i=1}^{n-1}|b_i|^{(2n+2)(-1)^{i-1}} \prod_{i=1}^{n-1}\left|1-\frac{z^{2n+2}}{b_i^{2n+2}}\right|^{(-1)^{i-1}}+|B_n|\\
\leq & \frac{(n+1)(1+s^{2n+1}/(n+1))}{1-ns^{n^2+n/2}}\,s^{3(n+1)/2}\prod_{i=1}^{n-1}\left(1+2\frac{|z|^{2n+2}}{|b_i|^{2n+2}}\right)+\frac{s^{2n+1}}{3n+3}<s^{n+1/2}.
\end{split}$$ It follows that $P_n(\overline{\mathbb{D}}_r)\subset\mathbb{D}_r$ for odd $n$, where $r=s^{n+1/2}$.
If $n$ is even, then $P_n$ maps a neighborhood of $0$ to that of $\infty$. For every $z$ such that $|z|\leq s^{n+1/2}$, we have $$\label{bound-lower-in-disk-even-parabolic-lp}
\begin{split}
|P_n(z)|
\geq &~\frac{(n+1)\,s^{-(n+1)/2}\,(1-s^{2n+1}/(n+1))}{1+ns^{n^2+n/2}}
\prod_{i=1}^{n-1}\left(1-2\frac{|z|^{2n+2}}{|b_i|^{2n+2}}\right)-\frac{s^{2n+1}}{3n+3} \\
> &~ n>1.
\end{split}$$ This ends the proof of (3). The proof is complete.
![The Julia set of $P_3$, which is a Cantor set of circles. The parameter $s$ is chosen small enough. The gray parts in the Figure denote the Fatou components which are iterated to the attracting Fatou component containing the origin, while the white parts denote the Fatou components iterated to the parabolic Fatou component whose boundary contains the parabolic fixed point $1$. Some equipotentials of Fatou coordinate have been drawn in the parabolic Fatou component and its preimages. Figure range: $[-1.6,1.6]\times[-1.2,1.2]$.[]{data-label="Fig_C-C-F"}](Cantor_Circle_Parabolic_Finally.png){width="100mm"}
*Proof of Theorem \[parameter-parabolic\]*. Let $A:=\overline{\mathbb{C}}\setminus (U_0\cup U_\infty)$. The Julia set of $P_n$ is equal to $\bigcap_{k\geq 0}P_n^{-k}(A)$. Note that $P_n$ is geometrically finite. The argument is completely similar to the proofs of Theorems \[parameter\] and \[non-hyper-cantor\]. The set of Julia components of $P_n$ is isomorphic to the one-sided shift on $n$ symbols $\Sigma_{n}:=\{0,1,\cdots,n-1\}^{\mathbb{N}}$. In particular, the Julia set of $P_n$ is homeomorphic to $\Sigma_{n}\times\mathbb{S}^1$, which is a Cantor set of circles, as desired (see Figure \[Fig\_C-C-F\]). We omit the details here. $\square$
Proof of Lemma \[key-lemma-complex\] {#sec-key-lemma}
====================================
This section will be devote to proving Lemma \[key-lemma-complex\], which is the key ingredient in the proof of Lemma \[lemma-want\] and hence in Theorem \[parameter-parabolic\].
Let $\widetilde{R}(z)=1/P_n(1/z)$, then Lemma \[key-lemma-complex\] reduces to proving $\widetilde{R}(\overline{\mathbb{D}})\subset \mathbb{D}\cup\{1\}$. Let $w=z^{n+1}$, by a straightforward calculation, we have $$R(w):=\widetilde{R}(z)=\frac{w+n}{n+1}\cdot\frac{1}{S(w)},$$ where $$\label{S-w-ori}
\begin{split}
S(w)= &~ A_n\,\prod_{i=1}^{n-1}(1-b_i^{2n+2}w^2)^{(-1)^{i-1}}+\frac{w+n}{n+1}B_n \\
= &~ 1+\frac{w-1}{1+(2n+2)C_n}\left(\frac{H(w)-1}{w-1}+2C_n\right)
\end{split}$$ and $$H(w)=\prod_{i=1}^{n-1}(1-b_i^{2n+2})^{(-1)^{i}}\prod_{i=1}^{n-1}(1-b_i^{2n+2}w^2)^{(-1)^{i-1}}.$$
Since $H(1)=1$, it follows that $H'(1)$ is a finite number. In fact, $$\label{I_w}
I(w):=\frac{H'(w)}{H(w)}=-2w\,\sum_{i=1}^{n-1}\frac{(-1)^{i-1}b_i^{2n+2}}{1-b_i^{2n+2}w^2}.$$ We know that $I(1)=H'(1)=-2C_n$. For every small enough $w-1$, we can write $S(w)$ as $$\label{S-w}
S(w)=1+\frac{(w-1)^2}{1+(2n+2)C_n}\cdot \frac{\frac{H(w)-1}{w-1}+2C_n}{w-1}=:1+\frac{(w-1)^2}{1+(2n+2)C_n}\cdot\Phi(w),$$ where $$\label{Phi-w}
\Phi(w)=\sum_{k\geq 2}\frac{H^{(k)}(1)}{k!}(w-1)^{k-2}.$$ The next step is to estimate $H^{(k)}(1)$ for every $k\geq 2$.
For every $k\geq 1$, let $$Y_k(w)=\sum_{i=1}^{n-1}(-1)^{i-1}\left(\frac{b_i^{2n+2}}{1-b_i^{2n+2}w^2}\right)^k.$$ In particular, $Y_1(1)=C_n$ and $$Y_k'(w)=2kw\,Y_{k+1}(w).$$ If $|w|=1$, we have $$|Y_k(w)|\leq \left|\frac{b_1^{2n+2}}{1-b_1^{2n+2}}\right|^k
\left(1+\sum_{i=2}^{n-1}\left|\frac{b_i^{2n+2}(1-b_1^{2n+2})}{b_1^{2n+2}(1-b_i^{2n+2})}\right|^k\right)
\leq \frac{11}{10}\,\left|\frac{b_1^{2n+2}}{1-b_1^{2n+2}}\right|^k.$$ Similarly, we have $|Y_k(w)|\geq \frac{9}{10}|{b_1^{2n+2}}/{(1-b_1^{2n+2})}|^k$. This means that $$\label{Y_k}
\left|\frac{Y_{k+1}(w)}{Y_k(w)}\right|\leq \frac{11}{9}\left|\frac{b_1^{2n+2}}{1-b_1^{2n+2}}\right|\leq 2s^{2n+2}<1/2.$$
We first claim that $|I^{(k)}(1)|\leq 2^{k+1}k!|C_n|$ for every $k\geq 0$. Since $I^{(0)}(w)=-2wY_1(w)$ and $I^{(1)}(w)=-2Y_1(w)-4w^2Y_2(w)$, it can be proved inductively that $I^{(k)}(w)$ can be written as $$\label{expansion}
I^{(k)}(w)=\sum_{j=1}^{2^k}Q_{k,j}(w)=\sum_{j=1}^{2^k}P_{k,j}(w)Y_{k,j}(w),$$ where $P_{k,j}(w)$ is a polynomial with degree at most $k+1$ and $Y_{k,j}=Y_l$ for some $1\leq l\leq k+1$. Note that some terms $Q_{k,j}$ may be equal to zero (the degree of corresponding polynomial $P_{k,j}$ is regarded as $-\infty$) and the formula can be simplified, but what we need is this ‘long’ expansion. In particular, without loss of generality, for $1\leq j\leq 2^k$, we require further that $$\label{dera}
P_{k+1,2j-1}(w)Y_{k+1,2j-1}(w)=P_{k,j}'(w)Y_{k,j}(w)~~\text{and}~~P_{k+1,2j}(w)Y_{k+1,2j}(w)=P_{k,j}(w)Y_{k,j}'(w).$$ Since $\deg (P_{k,j})\leq k+1$ and $Y_{k,j}=Y_l$ for some $1\leq l\leq k+1$, it follows that $$\label{deri-leq}
\begin{split}
&~ |P_{k+1,2j-1}(1)Y_{k+1,2j-1}(1)|+|P_{k+1,2j}(1)Y_{k+1,2j}(1)|\\
= &~ |P_{k,j}'(1)Y_{l}(1)|+|P_{k,j}(1)Y_{l}'(1)|\\
\leq &~ (k+1)|P_{k,j}(1)Y_{l}(1)|+2(k+1)|P_{k,j}(1)Y_{l+1}(1)|\\
\leq &~ 2(k+1)|P_{k,j}(1)Y_{k,j}(1)|
\end{split}$$ since $|Y_{l+1}(1)/Y_{l}(1)|\leq 1/2$ for every $l\geq 1$ by .
Denote $||I^{(k)}(1)||:=\sum_{j=1}^{2^k}|P_{k,j}(1)Y_{k,j}(1)|$, we have $||I^{(k)}(1)||\leq 2k||I^{(k-1)}(1)||$. This means that $$\label{bound-I-k}
|I^{(k)}(1)|\leq ||I^{(k)}(1)||\leq 2^k k!||I^{(0)}(1)||=2^{k+1}k!|C_n|.$$ This proves the claim $|I^{(k)}(1)|\leq 2^{k+1}k!|C_n|$ for every $k\geq 0$.
Secondly, we check by induction that $|H^{(k)}(1)|\leq 4^k k!|C_n|$ for $k\geq 1$. For $k=1$, we have $|H'(1)|=2|C_n|< 4|C_n|$. Assume that $|H^{(i)}(1)|\leq 4^i i!|C_n|$ for every $1\leq i\leq k$. By , we have $H'(w)=H(w)I(w)$. So $$\label{Deri-H-k}
\begin{split}
|H^{(k+1)}(1)|
\leq &~ |I^{(k)}(1)|+\sum_{i=1}^{k}\frac{k!}{i!(k-i)!}|H^{(i)}(1)|\cdot|I^{(k-i)}(1)|\\
\leq &~ 2^{k+1}k!|C_n|(1+2^{k+1}|C_n|)\leq 4^{k+1} (k+1)!|C_n|
\end{split}$$ since $|I^{(k-i)}(1)|\leq 2^{k-i+1}(k-i)!|C_n|$ and $|H^{(i)}(1)|\leq 4^i i!|C_n|$ for every $1\leq i\leq k$.
If $w=e^{i\theta}$ for $|\theta|\leq 1/20$, then $|w-1|<|\theta|\leq 1/20$. By and , we have $$\label{Phi-w-est}
|\Phi(w)|\leq \sum_{k\geq 2}4^k|C_n|(1/20)^{k-2}\leq 16|C_n|\sum_{k\geq 0}5^{-k}=20|C_n|.$$ By and , it follows that $$|S(w)|\geq 1-\frac{\theta^2}{1-(2n+2)|C_n|}20|C_n|\geq 1-\frac{s^{2n+1}}{n+1}\theta^2$$ since $n\geq 2$ and $|C_n|<s^{2n+1}/(8(n+1)^2)$ by .
On the other hand, if $w=e^{i\theta}$ for $0\leq|\theta|\leq \pi$, then $$\label{Q-z-est}
\left|\frac{w+n}{n+1}\right|=\left(1-\frac{4n}{(n+1)^2}\sin^2\frac{\theta}{2}\right)^{1/2}\leq \left(1-\frac{4n}{\pi^2(n+1)^2}\theta^2\right)^{1/2}\leq 1-\frac{2n}{(n+1)^2\pi^2}\theta^2$$ since $(1-x)^{1/2}\leq 1-x/2$ for $0\leq x< 1$. This means that if $w=e^{i\theta}$ for $|\theta|\leq 1/20$, then $$|R(w)|\leq (1-\frac{2n}{(n+1)^2\pi^2}\theta^2)(1-\frac{s^{2n+1}}{n+1}\theta^2)^{-1}\leq 1.$$ Moreover, $|R(w)|=1$ if and only if $w=1$.
If $w=e^{i\theta}$ for $|\theta|> 1/20$, by and Lemma \[para-fixed\](2), we have $$\label{S-w-est}
|S(w)|\geq (1-\frac{s^{2n+1}}{n+1})\prod_{i=1}^{n-1}(1-|b_i|^{2n+2})-\frac{s^{2n+1}}{3n+3}\geq 1-\frac{3s^{2n+1}}{n+1}.$$ By and , we have $$|R(w)|\leq (1-\frac{2}{20^2 (n+1)\pi^2})(1-\frac{3s^{2n+1}}{n+1})^{-1}< 1.$$ It follows that $R(w)$ maps the boundary of the unit disk into the unit disk except at $w=1$. Since $R(w)\neq \infty$ if $|w|\leq 1$, we know that $R(\overline{\mathbb{D}})\subset\mathbb{D}\cup\{1\}$. Therefore, $\widetilde{R}(\overline{\mathbb{D}})\subset\mathbb{D}\cup\{1\}$ and $\widetilde{R}$ maps $\{z\in\mathbb{C}:z^{n+1}=1\}$ onto 1. This ends the proof of Lemma \[key-lemma-complex\].
0.2cm
*Acknowledgements.* The authors would like to thank Guizhen Cui for discussions and the referees for their careful reading and comments. The first author was supported by the National Natural Science Foundation of China under grant No.11271074, and the third author was supported by the National Natural Science Foundation of China under grant No.11231009.
[99]{}
A. Beardon, *Iteration of rational functions*. Grad. Texts Math., **132**, Springer, New York, 1991.
G. Cui, Dynamics of rational maps, topology, deformation and bifurcation. Preprint, May 2002 (Early version: Geometrically finite rational maps with given combinatorics, 1997).
R. Devaney, D. Look and D. Uminsky, The Escape Trichotomy for Singularly Perturbed Rational Maps. *Indiana Univ. Math. J.*, **54**(6) (2005), 1621-1634.
P. Haïssinsky and K. Pilgrim, Quasisymmetrically inequivalent hyperbolic Julia sets. *Rev. Math. Iberoam.*, **28**(4) (2012), 1025-1034.
C. McMullen, Automorphisms of rational maps. In *Holomorphic Functions and Moduli I*, Math. Sci. Res. Inst. Publ. **10**, Springer, New York, 1988.
K. Pilgrim and L. Tan, Rational maps with disconnected Julia sets. *Astérisque*, **261** (2000), 349-383.
W. Qiu, X. Wang and Y. Yin, Dynamics of McMullen maps. *Adv. Math.*, **229**(4) (2012), 2525-2577.
W. Qiu and Y. Yin, Proof of the Branner-Hubbard conjecture on Cantor Julia sets. *Sci. China Ser. A*, **52**(1) (2009), 45-65.
N. Steinmetz, On the dynamics of the McMullen family $R(z)=z^m+\lambda/z^\ell$. *Conform. Geom. Dyn.*, **10** (2006) 159-183.
L. Tan and Y. Yin, Local connectivity of the Julia sets for geometrically finite rational maps. *Sci. China Ser. A*, **39**(1) (1996), 39-47.
| ArXiv |
---
abstract: 'Many kinds of algebraic structures have associated dual topological spaces, among others commutative rings with $1$ (this being the paradigmatic example), various kinds of lattices, boolean algebras, $C^*$-algebras, …. These associations are functorial, and hence algebraic endomorphisms of the structures give rise to continuous selfmappings of the dual spaces, which can enjoy various dynamical properties; one then asks about the algebraic counterparts of these properties. We address this question from the point of view of algebraic logic. The datum of a set of truth-values and a “conjunction” connective on them determines a propositional logic and an equational class of algebras. The algebras in the class have dual spaces, and the duals of endomorphisms of free algebras provide dynamical models for Frege deductions in the corresponding logic.'
address: |
Department of Mathematics\
University of Udine\
via delle Scienze 208\
33100 Udine, Italy
author:
- Giovanni Panti
title: |
Dynamical properties\
of logical substitutions
---
[^1]
Introduction
============
Everybody knows the classical truth-tables $$\begin{array}{c|cc}
\land & 0 & 1 \\
\hline
0 & 0 & 0 \\
1 & 0 & 1
\end{array}
\hspace{1cm}
\begin{array}{c|cc}
\lor & 0 & 1 \\
\hline
0 & 0 & 1 \\
1 & 1 & 1
\end{array}
\hspace{1cm}
\begin{array}{c|cc}
\to & 0 & 1 \\
\hline
0 & 1 & 1 \\
1 & 0 & 1
\end{array}
\hspace{1cm}
\begin{array}{c|c}
&\neg \\
\hline
0 & 1 \\
1 & 0
\end{array}$$ and uses them automatically. The [*classical propositional calculus*]{} studies the set of formulas that, when evaluated according to the truth-tables, always assume value $1$. One proceeds as follows:
1. a [*formula*]{} is a polynomial built up from the propositional variables $x_i$ using the connectives $\land,\lor,\to,\neg,0,1$;
2. a [*valuation*]{} is a function $p$, distributing over the connectives, from the set of formulas to ${\{0,1\}}$;
3. a formula $r$ is [*true*]{} if $p(r)=1$ for every valuation $p$;
4. a formula $r$ is [*deducible*]{} if:
- either is an element of a certain set $\Theta$ of basic axioms,
- or there exists a formula $s$ such that $s$ and $s\to r$ are deducible,
- or there exists a deducible formula $s$, propositional variables ${x_1,\ldots ,x_n}$, and formulas ${t_1,\ldots ,t_n}$, such that $r$ results from $s$ by substituting every $x_i$ that occurs in $s$ with the corresponding $t_i$;
5. the completeness theorem holds: a formula is true iff it is deducible.
The completeness theorem relates a semantical notion “the statement $r$ holds, regardless of the state of affairs $p$” with a computational notion “the statement $r$ can be deduced from certain statements using certain rules”. There are several computational procedures for which the completeness theorem holds: the ones sketched in (4) are known as [*substitutional Frege systems*]{}, and are the strongest —in terms of minimizing the number of steps required to prove a true statement— available proof systems [@cookrec79]. The two rules (4b) of [*Modus Ponens*]{} and (4c) of [*substitution*]{} have different flavors. The first rule is, in some sense, statical: if something is known “locally”, i.e., concerns certain propositional variables, then conclusions are drawn involving the same variables. On the other hand, the substitution rule adds dynamics to the picture: local knowledge can be moved around. This is of course just a vague heuristic, and in the course of these notes we will give a precise formal ground to it.
We will work at a level of generality broader than that of classical logic, enlarging the set of truth-values to include more than *true* and *false*; such logical systems are known as [*many-valued logics*]{}. Many-valued logic is an old discipline, going back to the twenties, and has recently been relived as a founding basis for fuzzy logic and fuzzy control; see [@hajek98], [@CignoliOttavianoMundici00], [@gottwald01] for detailed presentations and further references.
The key ideas of this work are the following: given a set of truth-values $M\supseteq{\{0,1\}}$, we introduce on it an algebraic structure, determined by the choice of a truth-table for the conjunction connective. We then consider the class ${\mathbf{V}}M$ of all algebras that are generated by $M$ in the sense of Universal Algebra, and we functorially associate a dual topological space to each object in ${\mathbf{V}}M$. Algebraic endomorphisms of certain objects of ${\mathbf{V}}M$ (the so-called free algebras) correspond to applications of the substitution rule in deductions in the logic determined by $M$. Moreover, such endomorphisms give rise to continuous selfmappings of the dual topological spaces. Any set $\Theta'\supseteq\Theta$ ($\Theta$ is a set of basic axioms as in (4a)) is associated to an open set $O_{\Theta'}$ in the dual, and the deduction of new formulas from $\Theta'$ corresponds to taking the union of the backwards translates of $O_{\Theta'}$ under the dynamics. Dynamical properties such as minimality or mixing have then logical consequences (see, e.g., Theorem \[ref10\], Theorem \[ref14\], and the discussion following Theorem \[ref23\]). It is worth remarking that the trade between the logical and the dynamical side may be beneficial to both: as an example, we obtain in Theorem \[ref22\] an intrinsic characterization of the differential of a piecewise-linear mapping, a concept introduced in [@Tsujii01].
A rather delicate point in our approach is the determination of the level of generality one should allow. Here we must really strike a balance: the stronger is the system (i.e., the more restrictions we put on $M$), the stronger are the results we obtain, and the more limited is the scope of the theory. The extreme case is in taking $M=\{0,1\}$, in which everything boils down to the Stone Duality. On the other extreme, one might relax the assumptions on $M$ to a bare minimum, even allowing cases in which the values $0$ and $1$ do not have a distinguished status: the only essential requirement seems to be that ${\mathbf{V}}M$ is a congruence-modular equational class. Of course, working at this level of generality requires a greater technical apparatus, and yields not easily visualizable results.
We stroke our balance by forcing $M$ to be a subset of the real unit interval ${[0,1]}$, and by insisting that the conjunction connective meets some natural restrictions. In the few places where we might have wished more elbow-room, we have added some Addenda to provide references for further developments. These Addenda are meant for people having some knowledge of Universal Algebra and lattice-ordered abelian groups, and may be safely skipped by the other readers.
Many-valued logic {#ref24}
=================
A [*t-norm*]{} is a continuous function ${\star}$ from ${[0,1]}^2$ to ${[0,1]}$ such that $({[0,1]},{\star},1)$ is a commutative monoid for which $a\le b$ implies $c{\star}a\le c{\star}b$. We have $a{\star}0=0$ for every $a$, since $a\le 1$ implies $0{\star}a\le 0{\star}1=0$. Every t-norm induces a binary operation $\to$ on ${[0,1]}$ via $$a\to b = \sup\{c:c{\star}a\le b\}.$$ Since ${\star}$ is continuous, the defining $\sup$ is really a $\max$. We call $\to$ the [*implication*]{} (or the [*residuum*]{}) induced by ${\star}$. One checks easily that the usual lattice operations on ${[0,1]}$ are definable from ${\star}$ and $\to$ via $a\land b=a{\star}(a\to b)$ and $a\lor b=\bigl((a\to b)\to b\bigl)\land
\bigl((b\to a)\to a\bigl)$. We also define $\neg a=a\to 0$.
The idea underlying these definitions is that ${\star}$ is a function on truth-values representing a “conjunction” operator. Once a conjunction has been fixed, it is natural to define the truth-value of the implication $a\to b$ as the weakest value $c$ such that the truth of the conjunction of $a$ and $c$ forces the truth of $b$. Note that “weakest” means “truest”, i.e., nearest to $1$: one should regard a more implausible assertion as a stronger one. The above interrelationship of ${\star}$ and $\to$ is usually expressed by saying that they constitute an [*adjoint pair*]{}.
\[ref3\]
1. $a{\star}b=a\land b$. One computes that $$a\to b=
\begin{cases}
1, & \text{if $a\le b$;} \\
b, & \text{otherwise;}
\end{cases}
\quad
\neg a=
\begin{cases}
1, & \text{if $a=0$;} \\
0, & \text{otherwise.}
\end{cases}$$ This t-norm is usually called the [*Gödel-Dummett conjunction*]{}.
2. $a{\star}b=ab$ (i.e., the ordinary product of $a$ and $b$). This is the [*product conjunction*]{}, and we have $$a\to b=
\begin{cases}
1, & \text{if $a\le b$;} \\
b/a, & \text{otherwise;}
\end{cases}
\quad
\neg a=
\begin{cases}
1, & \text{if $a=0$;} \\
0, & \text{otherwise.}
\end{cases}$$
3. $a{\star}b=\max(a+b-1,0)$. Then $$a\to b=
\begin{cases}
1, & \text{if $a\le b$;} \\
1-(a-b), & \text{otherwise;}
\end{cases}
\quad
\neg a=1-a.$$ These are the [*[Łukasiewicz]{} conjunction*]{}, [*implication*]{}, and [*negation*]{}
The above examples are in some sense exhaustive: by [@MostertShields57] every t-norm is obtainable as a combination of these three basic t-norms.
Fix a cardinal number $\kappa$, either finite or countable, and define the set of propositional variables to be $\{x_i:i<\kappa\}$ (then either $\kappa=n$ and the propositional variables are ${x_0,\ldots ,x_{n-1}}$, or $\kappa=\omega$ and the propositional variables are indexed by the natural numbers). Let $FORM_\kappa$ be the smallest set containing all propositional variables having index $<\kappa$, the constants $0$ and $1$, and such that, if $r,s\in FORM_\kappa$, then $(r{\star}s),(r\to s)\in FORM_\kappa$. A [*formula*]{} is an element $r$ of $FORM_\omega=\bigcup_{n<\omega}FORM_n$. We sometimes write $r(x_{i_1},\ldots,x_{i_n})$ to signify that all propositional variables occurring in $r$ are among $x_{i_1},\ldots,x_{i_n}$. We drop parentheses according to the usual conventions, and we write $r\land s$, $r\lor s$, and $\neg r$ as abbreviations for $r{\star}(r\to s)$, $\bigl((r\to s)\to s\bigl)\land
\bigl((s\to r)\to r\bigl)$, and $r\to 0$, respectively.
An [*algebra*]{} is a set $A$ on which two binary operations ${\star}_A,\to_A:A^2\to A$ and two elements $0_A,1_A\in A$ have been fixed. Given a formula $r({x_0,\ldots ,x_{n-1}})$ and elements ${a_0,\ldots ,a_{n-1}}\in A$, we write $r({a_0,\ldots ,a_{n-1}})$ for the element of $A$ obtained by replacing every $x_i$ with the corresponding $a_i$, and every operation symbol in $r$ with its realization in $A$ (the reader can easily supply a formal recursive definition). Given two formulas $r({x_0,\ldots ,x_{n-1}})$ and $s({x_0,\ldots ,x_{n-1}})$, we say that the identity $r=s$ is [*true*]{} in $A$, and we write $A\models r=s$, if for every ${a_0,\ldots ,a_{n-1}}\in A$ the elements $r({a_0,\ldots ,a_{n-1}})$ and $s({a_0,\ldots ,a_{n-1}})$ are equal.
\[ref4\]
1. Every singleton can be given the structure of an algebra in a unique trivial way; every identity is true in such an algebra.
2. Let $A={[0,1]}$, endowed with the Gödel-Dummett conjunction and implication, as in Example \[ref3\](1). Then $A\not\models\neg\neg x_0 = x_0$, so the double negation rule fails for the Gödel-Dummett connectives (and analogously for the product connectives). On the other hand, $\neg\neg x_0 = x_0$ holds true in ${[0,1]}$ endowed with the [Łukasiewicz]{} connectives.
Let $A,B$ be algebras. A mapping $\varphi:A\to B$ is a [*homomorphism*]{} if it commutes with the connectives (i.e., $\varphi(a{\star}_A b)=\varphi(a){\star}_B\varphi(b)$, $\varphi(0_A)=0_B$, and so on; in the following we will drop the subscripts). $A$ is \[isomorphic to\] a [*subalgebra*]{} of $B$ if there exists an injective homomorphism from $A$ to $B$. Let $\{A_j:j\in J\}$ be a family of algebras. The [*direct product*]{} of the family is the algebra whose base set is the cartesian product $\prod_jA_j$, and in which the operations are defined componentwise; if all factors are equal, say to $A$, then we write $A^J$. If $\varphi:A\to B$ is a homomorphism, then the [*epimorphic image*]{} $\varphi[A]$ of $A$ is a subalgebra of $B$.
Let ${\mathcal{A}}$ be a class of algebras; then ${\mathbf{H}}{\mathcal{A}}$ (respectively, ${\mathbf{S}}{\mathcal{A}}$ and ${\mathbf{P}}{\mathcal{A}}$) is the class of all epimorphic images (respectively, subalgebras and direct products) of algebras in ${\mathcal{A}}$. Note that we always work up to isomorphism, so we tacitly close every class we consider under isomorphic images.
A [*truth-value algebra*]{} is a subalgebra $M$ of some algebra $A$ of the form $A=({[0,1]},{\star},\to,0,1)$, where ${\star}$ and $\to$ are a t-norm and its residuum.
Truth-value algebras are our basic building blocks.
1. The set ${\{0,1\}}$ is always closed under the operations, regardless of the specific t-norm we choose. Moreover, all t-norms induce the same structure on ${\{0,1\}}$, namely that of the [*two-element boolean algebra*]{}, which we denote by ${\mathbf{2}}$.
2. $M=\{0,1/m,2/m,\ldots,(m-1)/m,1\}$ endowed either with the [Łukasiewicz]{} connectives or the Gödel-Dummett ones.
For any class ${\mathcal{A}}$ of algebras, let ${\mathbf{V}}{\mathcal{A}}$ be the [*equational class*]{} generated by ${\mathcal{A}}$, i.e., the class of all algebras in which are true all identities true in all algebras of ${\mathcal{A}}$. More explicitly, the algebra $B$ is in ${\mathbf{V}}{\mathcal{A}}$ iff, for every $r,s\in FORM_\omega$, if $A\models r=s$ for every $A\in{\mathcal{A}}$, then $B\models r=s$. Garrett Birkhoff’s completeness theorem [@burrissan81 Theorem II.11.9] says that ${\mathbf{V}}{\mathcal{A}}$ coincides with the class ${\mathbf{HSP}}{\mathcal{A}}$ of all epimorphic images of subalgebras of products of algebras in ${\mathcal{A}}$.
We will consider classes of algebras of the form ${\mathbf{V}}M={\mathbf{HSP}}M$, where $M$ is a truth-value algebra. We shall be concerned with two main cases:
- ${{\mathit{Boole}}}={\mathbf{V}}{\mathbf{2}}$. Elements of ${{\mathit{Boole}}}$ are called [*boolean algebras*]{};
- if $M={[0,1]}$ endowed with the [Łukasiewicz]{} connectives, then the elements of ${\mathbf{V}}M$ are called [*MV-algebras*]{} (MV stands for [*Many-Valued*]{}: the name is slightly misleading, since many-valued logic is not exhausted by [Łukasiewicz]{} logic, but it is firmly established; we accordingly write ${{\mathit{MV}}}$ for the equational class ${\mathbf{V}}M$).
A boolean algebra can be equivalently defined as a structure $A=(A,\land,\lor,\neg,0,1)$ such that $$\begin{gathered}
x\land 1=x\lor 0=x;\\
x\land\neg x=0;\quad x\lor\neg x=1;\\
\text{$\land$ and $\lor$ are commutative and mutually distributive}.\end{gathered}$$ Apart from the trivial change in the language ($\lor$ replaces $\to$), there is a theorem hidden in this equivalence, namely the fact that the above identities imply all other identities that hold in ${\mathbf{2}}$ [@Halmos63 p. 5].
An analogous alternative characterization of MV-algebras is obtained by adding a new connective $\oplus$ to the basic set $({\star},\to,0,1)$. We define $a\oplus b=\neg a\to b$, and directly compute that $\oplus$ is [*truncated addition*]{} on ${[0,1]}$, i.e., $a\oplus b=\min(a+b,1)$. Note that the basic set of connectives is equivalent to the set $(\oplus,\neg,0,1)$, since $a{\star}b=\neg(\neg a\oplus\neg b)$ and $a\to b=\neg a\oplus b$. Then, in terms of the new set, an MV-algebra is a structure $(A,\oplus,\neg,0,1)$ such that $(A,\oplus,0)$ is an abelian monoid and the identities $\neg\neg x=x$, $x\oplus 1=1$, $\neg(\neg x\oplus y)\oplus
y=\neg(\neg y\oplus x)\oplus x$ are satisfied [@mundicijfa §2], [@CignoliOttavianoMundici00].
Let\[ref5\] $M$ be a truth-value algebra, $A\in{\mathbf{V}}M$. Then:
- the operations $\land,\lor$ induce a lattice structure on $A$, with bottom element $0$ and top $1$;
- the lattice order in (i) is given by $a\le b$ iff $a\land b=a$ iff $a\to b=1$;
- $A\models r=s$ iff $A\models (r\to s)\land(s\to r)=1$.
A structure $(A,\land,\lor,0,1)$ is a lattice with bottom and top iff it satisfies a certain finite set of identities (see, e.g., [@burrissan81 p. 28]). Since $M$ is totally-ordered, these identities are satisfied in $M$, and hence in $A\in{\mathbf{V}}M$. The first equivalence in (ii) is just the definition of the lattice order on $A$. By definition of $\to$ in $M$, the identity $(x_0\land x_1)\to x_1=1$ is true in $M$, and hence in $A$. Therefore, if $a\land b=a$, then $a\to b=(a\land b)\to b=1$. On the other hand, if $a\to b=1$, then $a\land b=a{\star}(a\to b)=a{\star}1=a$. This proves (ii), and (iii) is then immediate.
By Lemma \[ref5\](ii) we can deal with the “less than” relation between formulas, thus writing $A\models r({x_0,\ldots ,x_{n-1}})\le s({x_0,\ldots ,x_{n-1}})$ for $A\models r\to s=1$; this just means that however we choose ${a_0,\ldots ,a_{n-1}}\in A$ we have $r({a_0,\ldots ,a_{n-1}})\le s({a_0,\ldots ,a_{n-1}})$. We then say that $r\le s$ is [*true*]{} in $A$.
Under\[ref6\] the same hypothesis as in Lemma \[ref5\], the following relations are true in $A$:
- $x_0{\star}x_1\le x_0\land x_1$;
- $x_0\le x_1\to(x_0{\star}x_1)$;
- $(x_0\to x_1){\star}(x_1\to x_2)\le x_0\to x_2$;
- $(x_0\to x_1){\star}(x_2\to x_3)\le (x_0{\star}x_2)\to
(x_1{\star}x_3)$;
- $(x_0\to x_1){\star}(x_2\to x_3)\le (x_1\to x_2)\to
(x_0\to x_3)$.
One just checks that for every t-norm with residuum $\to$ the above relations are true in $({[0,1]},{\star},\to,0,1)$. Hence they are true in $M$, and therefore in every algebra in ${\mathbf{V}}M$.
Fix now $\kappa$ and a truth-value algebra $M$. We want to construct an algebra $A$ in ${\mathbf{V}}M$ satisfying the following properties:
- $A$ is generated by a family $\{a_i:i<\kappa\}$ of elements indexed by $\kappa$;
- if $r(x_{i_1},\ldots,x_{i_n})\in FORM_\kappa$ is not true in $M$, then the element $r(a_{i_1},\ldots,a_{i_n})\in A$ is different from $1$.
Essentially, this means that the $a_i$’s satisfy only those algebraic relations they cannot avoid, namely those that hold in $M$. Therefore they behave “as freely as possible”, whence the name [*free algebra in ${\mathbf{V}}M$ over $\kappa$ generators*]{} for $A$. Such an algebra is unique up to isomorphism, and can be characterized by an appropriate universal property: see, e.g., [@burrissan81 II §10]. We write ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$ for $A$, and we construct it as follows: consider first $M^\kappa$, and let $a_i:M^\kappa\to M$ be the $i$-th projection. The $a_i$’s are elements of the algebra $M^{(M^\kappa)}$, and we define ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$ to be the subalgebra of $M^{(M^\kappa)}$ generated by them; the first condition is then automatically met. Suppose $M\not\models r$; then there exist elements $b_{i_1},\ldots,b_{i_n}\in M$ such that $r(b_{i_1},\ldots,b_{i_n})\not=1$. Choose an element $c\in M^\kappa$ such that $a_i(c)=b_i$ for every $i\in\{{i_1,\ldots ,i_n}\}$. Then the projection of $r(a_{i_1},\ldots,a_{i_n})\in M^{(M^\kappa)}$ onto the $c$-th component has value $r(a_{i_1}(c),\ldots,a_{i_n}(c))=r(b_{i_1},\ldots,b_{i_n})
\not=1$: therefore $r(a_{i_1},\ldots,a_{i_n})$ is different from $1$ in $M^{(M^\kappa)}$, and hence in ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$.
Although the above construction looks baroque, it really works trivially. Suppose, e.g., we want to construct $\operatorname{Free}_3{{\mathit{Boole}}}$. We first construct ${\mathbf{2}}^3$, which contains the eight elements $c_1=(0,0,0)$, $c_2=(0,0,1)$, …, $c_8=(1,1,1)$. Then we construct ${\mathbf{2}}^{({\mathbf{2}}^3)}$, which contains $2^8$ elements; three of these elements, namely $$\begin{aligned}
a_1 &= (0,0,0,0,1,1,1,1),\\
a_2 &= (0,0,1,1,0,0,1,1),\\
a_3 &= (0,1,0,1,0,1,0,1),\end{aligned}$$ correspond to the canonical projections to the first, second, and third component of the $c_j$’s (of course, the above explicit form for the $a_i$’s depends on how we listed the $c_j$’s). It is clear that if $r(x_1,x_2,x_3)$ is not $1$ in ${\mathbf{2}}$ for some choice of elements, then $r(a_1,a_2,a_3)\not=1=(1,1,1,1,1,1,1,1)$ in ${\operatorname{Free}_{3}({{\mathit{Boole}}})}$: we just tried all possible choices!
Often it is not trivial to determine, for a given $M$ and $\kappa$, which are the elements of ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$, i.e., which functions from $M^\kappa$ to $M$ are expressible as polynomials over the projections $a_i$. The case that most concerns us is the [Łukasiewicz]{} one, which we will treat in Theorem \[ref15\]. For the classical logic case, the answer is the following: give ${\mathbf{2}}$ the discrete topology, and ${\mathbf{2}}^\kappa$ the product topology. Then:
- an element $f\in{\mathbf{2}}^{({\mathbf{2}}^\kappa)}$ is in ${\operatorname{Free}_{\kappa}({{\mathit{Boole}}})}$ iff it is continuous as a function $f:{\mathbf{2}}^\kappa\to{\mathbf{2}}$.
Since continuous functions from a topological space $X$ to ${\mathbf{2}}$ correspond to clopen subsets of $X$, this amounts to saying that the clopen subsets of ${\mathbf{2}}^\kappa$ are exactly the boolean combinations of the sets of the form $a_i^{-1}[1]=\{p\in{\mathbf{2}}^\kappa:p_i=1\}$. With this hint, we leave the proof of (i) as an exercise for the reader. As a corollary we obtain:
- if $\kappa=n$ is finite, then ${\mathbf{2}}^n$ is a discrete space, and all functions $:{\mathbf{2}}^n\to{\mathbf{2}}$ are in ${\operatorname{Free}_{n}({{\mathit{Boole}}})}$, i.e., are expressible by $n$-variable formulas. This is sometimes called the [*functional completeness*]{} of the boolean connectives;
- if $\kappa=\omega$ is countably infinite, then ${\operatorname{Free}_{\omega}({{\mathit{Boole}}})}$ is the boolean algebra of all clopen subsets of the [*Cantor space*]{} ${\mathbf{2}}^\omega$, the latter being the only compact, totally disconnected, second countable space having no isolated points [@hockingyou §2.15].
Spectral spaces {#ref13}
===============
In the preceding section we have defined the equational class ${\mathbf{V}}M$ generated by a truth-value algebra $M$, and described the algebras ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$. In this section we functorially associate a dual topological space to each algebra in ${\mathbf{V}}M$; the duals of the free algebras are our main object of study.
We fix a truth-value algebra $M$; all algebras we consider are elements of ${\mathbf{V}}M$. A [*filter*]{} on $A\in{\mathbf{V}}M$ is the counterimage of $1$ under some homomorphism of domain $A$: sometimes filters are called [*dual ideals*]{} since ideals, as in ring theory, are counterimages of $0$.
A subset ${\mathfrak{f}}$ of $A$ is a filter iff it contains $1$ and is closed under Modus Ponens ($a,a\to b\in{\mathfrak{f}}$ implies $b\in{\mathfrak{f}}$). Every filter is closed under ${\star}$ and $\land$, and is upwards closed ($b\ge a\in{\mathfrak{f}}$ implies $b\in{\mathfrak{f}}$). Given two homomorphisms $\varphi:A\to B$ and $\psi:A\to C$, we have $\varphi^{-1}[1]=\psi^{-1}[1]$ iff there exists an isomorphism $\chi:\varphi[A]\to\psi[A]$ such that $\psi=\chi\circ\varphi$.
Let ${\mathfrak{f}}=\varphi^{-1}[1]$; then clearly $1\in{\mathfrak{f}}$. If $a,a\to b\in{\mathfrak{f}}$, then $\varphi(b)\ge\varphi(a\land b)=\varphi(a{\star}(a\to b))=\varphi(a){\star}\varphi(a\to b)=1$. Conversely, assume that ${\mathfrak{f}}$ is a subset of $A$ containing $1$ and closed under Modus Ponens. Then ${\mathfrak{f}}$ is upwards closed, since $a\le b$ implies $a\to b=1\in{\mathfrak{f}}$, and hence $a\in{\mathfrak{f}}$ implies $b\in{\mathfrak{f}}$. If $a\in{\mathfrak{f}}$, then $b\to(a{\star}b)\in{\mathfrak{f}}$ (by Lemma \[ref6\](ii)); hence, if $b\in{\mathfrak{f}}$ as well, then $a{\star}b,a\land b\in{\mathfrak{f}}$ (by Lemma \[ref6\](i)). Let now ${\mathfrak{f}}$ be a subset of $A$ containing $1$ and closed under MP. Define a relation $\sim$ on $A$ by $a\sim b$ iff $a\to b,b\to a\in{\mathfrak{f}}$. Then $\sim$ is an equivalence relation (transitivity follows from Lemma \[ref6\](iii)) which respects the operations. Indeed, if $a\sim b$ and $c\sim d$, then $a{\star}c\sim b{\star}d$ (by Lemma \[ref6\](iv)) and $a\to c\sim b\to d$ (by Lemma \[ref6\](v)). We can then form the quotient algebra $A/{\mathfrak{f}}$ in the natural way. If $a/{\mathfrak{f}}$ denotes the equivalence class of $a$ w.r.t. $\sim$, then the map $\rho(a)=a/{\mathfrak{f}}$ is a surjective homomorphism from $A$ to $A/{\mathfrak{f}}$. It is then straightforward to check that the map $\tau:A/{\mathfrak{f}}\to\varphi[A]$ defined by $\tau(a/{\mathfrak{f}})=\varphi(a)$ is an isomorphism. Since $\varphi=\tau\circ\rho$, our last claim follows by composing isomorphisms.
A filter ${\mathfrak{p}}$ is [*prime*]{} if it is proper (i.e., different from $A$) and for every two filters ${\mathfrak{f}},{\mathfrak{g}}$, if ${\mathfrak{p}}={\mathfrak{f}}\cap{\mathfrak{g}}$ then either ${\mathfrak{p}}={\mathfrak{f}}$ or ${\mathfrak{p}}={\mathfrak{g}}$. A filter is [*maximal*]{} if it is proper and not properly contained in any proper filter; clearly every maximal filter is prime. Although not difficult, the proof of the following lemma requires some knowledge of Universal Algebra; the reader can find a proof in [@pantigeneric Proposition 1.3].
The\[ref7\] following are equivalent:
1. ${\mathfrak{p}}$ is prime;
2. $A/{\mathfrak{p}}$ is totally-ordered;
3. ${\mathfrak{p}}=\varphi^{-1}[1]$, for some homomorphism $\varphi$ from $A$ to a totally-ordered algebra;
4. the set of filters $\supseteq{\mathfrak{p}}$ is totally-ordered by inclusion;
5. every filter $\supseteq{\mathfrak{p}}$ is prime;
6. if $a\lor b\in{\mathfrak{p}}$, then either $a\in{\mathfrak{p}}$ or $b\in{\mathfrak{p}}$.
Note that the only totally-ordered boolean algebra is ${\mathbf{2}}$ (if $a$ belongs to the totally-ordered boolean algebra $A$, then either $a\le\neg a$ or $\neg a\le a$, hence either $a\to\neg a=1$ or $\neg a\to a=1$; in the first case $a=0$, and in the second $a=1$), and therefore prime filters coincide with maximal ones. This simple fact distinguishes in a crucial way boolean algebras from MV-algebras and other algebras related to many-valued logics, as we will see later.
Let $A$ be an algebra, and let $\operatorname{Spec}A$ be the set of all prime filters of $A$. For every $a\in A$, let $O_a$ be the set of all ${\mathfrak{p}}\in\operatorname{Spec}A$ such that $a$ does not belong to ${\mathfrak{p}}$. Impose on $\operatorname{Spec}A$ the weakest topology in which all $O_a$’s are open (i.e., take the family of all $O_a$’s as an open subbasis). This is called the hull-kernel topology on $\operatorname{Spec}A$, and the resulting space is the [*spectral*]{} (or [*dual*]{}) [*space*]{} of $A$.
For every subset $D$ of $A$, let $O_D=\bigcup\{O_a:a\in D\}=
\{{\mathfrak{p}}:D\not\subseteq{\mathfrak{p}}\}$: it is an open set, and we will see in Theorem \[ref2\](ii) that every open set has this form. We write $F_a=(\operatorname{Spec}A)\setminus O_a$ and $F_D=(\operatorname{Spec}A)\setminus O_D$ for the corresponding closed sets.
The mapping $A\mapsto\operatorname{Spec}A$ is functorial. Indeed, let $\varphi:A\to B$ be any homomorphism, and define $\varphi^*:\operatorname{Spec}B\to\operatorname{Spec}A$ by $\varphi^*({\mathfrak{p}})=\varphi^{-1}[{\mathfrak{p}}]$. $\varphi^*({\mathfrak{p}})$ is a prime filter because, if ${\mathfrak{p}}=\psi^{-1}[1]$ for some homomorphism $\psi$ from $B$ to a totally-ordered algebra $C$, then $\varphi^*({\mathfrak{p}})$ is the kernel of $\psi\circ\varphi$, and the epimorphic image $(\psi\circ\varphi)[A]$ is totally-ordered, since it is a subalgebra of $C$. We have $(\varphi^*)^{-1}[O_D]=\{{\mathfrak{p}}\in\operatorname{Spec}B:\varphi^*({\mathfrak{p}})\in O_D\}=\{{\mathfrak{p}}:D\not\subseteq\varphi^{-1}[{\mathfrak{p}}]\}=\{{\mathfrak{p}}:\varphi[D]\not\subseteq{\mathfrak{p}}\}=O_{\varphi[D]}$, and hence $\varphi^*$ is continuous. Moreover, $(\psi\circ\varphi)^*=\varphi^*\circ\psi^*$, so $\operatorname{Spec}$ is a contravariant functor from ${\mathbf{V}}M$ (viewed as a category with the homomorphisms as arrows) to the category of topological spaces and continuous mappings.
Let\[ref2\] $A$ be an algebra.
- The open sets in $\operatorname{Spec}A$ are in 1–1 correspondence with the filters of $A$, and this correspondence is an isomorphism w.r.t. the $\subseteq$ relation.
- We have $O_a\cap O_b=O_{a\lor b}$ and $O_a\cup O_b=O_{a\land b}$. The defining subbasis is intersection-closed, and an open set is compact iff it is of the form $O_a$. $\operatorname{Spec}A$ is second countable iff $A$ is countable.
- $\operatorname{Spec}A$ is $T_0$, compact, and every closed irreducible set is the closure of a point.
The key point is that every filter ${\mathfrak{f}}$ is the intersection of all prime filters $\supseteq{\mathfrak{f}}$; this fact follows from a standard application of the Zorn Lemma. As a consequence, for every $D\subseteq A$, the intersection of all filters containing $D$ coincides with the intersection of all prime filters containing $D$. This intersection, namely $\bigcap F_D$, is the smallest filter containing $D$, and we denote it by ${\mathfrak{f}}(D)$. One verifies easily that ${\mathfrak{f}}(D)$ is the set of all $a\in A$ such that there exist ${a_1,\ldots ,a_r}\in D$ satisfying $a\ge a_1{\star}\cdots{\star}a_r$. Consider the mappings $$\begin{aligned}
A\supseteq D &\longmapsto F_D\in\text{sets closed in $\operatorname{Spec}A$} \\
\text{filters of $A$}\ni\textstyle{\bigcap} P
&\longleftarrow\!\mapstochar P\subseteq\operatorname{Spec}A\end{aligned}$$ They both reverse the $\subseteq$ relation. Their composition gives, on the left side, the mapping $D\mapsto \bigcap F_D={\mathfrak{f}}(D)$ that associates to a set the filter it generates, and on the right side the topological closure mapping $P\mapsto F_{\bigcap P}$ (Proof: ${\mathfrak{p}}$ belongs to the topological closure of $P$ iff $\forall a({\mathfrak{p}}\in O_a{\Rightarrow}O_a\cap P\not=\emptyset)$ iff $\forall a(P\subseteq F_a{\Rightarrow}a\in{\mathfrak{p}})$ iff ${\mathfrak{p}}\supseteq\bigcap P$). As a consequence, they induce an antiisomorphism between the lattice of filters of $A$ and the lattice of closed sets of $\operatorname{Spec}A$; this proves (i).
We leave (ii) as an exercise, and prove (iii). The $T_0$ property is clear, because the closure of ${\mathfrak{p}}$ is $F_{\bigcap\{{\mathfrak{p}}\}}=F_{\mathfrak{p}}=\{{\mathfrak{q}}:{\mathfrak{q}}\supseteq{\mathfrak{p}}\}$. Compactness follows from (ii) and the fact that $\operatorname{Spec}A=O_0$. Let $F_{\mathfrak{f}}$ be a closed irreducible set, i.e., a closed set that cannot be expressed nontrivially as the union of two closed sets; we must show that ${\mathfrak{f}}$ is prime, i.e., that $a\lor b\in{\mathfrak{f}}$ implies ($a\in{\mathfrak{f}}$ or $b\in{\mathfrak{f}}$). Assume $a\lor b\in{\mathfrak{f}}$; then $F_{a\lor b}\supseteq F_{\mathfrak{f}}$. Since $F_{a\lor b}=F_a\cup F_b$, we have $F_{\mathfrak{f}}=(F_{\mathfrak{f}}\cap F_a)\cup(F_{\mathfrak{f}}\cap F_b)$, which must be a trivial decomposition. Hence either $F_a\supseteq F_{\mathfrak{f}}$ or $F_b\supseteq F_{\mathfrak{f}}$, i.e., either $a\in{\mathfrak{f}}$ or $b\in{\mathfrak{f}}$.
A [*spectral space*]{} is a topological space in which the compact open sets form a basis closed under finite intersections, and such that the conditions in Theorem \[ref2\](iii) hold. By [@Hochster69], these are exactly the prime ideal spaces of commutative rings with $1$.
The spectral spaces of boolean algebras have a further property: the compact open sets are exactly the clopen sets (i.e., the sets which are both closed and open). Indeed, as we observed after Lemma \[ref7\], if $A\in{{\mathit{Boole}}}$ then every ${\mathfrak{p}}\in\operatorname{Spec}A$ is of the form ${\mathfrak{p}}=\varphi^{-1}[1]$ for some homomorphism $\varphi:A\to{\mathbf{2}}$. This immediately implies that $a\in{\mathfrak{p}}$ iff $\neg a\notin{\mathfrak{p}}$, i.e., $F_a=O_{\neg a}$. Therefore the $O_a$’s are clopen, and since every clopen is compact, there are no other clopens. Thus $A$ is in bijection with the clopen sets of $X=\operatorname{Spec}A$ via $a\mapsto F_a$, and since $F_{a\land b}=F_a\cap F_b$, $F_{\neg a}=X\setminus F_a$, $F_0=\emptyset$, and $F_1=X$, this bijection is a boolean algebra isomorphism. We have thus proved the [*Stone Representation Theorem*]{} [@Halmos63 §18]: every boolean algebra is isomorphic to the algebra of clopen subsets of its spectrum.
Spectral\[ref11\] spaces can be functorially introduced in any congruence-modular equational class. In general, filters should be substituted by congruences (unless the class turns out to be ideal-determined [@GummUrsini84]), and one introduces the notion of prime congruence by using the commutator product; the construction carries on smoothly [@agliano93]. The main trouble is with Theorem \[ref2\](i): the open sets of $\operatorname{Spec}A$ will now be in 1-1 correspondence only with the radical congruences of $A$, i.e., those congruences $\Phi$ such that, for every congruence $\Psi$, if $\Phi$ contains the commutator product of $\Psi$ with itself, then $\Phi$ already contains $\Psi$ (the spectrum of ${\mathbb{Z}}$ as a commutative ring is a typical example, the radical ideals being those generated by a squarefree integer). In our case there are no such problems, since equational classes are generated by truth-value algebras, in which a lattice structure is term-definable. All our classes are therefore congruence-distributive, and all congruences are radical.
Proofs and dynamics
===================
We can now make precise the heuristic in the Introduction about the dynamics in Frege proof systems.
A [*substitution*]{} on $FORM_\kappa$ is any mapping $\sigma:FORM_\kappa\to FORM_\kappa$ which distributes over the connectives (i.e., $\sigma(0)=0$, $\sigma(1)=1$, and $\sigma(r\circ s)=\sigma(r)\circ \sigma(s)$, for $\circ \in\{{\star},\to\}$). A substitution is therefore determined by an arbitrary assignment of formulas to propositional variables. Note that all variables must be substituted at the same time: e.g., if $r=x_1\to x_2$, $\sigma(x_1)=x_3{\star}x_2$, and $\sigma(x_2)=x_1$, then $\sigma(r)=(x_3{\star}x_2)\to x_1$.
Given a formula $r$ and a set of formulas $\Theta$, a [*deduction*]{} of $r$ from $\Theta$ is a finite sequence of formulas ${r_1,\ldots ,r_h}$ such that $r_h=r$ and for every $1\le j\le h$ we have:
- either $r_j\in\Theta$;
- or there exist $1\le k,m<j$ such that $r_m$ has the form $r_k\to r_j$ (we then say that $r_j$ follows from $r_k$ and $r_k\to r_j$ via Modus Ponens);
- or there exists $1\le k<j$ and a substitution $\sigma$ such that $r_j=\sigma(r_k)$.
An [*MP-deduction*]{} is a deduction in which the substitution rule (c) is never applied. For a fixed truth-value algebra $M$, it is often possible —although sometimes difficult— to find effectively a set of formulas $\Theta$ such that the formulas deducible from $\Theta$ are exactly the formulas which are true in $M$. If this happens, then we say that $\Theta$ provides an [*axiomatization*]{} of ${\mathbf{V}}M$.
Let $\Theta$ be the set of the following eight formulas: $$\begin{gathered}
\bigl((x_0\to x_1){\star}(x_1\to x_2)\bigr)\to(x_0\to x_2); \\
(x_0{\star}x_1)\to x_0; \quad
0\to x_0; \\
(x_0{\star}x_1)\to(x_1{\star}x_0); \quad
(x_0\land x_1)\to(x_1\land x_0); \\
\bigl(x_0\to(x_1\to x_2)\bigr)\to\bigl((x_0{\star}x_1)\to x_2\bigr);
\quad
\bigl((x_0{\star}x_1)\to x_2\bigr)\to\bigl(x_0\to(x_1\to x_2)\bigr); \\
\bigl[\bigl((x_0\to x_1)\to x_2\bigr){\star}\bigl((x_1\to x_0)\to x_2\bigr)\bigr]\to x_2.\end{gathered}$$ All of them —except perhaps the last one— have rather transparent meanings: the first one expresses transitivity of implication, the third is *ex falso quodlibet*, the fourth expresses commutativity of conjunction, and so on. We have [@hajek98]:
- $\Theta\cup\{\neg\neg x_0\to x_0\}$ axiomatizes ${{\mathit{MV}}}$;
- $\Theta\cup\{\neg\neg x_0\to\bigl((x_1{\star}x_0\to
x_2{\star}x_0)\to (x_1\to x_2)\bigr), \neg(x_0\land\neg x_0)\}$ axiomatizes the equational class generated by $M={[0,1]}$ endowed with the product connectives in Example \[ref3\](2);
- $\Theta\cup\{x_0\to(x_0{\star}x_0)\}$ axiomatizes the equational class generated by $M={[0,1]}$ endowed with the Gödel-Dummett connectives;
- $\Theta\cup\{x_0\lor\neg x_0\}$ axiomatizes ${{\mathit{Boole}}}$.
Fix a truth-value algebra $M$ and a cardinal $\kappa$. Let $\Theta_\kappa$ be the set of all formulas in $FORM_\kappa$ which are true in $M$. Hence $r(x_{i_1},\ldots,x_{i_n})\in\Theta_\kappa$ iff $r(a_{i_1},\ldots,a_{i_n})=1$ in ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$, where the $a_i$’s are the free generators. It is a customary abuse of notation to write $x_i$ for $a_i$, so the symbol $r$ may denote either an element of $FORM_\kappa$ or an element of ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$. This slight ambiguity is really sought for: it is exactly the ambiguity that results in working modulo $\Theta_\kappa$, or in identifying a formula with the function it induces on truth-values. Formally stated: $r,s\in FORM_\kappa$ are equal in ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$ iff $M\models r=s$ iff $r\to s,s\to r\in\Theta_\kappa$.
The set $\Theta_\omega$ is closed under Modus Ponens and substitution, so nothing new can be deduced from it. We let now $\Delta$ be any set of formulas, and raise two questions:
1. What can be deduced from $\Theta_\omega\cup\Delta$?
2. Which substitutions are needed for such a deduction?
If\[ref8\] $r$ is deducible from $\Theta_\omega\cup\Delta$, then there is a deduction involving only variables already appearing in $\{r\}\cup\Delta$.
By renaming variables we may assume that the variables appearing in $\{r\}\cup\Delta$ are exactly those variables with index $<\kappa$, for a certain $\kappa$. Let a deduction of $r$ from $\Theta_\omega\cup\Delta$ be given. By a standard argument [@church p. 149], we may transform the given deduction into an MP-deduction ${r_1,\ldots ,r_h}=r$ of $r$ from $\Theta_\omega\cup\{\sigma(t):\sigma\text{ is a substitution and }t\in\Delta\}$. Let $\tau$ be the substitution given by $\tau(x_j)=x_j$ if $j<\kappa$, and $\tau(x_j)=1$ otherwise. Then $\tau(r_1),\ldots,\tau(r_h)=r$ is an MP-deduction of $r$ from $\Theta_\kappa\cup\{\sigma(t):\sigma\text{ is a substitution, }t\in\Delta,\text{ and }\sigma(t)\in FORM_\kappa\}$, and hence a deduction of $r$ from $\Theta_\kappa\cup\Delta$ in which all formulas and all substitutions involve only variables with index $<\kappa$.
Identifying $FORM_\kappa$ with ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$, a substitution is nothing more than an endomorphism of ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$ (i.e., a homomorphism from ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$ to itself). The freeness of the generators says that however we choose elements $\{b_i:i<\kappa\}$ in ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$ there is precisely one endomorphism that maps $x_i$ to $b_i$.
We denote by $\Sigma_\kappa$ the monoid of all endomorphisms of ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$, and by $\Xi_\kappa\subseteq\Sigma_\kappa$ the group of all automorphisms of ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$ (i.e., of the invertible elements of $\Sigma_\kappa$). As explained before Theorem \[ref2\], to every $\sigma\in\Sigma_\kappa$ there corresponds a continuous selfmapping $\sigma^*$ of $X_\kappa=\operatorname{Spec}{\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$, which we call the [*dual*]{} of $\sigma$. Let $\Pi\subseteq\Sigma_\kappa$, and let $O$ be an open subset of $X_\kappa$. We define $(\Pi,O)$ to be the union of all backwards translates of $O$ under iteration of the substitutions in $\Pi$. Explicitly stated, $$(\Pi,O)=\bigcup\{(\sigma^*)^{-1}[O]:\sigma\text{ is in the submonoid of }\Sigma_\kappa\text{ generated by }\Pi\}.$$ If $(\Pi,O)=X_\kappa$ for every $O\not=\emptyset$, then we say that $\Pi$ acts [*minimally*]{} on $X_\kappa$: this is equivalent to saying that every point of $X_\kappa$ has a dense orbit under $\Pi$.
Let\[ref9\] $\{r\}\cup\Delta\subseteq FORM_\kappa$. Then:
- $r$ can be MP-deduced from $\Theta_\omega\cup\Delta$ iff $O_r\subseteq O_\Delta$ in $X_\kappa$;
- if $\Delta'\subseteq FORM_\kappa$ is another set of formulas, then $O_\Delta=O_{\Delta'}$ iff $\Theta_\omega\cup\Delta$ and $\Theta_\omega\cup\Delta'$ MP-deduce the same formulas;
- $r$ can be deduced from $\Theta_\omega\cup\Delta$ iff $O_r\subseteq(\Sigma_\kappa,O_\Delta)$.
\(i) follows from Theorem \[ref2\](i) and the proof of Lemma \[ref8\]: $r$ can be MP-deduced from $\Theta_\omega\cup\Delta$ iff $r$ can be MP-deduced from $\Theta_\kappa\cup\Delta$ iff $r$ belongs to the filter ${\mathfrak{f}}(\Delta)$ in ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$ iff ${\mathfrak{f}}(r)\subseteq{\mathfrak{f}}(\Delta)$ iff $O_r=O_{{\mathfrak{f}}(r)}\subseteq O_{{\mathfrak{f}}(\Delta)}=O_\Delta$. (ii) is clear: the sets $\Theta_\omega\cup\Delta$ and $\Theta_\omega\cup\Delta'$ MP-deduce the same formulas iff ${\mathfrak{f}}(\Delta)={\mathfrak{f}}(\Delta')$ iff $O_\Delta=O_{{\mathfrak{f}}(\Delta)}=O_{{\mathfrak{f}}(\Delta')}=O_{\Delta'}$. We prove (iii): assume that $r$ can be deduced from $\Theta_\omega\cup\Delta$. By Lemma \[ref8\], there exists a deduction ${r_1,\ldots ,r_h}=r$ such that every $r_j$ is in $\Theta_\kappa\cup\Delta$ and all substitutions applied are in $\Sigma_\kappa$. Working by induction on $h$ we assume that $$O_{r_1}\cup\cdots\cup O_{r_{h-1}}\subseteq (\Sigma_\kappa,O_\Delta).$$ If $r_h\in\Delta$, then of course we are through (note that $s\in\Theta_\kappa$ is equivalent to $O_s=\emptyset$). For every $s,t\in{\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$ we have $O_t\subseteq O_s\cup O_{s\to t}$. Indeed, if ${\mathfrak{p}}\notin O_s$ and ${\mathfrak{p}}\notin O_{s\to t}$, then $s,s\to t\in{\mathfrak{p}}$. Since filters are closed under MP, we have $t\in{\mathfrak{p}}$ and ${\mathfrak{p}}\notin O_t$. Therefore, if $r_h$ has been obtained via MP, then the induction hypothesis guarantees that $O_{r_h}\subseteq(\Sigma_\kappa,O_\Delta)$. Finally, if $r_h=\sigma(r_j)$ for some $\sigma\in\Sigma_\kappa$ and $1\le j<h$, then $O_{r_h}=O_{\sigma(r_j)}=
(\sigma^*)^{-1}[O_{r_j}]\subseteq(\sigma^*)^{-1}[(\Sigma_\kappa,
O_\Delta)]\subseteq (\Sigma_\kappa,O_\Delta)$. We leave the reverse implication as an exercise for the reader (Hint: $O_r$ is compact).
We say that $\Theta\subseteq FORM_\omega$ is [*equationally complete*]{} if, however we choose $r,s\in FORM_\omega$ with $r\notin\Theta$, the formula $s$ is deducible from $\Theta\cup\{r\}$.
The\[ref10\] set $\Theta_\omega$ is equationally complete iff $\Sigma_\omega$ acts minimally on $X_\omega$.
Let $O$ be a nonvoid open subset of $X_\omega$, and assume that $\Theta_\omega$ is equationally complete: we want to show that $(\Sigma_\omega,O)=X_\omega$. Let $r\in{\operatorname{Free}_{\omega}({\mathbf{V}}M)}$ be such that $\emptyset\not=O_r\subseteq O$; we then have $r\notin\Theta_\omega$. By assumption, the formula $0$ is deducible from $\Theta_\omega\cup\{r\}$, and hence we get $X_\omega=O_0\subseteq(\Sigma_\omega,O_r)\subseteq(\Sigma_\omega,O)$ from Lemma \[ref9\](iii). The same argument yields the reverse implication.
The equational completeness of $\Theta_\omega$ amounts to the lack of nontrivial equational subclasses of ${\mathbf{V}}M$. Among classes generated by truth-value algebras, the only one fulfilling this property is ${{\mathit{Boole}}}$ (easy proof, resting on the fact that ${\mathbf{2}}$ is a subalgebra of any $M$). Theorem \[ref10\] then implies that $\Sigma_\omega$ acts minimally on $X_\omega$ only in the case of boolean algebras. There are other cases in which an algebra $M$ (not a truth-value algebra in our sense) generates a congruence-distributive equational class having no nontrivial subclasses. A particularly interesting case is when $M$ is the set of integers equipped with its natural structure $({\mathbb{Z}},+,-,0,\lor,\land)$ of lattice-ordered group [@bkw], [@andersonfei]. The resulting equational class is the class of all lattice-ordered groups, and the above properties are fulfilled. Theorem \[ref10\] says then that the endomorphisms of the free lattice-ordered groups act minimally on the relative spectra: see [@pantiprime] for a description of such spaces.
For every $p=(\ldots,p_i,\ldots)\in M^\kappa$, the evaluation mapping at $p$, given by $r(x_{i_1},\ldots,x_{i_n})\mapsto
r(p_{i_1},\ldots,p_{i_n})$, is a homomorphisms $\varphi:{\operatorname{Free}_{\kappa}({\mathbf{V}}M)}\to M$. Since $M$ is totally-ordered, the kernel $\varphi^{-1}[1]$ is a prime filter ${\mathfrak{p}}$, hence an element of $X_\kappa$. We thus get a mapping $p\mapsto{\mathfrak{p}}$ from $M^\kappa$ to $X_\kappa$, which we denote by $\pi$. The map $\pi$ has dense range ($\emptyset\not= O_r$ ${\Rightarrow}$ $r\notin\Theta_\kappa$ ${\Rightarrow}$ $\exists p\in
M^\kappa\;r(p)\not=1$ ${\Rightarrow}$ $\exists p\; r\notin\pi(p)$ ${\Rightarrow}$ $\exists p\;\pi(p)\in O_r$), but it is not necessarily continuous; it is continuous in the two cases that most concern us, namely in classical logic (see Lemma \[ref25\]) and in [Łukasiewicz]{} logic (see the next section).
Let $\sigma:{\operatorname{Free}_{\kappa}({\mathbf{V}}M)}\to{\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$ be a substitution, $s_i=\sigma(x_i)$. Then the $\kappa$-tuple $(\ldots,s_i,\ldots)$ determines a function $S:M^\kappa\to M^\kappa$ via $p\mapsto(\ldots,s_i(p),\ldots)$. Since $\pi(S(p))=\{r\in{\operatorname{Free}_{\kappa}({\mathbf{V}}M)}:
r(\ldots,s_i(p),\ldots)=1\}=\{r:[\sigma(r)](p)=1\}=
\sigma^{-1}(\{t:t(p)=1\})=\sigma^*(\pi(p))$, the diagram $$\tag{$*$}
\begin{split}
\begin{xy}
\xymatrix{
{M^\kappa} \ar[r]^S \ar[d]_{\pi} & {M^\kappa} \ar[d]^{\pi} \\
{X_\kappa} \ar[r]_{\sigma^*} & {X_\kappa}
}
\end{xy}
\end{split}$$ commutes. As $\pi$ has dense range, $S$ determines the continuous function $\sigma^*$. We call $\sigma^*$ the [*dual*]{} of $\sigma$, and $S$ the [*mapping on truth-values*]{} induced by $\sigma$.
In\[ref25\] classical logic, the map $\pi:{\mathbf{2}}^\kappa
\to X_\kappa=\operatorname{Spec}{\operatorname{Free}_{\kappa}({{\mathit{Boole}}})}$ is a homeomorphism.
Recall from the end of Section \[ref24\] that ${\operatorname{Free}_{\kappa}({{\mathit{Boole}}})}$ is the boolean algebra of all clopen subsets of ${\mathbf{2}}^\kappa$, where the latter space is given the product topology. If $p$ and $q$ are distinct points of ${\mathbf{2}}^\kappa$, then there exists a clopen set $C$ containing $p$ and not containing $q$. Hence $C\in\pi(p)\setminus\pi(q)$ and $\pi$ is injective (as usual, we are identifying clopen subsets with their characteristic functions). Let ${\mathfrak{p}}\in X_\kappa$. Since ${\mathfrak{p}}$ is a proper filter, $\emptyset\notin{\mathfrak{p}}$ and in particular the intersection of any finite family of elements of ${\mathfrak{p}}$ is nonempty. By compactness, $\bigcap{\mathfrak{p}}$ contains a point $p\in{\mathbf{2}}^\kappa$. Since $C\in{\mathfrak{p}}$ implies $p\in C$, we have ${\mathfrak{p}}\subseteq\pi(p)$. But in a boolean algebra every prime filter is maximal, and hence ${\mathfrak{p}}=\pi(p)$. So $\pi$ is a bijection. Both in ${\mathbf{2}}^\kappa$ and in $X_\kappa$ the clopen sets generate the topology; moreover, as shown before Addendum \[ref11\], the mapping $C\mapsto F_C$ is a bijection between the two families of clopen sets. Since $p\in C$ iff $C\in\pi(p)$ iff $\pi(p)\in F_C$, the map $\pi$ is a homeomorphism.
Given a point $p$ and a nonvoid open subset $O$ of the Cantor space ${\mathbf{2}}^\omega$, one easily constructs a homeomorphism $S:{\mathbf{2}}^\omega\to{\mathbf{2}}^\omega$ such that $S(p)\in O$ (if $p=(p_0,p_1,\ldots)$ and $[{a_0,\ldots ,a_n}]=\{q\in{\mathbf{2}}^\omega:q_i=a_i\text{ for }i=0,\ldots,n\}$ is a block contained in $O$, then the mapping that exchanges $0$ with $1$ in those indices $i$ for which $p_i\not=a_i$ is such a homeomorphism). Hence not only $\Sigma_\omega$, but even $\Xi_\omega$ acts minimally on $\operatorname{Spec}{\operatorname{Free}_{\omega}({{\mathit{Boole}}})}$.
Of course we can do better than that, because there exist many minimal homeomorphisms of the Cantor space. The simplest example is obtained by identifying ${\mathbf{2}}^\omega$ with the topological group of $2$-adic integers ${\mathbb{Z}}_2$, and letting $S$ be the translation by $1$: $S(p)=p+1$. Let us compute the substitution $\sigma$ on ${\operatorname{Free}_{\omega}({{\mathit{Boole}}})}$ for which $S=\sigma^*$. If $x_i$ is the $i$-th free generator, then $F_{x_i}=\{p\in{\mathbf{2}}^\omega:p_i=1\}$, and $F_{\sigma(x_i)}=(\sigma^*)^{-1}[F_{x_i}]=\{p:S(p)\in F_{x_i}\}=\{p:(p+1)_i=1\}$. Since addition in ${\mathbb{Z}}_2$ is just addition in base $2$ with carry, we have that $p\in F_{\sigma(x_i)}$ iff
- either $p_i=1$ and $p_j=0$ for some $j<i$;
- or $p_i=0$ and $p_j=1$ for every $j<i$.
Therefore $$\begin{aligned}
F_{\sigma(x_i)} &=
\bigl[F_{x_i}\cap(O_{x_0}\cup\cdots\cup O_{x_{i-1}})\bigr] \\
&\quad \cup\bigl[O_{x_i}\cap F_{x_0}\cap\cdots\cap F_{x_{i-1}}\bigr] \\
&= \bigl[F_{x_i}\cap(F_{\neg x_0}\cup\cdots\cup
F_{\neg x_{i-1}})\bigr] \\
&\quad \cup\bigl[F_{\neg x_i}\cap F_{x_0}\cap\cdots\cap
F_{x_{i-1}}\bigr].\end{aligned}$$ Consider the following formulas: $$\begin{aligned}
s_0 &= \neg x_0 \\
s_i &= \bigl[x_i\land(\neg x_0\lor\cdots\lor\neg x_{i-1})\bigr]
\lor\bigl[\neg x_i\land x_0\land\cdots\land x_{i-1}\bigr] \\
&= x_i\triangle(x_0\land\cdots\land x_{i-1})
\quad\text{(for $i>0$)}\end{aligned}$$ ($\triangle$ is the boolean symmetric difference: $a\triangle b=(a\land\neg b)\lor(\neg a\land b)$). Then $F_{\sigma(x_i)}=F_{s_i}$ by the isomorphism cited before Addendum \[ref11\]. The required substitution is therefore the one defined by $\sigma(x_i)=s_i$. From the point of view of proof systems, we have thus obtained the following result.
From\[ref14\] the set of boolean tautologies plus any given non-tautology we can derive every formula using only Modus Ponens and the substitution $\sigma$ given above.
[Łukasiewicz]{} logic
=====================
In the rest of this paper we will concentrate on [Łukasiewicz]{} logic; we therefore fix $M={[0,1]}$ endowed with the connectives in Example \[ref3\](3). A key distinguishing feature of the [Łukasiewicz]{} connectives is their continuity with respect to the standard topology of ${[0,1]}$. As a matter of fact [Łukasiewicz]{} logic is the only t-norm based logic in which all connectives are continuous [@menupav].
A [*\[rational\] cellular complex over $M^n={[0,1]}^n$*]{} is a finite set $W$ of [*cells*]{} (i.e., compact convex polyhedrons), whose union is ${[0,1]}^n$, and such that:
1. every vertex of every cell of $W$ has rational coordinates;
2. if $C\in W$ and $D$ is a face of $C$, then $D\in W$;
3. every two cells intersect in a common face.
A [*McNaughton function*]{} is a continuous function $f:{[0,1]}^n\to{[0,1]}$ for which there exists a complex as above and affine linear functions with integer coefficients $F_j(\bar x)=a_j^1x_1+\cdots+a_j^nx_n+a_j^{n+1}$, in 1-1 correspondence with the $n$-dimensional cells $C_j$ of the complex, such that $f\restriction C_j=F_j$ for each $j$.
The\[ref15\] elements of ${\operatorname{Free}_{n}({{\mathit{MV}}})}$ (i.e., the functions from ${[0,1]}^n$ to ${[0,1]}$ induced by a formula of [Łukasiewicz]{} logic) are exactly the McNaughton functions.
Here are typical McNaughton functions, for $n=1$ and $n=2$: they are induced by the formulas $\neg x_0\lor\bigl((x_0\land\neg x_0)\oplus
(x_0\land\neg x_0)\bigr)$ and $(x_0\to x_1)\land(x_0\oplus x_0\oplus x_1\oplus x_1)$, respectively.
{width="3cm" height="3cm"} {width="3cm" height="3cm"}
Given a substitution $\sigma:{\operatorname{Free}_{n}({{\mathit{MV}}})}\to{\operatorname{Free}_{n}({{\mathit{MV}}})}$, to each function $s_i=\sigma(x_i)$ there corresponds a cellular complex $W_i$ such that $s_i$ is affine linear on each cell of $W_i$. Let $W$ be a complex that is a common refinement of ${W_0,\ldots ,W_{n-1}}$. Then on each cell $C_j$ of $W$ the function $S:{[0,1]}^n\to{[0,1]}^n$ defined before Lemma \[ref25\] is given by $$\tag{$**$}
\begin{pmatrix}
p_0 \\
\vdots \\
p_{n-1}
\end{pmatrix}
\mapsto
A_j\begin{pmatrix}
p_0 \\
\vdots \\
p_{n-1}
\end{pmatrix}
+B_j,$$ where $A_j$ is an $n\times n$ matrix and $B_j$ a column vector, both having integer coefficients. Conversely, every continuous selfmapping $S$ of ${[0,1]}^n$ which is piecewise affine linear with integer coefficients (i.e., is locally expressible in the form $(**)$, using finitely many $A_j$’s and $B_j$’s) is induced by some endomorphism $\sigma$ of ${\operatorname{Free}_{n}({{\mathit{MV}}})}$. We call such an $S$ a [*McNaughton mapping*]{}; if moreover $S$ is invertible we call it a [*McNaughton homeomorphism*]{}; McNaughton homeomorphisms on ${[0,1]}^n$ are exactly the mappings on truth-values induced by the automorphisms of ${\operatorname{Free}_{n}({{\mathit{MV}}})}$. Apart from its relevance in [Łukasiewicz]{} logic, the class of McNaughton mappings is quite interesting *per se*.
Consider the following complexes over ${[0,1]}^2$; they are both symmetric under a $\pi$ rotation about the centre of the square.
{height="3cm" width="3cm"} {height="3cm" width="3cm"}
The vertices of the lower inner triangle are $p_0=(1/4,1/4)$, $p_1=(1/2,1/4)$, $p_2=(1/4,1/2)$; for $0\le i\le 2$, let $p_i'$ be the vertex symmetric to $p_i$. Then there exists a unique homeomorphism $S$ such that:
1. $S(p_i)=p_{i+1\pmod{3}}$, and $S(p'_i)=p'_{i+1\pmod{3}}$;
2. every other vertex is fixed;
3. $S$ is affine linear on each cell.
In short, the first complex is mapped onto the second by “rotating counterclockwise” the two inner triangles, and distorting accordingly the border triangles. As a matter of fact, $S$ is topologically conjugate to the union of two twists [@pantibernoulli §5]. The data above determine the matrix $A_j$ and the column vector $B_j$ on each triangle $C_j$. One checks directly that all these matrices and vectors have integer entries; hence $S$ is a McNaughton homeomorphism. In doing computations, it is expedient to write $p=({p_0,\ldots ,p_{n-1}})\in{[0,1]}^n$ using projective coordinates $({a_0:\ldots :a_n})\sim({p_0:\ldots :p_{n-1}}:1)$. For example, if $C_1$ is the triangle ${\langle p_0,(1,0),p_1 \rangle}$, which is mapped to ${\langle p_1,(1,0),p_2 \rangle}$, then $A_1$ and $B_1$ are the upper left $2\times 2$ matrix and upper right $2\times 1$ column vector in the matrix $$\begin{pmatrix}
-1 & -5 & 2 \\
1 & 4 & -1 \\
0 & 0 & 1
\end{pmatrix}
=
\begin{pmatrix}
2 & 1 & 1 \\
1 & 0 & 2 \\
4 & 1 & 4
\end{pmatrix}
\begin{pmatrix}
1 & 1 & 2 \\
1 & 0 & 1 \\
4 & 1 & 4
\end{pmatrix}^{-1}.$$
If $S$ is induced by $\sigma$, then the action of $S$ on ${[0,1]}^n$ is just the surface of the action of the full dual $\sigma^*$ on $X_n$. Indeed, as proved in [@mundicijfa Proposition 8.1], in the case of [Łukasiewicz]{} logic the map $\pi$ in the diagram $(*)$ is a homeomorphic embedding of ${[0,1]}^n$ onto the subspace of maximal filters.
By Lemma \[ref7\], the points of $X_n$ (indeed, of any spectrum) form a forest under the [*specialization order*]{}: ${\mathfrak{p}}\le{\mathfrak{q}}$ iff ${\mathfrak{q}}$ is in the closure of $\{{\mathfrak{p}}\}$ iff ${\mathfrak{p}}\subseteq{\mathfrak{q}}$ (a [*forest*]{}, sometimes called a [*root system*]{}, is a poset in which the elements greater than any given element form a chain).
A full description of $X_n$ is given in [@pantiprime]. Since it is rather involved, here we limit ourselves to the cases $n=1$ and $n=2$.
Assume $n=1$. If $p\in(0,1)$ is rational, then there are two incomparable prime filters $\pi(p)^+$ and $\pi(p)^-$ properly contained in the maximal $\pi(p)$. Namely, $\pi(p)^-$ is the filter of all McNaughton functions $:{[0,1]}\to{[0,1]}$ that are $1$ in a left neighborhood of $p$, and analogously for $\pi(p)^+$ w.r.t. right neighborhoods. The only prime filter contained in $\pi(0)$ (respectively, in $\pi(1)$) is $\pi(0)^+$ (respectively, $\pi(1)^-$). If $p$ is irrational, then $\pi(p)$ is minimal in the specialization order.
Now assume $n=2$, $p=(p_0,p_1)\in(0,1)^2$. If $p_0,p_1,1$ are linearly independent over ${\mathbb{Q}}$, then $\pi(p)$ is a minimal —as well as maximal— prime filter. If $p_0,p_1,1$ satisfy exactly one (up to scalar multiples) nontrivial linear dependence over ${\mathbb{Q}}$, then there are two incomparable prime filters below $\pi(p)$, and both of them are minimal. Otherwise, consider the unit circle $S^1$ in the tangent space to $p$. For every $u\in S^1$ there is a prime filter ${\mathfrak{p}}_u$ contained in $\pi(p)$. If the line in the tangent space connecting the origin with $u$ does not hit any point with rational coordinates, then ${\mathfrak{p}}_u$ is minimal. Otherwise, ${\mathfrak{p}}_u$ contains two minimal prime filters ${\mathfrak{p}}_u^+$ and ${\mathfrak{p}}_u^-$. For $p$ along the border of the unit square, this description gets modified in the obvious way. The following picture may clarify the situation:
{width="4cm" height="2cm"}
Let $n$ (not necessarily $n=1,2$) be given, ${\mathfrak{p}}\in X_n$. Let ${\mathfrak{p}}={\mathfrak{p}}_0\subset {\mathfrak{p}}_1\subset\cdots
\subset{\mathfrak{p}}_t$ be the chain, of length $t$, of elements above ${\mathfrak{p}}$ in the specialization order. Given an endomorphism $\sigma$ of ${\operatorname{Free}_{n}({{\mathit{MV}}})}$, we have $$\frac{{\operatorname{Free}_{n}({{\mathit{MV}}})}}{\sigma^{-1}[{\mathfrak{p}}]} \simeq
\frac{\sigma[{\operatorname{Free}_{n}({{\mathit{MV}}})}]}{{\mathfrak{p}}\cap\sigma[{\operatorname{Free}_{n}({{\mathit{MV}}})}]} \subseteq
\frac{{\operatorname{Free}_{n}({{\mathit{MV}}})}}{{\mathfrak{p}}},$$ and hence the MV-algebra ${\operatorname{Free}_{n}({{\mathit{MV}}})}/\sigma^*({\mathfrak{p}})$ is a subalgebra of ${\operatorname{Free}_{n}({{\mathit{MV}}})}/{\mathfrak{p}}$. By [@pantiprime Theorem 4.7(i) and Corollary 4.9], this implies that the length of the chain above $\sigma^*({\mathfrak{p}})$ is less than or equal to $t$.
Since ${[0,1]}^n$ is dense in $X_n$, in principle we might reduce the study of $\sigma^*$ to the study of $S$. However, taking into consideration the action of $\sigma^*$ on the full spectrum gives us deeper insight. For example, it is possible to provide an intrinsic (i.e., coordinate-free) characterization of the differentials $T_pS$ of a McNaughton mapping $S$. Differentials of piecewise-linear maps have been constructed by Tsujii in [@Tsujii01]; we show here how Tsujii’s construction can be intrinsically described in purely algebraic terms.
It may be helpful for the reader to recall the coordinate-free description of the differentials of a morphism $S:X\to Y$ of differentiable varieties. Let $p$ be a point of $X$, $q=S(p)$, ${\mathcal{O}}_p$ and ${\mathcal{O}}_q$ the rings of germs of differentiable functions at $p$ and $q$, respectively. ${\mathcal{O}}_p$ and ${\mathcal{O}}_q$ are local rings: let ${\mathfrak{m}}$ and ${\mathfrak{n}}$ be the respective maximal ideals (i.e., ${\mathfrak{m}}=\{f\in{\mathcal{O}}_p:f(p)=0\}$, and analogously for ${\mathfrak{n}}$). The mapping $\sigma:{\mathcal{O}}_q\to{\mathcal{O}}_p$ defined by $\sigma(g)=g\circ S$ is a well-defined ring homomorphism, and $\sigma[{\mathfrak{n}}]\subseteq{\mathfrak{m}}$. Therefore, $\sigma$ induces a vector space homomorphism from ${\mathfrak{n}}/{\mathfrak{n}}^2$ to ${\mathfrak{m}}/{\mathfrak{m}}^2$, which we denote by $\bar{\sigma}$. The tangent spaces $T_pX$ and $T_qY$ are canonically isomorphic to the dual vector spaces $({\mathfrak{m}}/{\mathfrak{m}}^2)'$ and $({\mathfrak{n}}/{\mathfrak{n}}^2)'$, respectively, and under these isomorphisms the differential $T_pS$ corresponds to the dual mapping $\bar{\sigma}':({\mathfrak{m}}/{\mathfrak{m}}^2)'\to({\mathfrak{n}}/{\mathfrak{n}}^2)'$. Explicitly, if $T_pX\ni v:{\mathfrak{m}}/{\mathfrak{m}}^2\to{\mathbb{R}}$ is a tangent vector at $p$, then $(T_pS)(v)$ is the tangent vector at $q$ defined by $[(T_pS)(v)](g/{\mathfrak{n}}^2)=(v\circ\bar{\sigma})(g/{\mathfrak{n}}^2)=
v((g\circ S)/{\mathfrak{m}}^2)$.
We will develop an analogous description for piecewise-linear maps. Before doing so, we need a few more preliminaries; see [@bkw], [@andersonfei] for more details and unproved claims. A [*lattice-ordered abelian group*]{} ([*[$\ell$-group]{}*]{} for short) is a structure $(G,+,-,0,\land,\lor)$ such that $(G,+,-,0)$ is an abelian group, $(G,\land,\lor)$ is a lattice, and $+$ distributes over the lattice operations. [$\ell$-homomorphisms]{} of [$\ell$-groups]{} are groups homomorphisms that are also lattice homomorphisms. The class of all [$\ell$-groups]{} is equational and, as such, contains free objects. The free [$\ell$-group]{} over $n$ generators, $\operatorname{F\ell}(n)$, is the [$\ell$-group]{} (under pointwise operations) of all functions $g:{\mathbb{R}}^n\to{\mathbb{R}}$ that are continuous and piecewise-linear with integer coefficients (i.e., there exist finitely many homogeneous linear polynomials ${g_1,\ldots ,g_m}\in{\mathbb{Z}}[{x_1,\ldots ,x_n}]$ such that, for every $w\in{\mathbb{R}}^n$, $g(w)=g_j(w)$ for some $1\le j\le m$). The set $\operatorname{\ell-Hom}(\operatorname{F\ell}(n),{\mathbb{R}})$ of all [$\ell$-homomorphisms]{} from $\operatorname{F\ell}(n)$ to ${\mathbb{R}}$ is in 1-1 correspondence with ${\mathbb{R}}^n$, via the map that associates to $w\in{\mathbb{R}}^n$ the evaluation mapping $\varphi_w:g\mapsto g(w)$. A [*strong unit*]{} of the [$\ell$-group]{} $G$ is an element $0\le u\in G$ such that, for every $g\in G$, $g\le nu$ for some positive integer $n$. If $u$ is a strong unit of $G$, then the interval $[0,u]=
\{g\in G:0\le g\le u\}$ can be given the structure of an MV-algebra $\Gamma(G,u)=([0,u],\oplus,\neg,0,1)$ by setting $g\oplus h=(g+h)\land u$, $\neg g=u-g$, $0=0_G$, $1=u$. The mapping $(G,u)\mapsto\Gamma(G,u)$ is functorial, and determines a categorical equivalence between the category of [$\ell$-groups]{} with strong unit and the category of MV-algebras [@mundicijfa]. In particular, the filters of $\Gamma(G,u)$ are in natural 1-1 correspondence with the kernels of [$\ell$-homomorphisms]{} of domain $G$.
The preliminaries being over, let $S:{[0,1]}^n\to{[0,1]}^n$ be a McNaughton mapping, $p\in{[0,1]}^n$, $q=S(p)$. Then ${\mathfrak{m}}=\pi(p)$ and ${\mathfrak{n}}=\pi(q)$ are maximal filters of ${\operatorname{Free}_{n}({{\mathit{MV}}})}$. Given ${\mathfrak{p}}\in X_n$, the [*germinal filter*]{} corresponding to ${\mathfrak{p}}$ is the filter ${\mathfrak{g}}_{\mathfrak{p}}=\bigcap\{{\mathfrak{q}}\in X_n:{\mathfrak{q}}\subseteq{\mathfrak{p}}\}$. By [@bkw Proposition 10.5.3 and Definition 10.5.6], and using the properties of the $\Gamma$ functor, the quotient $A_p={\operatorname{Free}_{n}({{\mathit{MV}}})}/{\mathfrak{g}}_{\mathfrak{m}}$ is the MV-algebra of germs at $p$ of McNaughton functions; analogously for $A_q={\operatorname{Free}_{n}({{\mathit{MV}}})}/{\mathfrak{g}}_{\mathfrak{n}}$. The MV-algebra $A_p$ is [*local*]{}, i.e., has a unique maximal filter ${\mathfrak{m}}/{\mathfrak{g}}_{\mathfrak{m}}$. Let $\sigma$ be the endomorphism of ${\operatorname{Free}_{n}({{\mathit{MV}}})}$ that induces $S$.
Notation being as above, $\sigma[{\mathfrak{n}}]\subseteq{\mathfrak{m}}$ and $\sigma[{\mathfrak{g}}_{\mathfrak{n}}]\subseteq{\mathfrak{g}}_{\mathfrak{m}}$.
By the commutativity of the diagram $(*)$, $\sigma^{-1}[{\mathfrak{m}}]=
\sigma^*({\mathfrak{m}})={\mathfrak{n}}$, so the first statement is immediate. Let ${\mathfrak{p}}$ be a prime filter below ${\mathfrak{m}}$ in the specialization order. Since $\sigma^*$ is continuous and ${\mathfrak{m}}$ is in the closure of $\{{\mathfrak{p}}\}$, the maximal filter ${\mathfrak{n}}=\sigma^*({\mathfrak{m}})$ must be in the closure of $\{\sigma^*({\mathfrak{p}})\}$. Therefore $\sigma^*({\mathfrak{p}})$ is below ${\mathfrak{n}}$, hence ${\mathfrak{g}}_{\mathfrak{n}}\subseteq\sigma^*({\mathfrak{p}})=
\sigma^{-1}[{\mathfrak{p}}]$ and $\sigma[{\mathfrak{g}}_{\mathfrak{n}}]\subseteq{\mathfrak{p}}$.
As a consequence, $\sigma$ determines a homomorphism of [$\ell$-groups]{}$$\bar{\sigma}:{\mathfrak{n}}/{\mathfrak{g}}_{\mathfrak{n}}\to {\mathfrak{m}}/{\mathfrak{g}}_{\mathfrak{m}},$$ which plays the rôle of the codifferential on cotangent spaces. Since the composition of [$\ell$-homomorphisms]{} is an [$\ell$-homomorphism]{}, $\bar{\sigma}$ induces a dual mapping $\bar{\sigma}':\operatorname{\ell-Hom}({\mathfrak{m}}/{\mathfrak{g}}_{\mathfrak{m}},{\mathbb{R}})\to
\operatorname{\ell-Hom}({\mathfrak{n}}/{\mathfrak{g}}_{\mathfrak{n}},{\mathbb{R}})$ by $\bar{\sigma}'(\varphi)=\varphi\circ\bar{\sigma}$.
Under\[ref22\] the identification $w\mapsto\varphi_w$ of ${\mathbb{R}}^n$ with $\operatorname{\ell-Hom}(\operatorname{F\ell}(n),{\mathbb{R}})$ described above, the map $\bar{\sigma}'$ corresponds to Tsujii’s differential.
For simplicity’s sake, we assume that $p$ and $q=S(p)$ have rational coordinates and are in the topological interior of the $n$-cube: we will discuss in Addendum \[ref21\] how these assumptions can be discarded. Write $q$ in projective coordinates $({a_0:\ldots :a_n})$, with $a_n>0$, and let $Q=({a_0,\ldots ,a_n})\in{\mathbb{R}}^{n+1}$. Let ${\mathfrak{N}}$ be the kernel of $\varphi_Q$, and let ${\mathfrak{G}}_{\mathfrak{N}}$ be the germinal kernel associated to ${\mathfrak{N}}$ [@bkw Proposition 10.5.3]. Since $q$ has rational coordinates, $Q$ has rank $1$ according to [@pantiprime p. 188]. By [@pantiprime Theorem 4.8], the quotient ${\mathfrak{N}}/{\mathfrak{G}}_{\mathfrak{N}}$ is an [$\ell$-group]{}, which is [$\ell$-isomorphic]{} to $\operatorname{F\ell}(n)$ under the map $D_Q$ defined in [@pantiprime Definition 2.2]. By the properties of the $\Gamma$ functor, the [$\ell$-groups]{} ${\mathfrak{n}}/{\mathfrak{g}}_{\mathfrak{n}}$ and ${\mathfrak{N}}/{\mathfrak{G}}_{\mathfrak{N}}$ are [$\ell$-isomorphic]{} as well. We therefore obtain an [$\ell$-isomorphism]{} $D_q:{\mathfrak{n}}/{\mathfrak{g}}_{\mathfrak{n}}\to
\operatorname{F\ell}(n)$ which, by explicit computation, has the form $$\bigl[D_q(r/{\mathfrak{g}}_{\mathfrak{n}})\bigr](w)=
\lim_{h\to0^+}\frac{r(q+hw)-r(q)}{h}.$$ We have of course an analogous [$\ell$-isomorphism]{} $D_p:{\mathfrak{m}}/{\mathfrak{g}}_{\mathfrak{m}}\to\operatorname{F\ell}(n)$. Let $\mathcal{D}_pS$ denote the Tsujii differential of $S$ at $p$, and let $v\in{\mathbb{R}}^n$. Since $S$ is continuous and defined everywhere on ${[0,1]}^n$, the definition in [@Tsujii01 Eq. (13)] simplifies to $$(\mathcal{D}_pS)(v)=
\lim_{h\to0^+}\frac{S(p+hv)-S(p)}{h}.$$ We want to show that $\bar{\sigma}'(\varphi_v\circ D_p)=
\varphi_{(\mathcal{D}_pS)(v)}\circ D_q$. Setting $(\mathcal{D}_pS)(v)=w\in{\mathbb{R}}^n$, this amounts to the commutativity of the diagram $$\begin{xy}
\xymatrix{
{{\mathfrak{n}}/{\mathfrak{g}}_{\mathfrak{n}}} \ar[0,2]^{\bar{\sigma}} \ar[d]_{D_q} & &
{{\mathfrak{m}}/{\mathfrak{g}}_{\mathfrak{m}}} \ar[d]^{D_p} \\
{\operatorname{F\ell}(n)} \ar[r]_{\varphi_w} & {{\mathbb{R}}} &
{\operatorname{F\ell}(n)} \ar[l]^{\varphi_v}
}
\end{xy}$$ Choose $r/{\mathfrak{g}}_{\mathfrak{n}}\in{\mathfrak{n}}/{\mathfrak{g}}_{\mathfrak{n}}$. As remarked in [@Tsujii01 p. 358], the terms $h^{-1}\bigl(r(q+hw)-r(q)\bigr)$, $h^{-1}\bigl((r\circ S)(p+hv)-(r\circ S)(p)\bigr)$, and $h^{-1}\bigl(S(p+hv)-S(p)\bigr)$ take constant values for sufficiently small $h>0$. If $h_0$ is such an $h$, we get $$\begin{aligned}
\varphi_w\bigl(D_q(r/{\mathfrak{g}}_{\mathfrak{n}})\bigr) &=
\bigl(D_q(r/{\mathfrak{g}}_{\mathfrak{n}})\bigr)(w) \\
&= h_0^{-1}\bigl(r(q+h_0w)-r(q)\bigr) \\
&= h_0^{-1}\bigl(r(q+S(p+h_0v)-S(p))-r(q)\bigr) \\
&= h_0^{-1}\bigl((r\circ S)(p+h_0v)-(r\circ S)(p)\bigr) \\
&= \bigl(D_p((r\circ S)/{\mathfrak{g}}_{\mathfrak{m}})\bigr)(v) \\
&= \varphi_v\bigl(D_p(\bar{\sigma}(r/{\mathfrak{g}}_{\mathfrak{n}}))\bigr),\end{aligned}$$ as required.
In\[ref21\] the proof of Theorem \[ref22\] we assumed that $p$ and $q$ have rational coordinates and are in the topological interior of ${[0,1]}^n$. The first assumption is motivated by the fact that, by definition, McNaughton mappings have integer coefficients. This implies that the only tangent vectors, say at $q$, that can be algebraically recognized are those in the ${\mathbb{R}}$-span $V$ of the set of all $v\in{\mathbb{R}}^n$ such that the affine line through $q$ and $q+v$ is the intersection of affine hyperspaces having integer (equivalently, rational) coefficients. The [$\ell$-group]{} ${\mathfrak{n}}/{\mathfrak{g}}_{\mathfrak{n}}$ is then [$\ell$-isomorphic]{} to $\operatorname{F\ell}(k)$, for $k=\dim(V)\le n$, and the algebraic tangent space $({\mathfrak{n}}/{\mathfrak{g}}_{\mathfrak{n}})'$ is isomorphic to ${\mathbb{R}}^k$. According to the personal interests, one may either accept these underdimensional tangent spaces, or treat them drastically, by tensoring everything with ${\mathbb{R}}$. This means considering piecewise-linear functions with arbitrary real coefficients, so dropping the assumption that the cellular complexes involved have rational vertices, and passing from [$\ell$-groups]{} and MV-algebras to real vector lattices [@baker68] and their $\Gamma$ images. All quotients ${\mathfrak{n}}/{\mathfrak{g}}_{\mathfrak{n}}$ are then isomorphic to the free vector lattice over $n$ generators $\operatorname{FVL}(n)$ [@pantiprime Theorem 3.8]. The dual of $\operatorname{FVL}(n)$, i.e., the set of all ${\mathbb{R}}$-linear [$\ell$-homomorphisms]{} from $\operatorname{FVL}(n)$ to ${\mathbb{R}}$ is still in bijection with ${\mathbb{R}}^n$ via the evaluation mapping, and all dimensionality problems disappear.
About the other assumption: if $p$ or $q$ (say $p$) is on the boundary of ${[0,1]}^n$, then the quotient ${\mathfrak{m}}/{\mathfrak{g}}_{\mathfrak{m}}$ is [$\ell$-isomorphic]{}not to $\operatorname{F\ell}(n)$, but to a quotient of $\operatorname{F\ell}(n)$ by a principal kernel or, equivalently, to the [$\ell$-group]{} of restrictions of the elements of $\operatorname{F\ell}(n)$ to a polyhedral cone $W$. The dual $({\mathfrak{m}}/{\mathfrak{g}}_{\mathfrak{m}})'$ is then in bijection with $W$, in agreement with [@Tsujii01 Eq. (14)], and the proof of Theorem \[ref22\] carries on.
Chaotic actions
===============
Let $p=(\ldots,p_i,\ldots)\in{[0,1]}^\kappa$. We say that $p$ has [*finite denominator*]{} if all $p_i$’s are rational numbers, and there exists $0<d\in{\mathbb{Z}}$ such that $dp\in{\mathbb{Z}}^\kappa$. The least such $d$ is the [*denominator*]{} of $p$, written $\operatorname{den}(p)$.
Let\[ref16\] $p,q\in{[0,1]}^\kappa$.
- If $p$ has finite denominator and $\sigma\in\Sigma_\kappa$, then $S(p)$ has finite denominator and $\operatorname{den}(S(p))\mid\operatorname{den}(p)$.
- If $\operatorname{den}(q)\mid\operatorname{den}(p)$, then $S(p)=q$ for some $\sigma\in\Sigma_\kappa$.
- If $p$ does not have finite denominator, then the $\Sigma_\kappa$-orbit of $p$ is dense.
\(i) Let $d=\operatorname{den}(p)$, $S(p)=(\ldots,q_i,\ldots)$, $\sigma(x_i)=
s_i({x_{j_1},\ldots ,x_{j_n}})$. Then $q_i=s_i({p_{j_1},\ldots ,p_{j_n}})=
a^1p_{j_1}+\cdots+a^np_{j_n}+a^{n+1}$, for some $a^1,\ldots,a^{n+1}\in{\mathbb{Z}}$. Therefore $dq_i\in{\mathbb{Z}}$ and $dS(p)\in{\mathbb{Z}}^\kappa$, and our claim easily follows.(ii) Let $d=\operatorname{den}(p)$. The integers $\{dp_i:i<\kappa\}\cup\{d\}$ must be relatively prime (otherwise $\operatorname{den}(p)$ would be smaller than $d$). Therefore there exist indices ${j_1,\ldots ,j_m}$ and integer numbers $a^1,\ldots,a^m,a^{m+1}$ such that $d(a^1p_{j_1}+\cdots+
a^mp_{j_m}+a^{m+1})=1$. Let $n\le\kappa$ be greater than ${j_1,\ldots ,j_m}$. Then the affine linear polynomial $f_i=q_id(a^1x_{j_1}+\cdots+a^mx_{j_m}+a^{m+1})$ has integer coefficients, since $\operatorname{den}(q)\mid d$. The function $(f_i\lor 0)\land 1:
{[0,1]}^n\to{[0,1]}$ is a McNaughton function, hence by Theorem \[ref15\] it is expressible via a formula $s_i\in{\operatorname{Free}_{n}({{\mathit{MV}}})}$. Since $s_i({p_{j_1},\ldots ,p_{j_n}})=q_i$, the substitution $\sigma\in\Sigma_\kappa$ defined by $\sigma(x_i)=s_i$ satisfies our requirements.(iii) It suffices to show that, for every $0\le a<b\le 1$, there exists an element $s\in{\operatorname{Free}_{\kappa}({{\mathit{MV}}})}$ such that $a<s(p)<b$. Let $G$ be the additive subgroup of ${\mathbb{R}}$ generated by $\{p_i:i<\kappa\}\cup\{1\}$. Since $p$ does not have finite denominator, $G$ is dense in ${\mathbb{R}}$, and therefore there exist indices ${j_1,\ldots ,j_m}$ and integer numbers $a^1,\ldots,a^m,a^{m+1}$ such that $a<a^1p_{j_1}+\cdots+
a^mp_{j_m}+a^{m+1}<b$. One then argues as in (ii) above.
In the following we will tacitly identify via $\pi$ the $\kappa$-cube ${[0,1]}^\kappa$ with the subspace of $X_\kappa$ whose elements are the maximal filters.
Let\[ref17\] $\operatorname{Rat}_n$ be the set of rational points in ${[0,1]}^n$. Then $\operatorname{Rat}_n$ is a dense subset both of ${[0,1]}^n$ and of $X_n$. All the elements of $\operatorname{Rat}_n$ have a finite $\Sigma_n$-orbit. No point of $X_\omega$ has a finite $\Sigma_\omega$-orbit.
$\operatorname{Rat}_n$ coincides with the set of points in ${[0,1]}^n$ having finite denominator, and is dense in ${[0,1]}^n$. As the $n$-cube is dense in $X_n$, $\operatorname{Rat}_n$ is dense in $X_n$ as well. If $p\in\operatorname{Rat}_n$, then by Lemma \[ref16\] the $\Sigma_n$-orbit of $p$ is the set of points whose denominator divides $\operatorname{den}(p)$, and this set is finite. Since the submonoid of $\Sigma_\omega$ whose elements are all the substitutions $\sigma$ such that $\sigma(x_i)\in{\{0,1\}}$ has the cardinality of the continuum, our last claim is immediate.
By Lemma \[ref16\](i) $\Sigma_\kappa$ does not act minimally on $X_\kappa$. We are therefore lead to weaken the requirement of minimality to that of topological transitivity: we say that $\Pi\subseteq\Sigma_\kappa$ is [*topologically transitive*]{} on $X_\kappa$ if $(\Pi,O)$ is dense in $X_\kappa$, for every nonempty open set $O$. Using the fact that ${[0,1]}^\kappa$ is dense in $X_\kappa$, one shows easily that $\Pi$ is topologically transitive on $X_\kappa$ iff it is topologically transitive on the $\kappa$-cube. By standard arguments [@Walters82 Theorem 5.9], this amounts to the existence of a point in ${[0,1]}^\kappa$ (or a $G_\delta$ dense set of such points) whose $\Pi$-orbit is dense.
If $\Pi$ is topologically transitive and the set of points whose $\Pi$-orbit is finite is dense, then we say that $\Pi$ acts [*chaotically*]{}. By Lemma \[ref16\](iii) and Corollary \[ref17\], $\Sigma_n$ acts chaotically both on $X_n$ and on ${[0,1]}^n$. It is well known that a chaotic action on a space such as ${[0,1]}^n$ implies sensitive dependence on initial conditions, hence chaotic behaviour in the sense of Devaney [@Banksetal92].
One constructs easily chaotic elements of $\Sigma_n$. Indeed, the standard tent map on ${[0,1]}$ is a McNaughton function, expressible by the formula $s(x_0)=(x_0\land\neg x_0)\oplus(x_0\land\neg x_0)$. The substitution $\sigma:x_i\mapsto s(x_i)$, for $i<n$, induces therefore on ${[0,1]}^n$ the direct product of $n$ tent maps, which is mixing w.r.t. Lebesgue measure. Hence $\sigma$ acts in a topologically transitive and chaotic way. It is not so easy to construct elements of $\Xi_n$ which are chaotic; we will obtain such mappings for even $n$ in Corollary \[ref20\].
Let $\sigma\in\Xi_n$, let $S$ be the induced McNaughton homeomorphism of ${[0,1]}^n$, and let $W$ be a complex over the $n$-cube such that $S$ has the form $(**)$ on each $n$-dimensional cell $C_j$ of $W$. Then all the matrices $A_j$ have the same determinant, which is either $+1$ or $-1$.
Since the inverse of $S$ is expressible as in $(**)$ via matrices and vectors having integer entries, it is clear that all matrices $A_j$ are invertible and their inverses have integer entries. Therefore all $A_j$’s have determinant $\pm1$. Suppose by contradiction that the $n$-dimensional cells of $W$ are ${C_1,\ldots ,C_k}$ and that there is some $1\le r<k$ such that $\det(A_j)=+1$ for $1\le j\le r$, and $\det(A_j)=-1$ for $r<j\le k$. If $1\le j'\le r$ and $r<j''\le k$, then $C_{j'}$ and $C_{j''}$ cannot intersect in an $(n-1)$-dimensional face, because this would contradict the injectivity of $S$. Let $D$ be the topological interior of the $n$-cube, $E=D\cap(C_1\cup\cdots\cup C_r)$, $F=D\cap(C_{r+1}\cup\cdots\cup C_k)$. Then $E$ and $F$ are nonempty, and closed in the relative topology of $D$. By [@kelley Theorem 1.17], $(E\setminus F)\cup(F\setminus
E)=D\setminus(E\cap F)$ is not connected, and this contradicts [@hockingyou Theorem 3.61], since $E\cap F$ has topological dimension $\le n-2$.
For\[ref18\] every $n$ and every $\sigma\in\Xi_n$, the homeomorphism $S$ preserves the Lebesgue measure on ${[0,1]}^n$.
The only invertible substitutions on ${\operatorname{Free}_{1}({{\mathit{MV}}})}$ are the identity and the flip $x_0\mapsto \neg x_0$.
If $S$ is induced by $\sigma\in\Xi_1$, then either $S$ has the form $x_0+k_p$ (with $k_p\in{\mathbb{Z}}$) in every $p$ in which $S$ is differentiable, or the form $-x_0+k_p$. Since the range of $S$ is ${[0,1]}$, it must be $k_p=0$ in the first case, or $k_p=1$ in the second.
The following is the main result of [@pantibernoulli].
There\[ref19\] is an explicitly constructible family $\{\sigma_{lm}:1\le l,m\in{\mathbb{N}}\}$ of elements of $\Xi_2$ such that, for every $\sigma_{lm}$ in the family, the induced McNaughton homeomorphism $S_{lm}$ has the following properties:
- $S_{lm}$ fixes pointwise the boundary of ${[0,1]}^2$;
- $S_{lm}$ is ergodic with respect to the Lebesgue measure of the unit square;
- $S_{lm}$ is non-uniformly hyperbolic and Bernoulli.
We recall that a measure-preserving bijection is [*Bernoulli*]{} if it is measure-theoretically isomorphic to a Bernoulli $2$-sided full shift.
For \[ref20\] every even $n$ there exist Bernoulli McNaughton homeomorphisms of ${[0,1]}^n$. These mappings are mixing w.r.t. Lebesgue measure, topologically transitive, and chaotic in the sense of Devaney.
The Bernoulli property implies mixing, and is preserved under direct products [@CornfeldFomSi82 Ch. 10 §1]. Since nonvoid open subsets of ${[0,1]}^n$ have positive Lebesgue measure, mixing homeomorphisms are topologically transitive, and hence chaotic by Corollary \[ref17\].
It would be very interesting to construct a McNaughton homeomorphism of ${[0,1]}^3$ having the Bernoulli property: this would allow to extend Corollary \[ref20\] to all $n\ge 2$. Up to now, we can only prove the following result about the action of $\Xi_n$ as a group.
For\[ref23\] every $n\ge2$, $\Xi_n$ acts chaotically on ${[0,1]}^n$.
As discussed above, we have a stronger result for even $n$, so we assume $n$ odd. Let $S$ and $T$ be topologically transitive McNaughton homeomorphisms of ${[0,1]}^{n-1}$ and ${[0,1]}^2$, respectively. Consider the following direct products: $$\begin{aligned}
Q&=S\times(\text{identity map on the last coordinate});\\
R&=(\text{identity map on the first $n-2$ coordinates})
\times T.\end{aligned}$$ Both $Q$ and $R$ are McNaughton homeomorphisms of ${[0,1]}^n$, induced by elements of $\Xi_n$. For every $i<n$, let $0\le a_i<a'_i\le1$ and $0\le b_i<b'_i\le1$: we want to show that an appropriate composition of $Q$ and $R$ maps some point of the open box $A=\prod_{i<n}(a_i,a'_i)$ in the open box $B=\prod_{i<n}(b_i,b'_i)$. Since $S$ is topologically transitive, there exists $h\ge0$ such that the open set $$U=Q^h[A]\cap\bigl((b_0,b'_0)\times\cdots\times(b_{n-2},b'_{n-2})
\times(a_{n-1},a'_{n-1})\bigr)$$ is nonempty. Let $0\le c_i<c'_i\le1$ be such that the box $C=\prod_{i<n}(c_i,c'_i)$ is contained in $U$. The open box $D=\bigl(\prod_{i<n-1}(c_i,c'_i)\bigr)\times(b_{n_1},b'_{n-1})$ is contained in $B$ and, since $T$ is topologically transitive, there exists $k\ge0$ such that $R^k[C]\cap D\not=\emptyset$. Therefore $(R^k\circ Q^h)[A]\cap B\not=\emptyset$.
For every Borel probability measure $\mu$ on ${[0,1]}^n$, let $f_\mu(r)$ denote the integral of the formula $r({x_0,\ldots ,x_{n-1}})$ (viewed as a function $r:{[0,1]}^n\to{[0,1]}$) with respect to $\mu$. The number $f_\mu(r)$ may be thought of as the “average truth-value” of $r$ w.r.t. $\mu$. It is natural to restrict attention to measures which are [*faithful*]{} ($r\not=0$ implies $f_\mu(r)\not=0$) and [*automorphism-invariant*]{} ($f_\mu(r)=f_\mu(\sigma(r))$, for every $\sigma\in\Xi_n$). This latter property is particularly relevant: it says that the average truth-value of a formula should be intrinsic to the formula, and not depending on the particular embedding of it in ${\operatorname{Free}_{n}({{\mathit{MV}}})}$. Lebesgue measure $\lambda$ is faithful (clearly) and automorphism-invariant (by Corollary \[ref18\]). Let $\mu$ be another probability measure on ${[0,1]}^n$, absolutely continuous with respect to $\lambda$. We may assume that $n$ is even, possibly introducing a dummy variable. By Corollary \[ref20\], there exists an automorphism $\sigma$ of ${\operatorname{Free}_{n}({{\mathit{MV}}})}$ that induces a mixing homeomorphism $S$ on ${[0,1]}^n$. Therefore the push forward $S_*^k\mu$ of $\mu$ by $S^k$ converges to $\lambda$ in the weak${}^*$ topology [@Walters82 §4.9 and Theorem 6.12(ii)]. In particular, for $r$ as above we get $$\lim_{k\to\infty}f_\mu(\sigma^k(r))=
\lim_{k\to\infty}\int r\circ S^k\,d\mu=
\lim_{k\to\infty}\int r\,d(S_*^k)\mu=
\int r\,d\lambda=f_\lambda(r).$$ Hence the existence of mixing McNaughton homeomorphisms gives a distinguished status to $\lambda$. It appears plausible that the only ergodic $\Xi_n$-invariant measures on ${[0,1]}^n$ are $\lambda$ and the measures supported on finite orbits. We leave this as an open problem: since measures supported on finite orbits are not faithful, a positive answer would imply that the only reasonable averaging measure on truth-values in [Łukasiewicz]{} logic is Lebesgue measure.
[10]{}
P. Aglian[ò]{}. Prime spectra in modular varieties. , 30(4):581–597, 1993.
M. Anderson and T. Feil. . Reidel, Dordrecht, 1988.
K. Baker. Free vector lattices. , 20:58–66, 1968.
J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey. On [D]{}evaney’s definition of chaos. , 99(4):332–334, 1992.
A. Bigard, K. Keimel, and S. Wolfenstein. , volume 608 of [ *Lecture Notes in Math.*]{} Springer, 1977.
S. Burris and H. P. Sankappanavar. , volume 78 of [*Graduate Texts in Mathematics*]{}. Springer, 1981.
A. Church. . Princeton University Press, 1956.
R. Cignoli, I. D’Ottaviano, and D. Mundici. , volume 7 of [*Trends in logic*]{}. Kluwer, 2000.
S. A. Cook and R. A. Reckhow. The relative efficiency of propositional proof systems. , 44(1):36–50, 1979.
I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ. , volume 245 of [*Grundlehren der Mathematischen Wissenschaften*]{}. Springer, 1982.
S. Gottwald. , volume 9 of [*Studies in Logic and Computation*]{}. Research Studies Press Ltd., Baldock, 2001.
H. P. Gumm and A. Ursini. Ideals in universal algebras. , 19:45–54, 1984.
P. H[á]{}jek. , volume 4 of [*Trends in logic*]{}. Kluwer, 1998.
P. R. Halmos. . Van Nostrand Mathematical Studies, No. 1. D. Van Nostrand Co., Inc., Princeton, N.J., 1963.
M. Hochster. Prime ideal structure in commutative rings. , 142:43–60, 1969.
J. G. Hocking and G. S. Young. . Dover, 1988. First published in 1961 by Addison-Wesley.
J. L. Kelley. . Van Nostrand, 1955.
R. McNaughton. A theorem about infinite-valued sentential logic. , 16:1–13, 1951.
J. Menu and J. Pavelka. A note on tensor products on the unit interval. , 17:71–83, 1976.
P. S. Mostert and A. L. Shields. On the structure of semigroups on a compact manifold with boundary. , 65:117–143, 1957.
D. Mundici. Interpretation of [AF $C^*$]{}-algebras in [[Ł]{}]{}ukasiewicz sentential calculus. , 65:15–63, 1986.
D. Mundici. A constructive proof of [M]{}c[N]{}aughton’s theorem in infinite-valued logic. , 59:596–602, 1994.
G. Panti. Prime ideals in free $\ell$-groups and free vector lattices. , 219(1):173–200, 1999.
G. Panti. Generic substitutions. Available at `http://arXiv.org/abs/math/0308177`, to appear in the *J. of Symbolic Logic*, 2003.
G. Panti. Bernoulli automorphisms in many-valued logic. Available at `http://arxiv.org/abs/math/0408370`, submitted for publication, 2004.
M. Tsujii. Absolutely continuous invariant measures for expanding piecewise linear maps. , 143(2):349–373, 2001.
P. Walters. , volume 79 of [*Graduate Texts in Mathematics*]{}. Springer, 1982.
[^1]: *2000 Math. Subj. Class.*: 03B50; 37A05
| ArXiv |
---
abstract: 'Position-sensitive detectors for cold and ultra-cold neutrons (UCN) are in use in fundamental research. In particular, measuring the properties of the quantum states of bouncing neutrons requires micro-metric spatial resolution. To this end, a Charge Coupled Device (CCD) coated with a thin conversion layer that allows a real time detection of neutron hits is under development at LPSC. In this paper, we present the design and performance of a dedicated electronic board designed to read-out eight CCDs simultaneously and operating under vacuum.'
address:
- |
LPSC, Université Grenoble-Alpes, CNRS/IN2P3\
53, avenue des Martyrs, Grenoble, France
- |
ILL, Institut Laue Langevin\
71, avenue des Martyrs, Grenoble, France
author:
- 'O. Bourrion'
- 'B. Clement'
- 'D. Tourres'
- 'G. Pignol'
- 'Y. Xi'
- 'D. Rebreyend'
- 'V.V. Nesvizhevsky'
title: 'C2D8: An eight channel CCD readout electronics dedicated to low energy neutron detection.'
---
UCN, CCD readout, position sensitive detectors.
Introduction {#introSec}
============
Neutrons interact with matter mostly through strong nuclear interaction. When the neutron wavelength becomes commensurate with the inter-atomic spacing, only coherent scattering occurs. As a consequence, neutrons slowed down to kinetic energies below 100neV are totally reflected at any angle of incidence by most solid surfaces. These *ultra-cold neutrons* constitute a sensitive tool to study fundamental interactions and symmetries [@Dubbers2011]. In particular, ultra-cold neutrons are used to study gravity in a quantum context [@Nesvizhevsky2002; @Nesvizhevsky2010; @Jenke2011; @Pignol2015]. A neutron bouncing on top of an horizontal mirror realizes a simple one dimensional quantum well problem and the vertical motion of the neutron bouncer has discrete energy states. The wave functions associated to the stationary quantum states have a spatial extension governed by the parameter $z_0 = (\hbar^2/2m_n^2g)^{1/3} \approx 6$m. Therefore, observing the spatial structure of the quantum states requires position-sensitive neutron detectors with a micro-metric spatial resolution.
Semiconductor-based detectors, coated with adequate neutron converters, have been demonstrated to be well suited for this kind of measurements [@Jakubek2009; @Jakubek2009b; @Kawasaki2010; @Lauer2011]. The UCNBox (Ultra Cold Neutrons BOron piXels) detector has been recently developed as position sensitive sensor optimized to measure the wave-functions of the bouncing neutron in the GRANIT experiment [@Roulier2015]. The GRANIT facility uses a 30cm wide glass mirror as the surface where neutrons bounce. The setup is inside a vacuum chamber at $10^{-5}$mbar. The detector, composed of 8 Charge Coupled Devices (CCD), is designed to cover a sensitive area of $300 \, {\rm mm} \times 0.8 \, {\rm mm}$. Each CCD is an Hamamatsu S11071-1106N sensor (pixel size 14m $\times$ 14m and number of effective pixels $2048 \times 64$) coated with $^{10}$B, thanks to plasma assisted physical vapor deposition [@boronCCD].
Neutron capture on boron produces in most cases both a 1.5MeV $\alpha$ particle and a 0.8MeV $^{7}$Li nucleus. The CCD sensor is used as a pixelated silicon detector. The charge produced by the energy deposition migrates in the neighboring pixels allowing a precise reconstruction of the position using weighted average. To limit this migration of charges each CCD sensor must be read at approximately a 1Hz rate, with a dead-time as low as possible. In this project we aimed at dead-times below 1%. The UCN rate in the final experiment should not exceed 1Hz per sensor. Nevertheless, for calibration purposes, rates as high as 50Hz per sensor are desirable. Also, given the fact that no mechanical shutter system can be used to avoid neutron detection, the shortest possible readout time must be achieved for each CCD to avoid neutron detection during the CCD charge transfer, as it would corrupt the data. Consequently, the CCD must be read-out at the maximum speed specified by the manufacturer (10MHz), they can however be read one after another.
Typically, both detection-rate and number of hit pixels are low as all charges from one charged particle are collected within a $11\times11$ pixels matrix. For a maximum rate of 50Hz per sensor, only 5% of the sensor contains useful data. In normal conditions, this drops to 0.1%. This calls for the implementation of a data reduction system, that removes any pixel data below a discrimination threshold.
It must be noted, that an adjustable exposure time must be implemented to permit the use of bright light sources such as LED used to test and adjust CCD alignment.
Additionally, the readout system must be located very close to the sensor system in order to minimize signal integrity issues, but also to minimize the number of vacuum feed-through for the CCD signals (control and readout). This requirement, implies that the readout must be located inside the detector cell and thus withstand vacuum conditions. Consequently, special care must be taken on power usage and heat dissipation to ensure proper operation. This paper is organized as follows: section \[HardwareSec\] presents the hardware design, section \[FPGASec\] describes the firmware architecture. Eventually, a short summary is given in section \[SummarySec\].
Hardware description {#HardwareSec}
====================
To meet the requirements listed in section \[introSec\], we have opted for a solution based on two distinct modules: a front-end part composed of 8 boards, each holding a CCD; a back-end part for the control and readout circuits. As shown in figure \[c2d8Pic\], the front-end (FEB) and back-end boards are mounted on a common mechanical support. This support is designed such as to be able to finely adjust the position of each CCD board, thanks to dedicated screws. We have checked that this system allows for a relative alignment of the 8 CCDs within 10m, sufficient for our needs.
![Picture of the readout electronics mounted on its mechanical support. The electronic system is composed of two parts: the front-end composed of 8 carrier boards, each holding a CCD, and the back-end with the control and readout circuits. The boards are mounted on a mechanical support system equipped with adjustment screws to allow the adjustment of each CCD carrier board position. The four thermal sensors implemented on the board are indicated (T1 to T4). []{data-label="c2d8Pic"}](./figs/c2d8InstalledPic){width="90.00000%"}
Additionally, the electronics and the support system were designed to optimize the thermal coupling. Indeed, the support system is used to conduct a significant part of the heat flow to the vacuum vessel, while the remaining part of the heat is radiated in the chamber.
![Block diagram of the electronics system. Each FEB is connected to the back-end board with a Flexible Flat Cable (FFC). The back-end board is in charge of generating the CCD control signals (horizontal/ vertical shifts and reset gate), to perform the CCD signal digitization, to aggregate the data and to finally make them available for readout via a Universal Serial Bus (USB) interface. Each CCD signal digitization is done by a dedicated CCD signal processor [@ADDI7100].[]{data-label="elecDiag"}](./figs/hardBlockDiag){width="90.00000%"}
A block diagram of the electronics is shown in figure \[elecDiag\]. Each FEB is connected to the back-end board with a Flexible Flat Cable (FFC). The back-end board is in charge of generating the adequate CCD control signals (horizontal/ vertical shifts and reset gate); performing the CCD signal digitization; aggregating the data and finally making them available for readout via a Universal Serial Bus (USB) interface.
Each CCD signal digitization is carried-out by a dedicated CCD signal processor (Analog devices ADDI7100 [@ADDI7100]). This CCD processor can operate at 45MHz, which is significantly faster than the maximum readout speed of the Hamamatsu S11071-1106 CCD (10MHz), and has a digitization resolution of 12 bit with noise performance better than the CCD performance.
Indeed, the CCD processor is specified for having a system noise equivalent to 24.4e^-^ (0.8LSB rms with CDS gain set at +6dB for a typical CCD sensitivity of 8V/e- which corresponds to 30.5e^-^/ADU) while the typical CCD noise is composed of the readout noise (23e^-^ rms) and of the dark current integration (typically 50e^-^/pixel/s at 25C) resulting in a total of 73e^-^ for 1s of integration. The total expected system noise is thus dominated by the CCD and is about 77e^-^, which is compatible with the measurements that showed a system noise of 91.5$\pm$15e^-^.
The control signals required to read-out the CCD are generated by a Field Programmable Gate Array (FPGA). These signals, composed of Horizontal/Vertical (H/V) shifting signals and Reset Gate signals (RG), are amplified by dedicated drivers to accommodate the CCD load. The FPGA is also used to produce the signals necessary to operate the CCD signal processor, i.e. clamping and pre-blank signals, the correlated double sampler (CDS) signals and the serial control links.
The rationales for selecting the FPGA used in this design (Xilinx XC7A35-FGG484) were (i) its low power consumption; (ii) the possibility to precisely adjust the timing of the generated signals; (iii) the large amount of memory available. Indeed, the power had to be minimized by design as much as possible to permit the electronics operation under vacuum while avoiding a too fancy mechanical setup for thermalization (for instance: usage of the standby modes of the CCD processors during the exposure time). Additionally, this FPGA features high performance serializers in each of its input/output block. Thanks to these blocks, one can adjust output signals with a time resolution of about 2ns by using a high speed 480MHz clock (see section \[FPGASec\]). This makes it possible to conveniently set the CDS sampling times. The sizing of the memory was based on the criteria that its capacity should be at least half of the memory required to buffer the data generated by one CCD readout, i.e. $2048 \times 64 /2=65536$ 16-bits words, corresponding to more than 1Mbit of storage. This time equivalent buffering must be considered acknowledging the fact that, by specification, a new USB2 transaction can be placed every millisecond.
The total system power usage was measured to be 3W in full readout. In this budget, we estimate by combining specifications and measurements that about 140mW are used for each CCD (including about 32mW for the biasing and the line drivers losses); 600mW are used by the power converters (linear and switching) distributed on the board; about 1200mW are used by the FPGA. To asses the operating temperature of the system, dedicated measurements were performed under vacuum. A total of six temperatures were recorded over 25 hours, i.e. from the system power-up until system equilibrium (see figure \[ccdTemp\]). The temperature of the back-end electronics was recorded on four points thanks to the sensor circuits included in design (LM75B), see locations in figure \[elecDiag\] Additionally, the temperature of two CCD were recorded: one at the border and one in the middle of the detector plane with thermally coupled PT100 probes. We can see that the CCD highest temperature elevation from ambient is about 8C. The highest temperature elevation of 11C is measured for probe T2 which is located close to the FPGA.
![Plot of the system temperatures recorded from power-up to equilibrium. We can see that the CCD highest temperature elevation from ambient is about 8C. The highest temperature elevation of 11C is measured for probe T2 which is located close to the FPGA. []{data-label="ccdTemp"}](./figs_ext/CCDtemperature){width="80.00000%"}
Firmware description {#FPGASec}
====================
A block diagram of the FPGA firmware is shown in figure \[firmDiag\].
![Block diagram of the FPGA firmware. It is composed of six different blocks: clocking, USB interface, serializer, DAQ FIFO, DAQ manager and CCD timing. []{data-label="firmDiag"}](./figs/firmBlockDiag){width="90.00000%"}
It is composed of six different blocks: ‘clocking’, ‘USB interface’, ‘serializer’, ‘DAQ FIFO’, ‘DAQ manager’ and ‘CCD timing’. They are described hereafter.
The ‘clocking’ module is used to produce the CCD readout clock (10MHz) and the fast clocks required by the timing modules (respectively 80MHz and 240MHz). To achieve this, a Mixed Mode Clock Manager [@MMCM] (MMCM) uses the 50MHz reference clock, provided at the board level from a crystal oscillator, to perform the clock generation.
The ‘USB interface’ provides an interface between the USB micro-controller and the various configuration registers as well as an interface to read-out the acquisition FIFO (‘DAQ FIFO’).
The ‘DAQ FIFO’ is designed as a First Word Fall Through FIFO, it provides a buffering depth of 98305 words of 16-bit. This configuration was chosen to fully exploit the available memory in the selected FPGA and thus loosen the constraints on the acquisition software.
The ‘serializer’ module is used to interface the CCD processor serial configuration link, it is controlled either by the ‘USB interface’ or by the ‘DAQ manager’. When controlled by the ‘USB interface’, it allows the acquisition and control software to configure the CCD processor with the required parameters (gain settings, operation mode, ...). During acquisition mode, the ‘DAQ manager’ can use the ‘serializer’ to change the CCD processor operating mode (normal operation or full standby) of the active CCD. Hence, the CCD processors are in normal mode only when required, less than 1% of the time, and thus the power used is minimized.
The eight ‘CCD timing’ modules, one per CCD, are used to generate the CCD shifting signals (vertical, horizontal and reset gate) and the CCD processor control signals. These are the pedestal and data sampling control signals required for the correlated double sampler (respectively SHP and SHD), the preblank signal (PBLK) which is used to clear the processor output data during vertical shifting and the clamp optical black signal (CLOPB) used to remove residual offsets in the CCD processor signal chain. The CLOPB signal is supposed to be activated when the black pixels are being read-out. A detailed description of the ‘CCD timing’ module is given in section \[ccdTimingSec\].
The ‘DAQ manager’ module performs two main tasks. Its first role is to sequence the CCD acquisition by triggering the appropriate ‘CCD timing’ module. The CCD to read-out are individually selected by an eight bit selection mask (‘sel\_mask’). Its second purpose is to recover the data provided by the CCD processor associated with the CCD being accessed, to discriminate the data with respect to a threshold, to encapsulate the data and to eventually store them in the output buffer. More details about the ‘DAQ manager’ is given in section \[daqManagerSec\].
CCD timing module description {#ccdTimingSec}
-----------------------------
A block diagram detailing the internal architecture of the ‘CCD timing’ module is given in figure \[timingBlockDiag\].
![Block diagram of the CCD timing module.[]{data-label="timingBlockDiag"}](./figs/timingBlockDiag){width="90.00000%"}
The ‘horizontal timing’ module is in charge of controlling the four horizontal shifts (H shift) and the reset gate (RG) signals. Module operation is deactivated by the Horizontal Blank signal (HBLK), that is when the ‘timing core’ FSM is not in the horizontal shifting states (*preHor*, *validHor* and *postHor*). The horizontal shifting are done at the pixel clock speed, i.e. 10MHz. To cope with the various signal phases and widths required by the chosen CCD, the module is clocked eight times faster than the pixel clock.
The ‘vertical timing’ module controls the two vertical shifting signals. The module is designed to be able to shift several lines before signaling completion with *shiftDone*. The vertical shifting is requested by *startV* and the number of lines to move is determined by *shiftCount*. It may be noted that the CCD is operated in the Large Saturation Charge Mode [@LSCM].
The ‘timing core’ Finite State Machine (FSM) is used to coordinate the CCD control signals. It controls the horizontal and vertical timing modules and thus activates them when appropriate. The FSM is started either by the acquisition signal (*do\_acquire*) or by the flush request signal (*doFlush*), which is a delayed version of the *do\_acquire* signal. Note that by design, no data is recorded when the FSM is triggered by the *doFlush* signal. Given the fact that there is no mechanical shutter in the system, *doFlush* is used to implement an electronic shutter. Indeed, the effective exposure time of the CCD, is the amount of time elapsed between the last flush request and the new acquisition request.
Once started, the FSM moves to the *preVert* state, where it requests the removal of the optically covered lines to the ‘vertical timing’ module, thanks to the *startV* and *shiftCount* signals. As soon as the vertical shifting is done (*shiftDone*), the inner state machine loop processes each horizontal line (states *Vshift* to *postHor*). Likewise to the vertical shifting, optically covered pixels in the horizontal direction are removed before (*preH*) and after (*postH*) the optically effective zone (*validH*). Finally, when the full CCD processing is done, a *CCD\_done* signal is generated to inform the ‘DAQ manager’ module of the timing generation completion.
The PBLK signal, used to force the CCD processor data output at zero, is activated when the FSM is not moving on the inner loop, in other word, it is the complementary of HBLK. CLOPB, used to remove residual offsets in the CCD processor signal chain, is activated when reading the last optically covered pixels of each line (*postH* pixels). A *valid\_pixel* signal is constructed to ease the tasks of the acquisition manager module (see section \[daqManagerSec\]). This signal is activated only during the *validHor* state and if the line currently accessed is in the allowed range determined by *vertical start* and *vertical stop*. Indeed, the complete CCD sensitive zone can never be utilized for many reasons (alignment issues, exposed area, ...). The vertical selection was implemented to reduce the data volume by removing the meaningless lines.
The ‘ADC timing’ modules, that are used to precisely control the CCD processor sampling times, are activated at start-up time by *enable\_SHPD* and are free running. Each module is composed of a FSM operated at 80MHz and a fast Dual Data Rate serializer (OSERDES) operated at 240MHz. The FSM, which loops every eight 80MHz clock cycles (or steps), constructs and feeds the adequate 6 bit sub-step word to be serialized at 480Mbit/s at every steps. Thus, a tuning resolution of about 2ns (or 10MHz/48) can be reached. To set the start and stop times, 7 bit words are used. The three MSB of the word select during which step the signal must be activated, while the four LSB select at which sub-step the activation will take place.
Figure \[timingDiag\] gives an overview of the various signal timings involved in a CCD readout. The upper part of the diagram, depicts the behavior of the signals controlled by the ‘timing core’ inner FSM loop. The lower part of the diagram, shows the details of the ADC timing and the horizontal timing within a pixel clock cycle.
![Overview of the various signal timings involved in a CCD readout. The upper part of the diagram, depicts the behavior of the signals controlled by the ‘timing core’ inner FSM loop. The lower part of the diagram, shows the details of the ADC timing and the horizontal timing within a pixel clock cycle.[]{data-label="timingDiag"}](./figs/CCDtimings){width="90.00000%"}
DAQ manager module description {#daqManagerSec}
------------------------------
A block diagram of the ‘DAQ manager’ module is shown in figure \[daqDiag\]. The module is composed of two finite state machines (FSM): the ‘acquisition sequencer’ and the ‘data manager’.
![Block diagram of the ‘DAQ manager’ module. The module is composed of two finite state machines (FSM): the acquisition sequencer (l.h.s) and the data manager (r.h.s).[]{data-label="daqDiag"}](./figs/daqBlockDiag){width="90.00000%"}
The ‘acquisition sequencer’ FSM is restarted every second by a timer. Each CCD to be acquired (selection through the selection mask signal), is read-out one after another. At first, the ‘CCD processor’ is waken-up via a request to the ‘serializer’ module. This operation takes about 4ms. Then the ‘CCD timing’ module is started with the *do\_acquire* signal (duration of about 15ms) and the FSM moves directly in the *inter-CCD* waiting state. The waiting timer is set to 60ms, but given the fact that a CCD readout takes 15ms, the real waiting time with no operation is indeed 45ms. As soon as the CCD acquisition is done, a request is sent to the serializer to sleep the ‘CCD processor’.
The ‘data manager’ FSM is started at each CCD acquisition request. It first writes a data header, which contains useful information to determine if the data selection was used or not. If data selection is used the 15 LSB data only contain the CCD number, otherwise, they additionally contain the *vertical start* and *vertical stop* parameters. Indeed, when data selection is used, the valid pixel data words are not stored one after another but rather two data words per pixel, since their position in the payload do not reflect their coordinates anymore. In that latter case, the two words data results from the concatenation of the coordinate word composed of 17 bit and of the pixel digitized value (12 bit). Finally, once the full CCD payload is written in the memory, an End of Packet (EOP) is written in the FIFO memory. The EOP is composed of one or two words, again depending on the acquisition mode. In the full readout mode, the EOP word repeats the read-out CCD number, and the ‘vertical start’ and ‘vertical stop’ parameters. In the discrimination mode, the EOP two words contain the read-out CCD number and the number of pixels discriminated. Naturally, the use of the data selection mode makes sense only for situations where less than half of the pixels contains data above threshold.
Performances
============
Several tests of the detector were performed, first with $\alpha$ particles of varying energies to check the energy response of the system, then with a cold neutron beam at the PF1B beam line at the Institute Laue Langevin (ILL).
Particles are identified by looking at the pixel with the highest ADC value. On the $11\times11$ pixels matrix centered on that pixel, three quantities are reconstructed : the total sum of ADC values $\Sigma$ and the weighted averages $x$ and $y$ of the position in both directions. This procedure is then repeated on the next remaining highest ADC value pixels until all particles have been reconstructed
$\alpha$ measurement
--------------------
Energy measurement is not in itself necessary for the use of the UCNBox detector. Nevertheless, measuring the energy resolution and the linearity of the sensor is a test of the efficiency of the system. To this aim, an ^241^Am $\alpha$ source was used. The primary 5.48MeV particles were slowed down by a 12m aluminum foil. The energy was further decreased by changing the distance between the source and the sensor in air at 1bar, thus allowing to reach energies between 1 and 2MeV. This procedure widens significantly the energy distribution and reduces the precision of the measurement. For four different positions, the average energy was estimated using NIST tables [@NISTtable]. The resulting sum ADC spectra are presented in figure \[fig-calib\] with the resulting calibration curve. The response is found to be linear, within the precision of this measurement. The offset is set to zero, and the resulting fit gives a $\chi^2/Ndf = 2.13/3$, which is consistant with linearity. The slope translates into a collected charge of $7550\pm50$adu/MeV$ = 0.0369\pm0.0002$pC/MeV, whereas one would expect a created charge of $0.0443$pC/MeV in pure silicon. The difference can be accounted for by pair recombination within the CCD during the large exposition time (1s) and the clustering algorithm. It does not impact the final performance as $\alpha$ particles from 0.8MeV to 2MeV are clearly identified.
![Energy calibration and linearity: on the left the four energy spectra corresponding to different source-sensor distances and therefore to different peak energies; on the right the calibration curve obtained from the fit of the ADC peak.[]{data-label="fig-calib"}](./figs_ext/Calib_b_bw "fig:"){width="49.00000%"} ![Energy calibration and linearity: on the left the four energy spectra corresponding to different source-sensor distances and therefore to different peak energies; on the right the calibration curve obtained from the fit of the ADC peak.[]{data-label="fig-calib"}](./figs_ext/Calib_a_bw "fig:"){width="49.00000%"}
Neutron measurement
-------------------
The energy resolution was further investigated by exposing the detector to cold neutrons at the PF1B beam-line at ILL. For this experiment, the detector was in a dark room, not fully isolated from ambient light. Therefore the pixel threshold was set relatively high. Simulation of the attenuation of $\alpha$ particles produced in the boron layer was performed using SRIM [@SRIM]. The particle energy was determined using the previous calibration. The comparison between reconstructed data and simulation, shown in figure \[fig-neutron\], allows to extract an energy resolution of $58$keV. This value is sufficient for the purpose of the detector and will probably improve when lowering the pixel threshold.
![Energy spectrum of $\alpha$ particles produced in the boron layer by neutron capture. The blue lines are the result of SRIM simulations of the setup with (plain line) or without (dashed line) accounting for detector resolution.[]{data-label="fig-neutron"}](./figs_ext/EnergyAlpha){width="55.00000%"}
Summary {#SummarySec}
=======
To read-out a newly developed low energy neutron detector based on a set of 8 CCDs (sensitive area of $ \rm 300 \, mm \times 0.8 \, mm$), a dedicated electronics was designed. This electronics had to provide various features such as exposure time adjustment (0ms to 970ms), LED light calibration, embedded data-reduction, minimization of dead-time (< 1%) and low power usage (about 3W) to operate under vacuum. Additionally, the mechanical support had to provide a good thermal coupling to maintain the device at reasonable temperatures under vacuum and, at the same time, a precise adjustment mechanism to permit the relative height alignment of the CCDs. Using an $\alpha$ source of $^{241}$Am and a cold neutron beam, the performances of the full system (mechanical support, CCD and readout electronics) have been checked. In summary, this electronics is able to read-out simultaneously the eight CCDs at a rate of 1Hz and to meet all experimental requirements.
[99]{} D. Dubbers and M. G. Schmidt, Rev. Mod. Phys. [**83**]{} (2011) 1111.
V. V. Nesvizhevsky [*et al.*]{}, Nature [**415**]{} (2002) 297.
V. V. Nesvizhevsky, Physics Uspekhi [**53**]{} (2010) 645.
T. Jenke [*et al.*]{}, Nature Phys. [**7**]{} (2011) 468.
G. Pignol, Int. J. Mod. Phys. A [**24**]{} (2015) 1530048.
J. Jakubek [*et al.*]{}, Nucl. Instrum. Methods A [**600**]{} (2009) 651.
J. Jakubek [*et al.*]{}, Nucl. Instrum. Methods A [**607**]{} (2009) 45.
S. Kawasaki [*et al.*]{}, T. Sanuki c, S.Sonoda Nucl. Instrum. Methods A [**615**]{} (2010) 42.
T. Lauer [*et al.*]{}, Eur. Phys. J. A [**47**]{} (2011) 150.
D. Roulier [*et al.*]{}, Adv. High Energy Phys. [**2015**]{} (2015) 730437.
A. Bes et al., A position-sensitive neutron detector based on $^{10}$B coated CCD, in preparation.
[ADDI7100 datasheet on analog devices website.](http://www.analog.com/en/products/audio-video/cameracamcorder-analog-front-ends/addi7100.html)
UG472: 7 series FPGA Clocking ressources, Xilinx. [Image sensor handbook](http://www.hamamatsu.com/resources/pdf/ssd/e05_handbook_image_sensors.pdf), p25, Hamamatsu.
Berger, M.J., Coursey, J.S., Zucker, M.A., and Chang, J. (2005), ESTAR, PSTAR, and ASTAR: Computer Programs for Calculating Stopping-Power and Range Tables for Electrons, Protons, and Helium Ions(version 1.2.3). \[Online\] Available: <http://physics.nist.gov/Star>. National Institute of Standards and Technology, Gaithersburg, MD.
J. F. Ziegler, SRIM - The stopping and range of ions in matter (2010), [10.1016/j.nimb.2010.02.091](http://dx.doi.org/10.1016/j.nimb.2010.02.091) Nucl. Instrum. Methods Phys. Res. Sect. B, vol. 268, pp1818-1823
| ArXiv |
---
abstract: 'We consider distributed and dynamic caching of coded content at small base stations (SBSs) in an area served by a macro base station (MBS). Specifically, content is encoded using a maximum distance separable code and cached according to a time-to-live (TTL) cache eviction policy, which allows coded packets to be removed from the caches at periodic times. Mobile users requesting a particular content download coded packets from SBSs within communication range. If additional packets are required to decode the file, these are downloaded from the MBS. We formulate an optimization problem that is efficiently solved numerically, providing TTL caching policies minimizing the overall network load. We demonstrate that distributed coded caching using TTL caching policies can offer significant reductions in terms of network load when request arrivals are bursty. We show how the distributed coded caching problem utilizing TTL caching policies can be analyzed as a specific single cache, convex optimization problem. Our problem encompasses static caching and the single cache as special cases. We prove that, interestingly, static caching is optimal under a Poisson request process, and that for a single cache the optimization problem has a surprisingly simple solution.'
author:
- |
Jesper Pedersen, Alexandre Graell i Amat, , Jasper Goseling, ,\
Fredrik Brännström, , Iryna Andriyanova, , and Eirik Rosnes, [^1] [^2] [^3] [^4] [^5]
bibliography:
- 'confs-jrnls.bib'
- 'IEEEabrv.bib'
- 'library.bib'
title: Dynamic Coded Caching in Wireless Networks
---
Caching, content delivery networks, erasure correcting codes, TTL.
Introduction
============
Distributed wireless caching has attracted a significant amount of attention in the last few years as a promising technology to alleviate the load on backhaul links [@Boccardi2014]. Content may be cached in a distributed fashion across small base stations (SBSs) such that users can download requested content directly from them. For distributed caching, the use of erasure correcting codes has been shown to reduce the download delay as well as the network load [@Shanmugam2013; @Bioglio2015]. Content may also be cached directly in mobile devices such that users can download content from neighboring devices using device-to-device communication. Similar to the SBS caching case, the use of erasure correcting codes has been demonstrated to reduce the network load also for this scenario [@Pedersen2016; @Pedersen2019; @Wang2017]. Caching furthermore facilitates index-coded broadcasts to multiple users requesting different content, which has been shown to drastically reduce the amount of data that has to be transmitted over the SBS-to-device downlink [@Maddah-Ali2014]. All these works consider the cached content to be static for a period of time (e.g., a day) according to a given file popularity distribution.
Dynamic cache eviction policies, e.g., first-in-first-out (FIFO), least-recently-used (LRU), least-frequently-used (LFU), and random (RND), may be beneficial to use when the file library or file popularity profile is dynamic, or when users request content according to a renewal process [@Gelenbe1973]. Due to the complexity in analyzing such policies, timer-based policies that are significantly more tractable have been suggested. One such policy is time-to-live (TTL) where a request for a particular piece of content triggers it to be cached and then evicted after the expiration of a timer. The TTL policy has been shown to yield similar performance to FIFO, LRU, LFU, and RND policies in [@Che2002; @Fricker2012; @Bianchi2013; @Dehghan2019]. Goseling and Simeone extended the TTL policy to cache fractions of files, referred to as fractional TTL (FTTL), and showed that this can improve performance under a renewal request process [@Goseling2019]. Decreasing the fraction of a file that is cached over time, termed soft TTL (STTL), can further improve the performance. Optimal STTL caching policies are obtained through a convex optimization problem [@Goseling2019]. All previous works on TTL policies assume either a single cache or a number of caches, e.g., structured into lines or hierarchies, where users access a single cache. For these scenarios, coded caching does not bring any benefits. However, if users can access several caches, the use of erasure correcting codes can be beneficial. Hence, merging distributed coded caching with the TTL schemes in [@Goseling2019], which have both independently been shown to bring performance improvements, is an intriguing prospect.
In this paper, we generalize the TTL policies in [@Goseling2019] to a distributed coded caching scenario. Specifically, we consider the scenario where content is encoded using a maximum distance separable (MDS) code and cached in a distributed fashion across several SBSs. Coded content is evicted from the caches in accordance with the TTL policies in [@Goseling2019]. Users requesting a particular piece of content download coded packets from SBSs within communication range and, if necessary, download additional packets from a macro base station (MBS). We formulate a network load minimization problem, where the network load is defined as a sum of data rates over various network links, weighted by a cost representing, e.g., transmission delay or energy consumption of transmitting data over these links. We then rewrite the optimization problem as a mixed integer linear program (MILP) that is efficiently solved numerically. We furthermore prove that the distributed coded caching problem can equivalently be analyzed as a single cache problem with a specific decreasing and convex cost function. This is an important result because it shows that such a function, previously studied for the single cache case due to its analytical tractability [@Goseling2019], arises naturally in a distributed caching scenario. For SBSs deployed according to a Poisson point process [@Chiu2013 Ch. 2.3], we derive the cost function explicitly. We analyze two important special cases of the network load minimization problem. In particular, we show that our problem has the static coded caching problem where content is never updated (considered in, e.g., [@Shanmugam2013; @Bioglio2015]), as a special case. We furthermore prove that static coded caching is optimal under the assumption of a Poisson request process. Moreover, for the special case of users accessing a single cache, we prove that the STTL problem is a fractional knapsack problem with a greedy optimal solution. The performance of TTL, FTTL, and STTL, in terms of network load, is evaluated for a renewal process, specifically when the times between requests follow a Weibull distribution. We show that distributed coded caching using TTL caching policies can offer significant reductions in network load, especially for bursty renewal request processes.
Distributed caching of coded content utilizing TTL cache eviction policies was also investigated in [@Chen2019]. Compared to the problem studied in this paper, the work in [@Chen2019] is significantly different in a number of ways. Specifically, we consider an STTL policy with optimized TTL timers under a renewal request process, which was not considered in [@Chen2019]. Furthermore, a dynamic library of files with location-dependent popularity is considered in [@Chen2019], which is typically considered to be more general than a static file library and is not in the scope of our work. However, it is reasonable to consider scenarios where the file library remains fixed for a considerable amount of time, e.g., a day, and focus on an area with homogeneous file popularity.
System Model {#sec:model}
============
We consider an area served by an MBS that always has access to a file library of $N$ files, where file ${i}= 1, 2, \ldots, N$ has size $s_i$. Mobile users request files from the library according to independent renewal processes. Specifically, we denote the independent and identically distributed times between requests for file ${i}$ by $X_i$, the cumulative distribution function (CDF) of $X_i$ by $${F_{X_i}(t)} \triangleq \Pr(X_i \le t),$$ and the request rate of file ${i}$ by $$\omega_i \triangleq \operatorname*{\mathbb{E}}[X_i]^{-1}.$$ We let $p_i = \omega_i/\omega$, for some aggregate request rate in the area, $\omega = \sum_{i=1}^N \omega_i$. For a Poisson request process, i.e., exponentially distributed $X_i$, $p_i$ can be interpreted as the probability that file ${i}$ is requested. The request rates $\omega_i$ are assumed to be constant over a sufficiently long period of time, e.g., not changing during the course of one day. For such scenarios, file popularity predictions and content allocation optimization can be carried out during periods of low network traffic, e.g., during night time. ${B}$ SBSs are deployed in the area and each SBS has a cache with storage capacity $C$. We assume that a user can download content from an SBS if it is within a range ${r_\text{SBS}}$ and we denote by $\gamma_b$ the probability that a user is within range of $b$ SBSs at any given time. The model considered in this paper is illustrated in Fig. \[fig:model\].
Caching Policy {#sec:policy}
--------------
Each file ${i}$ of size $s_i$ is partitioned into $k_i$ packets, each of size $s_i/k_i$. The packets are encoded into $n_i$ coded packets (also of size $s_i/k_i$) using an $(n_i, k_i)$ MDS code. For analytical tractability, we assume that all SBSs cache the same content at all times, i.e., the caches are *synchronized*. With slight abuse of notation, we let $m_i(t) \le k_i$ denote the number of coded packets of file ${i}$ cached at each SBS at time $t$, where $t$ is the time since the last request for file ${i}$. We will use this interpretation of $t$ throughout the paper. The amount of file ${i}$ cached by each SBS at time $t$ is hence $m_i(t) s_i/k_i$. We normalize by the file size $s_i$ and let $$\mu_i(t) = m_i(t)/k_i$$ denote the fraction of file ${i}$ cached at time $t$. Similar to [@Goseling2019], we refer to $\mu_i(t)$ as the *caching policy*.
We adopt an STTL cache eviction policy, shown to increase the amount of content that can be downloaded from a single cache under a renewal request process in [@Goseling2019]. Hence, coded packets of file ${i}$ may be evicted from the caches at periodic times with period $T$ after the last request for file ${i}$. We allow $K$ potential updates within a total time equal to $KT$, which we refer to as the *update window* length. For $K=0$, the caches are never updated. This corresponds to *static caching*, which is the type of caching considered in a big part of the literature [@Shanmugam2013; @Bioglio2015; @Pedersen2016; @Pedersen2019; @Wang2017; @Maddah-Ali2014]. The caching policies $\mu_i(t)$ are decreasing functions of $t$ given by [@Goseling2019] $$\mu_i(t) = \begin{cases}
\mu_{i,0}, & \text{if}~t < T,\\
\mu_{i,j}, & \text{if}~jT \le t < (j+1)T,~j = 1, 2, \ldots, K-1,\\
\mu_{i,K}, & \text{if}~t \ge KT,
\end{cases}
\label{eq:mu}$$ where $$1 \ge \mu_{i,0} \ge \mu_{i,1} \ge \cdots \ge \mu_{i,K} \ge 0.$$ In practice, each $\mu_{i,j}$ has to be quantized to correspond to valid code parameters $k_i$ and $n_i$. In this work, we will assume that $\mu_{i,j} \in {\mathbb{R}}$ for simplicity. We refer to $f = 1/T$ as the update frequency and remark that static caching ($K=0$) corresponds to $f=0$.
Content Download {#sec:down}
----------------
For an MDS code, any $k_i$ coded packets of file ${i}$ suffice to decode the file. All files requested by users can be decoded by downloading packets available at SBSs within communication range and, if necessary, retrieving additional packets from the MBS. Specifically, a user requesting file ${i}$ at time $t$ downloads $m_i(t)$ packets from the $b$ SBSs within communication range. If $bm_i(t) \ge k_i$, the user can decode the file. If $b m_i(t) < k_i$, the additional $k_i-bm_i(t)$ coded packets required to decode the file are downloaded from the MBS. Consequently, the fraction of file ${i}$ downloaded from SBSs can be expressed as $$\min\{1, b\mu_i(t)\}
\label{eq:sbsfrac}$$ and the fraction of file ${i}$ downloaded from the MBS as $$\max\{0, 1-b\mu_i(t)\}.
\label{eq:mbsfrac}$$ We assume that downloading one bit of data from the MBS and the SBSs comes at a cost ${{\theta}_\text{MBS}}$ and ${{\theta}_\text{SBS}}$ per bit, respectively. The cost represents, e.g., the transmission delay or energy consumption of transmitting one bit. Finally, the cost to send data to the caches, referred to as the cache update cost, is denoted by ${{\theta}_\text{C}}$.
Preliminaries {#sec:prel}
=============
The caching policies in correspond to STTL [@Goseling2019]. FTTL policies are obtained as a special case of , where the same fraction $\nu_i$ of file ${i}$ is cached for a time $LT$, defined by an integer $0 \le L \le K$, i.e., $\mu_{i,0} = \mu_{i,1} = \ldots = \mu_{i,L} = \nu_i$ and $\mu_{i,L+1} = \mu_{i,L+2} = \ldots = \mu_{i,K} = 0$ [@Goseling2019]. Furthermore, letting $\nu_i = 1$ we obtain TTL caching policies. In [@Goseling2019], the caching problem is framed as a utility maximization problem where a strictly concave and increasing utility function $g_i(\mu)$ measures the utility resulting from caching a fraction $\mu$ when file ${i}$ is requested. The choice of letting $g_i(\mu)$ be a strictly concave and increasing function of $\mu$ is due to analytical tractability. In practice, a linear utility function is more reasonable [@Neglia2018].
For the case of a single cache, the sum utility maximization solved in [@Goseling2019] is $$\begin{aligned}
\underset{\substack{\mu_{i,j}, \nu_i \in {\mathbb{R}}\\ \beta_{i,j} \in \{0,1\}}}{\text{maximize}}~ & \sum_{i=1}^N \omega_i \sum_{j=0}^K g_i(\mu_{i,j}) F_{i,j}, \label{obj:goseling}\\
\text{subject to}~ & \sum_{i=1}^N \omega_i s_i \sum_{j=0}^K \mu_{i,j} A_{i,j} \le C, \label{cnstr:cache}\\
& 1 \ge \mu_{i,0} \ge \mu_{i,1} \ge \cdots \ge \mu_{i,K} \ge 0, \label{cnstr:mu}\\
& 0 \le \nu_i \le 1, \label{cnstr:nu}\\
& -\beta_{i,j} \le \mu_{i,j}\le \beta_{i,j}, \label{cnstr:beta1}\\
& \beta_{i,j}-1 \le \mu_{i,j}-\nu_i \le 1-\beta_{i,j}, \label{cnstr:beta2}\end{aligned}$$ where is a long-term average cache capacity constraint [@Goseling2019 Lem. 1], $$\label{eq:F}
F_{i,j} = \begin{cases}
{F_{X_i}((j+1)T)}-{F_{X_i}(jT)}, & \text{if}~j = 0,\ldots, K-1,\\
1-{F_{X_i}(KT)}, & \text{if}~j = K
\end{cases}$$ is the probability that file $i$ is requested in time-slot $j$, and $$\label{eq:A}
A_{i,j} = \begin{cases}
\displaystyle{\int}_{jT}^{(j+1)T} 1-{F_{X_i}(t)} {~\mathrm{d}}t, & \text{if}~j = 0,\ldots, K-1,\\[1em]
\displaystyle{\int}_{KT}^\infty 1-{F_{X_i}(t)} {~\mathrm{d}}t, & \text{if}~j = K.
\end{cases}$$ The ratio $F_{i,j}/A_{i,j} \approx h_i(jT)$, where $h_i(\cdot)$ is the hazard function of the request process, represents the probability to observe a request given the time since the last request, and the approximation follows by considering the continuous limit $T \to 0$ [@Goseling2019]. For the remainder of this paper, $h_i(jT)$ is assumed to be decreasing in $j$. The solution to – provides optimal FTTL caching policies [@Goseling2019]. Optimal TTL caching policies are achieved by letting $\nu_i = 1$ and removing the constraint , while STTL policies are achieved by removing the constraints – [@Goseling2019].
Distributed Coded TTL Caching {#sec:analysis}
=============================
In this section, we formulate the average rate at which data is sent through the network described in Section \[sec:model\] and an optimization problem to minimize the network load for coded TTL caching. In particular, we generalize the optimization problem – to a distributed coded caching scenario, utilizing TTL caching policies. We propose an equivalent, more tractable formulation of the optimization problem that is efficiently solved numerically. The average rate at which data is downloaded from the SBSs and the MBS is denoted by ${R_\text{SBS}}$ and ${R_\text{MBS}}$, respectively. We choose the utility function $g_i(\mu_i(t)) = s_i \min\{1, b \mu_i(t)\}$, representing the amount of data that a user requesting file $i$ at time $t$ can download from the $b$ SBSs within communication range (see ). Using and also averaging over the number of SBSs within range of a user requesting a particular content in , we obtain $${R_\text{SBS}}= \sum_{b=0}^{{B}} \gamma_b \sum_{i=1}^N \omega_i s_i \sum_{j=0}^K \min\{1, b\mu_{i,j}\} F_{i,j}.
\label{eq:ratesbs}$$ Similarly, substituting in , the MBS download rate is $$\label{eq:ratembsbeta}
{R_\text{MBS}}= \sum_{b=0}^{{B}} \gamma_b \sum_{i=1}^N \omega_i s_i \sum_{j=0}^K \max\{0, 1-b\mu_{i,j}\} F_{i,j}.$$ Note that $\max\{0, 1-b\mu_{i,j}\} = 1-\min\{1, b\mu_{i,j}\}$ and that, using together with ${F_{X_i}(0)} = 0$, $$\label{eq:sumF}
\sum_{j=0}^K F_{i,j} = {F_{X_i}(KT)}-{F_{X_i}(0)}+1-{F_{X_i}(KT)} = 1.$$ Hence, we may rewrite as $$\begin{aligned}
{R_\text{MBS}}& = \sum_{b=0}^{{B}} \gamma_b \sum_{i=1}^N \omega_i s_i \sum_{j=0}^K (1-\min\{1, b\mu_{i,j}\}) F_{i,j}\nonumber\\
& = \sum_{i=1}^N \omega_i s_i - {R_\text{SBS}},
\label{eq:ratembs}\end{aligned}$$ i.e., all requested content not downloaded from SBSs is downloaded from the MBS. The average data rate at which the SBSs caches are updated, denoted by ${R_\text{C}}$, is $${R_\text{C}}= {B}\sum_{i=1}^N \omega_i s_i \sum_{j=0}^K (\mu_{i,0}-\mu_{i,j}) F_{i,j}.$$ The above expression assumes that all caches are updated at each request in the area, which we refer to as *synchronous* updates. This simplification is due to analytical tractability. Obtaining optimal caching policies under *asynchronous* updates appears to be a formidable task. In Section \[sec:results\], we nonetheless simulate caching policies that are optimal under synchronous updates for an asynchronous cache updating scenario.
We define the *network load* as $$W = {{\theta}_\text{MBS}}{R_\text{MBS}}+{{\theta}_\text{SBS}}{R_\text{SBS}}+{{\theta}_\text{C}}{R_\text{C}},
\label{eq:W}$$ where $${{\theta}_\text{MBS}}{R_\text{MBS}}+{{\theta}_\text{SBS}}{R_\text{SBS}}= {{\theta}_\text{MBS}}\sum_{i=1}^N \omega_i s_i - ({{\theta}_\text{MBS}}-{{\theta}_\text{SBS}}) {R_\text{SBS}}\label{eq:sumrate}$$ using . We want to minimize the network load over the caching policies $\mu_i(t)$ under the constraints –. The first term in is independent of $\mu_i(t)$. Hence, minimizing is equivalent to minimizing $${{\theta}_\text{C}}{R_\text{C}}- ({{\theta}_\text{MBS}}-{{\theta}_\text{SBS}}) {R_\text{SBS}}.$$ Thus, minimizing the network load corresponds to the optimization problem $$\begin{aligned}
\underset{\substack{\mu_{i,j}, \nu_i \in {\mathbb{R}}\\ \beta_{i,j} \in \{0,1\}}}{\text{minimize}}~ & {{\theta}_\text{C}}{R_\text{C}}- ({{\theta}_\text{MBS}}-{{\theta}_\text{SBS}}) {R_\text{SBS}}, \label{obj:min}\\
\text{subject to}~ & \text{\eqref{cnstr:cache}--\eqref{cnstr:beta2}}. \nonumber\end{aligned}$$ Consider briefly the case of zero cache update cost, i.e., ${{\theta}_\text{C}}= 0$. For ${{\theta}_\text{MBS}}> {{\theta}_\text{SBS}}$, we see that represents a maximization of the SBS download rate ${R_\text{SBS}}$ and for ${{\theta}_\text{MBS}}\le {{\theta}_\text{SBS}}$, has a trivial solution $\mu_{i,j} = 0$, i.e., caching at the SBSs is turned off (${R_\text{SBS}}= 0$) and all data is fetched from the MBS, for which $$W = {{\theta}_\text{MBS}}\sum_{i=1}^N \omega_i s_i$$ using and .
Next, we reformulate the optimization problem in a way that is more tractable. Using the epigraph formulation [@Boyd2009 Ch. 3.1.7], we introduce the auxiliary optimization variables $\xi_{b,i,j} \in {\mathbb{R}}$ and the constraints $$\begin{aligned}
\xi_{b,i,j} & \le 1, \label{cnstr:xi1}\\
\xi_{b,i,j} & \le b \mu_{i,j}. \label{cnstr:xi2}\end{aligned}$$ Expressing the SBS download rate in as $$\label{eq:ratesbstilde}
{\tilde{R}_\text{SBS}}= \sum_{b=0}^B \gamma_b \sum_{i=1}^N \omega_i s_i \sum_{j=0}^K \xi_{b,i,j} F_{i,j},$$ with the notation ${\tilde{R}_\text{SBS}}$ to emphasize that it corresponds to the download rate of the epigraph formulation, we formulate the MILP $$\begin{aligned}
\underset{\substack{\mu_{i,j}, \nu_i, \xi_{b,i,j} \in {\mathbb{R}}\\ \beta_{i,j} \in \{0,1\}}}{\text{minimize}}~ & {{\theta}_\text{C}}{R_\text{C}}- ({{\theta}_\text{MBS}}-{{\theta}_\text{SBS}}) {\tilde{R}_\text{SBS}}, \label{obj:epi}\\
\text{subject to}~ & \text{\eqref{cnstr:cache}--\eqref{cnstr:beta2}, \eqref{cnstr:xi1}, \eqref{cnstr:xi2}},\nonumber\end{aligned}$$ which is equivalent to and efficiently solved using, e.g., Gurobi [@Gurobi2018]. The MILP provides optimal FTTL caching policies for the distributed coded caching scenario. Optimal coded TTL policies are achieved by letting $\nu_i = 1$ and removing the constraint , while coded STTL policies are attained by removing the constraints –. Note that the STTL optimization problem is a linear program.
Analysis as Single Cache TTL {#sec:transform}
----------------------------
In this subsection, we will show that the distributed coded caching problem using TTL caching policies in can equivalently be analyzed as a single cache TTL problem using a particular decreasing and convex cost function. We also show how our distributed caching problem maps to the sum utility maximization for the single cache case. Let the random variable $Y$ denote the number of SBSs within range of a user, with $\Pr(Y = b) = \gamma_b$, $b = 0, 1, \ldots, {B}$. Changing the order of summation in yields $$\begin{aligned}
{R_\text{SBS}}& = \sum_{i=1}^N \omega_i s_i \sum_{j=0}^K F_{i,j} \operatorname*{\mathbb{E}}[\min\{1, \mu_{i,j} Y\}].\label{eq:sumb}\end{aligned}$$ Regarding the expectation in , we will need the following lemma in subsequent theorems.
\[lem:exp\] For a nonnegative random variable $Y$ and $\mu \ge 0$, $$\operatorname*{\mathbb{E}}[\min\{1, \mu Y\}] = \int_0^1 1 - F_Y(z/\mu) {~\mathrm{d}}z.$$
See Appendix \[prf:lemexp\].
The following theorem gives some important properties of the expectation in , as a function of the caching policy $\mu_{i,j}$.
\[th:exp\] For a nonnegative random variable $Y$, the expectation $\operatorname*{\mathbb{E}}[\min\{1, \mu Y\}]$ is an increasing and concave function of $\mu \ge 0$.
See Appendix \[prf:exp\].
The result of Theorem \[th:exp\] is interesting because it proves that is convex. Furthermore, it shows how the cost minimization with link costs ${{\theta}_\text{MBS}}= 1$, ${{\theta}_\text{SBS}}= 0$, and no cache update cost (${{\theta}_\text{C}}= 0$), which corresponds to a distributed caching scenario, maps to the utility maximization , which assumes a single cache. The following theorem considers the important special case of SBSs distributed in an area according to a Poisson point process, in which case $Y$ corresponds to a Poisson random variable [@Chiu2013 Ch. 2.3].
\[th:poissexp\] For $Y\sim\text{Poisson}(\lambda)$, $$\label{eq:poissexp}
\operatorname*{\mathbb{E}}[\min\{1, \mu Y\}] = 1+(\lambda\mu-1) Q({\lceil1/\mu\rceil}, \lambda)-\frac{{\mathrm{e}}^{-\lambda} \lambda^{{\lceil1/\mu\rceil}} \mu}{\Gamma({\lceil1/\mu\rceil})},$$ where $Q(\cdot, \cdot)$ is the regularized Gamma function and $\Gamma(\cdot)$ is the Gamma function.
See Appendix \[prf:poissexp\].
The expression is an increasing and concave function of $\mu$, according to Theorem \[th:exp\]. Due to the ceiling function ${\lceil1/\mu\rceil}$ in , we see that we should set $1/\mu \in {\mathbb{N}}$ in order to minimize while not wasting cache capacity resources (see ).
Special Cases
=============
The distributed coded caching problem utilizing TTL caching policies has two interesting problems as special cases; static caching ($K = 0$), studied in [@Shanmugam2013; @Bioglio2015; @Pedersen2019; @Wang2017] for MDS codes, and single cache ($\gamma_1 = 1$), investigated in [@Goseling2019]. In this section, we show the connection between our problem and the special case of static caching, which we prove is optimal under a Poisson request process, and the special case of a single cache, which we prove has a particularly simple optimal solution.
Static Coded Caching
--------------------
Before showing that includes static caching as a special case, we have the following theorem.
\[th:static\] For a Poisson request process, static caching minimizes .
See Appendix \[prf:static\].
Under static caching, FTTL and STTL are identical as only the updates distinguish the two caching policies. In the following, we assume that all files are of equal size, i.e., $s_i = s$, and study the nontrivial case ${{\theta}_\text{MBS}}> {{\theta}_\text{SBS}}$. For static caching ($K=0$), reduces to $$F_{i,j} = F_{i,0} = 1 - {F_{X_i}(0)} = 1$$ and the objective function becomes $$\begin{aligned}
-{R_\text{SBS}}& = -\sum_{b=0}^{{B}} \gamma_b \sum_{i=1}^N \omega_i s_i \min\{1, b\mu_{i,0}\} F_{i,0}\nonumber\\
& = -\omega s \sum_{b=0}^{{B}} \gamma_b \sum_{i=1}^N p_i \min\{1, b\mu_{i,0}\}.\label{obj:bioglio}\end{aligned}$$ Also, is $$A_{i,j} = A_{i,0} = \int_0^\infty 1-{F_{X_i}(t)} {~\mathrm{d}}t = \operatorname*{\mathbb{E}}[X_i] = \omega_i^{-1}.$$ Hence, the constraint simplifies to $$\sum_{i=1}^N \omega_i s_i \sum_{j=0}^K \mu_{i,j} A_{i,j} = s \sum_{i=1}^N \mu_{i,0} \le C.
\label{cnstr:bioglio}$$ Using in , under constraints and , the optimization problem is precisely the static caching problem considered in [@Bioglio2015], apart from additive and multiplicative constants. Hence, the static caching problem explored in [@Bioglio2015] is a special case of .
Single Cache TTL
----------------
We proceed with the other interesting special case, i.e., the single cache problem. Letting $\gamma_1 = 1$ in , i.e., users access a single cache with probability 1, we see that the average rate at which data is downloaded from the SBSs is $$\begin{aligned}
{R_\text{SBS}}& = \sum_{i=1}^N \omega_i s_i \sum_{j=0}^K \min\{1, \mu_{i,j}\} F_{i,j} \nonumber\\
& = \sum_{i=1}^N \omega_i s_i \sum_{j=0}^K \mu_{i,j} F_{i,j}, \label{obj:minsingle}\end{aligned}$$ since $\mu_{i,j} \le 1$ using . For the nontrivial case of ${{\theta}_\text{MBS}}> {{\theta}_\text{SBS}}$, and update cost ${{\theta}_\text{C}}= 0$, the minimization problem is equivalent to a maximization problem of the objective function . In particular, the STTL problem has a surprisingly simple solution given by the following theorem.
\[th:fracknap\] For users accessing a single cache, link costs ${{\theta}_\text{MBS}}> {{\theta}_\text{SBS}}$ and ${{\theta}_\text{C}}= 0$, the STTL optimization problem, i.e., maximizing under constraints and , is a fractional knapsack problem with a greedy optimal solution equivalent to FTTL and TTL.
See Appendix \[prf:fracknap\].
A similar result was proved in [@Goseling2019] for a single file, i.e., $N=1$.
Numerical Results {#sec:results}
=================
In the following, we will assume that the times between requests are distributed according to the Weibull distribution, which has been shown to accurately estimate inter-request times [@Costa2004], i.e., $X_i \sim \text{Weibull}(a, b_i)$, where $a$, $0 < a \le 1$, and $b_i$ are the shape and scale parameters of the distribution, respectively. Its CDF is $${F_{X_i}(t)} = 1 - \exp\left[-\left(\frac{t}{b_i}\right)^{a}\right].$$ Also, $$\omega_i^{-1} = \operatorname*{\mathbb{E}}[X_i] = b_i \Gamma(1+a^{-1}),$$ which implies $$b_i = \frac{1}{\omega_i \Gamma(1+a^{-1})}.$$ We assume the aggregate request rate per hour $\omega = 100$ and $$p_i = \frac{1/i^\alpha}{\sum_{\ell=1}^N 1/\ell^\alpha},$$ which is the Zipf probability mass function with parameter $\alpha \ge 0$. We remind that $p_i$ has the interpretation of file popularity under a Poisson request process.
The area is defined by the communication range of the MBS, which is denoted by ${r_\text{MBS}}$ and assumed to be ${r_\text{MBS}}= 800$ meters (m), i.e., the considered area is $\pi {r_\text{MBS}}^2 \approx 2$ square kilometers. The SBSs are deployed in the area according to a Poisson point process. Let $\rho$ be the density of SBSs per square kilometer (km$^{-2}$), i.e., $\rho = {B}/(\pi {r_\text{MBS}}^2)$. The probability that a user is within range of $b$ SBSs is [@Chiu2013 Ch. 2.3] $$\gamma_b = {\mathrm{e}}^{-\lambda} \frac{\lambda^b}{b!},$$ where $\lambda = \rho \pi {r_\text{SBS}}^2 = {B}({r_\text{SBS}}/{r_\text{MBS}})^2$. Unless stated otherwise, we will assume the following setup for the remainder of this section. The library holds $N = 100$ files, each of normalized size $s_i = 1$. We set the Weibull shape $a = 0.6$, which is within the range specified in [@Costa2004]. Also, we set $\alpha = 0.7$, which has been shown to accurately capture the popularity of Youtube videos [@Cheng2008]. We assume that there are $B = 100$ SBSs in the area, corresponding to a density $\rho \approx 50$ km$^{-2}$ and that users can download content from SBSs within a range of ${r_\text{SBS}}= 100$ m. Each SBS has the capacity to cache $C = 10$ files or $10\%$ of the file library. We assume the link costs ${{\theta}_\text{SBS}}= 0$ and ${{\theta}_\text{MBS}}= 1$. Furthermore, we assume that ${{\theta}_\text{C}}\ll {{\theta}_\text{MBS}}$, which is a reasonable assumption since data can be transmitted to caches over high capacity fiber-optical or highly directional wireless backhaul links, while the MBS serves a large number of users over potentially large distances. Finally, we consider an update window length of $K/f = 1$ hour and update frequencies $f = 6$ per hour.
![The fraction of data downloaded from the MBS as a function of the SBS density $\rho$.[]{data-label="fig:density"}](density.pdf){width="\columnwidth"}
We obtain optimal TTL, FTTL, and STTL caching policies by solving and plot the network load normalized by the aggregate request rate $\omega$. For ${{\theta}_\text{C}}= 0$, the network load is interpreted as the fraction of data downloaded from the MBS. Fig. \[fig:density\] shows this fraction as a function of the SBS density $\rho$ for no cache updates, i.e., $f = 0$ implying $\mu_{i,j} = \mu_{i,0}$, and cache update frequency per hour $f = 6$. The network load using FTTL and STTL overlap for $f = 0$, which is expected since only the cache updates distinguish the two policies. We also see that there is a reduction in network load when choosing the FTTL or STTL caching policies over the TTL policy and that the network load decreases with increasing SBS density. The reason for the performance loss when using the TTL caching policy is that users within range of $b>1$ SBSs will download superfluous data, which correspond to a wasteful use of cache memory resources. The gain for the static caching scenario ($f = 0$) was observed already in [@Bioglio2015]. Finally, we observe that, for $f=6$, there is only a small reduction in network load for STTL as compared to FTTL, but the load reduction is increasing for increasing $\rho$.
![The fraction of data downloaded from the MBS as a function of the Weibull shape $a$ for an update frequency $f=6$.[]{data-label="fig:shape"}](shape.pdf){width="\columnwidth"}
Fig. \[fig:shape\] shows the fraction of data downloaded from the MBS versus the Weibull shape $a$, for update frequency per hour $f=6$, and no cache update cost (${{\theta}_\text{C}}=0$). We also include curves for static caching ($f=0$), which do not depend on $a$ as is shown in and , for comparison. For bursty request arrivals, i.e., small values of $a$, we see that the use of TTL caching policies reduces the fraction of data downloaded from the MBS significantly with respect to static caching. Furthermore, we observe that, for very small values of $a$, all TTL policies have similar performance. This is because, with high probability, the times between requests are less than the period $T$, i.e., $F_{i,0} \approx 1$ for all $i$, and TTL is an optimal caching policy. For $a=1$, corresponding to a Poisson request process, FTTL and STTL yield the same network load as proved in Theorem \[th:static\], which is, however, lower than the network load using TTL. A similar effect was shown in [@Goseling2019] for the single cache case.
Fig. \[fig:updatefreq\] shows the normalized minimum network load as a function of the update frequency $f$ for the case of no cache update cost (${{\theta}_\text{C}}= 0$) and ${{\theta}_\text{C}}= 10^{-3}$. For both cases, updating content on the caches is seen to be beneficial for all TTL policies. For example, using STTL and assuming ${{\theta}_\text{C}}= 0$, the reduction is roughly 10% as compared to static caching. We observe that the decrease in network load for an increase in update frequency saturates for moderately large $f$. Hence, cache updates need not be very frequent to reap the benefits of the TTL, FTTL, and STTL caching policies. The sufficient update frequency of course depends on several parameter values, in particular, the Weibull shape $a$ is a key parameter when deciding update frequencies.
Finally, in Fig. \[fig:updatecost\], the normalized minimum network load is plotted versus the cache update cost ${{\theta}_\text{C}}$ for an update frequency per hour $f=6$. The load for static caching ($f=0$) is also shown in the figure. As previously described, the network load when using FTTL and STTL is the same for static caching. It is interesting to note that all TTL policies revert to static caching for sufficiently large values of ${{\theta}_\text{C}}$. Also included in Fig. \[fig:updatecost\] is a simulation of the optimal (under synchronous cache updates) STTL and TTL caching policies for asynchronous cache updates, i.e., only the SBSs within range of a user placing a request update cached content. The considered caching policies do not exhibit a better performance under asynchronous updates, which is to be expected for two reasons. Firstly, since the file request process is homogenous over the considered area, the spatial average cached content is important and the same average cached content can be achieved by both synchronous and asynchronous cache updates. Secondly, the request rate within the communication range of an SBS is smaller than the request rate in the entire area, implying less content to be cached over time using asynchronous updates, i.e., the caches are underutilized.
![Normalized network load as a function of the cache update frequency $f$.[]{data-label="fig:updatefreq"}](updatefreq.pdf){width="\columnwidth"}
Conclusion
==========
We optimized time-to-live (TTL) caching policies with periodic eviction of coded content to minimize the overall network load for a scenario where content is encoded using a maximum distance separable code and cached in a distributed fashion across small base stations. The proposed optimization problem is efficiently solved numerically. Interestingly, we show that the problem can equivalently be analyzed as a single cache optimization problem under a specific decreasing and convex cost function. For small base stations deployed according to a Poisson point process, we provide the cost function explicitly. The analyzed scenario encompasses static caching and single caching as important special cases. We proved that, interestingly, static caching is optimal under a Poisson request process. We also proved that the single cache problem has a simple greedy solution. We showed that TTL caching policies can offer substantial reductions in network load compared with static caching under a request renewal process, in particular when the request process is bursty. Conversely, for sufficiently large cache update cost, dynamic caching is futile, i.e., static caching is optimal. Finally, although we consider a wireless network scenario, the results are general in the sense that they can be applied to any distributed caching scenario.
![Normalized network load versus the update cost ${{\theta}_\text{C}}$ when using the various caching policies under synchronous and asynchronous cache updates.[]{data-label="fig:updatecost"}](updatecost.pdf){width="\columnwidth"}
Proof of Lemma \[lem:exp\] {#prf:lemexp}
==========================
We represent $1$ as a random variable with degenerate distribution $\delta(z-1)$, where $\delta(\cdot)$ is the Dirac delta function, and let $Z = \min\{1, \mu Y\}$, for which the CDF of $Z$ is $$\begin{aligned}
F_Z(z) & \triangleq \Pr(Z \le z)\\
& = \Pr(\min\{1, \mu Y\} \le z)\\
& = 1-\Pr(1>z, \mu Y > z)\\
& = 1-H(1-z) (1-F_Y(z/\mu)),\end{aligned}$$ where $H(\cdot)$ is the heavyside function. The expected value of $Z$ is $$\begin{aligned}
\operatorname*{\mathbb{E}}[Z] & = \int_0^\infty 1-F_Z(z) {~\mathrm{d}}z\\
& = \int_0^\infty H(1-z) (1-F_Y(z/\mu)) {~\mathrm{d}}z\\
& = \int_0^1 1- F_Y(z/\mu) {~\mathrm{d}}z.\end{aligned}$$
Proof of Theorem \[th:exp\] {#prf:exp}
===========================
Since $F_Y(y)$ is an increasing function of $y$, $1-F_Y(z/\mu)$ is an increasing function of $\mu$, and $$\int_0^1 1-F_Y(z/\mu) {~\mathrm{d}}z$$ is an increasing function of $\mu$. Using Lemma \[lem:exp\], the expectation $\operatorname*{\mathbb{E}}[\min\{1, \mu Y\}]$ is an increasing function of $\mu$.
For $\mu_1 \ge 0$, $\mu_2 \ge 0$, $Y \ge 0$, and $0 \le \alpha \le 1$, the following inequalities hold, $$\begin{aligned}
(1-\alpha) \mu_1 Y & \ge (1-\alpha) \min\{1, \mu_1 Y\},\\
\alpha \mu_2 Y & \ge \alpha \min\{1, \mu_2 Y\}.\end{aligned}$$ Hence, $$((1-\alpha) \mu_1 + \alpha \mu_2) Y \ge (1-\alpha) \min\{1, \mu_1 Y\} + \alpha \min\{1, \mu_2 Y\}.
\label{eq:conv1}$$ Similarly, using $$\begin{aligned}
(1-\alpha) & \ge (1-\alpha) \min\{1, \mu_1 Y\},\\
\alpha & \ge \alpha \min\{1, \mu_2 Y\},\end{aligned}$$ we have that $$1 = 1-\alpha+\alpha \ge (1-\alpha) \min\{1, \mu_1 Y\} + \alpha \min\{1, \mu_2 Y\}.
\label{eq:conv2}$$ Using and , we get $$\begin{aligned}
& \min\{1, ((1-\alpha) \mu_1 + \alpha \mu_2) Y\}\\
& \hspace{4em} \ge (1-\alpha) \min\{1, \mu_1 Y\} + \alpha \min\{1, \mu_2 Y\}.\end{aligned}$$ Taking the expectation of both sides yields $$\begin{aligned}
& \operatorname*{\mathbb{E}}[\min\{1, ((1-\alpha) \mu_1 + \alpha \mu_2) Y\}]\\
& \hspace{4em} \ge \operatorname*{\mathbb{E}}[(1-\alpha) \min\{1, \mu_1 Y\} + \alpha \min\{1, \mu_2 Y\}]\\
& \hspace{4em} = (1-\alpha) \operatorname*{\mathbb{E}}[\min\{1, \mu_1 Y\}] + \alpha \operatorname*{\mathbb{E}}[\min\{1, \mu_2 Y\}],\end{aligned}$$ which concludes the proof.
Proof of Theorem \[th:poissexp\] {#prf:poissexp}
================================
For a Poisson random variable $Y$ with rate $\lambda$, $$\label{eq:ycdf}
F_Y(y) = \sum_{i=0}^{{\lfloory\rfloor}} \frac{\lambda^i}{i!} {\mathrm{e}}^{-\lambda}.$$ For a positive integer $x$, let $\Gamma(x)$ and $Q(x, \lambda)$ denote the Gamma function and the regularized Gamma function, i.e., $\Gamma(x) = (x-1)!$ and $$\label{eq:qrec}
Q(x+1, \lambda) = \int_\lambda^\infty \frac{t^x {\mathrm{e}}^{-t}}{\Gamma(x+1)} {~\mathrm{d}}t \overset{(a)}{=} \frac{\lambda^x {\mathrm{e}}^{-\lambda}}{\Gamma(x+1)} + Q(x, \lambda),$$ respectively, where $(a)$ is obtained after integration by parts. Unfolding the recursion in , using $Q(1, \lambda) = {\mathrm{e}}^{-\lambda}$ yields $$\label{eq:qsum}
Q(x+1, \lambda) = \sum_{i=0}^x \frac{\lambda^i}{\Gamma(i+1)} {\mathrm{e}}^{-\lambda} = \sum_{i=0}^x \frac{\lambda^i}{i!} {\mathrm{e}}^{-\lambda}$$ and $$\label{eq:qcdf}
Q({\lfloory\rfloor}+1, \lambda) = F_Y(y),$$ using .
Using , $$\begin{aligned}
& \int_0^1 F_Y(y/\mu) {~\mathrm{d}}y = \int_0^1 Q({\lfloory/\mu\rfloor}+1, \lambda) {~\mathrm{d}}y\nonumber\\
& \hspace{2em} = \mu \sum_{i=1}^{{\lfloor1/\mu\rfloor}} Q(i, \lambda) + (1-{\lfloor1/\mu\rfloor}\mu) Q({\lfloor1/\mu\rfloor}+1, \lambda),\label{eq:cdfint}\end{aligned}$$ where the integral is a summation due to the floor function in the argument of $Q(\cdot, \cdot)$. Applying the recursion repeatedly yields $$\begin{aligned}
\sum_{i=1}^{{\lfloor1/\mu\rfloor}} Q(i, \lambda) & = {\lfloor1/\mu\rfloor} Q({\lfloor1/\mu\rfloor}+1, \lambda) - \sum_{i=1}^{{\lfloor1/\mu\rfloor}} i \frac{\lambda^i {\mathrm{e}}^{-\lambda}}{\Gamma(i+1)}\\
& = {\lfloor1/\mu\rfloor} Q({\lfloor1/\mu\rfloor}+1, \lambda) - \lambda \sum_{j=0}^{{\lfloor1/\mu\rfloor}-1} \frac{\lambda^{j} {\mathrm{e}}^{-\lambda}}{\Gamma(j+1)}\\
& \overset{(b)}{=} {\lfloor1/\mu\rfloor} Q({\lfloor1/\mu\rfloor}+1, \lambda) - \lambda Q({\lfloor1/\mu\rfloor}, \lambda),\end{aligned}$$ where we have used in $(b)$. Inserting this expression in , one obtains $$\begin{aligned}
\int_0^1 F_Y(y/\mu) {~\mathrm{d}}y & = Q({\lfloor1/\mu\rfloor}+1, \lambda) - \mu \lambda Q({\lfloor1/\mu\rfloor}, \lambda)\nonumber\\
& \overset{(c)}{=} (1-\lambda\mu) Q({\lceil1/\mu\rceil}, \lambda) + \frac{\lambda^{{\lceil1/\mu\rceil}} \mu {\mathrm{e}}^{-\lambda}}{\Gamma({\lceil1/\mu\rceil})},\label{eq:cdfint2}\end{aligned}$$ where we have used and $${\lceil1/\mu\rceil}-{\lfloor1/\mu\rfloor} = \begin{cases}
0, & \text{if}~1/\mu \in \mathbb{Z}\\
1, & \text{if}~1/\mu \notin \mathbb{Z}
\end{cases}$$ in $(c)$. Combining with the result of Lemma \[lem:exp\] yields the desired result.
Proof of Theorem \[th:static\] {#prf:static}
==============================
For proving that static caching minimizes (and hence ) it is sufficient to show that it maximizes ${\tilde{R}_\text{SBS}}$ (see ), as for static caching $R_C = 0$. Maximizing under constraints , , , and , is equivalent to $$\begin{aligned}
\underset{\mu_{i,j}, \xi_{b,i,j}, C_i \in {\mathbb{R}}}{\text{maximize}}~ & \sum_{b=0}^B \gamma_b \sum_{i=1}^N \omega_i s_i \sum_{j=0}^K \xi_{b,i,j} F_{i,j},\label{obj:sbs}\\
\text{subject to}~ & s_i \sum_{j=0}^K \mu_{i,j} F_{i,j} \le C_i,\label{cnstr:cache1}\\
& \sum_{i=1}^N C_i = C,\label{cnstr:cache2}\\
& \mu_{i,j} - \mu_{i,j-1} \le 0,~\mu_{i,-1} = 1,\\
& -\mu_{i,j} \le 0,\\
& \xi_{b,i,j} \le 1,\\
& \xi_{b,i,j} \le b \mu_{i,j},\label{cnstr:xi2sbs}\end{aligned}$$ where $C_i$ may be regarded as the size of the cache partition reserved for file $i$, and we used the fact that for exponentially distributed inter-request times $F_{i,j}/A_{i,j} = \omega_i$ (from and ), i.e., the hazard function is constant for a Poisson request process. For fixed $C_i$’s, the maximization problem is separable in $i$. Thus, we can consider the following optimization problem $$\begin{aligned}
\underset{\mu_{i,j}, \xi_{b,i,j} \in {\mathbb{R}}}{\text{maximize}}~ & \sum_{b=0}^B \gamma_b \sum_{j=0}^K \xi_{b,i,j} F_{j}, \label{obj:single}\\
\text{subject to}~ & s_i \sum_{j=0}^K \mu_{i,j} F_{i,j} \le C_i, \label{cnstr:cachesingle}\\
& \mu_{i,j} - \mu_{i,j-1} \le 0,~\mu_{i,-1} = 1, \label{cnstr:mu1single}\\
& -\mu_{i,j} \le 0, \label{cnstr:mu2single}\\
& \xi_{b,i,j} \le 1, \label{cnstr:xi1single}\\
& \xi_{b,i,j} \le b \mu_{i,j}, \label{cnstr:xi2single}\end{aligned}$$ for each file $i=1, \ldots, N$ separately. We can now prove the following lemma.
\[lem:kkt\] Static caching is an optimal solution to –.
For ease of exposition, we drop the subindex $i$ in the proof. Introducing the dual variables $\lambda$, $\phi_j$, $\psi_j$, $\delta_{b,j}$, and $\epsilon_{b,j}$, the Karush-Kuhn-Tucker (KKT) conditions of – are $$\begin{aligned}
-\gamma_b F_j + \delta_{b,j} + \epsilon_{b,j} & = 0,\\
\lambda s F_j + \phi_j - \phi_{j+1} - \psi_j - \sum_{b=0}^B \epsilon_{b,j} b & = 0,~\phi_{K+1} = 0,\\
\lambda \left( - C + s \sum_{j=0}^K \mu_j F_j \right) & = 0,\\
\phi_j (\mu_j-\mu_{j-1}) & = 0,~\mu_{-1} = 1,\\
\psi_j (-\mu_j) & = 0,\\
\delta_{b,j} (\xi_{b,j}-1) & = 0,\\
\epsilon_{b,j} (\xi_{b,j}-b\mu_j) & = 0,\\
\lambda & \ge 0,\\
\phi_j & \ge 0,\\
\psi_j & \ge 0,\\
\delta_{b,j} & \ge 0,\\
\epsilon_{b,j} & \ge 0,\end{aligned}$$ and –. Let $$\begin{aligned}
\mu_j & = C/s,\\
\xi_{b,j} & = \min\{1, b C/s\},\end{aligned}$$ which corresponds to static caching utilizing completely the given cache partition, and largest possible values of the variables $\xi_{b,j}$. Furthermore, let $$\begin{aligned}
\phi_j & = 0,\\
\psi_j & = 0,\\
\delta_{b,j} & = \begin{cases}
0, & \text{if}~b \le s/C\\
\gamma_b F_j, & \text{if}~b > s/C
\end{cases},\\
\epsilon_{b,j} & = \begin{cases}
\gamma_b F_j, & \text{if}~b \le s/C\\
0, & \text{if}~b > s/C
\end{cases},\label{eq:epsilon}\\
\lambda & = \frac{1}{s} \sum_{j=0}^K \sum_{b=0}^B \epsilon_{b,j} b \overset{(a)}{=} \frac{1}{s} \sum_{b=0}^{{\lfloors/C\rfloor}} \gamma_b b,\end{aligned}$$ where we have used and in $(a)$. It is readily verified that the choice of optimization and dual variables satisfy the KKT conditions and are hence optimal since the problem is convex [@Boyd2009 Ch. 5.5.3]. Therefore, static caching maximizes .
It remains to optimize over the $C_i$, but since, by Lemma \[lem:kkt\], static caching is optimal for any assignment of $C_i$’s, it is optimal for –.
Proof of Theorem \[th:fracknap\] {#prf:fracknap}
================================
Letting $\gamma_1 = 1$, ${{\theta}_\text{C}}= 0$, and ${{\theta}_\text{MBS}}> {{\theta}_\text{SBS}}$, the STTL problem, i.e., maximizing under constraints and , is equivalent to $$\begin{aligned}
\underset{\mu_{i,j} \in {\mathbb{R}}}{\text{maximize}}~ & \sum_{i=1}^N \omega_i s_i \sum_{j=0}^K \mu_{i,j} F_{i,j},\\
\text{subject to}~ & \sum_{i=1}^N \omega_i s_i \sum_{j=0}^K \mu_{i,j} A_{i,j} \le C,\\
& 1 \ge \mu_{i,0} \ge \mu_{i,1} \ge \cdots \ge \mu_{i,K} \ge 0.\label{cnstr:mufracknap}\end{aligned}$$ Relaxing the constraint , replacing it with $0 \le \mu_{i,j} \le 1$, and letting $x_{i,j} = \omega_i s_i \mu_{i,j} A_{i,j}$, we obtain $$\begin{aligned}
\underset{x_{i,j} \in {\mathbb{R}}}{\text{maximize}}~ & \sum_{i=1}^N \sum_{j=0}^K x_{i,j} \frac{F_{i,j}}{A_{i,j}},\\
\text{subject to}~ & \sum_{i=1}^N \sum_{j=0}^K x_{i,j} \le C,\\
& 0 \le x_{i,j} \le \omega_i s_i A_{i,j},\end{aligned}$$ which is recognized as the fractional knapsack problem [@Danzig1957]. The optimal solution to this problem is obtained by setting $x_{i,j} = \omega_i s_i A_{i,j}$, i.e., $\mu_{i,j} = 1$, greedily with respect to the fractions $F_{i,j}/A_{i,j}$ [@Danzig1957]. We observe that, since $F_{i,j}/A_{i,j}$ is decreasing in $j$ as explained in Sec. \[sec:prel\], the constraint is met and we have a valid STTL caching policy. Apart from the possibility that one $\mu_{i,j}<1$ depending on the value of $C$, the constraints – are also satisfied by this policy and, hence, the optimal STTL caching policy is equivalent to the optimal FTTL and TTL caching policies.
[^1]: This work was funded by the Swedish Research Council under grant 2016-04253 and by the National Center for Scientific Research in France under grant CNRS-PICS-2016-DISCO.
[^2]: J. Pedersen, A. Graell i Amat, and F. Brännström are with the Department of Electrical Engineering, Chalmers University of Technology, SE-41296 Gothenburg, Sweden (e-mail: {jesper.pedersen, alexandre.graell, fredrik.brannstrom}@chalmers.se).
[^3]: J. Goseling is with the Department of Applied Mathematics, University of Twente, 7522 Enschede, The Netherlands (e-mail: [email protected]).
[^4]: I. Andriyanova is with the ETIS-UMR8051 group, ENSEA/University of Cergy-Pontoise/CNRS, 95015 Cergy, France (e-mail: [email protected]).
[^5]: E. Rosnes is with Simula UiB, N-5020 Bergen, Norway (e-mail: [email protected]).
| ArXiv |
---
abstract: 'Topological measures and deficient topological measures are defined on open and closed subsets of a topological space, generalize regular Borel measures, and correspond to (non-linear in general) functionals that are linear on singly generated subalgebras or singly generated cones of functions. They lack subadditivity, and many standard techniques of measure theory and functional analysis do not apply to them. Nevertheless, we show that many classical results of probability theory hold for topological and deficient topological measures. In particular, we prove a version of Aleksandrov’s Theorem for equivalent definitions of weak convergence of deficient topological measures. We also prove a version of Prokhorov’s Theorem which relates the existence of a weakly convergent subsequence in any sequence in a family of topological measures to the characteristics of being a uniformly bounded in variation and uniformly tight family. We define Prokhorov and Kantorovich-Rubenstein metrics and show that convergence in either of them implies weak convergence of (deficient) topological measures on metric spaces. We also generalize many known results about various dense and nowhere dense subsets of deficient topological measures. The present paper constitutes a first step to further research in probability theory and its applications in the context of topological measures and corresponding non-linear functionals.'
author:
- 'S. V. Butler, University of California, Santa Barbara'
date: 'May 20, 2020'
title: Weak convergence of topological measures
---
Introduction
============
The origins of the theory of quasi-linear functionals and topological measures lie in mathematical axiomatization and interpretations of quantum physics ([@vonN], [@MackeyPaper], [@MackeyBook], [@Kadison]). In J. von Neumann’s axiomatization of quantum mechanics, physical observables can be represented by the space $\mathcal{L}$ of Hermitian operators on a complex Hilbert space. The state of a physical system is represented by a positive normalized linear functional on $\mathcal{L}$. Some physicists, however, argued that the linearity of the functional, $\rho(A + B) = \rho(A) + \rho(B), \, A, B \in \mathcal{L} $, makes sense if observables $A$ and $B$ are simultaneously measurable, which means that $A,B$ are polynomials of the same $C \in \mathcal{L} $, so $A,B$ belong to the subalgebra of $\mathcal{L} $ generated by $C$. Mathematical interpretations of quantum physics by G. W. Mackey and R. V. Kadison led to very interesting mathematical problems, including the extension problem for probability measures in von Neumann algebras. This extension problem may be regarded as a special case of the linearity problem for physical states, which is closely related to the existence of quasi-linear functionals. J. F. Aarnes [@Aarnes:TheFirstPaper] introduced quasi-linear functionals (that are not linear) on $C(X) $ for a compact Hausdorff space $X$ and corresponding set functions, generalizing measures (initially called quasi-measures, now topological measures). He connected the two by establishing a representation theorem. Aarnes’s quasi-linear functionals are functionals that are linear on singly generated subalgebras, but (in general) not linear. For more information about physical interpretation of quasi-linear functionals see [@EntovPolterovich], [@EnPoltZap], [@Entov], [@PoltRosenBook], [@Aarnes:PhysStates69], [@Aarnes:QuasiStates70], [@Aarnes:TheFirstPaper].
M. Entov and L. Polterovich first linked the theory of quasi-linear functionals to symplectic topology. They introduced symplectic quasi-states and partial symplectic quasi-states ([@EntovPolterovich]), which are subclasses of quasi-linear functionals. (On a symplectic manifold that is a closed oriented surface every normalized quasi-linear functional is a symplectic quasi-state, see [@PoltRosenBook Chapter 5]). Article [@EntovPolterovich] was followed by numerous papers and a monograph [@PoltRosenBook], and many authors have investigated and used various aspects of symplectic quasi-states and topological measures: their properties, their connection to spectral numbers and homogeneous quasi-morphisms, ways of constructing and approximating symplectic quasi-states, etc. Symplectic quasi-states can be used as a measurement of Poisson commutativity, and topological measures can be used to distinguish Lagrangian knots that have identical classical invariants ([@EntovPolterovich Chapters 4,6]). Symplectic quasi-states and topological measures play an important role in function theory on symplectic manifolds.
Deficient topological measures are generalizations of topological measures. They were first defined and used by A. Rustad and O. Johansen ([@OrjanAlf:CostrPropQlf]) and later independently reintroduced and further developed by M. Svistula ([@Svistula:Signed], [@Svistula:DTM]). Deficient topological measures are not only interesting by themselves, but also provide an essential framework for studying topological measures and quasi-linear functionals. Topological measures and deficient topological measures generalize regular Borel measures and correspond to functionals that are linear on singly generated subalgebras or singly generated cones of functions. These non-linear functionals can be described in several ways, including symmetric and asymmetric Choquet integrals, see [@DD pp. 62, 87] and [@Butler:ReprDTM Corollary 8.5, Theorem 8.7, Remark 8.11]. Deficient topological measures are not supermodular, and their domains are not closed under intersection and union; for these and other reasons, results of Choquet theory do not automatically translate for functionals representing deficient topological measures. It is interesting that, with different proof methods, one may obtain results that are typical for, stronger than, or strikingly different from Choquet theory results.
Topological measures and deficient topological measures are defined on open and closed subsets of a topological space, which means that there is no algebraic structure on the domain. They lack subadditivity and other properties typical for measures, and many standard techniques of measure theory and functional analysis do not apply to them. Nevertheless, we show that many classical results of probability theory hold for topological and deficient topological measures. In particular, we prove versions of Aleksandrov’s Theorem for equivalent definitions of weak convergence of topological and deficient topological measures. We also prove a version of Prokhorov’s Theorem which relates the existence of a weakly convergent subsequence in any sequence in a family of topological measures to the characteristics of being a uniformly bounded in variation and uniformly tight family. We define Prokhorov and Kantorovich-Rubenstein metrics and show that convergence in either of them implies weak convergence of deficient topological measures. We also generalize many known results about various dense and nowhere dense subsets of deficient topological measures.
The present paper constitutes a first step to further research in probability theory and its applications in the context of topological measures and corresponding non-linear functionals.
In this paper $X$ is a locally compact space Hausdorff space. By $C(X)$ we denote the set of all real-valued continuous functions on $X$ with the uniform norm, by $C_0(X)$ the set of continuous functions on $X$ vanishing at infinity, by $C_c(X)$ the set of continuous functions with compact support, and by $C_0^+(X)$ the collection of all nonnegative functions from $C_0(X)$. When we consider maps into extended real numbers we assume that any such map is not identically $\infty$.
We denote by $\overline E$ the closure of a set $E$, and by $ \bigsqcup$ a union of disjoint sets. A set $A \subseteq X$ is called bounded if $\overline A$ is compact. We denote by $id$ the identity function $id(x) = x$, and by $1_K$ the characteristic function of a set $K$. By $ supp \, f $ we mean $ \overline{ \{x: f(x) \neq 0 \} }$. We say that $Y$ is dense in $Z$ if $Z \subseteq \overline Y$.
Several collections of sets are used often. They include: $\mathscr{O}(X)$; $\mathscr{C}(X)$; and $\mathscr{K}(X)$– the collection of open subsets of $X$; the collection of closed subsets of $X $; and the collection of compact subsets of $X $, respectively.
\[MDe2\] Let $X$ be a topological space and $\nu$ be a set function on a family $\mathcal{E}$ of subsets of $X$ that contains $\mathscr{O}(X) \cup \mathscr{C}(X)$ with values in $[0, \infty]$. We say that
- $\nu$ is compact-finite if $ \nu(K) < \infty$ for any $ K \in \mathscr{K}(X)$;
- $\nu$ is simple if it only assumes values $0$ and $1$;
- $ \nu$ is finite if $ \nu(X) < \infty$;
- $\nu$ is inner regular (or inner compact regular) if $\nu(A) = \sup \{ \nu(C) : C \subseteq A, C \in \mathscr{K}(X)\}$ for $A \in \mathcal{E}$;
- $\nu$ is inner closed regular if $\nu(A) = \sup \{ \nu(C) : C \subseteq A, C \in \mathscr{C}(X) \}$ for $A \in \mathcal{E}$;
- $\nu$ is outer regular if $\nu(A) = \inf \{ \nu(U) : A \subseteq U, U \in \mathscr{O}(X) \}$ for $A \in \mathcal{E}$.
A measure on $X$ is a countably additive set function on a $\sigma$-algebra of subsets of $X$ with values in $[0, \infty]$. A Borel measure on $X$ is a measure on the Borel $\sigma$-algebra on $X$. A Radon measure $m$ on $X$ is a compact-finite Borel measure that is outer regular on all Borel sets, and inner regular on all open sets, i.e. $m(K) < \infty$ for every compact $K$, $ m(E) = \inf \{ m(U): E \subseteq U, U \text{ is open} \} $ for every Borel set $E$, and $m(U) = \sup \{ m(K): K \subseteq U, K \text{ is compact} \}$ for every open set $U$. For a Borel measure $m$ that is inner regular on all open sets (in particular, for a Radon measure) we define $supp \ m$, the support of $m$, to be the complement of the largest open set $W$ such that $m(W) =0$.
For the following fact see, for example, [@Dugundji Chapter XI, 6.2] and [@Butler:TMLCconstr Lemma 7].
\[easyLeLC\] Let $K \subseteq U, \ K \in \mathscr{K}(X), \ U \in \mathscr{O}(X)$ in a locally compact space $X$. Then there exists a set $V \in \mathscr{O}(X)$ such that $ C = \overline V$ is compact and $ K \subseteq V \subseteq \overline V \subseteq U. $ If $X$ is also locally connected, and either $K$ or $U$ is connected, then $V$ and $C$ can be chosen to be connected.
\[DTM\] A deficient topological measure on a locally compact space $X$ is a set function $\nu: \mathscr{C}(X) \cup \mathscr{O}(X) \longrightarrow [0, \infty]$ which is finitely additive on compact sets, inner compact regular, and outer regular, i.e. :
1. \[DTM1\] if $C \cap K = \emptyset, \ C,K \in \mathscr{K}(X)$ then $\nu(C \sqcup K) = \nu(C) + \nu(K)$;
2. \[DTM2\] $ \nu(U) = \sup\{ \nu(C) : \ C \subseteq U, \ C \in \mathscr{K}(X) \} $ for $U\in\mathscr{O}(X)$;
3. \[DTM3\] $ \nu(F) = \inf\{ \nu(U) : \ F \subseteq U, \ U \in \mathscr{O}(X) \} $ for $F \in \mathscr{C}(X)$.
Clearly, for a closed set $F$, $ \nu(F) = \infty$ iff $ \nu(U) = \infty$ for every open set $U$ containing $F$. If two deficient topological measures agree on compact sets (or on open sets) then they coincide.
\[TMLC\] A topological measure on $X$ is a set function $\mu: \mathscr{C}(X) \cup \mathscr{O}(X) \longrightarrow [0,\infty]$ satisfying the following conditions:
1. \[TM1\] if $A,B, A \sqcup B \in \mathscr{K}(X) \cup \mathscr{O}(X) $ then $
\mu(A\sqcup B)=\mu(A)+\mu(B);
$
2. \[TM2\] $
\mu(U)=\sup\{\mu(K):K \in \mathscr{K}(X), \ K \subseteq U\}
$ for $U\in\mathscr{O}(X)$;
3. \[TM3\] $
\mu(F)=\inf\{\mu(U):U \in \mathscr{O}(X), \ F \subseteq U\}
$ for $F \in \mathscr{C}(X)$.
By $ \mathbf{DTM}(X)$ and $ \mathbf{TM}(X)$ we denote, respectively, the collections of all finite deficient topological measures and all finite topological measures on $X$.
The following two theorems from [@Butler:DTMLC Section 4] give criteria for a deficient topological measure to be a topological measure or a measure.
\[DTMtoTM\] Let $X$ be compact, and $\nu$ a deficient topological measure. The following are equivalent:
1. $\nu$ is a real-valued topological measure;
2. $\nu(X) = \nu(C) + \nu(X \setminus C), \quad C \in \mathscr{C}(X);$
3. $\nu(X) \le \nu(C) + \nu(X \setminus C),\quad C \in \mathscr{C}(X).$
Let $X$ be locally compact, and $\nu$ a deficient topological measure. The following are equivalent:
1. $\nu$ is a topological measure;
2. $\nu(U) = \nu(C) + \nu(U \setminus C), \quad C \in \mathscr{K}(X),\quad U \in \mathscr{O}(X);$
3. $\nu(U) \le \nu(C) + \nu(U \setminus C),\quad C \in \mathscr{K}(X),\quad U \in \mathscr{O}(X).$
\[subaddit\] Let $\mu$ be a deficient topological measure on a locally compact space $X$. The following are equivalent:
- If $C, K$ are compact subsets of $X$, then $\mu(C \cup K ) \le \mu(C) + \mu(K)$.
- If $U, V$ are open subsets of $X$, then $\mu(U \cup V) \le \mu(U) + \mu(V)$.
- $\mu$ admits a unique extension to an inner regular on open sets, outer regular Borel measure $m$ on the Borel $\sigma$-algebra of subsets of $X$. $m$ is a Radon measure iff $\mu$ is compact-finite. If $\mu$ is finite then $m$ is an outer regular and inner closed regular Borel measure.
\[Vloz\] Let $X$ be locally compact, and let $ \mathscr{M}$ be the collection of all Borel measures on $X$ that are inner regular on open sets and outer regular on all Borel sets. Thus, $ \mathscr{M}$ includes regular Borel measures and Radon measures. We denote by $M(X)$ the restrictions to $\mathscr{O}(X) \cup \mathscr{C}(X)$ of measures from $ \mathscr{M}$, and by $ \mathbf{M}(X)$ the set of all finite measures from $M(X)$. We have: $$\begin{aligned}
\label{incluMTD}
M(X) \subsetneqq TM(X) \subsetneqq DTM(X).\end{aligned}$$ The inclusions follow from the definitions. When $X$ is compact, there are examples of topological measures that are not measures and of deficient topological measures that are not topological measures in numerous papers, beginning with [@Aarnes:TheFirstPaper], [@OrjanAlf:CostrPropQlf], and [@Svistula:Signed]. When $X$ is locally compact, see [@Butler:TechniqLC], Sections 5 and 6 in [@Butler:DTMLC], and Section 9 in [@Butler:TMLCconstr] for more information on proper inclusion in (\[incluMTD\]), criteria for a deficient topological measure to be a measure from $ M(X)$, and various examples.
\[tausm\] In [@Butler:DTMLC Section 3] we show that a deficient topological measure $ \nu$ is $\tau$-smooth on compact sets (i.e. if a net $K_\alpha \searrow K$ , where $ K_\alpha, K \in \mathscr{K}(X)$ then $\mu(K_\alpha) \rightarrow \mu(K)$), and also $\tau$-smooth on open sets (i.e. if a net $U_\alpha \nearrow U$, where $U_\alpha, U \in \mathscr{O}(X)$ then $\mu(U_\alpha) \rightarrow \mu(U)$). In particular, a deficient topological measure is additive on open sets. A deficient topological measure $ \nu$ is also superadditive, i.e. if $ \bigsqcup_{t \in T} A_t \subseteq A, $ where $A_t, A \in \mathscr{O}(X) \cup \mathscr{C}(X)$, and at most one of the closed sets (if there are any) is not compact, then $\nu(A) \ge \sum_{t \in T } \nu(A_t)$. If $ F \in \mathscr{C}(X)$ and $C \in \mathscr{K}(X)$ are disjoint, then $ \nu(F) + \nu(C) = \nu ( F \sqcup C)$. One may consult [@Butler:DTMLC] for more properties of deficient topological measures on locally compact spaces.
\[SDTMnorDe\] For a deficient topological measure $\mu$ we define $ \| \mu \| = \mu(X) = \sup \{ \mu(K): K \in \mathscr{K}(X) \}$.
\[cqlf\] We call a functional $\rho$ on $C_0(X)$ with values in $[ -\infty, \infty]$ (assuming at most one of $\infty, - \infty$) and $| \rho(0) | < \infty$ a p-conic quasi-linear functional if
1. If $f\, g=0, f, g \ge 0$ then $ \rho(f+ g) = \rho(f) + \rho(g)$.
2. If $0 \le g \le f$ then $\rho(g) \le \rho(f)$.
3. For each $f$, if $g,h \in A^+(f), \ a,b \ge 0$ then $\rho(a g + bh) = a \rho(g) + b \rho(h)$. Here $ A^+(f) = \{ \phi \circ f: \ \phi \in C(\overline{f(X)}), \phi \mbox{ is non-decreasing}\} $, (with $ \phi(0) = 0 $ if $X$ is non-compact) is a cone generated by $f$.
For a functional $\rho$ on $C_0(X)$ we consider $\| \rho \| = \sup \{ | \rho(f) | : \ \| f \| \le 1 \} $ and we say $\rho$ is bounded if $\| \rho \| < \infty$. Let $\mathbf{\Phi^+}(C_0^+(X))$ be the set of all bounded p-conic quasi-linear $\rho$ functionals on $C_0^+(X)$.
A real-valued map $\rho$ on $C_0(X)$ is a quasi-linear functional (or a positive quasi-linear functional) if
1. \[QIpositLC\] $ f \ge 0 \Longrightarrow \rho(f) \ge 0.$
2. \[QIconsLC\] $\rho(a f) = a \rho(f)$ for $ a \in \mathbb{R}.$
3. \[QIlinLC\] For each $f $, if $g,h \in B(f)$, then $\rho(h + g) = \rho (h) + \rho (g)$. Here $B(f) = \{ \phi \circ f : \, \phi \in C(\overline{f(X)}) \} $ (with $ \phi(0) = 0 $ if $X$ is non-compact) is a subalgebra generated by $f$.
\[RemBRT\]
There is an order-preserving bijection between $\mathbf{DTM}(X)$ and $\mathbf{\Phi^+}(C_0^+(X))$. See [@Butler:ReprDTM Section 8]. In particular, there is an order-preserving isomorphism between finite topological measures on $X$ and quasi-linear functionals on $C_0(X)$ of finite norm, and $\mu$ is a measure iff the corresponding functional is linear (see [@Butler:ReprDTM Theorem 8.7], [@Alf:ReprTh Theorem 3.9], and [@Svistula:DTM Theorem 15]). We outline the correspondence.
1. \[prt1\] Given a finite deficient topological measure $\mu$ on a locally compact space $X$ and $f \in C_b(X)$, define functions on $\mathbb{R}$: $$R_1 (t) = R_{1, \mu, f} (t) = \mu(f^{-1} ((t, \infty) )),$$ $$R_2 (t) = R_{2, \mu, f} (t) =\mu(f^{-1} ([t, \infty) )).$$ Let $r$ be the Lebesque-Stieltjes measure associated with $-R_1$, a regular Borel measure on $ \mathbb{R}$. The $ supp \ r \subseteq \overline{f(X)}$. We define a functional on $C_b(X)$ (in particular, a functional on $C_0(X)$): $$\begin{aligned}
\label{rfform}
\mathcal{R} (f) & = \int _{\mathbb{R}} id \, dr = \int_{[a,b]} id \, dr = \int_a^b R_1 (t) dt + a \mu(X) = \int_a^b R_2 (t) dt + a \mu(X). \end{aligned}$$ where $[a,b]$ is any interval containing $f(X)$. If $f(X) \subseteq [0,b]$ we have: $$\begin{aligned}
%\label{rfformp}
\mathcal{R} (f) = \int_{[0,b]} id \, dr = \int_0^b R_1 (t) dt = \int_0^b R_2 (t) dt.\end{aligned}$$ We call the functional $\mathcal{R}$ a quasi-integral (with respect to a deficient topological measure $ \mu$) and write: $$\begin{aligned}
%\label{RFint}
\int_X f \, d\mu = \mathcal{R}(f) = \mathcal{R}_{\mu} (f) = \int _{\mathbb{R}} id \, dr.\end{aligned}$$
2. \[RHOsvva\] Functional $\mathcal{R} $ is non-linear. By [@Butler:ReprDTM Lemma 7.7, Theorem 7.10, Lemma 3.6, Lemma 7.12] we have:
1. $\mathcal{R} (f) $ is positive-homogeneous, i.e. $\mathcal{R} (cf) = c \mathcal{R} (f) $ for $c \ge 0$ and $ f \in C_b(X)$.
2. $\mathcal{R} (0) =0$.
3. $\mathcal{R}$ is monotone, i.e. if $ f \le g$ then $\mathcal{R} (f) \le \mathcal{R} (g)$ for $f, g \in C_b(X)$.
4. $ \mu(X) \cdot \inf_{x \in X} f(x) \le \mathcal{R}(f) \le \mu(X) \cdot \sup _{x \in X} f(x) $ for $f \in C_b(X)$.
5. If $f g = 0 $, where $ f, g \ge 0$ then $\mathcal{R} (f+g) = \mathcal{R} (f) + \mathcal{R} (g)$ for $f, g \in C_b(X)$;\
if $f g = 0 $, where $f \ge 0, g \le 0$ or $ f, g \ge 0$, then $\mathcal{R} (f+g) = \mathcal{R} (f) + \mathcal{R} (g)$ for $f, g \in C_0(X)$.
3. \[mrDTM\] A functional $\rho$ with values in $[ -\infty, \infty]$ (assuming at most one of $\infty, - \infty$) and $| \rho(0) | < \infty$ is called a d-functional if on nonnegative functions it is positive-homogeneous, monotone, and orthogonally additive, i.e. for $f, g \in D(\rho)$ (the domain of $ \rho$) we have: (d1) $f \ge 0, \ a > 0 \Longrightarrow \rho (a f) = a \rho(f)$; (d2) $0 \le g \le f \Longrightarrow \rho(g) \le \rho(f) $; (d3) $f \cdot g = 0, f,g \ge 0 \Longrightarrow \rho(f + g) = \rho(f) + \rho(g)$.
Let $\rho$ be a d-functional with $ C_c^+(X) \subseteq D(\rho) \subseteq C_b(X)$. In particular, we may take functional $ \mathcal{R}$ on $ C_0^+(X)$. The corresponding deficient topological measure $ \mu = \mu_{\rho}$ is given as follows:
If $U$ is open, $ \mu_{\rho}(U) = \sup\{ \rho(f): \ f \in C_c(X), 0\le f \le 1, supp \, f\subseteq U \},$
if $F$ is closed, $ \mu_{\rho}(F) = \inf \{ \mu_{\rho}(U): \ F \subseteq U, U \in \mathscr{O}(X) \}$.
If $K$ is compact, $ \mu_{\rho}(K) = \inf \{ \rho(g): \ g \in C_c(X), g \ge 1_K \}
= \inf \{ \rho(g): \ g \in C_c(X), 1_K \le g \le 1 \}. $ (See [@Butler:ReprDTM Section 5].)
If given a finite deficient topological measure $\mu$, we obtain $ \mathcal R$, and then $\mu_{ \mathcal R}$, then $ \mu = \mu_{ \mathcal R}$.
\[LipQLF\] Integrals with respect to (deficient) topological measures on a locally compact space $X$ have Lipschitz property: If $\mu$ is a finite deficient topological measure, $f, g \in C_c(X), \, f,g \ge 0, \, supp \, f, supp \, g \subseteq K$ where $K$ is compact, then $$| \mathcal{R}(f) - \mathcal{R}(g) | = | \int_X f \, d\mu - int_X g \, d\mu | \le \| f - g \| \, \mu(K).$$ If $ \mu$ is a finite topological measure, $f, g \in C_0(X)$ then $$| \int_X f \, d\mu - int_X g \, d\mu | \le 2 \| f - g \| \, \mu(X).$$ See [@Butler:ReprDTM Lemma 7.12] and [@Butler:DTMLC Corollary 53].
We would like to give some examples.
A set $A$ is bounded if $\overline{A}$ is compact. If $X$ is locally compact, non-compact, a set $A$ is solid if $A$ is connected, and $X \setminus A$ has only unbounded connected components. If $X$ is compact, a set $A$ is solid if $A$ and $X \setminus A$ are connected.
Many examples of topological measures that are not measures are obtained in the following way. Define a so-called solid-set function on bounded open solid and compact solid sets in a locally compact, connected, locally connected, Hausdorff space. A solid set function extends to a unique topological measure. See [@Aarnes:ConstructionPaper Definition 2.3, Theorem 5.1], [@Butler:TMLCconstr Definition 39, Theorem 48].
\[ExDan2pt\] Suppose that $ \lambda$ is the Lebesgue measure on $X = \mathbb{R}^2$, and the set $P$ consists of two points $p_1 = (0,0)$ and $p_2 = (2,0)$. For each bounded open solid or compact solid set $A$ let $ \nu(A) = 0$ if $A \cap P = \emptyset$, $ \nu(A) = \lambda(A) $ if $A$ contains one point from $P$, and $ \nu(A) = 2 \lambda(X)$ if $A$ contains both points from $P$. Then $\nu$ is a solid-set function (see [@Butler:TMLCconstr Example 61]), and $\nu$ extends to a unique topological measure on $X$. Let $K_i$ be the closed ball of radius $1$ centered at $p_i$ for $i=1,2$. Then $K_1, K_2$ and $ C= K_1 \cup K_2$ are compact solid sets, $\nu(K_1) = \nu(K_2) = \pi, \, \nu(C) = 4 \pi$. Since $\nu$ is not subadditive, it can not be a measure. The quasi-linear functional corresponding to $ \nu$ is not linear.
\[nvssf\] Let $X = \mathbb{R}^2$ or a square, $n$ be a natural number, and let $P$ be a set of distinct $2n+1$ points. For each bounded open solid or compact solid set $A$ let $ \nu(A) = i/n$ if $ A$ contains $2i$ or $2i+1$ points from $P$. The set function $ \nu$ defined in this way is a solid-set function, and it extends to a unique topological measure on $X$ that assumes values $0, 1/n, \ldots, 1$. See [@Aarnes:Pure Example 2.1], [@QfunctionsEtm Examples 4.14, 4.15], and [@Butler:TMLCconstr Example 65]. The resulting topological measure is not a measure. For instance, when $X$ is the square and $n=3$, it is easy to represent $X = A_1 \cup A_2 \cup A_3$, where each $A_i$ is a compact solid set containing one point from $P$. Then $\nu(A_i) =0$ for $i=1,2,3$, while $\nu(X) = 1$. Since $\nu$ is not subbadditive, it is not a measure, and the quasi-linear functional $\rho$ corresponding to $ \nu$ is not linear. In [@Butler:QLFLC Example 56] we take $n=5$ and show that there are $f,g \ge 0$ such that $ \rho(f+g) \neq \rho(f) + \rho(g)$. If $X$ is locally compact, non-compact, for the functional $\rho$ we consider a new functional $ \rho_g$ defined by $\rho_g(f) = \rho(gf)$, where $g \ge 0$. The new functional $\rho_g$ corresponds to a deficient topological measure obtained by integrating $g$ over closed and open sets with respect to a topological measure $\nu$. We can choose $ g \ge 0$ or $ g >0$ so that $\rho_g$ is no longer linear on singly generated subalgebras, but only linear on singly generated cones. See [@Butler:Integration Example 32, Theorem 40] for details.
Let $X$ be locally compact, and let $D$ be a connected compact subset of $X$. Define a set function $\nu$ on $\mathscr{O}(X) \cup \mathscr{C}(X)$ by setting $\nu(A) = 1$ if $ D \subseteq A$ and $\nu(A) = 0$ otherwise, for any $A \in \mathscr{O}(X) \cup \mathscr{C}(X)$. If $D$ has more than one element, then $\nu$ is a deficient topological measure, but not a topological measure. See [@Butler:DTMLC Example 46] and [@Svistula:DTM Example 1, p.729] for details.
For more examples of topological measures and quasi-integrals on locally compact spaces see [@Butler:TechniqLC] and the last sections of [@Butler:TMLCconstr] and [@Butler:QLFLC]. For more examples of deficient topological measures see [@Butler:DTMLC] and [@Svistula:DTM].
Aleksandrov’s Theorem for deficient topological measures
========================================================
\[defwk\] The weak topology on $ \mathbf{DTM}(X)$ is the coarsest (weakest) topology for which maps $ \mu \longmapsto \mathcal{R}_{\mu} (f), f \in C_0^+(X) $ are continuous.
The basic neighborhoods for the weak topology have the form $$\begin{aligned}
\label{wstRho}
N(\nu, f_1, \ldots, f_n, \epsilon) = \{ \mu \in \mathbf{DTM}(X): \ |\mathcal{R}_{\mu}(f_i) - \mathcal{R}_{\nu} (f_i) | &< \epsilon, \, f_i \in C_0^+(X), \\
&i=1, \ldots, n \notag \}.\end{aligned}$$
Let $\mu_{\alpha}$ be a net in $\mathbf{DTM}(X)$, $\mu \in \mathbf{DTM}(X)$. The net $ \mu_{\alpha} $ converges weakly to $ \mu$ (and we write $ \mu_{\alpha} \Longrightarrow \mu$) iff $\mathcal{R}_{\mu_{\alpha}} (f) \rightarrow \mathcal{R}_{\mu} (f)$ for every $ f \in C_0^+(X)$, i.e. $\int f \, d \mu_{\alpha} \rightarrow \int f \, d \mu$ for every $ f \in C_0^+(X)$.
By [@Butler:ReprDTM Theorem 8.7], $\mathbf{DTM}(X)$ with weak convergence is homeomorphic to $\mathbf{\Phi^+}(C_0^+(X))$ with pointwise convergence, and $\mathbf{TM}(X)$ is homeomorphic to the space of quasi-linear functionals with pointwise convergence.
\[wkwk\*\] Our definition of weak convergence corresponds to one used in probability theory. It is the same as a functional analytical definition of $wk*$ convergence on $\mathbf{DTM}(X)$ (respectively, on $\mathbf{TM}(X)$), which is justified by the fact that this topology agrees with the weak$^*$ topology induced by p-conic quasi-linear functionals (respectively, quasi-linear functionals). In many papers the term “$wk^*$-topology” is used.
\[muContSet\] Let $\mu$ be a deficient topological measure. A set $A$ is called a $\mu$-continuity set if $ \mu (\overline A) = \mu(A^o)$.
In probability theory, with $\mu$ a measure, a set $A$ is called a $\mu$-continuity set if $\mu( \partial A) =0$. If $\mu$ is a measure (or $\mu$ is a topological measure and $\overline A$ is compact) this definition is equivalent to Definition \[muContSet\]. If $ \mu$ is a deficient topological measure, then by superadditivity $ \mu( \overline A )\ge \mu(A^o) + \mu( \partial A)$, so for any $\mu$-continuity set $A$ we have $\mu( \partial A) =0$.
We have the following generalizations of Aleksandrov’s well-known theorem for weak convergence of measures. (Aleksandrov’s Theorem is often incorrectly called the “Portmanteau theorem”, a usage apparently deliberately started by Billingsley, who in [@BillingsleySv] cited a paper of the non-existent mathematician Jean-Pierre Portmanteau, “published” in a non-existent issue of the Annals of non-existent university; see [@Pfanzagl p.130] and [@Simon p.313].) This theorem gives equivalent definitions of weak convergence.
\[AleksandrovLC\] Let $X$ be locally compact, and let $\mu, \mu_{\alpha}$ be deficient topological measures. The following are equivalent:
1. \[portm1\] $\int f \, d \mu_{\alpha} \rightarrow \int f \, d \mu$ ( i.e. $\mathcal{R}_{\mu_{\alpha}} (f) \rightarrow \mathcal{R}_{\mu} (f)$ ) for every $ f \in C_0^+(X)$.
2. \[portm3\] $\liminf \mu_{\alpha} (U) \ge \mu(U)$ for any $ U \in \mathscr{O}(X)$ and $ \limsup \mu_{\alpha} (K) \le \mu(K)$ for any $ K \in \mathscr{K}(X)$.
3. \[portm4\] $ \mu_{\alpha} (A) \rightarrow \mu(A)$ for any compact or open bounded $\mu$-continuity set $A$.
4. \[portm5\] If $ f \in C_0^+(X)$ then $R_{2, \mu_{\alpha}, f} (t) \rightarrow R_{2, \mu, f} (t) $ and $R_{1, \mu_{\alpha}, f} (t) \rightarrow R_{1, \mu, f} (t) $ for each point $t$ at which $R_{2, \mu, f}$ is continuous.
\[portm1\] $ \Rightarrow $ \[portm3\]. Let $ U \in \mathscr{O}(X), \ K \in \mathscr{K}(X), \epsilon >0$. By part \[mrDTM\] of Remark \[RemBRT\] choose $f, g \in C_c(X)$ such that $supp f \subseteq U, \ K \subseteq supp\, g$ and $ \mathcal{R}_{\mu} (f) > \nu(U) - \epsilon, \ \ \mathcal{R}_{\mu} (g) < \nu(K) + \epsilon.$ Choose $ \alpha_0$ such that $ | \mathcal{R}_{\mu_{\alpha}} (f) - \mathcal{R}_\mu (f) | < \epsilon $ and $ | \mathcal{R}_{\mu_{\alpha}} (g) - \mathcal{R}_\mu (g) | < \epsilon $ for all $ \alpha > \alpha_0$. Then $$\begin{aligned}
%\label{muale}
\mu_{\alpha} (U) \ge \mathcal{R}_{\mu_{\alpha}} (f) > \mathcal{R}_{\mu} (f) - \epsilon > \mu(U) - 2\epsilon , \\
\mu_{\alpha} (K) \le \mathcal{R}_{\mu_{\alpha}} (g) < \mathcal{R}_{\mu}(g) + \epsilon < \mu(K) + 2 \epsilon,\end{aligned}$$ and it is easy to see that $\liminf \mu_{\alpha} (U) \ge \mu(U)$ and $\limsup \mu_{\alpha} (K) \le \mu(K)$.\
\[portm3\] $ \Rightarrow $ \[portm4\]. We have: $\mu(A^o) \le \liminf \mu_{\alpha} (A^o) \le \liminf \mu_{\alpha} (A) \le \limsup \mu_{\alpha} (A) \le \limsup \mu_{\alpha} (\overline A) \le \mu(\overline A). $ If $A$ is an $\mu$-continuity set (whether $A$ is compact or open bounded), we then see that $\lim \mu_{\alpha} (A) = \mu(A)$.\
\[portm4\] $ \Rightarrow $ \[portm5\]. If $ t$ is a point of continuity of $ R_{2, \mu, f}$ then from [@Butler:ReprDTM Lemma 6.3 (III)] it follows that the sets $ f^{-1}((t, \infty))$ and $ f^{-1}([t, \infty))$ are $\mu$-continuity sets. The statement follows from \[portm4\].\
\[portm5\] $ \Rightarrow $ \[portm1\]. By [@Butler:ReprDTM Lemma 6.3] $R_{2, \mu, f}$ has at most countably many points of discontinuity; the statement follows from formulas (\[rfform\]) and \[portm5\].
If $\mu, \mu_{\alpha}$ are finite topological measures on a compact space $X$, and $ \lim \mu_{\alpha} (X) = \mu(X)$, then from part \[TM1\] of Definition \[TMLC\] it follows that $\liminf \mu_{\alpha} (U) \ge \mu(U)$ for any $ U \in \mathscr{O}(X)$ iff $ \limsup \mu_{\alpha} (D) \le \mu(D)$ for any $ D \in \mathscr{C}(X)$. Therefore, we have the following version of Aleksandrov’s Theorem:
\[AleksandrovLCtm\] Let $X$ be compact, and let $\mu, \mu_{\alpha}$ be finite topological measures. TFAE:
1. \[portm1tm\] $\int f \, d \mu_{\alpha} \rightarrow \int f \, d\mu$ ( i.e. $\mathcal{R}_{\mu_{\alpha}} (f) \rightarrow \mathcal{R}_{\mu} (f)$ ) for every $ f \in C(X)$.
2. \[portm3tm\] $\liminf \mu_{\alpha} \ge \mu(U)$ for any $ U \in \mathscr{O}(X)$ and $ \lim \mu_{\alpha} (X) = \mu(X)$.
3. \[portm3atm\] $ \limsup \mu_{\alpha} (D) \le \mu(D)$ for any $ D \in \mathscr{C}(X)$ and $ \lim \mu_{\alpha} (X) = \mu(X)$.
4. \[portm4tm\] $ \mu_{\alpha} (A) \rightarrow \mu(A)$ for any $\mu$-continuity set $A$.
5. \[portm5tm\] If $ f \in C_0^+(X)$ then $R_{2, \mu_{\alpha}, f} (t) \rightarrow R_{2, \mu, f} (t) $ and $R_{1, \mu_{\alpha}, f} (t) \rightarrow R_{1, \mu, f} (t) $ for each point $t$ at which $R_{2, \mu, f}$ is continuous.
\[wkBase\] The weak topology on $\mathbf{DTM}(X)$ is given by basic neighborhoods of the form $$\begin{aligned}
%\label{WkNbd}
W( &\nu, U_1, \ldots, U_n, C_1, \ldots, C_m, \epsilon) = \{ \mu \in DTM: \ \mu(U_i) > \nu(U_i) - \epsilon, \ \mu(C_j) < \nu(C_j) + \epsilon, \\
& i=1, \ldots, n, \ j=1, \ldots m \} \end{aligned}$$ where $\nu \in \mathbf{DTM}(X), U_i \in \mathscr{O}(X), C_j \in \mathscr{K}(X), \epsilon >0, n, m \in \mathbb{N}$.
The weak topology is the topology $\tau_N$ given by basic neighborhoods of the form (\[wstRho\]). It is easy to see that the sets $ W(\nu, U_1, \ldots, U_n, C_1, \ldots, C_m, \epsilon)$ are basic neighborhoods for some topology $\tau_W$ on $\mathbf{DTM}(X)$. Consider a basic neighborhood $ W(\nu, U, C, \epsilon)$. Given $\epsilon >0$, by part \[mrDTM\] of Remark \[RemBRT\] choose $f, g \in C_c(X)$ such that $ supp f \subseteq U, g \ge 1_K$ and $$\mathcal{R}_{\nu} (f) > \nu(U) - \frac{\epsilon}{2}, \ \ \mathcal{R}_{\nu} (g) < \nu(C) + \frac{\epsilon}{2}.$$ Let $ \mu \in N(\nu, f,g, \epsilon/2)$ as in (\[wstRho\]). We have: $$\mu(U) > \mathcal{R}_{\mu} (f) > \mathcal{R}_{\nu} (f) - \frac{\epsilon}{2} > \nu(U) - \epsilon,$$ $$\mu(C) \le \mathcal{R}_{\mu}(g) < \mathcal{R}_{\nu}(g) + \frac{\epsilon}{2} < \nu(C) + \epsilon.$$ Therefore, $ N(\nu, f, g, \epsilon/2) \subseteq W(\nu, U, C, \epsilon)$. We see that $\tau_W \subseteq \tau_N$, i.e. $\tau_W$ is a coarser topology than $\tau_N$. If $ \mu_{\alpha} \rightarrow \mu$ in the topology $\tau_W$ then it is easy to see that $\liminf \mu_{\alpha}(U) \ge \mu(U)$ for any open set $U$, and that $\limsup \mu_{\alpha}(K) \ge \mu(K)$ for any compact set $K$. By Theorem \[AleksandrovLC\] $\int f \, d \mu_{\alpha} \rightarrow \int f \, d\mu$ for every $f \in C_0^+(X)$. The weak topology $\tau_N$ is the coarsest topology with this property, thus, $ \tau_N = \tau_W$.
\[basicTP\] The space $\mathbf{DTM}(X)$ is Hausdorff and locally convex. Every set of the form $\{ \mu \in \mathbf{DTM}(X): \mu(X) \le c\} = \{ \mathcal{R}: \| \mathcal{R} \| \le c\} , c>0$ is compact. If $X$ is compact then $\mathbf{DTM}(X)$ is locally compact.
First we shall show that $\mathbf{DTM}(X)$ is Hausdorff. Suppose $ \mu \neq \nu$, then there is $ K \in \mathscr{K}(X)$ such that $ \nu(K) \neq \mu(K)$. Let $ | \mu(K) - \nu(K) | = 5 \epsilon >0$. By part \[mrDTM\] of Remark \[RemBRT\] find $g, h \in C_c(X)$ such that $ \mathcal{R}_{\mu}(g) - \mu(K) < \epsilon, \, \mathcal{R}_{\nu}(h) - \nu(K) < \epsilon$. Let $f = g \wedge h$, so $ \mathcal{R}_{\mu}(f) - \mu(K) < \epsilon, \, \mathcal{R}_{\nu}(f) - \nu(K) < \epsilon$. Then $N(\mu, f, \epsilon)$ and $N(\nu, f , \epsilon)$ as in formula (\[wstRho\]) are disjoint neighborhoods of $ \mu$ and $ \nu$: otherwise, if $ \lambda \in N(\mu, f, \epsilon) \cap N(\nu, f, \epsilon) $ then $ | \mu(K) - \nu(K) | \le | \mu(K) - \mathcal{R}_{\mu}(f)| + | \mathcal{R}_{\mu}(f) - \mathcal{R}_{\lambda} (f) |
+ | \mathcal{R}_{\lambda}(f) - \mathcal{R}_{\nu} (f) | + | \mathcal{R}_{\nu} (f) - \nu(K)| < 4\epsilon
< | \mu(K) - \nu(K) | $, which is a contradiction.
One can also see that $\mathbf{DTM}(X)$ is Hausdorff because a homeomorphic space $\mathbf{\Phi^+}(C_0^+(X))$ is Hausdorff. The basic open set in $\mathbf{\Phi^+}(C_0^+(X))$ is of the form $W = \{ \mathcal{R}: \mathcal{R}(f_i) \in O_i, \, O_ i
\mbox{ are open in } \mathbb{R}, \, f_i \in C_0^+(X), \, i=1, \ldots, n, \}$. If $ \mathcal{R} $ and $\rho$ are in $W$, then their convex combination is also in $W$. Thus, $\mathbf{DTM}(X)$ is locally convex.
Let $ c>0$ and $ P= \{ \mu \in \mathbf{DTM}(X): \mu(X) \le c\}$. Consider the product space $$\begin{aligned}
Y = \prod_{f \in C_0^+(X)} [ -c \| f \| , c \| f \| \ ] \end{aligned}$$ and the function $ T : P \longrightarrow Y$ defined by $( T(\mu) )_f = \rho_{\mu} (f) = \int f \, d\mu $. The function $T$ is continuous , since each of the maps $ \mu \longmapsto \rho_{\mu} (f) $ is continuous. $ T$ is $ 1-1 $ which follows from Remark \[RemBRT\]. Also $ T : P \longrightarrow T(P) $ is a homeomorphism, because $ T(\mu_{\gamma}) \longrightarrow T(\mu_0) $ implies $ \mu_{\gamma} \longrightarrow \mu_0 $. To show that $P$ is compact it is enough to show that $ T(P) $ is closed in $Y$. Let $ T(\mu_{\alpha}) \longrightarrow L $ in $ Y $. Define $ \rho(f) = L_f, \, f \in C_0^+(X) $. Then $\rho$ is a p-conic quasi-linear functional, and by Remark \[RemBRT\] there exists a finite deficient topological measure $ \mu_0 $ such that $ \rho = \rho_{\mu_0} $. Then $ L_f= \rho_f=(\rho_{\mu_0})_f = (T(\mu_0))_f $, i.e. $ L=T(\mu_0) $.
If $X$ is compact, then for $\nu \in \mathbf{DTM}(X)$ and $W = \{ \mu : \mu(X) < \nu(X) + \epsilon\}$ we have: $ \nu \in W \subseteq \{ \mu: \mu(X) \le \nu(X) + \epsilon \}$, and the last set is compact.
Prokhorov’s Theorem for topological measures
============================================
In this section we show that several classical results of probability theory hold for deficient topological measures or topological measures.
If each sequence $\{\mu_{n_i}\} $ of $\{\mu_{n}\} $, where $\mu_n$ are deficient topological measures, contains a further subsequence $\{\mu_{n_{i_j}}\} $ such that $\mu_{n_{i_j}}$ converges weakly to a deficient topological measure $\mu$, then $\mu_n$ converges weakly to $\mu$.
If $\mu_n$ does not converge weakly to $\mu$, then there is $f \in C_0^+(X)$ such that $| \int f \, d \mu_{n_i} - \int f \, d\mu | \ge \epsilon$ for some $ \epsilon>0$ and all $\mu_{n_i}$ in some subsequence. But then no subsequence of $ \{ \mu_{n_i} \}$ can converge weakly to $\mu$.
We clearly have
\[L6.1\] $X$ is homeomorphic to the (topological) subset $D = \{ \delta_x: \, x \in X\}$ of $ \mathbf{DTM}(X)$ (equipped with the weak topology).
\[metrization\] Let $c\ge 0$. Then $ P = \{ \mu \in \mathbf{DTM}(X): \mu(X) \le c\} $ can be metrized as a separable metric space iff $X$ is a separable metric space.
Suppose $X$ is a separable metric space. By Urysohn’s metrization theorem (see [@Kelly p.125]) $X$ can be topologically embedded in a countable product of unit intervals. Consequently, there exists an equivalent totally bounded metrization on $X$. We will consider this metric on $X$. From [@Parthasarathy Lemma 6.3] $C_b(X)$ is separable. Let $\{ f_1, f_2, \ldots \}$ be a countable dense subset of $C_b(X)$.
Let $Y$ be a countable product of $\mathbb{R}$. Define a map $T : P \longrightarrow Y$ as in Theorem \[basicTP\], i.e. $T(\mu) = (\int f_1 \, d \mu, \int f_2 \, d \mu, \ldots)$. We will show that $T$ is a homeomorphism on $P$. First, $T$ is $1-1$. (If $T(\mu) = T(\nu)$ then $\int f_i \, d \mu = \int f_i \, d \nu$ for all $i$, and, hence, $\int f \, d \mu = \int f \, d \nu$ for all $ f \in C_0^+(X)$. By Remark \[RemBRT\], $ \mu = \nu$.) Second, $T$ and $T^{-1}$ are continuous, as in the proof of Theorem \[basicTP\]. Since $Y$ is a separable metric space, and $P$ is homeomorphic to a subset of $Y$, it follows that $P$ is a separable metric space.
Conversely, suppose $P$ is a separable metric space. By Lemma \[L6.1\] $X$ is homeomorphic to $ D =\{ \delta_x: \, x \in X\}$. $D$ is a separable metric space, and then so is $X$.
\[UnTight\] Let $X$ be locally compact. A family $\mathcal{M} \subseteq \mathbf{DTM}(X)$ is uniformly tight if for every $ \epsilon >0$ there exists a compact set $ K_\epsilon$ such that $\mu(K_\epsilon) > \epsilon $ for each $\mu \in \mathcal{M}$. A family $\mathcal{M} \subseteq \mathbf{DTM}(X)$ is uniformly bounded in variation if there is a positive constant $M$ such that $ \| \mu \| \le M$ for each $\mu \in \mathcal{M} $.
One uniformly bounded in variation family that is the often used is the collection of all normalized (i.e. satisfying condition $\mu(X) = 1$) topological measures on a compact space.
\[wkseqBd\] Suppose $X$ is locally compact. If a sequence $(\mu_n) \in \mathbf{DTM}(X)$ is weakly fundamental (i.e. $ \int f \, d\mu_n$ is a fundamental sequence for each $ f \in C_0^+(X)$) then it is uniformly bounded in variation.
If not, then there is a subsequence $ (\mu_{n_k})$ such that $ \| \mu_{n_k} \| > k 2^k$ for each $k$; and by part \[mrDTM\] of Remark \[RemBRT\] there are functions $f_{n_k} \in C_c(X), 0 \le f_{n_k} \le 1$ such that $ \int_X f_{n_k} \, d\mu_{n_k} > k 2^k$. Then the function $ f = \sum_{k=1}^\infty \frac{f_{n_k} }{2^k} \in C_0^+(X), \, 0 \le f \le 1$, and $ \int_X f \, d\mu_{n_k} \ge k$ for each $k$. This contradicts the fact that the sequence $( \int f \, d\mu_n)$ is Cauchy, hence, bounded.
\[wkfamBd\] Suppose $\mathcal{M} \subseteq \mathbf{DTM}(X)$ is a family of finite deficient topological measures such that every sequence in $\mathcal{M} $ contains a weakly convergent subsequence. Then $\mathcal{M}$ is uniformly bounded in variation.
If not, then there is a sequence $\mu_n \subseteq \mathcal{M} $ such that $ \| \mu_n \| > n $ for every natural $n$. Let $ m_{n_k}$ be its weakly convergent subsequence. Then $\| m_{n_k} \| > n_k$, while by Proposition \[wkseqBd\] this subsequence must be uniformly bounded in variation.
\[wkfamTgt\] Suppose $X$ is locally compact. Suppose $\mathcal{M} \subseteq \mathbf{TM}(X)$ is a family of finite topological measures such that every sequence in $\mathcal{M} $ contains a weakly convergent subsequence. Then $\mathcal{M} $ is uniformly tight.
Suppose $\mathcal{M} $ is not uniformly tight. Then there exists $ \epsilon>0$ such that for every compact $K$ one can find $\mu^K \in \mathcal{M}$ with $$\begin{aligned}
\label{vMuKe}
\mu^K (X \setminus K) > \epsilon.\end{aligned}$$ Take $\mu_1$ to be any topological measure with $ \| \mu_1 \| > \epsilon$, and let $ K_1 \in \mathscr{K}(X)$ be such that $ \mu(K_1) > \epsilon$. Then by Lemma \[easyLeLC\] there is $V_1 \in \mathscr{O}(X) $ with compact closure such that $ K_1 \subseteq V_1$ and so $ \mu_1(\overline{V_1}) > \epsilon$. By (\[vMuKe\]) find $ \mu_2$ satisfying $\mu_2(X \setminus \overline{V_1}) >\epsilon$, and let $K_2 \in \mathscr{K}(X)$ be such that $ K_2 \subseteq X \setminus \overline{V_1} $ and $ \mu(K_2) > \epsilon$. Find $V_2 \in \mathscr{O}(X)$ with compact closure such that $K_2 \subseteq V_2 \subseteq \overline{V_2} \subseteq X \setminus \overline{V_1} $, so $ \mu_2(\overline{V_2}) > \epsilon$. Find a topological measure $\mu_3$ with $\mu_3( X \setminus (\overline{V_1} \sqcup \overline{V_2}) > \epsilon$, and so on. By induction we find a sequence of compact sets $ K_j$, a sequence of open sets $V_j$ with compact closure, and a sequence of topological measures $\mu_j \in \mathcal{M} $ with the following properties: $ K_j \subseteq V_j \subseteq \overline{V_j}$, $ \overline{V_j}$ are pairwise disjoint, and $$\quad \mu_j(\overline{V_j}) \ge \mu_j(K_j) >
\epsilon, \quad \quad K_{j+1} \subseteq \overline{V_{j+1}} \subseteq X \setminus \bigsqcup_{i=1}^j \overline{V_i}.$$ By part \[mrDTM\] of Remark \[RemBRT\] find functions $f_j \in C_c(X), 1_K \le f_j \le 1, supp f_j \subseteq V_j, $ with $ \int_X f_j \, d \mu_j > \epsilon$. By our assumption the sequence $ (\mu_j) $ contains a weakly convergent subsequence. For notational simplicity, assume that $ (\mu_j) $ is weakly convergent.
By Lemma \[wkfamBd\] we may assume that $\mathcal{M}$ is uniformly bounded in variation by $M$. We let $$a_n^i = \int_X f_i \, d \mu_n.$$ Then $a_n:= (a_n^1, a_n^2, \ldots,)$ belongs to $l^1$, because for each $m \in \mathbb{N}$, $f_1 \cdot f_2 \cdot \ldots \cdot f_m = 0, f_1 + \ldots + f_m \in C_c(X), 0 \le f_1 + \ldots + f_m \le 1$, and so by part \[mrDTM\] of Remark \[RemBRT\] each partial sum $\sum_{i=1}^m a_n^i = \int_X (f_1 + f_2 + \ldots f_m) \, d \mu_n \le \|\mu_n \| \le M$. With $$b_n =\sum_{i=1}^\infty \int_X f_i \, d \mu_n = \| a_n \|_1 \le M,$$ the sequence $( b_n)$ is bounded, and we may chose a convergent subsequence. To simplify notations, we assume that $(b_n)$ itself converges.
Let $\lambda = (\lambda_i) \in l^\infty$. Since $ |\langle \lambda, a_n \rangle | \le \| \lambda \|_{\infty} \, \| a_{n} \|_{1} \le \| \lambda \| M,$ we see that the sequence of inner products $\langle \lambda, a_n \rangle$ is bounded, hence, contains a convergent subsequence. Again, for notational simplicity we assume the sequence itself converges.
By [@Bogachev:WkConv Lemma 1.3.7] the sequence $(a_n)$ converges in $l_1-$norm. Then $\lim_{n \rightarrow \infty} a_n^n = 0$, which contradicts our choice of $f_n$.
\[wkfuncon\] Let $X$ be locally compact. If $(\mu_n)$ is a weakly fundamental sequence of finite deficient topological measures which is also uniformly bounded in variation, then $ \mu_n$ converges weakly to some finite deficient topological measure $ \mu$.
Consider functional $L$ on $ C_0^+(X)$ defined as $L(f) = \lim_n \int_X f \, d\mu_n$. It is easy to check that $L$ is a p-conic quasi-linear functional. Say, $(\mu_n)$ is uniformly bounded in variation by $M$. Since $ L(f) \le \| \mu_n\| \le M$ for any $ f\in C_0^+(X), 0 \le f \le1$, we see that $L \in \mathbf{\Phi^+}(C_0^+(X))$, and by Remark \[RemBRT\] there is a finite deficient topological measure $ \mu$ such that $L(f) = \int_X f \, d \mu$.
\[EbSmul\] Suppose $X$ is a locally compact space such that $C_0^+(X)$ is separable. Then every uniformly bonded in variation sequence of finite topological measures has a subsequence which is weakly fundamental.
Suppose $(\mu_n) \in \mathbf{DTM}(X)$ and $ \| \mu_n \| \le M$ for each $n$. Let $g \in C_0^+(X)$, so $0 \le g \le b$ for some $b$. Each of the functions $R_{2, \mu_n, g} (t) $ is monotone and bounded above by $M$ on $[0,b]$. By the Helly-Bray theorem (see [@Bogachev:WkConv Theorem 1.4.6]), there is pointwise convergent subsequence $R_{2, \mu_{n_i}, g} $. Then the sequence of integrals $ \int_X g \, d \mu_{n_i} = \int_0^b R_{2, \mu_{n_i}, g}(t) dt $ converges, hence, is fundamental.
If $G$ is a countable dense set in $C_0^+(X)$, we pick a first subsequence of $ (n_i)$ such that $ ( \int_X g_1 \, d \mu_{n_i}) $ is fundamental for the first function $g_1 \in G$, then we choose a further subsequence $(n_{i_j})$ for which $ ( \int_X g_2 \, d \mu_{n_{i_j}}) $ is fundamental for the function $g_2 \in G$, and so on. By diagonal process we obtain a subsequence of $(\mu_n)$ for which the sequence of integrals is fundamental for each $ g \in G$. For notational simplicity, let us assume that $(\mu_n)$ is such a subsequence, i.e. $ ( \int_X g \, d \mu_n) $ is fundamental for each function $g \in G$.
For arbitrary $ f \in C_0^+(X)$ and $ \epsilon >0$ choose $ g \in G$ such that $ \| f - g \| \le \epsilon$ and $n_0$ such that $ | \int_X g \, d\mu_n - \int_X g \, d \mu_i | < \epsilon$ for $n , i \ge n_0$. Then using [@Butler:QLFLC Corollary 53] we have: $$\begin{aligned}
| &\int_X f \, d\mu_n - \int_X f \, d \mu_i | \\
&\le | \int_X f \, d\mu_n - \int_X g \, d \mu_n | + | \int_X g \, d\mu_n - \int_X g \, d \mu_i | +
| \int_X g \, d\mu_i - \int_X f \, d \mu_i | \\
&\le \| f - g \| \| \mu_n \| + \epsilon + \| f - g \| \| \mu_i \| \le 2 \epsilon M + \epsilon,\end{aligned}$$ and the sequence of integrals $ ( \int_X f \, d \mu_n) $ is fundamental. Thus, $ ( \mu_n)$ is weakly fundamental.
If $X$ is a locally compact Hausdorff space which is second countable or satisfies any of the other equivalent conditions of [@Kechris Theorem 5.3, p.29], then $\hat{X}$, the Aleksandrov one-point compactification of $X$, is a compact metrizable (hence, a second countable) space. Then $C(\hat{X})$ is separable, and $C_0(X)$ is also separable as as a subspace of a separable metric space.
For topological measures we have the following version of Prokhorov’s well-known theorem.
\[Proh2\] Suppose $X$ is a locally compact space such that $C_0^+(X)$ is separable. Suppose $\mathcal{M}$ is a family of finite topological measures on $X$. The the following are equivalent:
1. \[ProhUsl1\] If every sequence from $\mathcal{M}$ contains a weakly convergent subsequence then $\mathcal{M}$ is uniformly tight and uniformly bounded in variation.
2. \[ProhUsl2\] If $\mathcal{M}$ is uniformly bounded in variation then every sequence from $\mathcal{M}$ contains a weakly convergent subsequence.
(\[ProhUsl1\]) follows from Theorem \[wkfamBd\] and Theorem \[wkfamTgt\]. (\[ProhUsl2\]) follows from Theorem \[EbSmul\] and Lemma \[wkfuncon\].
Prokhorov and Kantorovich-Rubenstein metrics
============================================
It is clear that $d_o(\mu, \nu) = \sup \{| \int_X f \, d\mu - \int_X f \, d\nu|: f \in C_0^+(X) \} $ is a metric on $ \mathbf{DTM}(X)$, and the topology induced by this metric is the weak topology.
For the rest of this section let $(X, d)$ be a locally compact metric space. We shall consider two other metrics on $ \mathbf{DTM}(X)$.
Let $A^t = \{ x \in X: d(x, A) < t \}$ for $ A \in \mathscr{O}(X) \cup \mathscr{C}(X), A \ne \emptyset$, and $\emptyset^t = \emptyset$ for all $ t >0$. Each $A^t$ is an open set. Consider the Prokhorov metric $d_{\text{P}}$ on $ \mathbf{DTM}(X)$: $$\begin{aligned}
d_{\text{P}}(\mu, \nu) &= \inf\{ t >0: \, \mu(A) \le \nu(A^t) + t, \ \nu(A) \le \mu(A^t) + t, \\
&\forall A \in \mathscr{O}(X) \cup \mathscr{K}(X) \}.\end{aligned}$$ Taking $ t = \| \mu\| + \| \nu\| $ we see that $\inf$ is well defined.
Note that if $\mu$ and $ \nu$ are Borel measures and $A$ is a Borel set, then we obtain the usual definition of Prokhorov’s metric (sometimes also called Lévy-Prokhorov metric).
\[dPmetric\] $d_{\text{P}}$ is a metric on $ \mathbf{DTM}(X)$.
It is clear that $d_{\text{P}} \ge 0$ and $ d_{\text{P}}(\mu, \nu) = d_{\text{P}}( \nu, \mu)$. For any $ A \in \mathscr{O}(X) \cup \mathscr{C}(X)$ we have $ \mu(A) \le \mu(A^t )+ t$ for all $ t >0$, so $d_{\text{P}}(\mu, \mu) = 0$. Suppose $ d_{\text{P}}(\mu, \nu) = 0$.Then there is $ t_n \searrow 0$ such that $ \mu(K) \le \nu(K^{t_n}) + t_n$ and $ \nu(K) \le \mu(K^{t_n}) + t_n$ for all $ K \in \mathscr{K}(X)$. For $K \in \mathscr{K}(X)$ and $ \epsilon >0$ choose $U \in \mathscr{O}(X)$ such that $ K \subseteq U$ and $\nu(U) < \nu(K) + \epsilon$. There exists $ r>0$ such that $K^r\subseteq U$. Then for $ t_n < r$ $$\mu(K) \le \nu(K^{t_n}) + t_n \le \nu(U) + t_n \le \nu(K) + \epsilon + t_n.$$ It follows that $ \mu(K) \le \nu(K)$, and, similarly, $ \nu(K) \le \mu(K)$. Then $ \mu = \nu$ on $ \mathscr{K}(X)$, so $ \mu = \nu$.
Now we shall show the triangle inequality. Suppose that for all $A \in \mathscr{O}(X) \cup \mathscr{K}(X)$ $$\mu(A) \le \lambda(A^t) + t, \quad \lambda(A) \le \mu(A^t) + t,$$ $$\lambda(A) \le \nu(A^r) + r, \quad \nu(A) \le \lambda(A^r) + r.$$ Since $(A^t)^r \subseteq A_{t + r}$ and $ (A^r)^t \subseteq A_{t + r}$, we have: $$\mu(A) \le \lambda(A^t) + t \le \nu(A^t)^r + t + r \le \nu(A_{t + r}) + t + r,$$ and, similarly, $ \nu(A) \le \mu(A_{t + r}) + t + r$. Thus, $ d_{\text{P}}(\mu, \nu) \le t + r$. It follows that $ d_{\text{P}}(\mu, \nu) \le d_{\text{P}}( \mu, \lambda) + d_{\text{P}}(\lambda, \nu)$.
\[dPwkconv\] Let $(X, d)$ be a locally compact metric space. Suppose $ d_{\text{P}}( \mu_{\alpha}, \mu) \rightarrow 0$ for a net $(\mu_{\alpha})$; $ \mu_{\alpha}, \mu \in \mathbf{DTM}(X)$. Then $ \mu_{\alpha} \Longrightarrow \mu$.
Suppose $ d_{\text{P}}( \mu_{\alpha}, \mu) \rightarrow 0$.
Let $K \in \mathscr{K}(X)$ and $\epsilon >0$. Choose $ U \in \mathscr{O}(X)$ such that $ K \subseteq U$ and $ \mu(U) < \mu(K) + \epsilon $. There exists $ r>0$ such that $K^t \subseteq U$ for all $t \le r$. For $ \delta = min \{r, \epsilon \}$ let $ \alpha_0$ be such that $d_{\text{KR}}( \mu_{\alpha}, \mu) < \delta$ for each $ \alpha \ge \alpha_0$. Then for each $ \alpha \ge \alpha_0$ there exists $t_{\alpha} < \delta$ such that $ \mu_{\alpha}(K) \le \mu(K^{t_{\alpha}}) + t_{\alpha} \le \mu(U) + \epsilon \le \mu(K) + 2 \epsilon$. Then $$\limsup \mu_{\alpha}(K) \le \mu(K) + 2 \epsilon.$$ It follows that $ \limsup \mu_n(K) \le \mu(K)$.
Now let $U \in \mathscr{O}(X)$ and $\epsilon >0$. Choose $ K \in \mathscr{K}(X)$ such that $ K \subseteq U$ and $ \mu(K) > \mu(U) - \epsilon $. Let $r, \delta$ and $ \alpha_0$ be as above. Then for each $ \alpha \ge \alpha_0$ there exists $t_{\alpha} < \delta$ such that $ \mu(K) \le \mu_{\alpha}(K^{t_{\alpha}}) + t_{\alpha} \le \mu_{\alpha} (U) + \epsilon$. Then $$\liminf \mu_{\alpha}(U) \ge \mu(K) - \epsilon \ge \mu(U) - 2 \epsilon.$$ It follows that $ \liminf(U) \ge \mu(U)$.
By Theorem \[AleksandrovLC\] $ \mu_{\alpha} \Longrightarrow \mu$.
Let family $\mathcal{M} \subseteq \mathbf{TM}(X)$ be uniformly bounded in variation. We consider the Kantorovich-Rubinstein metric $d_{\text{KR}}$ on $\mathcal{M}$. $$\begin{aligned}
\label{KRmetric}
d_{\text{KR}}(\mu, \nu) &= \sup \{| \int_X f \, d\mu - \int_X f \, d\nu|: f \in Lip_1(X,d) \cap C_c(X), \, \| f \| \le 1 \} \end{aligned}$$ where $Lip_1(X) = \{f:X \Longrightarrow \mathbb{R}: | f(x) - f(y) | \le d(x,y) \ \forall x,y \in X\} $.
Our definition is related to the definition of the Kantorovich-Rubinstein metric for Borel measures, which is obtained from the Kantorovich-Rubinstein norm $$\| \mu \|_{KR} = \sup \{ \int_X f \, d\mu: f \in Lip_1(X,d), \, \| f \| \le 1 \}.$$ This metric is sometimes is also called the Wasserstein metric $W(\mu, \nu)$, although there is no author with this name. See [@Bogachev pp. 453-454, Comments to Ch.8] for a good note on the history and use of this metric.
Our use of $f \in Lip_1(X,d) \cap C_c(X)$ in (\[KRmetric\]) is dictated, on one hand, by relation to Kantorovich-Rubinstein metric for Borel measures and, on the other hand, by the role of $C_c(X)$ in the theory of (p-conic) quasi-linear functionals. Note that by [@Andreou Theorem 2] Lipschitz functions with compact support are dense in $C_0(X)$.
$d_{\text{KR}}$ is a metric on a uniformly bounded in variation family $\mathcal{M}$.
We shall show that $d_{\text{KR}}(\mu, \nu) = 0$ implies $\mu = \nu$; the remaining properties are obvious. Let $M$ be such that $ \| \mu \| \le M$ for each $\mu \in \mathcal{M}$. Take $f \in C_0(X)$. Given $ \epsilon >0$, choose a Lipschitz function $g$ with compact support so that $ \| f - g \| < \epsilon$. Since $d_{\text{KR}}(\mu, \nu) = 0$, we see that $ | \int_X g \, d\mu - \int_X g \, d \nu | = 0$. Using also Remark \[LipQLF\] we have: $$\begin{aligned}
| &\int_X f \, d\mu - \int_X f \, d \nu | \\
&\le | \int_X f \, d\mu - \int_X g \, d \mu | + | \int_X g \, d\mu - \int_X g \, d \nu | +
| \int_X g \, d \nu - \int_X f \, d \nu | \\
&\le \| f - g \| \mu(X) + \| f - g \| \nu(X) \le 2 \epsilon M.\end{aligned}$$ Thus, $\int_X f \, d\mu = \int_X f \, d \nu$ for every $f \in C_0(X)$. By Remark \[RemBRT\] $ \mu = \nu$.
\[dKRwkconv\] Let $X$ be a locally compact metric space. In either of the following situations:
1. a family $\mathcal{M} \subseteq \mathbf{TM}(X)$ is uniformly bounded in variation;
2. given $M >0$, a family $\mathcal{M} \subseteq \mathbf{DTM}(X)$ is the family of deficient topological measures corresponding to functionals $\mathcal{R}$ on $C_c^+(X)$ with $ \| \mathcal{R} \| \le M$;
if a net $(\mu_{\alpha}) \in \mathcal{M} $, $\mu \subseteq \mathcal{M} $, and $ d_{\text{KR}} ( \mu_{\alpha}, \mu) \rightarrow 0$, then $ \mu_{\alpha} \Longrightarrow \mu$.
1. Let $f \in C_0(X)$. Given $ \epsilon >0$, choose a Lipschitz function with compact support $g$ so that $ \| f - g \| < \epsilon$. Since$ | \int_X g \, d \mu_{\alpha} - \int_X g, d \mu | \le d_{\text{KR}} ( \mu_{\alpha}, \mu) \, \| g \|_{Lip} \, \| g \|$, say, $ | \int_X g \, d \mu_{\alpha} - \int_X g, d \mu | \le \epsilon $ for all $ \alpha \ge \alpha_0$. Then for all $ \alpha \ge \alpha_0$ using Remark \[LipQLF\] we have: $$\begin{aligned}
| &\int_X f \, d\mu_{\alpha} - \int_X f \, d \mu | \\
&\le | \int_X f \, d\mu_{\alpha}- \int_X g \, d\mu_{\alpha}| + | \int_X g \, d\mu_{\alpha}- \int_X g \, d \mu | +
| \int_X g \, d\mu - \int_X f \, d \mu | \\
&\le \| f - g \| \mu_{\alpha}(X) + \epsilon + \| f - g \| \mu(X) \le 2 \epsilon M + \epsilon,\end{aligned}$$ so $\int_X f \, d\mu_{\alpha} \longrightarrow \int_X f, d\mu$. It follows that $ \mu_{\alpha} \Longrightarrow \mu$.
2. If a deficient topological measure corresponds to $\mathcal{R}$ then $ \| \mu \| \le M$. Thus, the family $\mathcal{M} $ is uniformly bounded in variation, and we may use the same argument as in previous part.
Let $X$ be a compact metric space. Given $M >0$, let $ \mathcal{M} = \{ \mu \in \mathbf{DTM}(X): \| \mu \| \le M \}$. Then the topology on $ \mathcal{M}$ induced by the metric $ d_{\text{KR}}$ is the weak topology.
By Theorem \[dKRwkconv\] if a net $(\mu_{\alpha})$ converges to $\mu$ in the metric $d_{\text{KR}}$ then it also converges to $\mu$ weakly. For $ \mathcal{M} = \{ \mu \in \mathbf{TM}(X): \| \mu \| \le 1 \}$ and a slightly different metric the result was first shown in [@DickZap Proposition 1.10], and our proof of Theorem \[dKRwkconv\] follows the argument in that paper. Because of Remark \[LipQLF\] and the fact that the family of functions in (\[KRmetric\]) is compact by the Arzela-Ascoli theorem, one can basically repeat an argument from [@DickZap Proposition 1.10] to show that the weak convergence of $(\mu_{\alpha})$ to $\mu$ implies convergence in the metric $d_{\text{KR}}$.
Density theorems
================
\[properSDTM\] A deficient topological measure $\nu$ is called proper if from $m \le \nu $, where $m$ is a Radon measure it follows that $m = 0$.
\[properDtm\] From [@Butler:Decomp Theorem 4.3] it follows that a finite deficient topological measure can be written as a sum of a finite Radon measure and a proper finite deficient topological measure. The sum of two proper deficient topological measures is proper (see [@Butler:Decomp Theorem 4.5]).
A finite Radon measure on a compact space is a regular Borel measure, so our definition (which is given in [@Butler:Decomp]) of a proper deficient topological measure coincides with definitions in papers prior to [@Butler:Decomp].
In what follows, $p\mathbf{DTM}(X)$ and $p\mathbf{TM}(X)$ denote, respectively, the family of proper finite deficient topological measures and the family of finite topological measures.
Let $X$ be a locally compact non-compact space. A set $A$ is called solid if $A$ is connected, and $X \setminus A$ has only unbounded connected components. When $X$ is compact, a set is called solid if it and its complement are both connected. For a compact space $X$ we define a certain topological characteristic, genus. See [@Aarnes:ConstructionPaper] for more information about genus $g$ of the space. A compact space has genus 0 iff any finite union of disjoint closed solid sets has a connected complement. Intuitively, $X$ does not have holes or loops. In the case where $X$ is locally path connected, $g=0$ if the fundamental group $\pi_1(X)$ is finite (in particular, if $X$ is simply connected). Knudsen [@Knudsen] was able to show that if $H^1(X) = 0 $ then $g(X) = 0$, and in the case of CW-complexes the converse also holds.
\[notTM\] From Theorem \[DTMtoTM\] it is easy to see that if $ \mu, \nu$ are deficient topological measures, and $\nu$ is not a topological measure, then $ \mu + \nu$ is a deficient topological measure which is not a topological measure.
\[messyTh\]
1. \[paforDTM\] (Proper simple deficient topological measures that are not topological measures are dense in the set of all point-masses) $\Longrightarrow$ ($p\mathbf{DTM}(X) \setminus \mathbf{TM}(X)$ is dense in $\mathbf{M}(X)$) $\Longleftrightarrow$ ($p\mathbf{DTM}(X) \setminus \mathbf{TM}(X)$ is dense in $\mathbf{DTM}(X) \setminus \mathbf{TM}(X)$) $\Longrightarrow$ ($p\mathbf{DTM}(X) $ is dense in $\mathbf{DTM}(X)$) $\Longleftrightarrow$ ($p\mathbf{DTM}(X) $ is dense in $\mathbf{M}(X)$).
2. \[paforTM\] (Proper simple $\mathbf{TM}(X)$ are dense in the set of all point-masses) $\Longrightarrow$ ($p\mathbf{TM}(X)$ is dense in $\mathbf{M}(X)$) $\Longleftrightarrow$ ($p\mathbf{TM}(X)$ is dense in $\mathbf{TM}(X)$) $\Longrightarrow$ ($p\mathbf{DTM}(X) $ is dense in $\mathbf{DTM}(X)$).
We shall prove the first part; the proof of the second part is similar, but simpler.
1. \[imA\] We shall show the first implication. Any measure is approximated by convex combinations of point-masses, so by assumption, it is approximated by convex combinations of proper simple deficient topological measures that are not topological measures. By Remark \[properDtm\] and Remark \[notTM\] the latter combinations are in $p\mathbf{DTM}(X) \setminus \mathbf{TM}(X)$.
2. \[imB\] ($p\mathbf{DTM}(X) \setminus \mathbf{TM}(X)$ is dense in $\mathbf{M}(X)$) $\Longrightarrow$ ($p\mathbf{DTM}(X) \setminus \mathbf{TM}(X)$ is dense in $\mathbf{DTM}(X) \setminus \mathbf{TM}(X)$): Suppose $ \mu \in \mathbf{DTM}(X) \setminus \mathbf{TM}(X)$. By Remark \[properDtm\] write $ \mu = m + \mu'$, where $ \mu'$ is a proper deficient topological measure, and $ m$ is a measure from $\mathbf{M}(X)$. By assumption, $m$ is approximated by $\nu \in p\mathbf{DTM}(X) \setminus \mathbf{TM}(X)$. Then $\mu$ is approximated by $\nu + \mu'$, where by Remark \[properDtm\] and Remark \[notTM\] $\nu + \mu' $ is in $p\mathbf{DTM}(X) \setminus \mathbf{TM}(X)$.
3. \[imC\] ($p\mathbf{DTM}(X) \setminus \mathbf{TM}(X)$ is dense in $\mathbf{DTM}(X) \setminus \mathbf{TM}(X)$) $\Longrightarrow$ ($p\mathbf{DTM}(X) \setminus \mathbf{TM}(X)$ is dense in $\mathbf{M}(X)$): Suppose to the contrary that there exists a measure $m \in \mathbf{M}(X) $ and its neighborhood $N$ which contains no elements of $p\mathbf{DTM}(X) \setminus p\mathbf{TM}(X)$. Take $\lambda \in \mathbf{DTM}(X) \setminus \mathbf{TM}(X)$. Then for any deficient topological measure $ \nu \in N$ we see that $ \lambda + \nu$ is a deficient topological measure that is not a topological measure and is not proper. Thus, a neighborhood $\lambda + N \subseteq\mathbf{DTM}(X) \setminus \mathbf{TM}(X)$ contains no elements of $p\mathbf{DTM}(X) \setminus \mathbf{TM}(X)$, which contradicts the assumption.
4. \[imD\] ($p\mathbf{DTM}(X) \setminus \mathbf{TM}(X)$ is dense in $\mathbf{DTM}(X) \setminus \mathbf{TM}(X)$) $\Longrightarrow$ ($p\mathbf{DTM}(X) $ is dense in $\mathbf{DTM}(X)$): Let $\nu \in \mathbf{DTM}(X)$. If $ \nu \in \mathbf{DTM}(X) \setminus p\mathbf{TM}(X)$ then the statement follows from the assumption, and if $ \nu \in \mathbf{DTM}(X) \cap p\mathbf{TM}(X)$ then the statement is obvious.
5. \[imE\] ($p\mathbf{DTM}(X) $ is dense in $\mathbf{DTM}(X)$) $\Longrightarrow$ ($p\mathbf{DTM}(X) $ is dense in $\mathbf{M}(X)$): obvious.
6. \[imF\] ($p\mathbf{DTM}(X) $ is dense in $\mathbf{M}(X)$) $\Longrightarrow$ ($p\mathbf{DTM}(X) $ is dense in $\mathbf{DTM}(X)$): follows from Remark \[properDtm\] and Remark \[notTM\] in a manner similar to the one in part \[imB\].
\[Kcden\] Suppose any open set in a locally compact space $X$ contains a compact connected subset that is not a singleton. Then $p\mathbf{DTM}(X)$ is dense in $\mathbf{DTM}(X)$.
If we shall show that proper simple $\mathbf{DTM}(X) \setminus \mathbf{TM}(X)$ are dense in the set of point-masses, then the statement will follow from Theorem \[messyTh\]. Let $\delta_a$ be a point-mass at $a$. Let $ \{ V \in \mathscr{O}(X): a \in V\}$ be ordered by reverse inclusion. For each $V$, let $K_V \subseteq V$ be the non-singleton connected compact set. Consider $ \lambda^V$ defined on $\mathscr{O}(X) \cup \mathscr{C}(X)$ as follows: $ \lambda^V (A) =1$ if $ K_V \subseteq A$ and $ \lambda^V (A) =0$ otherwise. By [@Butler:DTMLC Example 46] $ \lambda^V$ is simple and $ \lambda^V \in \mathbf{DTM}(X) \setminus \mathbf{TM}(X)$. If $ U \in \mathscr{O}(X) $ and $ \delta_a(U) = 1$, then $ a \in U$ and for all $ V \subseteq U, V \in \mathscr{O}(X)$ we have $ K_V \subseteq U$, so $ \lambda^V (U) = 1$. Then $ \liminf \lambda^V (U) = 1 = \delta_a(U)$. If $C \in \mathscr{K}(X)$ and $ \delta_a(C) = 0$, then $a \notin C$ and we may find $U \in \mathscr{O}(X) $ such that $a \in U, U \cap C = \emptyset$. Then for each $ V \subseteq U, V \in \mathscr{O}(X)$ we have $K_V \cap C = \emptyset$ and $\lambda^V(C) = 0$. Then $ \limsup \lambda^V(C) = 0 = \delta_a(C)$. By Theorem \[AleksandrovLC\] the net $(\lambda^V)$ converges weakly to $ \delta_a$.
\[KcdenRe\] Among spaces that satisfy the condition of the previous theorem are: non-singleton locally compact spaces that are locally connected or weakly locally connected; manifolds; CW complexes.
\[singDen\] Suppose $X$ is a non-singleton connected, locally connected, locally compact space with no cut points and such that the Aleksandrov one-point compactification of $X$ has genus $0$. Then $p\mathbf{TM}(X)$ is dense in $\mathbf{TM}(X)$, and $p\mathbf{DTM}(X)$ is dense in $\mathbf{DTM}(X)$.
We shall give the proof for the case when $X$ is not compact. (When $X$ is compact the proof is similar but simpler; also, one may use [@Svistula:Integrals Theorem 4.9].) We shall show that proper simple topological measures are dense in the set of simple measures, and the statements will follow from part (\[paforTM\]) of Theorem \[messyTh\].
Let $\delta_a$ be a point-mass. It is enough to show that a neighborhood of the form $ W(\delta_a, U, C, \epsilon) $ as in Theorem \[wkBase\] contains a simple proper topological measure.
Suppose first $a \in U \in \mathscr{O}(X), a \notin C$. We may assume that $ U \cap C = \emptyset$. Since $a \in U \in \mathscr{O}(X)$, by Lemma \[easyLeLC\] there is a bounded open connected set $V$ and a compact connected set $D$ such that $ a \in V \subseteq D \subseteq U$. Since $X$ is connected and non-singleton, $ a \subsetneq V $, and we may choose 3 different points in $D$. Let $\lambda$ be a simple topological measure on $X$ given by [@Butler:TMLCconstr Example 46], so $\lambda(A) = 1$ if a bounded solid set $A$ contains two or three of the chosen points, and $\lambda(A) = 0$ if a bounded solid set $A$ contains no more than one of the chosen points. Since the solid hull of $D$ (a compact solid set) contains all three points, and each bounded component of $X \setminus D$ (a bounded open solid set) contains none of the three points, by [@Butler:TMLCconstr Definition 41] we compute $\lambda(D) = 1$. Then $ \lambda(U) = 1$. Since $C$ is disjoint from $U$, and $ \lambda(X) = 1$, by superadditivity we have $ \lambda(C) = 0$. Thus, $ \lambda \in W(\delta_a, U, C, \epsilon) $.
We shall show that $ \lambda$ is proper. Let $x \in X$. Since $X \setminus \{x\}$ is connected, by Lemma \[easyLeLC\] there is a compact connected set $B \subseteq X \setminus \{x\} $ such that $B$ contains at least two of the three chosen points. Argument as above shows that $\lambda(B) = 1$. Then $ \lambda(\{x\}) \le \lambda(X \setminus B) = \lambda(X) - \lambda(B) =0$. Thus, $\lambda(\{x\}) =0$ for any $x \in X$, and by [@Butler:Decomp Lemma 4.12] $\lambda$ is proper.
The remaining three cases are easy. For example, if $ a \in U, a \in C$ then $\lambda$ as above will do.
\[inftsumD\] Suppose $X$ is locally compact, $\sum_{i=1}^\infty \mu_i(X) < \infty$ where each $\mu_i$ is a deficient topological measure. Then $\mu = \sum_{i=1}^\infty \mu_i$ is a finite deficient topological measure. If each $\mu_i$ is a topological measure, then $\mu$ is a finite topological measure.
Let $\mu = \sum_{i=1}^\infty \mu_i$ on $ \mathscr{O}(X) \cup \mathscr{C}(X)$. It is easy to see that $\mu$ is finitely additive on compact sets. For $ \epsilon >0$ let $j$ be such that $\sum_{i=j+1}^\infty \mu_i(X) < \epsilon$, and let $\lambda = \sum_{i=1}^j \mu_i$. Then $\lambda $ is a finite deficient topological measure. For $U \in \mathscr{O}(X)$ there exists $K \in \mathscr{K}(X)$ such that $\lambda(U)< \lambda(K) + \epsilon$. Then $ \mu(U) < \lambda(U) + \epsilon< \lambda(K) + 2\epsilon <\mu(K) + 2\epsilon$, and the inner regularity of $\mu$ follows. Similarly, $\mu$ is outer regular. Thus, $\mu$ is a deficient topological measure; clearly, $\mu$ is finite. If each $\mu_i$ is a topological measure, it is easy to check additivity of $ \mu$ on $\mathscr{O}(X) \cup \mathscr{K}(X)$, so condition \[TM1\] of Definition \[TMLC\] holds, and $\mu$ is a topological measure.
\[infprsum\] Suppose $X$ is locally compact, $\sum_{i=1}^\infty \mu_i(X) < \infty$ where each $\mu_i$ is a proper deficient topological measure (respectively, a proper topological measure). Then $\mu = \sum_{i=1}^\infty \mu_i$ is a finite proper deficient topological measure (respectively, a finite proper topological measure).
By Lemma \[inftsumD\] $\mu$ is a finite deficient topological measure (respectively, a finite topological measure). We need to show that $\mu$ is proper. By Remark \[properDtm\] write $ \mu = m + \mu'$, where $m$ is a finite Radon measure and $ \mu'$ is a proper deficient topological measure. We shall show that $ m=0$.
Let $K \in \mathscr{K}(X)$. For $ \epsilon >0$ let $N$ be such that $\sum_{i=N+1}^\infty \mu_i(X) < \epsilon$, and let $\mu^N = \sum_{i=1}^N \mu_i$.
By Remark \[properDtm\] $\mu^N$ is a proper deficient topological measure. By [@Butler:Decomp Theorem 4.4] there are compact sets $K_1, \ldots, K_n$ such that $K = \cup K_j$ and $ \sum_{j=1}^n \mu^N(K_j) < \epsilon$. Let $E_1, \ldots, E_n $ be disjoint Borel sets such that $E_j \subseteq K_j$ and $ \bigsqcup_{j=1}^n E_i = \bigcup_{j=1}^n K_j$. Since $m$ is finite, outer regularity of $m$ is equivalent to inner closed regularity of $m$. Find disjoint sets $C_j, C_j \subseteq E_j \subseteq K_j, j=1, \ldots, n$ such that $C_j$ are closed (hence, compact) and $m(C_j) > m(E_j) - \frac{\epsilon}{n}$. Then $$\begin{aligned}
m&(K) = \sum_{j=1}^n m(E_j) \le \epsilon + \sum_{j=1}^n m(C_i) \le \epsilon + \mu(C_1 \sqcup \ldots \sqcup C_n) \\
&\le \epsilon + \mu^N (C_1 \sqcup \ldots \sqcup C_n) + \epsilon = 2 \epsilon + \sum_{j=1}^n \mu^N(C_j ) \le 2 \epsilon + \sum_{j=1}^n \mu^N (K_i) \le 3 \epsilon.\end{aligned}$$ It follows that $ m(K) = 0$ for any $ K \in \mathscr{K}(X)$. Thus, $m=0$, and $\mu$ is proper.
\[poXi\] Let $X$ be locally compact. Suppose $X = \bigcup_{i=1}^\infty X_i$, where each $X_i $ is a compact subset of $X$.
1. \[poXi1\] If $p\mathbf{DTM}(X_i)$ is dense in $\mathbf{M}(X_i), \, i \in \mathbb{N}$ then $p\mathbf{DTM} (X)$ is dense in $\mathbf{M} (X)$.
2. \[poXi2\] If $p\mathbf{TM} (X_i)$ is dense in $\mathbf{M}(X_i), \, i \in \mathbb{N}$ then $p\mathbf{TM} (X)$ is dense in $\mathbf{M} (X)$.
Note that each $X_i$ is a locally compact space with respect to the subspace topology. We shall prove the first part. Let $m \in \mathbf{M}(X)$. We shall show that every neighborhood $W$ of $m$ as in Theorem \[wkBase\] contains a proper deficient topological measure. To simplify notation, we consider $W(m, U, C, \epsilon)$ where $U \in \mathscr{O}(X), C \in \mathscr{K}(X), \epsilon >0$. Take Borel subsets $Y_i$ of $X$ such that $ Y_i \subseteq X_i$ and $\bigsqcup_{i=1}^\infty Y_i = X$. Consider $m_i(B) = m(B \cap Y_i)$, where $B$ is a Borel set in $X_i$, $i \in \mathbb{N}$. It is easy to see that $m_i \in \mathbf{M}(X_i)$.
Let $ \epsilon >0$. Let $U_i = U \cap X_i, C_i = C \cap X_i, \epsilon_ i = \epsilon 2^{-i}$ for $ i \in \mathbb{N}$, so $U_i$ is open in $X_i$ and $ C_i$ is compact in $X_i$. By assumption, there is $\lambda_i \in p\mathbf{DTM}(X_i)$ such that $\lambda_i \in W(m_i; U_i, C_i, X_i, \epsilon_i)$. Let $\nu_i$ be the extension of $\lambda_i$ to $\mathscr{O}(X) \cup \mathscr{C}(X)$ given by $\nu_i(A) = \lambda_i(A \cap X_i)$ for $ A \in \mathscr{O}(X) \cup \mathscr{C}(X)$. It is easy to see that $\nu_i $ is a deficient topological measure, and $ \nu_i(X) = \lambda_i(X_i) < \infty$. Since $\lambda_i$ is proper, by [@Butler:Decomp Theorem 4.4] given $ \delta >0$ there are sets of the form $ V_j \cap X_i, V_j \in \mathscr{O}(X), j=1, \ldots, n$ such that they cover $X_i$ and $\sum_{j=1}^n \lambda_i(V_j \cap X_i) < \delta$. Then open sets $V_1, \ldots, V_n, X \setminus X_i$ cover $X$ and $\sum_{j=1}^n \nu_i (V_j) + \nu_i(X \setminus X_i) = \sum_{j=1}^n \lambda_i(V_j \cap X_i) < \delta$, and so $\nu_i$ is proper. Thus, $\nu_i \in p\mathbf{DTM}(X)$ by [@Butler:Decomp Theorem 4.4].
Since $ \sum_{i=1}^\infty \nu_i(X) = \sum_{i=1}^\infty \lambda_i(X_i) \le \sum_{i=1}^\infty (m_i(X_i) + \epsilon_i) = m(X) + \epsilon < \infty$, by Lemma \[infprsum\] $\nu = \sum_{i=1}^\infty \nu_i$ is a finite proper deficient topological measure. We have: $$\nu(U) = \sum_{i=1}^\infty \nu_i(U) = \sum_{i=1}^\infty \lambda_i(U \cap X_i) >\sum_{i=1}^\infty (m_i(U \cap X_i) - \epsilon_i) = m(U) - \epsilon,$$ $$\nu(C) = \sum_{i=1}^\infty \nu_i(C) = \sum_{i=1}^\infty \lambda_i(C \cap X_i) < \sum_{i=1}^\infty (m_i(C \cap X_i) + \epsilon_i) = m(C) + \epsilon.$$ Thus, $\nu \in W(m, U, C, \epsilon)$.
The proof of the second part is the same, taking into account that $\lambda_i, \nu_i, \nu$ are proper topological measures.
\[poxiCor\] Let $X = \cup_{i=1}^\infty X_i$, where each $ X_i $ as in Theorem \[singDen\]. Then $p\mathbf{TM}(X) $ is dense in $\mathbf{TM}(X)$, and $p\mathbf{DTM}(X) $ is dense in $\mathbf{DTM}(X)$.
By part \[paforTM\] of Theorem \[messyTh\] it is enough to show that $p\mathbf{TM}(X) $ is dense in $\mathbf{M}(X)$. By Theorem \[singDen\], $p\mathbf{TM}(X_i) $ is dense in $\mathbf{M}(X_i)$ for each $i$, and we apply part \[poXi2\] of Theorem \[poXi\].
In Corollary \[poxiCor\] one may take, for example, a compact n-manifold, $n \ge 2$ as $X$, or $X$ that is covered by countably many sets homeomorphic to balls $B^n$ with varying $n \ge 2$.
$\mathbf{TM}(X) $ is a closed subset of $\mathbf{DTM}(X)$, and $\mathbf{M}(X) $ is a closed subset of $\mathbf{DTM}(X)$.
By Remark \[RemBRT\] $ \mu \in \mathbf{TM}(X) $ iff $ \rho$ is a quasi-linear functional on $C_0(X)$, and $\mu \in \mathbf{M}(X)$ iff $\rho$ is a linear functional on $C_0(X)$, where $ \rho(f) = \mathcal{R}_{\mu} (f^+) - \mathcal{R}_{\mu} (f^-) $. Using basic open sets in Definition \[defwk\] it is easy to check that $\mathbf{TM}(X) $ is a closed subset of $\mathbf{DTM}(X)$, and $\mathbf{M}(X) $ is a closed subset of $\mathbf{DTM}(X)$.
\[densEq\] Suppose $X$ is locally compact. The following are equivalent:
1. \[densEq1\] $\mathbf{M}(X) $ is nowhere dense in $\mathbf{DTM}(X)$ (or in $\mathbf{TM}(X)$).
2. \[densEq2\] There exists a finite deficient topological measure (respectively, a finite topological measure) that is not a measure.
3. \[densEq3\] There exists a nonzero finite proper deficient topological measure (respectively, nonzero finite proper topological measure).
(\[densEq1\]) $\Longrightarrow$ (\[densEq2\]) is obvious. (\[densEq2\]) $\Longrightarrow$ (\[densEq3\]): Let $\mu$ be a deficient topological measure that is not a measure. By Remark \[properDtm\] write $ \mu = m + \mu'$ where $m$ is a measure and $\mu' $ is a proper deficient topological measure. Then $ \mu' \neq 0$. (\[densEq3\]) $\Longrightarrow$ (\[densEq1\]): Suppose $\nu \neq 0$ is a proper finite deficient topological measure. Let $ m \in \mathbf{M}(X) $. Consider a set functions $\mu_n$ on $\mathscr{O}(X) \cup \mathscr{C}(X)$ given by $$\mu_n(A) = \frac1n \frac{1}{\nu(X)} \nu(A) + (1 - \frac1n) m(A).$$ Then each $\mu_n$ is a deficient topological measure that is not a measure, and $ \mu_n \Longrightarrow m$ by Theorem \[AleksandrovLC\]. Thus, $\mathbf{DTM} (X) \setminus \mathbf{M}(X)$ is dense in $ \mathbf{M}(X)$, and since $ \mathbf{M}(X)$ is a closed subset of $\mathbf{DTM}(X)$, we see that $ \mathbf{M}(X)$ is nowhere dense in $\mathbf{DTM}(X)$. The proof for topological measures is similar.
\[ExDensi\] Suppose $X$ is locally compact. If $X$ contains a non-singleton compact connected set, then $\mathbf{M}(X) $ is nowhere dense in $\mathbf{DTM}(X)$. If $X$ contains an open (or closed) locally connected, connected, non-singleton subset whose Aleksandrov one-point compactification has genus $0$ then $\mathbf{M}(X) $ is nowhere dense in $\mathbf{TM}(X)$.
Use part (\[densEq2\]) of Theorem \[densEq\]. For the first statement, as an example of a finite deficient topological measure that is not a topological measure (hence, not a measure) one may use [@Butler:DTMLC Example 46], For the second statement, as an example of a finite topological measure that is not a measure one may take [@Butler:TMLCconstr Example 61].
The proof of the next Theorem and Corollary are similar to the proof of Theorem \[densEq\] and Corollary \[ExDensi\].
\[densEqT\] Suppose $X$ is locally compact. The following are equivalent:
1. \[densEq1a\] $\mathbf{TM}(X) $ is nowhere dense in $\mathbf{DTM}(X)$.
2. \[densEq2a\] There exists a finite deficient topological measure that is not a topological measure.
3. \[densEq3a\] There exists a nonzero finite proper deficient topological measure that is not a topological measure.
\[CdensEqT\] If a locally compact space $X$ contains a non-singleton compact connected set, then $\mathbf{TM}(X) $ is nowhere dense in $\mathbf{DTM}(X)$.
When the space is compact, the equivalence of the first two conditions in Theorem \[AleksandrovLC\] and of first three conditions in Theorem \[AleksandrovLCtm\] was first given in [@Svistula:Integrals Corollary 4.4, 4.5]. When $X$ is compact Theorem \[wkBase\] was proved in [@Svistula:Integrals], but the method there does not work for a locally compact non-compact space, as the set $f^{-1}([0, \infty)) = X$ is not compact. Theorem \[basicTP\] generalizes results from several papers, including [@Aarnes:QuasiStates70], [@GrubbLaBerge:Spaces], and [@Svistula:Integrals]. Theorem \[metrization\] is an adaptation of [@Parthasarathy Theorem 6.2]. Our proof of Theorem \[wkfamTgt\] is adapted from a nice proof in [@Bogachev:WkConv Theorem 2.3.4]. In the last section we generalize results from [@Svistula:Integrals Section 4] and [@Density] from a compact space to a locally compact one.
[**[Acknowledgments]{}**]{}: The author would like to thank the Department of Mathematics at the University of California Santa Barbara for its supportive environment.
Conflict of interest {#conflict-of-interest .unnumbered}
====================
The author declares no conflict of interest.
[widest]{}
Aarnes, J. F.: Physical States on C\*-algebra. Acta Math. 122, 161–172 (1969)
Aarnes, J. F.: Quasi-states on $C^*$-algebras. Trans. Amer. Math. Soc. 149, 601–625 (1970)
Aarnes, J. F.: Quasi-states and quasi-measures. Adv. Math. 86 (1), 41–67 (1991)
Aarnes, J. F.: Pure quasi-states and extremal quasi-measures. Math. Ann. 295, 575–588 (1993)
Aarnes, J. F.: Construction of non-subadditive measures and discretization of Borel measures. Fund. Math. 147, 213–237 (1995)
Andreou, T.: Density of Lipschitz functions. Proceedings of The Conference of Applied Differential Geometry-General Relativity and The Workshop on Global Analysis, Differential Geometry and Lie Algebras 2001, Balkan Society of Geometers, Geometry Balkan Press, 1–4 (2004).
Billingsley, P.: Convergence of Probability Measures, 2nd edition. John Wiley and Sons, Inc., New York (1999)
Bogachev, V. I. : Measure Theory. vol. 2: Regular and Chaotic Dynamics, Izhevsk (2003). English transl.: Springer-Verlag, Berlin (2007).
Bogachev, V.I. : Weak convergence of measures. AMS Series: Mathematical surveys and monographs, vol. 234, Providence RI (2018)
Butler, S. V.: Density in the space of topological measures. Fund. Math. 174, 239–251 (2002)
Butler, S. V.: q-Functions and extreme topological measures. J. Math. Anal. Appl. 307, 465–479 (2005)
Butler, S. V.: Ways of obtaining topological measures on locally compact spaces. Bull. of Irkutsk State Univ., Series “Mathematics” (Izvestiya Irkutskogo Gosudarstvennogo Universiteta, Seriya “Matematika”) 25, 33–45 (2018)
Butler, S. V.: Solid-set functions and topological measures on locally compact spaces. Preprint arXiv: 1902.01957
Butler, S. V.: Non-linear functionals, deficient topological measures, and representation theorems on locally compact spaces. Banach J. Math. Anal., ISSN 2662-2033, Doi 10.1007/s43037-019-00034-0 (2020)
Butler, S. V.: Quasi-linear functionals on locally compact spaces. Preprint arXiv: 1902.03358
Butler, S. V.: Deficient topological measures on locally compact spaces. Accepted by Mathematische Nachrichten. Preprint arXiv: 1902.02458
Butler, S. V.: Decompositions of signed deficient topological measures. Letters in Math. Sciences 1 (1), 61–77 (2019)
Butler, S. V.: Integration with respect to deficient topological measures on locally compact spaces. Accepted by Mathematica Slovaca. Preprint arXiv:1902.08372
Denneberg, D.: Non-additive measure and integral. Kluwer Academic Publishers, Dordrecht (1994)
Dickstein, A., Zapolsky, F.: Approximation of quasi-states on manifolds. J. of Appl. and Computational Top. 3, 221–248 (2019)
Dugundji, J.: Topology. Allyn and Bacon, Inc., Boston (1966)
Entov, M., Polterovich, L.: Quasi-states and symplectic intersections. Comment. Math. Helv. 81, 75–99 (2006)
Entov, M., Polterovich, L., Zapolsky, F.: Quasi-morphisms and the Poisson Bracket. Pure and Appl. Math. Quarterly 4 (3), 1037–1055 (2007)
Entov, M.: Quasi-morphisms and quasi-states in symplectic topology. Proceedings of the International Congress of Mathematicians, Seoul, 1147–1171 (2014)
Grubb, D. J., Laberge, T.: Spaces of quasi-measures. Canad. Math. Bull. 42 (3), 291–297 (1999)
Johansen, Ø., Rustad, A.: Construction and Properties of quasi-linear functionals. Trans. Amer. Math. Soc. 358 (6), 2735–2758 (2006)
Kadison, R. V.: Transformation of states in operator theory and dynamics. Topology 3, 177–198 (1965)
Kechris, A. S.: Classical Descriptive Set Theory. Springer-Verlag, New York (1995)
Kelley, J. L.: General Topology. Springer-Verlag, reprint of Van Nostrand (1955)
Knudsen, F.: Topology and the construction of Extreme Quasi-measures. Adv. Math. 120 (2), 302–321 (1996)
Mackey, G. W.: Quantum mechanics and Hilbert space. Amer. Math. Monthly 64, 45–57 (1957)
Mackey, G. W.: The Mathematical Foundations of Quantum Mechanics. Benjamin, New York (1963)
Parthasarathy, K. R.: Probability measures on metric spaces. Academic Press, New York (1967)
Pfanzagl, J.: Mathematical Statistics: Essays on History and Methodology. Springer, Berlin (2017)
Polterovich, L., Rosen, D.: Function theory on symplectic manifolds. CRM Monograph series, vol. 34. American Mathematical Society, Providence, RI (2014)
Rustad, A. B.: Unbounded quasi-integrals. Proc. Amer. Math. Soc. 129 (1), 165–172 (2000)
Simon, B.: Real Analysis: A Comprehensive Course in Analysis, Part I. American Mathematical Society, USA (2015)
Svistula, M. G.: A Signed quasi-measure decomposition. Vestnik Samara Gos. Univ. Estestvennonauchn. 62 (3), 192–207 (Russian) (2008)
Svistula, M. G.: Deficient topological measures and functionals generated by them. Sbornik: Mathematics, 204 (5), 726–761 (2013)
Svistula, M. G.: On integration with respect to a DT-measure. Positivity 20 (3), 579–598 (2016)
von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton (1955). (Translation of Mathematische Grundlagen der Quantenmechanik. Springer, Berlin (1932).)
| ArXiv |
---
abstract: |
This paper deals with the problem of estimating predictive densities of a matrix-variate normal distribution with known covariance matrix. Our main aim is to establish some Bayesian predictive densities related to matricial shrinkage estimators of the normal mean matrix. The Kullback-Leibler loss is used for evaluating decision-theoretical optimality of predictive densities. It is shown that a proper hierarchical prior yields an admissible and minimax predictive density. Also, superharmonicity of prior densities is paid attention to for finding out a minimax predictive density with good numerical performance.
[*AMS 2010 subject classifications:*]{} Primary 62C15, 62C20; secondary 62C10.
[*Key words and phrases:*]{} Admissibility, Gauss’ divergence theorem, generalized Bayes estimator, inadmissibility, Kullback-Leibler loss, minimaxity, shrinkage estimator, statistical decision theory.
author:
- 'Hisayuki Tsukuma[^1] and Tatsuya Kubokawa[^2]'
title: 'Proper Bayes and Minimax Predictive Densities for a Matrix-variate Normal Distribution'
---
Introduction {#sec:intro}
============
The problem of predicting a density function for future observation is an important field in practical applications of statistical methodology. Since predictive density estimation has been revealed to be parallel to shrinkage estimation for location parameter, it has extensively been studied in the literature. Particularly, the Bayesian prediction for a multivariate (vector-valued) normal distribution has been developed by Komaki (2001), George et al. (2006) and Brown et al. (2008). See George et al. (2012) for a broad survey including a clear explanation of parallelism between density prediction and shrinkage estimation.
This paper addresses Bayesian predictive density estimation for a matrix-variate normal distribution. Denote by $\Nc_{a\times b}(M,\Psi\otimes\Si)$ the $a\times b$ matrix-variate normal distribution with mean matrix $M$ and positive definite covariance matrix $\Psi\otimes\Si$, where $M$, $\Psi$ and $\Si$ are, respectively, $a\times b$, $a\times a$ and $b\times b$ matrices of parameters and $\Psi\otimes\Si$ represents the Kronecker product of the positive definite matrices $\Psi$ and $\Si$. Let $A^\top$ be the transpose of a matrix $A$ and let $\tr A$ and $|A|$ be, respectively, the trace and the determinant a square matrix $A$. Also, let $A^{-1}$ be the inverse of a nonsingular matrix $A$. If an $a\times b$ random matrix $Z$ is distributed as $\Nc_{a\times b}(M,\Psi\otimes\Si)$, then $Z$ has density of the form $$(2\pi)^{-ab/2}|\Psi|^{-b/2}|\Si|^{-a/2}\exp[-2^{-1}\tr\{\Psi^{-1}(Z-\Th)\Si^{-1}(Z-\Th)^\top\}].$$ For more details of matrix-variate normal distribution, see Muirhead (1982) and Gupta and Nagar (1999).
It is assumed in this paper that the covariance matrix of a matrix-variate normal distribution is known. Then the prediction problem is more precisely formulated as follows: Let $X|\Th\sim\Nc_{r\times q}(\Th, v_xI_r\otimes I_q)$ and $Y|\Th\sim\Nc_{r\times q}(\Th, v_yI_r\otimes I_q)$, where $\Th$ is a common $r\times q$ matrix of unknown parameters, $v_x$ and $v_y$ are known positive values and $I_r$ stands for the identity matrix of order $r$. Assume that $q\geq r$ and $X$ and $Y$ are independent. Let $p(X\mid \Th)$ and $p(Y\mid \Th)$ be the densities of $X$ and $Y$, respectively. Consider here the problem of estimating $p(Y\mid \Th)$ based only on the observed $X$. Denote by $\ph=\ph(Y\mid X)$ an estimated density for $p(Y\mid \Th)$ and hereinafter $\ph$ is referred to as a predictive density of $Y$. Define the Kullback-Leibler (KL) loss as $$\begin{aligned}
\label{eqn:loss}
L_{KL}(\Th,\ph)
&=\Er^{Y|\Th}\bigg[\log {p(Y\mid\Th)\over\ph(Y\mid X)}\bigg] \non\\
&=\int_{\Re^{r\times q}} p(Y\mid \Th)\log {p(Y\mid\Th)\over\ph(Y\mid X)}\dd Y.\end{aligned}$$ The performance of a predictive density $\ph$ is evaluated by the risk function with respect to the KL loss (\[eqn:loss\]), $$\begin{aligned}
R_{KL}(\Th,\ph)&=\Er^{X|\Th}[L_{KL}(\Th,\ph)]\\
&=\int_{\Re^{r\times q}}\int_{\Re^{r\times q}}p(X\mid\Th)p(Y\mid\Th)\log {p(Y\mid\Th)\over\ph(Y\mid X)}\dd Y\dd X.\end{aligned}$$
Let $\pi(\Th)$ be a proper/improper density of prior distribution for $\Th$, where we assume that the marginal density of $X$, $$m_\pi(X;v_x)=\int_{\Re^{r\times q}} p(X\mid \Th)\pi(\Th)\dd\Th,$$ is finite for all $X\in\Re^{r\times q}$. Denote the Frobenius norm of a matrix $A$ by $\Vert A\Vert=\sqrt{\tr AA^\top}$. Let $$p_\pi(X,Y)=\int_{\Re^{r\times q}} p(X\mid \Th) p(Y\mid \Th)\pi(\Th)\dd\Th.$$ Note that $p_\pi(X,Y)$ is finite if $m_\pi(X;v_x)$ is finite. Here $p_\pi(X,Y)$ can be rewritten as $$\begin{aligned}
p_\pi(X,Y)&={1\over (2\pi v_s)^{qr/2}}e^{-\Vert Y-X\Vert^2/2v_s} \times
\int_{\Re^{r\times q}} {1\over (2\pi v_w)^{qr/2}}e^{-\Vert W-\Th\Vert^2/2v_w}\pi(\Th)\dd\Th \\
&\equiv \ph_U(Y\mid X) \times m_\pi(W;v_w),\end{aligned}$$ where $v_s=v_x+v_y$ and $$W=v_w(X/v_x+Y/v_y)\mid \Th \sim\Nc_{r\times q}(\Th, v_wI_r\otimes I_q)$$ with $v_w=(1/v_x+1/v_y)^{-1}$. From Aitchison (1975), a Bayesian predictive density relative to the KL loss (\[eqn:loss\]) is given by $$\label{eqn:BPD}
\ph_\pi(Y\mid X)={p_\pi(X,Y) \over m_\pi(X; v_x)}
={m_\pi(W;v_w)\over m_\pi(X;v_x)}\,\ph_U(Y\mid X).$$ See George et al. (2006, Lemma 2) for the multivariate (vector-valued) normal case.
It is noted that $\ph_U(Y\mid X)$ is the Bayesian predictive density with respect to the uniform prior $\pi_U(\Th)=1$. Under the predictive density estimation problem relative to the KL loss (\[eqn:loss\]), $\ph_U(Y\mid X)$ is the best invariant predictive density with respect to a location group. Using the same arguments as in George et al. (2006, Corollary 1) gives that, for any $r$ and $q$, $\ph_U(Y\mid X)$ is minimax relative to the KL loss (\[eqn:loss\]) and has a constant risk.
Recently, Matsuda and Komaki (2015) constructed an improved Bayesian predictive density on $\ph_U(Y\mid X)$ by using a prior density of the form $$\label{eqn:pr_em}
\pi_{EM}(\Th)=|\Th\Th^\top|^{-\al^{EM}/2},\quad \al^{EM}=q-r-1.$$ The prior (\[eqn:pr\_em\]) is interpreted as an extension of Stein’s (1973, 1981) harmonic prior $$\label{eqn:pr_js}
\pi_{JS}(\Th)=\Vert\Th\Vert^{-\be^{JS}}=\{\tr(\Th\Th^\top)\}^{-\be^{JS}/2},\quad \be^{JS}=qr-2.$$ In the context of Bayesian estimation for mean matrix, (\[eqn:pr\_em\]) yields a matricial shrinkage estimator, while (\[eqn:pr\_js\]) does a scalar shrinkage one. Note that, when $X\sim\Nc_{r\times q}(\Th, v_x I_r\otimes I_q)$, typical examples of the matricial and the scalar shrinkage estimators for $\Th$ are, respectively, the Efron-Morris (1972) estimator $$\label{eqn:EM}
\Thh_{EM}=\{I_r-\al^{EM}v_x(XX^\top)^{-1}\}X
\quad \textup{for $\al^{EM}\geq 1$ (i.e., $q\geq r+2$)}$$ and the James-Stein (1961) like estimator $$\label{eqn:JS}
\Thh_{JS}=\Big\{1-\frac{\be^{JS}v_x}{\tr(XX^\top)}\Big\}X
\quad \textup{for $\be^{JS}\geq 1$ (i.e., $qr\geq 3$)}.$$ The two estimators $\Thh_{EM}$ and $\Thh_{JS}$ are minimax relative to a quadratic loss. Also, $\Thh_{EM}$ and $\Thh_{JS}$ are characterized as empirical Bayes estimators, but they are not generalized Bayes estimators which minimize the posterior expected quadratic loss.
The purposes of this paper are to construct some Bayesian predictive densities with different priors from (\[eqn:pr\_em\]) and (\[eqn:pr\_js\]) and to discuss their decision-theoretic properties such as admissibility and minimaxity. Section \[sec:preliminaries\] first lists some results on the Kullback-Leibler risk and the differentiation operators. Section \[sec:properminimax\] applies an extended Faith’s (1978) prior to our predictive density estimation problem and provides sufficient conditions for minimaxity of the resulting Bayesian predictive densities. Also, an admissible and minimax predictive density is obtained by considering a proper hierarchical prior. In Section \[sec:superharmonic\], we utilize Stein’s (1973, 1981) ideas for deriving some minimax predictive densities with superharmonic priors. Section \[sec:MCstudies\] investigates numerical performance in risk of some Bayesian minimax predictive densities.
Preliminaries {#sec:preliminaries}
=============
The Kullback-Leibler risk
-------------------------
First, we state some useful lemmas in terms of the Kullback-Leibler (KL) risk. The lemmas are based on Stein (1973, 1981), George et al. (2006) and Brown et al. (2008) and play important roles in studying decision-theoretic properties of a Bayesian predictive density.
From George et al. (2006, Lemma 3), we observe that $m_\pi(W;v_w)<\infi$ for all $W\in\Re^{r\times q}$ if $m_\pi(X;v_x)<\infi$ for all $X\in\Re^{r\times q}$. Note also that $\int_{\Re^{r\times q}}\ph_\pi(Y\mid X)\dd Y=1$ and $$\int_{\Re^{r\times q}}Y\ph_\pi(Y\mid X)\dd Y
=\frac{\int_{\Re^{r\times q}} \Th p(X\mid \Th)\pi(\Th)\dd\Th}{\int_{\Re^{r\times q}} p(X\mid \Th)\pi(\Th)\dd\Th},$$ namely, the mean of a predictive distribution for $Y$ is the same as the posterior mean of $\Th$ given $X$ or, equivalently, the generalized Bayes estimator relative to a quadratic loss for a mean of $X$.
Hereafter denote by $p(W|\Th)$ a density of $W|\Th\sim\Nc_{r\times q}(\Th, vI_r\otimes I_q)$ with a positive value $v$. In order to prove minimaxity of a Bayesian predictive density, we require the following lemma, which implies that our Bayesian prediction problem can be reduced to the Bayesian estimation problem for the normal mean matrix relative to a quadratic loss.
\[lem:identity\] The KL risk difference between $\ph_U(Y\mid X)$ and $\ph_\pi(Y\mid X)$ can be written as $$R_{KL}(\Th,\ph_U) - R_{KL}(\Th,\ph_\pi)
=\frac{1}{2}\int_{v_w}^{v_x}\frac{1}{v^2}\{\Er^{W|\Th}[\Vert W-\Th\Vert^2]-\Er^{W|\Th}[\Vert\Thh_\pi-\Th\Vert^2]\}\dd v,$$ where $\Er^{W|\Th}$ stands for expectation with respect to $W$ and $$\Thh_\pi=\Thh_\pi(W)=\frac{\int_{\Re^{r\times q}} \Th p(W\mid \Th)\pi(\Th)\dd\Th}{\int_{\Re^{r\times q}} p(W\mid \Th)\pi(\Th)\dd\Th}.$$
[**Proof.**]{} This is verified by the same arguments as in Brown et al. (2008, Theorem 1 and its proof). $\Box$
Let $\nabla_W=(\partial/\partial w_{ij})$ be an $r\times q$ matrix of differentiation operators with respect to an $r\times q$ matrix $W=(w_{ij})$ of full row rank. For a scalar function $g(W)$ of $W$, the operation $\nabla_W g(W)$ is defined as an $r\times q$ matrix whose $(i,j)$-th element is $\partial g(W)/\partial w_{ij}$. Also, for a $q\times a$ matrix-valued function $G(W)=(g_{ij})$ of $W$, the operation $\nabla_WG(W)$ are defined as an $r\times a$ matrix whose $(i,j)$-th element of $\nabla_WG(W)$ is $\sum_{k=1}^q \partial g_{kj}/\partial w_{ik}$.
Stein (1973) showed that for a $q\times r$ matrix $G(W)$ $$\int_{\Re^{r\times q}} \tr\{(W-\Th)G(W)\}p(W|\Th)\dd W = v \int_{\Re^{r\times q}}\tr\{\nabla_WG(W)\}p(W|\Th)\dd W,$$ namely, $$\Er^{W|\Th}[\tr\{(W-\Th)G(W)\}] = v \Er^{W|\Th}[\tr\{\nabla_WG(W)\}].$$ This identity is referred to as the Stein identity in the literature. Using the Stein identity, we can easily obtain the following lemma.
\[lem:identity2\] Use the same notation as in Lemma \[lem:identity\]. Then we obtain $$\begin{aligned}
&R_{KL}(\Th,\ph_U) - R_{KL}(\Th,\ph_\pi)\\
&=-\int_{v_w}^{v_x}\Er^{W|\Th}\bigg[2\frac{\tr[\nabla_W\nabla_W^\top m_\pi(W;v)]}{m_\pi(W;v)}-\frac{\Vert\nabla_W m_\pi(W;v)\Vert^2}{\{m_\pi(W;v)\}^2}\bigg]\dd v.\end{aligned}$$
[**Proof.**]{} This lemma can be shown by the same arguments as in Stein (1973, 1981). We provide only an outline of proof.
Note from Brown (1971) that $\Thh_\pi$, given in Lemma \[lem:identity\], can be represented as $$\Thh_\pi=W+v\frac{\nabla_W m_\pi(W;v)}{m_\pi(W;v)}=W+v\nabla_W\log m_\pi(W;v).$$ By some manipulation after using the Stein identity, we have $$\begin{aligned}
&\frac{1}{v}\{\Er^{W|\Th}[\Vert W-\Th\Vert^2]-\Er^{W|\Th}[\Vert\Thh_\pi-\Th\Vert^2]\} \\
&=-v\Er^{W|\Th}\bigg[2\frac{\tr[\nabla_W\nabla_W^\top m_\pi(W;v)]}{m_\pi(W;v)}-\frac{\Vert\nabla_W m_\pi(W;v)\Vert^2]}{\{m_\pi(W;v)\}^2}\bigg].\end{aligned}$$ Combining this identity and Lemma \[lem:identity\] completes the proof. $\Box$
Using Lemma \[lem:identity2\] immediately establishes the following proposition.
\[prp:cond\_mini\] $\ph_\pi(Y|X)$ is minimax relative to the KL loss (\[eqn:loss\]) if $$2\tr[\nabla_W\nabla_W^\top m_\pi(W;v)]-\frac{\Vert\nabla_W m_\pi(W;v)\Vert^2}{m_\pi(W;v)}\leq 0$$ for $v_w\leq v\leq v_x$.
Differentiation of matrix-valued functions
------------------------------------------
Next, some useful formulae are listed for differentiation with respect to a symmetric matrix. The formulae are applied to evaluation of the Kullback-Leibler risks of our Bayesian predictive densities.
Let $S=(s_{ij})$ be an $r\times r$ symmetric matrix of full rank. Let $\Dc_S$ be an $r\times r$ symmetric matrix of differentiation operators with respect to $S$, where the $(i,j)$-th element of $\Dc_S$ is $$\{\Dc_S\}_{ij}=\frac{1+\de_{ij}}{2}\frac{\partial}{\partial s_{ij}}$$ with the Kronecker delta $\de_{ij}$.
Let $g(S)$ be a scalar-valued and differentiable function of $S=(s_{ij})$. Also let $G(S)=(g_{ij}(S))$ be an $r\times r$ matrix, where all the elements $g_{ij}(S)$ are differentiable functions of $S$. The operations $\Dc_S g(S)$ and $\Dc_S G(S)$ are, respectively, $r\times r$ matrices, where the $(i,j)$-th elements of $\Dc_S g(S)$ and $\Dc_S G(S)$ are defined as, respectively, $$\{\Dc_S g(S)\}_{ij}=\frac{1+\de_{ij}}{2}\frac{\partial g(S)}{\partial s_{ij}},\quad
\{\Dc_S G(S)\}_{ij}=\sum_{k=1}^r\frac{1+\de_{ik}}{2}\frac{\partial g_{kj}(S)}{\partial s_{ik}}.$$
First, the product rule in terms of $\Dc_S$ is expressed in the following lemma due to Haff (1982).
\[lem:diff1\] Let $G_1$ and $G_2$ be $r\times r$ matrices such that all the elements of $G_1$ and $G_2$ are differentiable functions of $S$. Then we have $$\Dc_S (G_1G_2)=(\Dc_S G_1)G_2+(G_1^\top \Dc_S)^\top G_2.$$ In particular, for differentiable scalar-valued functions $g_1(S)$ and $g_2(S)$, $$\Dc_S \{g_1(S)g_2(S)\}=g_2(S)\Dc_S g_1(S)+g_1(S) \Dc_S g_2(S).$$
Denote by $S=HLH^\top$ the eigenvalue decomposition of $S$, where $H=(h_{ij})$ is an orthogonal matrix of order $r$ and $L=\diag(\ell_1,\ldots,\ell_r)$ is a diagonal matrix of order $r$ with $\ell_1\geq \cdots\geq \ell_r$. The following lemma is provided by Stein (1973).
\[lem:diff2\] Define $\Psi(L)=\diag(\psi_1,\ldots,\psi_r)$, whose diagonal elements are differentiable functions of $L$. Then we obtain
1. $\{\Dc_S\}_{ij} \ell_k=h_{ik}h_{jk}$ $(k=1,\ldots,r)$,
2. $\Dc_S H\Psi(L)H^t=H\Psi^*(L)H^t$, where $\Psi^*(L)=\diag(\psi_1^*,\ldots,\psi_r^*)$ with $$\psi_i^*=\frac{\partial \psi_i}{\partial\ell_i}+\frac{1}{2}\sum_{j\ne i}^r\frac{\psi_i-\psi_j}{\ell_i-\ell_j}.$$
\[lem:diff3\] Let $a$ and $b$ be constants and let $C$ be a symmetric constant matrix $C$. Then it holds that
1. $\Dc_S \tr(S C)=C$,
2. $\displaystyle \Dc_S S=\frac{r+1}{2}I_r$,
3. $\displaystyle \Dc_S S^2=\frac{r+2}{2}S+\frac{1}{2}(\tr S)I_r$.
4. $\Dc_S |aI_r+bS|=b|aI_r+bS|(aI_r+bS)^{-1}$ if $aI_r+bS$ is nonsingular.
[**Proof.**]{} For proofs of Parts (i), (ii) and (iii), see Haff (1982) and Magnus and Neudecker (1999). Using (i) of Lemma \[lem:diff2\] gives that $$\begin{aligned}
\{\Dc_S |aI_r+bS|\}_{ij} &= \{\Dc_S\}_{ij} \prod_{k=1}^r(a+b\ell_k) \\
&=b\sum_{c=1}^r h_{ic}h_{jc}\prod_{k\ne c}^r(a+b\ell_k) \\
&=b|aI_r+bS|\sum_{c=1}^r h_{ic}h_{jc}(a+b\ell_c)^{-1} \\
&=b|aI_r+bS|\{(aI_r+bS)^{-1}\}_{ij},\end{aligned}$$ which implies Part (iv).
Let $\nabla_W$ be the same $r\times q$ differentiation operator matrix as in the preceding subsection. If $S=WW^\top$, then we have the following lemma, where the proof is referred to in Konno (1992).
\[lem:diff4\] Let $G$ be an $r\times r$ symmetric matrix, where all the elements of $G$ are differentiable function of $S=WW^\top$. Then it holds that
1. $\nabla_W^\top G=2W^\top \Dc_S G$,
2. $\tr(\nabla_W W^\top G)=(q-r-1)\tr G+2\tr(\Dc_S SG)$.
Admissible and minimax predictive densities {#sec:properminimax}
===========================================
In this section, we consider a class of hierarchical priors inspired by Faith (1978) and derive a sufficient condition for minimaxity of the resulting Bayesian predictive density. Also, a proper Bayes and minimax predictive density is provided.
A class of hierarchical prior distributions
-------------------------------------------
Let $\Sc_r$ be the set of $r\times r$ symmetric matrices. For $A$ and $B\in\Sc_r$, write $A\prec(\preceq) B$ or $B\succ(\succeq) A$ if $B-A$ is a positive (semi-)definite matrix. The set $\Rc_r$ is defined as $$\Rc_r=\{ \La\in\Sc_r \mid 0_{r\times r}\prec \La \prec I_r\},$$ where $0_{r\times r}$ is the $r\times r$ zero matrix. Denote the boundary of $\Rc_r$ by $\partial\Rc_r$. It is noted that if $\Om\in\partial\Rc_r$ then $0_{r\times r}\preceq \Om \preceq I_r$ and also then $|\Om|=0$ or $|I_r-\Om|=0$.
Consider a proper/improper hierarchical prior $$\pi_H(\Th)=\int_{\Rc_r}\pi_1(\Th|\Om)\pi_2(\Om)\dd\Om.$$ The priors $\pi_1(\Th|\Om)$ and $\pi_2(\Om)$ are specified as follows: Assume that a prior distribution of $\Th$ given $\Om$ is $\Nc_{r\times q}(0_{r\times q},v_0\Om^{-1}(I_r-\Om)\otimes I_q)$, where $v_0$ is a known constant satisfying $$v_0\geq v_x.$$ Then the first-stage prior density $\pi_1(\Th|\Om)$ can be written as $$\label{eqn:pr_Th}
\pi_1(\Th|\Om)=(2\pi v_0)^{-qr/2}|\Om(I_r-\Om)^{-1}|^{q/2}\exp\Big[-\frac{1}{2v_0}\tr\{\Om(I_r-\Om)^{-1}\Th\Th^\top\}\Big].$$ Assume also that $\pi_2(\Om)$, a second-stage prior density for $\Om$, is a differentiable function on $\Rc_r$.
Denote by $\ph_H=\ph_H(Y|X)$ the resulting Bayesian predictive density with respect to the hierarchical prior $\pi_H(\Th)$. Assume that a marginal density of $W$ with respect to $\pi_H(\Th)$ is finite when $v=v_x$. The marginal density is given by $$\begin{aligned}
\label{eqn:m(W)}
m(W)&=\int_{\Re^{r\times q}} p(W|\Th)\pi_H(\Th)\dd\Th \non\\
&=\int_{\Rc_r}\int_{\Re^{r\times q}} \pi(\Th|\Om,W) \dd\Th \pi_2(\Om)\dd\Om,\end{aligned}$$ where $\pi(\Th|\Om,W)=p(W|\Th)\pi_1(\Th|\Om)$ is a posterior density of $\Th$ given $\Om$ and $W$. To make it easy to derive sufficient conditions that $\ph_H$ is minimax, we show the following lemma.
\[lem:alter\_m(W)\] The marginal density $m(W)$ can alternatively be represented as $$m(W)=\int_{\Rc_r}f_\pi(\La;W)\dd\La,$$ where $$f_\pi(\La;W)=(2\pi v)^{-qr/2}|\La|^{q/2}\pi_2^J(\La)\exp\Big[-\frac{1}{2v}\tr(\La WW^\top)\Big]$$ with $$\pi_2^J(\La)
=v_1^{r(r+1)/2}|v_1I_r+(1-v_1)\La|^{-r-1}\pi_2[\La\{v_1I_r+(1-v_1)\La\}^{-1}].$$
[**Proof.**]{} Let $$\La(I_r-\La)^{-1}=v_1\Om(I_r-\Om)^{-1},\quad v_1=\frac{v}{v_0},$$ where $0_{r\times r}\prec\La\prec I_r$. Since $v^{-1}(I_r-\La)^{-1}=v^{-1}I_r+v_0^{-1}\Om(I_r-\Om)^{-1}$, we observe that $$\begin{aligned}
&\frac{1}{v}\Vert W-\Th\Vert^2+\frac{1}{v_0}\tr\{\Om(I_r-\Om)^{-1}\Th\Th^\top\}\\
&=\frac{1}{v}\tr\Big[(I_r-\La)^{-1}\{\Th-(I_r-\La)W\}\{\Th-(I_r-\La)W\}^\top\Big]
+\frac{1}{v}\tr(\La WW^\top),\end{aligned}$$ so $\pi(\Th|\Om,W)$ is proportional to $$\pi(\Th|\Om,W)
\propto \exp\Big[-\frac{1}{2v}\tr\Big[(I_r-\La)^{-1}\{\Th-(I_r-\La)W\}\{\Th-(I_r-\La)W\}^\top\Big]\Big],$$ namely, $\Th|\Om,W\sim\Nc_{r\times q}((I_r-\La)W,v(I_r-\La)\otimes I_q)$. Integrating out (\[eqn:m(W)\]) with respect to $\Th$ gives that $$\label{eqn:m(W)-1}
m(W)=(2\pi v)^{-qr/2}\int_{\Rc_r} |\La|^{q/2}\pi_2(\Om)\exp\Big[-\frac{1}{2v}\tr(\La WW^\top)\Big]\dd\Om.$$ Note that $\Om=\La\{v_1I_r+(1-v_1)\La\}^{-1}$ and the Jacobian of the transformation from $\Om$ to $\La$ is given by $$J[\Om\to \La]=v_1^{r(r+1)/2}|v_1I_r+(1-v_1)\La)|^{-r-1}.$$ Hence making the transformation from $\Om$ to $\La$ for (\[eqn:m(W)-1\]) completes the proof.
Let $\Dc_\La$ be an $r\times r$ symmetric matrix of differentiation operators with respect to $\La=(\la_{ij})$, where the $(i,j)$-th element of $\Dc_\La$ is $$\{\Dc_\La\}_{ij}=\frac{1+\de_{ij}}{2}\frac{\partial}{\partial\la_{ij}}.$$ Proposition \[prp:cond\_mini\] and Lemma \[lem:alter\_m(W)\] are utilized to get sufficient conditions for minimaxity of $\ph_H$.
\[thm:faith\] Let $f_\pi(\La;W)$ and $\pi_2^J(\La)$ be defined as in Lemma \[lem:alter\_m(W)\]. Let $$M=M(W)=\int_{\Rc_r}\La f_\pi(\La;W)\dd\La.$$ Assume that $$f_\pi(\La;W)=0 \quad \textup{for all $\La\in\partial\Rc_r$}.$$ Then $\ph_H$ is minimax relative to the KL loss $(\ref{eqn:loss})$ if $\De(W;\pi_2^J)\leq 0$, where $$\De(W;\pi_2^J)=\De_1(W;\pi_2^J)-\De_2(W;\pi_2^J)-(q-3r-3)\tr M$$ with $$\begin{aligned}
\De_1(W;\pi_2^J)&=4\int_{\Rc_r}\frac{1}{\pi_2^J(\La)}\tr\{\La^2\Dc_\La \pi_2^J(\La)\}f_\pi(\La;W)\dd\La,\\
\De_2(W;\pi_2^J)&=\frac{2}{m(W)}\int_{\Rc_r}\frac{1}{\pi_2^J(\La)}\tr\{\La M\Dc_\La \pi_2^J(\La)\}f_\pi(\La;W)\dd\La,\end{aligned}$$ provided all the integrals are finite.
[**Proof.**]{} From Proposition \[prp:cond\_mini\], $\ph_H$ is minimax when $$\De=2\tr[\nabla_W\nabla_W^\top m(W)]-\frac{\Vert\nabla_W m(W)\Vert^2}{m(W)}\leq 0.$$
It is seen from Lemma \[lem:alter\_m(W)\] that $$\nabla_W f_\pi(\La;W)=-\frac{1}{v}\La W f_\pi(\La;W)$$ and $$\nabla_W\nabla_W^\top f_\pi(\La;W)=\Big(\frac{1}{v^2}\La WW^\top\La-\frac{q}{v}\La\Big)f_\pi(\La;W).$$ Hence we obtain $$\label{eqn:d2_mw1}
\De=\frac{1}{v}[2E_1(W)-\{m(W)\}^{-1}E_2(W)],$$ where $$\begin{aligned}
E_1(W)&=\int_{\Rc_r}\Big[\frac{1}{v}\tr(WW^\top\La^2)-q\tr\La\Big]f_\pi(\La;W)\dd\La,\\
E_2(W)&=\frac{1}{v}\tr\bigg[WW^\top\bigg\{\int_{\Rc_r}\La f_\pi(\La;W)\dd\La\bigg\}^2\bigg] \\
&=\frac{1}{v}\int_{\Rc_r}\tr(MWW^\top\La)f_\pi(\La;W)\dd\La.\end{aligned}$$
Using Lemmas \[lem:diff1\] and \[lem:diff3\] yields that $$\Dc_\La f_\pi(\La;W)=\frac{1}{2}\Big[q\La^{-1}+\frac{2}{\pi_2^J(\La)}\Dc_\La \pi_2^J(\La)-\frac{1}{v}WW^\top\Big]f_\pi(\La;W),$$ so that $$\begin{aligned}
\tr[\Dc_\La\{f_\pi(\La;W)\La^2\}]&=\tr[\La^2\Dc_\La f_\pi(\La;W)]+f_\pi(\La;W)\tr[\Dc_\La\La^2]\\
&=\frac{1}{2}\Big[\frac{2}{\pi_2^J(\La)}\tr[\La^2\Dc_\La \pi_2^J(\La)]-\Big\{\frac{1}{v}\tr(WW^\top\La^2)-q\tr\La\Big\} \\
&\qquad +2(r+1)\tr\La \Big]f_\pi(\La;W).\end{aligned}$$ Thus $E_1(W)$ can be expressed as $$\begin{aligned}
\label{eqn:E1}
E_1(W)
&=2(r+1)\tr M+\frac{1}{2}\De_1(W;\pi_2^J) -2\int_{\Rc_r}\tr[\Dc_\La\{f_\pi(\La;W)\La^2\}]\dd\La.\end{aligned}$$
Similarly, we observe that from Lemmas \[lem:diff1\] and \[lem:diff3\] $$\begin{aligned}
\tr[\Dc_\La\{f_\pi(\La;W)\La\}M]
&=\tr[\La M\Dc_\La f_\pi(\La;W)]+f_\pi(\La;W)\tr[M\Dc_\La\La]\\
&=\frac{1}{2}\Big[(q+r+1)\tr M+\frac{2}{\pi_2^J(\La)}\tr[\La M\Dc_\La \pi_2^J(\La)] \\
&\qquad -\frac{1}{v}\tr(MWW^\top\La)\Big]f_\pi(\La;W),\end{aligned}$$ which leads to $$\begin{aligned}
\label{eqn:E2}
E_2(W)
&=(q+r+1)m(W)\tr M +m(W)\De_2(W;\pi_2^J) \non\\
&\qquad -2\int_{\Rc_r}\tr[\Dc_\La\{f_\pi(\La;W)\La\}M]\dd\La .\end{aligned}$$ Combining (\[eqn:d2\_mw1\]), (\[eqn:E1\]) and (\[eqn:E2\]) gives that $$\begin{aligned}
\label{eqn:d2_mw2}
\De
&= \frac{\De(W;\pi_2^J)}{v} -\frac{4}{v}\int_{\Rc_r}\tr[\Dc_\La\{f_\pi(\La;W)\La^2\}]\dd\La \non\\
&\qquad +\frac{2}{vm(W)}\int_{\Rc_r}\tr[\Dc_\La\{f_\pi(\La;W)\La\}M]\dd\La.\end{aligned}$$ If we can show that two integrals in (\[eqn:d2\_mw2\]) are, respectively, equal to zero, then the proof is complete.
Let $G=(g_{ij})$ be an $r\times r$ symmetric matrix such that all the elements of $G$ are differentiable functions of $\La\in\Rc_r$. Denote $$\vec(G)=(g_{11},g_{12},\ldots,g_{1r},g_{22},g_{23},\ldots,g_{r-1,r-1},g_{r-1,r},g_{rr})^\top,$$ which is a $\{2^{-1}r(r+1)\}$-dimensional column vector. Denote an outward unit normal vector at a point $\La$ on $\partial\Rc_r$ by $$\nu=\nu(\La)=(\nu_{11},\nu_{12},\ldots,\nu_{1r},\nu_{22},\nu_{23},\ldots,\nu_{r-1,r-1},\nu_{r-1,r},\nu_{rr})^\top.$$ If $\tr(\Dc_\La G)$ is integrable on $\Rc_r$ then it is seen that $$\int_{\Rc_r}\tr(\Dc_\La G)\dd\La
=\int_{\Rc_r}\sum_{i=1}^r\sum_{j=1}^r\frac{1+\de_{ij}}{2}\frac{\partial g_{ji}}{\partial\la_{ij}}\dd\La
=\int_{\Rc_r}\sum_{i=1}^r\sum_{j=i}^r\frac{\partial g_{ij}}{\partial\la_{ij}}\dd\La$$ by symmetry of $\La$ and $G$. From the Gauss divergence theorem, we obtain $$\int_{\Rc_r}\tr(\Dc_\La G)\dd\La=\int_{\partial\Rc_r}\sum_{i=1}^r\sum_{j=i}^r\nu_{ij} g_{ij}\dd\si=\int_{\partial\Rc_r}\nu^\top\vec(G)\dd\si,$$ where $\si$ stands for Lebesgue measure on $\partial\Rc_r$.
Note that $$\begin{aligned}
\tr[\Dc_\La\{f_\pi(\La;W)\La\}M]=\tr[\Dc_\La\{f_\pi(\La;W)\La M\}]=\tr[\Dc_\La\{f_\pi(\La;W)M\La\}]\end{aligned}$$ because $M=M(W)$ is symmetric and does not depend on $\La$. It is observed that $\La^2$ and $\La M+M\La$ are symmetric for $\La\in\Rc_r$, so that $$\label{eqn:gdt1}
\int_{\Rc_r}\tr[\Dc_\La\{f_\pi(\La;W)\La^2\}]\dd\La=\int_{\partial\Rc_r}\nu^\top\vec(\La^2)f_\pi(\La;W)\dd\si,$$ and $$\begin{aligned}
&\int_{\Rc_r}\tr[\Dc_\La\{f_\pi(\La;W)\La\}M]\dd\La \non\\
&=\frac{1}{2}\int_{\Rc_r}\tr[\Dc_\La\{f_\pi(\La;W)\La M+f_\pi(\La;W)M\La\}]\dd\La \non\\
&=\frac{1}{2}\int_{\partial\Rc_r}\nu^\top \vec(\La M+M\La)f_\pi(\La;W)\dd\si.
\label{eqn:gdt2}\end{aligned}$$ Recall that $M$ is finite and $0_{r\times r} \preceq \La \preceq I_r$ for $\La\in\partial\Rc_r$, so that $\nu^\top\vec(\La^2)$ and $\nu^\top \vec(\La M+M\La)$ are bounded. Since $f_\pi(\La;W)=0$ for any $\La\in\partial\Rc_r$, (\[eqn:gdt1\]) and (\[eqn:gdt2\]) are, respectively, equal to zero, which completes the proof. $\Box$
Proper Bayes and minimax predictive densities
---------------------------------------------
Define a second-stage prior density for $\Om$ as $$\label{eqn:pr_GB}
\pi_{GB}(\Om)=K_{a,b}|\Om|^{a/2-1}|I_r-\Om|^{b/2-1},\qquad 0_{r\times r}\prec \Om\prec I_r,$$ where $a$ and $b$ are constants and $K_{a,b}$ is a normalizing constant. The hierarchical prior (\[eqn:pr\_Th\]) with (\[eqn:pr\_GB\]) is a generalization of Faith (1978) in Bayesian minimax estimation of a normal mean vector. Faith’s (1978) prior has also been discussed in detail by Maruyama (1998). When $a>0$ and $b>0$, $\pi_{GB}(\Om)$ is proper and the distribution of $\Om$ is often called the matrix-variate beta distribution. Konno (1988) showed that $$\begin{aligned}
&\int_{\Rc_r} \Om \pi_{GB}(\Om)\dd\Om=\frac{a+r-1}{a+b+2r-2}I_r \qquad \textup{for $a>0$ and $b>0$},\\
&\int_{\Rc_r} \Om(I_r-\Om)^{-1} \pi_{GB}(\Om)\dd\Om=\frac{a+r-1}{b-2}I_r \qquad \textup{for $a>0$ and $b>2$}.\end{aligned}$$ For other properties of the matrix-variate beta distribution, see Muirhead (1982) and Gupta and Nagar (1999).
Let $\ph_{GB}(Y\mid X)$ be the generalized Bayesian predictive density with respect to (\[eqn:pr\_Th\]) and (\[eqn:pr\_GB\]). A sufficient condition for minimaxity of $\ph_{GB}(Y\mid X)$ is given as follows.
\[prp:mini\_GB\] Assume that $q-r-1>0$. Then $\ph_{GB}(Y\mid X)$ is minimax relative to the KL loss [(\[eqn:loss\])]{} if $$\label{eqn:upper0}
a>-q+2,\quad b>2,\quad a+b \leq (q-r-1)/(2-v_w/v_0)-2r+2.$$ There exist constants $a$ and $b$ satisfying (\[eqn:upper0\]) if $
q-r-1+(2-v_w/v_0)(q-2r-2)>0.
$
Recall that $\ph_U(Y\mid X)$ is minimax and has a constant risk relative to the KL loss (\[eqn:loss\]). Hence if $\ph_{GB}(Y\mid X)$ is proper Bayes then it is admissible.
Assume that $q-r-1>0$. Then $\ph_{GB}(Y\mid X)$ is admissible and minimax relative to the KL loss [(\[eqn:loss\])]{} if $$\label{eqn:upper1}
a>0,\quad b>2, \quad a+b \leq (q-r-1)/(2-v_w/v_0)-2r+2.$$ Thus, there exist constants $a$ and $b$ satisfying (\[eqn:upper1\]) if $(-q+5r+1)/(2r)<v_w/v_0 <1$.
Since $0<v_w<v_x\leq v_0$, it is observed that $$(q-r-1)/(2-v_w/v_0)-2r+2>(q-r-1)/2-2r+2=(q-5r+3)/2.$$ We also obtain the following corollary.
\[cor:proper2\] Assume that $q-5r-1>0$. Then, for any $v_x$, $v_y$ and $v_0\ (\geq v_x)$, $\ph_{GB}(Y\mid X)$ is admissible and minimax relative to the KL loss [(\[eqn:loss\])]{} if $$a>0,\quad b>2, \quad a+b \leq (q-5r+3)/2.$$
[**Proof of Proposition \[prp:mini\_GB\].**]{} Using Theorem \[thm:faith\], we will derive a sufficient condition for minimaxity of $\ph_{GB}(Y\mid X)$.
Denote $$c=a+b+2r-2.$$ Let $$\begin{aligned}
\pi_{GB}^J(\La)&=v_1^{r(r+1)/2}|v_1I_r+(1-v_1)\La|^{-r-1}\pi_{GB}[\La\{v_1I_r+(1-v_1)\La\}^{-1}]\\
&=K_{a,b}v_1^{r(r+b-1)/2}|\La|^{a/2-1}|I_r-\La|^{b/2-1}|v_1I_r+(1-v_1)\La|^{-c/2}.\end{aligned}$$ When $$\label{eqn:bound-cond}
q+a>2 \quad\textup{and}\quad b>2,$$ it follows that for any $\La\in\partial\Rc_r$ $$f_{GB}(\La;W)=(2\pi v)^{-qr/2}|\La|^{q/2}\pi_{GB}^J(\La)\exp\Big[-\frac{1}{2v}\tr(\La WW^\top)\Big]=0.$$
Define $$\begin{aligned}
m_{GB}&=m_{GB}(W)=\int_{\Rc_r} f_{GB}(\La;W)\dd\La, \\
M_{GB}&=M_{GB}(W)=\int_{\Rc_r}\La f_{GB}(\La;W)\dd\La.\end{aligned}$$ Since $0<v_1\leq 1$ and $\La\in\Rc_r$, it holds that $$|v_1I_r+(1-v_1)\La|^{-c/2}\leq \max\big(1,v_1^{-rc/2}\big),$$ which implies that $$f_{GB}(\La;W)\leq \textup{const.}\times |\La|^{(q+a)/2-1}|I_r-\La|^{b/2-1}.$$ Thus if $q+a>0$ and $b>0$ then $m_{GB}$ and $M_{GB}$ are finite.
It is seen from Lemmas \[lem:diff1\] and \[lem:diff3\] that $$\frac{\Dc_\La \pi_{GB}^J(\La)}{\pi_{GB}^J(\La)}
=\frac{1}{2}\big[(a-2)\La^{-1}-(b-2)(I_r-\La)^{-1}-(1-v_1)c\{v_1I_r+(1-v_1)\La\}^{-1}\Big],$$ so that $$\begin{aligned}
\De_1(W;\pi_{GB}^J)
&=4\int_{\Rc_r}\frac{1}{\pi_{GB}^J(\La)}\tr[\La^2\Dc_\La \pi_{GB}^J(\La)]f_{GB}(\La;W)\dd\La\\
&=2(a-2)\tr M_{GB}-2(b-2)\int_{\Rc_r}\tr[(I_r-\La)^{-1}\La^2]f_{GB}(\La;W)\dd\La\\
&\quad -2(1-v_1)c\int_{\Rc_r}\tr[\{v_1I_r+(1-v_1)\La\}^{-1}\La^2]f_{GB}(\La;W)\dd\La.\end{aligned}$$ Note that $$\tr[(I_r-\La)^{-1}\La^2]=-\tr\La+\tr[(I_r-\La)^{-1}\La],$$ which leads to $$\begin{aligned}
\label{eqn:De1}
&\De_1(W;\pi_{GB}^J) \non\\
&=2(a+b-4)\tr M_{GB}-2(b-2)\int_{\Rc_r}\tr[(I_r-\La)^{-1}\La]f_{GB}(\La;W)\dd\La \non\\
&\quad -2(1-v_1)c\int_{\Rc_r}\tr[\{v_1I_r+(1-v_1)\La\}^{-1}\La^2]f_{GB}(\La;W)\dd\La.\end{aligned}$$ Similarly, Lemmas \[lem:diff1\] and \[lem:diff3\] are used to see that $$\begin{aligned}
\frac{2\tr[\La M_{GB}\Dc_\La \pi_{GB}^J(\La)]}{\pi_{GB}^J(\La)}
&=(a-2)\tr M_{GB}-(b-2)\tr[M_{GB}(I_r-\La)^{-1}\La] \\
&\quad -(1-v_1)c\tr[M_{GB}\{v_1I_r+(1-v_1)\La\}^{-1}\La],\end{aligned}$$ which yields that $$\begin{aligned}
\label{eqn:De2}
\De_2(W;\pi_{GB}^J)
&= \frac{2}{m_{GB}}\int_{\Rc_r}\frac{\tr[\La M_{GB}\Dc_\La \pi_{GB}^J(\La)]}{\pi_{GB}^J(\La)}f_{GB}(\La;W)\dd\La \non\\
&= (a-2)\tr M_{GB} -\frac{b-2}{m_{GB}}\int_{\Rc_r}\tr[M_{GB}(I_r-\La)^{-1}\La]f_{GB}(\La;W)\dd\La \non\\
&\quad -\frac{(1-v_1)c}{m_{GB}}\int_{\Rc_r}\tr[M_{GB}\{v_1I_r+(1-v_1)\La\}^{-1}\La]f_{GB}(\La;W)\dd\La.\end{aligned}$$ Hence combining (\[eqn:De1\]) and (\[eqn:De2\]) gives that $$\begin{aligned}
\label{eqn:De0}
\De(W;\pi_{GB}^J)&=\De_1(W;\pi_{GB}^J)-\De_2(W;\pi_{GB}^J)-(q-3r-3)\tr M_{GB} \non\\
&= -(q-3r+3-a-2b)\tr M_{GB} +\De_3+\De_4,\end{aligned}$$ where $$\begin{aligned}
\De_3 &= -2(b-2)\int_{\Rc_r}\tr[(I_r-\La)^{-1}\La]f_{GB}(\La;W)\dd\La \\
&\qquad +\frac{b-2}{m_{GB}}\int_{\Rc_r}\tr[M_{GB}(I_r-\La)^{-1}\La]f_{GB}(\La;W)\dd\La, \\
\De_4 &=-2(1-v_1)c\int_{\Rc_r}\tr[\{v_1I_r+(1-v_1)\La\}^{-1}\La^2]f_{GB}(\La;W)\dd\La \\
&\qquad +\frac{(1-v_1)c}{m_{GB}}\int_{\Rc_r}\tr[M_{GB}\{v_1I_r+(1-v_1)\La\}^{-1}\La]f_{GB}(\La;W)\dd\La.\end{aligned}$$ Here, it can easily be verified that $\De(W;\pi_{GB}^J)$ is finite for $q+a>0$ and $b>2$.
For notational simplicity, we use the notation $$\Er_\La[g(\La)]=\int_{\Rc_r}g(\La)f_{GB}(\La;W)\dd\La \Big/\int_{\Rc_r}f_{GB}(\La;W)\dd\La$$ for an integrable function $g(\La)$. Then from (\[eqn:De0\]), $$\begin{aligned}
\label{eqn:De00}
{\De(W;\pi_{GB}^J)\over m_{GB}}
=& (c-q+r+b-1)\tr \Er_\La(\La) \non\\
&+ (b-2)\Big[ \tr\big[ \Er_\La(\La)\Er_\La\{(I_r-\La)^{-1}\La\} \big] - 2\tr\big[ \Er_\La\{(I_r-\La)^{-1}\La\}\big]\Big]\non\\
&+(1-v_1)c\Big[ \tr\big[ \Er_\La(\La)\Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\La\}\big]
\non\\
&\qquad\qquad\qquad -2\tr\big[ \Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\La^2\}\big]\Big]\end{aligned}$$ for $c=a+b+2r-2$. Note that $0_{r\times r} \preceq \La \preceq I_r$ and $I_r \preceq (I_r-\La)^{-1}$. Since $\Er_\La(\La) \preceq I_r$ and $\tr\big[ (I_r-\La)^{-1}\La \big] \geq \tr\La$, the second term in the r.h.s. of (\[eqn:De00\]) is evaluated as $$\begin{aligned}
\tr\big[ &\Er_\La(\La)\Er_\La\{(I_r-\La)^{-1}\La\}\big] - 2\tr\big[ \Er_\La\{(I_r-\La)^{-1}\La\}\big]
\\
&\leq - \tr\big[ \Er_\La\{(I_r-\La)^{-1}\La\}\big] \leq - \tr \Er_\La(\La).\end{aligned}$$ Since $b>2$, we have $$\begin{aligned}
\label{eqn:De01}
{\De(W;\pi_{GB}^J)\over m_{GB}}
\leq & (c-q+r+1)\tr \Er_\La(\La) \non\\
&+(1-v_1)c\Big[ \tr\big[ \Er_\La(\La)\Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\La\}\big]
\non\\
&\qquad\qquad\qquad -2\tr\big[ \Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\La^2\}\big]\Big].\end{aligned}$$ It is here observed that $$\begin{aligned}
&(1-v_1)\{v_1I_r+(1-v_1)\La\}^{-1}\La \\
&\qquad = \{v_1I_r+(1-v_1)\La\}^{-1} \{ v_1I_r+(1-v_1)\La - v_1I_r\}\\
&\qquad= I_r - v_1 \{v_1I_r+(1-v_1)\La\}^{-1},
\\
&(1-v_1)\{v_1I_r+(1-v_1)\La\}^{-1}\La^2 \\
&\qquad= \La - v_1 \{v_1I_r+(1-v_1)\La\}^{-1}\La,\end{aligned}$$ which is used to get $$\begin{aligned}
&(1-v_1)c\Big[ \tr\big[ \Er_\La(\La)\Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\La\}\big] \\
&\qquad\qquad -2\tr\big[ \Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\La^2\}\big]\Big] \\
&\qquad =-c\tr \Er_\La(\La)\\
&\qquad\qquad + cv_1 \Big[ 2 \tr\big[ \Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\La\}\big] \\
&\qquad\qquad\qquad-\tr\big[ \Er_\La(\La) \Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\}\big]\Big].\end{aligned}$$ Substituting this quantity into (\[eqn:De01\]) gives $$\begin{aligned}
\label{eqn:De02}
{\De(W;\pi_{GB}^J)\over m_{GB}}
\leq & -(q-r-1)\tr \Er_\La(\La) \non\\
&+ cv_1 \Big[ 2 \tr\big[ \Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\La\}\big] \non\\
&\qquad\qquad -\tr\big[ \Er_\La(\La) \Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\}\big]\Big].\end{aligned}$$
To evaluate the second term in the r.h.s. of (\[eqn:De02\]), note that $$I_r \preceq \{v_1I_r+(1-v_1)\La\}^{-1} \preceq v_1^{-1}I_r.
\label{eqn:inq}$$ In the case of $c\geq 0$, it is seen from (\[eqn:inq\]) that $$\begin{aligned}
cv_1 &\Big[ 2 \tr\big[ \Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\La\}\big]-\tr\big[ \Er_\La(\La) \Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\}\big]\Big]
\\
&\leq cv_1 \Big\{ {2\over v_1}\tr \Er_\La(\La) - \tr \Er_\La(\La)\Big\}
= c (2-v_1) \tr \Er_\La(\La),\end{aligned}$$ which implies that $$\De(W;\pi_{GB}^J)/m_{GB}
\leq \{ - (q-r-1) + c(2-v_w/v_0)\}\tr \Er_\La(\La),$$ because $1>v_1=v/v_0\geq v_w/v_0>0$. It is noted that $c=a+b+2r-2>-q + 2r +2=- (q-r-1) + r+1$ because $a>2-q$ and $b>2$. Thus, one gets a sufficient condition given by $$\max \{0, - (q-r-1) + r+1\} \leq c \leq (q-r-1)/(2-v_w/v_0).
\label{eqn:sc1}$$ In the case of $c\leq 0$, it is seen from (\[eqn:inq\]) that $$\begin{aligned}
cv_1 &\Big[ 2 \tr\big[ \Er_\La\{(v_1I_r+(1-v_1)\La)^{-1}\La\}\big]-\tr\big[ \Er_\La(\La) \Er_\La(v_1I_r+(1-v_1)\La)^{-1}\big]\Big]
\\
&\leq cv_1 \Big\{ 2\tr \Er_\La(\La) - {1\over v_1}\tr \Er_\La(\La)\Big\}
= c (2v_1-1) \tr \Er_\La(\La),\end{aligned}$$ which implies that $$\De(W;\pi_{GB}^J)/m_{GB}
\leq \{ - (q-r-1) + c(2v_w/v_0-1)\}\tr \Er_\La(\La).$$ Hence, it holds true that $ - (q-r-1) + c(2v_w/v_0-1)\leq 0$ if $$\min \{0, - (q-r-1) + r+1\} \leq c \leq 0.
\label{eqn:sc2}$$ Combining (\[eqn:sc1\]) and (\[eqn:sc2\]) yields the condition $- (q-r-1) + r+1 \leq c \leq (q-r-1)/(2-v_w/v_0)$, namely, $-q+4\leq a+b\leq (q-r-1)/(2-v_w/v_0)-2r+2$. From (\[eqn:bound-cond\]), the sufficient conditions on $(a,b)$ for minimaxity can be written as $a>2-q$, $b>2$ and $a+b\leq (q-r-1)/(2-v_w/v_0)-2r+2$ if $$\begin{aligned}
&\{(q-r-1)/(2-v_w/v_0)-2r+2\}-\{-q+4\}\\
&=\{q-r-1+(2-v_w/v_0)(q-2r-2)\}/(2-v_w/v_0)>0.\end{aligned}$$ Thus the proof is complete. $\Box$
Take $v_x=v_w=v_0=1$. Let $X|\Th\sim\Nc_{r\times q}(\Th,I_r\otimes I_q)$. Consider the problem of estimating the mean matrix $\Th$ under the squared Frobenius norm loss $\Vert\Thh-\Th\Vert^2$. The Bayesian estimator with respect to (\[eqn:pr\_Th\]) and (\[eqn:pr\_GB\]) is expressed as $$\Thh_{GB}=\bigg[I_r-\frac{\int_{\Rc_r}\Om|\Om|^{(q+a)/2-1}|I_r-\Om|^{b/2-1}\exp[-\tr(\Om XX^\top)/2]\dd\Om}{\int_{\Rc_r}|\Om|^{(q+a)/2-1}|I_r-\Om|^{b/2-1}\exp[-\tr(\Om XX^\top)/2]\dd\Om}\bigg]X.$$ Then the same arguments as in this section yield that $\Thh_{GB}$ is proper Bayes and minimax if $a>0$, $b>2$, $q>3r+1$ and $ 2 < a+b \leq q-3r+1$. $\Box$
Superharmonic priors for minimaxity {#sec:superharmonic}
===================================
In estimation of the normal mean vector, Stein (1973, 1981) discovered an interesting relationship between superharmonicity of prior density and minimaxity of the resulting generalized Bayes estimator. The relationship is very important and useful in Bayesian predictive density estimation. In this section we derive some Bayesian minimax predictive densities with superharmonic priors.
Let $\ph_\pi=\ph_\pi(Y|X)$ be a Bayesian predictive density with respect to a prior $\pi(\Th)$, where $\pi(\Th)$ is twice differentiable and the marginal density $m_\pi(X;v_x)$ is finite. All the results in this section are based on the following key lemma.
\[lem:superharmonic\] Denote by $\nabla_\Th=(\partial/\partial\th_{ij})$ the $r\times q$ differentiation operator matrix with respect to $\Th$. Then $\ph_\pi$ is minimax relative to the KL loss $(\ref{eqn:loss})$ if $\pi(\Th)$ is superharmonic, namely, $$\tr[\nabla_\Th\nabla_\Th^\top \pi(\Th)]=\sum_{i=1}^r\sum_{j=1}^q\frac{\partial^2 \pi(\Th)}{\partial \th_{ij}^2}\leq 0.$$
[**Proof.**]{} This lemma can be proved along the same arguments as in Stein (1981). See also George et al. (2006) and Brown et al. (2008). $\Box$
Define a class of prior densities as $$\pi(\Th)=g(\Si),\quad \Si=\Th\Th^\top,$$ where $g$ is twice differentiable with respect to $\Si$. Let $\Dc_\Si$ be an $r\times r$ matrix of differential operator with respect to $\Si=(\si_{ij})$ such that the $(i,j)$ element of $\Dc_\Si$ is $$\{\Dc_\Si\}_{ij}=\frac{1+\de_{ij}}{2}\frac{\partial}{\partial \si_{ij}},$$ where $\de_{ij}$ stands for the Kronecker delta. Let $$G=(g_{ij})=G(\Si)=\Dc_\Si g(\Si),$$ namely, $G$ is an $r\times r$ symmetric matrix such that $g_{ij}=\{\Dc_\Si\}_{ij} g(\Si)$.
\[lem:condition1\] $\ph_\pi$ with respect to $\pi(\Th)=g(\Si)$ is minimax relative to the KL loss $(\ref{eqn:loss})$ if $$\tr[\nabla_\Th\nabla_\Th^\top\pi(\Th)]=2[(q-r-1)\tr(G)+2\tr(\Dc_\Si \Si G)]\leq 0,$$ where $G=\Dc_\Si g(\Si)$.
[**Proof.**]{} Using (i) and (ii) in Lemma \[lem:diff4\] gives that $$\begin{aligned}
\tr[\nabla_\Th\nabla_\Th^\top \pi(\Th)]
&=2\tr(\nabla_\Th \Th^\top \Dc_\Si g(\Si))=2\tr(\nabla_\Th \Th^\top G)\\
&=2\big[(q-r-1)\tr(G)+2\tr(\Dc_\Si \Si G)\big].\end{aligned}$$ From Lemma \[lem:superharmonic\], the proof is complete.
Let $\la_1,\ldots,\la_r$ be ordered eigenvalues of $\Si=\Th\Th^\top$, where $\la_1\geq\cdots\geq\la_r$, and let $\La=\diag(\la_1,\ldots,\la_r)$. Denote by $\Ga=(\ga_{ij})$ an $r\times r$ orthogonal matrix such that $\Ga^\top\Si \Ga=\La$. Assume that $g(\Si)$ is orthogonally invariant, namely, $g(\Si)=g(P\Si P^\top)$ for any orthogonal matrix $P$. Then, we can assume that $g(\Si)=g(\La)$ without loss of generality.
\[prp:condition2\] Assume that $g(\Si)=g(\La)$ and $g(\La)$ is a twice differentiable function of $\La$. Then $\ph_\pi$ with $\pi(\Th)=g(\La)$ is minimax relative to the KL loss $(\ref{eqn:loss})$ if $$\begin{aligned}
&\tr[\nabla_\Th\nabla_\Th^\top\pi(\Th)]\\
&=2\sum_{i=1}^r\bigg\{(q-r+1)\phi_i(\La)+\sum_{j\ne i}^r\frac{\la_i\phi_i(\La)-\la_j\phi_j(\La)}{\la_i-\la_j}+2\la_i\frac{\partial\phi_i(\La)}{\partial\la_i}\bigg\}\leq 0,\end{aligned}$$ where $\phi_i(\La)=\partial g(\La)/\partial\la_i$.
[**Proof.**]{} Since from (i) of Lemma \[lem:diff2\] $$\{\Dc_\Si\}_{ij}\la_k=\ga_{ik}\ga_{jk},$$ it is observed that by the chain rule $$\{\Dc_\Si\}_{ij} g(\La)=\sum_{k=1}^r \frac{\partial g(\La)}{\partial\la_k}\{\Dc_\Si\}_{ij}\la_k
=\{\Ga\Phi(\La)\Ga^\top\}_{ij},$$ where $\Phi(\La)=\diag(\phi_1(\La),\ldots,\phi_r(\La))$. Using Lemma \[lem:condition1\] and (ii) of Lemma \[lem:diff2\] gives that $$\begin{aligned}
&\tr[\nabla_\Th\nabla_\Th^\top\pi(\Th)] \\
&=2[(q-r-1)\tr\{\Ga\Phi(\La)\Ga^\top\}+2\tr\{\Dc_\Si \Ga\La\Phi(\La)\Ga^\top\}] \\
&=2\sum_{i=1}^r\bigg[(q-r-1)\phi_i(\La)+\sum_{j\ne i}^r\frac{\la_i\phi_i(\La)-\la_j\phi_j(\La)}{\la_i-\la_j}+2\frac{\partial}{\partial\la_i}\{\la_i\phi_i(\La)\}\bigg]\\
&=2\sum_{i=1}^r\bigg\{(q-r+1)\phi_i(\La)+\sum_{j\ne i}^r\frac{\la_i\phi_i(\La)-\la_j\phi_j(\La)}{\la_i-\la_j}+2\la_i\frac{\partial\phi_i(\La)}{\partial\la_i}\bigg\}.\end{aligned}$$ Hence the proof is complete.
Using Proposition \[prp:condition2\], we give some examples of Bayesian predictive densities with respect to superharmonic priors. Consider a class of shrinkage prior densities, $$\pi_{SH}(\Th) = \{\tr(\Th\Th^\top)\}^{-\be/2}\prod_{i=1}^r \la_i^{-\al_i/2}
=\bigg\{\sum_{i=1}^r \la_i\bigg\}^{-\be/2}\prod_{i=1}^r \la_i^{-\al_i/2},$$ where $\al_1,\ldots,\al_r$ and $\be$ are nonnegative constants. The class $\pi_{SH}(\Th)$ includes both harmonic priors $\pi_{EM}(\Th)$ and $\pi_{JS}(\Th)$, which are given in (\[eqn:pr\_em\]) and (\[eqn:pr\_js\]), respectively. Indeed, $\pi_{SH}(\Th)$ is the same as $\pi_{EM}(\Th)$ if $\al_1=\cdots=\al_r=\al^{EM}$ and $\be=0$ and as $\pi_{JS}(\Th)$ if $\al_1=\cdots=\al_r=0$ and $\be=\be^{JS}$.
It is noted that $$\begin{aligned}
\frac{\partial}{\partial \la_k}\pi_{SH}(\Th) &= -\frac{1}{2}\Big(\frac{\al_k}{\la_k}+\frac{\be}{\sum_{i=1}^r \la_i}\Big)\pi_{SH}(\Th), \label{eqn:d_pi_g} \\
\frac{\partial^2}{\partial \la_k^2}\pi_{SH}(\Th) &=\frac{1}{2}\bigg\{\Big(\frac{\al_k}{\la_k^2}+\frac{\be}{(\sum_{i=1}^r \la_i)^2}\Big)+\frac{1}{2}\Big(\frac{\al_k}{\la_k}+\frac{\be}{\sum_{i=1}^r \la_i}\Big)^2\bigg\}\pi_{SH}(\Th) . \label{eqn:dd_pi_g}\end{aligned}$$ Combining (\[eqn:d\_pi\_g\]), (\[eqn:dd\_pi\_g\]) and Proposition \[prp:condition2\], we obtain $$\begin{aligned}
&\tr[\nabla_\Th\nabla_\Th^\top \pi_{SH}(\Th)] \non\\
&=\pi_{SH}(\Th)\sum_{i=1}^r\bigg[\{\al_i^2-(q-r-1)\al_i\}\frac{1}{\la_i}-2\sum_{j>i}^r\frac{\al_i-\al_j}{\la_i-\la_j}+\frac{2\al_i\be}{\tr(\Th\Th^\top)}\bigg] \non\\
&\qquad +\pi_{SH}(\Th)\frac{\be^2-(qr-2)\be}{\tr(\Th\Th^\top)}.
\label{eqn:dd-pi_g}\end{aligned}$$
\[exm:1\] Let $$\pi_{ST}(\Th)=\prod_{i=1}^r \la_i^{-\al_i/2},$$ where $\al_1,\ldots,\al_r$ are nonnegative constants. Assume that $\al_1\geq\cdots\geq\al_r$. Note that $$\begin{aligned}
\sum_{i=1}^r\sum_{j>i}^r\frac{\al_i-\al_j}{\la_i-\la_j}
&=\sum_{i=1}^r\sum_{j>i}^r\frac{1}{\la_i}\frac{\la_i-\la_j+\la_j}{\la_i-\la_j}(\al_i-\al_j)\\
&=\sum_{i=1}^r(r-i)\frac{\al_i}{\la_i}-\sum_{i=1}^r\frac{1}{\la_i}\sum_{j>i}^r\al_j+\sum_{i=1}^r\sum_{j>i}^r\frac{\la_j}{\la_i}\frac{\al_i-\al_j}{\la_i-\la_j}\\
&\geq \sum_{i=1}^r(r-i)\frac{\al_i}{\la_i}-\sum_{i=1}^r\frac{1}{\la_i}\sum_{j>i}^r\al_j.\end{aligned}$$ From (\[eqn:dd-pi\_g\]), it is seen that $$\begin{aligned}
&\tr[\nabla_\Th\nabla_\Th^\top \pi_{ST}(\Th)] \\
&\leq\pi_{ST}(\Th)\sum_{i=1}^r\bigg\{\al_i^2-(q-r-1)\al_i-2(r-i)\al_i+2\sum_{j>i}^r\al_j\bigg\}\frac{1}{\la_i}\\
&=\pi_{ST}(\Th)\sum_{i=1}^r\bigg\{\al_i^2-(q+r-2i-1)\al_i+2\sum_{j>i}^r\al_j\bigg\}\frac{1}{\la_i}.\end{aligned}$$ Here, assume additionally that $\al_i\leq\al_i^{ST}/2$ with $\al_i^{ST}=q+r-2i-1$ for $i=1,\ldots,r$. For each $i$ we observe that $$\begin{aligned}
&\al_i^2-(q+r-2i-1)\al_i+2\sum_{j>i}^r\al_j \\
&\leq \al_{i+1}^2-(q+r-2i-1)\al_{i+1}+2\sum_{j>i}^r\al_j\\
&= \al_{i+1}^2-\{q+r-2(i+1)-1\}\al_{i+1}+2\sum_{j>i+1}^r\al_j\\
&\leq \cdots\\
&\leq \al_r^2-(q-r-1)\al_r\leq0,\end{aligned}$$ which implies that $\tr[\nabla_\Th\nabla_\Th^\top \pi_{ST}(\Th)]\leq 0$ if $\al_1\geq\cdots\geq\al_r$ and $\al_i\leq\al_i^{ST}/2$ for each $i$. Then the resulting Bayesian predictive density is minimax under the KL loss (\[eqn:loss\]). $\Box$
\[exm:2\] Consider a prior density of the form $$\label{eqn:pr_MS1}
\pi_{MS1}(\Th)=\{\tr(\Th\Th^\top)\}^{-\be^{MS}/2}\prod_{i=1}^r \la_i^{-\al_i^{ST}/4},$$ where $\be^{MS}=2(r-1)$. Combining Example \[exm:1\] and (\[eqn:dd-pi\_g\]) gives that $$\begin{aligned}
&\tr[\nabla_\Th\nabla_\Th^\top \pi_{MS1}(\Th)]\\
&\leq\pi_{MS1}(\Th)\sum_{i=1}^r\frac{\al_i^{ST}\be^{MS}}{\tr(\Th\Th^\top)}
+\pi_{MS1}(\Th)\frac{(\be^{MS})^2-(qr-2)\be^{MS}}{\tr(\Th\Th^\top)}=0.\end{aligned}$$ Hence the Bayesian predictive density with respect to $\pi_{MS1}(\Th)$ is minimax relative to the KL loss (\[eqn:loss\]). $\Box$
In the literature, many shrinkage estimators have been developed in estimation of a normal mean matrix. It is worth pointing out that the Bayesian predictive densities with superharmonic prior $\pi_{SH}(\Th)$ correspond to such shrinkage estimators.
Let $X|\Th\sim\Nc_{r\times q}(\Th, v_x I_r\otimes I_q)$ and denote an estimator of $\Th$ by $\Thh$. Consider the problem of estimating the mean matrix $\Th$ relative to quadratic loss $L_Q(\Thh,\Th)=\Vert\Thh-\Th\Vert^2$. Then the generalized Bayes estimator of $\Th$ with the prior density $\pi_{SH}$ is expressed as $$\begin{aligned}
\Thh_{SH}
&=\frac{\int_{\Re^{r\times q}} \Th \exp(-\Vert X-\Th \Vert^2/(2v_x))\pi_{SH}(\Th)\dd\Th}{\int_{\Re^{r\times q}} \exp(-\Vert X-\Th \Vert^2/(2v_x))\pi_{SH}(\Th)\dd\Th}.\end{aligned}$$ If $\pi_{SH}$ is superharmonic then $\Thh_{SH}$ is minimax relative to the quadratic loss $L_Q$.
Since $v_x\nabla_\Th \exp(-\Vert X-\Th \Vert^2/(2v_x))=-(\Th-X)\exp(-\Vert X-\Th \Vert^2/(2v_x))$, the integration by parts gives that $$\begin{aligned}
\Thh_{SH}
&=X-v_x\frac{\int_{\Re^{r\times q}} [\nabla_\Th \exp(-\Vert X-\Th \Vert^2/(2v_x))]\pi_{SH}(\Th)\dd\Th}{\int_{\Re^{r\times q}} \exp(-\Vert X-\Th \Vert^2/(2v_x))\pi_{SH}(\Th)\dd\Th} \\
&=X+v_x \frac{\int_{\Re^{r\times q}} \exp(-\Vert X-\Th \Vert^2/(2v_x))[\nabla_\Th\pi_{SH}(\Th)]\dd\Th}{\int_{\Re^{r\times q}} \exp(-\Vert X-\Th \Vert^2/(2v_x))\pi_{SH}(\Th)\dd\Th}.\end{aligned}$$ Here using (i) of Lemma \[lem:diff4\] and (i) of Lemma \[lem:diff2\] gives that $$\begin{aligned}
\nabla_\Th^\top \pi_{SH}(\Th)
&= 2\Th^\top \Dc_\Si \pi_{SH}(\Th) \non\\
&=- \Th^\top \bigg\{\Ga\diag\Big(\frac{\al_1}{\la_1},\ldots,\frac{\al_r}{\la_r}\Big)\Ga^\top+\frac{\be}{\tr(\Si)}I_r \bigg\} \pi_{SH}(\Th),\end{aligned}$$ which leads to $$\label{eqn:Th_MS}
\Thh_{SH}
=X-v_x\Er^{\Th|X}\bigg[ \bigg\{\Ga\diag\Big(\frac{\al_1}{\la_1},\ldots,\frac{\al_r}{\la_r}\Big)\Ga^\top+\frac{\be}{\tr(\Si)}I_r \bigg\} \Th\bigg],$$ where $\Er^{\Th|X}$ stands for the posterior expectation with respect to a density proportional to $\exp(-\Vert \Th-X \Vert^2/(2v_x))\pi_{SH}(\Th)$.
Denote by $XX^\top=HLH^\top$ the eigenvalue decomposition of $XX^\top$, where $H=(h_{ij})$ is an orthogonal matrix of order $r$ and $L=\diag(\ell_1,\ldots,\ell_r)$ is a diagonal matrix of order $r$ with $\ell_1\geq \cdots\geq \ell_r$. Substituting $(X,H,L)$ for $(\Th,\Ga,\La)$ in the second term of the r.h.s. of (\[eqn:Th\_MS\]), we obtain an empirical Bayes shrinkage estimator $$\Thh_{MS}=X-v_x\bigg\{H\diag\Big(\frac{\al_1}{\ell_1},\ldots,\frac{\al_r}{\ell_r}\Big)H^\top+\frac{\be}{\tr(XX^\top)}I_r \bigg\}X.$$
The shrinkage estimator $\Thh_{MS}$ is equivalent to $\Thh_{JS}$, given in (\[eqn:JS\]), when $\al_1=\cdots=\al_r=0$ and $\be=\be^{JS}$, and to $\Thh_{EM}$, given in (\[eqn:EM\]), when $\al_1=\cdots=\al_r=\al^{EM}$ and $\be=0$. In estimation of the normal mean matrix relative to the quadratic loss $L_Q$, $\Thh_{JS}$ and $\Thh_{EM}$ are minimax.
If $\Thh_{MS}$ with certain specified $\al_1,\ldots,\al_r$ and $\be$ has good performance, the prior density $\pi_{SH}$ with the same $\al_1,\ldots,\al_r$ and $\be$ would produce a good Bayesian predictive density. From Tsukuma (2008), $\Thh_{MS}$ is a minimax estimator dominating $\Thh_{EM}$ when $\al_i=\al_i^{ST}$ for $i=1,\ldots,r$ and $0\leq \be\leq 4(r-1)$. A reasonable choice for $\be$ is $\be^{MS}=2(r-1)$ and this suggests that we should consider a prior density of the form $$\label{eqn:pr_MS2}
\pi_{MS2}(\Th)
=\{\tr(\Th\Th^\top)\}^{-\be^{MS}/2}\prod_{i=1}^r \la_i^{-\al_i^{ST}/2}
=\pi_{MS1}(\Th)\prod_{i=1}^r \la_i^{-\al_i^{ST}/4}.$$ The prior density $\pi_{MS2}(\Th)$ is not superharmonic, and it is not known whether the resulting Bayesian predictive density is minimax or not. In the next section, we verify risk behavior of the Bayesian predictive density with respect to $\pi_{MS2}(\Th)$ through Monte Carlo simulations.
Monte Carlo studies {#sec:MCstudies}
===================
This section briefly reports some numerical results so as to compare performance in risk of some Bayesian predictive densities for $r=2$ and $q=15$.
First we investigate risk behavior of generalized Bayes predictive densities $\ph_{GB}(Y|X)$ with $v_0=1$ in the following six cases: $$(a,\, b)=(-11,\, 3),\ (-11,\, 9),\ (-11,\, 15),\ (-5,\, 3),\ (-5,\, 9),\ (1,\, 3)$$ for the second-stage prior (\[eqn:pr\_GB\]). When $r=2$ and $q=15$, $\ph_{GB}(Y|X)$ with the above six cases are minimax and, in particular, $\ph_{GB}(Y|X)$ with $(a,\, b)=(1,\,3)$ is proper Bayes for any $v_x$ and $v_y$ (see Corollary \[cor:proper2\]).
The risk has been simulated by 100,000 independent replications of $X$ and $Y$, where $X|\Th\sim\Nc_{r\times q}(\Th, v_xI_r\otimes I_q)$ and $Y|\Th\sim\Nc_{r\times q}(\Th, v_yI_r\otimes I_q)$ with $(v_x,v_y)=(0.1,\, 1),\ (1,\, 1)$ and $(1,\, 0.1)$. It has been assumed that a pair of the maximum and the minimum eigenvalues of $\Th\Th^\top$ is $(0,\, 0),\ (24,\, 0)$ or $(24,\, 24)$. Note that the best invariant predictive density $\ph_U(Y|X)$ has a constant risk and its risk is approximately given by $$R(\ph_U,\Th)=\frac{rq}{2}\log\frac{v_s}{v_y}\approx\begin{cases}
1.42 & \textup{for $(v_x,v_y)=(0.1,\ 1)$},\\
10.4 & \textup{for $(v_x,v_y)=(1,\ 1)$},\\
36.0 & \textup{for $(v_x,v_y)=(1,\ 0.1)$},
\end{cases}$$ when $r=2$ and $q=15$.
Denote by $\Bc(a,b)$ the matrix-variate beta distribution having the density (\[eqn:pr\_GB\]). Using (\[eqn:m(W)-1\]) with $\La=v_1\Om\{I_r-(1-v_1)\Om\}^{-1}$ and $v_1=v/v_0$, we can rewrite $\ph_{GB}(Y|X)$ as $$\ph_{GB}(Y|X)=\frac{\Er^{\Om}[g_{v_w}(\Om|W)]}{\Er^{\Om}[g_{v_x}(\Om|X)]}\ph_U(Y|X),$$ where $\Er^{\Om}$ indicates expectation with respect to $\Om\sim \Bc(a+q,b)$ and $$g_{v}(\Om|Z)=\Big|I_r-\Big(1-\frac{v}{v_0}\Big)\Om\Big|^{-q/2}\exp\Big[-\frac{1}{2v_0}\tr\Big[\Om\Big\{I_r-\Big(1-\frac{v}{v_0}\Big)\Om\Big\}^{-1}ZZ^\top\Big]\Big]$$ for an $r\times q$ matrix $Z$. Hence in our simulations, the expectation $\Er^{\Om}[g_{v}(\Om|Z)]$ was estimated by $j_0^{-1}\sum_{j=1}^{j_0} g_{v}(\Om_j|Z)$, where $j_0=100,000$ and the $\Om_j$ are independent replications from $\Bc(a+q,b)$.
$
\begin{array}{ccccccccc}
\hline
(v_x, v_y) & {\rm Eigenvalues}&{\rm Minimax}&\multicolumn{6}{c}{(a,b)}\\
\cline{4-9}
& {\rm of}\ \Th\Th^\top &{\rm risk}&(-11,3)&(-11,9)&(-11,15)&(-5,3)&(-5,9)&(1,3) \\
\hline
(0.1,1) &(\ 0,\ 0)&1.42& 0.47 & 0.96 & 1.20 & 0.38 & 0.80 & 0.33 \\
&(24,\ 0) & & 0.91 & 1.17 & 1.30 & 0.87 & 1.09 & 0.84 \\
&(24,24) & & 1.39 & 1.39 & 1.40 & 1.37 & 1.37 & 1.39 \\
[6pt]
(1, 1) &(\ 0,\ 0)&10.4& 5.3 & 6.9 & 7.7 & 2.8 & 4.6 & 1.6 \\
&(24,\ 0) & & 6.9 & 8.0 & 8.4 & 5.3 & 6.3 & 4.6 \\
&(24,24) & & 8.7 & 9.0 & 9.2 & 7.9 & 8.0 & 8.4 \\
[6pt]
(1,0.1)&(\ 0,\ 0)&36.0& 15.2 & 24.0 & 28.2 & 9.6 & 17.7 & 6.8 \\
&(24,\ 0) & & 23.6 & 28.5 & 30.9 & 20.1 & 24.5 & 18.7 \\
&(24,24) & & 32.7 & 33.2 & 33.5 & 31.0 & 31.4 & 32.4 \\
\hline
\end{array}
$
The simulated results for risk of $\ph_{GB}(Y|X)$ are given in Table \[tab:1\]. When the pair of eigenvalues of $\Th\Th^\top$ is $(0,\, 0)$, our simulations suggest that the risk of $\ph_{GB}(Y|X)$ decreases as $a$ increases under which $b$ is fixed or under which $a+b$ is fixed and also that the risk of $\ph_{GB}(Y|X)$ increases as $b$ increases under which $a$ is fixed. It is observed that $\ph_{GB}(Y|X)$ with $(a, b)=(1, 3)$ is superior to others.
When the pair of eigenvalues of $\Th\Th^\top$ is $(24,\, 24)$, $\ph_{GB}(Y|X)$ with $(a, b)=(-5, 3)$ or $(-5, 9)$ is best, but the improvement over $\ph_U(Y|X)$ is little. When the pair of eigenvalues of $\Th\Th^\top$ is $(24,\, 0)$, $\ph_{GB}(Y|X)$ with $(a, b)=(1, 3)$ is best.
Next, we investigate the risk of Bayesian predictive densities based on superharmonic priors when $r=2$ and $q=15$. If $\pi_s(\Th)$ is a superharmonic prior, then the Bayesian predictive density (\[eqn:BPD\]) can be expressed as $$\ph_{\pi_s}(Y|X)=\frac{\Er^{\Th|W}[\pi_s(\Th)]}{\Er^{\Th|X}[\pi_s(\Th)]}\ph_U(Y|X),$$ where $\Er^{\Th|W}$ and $\Er^{\Th|X}$ stand, respectively, for expectations with respect to $\Th|W\sim\Nc_{r\times q}(W, v_w I_r\otimes I_q)$ and $\Th|X \sim\Nc_{r\times q}(X, v_x I_r\otimes I_q)$. In our simulations, $\ph_{\pi_s}(Y|X)$ was estimated by means of $$\ph_{\pi_s}(Y|X)\approx \frac{\sum_{i=1}^{i_0}\pi_s(\Th_i)}{\sum_{i=1}^{i_0}\pi_s(\Th_i)}\ph_U(Y|X),$$ where $i_0=100,000$ and the $\Th_i$ and the $\Th_j$ are, respectively, independent replications from $\Nc_{r\times q}(W, v_w I_r\otimes I_q)$ and $\Nc_{r\times q}(X, v_x I_r\otimes I_q)$.
$
\begin{array}{cccc@{\hspace{20pt}}ccccc}
\hline
v_x & v_y & {\rm Eigenvalues} &{\rm Minimax}& $GB$ & $JS$\ & $EM$\, & $MS1$ & $MS2$ \\
&& {\rm of}\ \Th\Th^\top &{\rm risk}&&&&&\\
\hline
0.1& 1 &(\ 0,\ 0)&1.42& 0.33 & 0.09 & 0.28 & 0.71 & 0.09 \\
& &(24,\ 0) & & 0.84 & 1.30 & 0.83 & 1.11 & 0.82 \\
& &(24,\ 4) & & 1.27 & 1.31 & 1.27 & 1.30 & 1.26 \\
& &(24,\ 8) & & 1.32 & 1.33 & 1.33 & 1.34 & 1.32 \\
& &(24, 12) & & 1.35 & 1.34 & 1.35 & 1.36 & 1.35 \\
& &(24, 24) & & 1.39 & 1.36 & 1.37 & 1.38 & 1.37 \\
[6pt]
1 & 1 &(\ 0,\ 0)&10.4& 1.6 & 0.7 & 2.1 & 5.2 & 0.7 \\
& &(24,\ 0) & & 4.6 & 5.5 & 4.9 & 7.0 & 4.4 \\
& &(24,\ 4) & & 5.7 & 5.9 & 5.9 & 7.4 & 5.4 \\
& &(24,\ 8) & & 6.6 & 6.3 & 6.6 & 7.7 & 6.2 \\
& &(24, 12) & & 7.2 & 6.6 & 7.1 & 8.0 & 6.7 \\
& &(24, 24) & & 8.4 & 7.3 & 7.9 & 8.4 & 7.6 \\
[6pt]
1 &0.1&(\ 0,\ 0)&36.0& 6.8 & 2.4 & 7.2 & 18.0 & 2.4 \\
& &(24,\ 0) & & 18.7 & 25.8 & 18.9 & 26.1 & 18.2 \\
& &(24,\ 4) & & 25.5 & 26.8 & 25.7 & 28.9 & 25.0 \\
& &(24,\ 8) & & 28.1 & 27.7 & 28.2 & 30.2 & 27.6 \\
& &(24, 12) & & 29.7 & 28.4 & 29.5 & 30.9 & 28.9 \\
& &(24, 24) & & 32.4 & 30.0 & 31.3 & 32.0 & 30.8 \\
\hline
\end{array}
$
The risk is based on 100,000 independent replications of $X$ and $Y$ for some pairs of two eigenvalues of $\Th\Th^\top$. The simulation results are provided in Table \[tab:2\], where GB, JS, EM, MS1 and MS2 are the Bayesian predictive densities with the following priors.
[ ]{}
(\[eqn:pr\_Th\]) and (\[eqn:pr\_GB\]) with $a=1$, $b=3$ and $v_0=1$,
(\[eqn:pr\_js\]),
(\[eqn:pr\_em\]),
(\[eqn:pr\_MS1\]),
(\[eqn:pr\_MS2\]).
Note that GB, JS, EM and MS1 are minimax, while MS2 has not been shown to be minimax.
When the pair of eigenvalues of $\Th\Th^\top$ is $(0,\, 0)$, JS and MS2 are superior. When the pair of eigenvalues of $\Th\Th^\top$ is $(24,\, 24)$, JS has nice performance but it is bad if the two eigenvalues of $\Th\Th^\top$ are much different.
Our simulations suggest that MS2 is better than EM and MS1. When the two eigenvalues of $\Th\Th^\top$ are much different, namely they are $(24,\, 0)$ and $(24,\, 4)$, MS2 is best and GB or EM is second-best.
[**Acknowledgments.**]{} The research of the first author was supported by Grant-in-Aid for Scientific Research (15K00055) from Japan Society for the Promotion of Science (JSPS). The research of the second author was supported in part by Grant-in-Aid for Scientific Research (15H01943 and 26330036) from JSPS.
[100]{}
Aitchison, J. (1975). Goodness of prediction fit, [*Biometrika*]{}, [**62**]{}, 547–554.
Brown, L.D. (1971). Admissible estimators, recurrent diffusions, and insoluble boundary value problems, [*Ann. Math. Statist.*]{}, [**42**]{}, 855–903.
Brown, L.D., George, E.I. and Xu, X. (2008). Admissible predictive density estimation, [*Ann. Statist.*]{}, [**36**]{}, 1156–1170.
Efron, B. and Morris, C. (1972). Empirical Bayes on vector observations: An extension of Stein’s method, [*Biometrika*]{}, [**59**]{}, 335–347.
Faith, R.E. (1978). Minimax Bayes estimators of a multivariate normal mean, [*J. Multivariate Anal.*]{}, [**8**]{}, 372–379.
George, E.I., Liang, F. and Xu, X. (2006). Improved minimax predictive densities under Kullback-Leibler loss, [*Ann. Statist.*]{}, [**34**]{}, 78–91.
George, E.I., Liang, F. and Xu, X. (2012). From minimax shrinkage estimation to minimax shrinkage prediction, [*Statist. Sci.*]{}, [**27**]{}, 82–94.
Gupta, A.K. and Nagar, D.K. (1999). [*Matrix Variate Distributions*]{}, Chapman & Hall/CRC, New York.
Haff, L.R. (1982). Identities for the inverse Wishart distribution with computational results in linear and quadratic discrimination. [*Sankhyā*]{}, [**44**]{}, Series B, 245–258.
James, W. and Stein, C. (1961). Estimation with quadratic loss, In [*Proc. Fourth Berkeley Symp. Math. Statist. Probab.*]{}, [**1**]{}, 361–379, Univ. of California Press, Berkeley.
Komaki, F. (2001). A shrinkage predictive distribution for multivariate normal observables, [*Biometrika*]{}, [**88**]{}, 859–864.
Konno, Y. (1988). Exact moments of the multivariate F and beta distributions, [*J. Japan Statist. Soc.*]{}, [**18**]{}, 123–130.
Konno, Y. (1992). Improved estimation of matrix of normal mean and eigenvalues in the multivariate $F$-distribution, Doctoral dissertation, Institute of Mathematics, University of Tsukuba. This is downloadable from his website (http://mcm-www.jwu.ac.jp/\~konno/).
Magnus, J.R. and Neudecker, H. (1999). [*Matrix differential calculus with applications in statistics and econometrics, 2nd ed.*]{}, Wiley, New York.
Maruyama, Y. (1998). A unified and broadened class of admissible minimax estimators of a multivariate normal mean [*J. Multivariate Anal.*]{}, [**64**]{}, 196–205.
Matsuda, T. and Komaki, F. (2015). Singular value shrinkage priors for Bayesian prediction, [*Biometrika*]{}, [**102**]{}, 843–854.
Muirhead, R.J. (1982). [*Aspects of Multivariate Statistical Theory*]{}, John Wiley & Sons, New York.
Stein, C. (1973). Estimation of the mean of a multivariate normal distribution, In: [*Proc. Prague Symp. Asymptotic Statist.*]{}, 345-381.
Stein, C. (1981). Estimation of the mean of a multivariate normal distribution, [*Ann. Statist.*]{}, [**9**]{}, 1135–1151.
Tsukuma, H. (2008). Admissibility and minimaxity of Bayes estimators for a normal mean matrix, [*J. Multivariate Anal.*]{}, [**99**]{}, 2251–2264.
[^1]: Faculty of Medicine, Toho University, 5-21-16 Omori-nishi, Ota-ku, Tokyo 143-8540, Japan, E-Mail: [email protected]
[^2]: Faculty of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan, E-Mail: [email protected]
| ArXiv |
---
author:
- 'Junya [Hashida]{} and Yuichiro [Kiyo]{}'
title: More on Large $Q^2$ Events with Polarized Beams
---
In 1997, an event excess in the neutral current process $e^{+}p\rightarrow e^{\prime +} X$ in the region of high momentum transfer $Q^2 \geq 15,000$ GeV$^2$ was reported by H1 and ZEUS at HERA [@H1ZEUS]. The observed cross section was $0.71^{+0.14}_{-0.12}$ pb, whereas the standard model (SM) predicts $0.49$ pb. The new data [@TRAPE] analyzed in 1998 are in agreement with the SM up to $Q^2\simeq 10,000$ GeV$^2$. The excess at $Q^2 \geq 20,000$ GeV$^2$ is not confirmed by the new data but is still present. The present situation is rather vague, [@TRAPE; @ALTA1; @VALE] and it is still an open question whether this is really an anomalous event or if it just results from statistical fluctuation. If the excess is not just a statistical fluctuation, it must be an indication of new interactions beyond the SM, because it appears to be very difficult to explain the data in the framework of the SM.
There have appeared many proposals and analyses of this problem. New contact interactions (CI) stemming from high energy scale physics have been analyzed, [@ALTA; @BARG; @DESH] and supersymmetric (SUSY) models with R-parity violating ($R_{p}\hspace{-11pt}/~~$) interactions have also been discussed. [@ALTA] The two-stop scenario, [@KON] left stop $\tilde{t}_{L}$ is a mixture of the almost degenerate mass eigenstates of $\tilde{t}_{1}$ and $\tilde{t}_{2}$, with $R_{p}\hspace{-11pt}/~~$ interactions was proposed as one of the candidates to explain broad mass distribution in the data.
HERA will begin a polarized experiment, [@EXP] polarized proton $p(\uparrow/\downarrow)$ and lepton (positron in our discussion) scattering, in the near future. The polarized experiment is important because the polarization of the proton and lepton beams make it possible to test the chiral structure of the interactions. [@VIRE] Thus it is interesting to ask what HERA will teach us about the models in the future polarized program.
In this paper, we examine two scenarios, the CI and the two-stop scenarios in the context of the large $Q^{2}$ events at the polarized HERA. Our interest is in determining how we can examine these scenarios and what the characteristics of the models are. Thus we discuss these scenarios with regard to the future polarized experiment $e^{+} p(\uparrow/\downarrow) \rightarrow e^{+ \prime}X$. After giving the model Lagrangians, we calculate the parton level cross sections which will be convoluted with parton distributions to form the physical cross section.
The Lagrangian for the CI [@ALTA; @BARG; @DESH] assumes the form $$\begin{aligned}
L_{CI}
&=&
\frac{4 \pi}{\Lambda^2}
\sum_{\stackrel{q=u,d}{a,b=L,R}}
\eta^q_{ab}
\left(
\bar{e}_{a}\gamma^\mu e_{a}
\right)
\left(
\bar{q}_{b}\gamma_\mu q_{b}
\right),\end{aligned}$$ which is the effective interaction of a certain underlying high energy physics describing low energy phenomena in the neutral current process. The subscript $R (L)$ denotes the chirality of the fields, $\eta^q_{ab}=\pm 1, 0$, and $\Lambda$ is the mass scale of a heavy particle which might be exchanged among quarks and leptons. Thus these interactions are suppressed by the mass scale of the new physics, and some constraints [@TRAPE; @GCHO] have been obtained for $\Lambda$ in many experiments. The superpotential, for the stop scenario with $R_{p}\hspace{-11pt}/~~$ interaction [@ALTA; @KON], is given by $$W_{R\hspace{-6pt}/~}
=
\lambda^{\prime}_{131}
L_{1} Q_{3} D^{c}_{1},$$ where $L_{1}$ and $Q_{3}$ are the superfields of the $SU(2)_L$ lepton and quark doublet, respectively, and $D^{c}_{1}$ is the singlet down type quark. Here the subscripts 1, 2 and 3 are the generation indices. The interaction Lagrangian can be obtained from the superpotential $$L_{\lambda^\prime}
=
\lambda^{\prime}_{131}
\left(
\tilde{t}_{L} \bar{d}P_{L}e +
\tilde{e}_{L} \bar{d}P_{L}t +
\bar{\tilde{d}}_{R}\bar{e}^{c}P_{L}t
-
\tilde{b}_{L}\bar{d}P_{L}\nu_{e}-
\tilde{\nu}_{L}\bar{d}P_{L}b -
\bar{\tilde{d}}_{R}\bar{\nu}^{c}_{e} P_{L}b
\right)
+ h.c.$$ For the scalar fields, $R (L)$ denotes the chirality of their superpartners. We discuss the proton-positron scattering, so only the first term $\tilde{t}_{L} \bar{d}P_{L}e + h.c.$ is relevant. In the two-stop scenario, the left stop $\tilde{t}_L$ is the superposition of the two mass eigenstates $\tilde{t}_1$ and $\tilde{t}_2$ with the mixing angle $\theta_t$; namely $\tilde{t}_L= \tilde{t}_1 \cos\theta_t - \tilde{t}_2 \sin\theta_t$. The stop $\tilde{t}_{L}$ can couple only to the left handed lepton field $e_{L}$ and the right handed down quark $d_{R}$. This is an important point in our discussion, because the polarized experiment can distinguish the chiral structure of the interactions in the parton-lepton scattering.
The partonic cross sections $\hat{\sigma}$ for the models are given by $$\begin{aligned}
\frac{ d \hat{\sigma}(e^{+}_{I} f_{J}) }{dx_B dQ^{2}}
&=&
\delta(x_B - x)\frac{(4\pi \alpha_{e})^{2}}{8 \pi \left( \hat{s} ~ Q^{2} \right)^{2} }
\left[
(1+ I \cdot J )\hat{s}^{2} + (1- I\cdot J) \hat{u}^{2}
\right]
\nonumber \\
&\times&
\left|
Q_{\gamma}(e)Q_{\gamma}(f)
+ \frac{Q_{Z}^{-I}(e)Q_{Z}^{J}(f)
}{\sin^2\theta_W}
\frac{Q^{2}}{Q^{2}+M_{Z}^{2}}
+\Delta
\right|^{2},\end{aligned}$$ where $I(J)=\pm$ correspond to the helicities $\pm 1/2$ of the positron (quark), $x_B$ is the Bjorken variable and $x$ is the momentum fraction of the parton, $\alpha_{e}=e^{2}/(4\pi)$, $\theta_W$ is the electro-weak angle, and $\hat{s}$ and $\hat{u}$ are the Mandelstam variables with respect to the parton-positron system, which are defined by $\hat{s}=x s$ and $\hat{u}=x u$. $\Delta$ is the contribution from the CI or $R_{p}\hspace{-11pt}/~~$ interaction. We neglect the masses of the quarks and positron in this paper. The coupling constants of the electron and up and down quarks to the photon and Z boson are given by $$\begin{aligned}
Q_{\gamma}(e)&=&-1,~~~
Q_{Z}^{+}(e)=
\frac{\sin^2\theta_W}{\cos^2\theta_W},~~~
Q_{Z}^{-}(e)=
\frac{2 \sin^2\theta_W-1}{2 \cos\theta_W},~~~
\\
%
Q_{\gamma}(u)&=&\frac{2}{3},~~~
Q_{Z}^{+}(u)=
\frac{-2 \sin^2\theta_W}{3 \cos\theta_W},~~~
Q_{Z}^{-}(u)=
\frac{3- 4 \sin^2\theta_W}{6 \cos\theta_W},~~~
\\
%
Q_{\gamma}(d)&=&\frac{-1}{3},~~~
Q_{Z}^{+}(d)=
\frac{\sin^2\theta_W}{3 \cos^2\theta_W},~~~
Q_{Z}^{-}(d)=
\frac{-3+ 2 \sin^2\theta_W}{6 \cos\theta_W}.\end{aligned}$$ For the CI scenario, $\Delta$ is given by $$\Delta(Q^2)=-\frac{Q^2 \eta^q_{-IJ} }{\alpha_e \Lambda^2},$$ where the subscripts $+$ and $-$ of the $\eta^q_{ab}$ correspond respectively to $R$ and $L$. The stop exchange with the $R_{p}\hspace{-11pt}/~~$ interaction yields the following contribution $$\Delta(\hat{s},Q^2)
=
- \frac{ \alpha_{131} Q^2
}{2 \alpha_{e}}
\left( \frac{\cos^{2}\theta_{t}
}{\hat{s}-\tilde{m}_{1}^{2}+i\tilde{m}_{1}\Gamma_{\tilde{t}_{1}}}
+
\frac{\sin^{2}\theta_{t}
}{\hat{s}-\tilde{m}_{2}^{2}+i\tilde{m}_{2}\Gamma_{\tilde{t}_{2}}}
\right)$$ for the $I=J=+$ channel and $f=d$. Otherwise $\Delta=0$ in the stop scenario. Here $\alpha_{131}=\lambda^{\prime 2}_{131}/(4\pi)$, $\tilde{m}_{1,2}$ and $\Gamma_{ \tilde{t}_{1,2}}$ are the masses and widths of $\tilde{t}_{1,2}$ respectively.
The cross section for the polarized proton-positron scattering is obtained by convoluting the partonic cross sections with the polarized parton distribution functions. The cross section $\sigma(e^{+}p(\uparrow))$ for the longitudinally polarized proton $p(\uparrow)$ and positron scattering can be written: $$\frac{ d \sigma }{dx_B dQ^{2}}(e^{+}p(\uparrow) )
=
\int^{1}_{0} d x
\sum_{f}
\left(
\frac{d\hat{\sigma} \left( e^{+} f_{+} \right)
}{dx_B dQ^{2}}
f_{+/\uparrow}(x)
+
\frac{d\hat{\sigma} \left( e^{+} f_{-} \right)
}{dx_B dQ^{2}}(x)
f_{-/\uparrow}
\right),$$ where $f_{\pm/\uparrow}(x)$ is the polarized parton distribution function for the flavor $f$ parton with momentum fraction $x$ and helicity $\pm1/2$ in the proton $p(\uparrow)$.
We are interested in the region which is characterized by the two variables $Q^{2}$ and the invariant mass $M = \sqrt{\hat{s}}$, with $Q^{2} \geq 20,000$ GeV$^2$ and $M \sim 200$ GeV. This corresponds to the region in which the partons in the proton have a momentum fraction $x \sim 0.4$. Thus we can safely neglect the contribution from the sea quarks, because their distribution is quite small in that region, and contributions from gluons are next to leading order in the QCD coupling constant.
In Fig.\[fig:pd\], we show the polarized parton distributions $x u_{\pm/\uparrow}(x),x d_{\pm/\uparrow}(x)$ in the proton as a function of the momentum fraction $x$. The scale of the distributions is taken to be $Q^{2}=20,000$ GeV$^2$ using the parameterization of Refs. [@GS] and [@MRS]. Our numerical estimation has shown that the effects of $Q^2$ evolutions to the parton distributions were tiny in the region considered in this paper. This is reasonable because the change in the QCD coupling constant is small for large $Q^2$.
One can see that most of the down quarks are oppositely polarized ($x d_{+}(x) \leq x d_{-}(x) $) with respect to the proton spin, while the up quarks are polarized along the proton spin ($x u_{-}(x) \leq x u_{+}(x) $).
The largest component of the proton in the region of interest is the up quark with helicity $+1/2$. In the CI scenario, the up quarks might contribute to the large $Q^2$ excess if $\eta_{ab}^u$ is sufficiently large. However, in the stop scenario, they can not contribute, because there is no coupling between the up quark and stop. The next large component is the down quark with helicity $-1/2$. This also does not couple to the stop. The situation changes if we use an oppositely polarized proton beam $p(\downarrow)$, because in this case the down quarks with helicity $+1/2$ represent the next largest component in the proton $p(\downarrow)$. Hence the cross section for $e^{+} p(\downarrow)\rightarrow e^{+\prime}X$ is larger than that for $e^{+} p(\uparrow)\rightarrow e^{+\prime}X$ in the stop scenario.
To begin with, we give numerical results for the unpolarized case. Figure \[fig:unpol\] shows how the CI and stop scenarios explain the unpolarized HERA data. [@TRAPE] The following three parameter sets for the CI scenario have been employed:
$
(1) ~~
VA+ :
(\eta^{u/d}_{LL},\eta^{u/d}_{LR},\eta^{u/d}_{RL},\eta^{u/d}_{RR})
=(+,-,+,-)
$ with $\Lambda=2.8$ TeV,\
$
(2) ~~
VA- :
(\eta^{u/d}_{LL},\eta^{u/d}_{LR},\eta^{u/d}_{RL},\eta^{u/d}_{RR})
=(-,+,-,+)
$ with $\Lambda=2.8$ TeV,\
$
(3) ~~~
X6- :
(\eta^{u/d}_{LL},\eta^{u/d}_{LR},\eta^{u/d}_{RL},\eta^{u/d}_{RR})
=(0,0,-,+)
$ with $\Lambda=1.9$ TeV,\
where the 95% confidence level limits [@TRAPE] on $\Lambda$ obtained by the ZEUS and H1 collaborations were used. The value of $\Lambda$ for the other parameter sets are very strongly constrained by the results of other experiments. [@GCHO] For the two-stop scenario, we used the values $\lambda_{131}^\prime=0.07, 0.05, 0.03$, $\tilde{m}_{1,2}=200, 230$ GeV, and the stop mixing angle $\cos\theta_{t}=0.5$. The branching ratios $Br(\tilde{t}_{1,2} \rightarrow e^+ d )$ are 0.65 and 1.0 for $\tilde{t}_{1}$ and $\tilde{t}_{2}$, respectively. [@KON; @ASAK] The parameter sets of the CI and the stop scenarios account for the unpolarized experimental data quite well. It is also seen that the behavior of the unplarized cross sections are similar in both the scenarios.
Next, we discuss the polarized case. For our purposes, it is useful to introduce the spin asymmetry $\cal{A}$, which is defined by $$\begin{aligned}
{\cal A}(Q_{0}^2)
&\equiv&
\frac{\int \sigma_{\uparrow}-\int \sigma_{\downarrow}
}{\int \sigma_{\uparrow}+\int \sigma_{\downarrow}},\end{aligned}$$ where $\int \sigma_{\uparrow/\downarrow}$ is the integrated cross section for the polarized proton $p(\uparrow/\downarrow)$ and positron scattering. We have $$\begin{aligned}
\int \sigma_{\uparrow/\downarrow}
&=&
\int_{Q_{0}^2}^{Q_{max}^2} dQ^2 dx_{B}
~\sigma(p(\uparrow/\downarrow)e^+\rightarrow e^{\prime +}X),\end{aligned}$$ where $Q_{max}^{2}=90,000$ GeV$^2$.
In Fig.\[fig:asy\], we plot the spin asymmetry for the CI and two-stop scenarios. One can see that $\cal A$ in the stop scenario has a large negative value in the large $Q_0^2$ region, while the SM prediction is positive in the region $Q^2_0 \geq 2,500 $ GeV$^2$. This is because the proton $p(\uparrow)$ contains more down quarks with helicity $-1/2$ than with helicity $+1/2$ and only the down quark with helicity $+1/2$ can couple to the stops which produce the large contribution to the cross sections. This is a characteristic feature for the stop scenario with the $R_{p}\hspace{-11pt}/~~$ interaction. A different choice for the parameters in the stop scenario does not change the results appreciably. The asymmetry for the parameter set VA+ in the CI scenario has a negative value and is similar to that in the stop scenario. However, the asymmetry for VA+ has negative value even when $Q^2_0 \sim 2,500$ GeV$^2$, where the asymmetry in the stop scenario is nearly zero. The asymmetries for the CI scenario are very different from that for the SM, even at $Q^2_0 \sim 2,500$ GeV$^2$. Hence, observing these behavior, the two scenarios may be considered distinguishable.
In summary, we discussed polarized proton-positron scattering in the context of the excess of large $Q^2$ events at HERA. For the CI scenario, the asymmetries $\cal{A}$ exhibit distinctive behavior for $Q^2_0\sim 2,500$ GeV$^2$, and the value depends on the parameter sets. For the two-stop scenario, there is a characteristic dependence on $Q_0^2$: the value changes from zero to near -1 as $Q_0^2$ becomes larger. Studying this behavior at future polarized HERA will provide a good test for these models.
The authors thank to T. Nasuno, H. Tochimura and Y. Yasui for helpful discussions, and to J. Kodaira for useful comments and reading manuscript.
[99]{} H1 Collaboration, .\
ZEUS Collaboration, .\
H1 and ZEUS Collaboration, B. Straub, in the Proceedings of “Lepton-Photon 97”, Hamburg, 28/07/97. U. F. Katz, representing the ZEUS and H1 Collaborations,\
SLAC Topical Conference, Stanford, 12-14 August 1998.\
http://www-h1.desy.de:80/h1/www/publications/hiq2/hiq2.html\
http://www-zeus.desy.de/$\tilde{~}$ukatz/ZEUS$\underbar{ }$PUBLIC/hqex/hqex$
\underbar{ }$papers.html$\sharp$Talks. G. Altarelli, . V. A. Noyes, [*preprint*]{} hep-ex/9707037. G. Altarelli, J. Ellis, G. F. Giudice, S. Lola and M. L. Mangano, . V. Barger, K. Cheung, K. Hagiwara and D. Zeppenfeld, . N. G. Deshpande, B. Dutta and X. He, . T. Kon and T. Kobayashi, . See, for example, A. D. Roeck and T. Gehrmann, in the proceedings of the Workshop on Deep Inelastic Scattering off Polarized Targets: Theory Meets Experiment (DESY 97-200), 523, and references therein. J. M. Virey, hep-ph/9710423, in the proceedings of Workshop on Deep Inelastic Scattering off Polarized Targets: Theory Meets Experiment (DESY 97-200), 654. G. Cho, K. Hagiwara and S. Matsumoto, .\
CDF Coll., F. Abe et al., [*Phys. Rev. Lett*]{} [**79**]{} (1997), 2198.\
OPAL Coll., G. Abbiendi et al., CERN-EP/98-108 (ICHEP98, Abs.264).\
ALEPH Coll., ALEPH 98-060(19998) (ICHEP98, Abs. 906).\
L3 Coll., M. Acciarri et al., CERN-EP/98-31(1998).\
A. Deandrea, Phys. Lett. [**B409**]{} (1997), 277 and references therein. T. Gehrmann and W. J. Stirling, . A. D. Martin, W. J. Stirling and R. G. Roberts, . E. Asakawa, J. Kamoshita and A. Sugamoto. hep-ph/9803321.
| ArXiv |
---
abstract: 'In this paper, we present a new supervised learning algorithm that is based on the Hebbian learning algorithm in an attempt to offer a substitute for back propagation along with the gradient descent for a more biologically plausible method. The best performance for the algorithm was achieved when it was run on a feed-forward neural network with the MNIST handwritten digits data set reaching an accuracy of 70.4% on the test data set and 71.48% on the validation data set.'
author:
- |
Rafi Qumsieh\
[email protected]
title: 'A Supervised Modified Hebbian Learning Method On Feed-forward Neural Networks'
---
Introduction and Motivation
===========================
Back propagation along with gradient descent are the primary algorithms used in training state-of-the-art neural network models. They have been very successful in producing very efficient models. They work by attempting to minimize the cost function by rolling down the cost function using gradient descent which depends on calculating partial derivatives with respect to the weights and biases through what is known as the back propagation process. A couple of the concerns regarding these algorithms are:
1. They are not biologically plausible as we have no evidence of back propagation happening in neural systems.
2. They are computationally expensive as they need to calculate a potentially large number of partial derivatives with respect to weights and biases for each data point.
Even Geoffrey Hinton, one of the founders of modern Artificial Intelligence remarked that he is suspicious of back propagation and that we ought to start over. \[1\]
Donald Hebb, a Canadian psychologist, postulated that the brain is plastic and it learns through changing the synaptic connections strengths between neurons depending on whether the input signal caused the output neuron to fire or not, and how large the signals are \[2\]. If a neuron causes another neuron to fire, their connection is strengthened in what is known as the long-term potentiation process. If a neuron fires, but does not cause the other neuron to fire, the connection weakens in what is known as the long-term depression process. Mathematically, we can describe Hebb’s postulate as the following: $$\Delta w = \eta x y$$ Where $w$ is the connection weight: a numerical value that indicates the strength of the connection between the two neurons, $\eta$ is the learning rate: a small positive number that indicates how much the weight will change at each iteration, $x$ is the input signal, $y$ is the output of the neuron. This formula captures some parts of Hebb’s hypothesis as it increases the weight if both $x$ and $y$ are large. It does not change the weight if any or both of them are zero. It might have a couple of problems:
1. If the input $x$ and output $y$ are large, the weight change can grow indefinitely. This is something that is not plausible in nature.
2. If one of them is zero while the other is not, the formula returns zero change for the weight change, but evidence from the long-term depression process shows that if the input is large but the neuron does not cause an output signal in the following neuron, a metabolic process that reduces the synaptic weight will take place.
A solution to the first problem was introduced by Oja in what is known as Oja’s rule \[3\]. It is a modified version of Hebb’s rule where the weight vector to each neuron does not change its magnitude, only its direction. One solution to the second problem is to treat -1 values as zeroes. But instead of relying on one formula to capture a set of potentially complex processes in nature, namely long-term potentiation and long-term depression, we will write an algorithm that handles the different scenarios that the neurons face while learning and how they react to them.\
In this paper, we discuss a feed-forward neural network that is trained on the MNIST handwritten digits data set using a modified Hebbian learning algorithm.
The Modified Hebbian Learning Algorithm
=======================================
The algorithm is used to train a neural network using the training data set by taking each training data point $(x_{train},y_{train})$ and running the following:
1. Feed the signal forward by calculating the activation of each neuron in the network using a rectified version of the hyperbolic tangent activation function $ y = \tanh_{rec}(\Sigma x_i w_i - b_i) $. This $\tanh_{rec}$ function takes a hyper parameter $c_{activation} $ that is the coefficient of $x$ and it controls the curve of $\tanh$. At the every layer, the activation of the layer is appended to an activations list. At the last layer, the desired output vector is forcefully appended to the activations list (in a winner-takes-all manner) so that we ensure a supervised learning of the pattern. The activation function of the last output layer is the $ReLU$ function to make distinguishing values easier (instead of squashing the values). The $\tanh_{rec}$ function has the following definition:
$$\tanh_{rec}(x, c) = \left\{\begin{array}{lr}
\frac{e^{cx}-e^{-cx}}{e^{cx}+e^{-cx}}, & \text{if } x > 0 \\ \\
0 & \text{if } x \leq 0 \\ \\
\end{array}\right\}$$
2. After obtaining all activations for the data point, for each weight, we evaluate the modified Hebbian weight update function asynchronously (with respect to each layer), starting from the first layer:
$$\Delta w(x,y) = \left\{\begin{array}{lr}
+\eta_{ltp} x y, & \text{if } x * y \geq T, w \neq 0 \\ \\
-\eta_{ltp} x y, & \text{if } x * y < T, w \neq 0 \\ \\
0.50, & \text{if } w = 0 \\ \\
\end{array}\right\}$$
Where $T$ is a positive number that represents the threshold at which the weight will change for each neuron. $x$ and $y$ have values between 0 and 1. Sometimes, we modify this weight update function to handle additional cases. The above weight update function is a compressed version that captures enough information for the neuron to learn using the long-term potentiation and long-term depression processes. We can expand the weight update function to include more cases as we try model more processes / scenarios from biology. Below is an extended version of the function: $$\Delta w(x,y) = \left\{\begin{array}{lr}
+\eta_{ltp} x y, & \text{if } x > 0 ,\space y > 0,\space w > 0 \\ \\
-\eta_{ltp} x y, & \text{if } x > 0 ,\space y > 0, \space w < 0 \\ \\
-\eta_{ltd} x, & \text{if } x>0, \space y=0,\space w > 0 \\ \\
+\eta_{ltd} x, & \text{if } x>0, \space y=0,\space w < 0 \\ \\
+\eta_{ltp2} y, & \text{if } x=0, \space y>0,\space w > 0 \\ \\
-\eta_{ltp2} y, & \text{if } x=0, \space y>0,\space w < 0 \\ \\
+0.50 & \text{if } x*y\geq T,\space w=0 \\ \\
0, & \text{Otherwise} \\ \\
\end{array}\right\}$$
3. Calculate the new updated weights using the rectified linear unit function $ReLU(x)$ as follows: $w_{new} = ReLU(w_{old} + \Delta w) $. If $w_{old} > 0$ but $w_{new} < 0 $, then reset $w$ to zero and vice versa. The rationale here is that a neuron cannot change its type from being excitatory to being inhibitory, or the other way around. The rectified linear unit function is defined as: $$ReLU(x) = \left\{\begin{array}{lr}
x, & \text{if } x > 0 \\ \\
0 & \text{if } x \leq 0 \\ \\
\end{array}\right\}$$
Note that these rules come from assumptions and attempts to model some well known mechanisms on how plasticity works. We can always modify these rules as we discover more information on how plasticity, long-term potentiation, and long-term depression work.
The Networks
============
The Shallow Network
-------------------
### Structure Of The Network
First, let us start with describing the problem. We are trying to classify a grayscale image of size 28 by 28 pixels of a handwritten digit to one of the digits between 0 and 9. One way to look at the problem is that we are trying to find a classifier map $C$ that can be defined as follows: $$C : [0,1]^{784} \to \{0,1\}^{10}$$
We will represent the desired map by creating a feedforward neural network with two layers: An input layer of size 784 and an output layer of size 10 and no hidden layers. In this paper, a feed-forward neural network written in Python by Michael Nielsen \[4\] was used with the following layers:
1. Input layer: This layer has 784 neurons corresponding to the cells in the input of the 28 by 28 pixel gray-scale images used in the MNIST data set. Each neuron’s output is a real number between 0 and 1.
2. Output layer: This layer has 10 neurons. Each neuron corresponds to a digit from 0 to 9. The neuron that outputs the highest value is interpreted as being the classified neuron.
### Weights And Biases Initialization
All neurons in the network had a fixed bias of 0.95 that was adjusted along with the other hyper parameters. The idea behind having a fixed bias is that, as far as we know, neurons have a fixed threshold voltage of -70 mV, so our assumption brings us closer to nature. As for the weights, the connection weights have the following configuration:
1. Input-to-Output layer: All weights are initialized to a preset value. In our case, we can start the weight with value $ w = 0$. Having all weights equal zero gives the network no prejudice towards any output as it starts learning.
### Properties Of All The Variables And Parameters
1. $x$ is the input signal for each neuron. $x \in [0,1]$. Where a value of 0 indicates no signal, and a value of 1 indicates maximum signal.
2. $y$ is the output signal for each neuron. $y \in [0,1]$. Where a value of 0 indicates no signal, and a value of 1 indicates maximum output signal.
3. $w$ is the connection weight value for each pair of neurons from a layer to a consecutive layer. $w \in [-1,1]$. Where a negative value indicates that the connection is inhibitory and a positive value indicates an excitatory connection. A value closer to zero indicates a weaker connection. Anytime an adjusted weight exceeds the value of 1, it will be either passed through a squashing function or hard-resetted to the value of 0.90, and similarly, anytime an adjusted weight value decreases beyond -1, it will be either passed through a squashing function or hard-resetted to -0.90. This technique is used to ensure that not all weights reach complete saturation, and thus will hinder learning. Even though the weights can have negative values, we restricted the values to positive ones in the experiments.
The Medium Network
------------------
The previous shallow network can do a pretty decent job at mapping input vectors to output vectors. It faces the problem of needing to learn a relatively large number of connections between the input and the output layers (the input is large), and this will cause a long learning time as we will see in the results section. We can add a pooling / dimensionality reduction layer that extracts important features from the input, and then learn the new reduced input. This is the purpose of using the medium network.
### Structure Of The Network
The network has the following layers:
1. Input layer: This layer has 784 neurons corresponding to the cells in the input of the 28 by 28 pixel gray-scale images used in the MNIST data set.
2. Hidden layer: This layer can have either 196 or 49 neurons depending on how much we want to reduce the input by.
3. Output layer: This layer has 10 neurons. Each neuron corresponds to a digit from 0 to 9.
### Weights And Biases Initialization
All neurons in the same layer had a fixed bias that was adjusted along with the other hyper parameters. As for the weights, the connection weights have the following configuration:
1. Input-to-Hidden layer: The weights are initialized to values $ w \in [-1, 1]$ with any desired probability distribution, and the connections are set up in a configuration that forces dimensionality reduction on the input data through a form of pooling. More on this in the Appendix A. In the experiments, this layer was not set to learn and so its weights did not change.
2. Hidden-to-Output layer: The weights can start with any fixed value in $ w \in [-1, 1]$ that can be changed along with other hyper parameters, and the connections are set up in a way that resembles a fully connected configuration.
One way to view the structure of this network is that the weights are set up in a manner that resembles a pooling map that reduces dimensionality, followed by a map of association between the reduced input and the desired output.
### Properties Of All The Variables And Parameters
1. $x$ is the input signal for each neuron. $x \in [0,1]$. Where a value of 0 indicates no signal, and a value of 1 indicates maximum signal.
2. $y$ is the output signal for each neuron. $y \in [0,1]$. Where a value of 0 indicates no signal, and a value of 1 indicates maximum output signal.
3. $w$ is the connection weight value for each pair of neurons from a layer to a consecutive layer. $w \in [-1,1]$. Where a negative value indicates that the connection is inhibitory and a positive value indicates an excitatory connection. A value closer to zero indicates a weaker connection. Anytime an adjusted weight exceeds the value of 1, it will be either passed through a squashing function or hard-resetted to the value of 0.90, and similarly, anytime an adjusted weight value decreases beyond -1, it will be either passed through a squashing function or hard-resetted to -0.90. This technique is used to ensure that not all weights reach complete saturation, and thus will hinder learning.
The Deeper Network
------------------
The previous medium (3-layered) network reduces the dimensionality of the input which theoretically should speed up the learning process. We wanted to see if we can reduce the learning time even further while allowing the network to learn more. This is the purpose of this network.
### Structure Of The Network
The network has the following layers:
1. Input layer: This layer has 784 neurons corresponding to the cells in the input of the 28 by 28 pixel gray-scale images used in the MNIST data set.
2. Hidden layer 1: This layer can have either 196 or 49 neurons depending on how much we want to reduce the input by.
3. Hidden layer 2: This layer can have either 196 or 49 neurons depending on how much we want to reduce the input by. We prefer this layer to have a smaller number of neurons than hidden layer 1.
4. Output layer: This layer has 10 neurons. Each neuron corresponds to a digit from 0 to 9.
### Weights And Biases Initialization
All neurons in each layer have a fixed bias value that can be adjusted along with other hyper parameters. As for the weights, the connection weights have the following configuration:
1. Input-to-Hidden 1 layer : The weights are initialized to values $ w \in [-1, 1]$ with any desired probability distribution, and the connections are set up in a configuration that forces dimensionality reduction on the input data through a form of pooling. More on this in the Appendix A. In the experiments, this layer was not set to learn and so its weights did not change.
2. Hidden 1-to-Hidden 2 layer : The weights are initialized to values $ w \in [-1, 1]$ with any desired probability distribution, and the connections are set up in a configuration that forces dimensionality reduction on the input data through a form of pooling. More on this in the Appendix A. In the experiments, this layer was not set to learn and so its weights did not change.
3. Hidden 2-to-Output layer: The weights can start with any fixed value in $ w \in [-1, 1]$ that can be adjusted along with other hyper parameters, and the connections are set up in a way that resembles a fully connected configuration.
One way to view the structure of this network is that the weights are set up in a manner that resembles a pooling map that reduces dimensionality, followed by another pooling map that reduces dimensionality and extracts important features, followed by a map of association between the final reduced input and the desired output.
### Properties Of All The Variables And Parameters
1. $x$ is the input signal for each neuron. $x \in [0,1]$. Where a value of 0 indicates no signal, and a value of 1 indicates maximum signal.
2. $y$ is the output signal for each neuron. $y \in [0,1]$. Where a value of 0 indicates no signal, and a value of 1 indicates maximum output signal.
3. $w$ is the connection weight value for each pair of neurons from a layer to a consecutive layer. $w \in [-1,1]$. Where a negative value indicates that the connection is inhibitory and a positive value indicates an excitatory connection. A value closer to zero indicates a weaker connection. Anytime an adjusted weight exceeds the value of 1, it will be either passed through a squashing function or hard-resetted to the value of 0.90, and similarly, anytime an adjusted weight value decreases beyond -1, it will be either passed through a squashing function or hard-resetted to -0.90. This technique is used to ensure that not all weights reach complete saturation, and thus will hinder learning.
Setup
=====
The experiments were performed on an HP^^ Pavilion laptop with an i5 processor and 8 GB RAM memory on Windows^^. Everything was written using Python 3.7 on top of Anaconda. The code was imported and modified from Michael Nielsen’s neural network book’s code \[2\]. All of the code involved in this experiment can be found using reference \[5\].
The Experiments And Results
===========================
There were two objectives in the experiments. The first objective was to achieve high accuracies using a very small number of images per digit (Low-shot learning), the second objective was to achieve the highest possible accuracy. The experiments involved tuning many parameters, but below are the optimal hyper parameters found for each type of network used:
The Shallow Network
-------------------
### Low-Shot Learning
Using only 2 images per digit over 1 epoch, the network scored an accuracy of 47.62% on the test data.
### Highest Accuracy
For this part, the optimal hyper parameters that resulted in the highest prediction accuracy on the test and validation data are:
1. The feed-forward neural network had 3 layers of sizes 784, 196, 10 respectively.
2. $ \eta_{ltp} = 0.001 $ and $ \eta_{ltd} = 0.0001 $.
3. $ c_{output} = 0.05 $ and $ c_{weights} = 0.5 $.
4. Biases of neurons in each layer = \[0.95\]
5. Initial weights of neurons in each layer = \[0.00\]
6. Weight creation value is 0.50 at threshold $x*y \geq 0.25$
7. Number of images per digit = 60 images per digit which yields 600 unique images.
8. Number of epochs = 1
Using this configuration, the network gave about 81%, 69%, 71.05% on the training, test, validation datasets respectively.
### Overall trend
The network needed a training time of about 60.76 seconds for the 600 training data points, and its accuracy vs. IPD relationship is graphed below:
![The accuracy of the shallow network as a function of the number of images-per-digit (IPD).[]{data-label="fig:shallownetwork"}](Figure_1.png){width="\linewidth"}
It can be seen that there is an overall decreasing trend for the accuracy as we increase the IPD value.
The Medium Network
------------------
### Low-Shot Learning
Using only 1 image per digit over 1 epoch, the network scored an accuracy of 36.25% on the test data, and using 2 images per digit over 1 epoch, the network gave an accuracy of 43.63% on the test data.
### Highest Accuracy
For this part, the optimal hyper parameters that resulted in the highest prediction accuracy on the test and validation data are:
1. The feed-forward neural network had 3 layers of sizes 784, 196, 10 respectively.
2. $ \eta_{ltp} = 0.01 $ and $ \eta_{ltd} = 0.0005 $.
3. $ c_{output} = 0.50 $ and $ c_{weights} = 0.50 $.
4. biases of neurons in each layer = \[0.95, 0.00\]
5. Initial weights of neurons in each layer = \[0.50, 0.00\]
6. Weight creation value is 0.50 at threshold $x*y \geq 0.25$
7. Number of images per digit = 60 images per digit which yields 600 unique images.
8. Number of epochs = 1
Using this configuration, the network gave about 79.0%, 70.4%, 71.48% on the training, test, validation datasets respectively.
### Overall Trend
The network needed a training time of about 14.18 seconds to train on 600 data points, and its accuracy vs. IPD relationship is graphed below:
![The accuracy of the medium network as a function of the number of images-per-digit (IPD).[]{data-label="fig:med"}](Figure_2.png){width="\linewidth"}
The Deeper Network
------------------
### Low-Shot Learning
Using only 1 image per digit over 1 epoch, the network scored an accuracy of 36.33% on the test data.
### Highest Accuracy
For this part, the optimal hyper parameters that resulted in the highest prediction accuracy on the test and validation data are:
1. The feed-forward neural network had 3 layers of sizes 784, 196, 10 respectively.
2. $ \eta_{ltp} = 0.001 $ and $ \eta_{ltd} = 0.0001$.
3. $ c_{output} = 1.0 $ and $ c_{weights} = 0.5 $.
4. biases of neurons in each layer = \[0.35, 0.05, 0.00\]
5. Initial weights of neurons in each layer = \[0.50, 0.60, 0.00\]
6. Weight creation value is 0.50 at threshold $x*y \geq 0.25$
7. Connectivity factors in each layer = \[0.75, 0.25\]
8. Number of images per digit = 200 images per digit which yields 2,000 unique images.
9. Number of epochs = 1
Using this configuration, the network gave about 58.33%, 48.17%, 50.27% on the training, test, validation datasets respectively.
### Overall Trend
The network needed a training time of about 7.72 seconds to train on 600 data points, and its accuracy vs. IPD relationship is graphed below:
![The accuracy of the deeper network as a function of the number of images-per-digit (IPD).[]{data-label="fig:deeper"}](Figure_3.png){width="\linewidth"}
Discussion and Conclusion
=========================
Testing the learning algorithm across the different types of networks and across the different test sets shows that there is some form of learning taking place in those networks. The networks seem to reach a saturation point where adding more training data does not increase the accuracy, but decrease it instead. The highest achieved accuracy is about 70% for the test data on the medium (3-layered) network. Increasing the number of epochs/recitations seems to increase the accuraries only for the smallest number of IPD (1 - 10), but seems to decrease the accuracies for larger numbers of IPD. One possible reason for the saturation is that none of the networks had any inhibitory neurons (negative weights) which can help regulate the network and prevent it from reaching saturation with a small number of training data points. Another thing that is worth mentioning is that using the detailed version of the algorithm versus the compressed version did not seem to have a large impact on the accuracy measurements.\
One way to increase the accuracies even further is to use what can be referred to as “assisted learning”. Assisted learning basically involves looking at which labels were poorly classified by the algorithm, and offering more data points for the labels that were most poorly classified. So for example, if our model did not classify the digit ’2’ well, we would add more data points for the label ’2’ during training. This is similar to helping a student who is struggling with a certain concept by having them work on more exercises that involve that topic until they master it. Keep in mind that the method of assisted learning is not specific to this algorithm and can probably be applied to different models.\
A potential improvement to the network might lie in adding another layer that takes the reduced input and maps it to a higher dimension in a way that preserves its main features but makes differentiation between inputs easier for the final fully connected layer.\
The algorithm discussed in this paper has the following advantages:
1. This learning algorithm attempts to mimic our current understanding of the processes that underlie plasticity. This is especially beneficial for neuromorphic chips that use memristors which work in a similar manner.
2. The algorithm does not require the model to have random values for weights and biases, which is something that is more likely not found in nature as cells are created using specific instructions.
3. The learning occurs simultaneously with the feedforward process in a less computationally expensive way.
4. It can be highly modified to handle more cases and scenarios as we learn more about plasticity.
On the other hand, the algorithm at this point has the following disadvantages and caveats:
1. It needs to be more formally defined.
2. It needs thorough testing and evaluating.
3. Inhibitory neurons / negative weights should be incorporated into the model as a way control information flow.
4. Learning only occurs at the last layer where we force the output to be the desired output. If learning can be extended to more layers, it will probably give the network a larger capacity to store more information.
There are still many questions and issues that need to be addressed to further the understanding of the algorithm. This paper serves as a pilot to pique interest in the algorithm and its potential applications.
References
==========
1. Jeoffrey Hinton’s interview with Axios - Axios - https://www.axios.com/artificial-intelligence-pioneer-says-we-need-to-start-over-1513305524-f619efbd-9db0-4947-a9b2-7a4c310a28fe.html
2. The Organization of Behavior - Donald Hebb - Wiley & Sons.
3. Simplified neuron model as a principal component analyzer - Erkki Oja - Journal of Mathematical Biology. 15 (3): 267–273
4. Neural Networks and Deep Learning - Michael Nielsen - http://neuralnetworksanddeeplearning.com
5. Algorithm’s Code Repository - Rafi Qumsieh - https://github.com/rqumsieh0/supervised-hebbian
Appendix A - Details Of The Pooling Map
=======================================
Let us assume we have a layer (set) of cells of size $n$, and let us call this set $C^n$. Now, let us define a new layer (set) of cells of size $m$, and let us call it $C^m$. Associated with each cell in these sets a value $b$ that denotes its bias to firing given an input. Let the following assumptions be true about our system:
1. $n$ is a perfect square, and $m$ is a perfect square. Furthermore, $m = \sqrt{n}$
2. The first layer is the input layer, and that information flows uni-directionally from the first layer to the second layer only.
3. The cells in each layer are not connected to each other.
It is important to notice that any of the layers can be arranged into a square matrix. For example, $C^n$ can be arranged into a square matrix of size $m \times m$, and the row and column indices range from $0,....,m-1 $. The function we can use to decompose any layer index $x$ in $C^n$ into its row index $i$ and column index $j$ is using this formula:\
$$i = \text{floor}(n/x)$$ $$j = n\% x$$
Same formula can be used to decompose the index of any layer into its row and column indices on the condition that the layer’s size is a perfect square. The concept can be shown in the figure below:
![An example of the pooling map from a 4 by 4 layer to a 2 by 2 layer with indices. Same colors means the neurons are connected between the two layers. []{data-label="fig:mapneuron"}](MapNeuron.png){width="\linewidth"}
Defining the weights
--------------------
Now let us define the set of weights as the following map $W : C^n \times C^m \to [0,1]$. Take an element of $C^n$ and call it by its index $x$. $x$ can be decomposed into $(i_1,j_1)$ row and column indices. Similarly, take an element of $C^m$ and call it by its index $y$. $y$ can be decomposed into $(i_2,j_2)$ row and column indices. Define a new value as the ratio $ r = \frac{\sqrt{n}}{\sqrt{m}} $. We will define the map element-wise as follows:
$$w(x,y) = \left\{\begin{array}{lr}
c, & \text{if } \text{floor}(i_1 / r) = i_2, \text{floor}(j_1 / r) = j_2 \\
0, & \text{if } \text{otherwise} \\
\end{array}\right\}$$
The map can be generalized to include more overlapping connections between neurons by the following modification:
$$w(x,y) = \left\{\begin{array}{lr}
c, & \text{if } |\text{floor}(i_1 / r) - i_2| \leq v, |\text{floor}(j_1 / r) - j_2| \leq v \\
0, & \text{if } \text{otherwise} \\
\end{array}\right\}$$
Where $c$ is just some constant weight value, and $v$ can be called the connectivity factor and is some integer less than $\sqrt{n} - 1 $, and controls the spread of the connection to each neuron. This can be shown using the following figure:\
Using this weight connection configuration, we can reduce the dimensions of the input while preserving some form of a distance metric in a process similar to pooling.
| ArXiv |
---
abstract: |
We present an analysis of multi-timescale variability in line-of-sight X-ray absorbing gas as a function of optical classification in a large sample of Seyfert active galactic nuclei (AGN) to derive the first X-ray statistical constraints for clumpy-torus models. We systematically search for discrete absorption events in the vast archive of *Rossi X-ray timing Explorer* monitoring of dozens of nearby type I and Compton-thin type II AGN. We are sensitive to discrete absorption events due to clouds of full-covering, neutral or mildly ionized gas with columns $\ga 10^{22-25}$ cm$^{-2}$ transiting the line of sight.
We detect 12 eclipse events in 8 objects, roughly tripling the number previously published from this archive. Peak column densities span $\sim4-26 \times 10^{22}$ cm$^{-2}$, i.e., there are no full-covering Compton-thick events in our sample. Event durations span hours to months. The column density profile for an eclipsing cloud in NGC 3783 is doubly spiked, possibly indicating a cloud that is being tidally sheared.
We infer the clouds’ distances from the black hole to span $\sim 0.3
- 140 \times 10^{4} R_{\rm g}$. In seven objects, the clouds’ distances are commensurate with the outer portions of Broad Line Regions (BLR), or outside the BLR by factors up to $\sim10$ (the inner regions of infrared-emitting dusty tori). We discuss implications for cloud distributions in the context of clumpy-torus models. Eight monitored type II AGN show X-ray absorption that is consistent with being constant over timescales from 0.6 to 8.4 yr. This can either be explained by a homogeneous medium, or by X-ray-absorbing clouds that each have $N_{\rm H} \ll 10^{22}$ cm$^{-2}$. The probability of observing a source undergoing an absorption event, independent of constant absorption due to non-clumpy material, is $0.006^{+0.160}_{-0.003}$ for type Is and $0.110^{+0.461}_{-0.071}$ for type IIs.
author:
- |
A. G. Markowitz$^{1,2,3}$[^1], M. Krumpe$^{4,1}$, R. Nikutta$^{5}$\
$^{1}$University of California, San Diego, Center for Astrophysics and Space Sciences, 9500 Gilman Dr., La Jolla, CA 92093-0424, USA\
$^{2}$Dr. Karl Remeis Sternwarte, Sternwartstrasse 7, D-96049 Bamberg, Germany\
$^{3}$Alexander van Humboldt Fellow\
$^{4}$European Southern Observatory, Karl-Schwarzschild-Strasse 2, D-85748 Garching bei München, Germany\
$^{5}$Departamento de Ciencias Fisicas, Universidad Andrés Bello, Av. República 252, Santiago, Chile
date: 'Accepted 2013 December 24. Received 2013 November 29; in original form 2013 September 3'
title: 'First X-ray-Based Statistical Tests for Clumpy-Torus Models: Eclipse Events from 230 Years of Monitoring of Seyfert AGN'
---
\[firstpage\]
galaxies: active – X-rays: galaxies – galaxies: Seyfert
INTRODUCTION
============
While it is generally accepted that active galactic nuclei (AGN) are powered by accretion of gas on to supermassive ($10^{6-9}{\ensuremath{\rm M_\odot}\xspace}$) black holes, the exact geometry and mechanisms by which material gets funneled from radii of kpc/hundreds of pc down to the accretion disk at sub-pc scales remains unclear. This unsolved question is, however, linked to the overall efficiency of how supermassive black holes are fed and to AGN duty cycles.
Clues to the morphology of the circumnuclear gas come from different studies across the electromagnetic spectrum. For instance, Seyfert AGN are generally classified into two broad categories based on the detection or lack of broad optical/UV emission lines (type I or II, respectively), with intermediate subclasses (1.2, 1.5, 1.8, 1.9) depending on broad lines’ strengths (e.g., Osterbrock 1981). Unification theory posits that all Seyferts host the same central engine, but observed properties depend on orientation due to the presence of dusty circumnuclear gas blocking the line of sight to the central illuminating source in type II Seyferts. The classical model holds that a primary component of this obscuring gas is a pc-scale, equatorial, dusty torus (Antonucci & Miller 1985; Urry & Padovani 1995), where “torus” generally denotes a donut-shaped morphology, supplying the accretion disk lying co-aligned inside it. However, some recent papers (e.g., Pott et al. 2010) use the term “torus” to denote simply the region where circumnuclear gas can exist, with the precise morphology of that region still to be determined; we adopt that notation in this paper.
A donut morphology can explain the observed bi-conical morphology of Narrow Line Regions (NLR), e.g., via collimation of ionizing radiation (e.g., Evans [et al.]{} 1994), and also why most type II Seyferts show evidence for X-ray obscuration along the line of sight. However, the relation between optical/UV-reddening dust and X-ray absorbing gas is not straightforward. Very roughly 5 per cent of all AGN have differing X-ray and optical obscuration indicators (e.g., X-ray-absorbed type Is, Perola [et al.]{} 2004; Garcet [et al.]{} 2007); a complication is that optical obscuration probes only dusty gas, while X-ray absorption probes both dusty and dust-free gas, and X-ray columns can exceed inferred optical-reddening columns by factors of 3–100 (Maiolino [et al.]{} 2001). Lutz et al. (2004) and Horst et al. (2006) demonstrated that the infrared (IR) emission by dust, which thermally re-radiates higher energy radiation, is isotropic, with type I and II Seyferts having similar ratios of X-ray to mid-IR luminosity. This is not expected from classical unification models, since the classical donut-shaped torus both absorbs and re-emits anisotropically. Furthermore, if the torus is comprised of primarily Compton-thick gas, i.e., with X-ray-absorption column densities $N_{\rm H}$ above $1.5 \times 10^{24}$ cm$^{-2}$, then it is not clear how Compton-thin columns can be attained unless lines of sight happen to “graze” the outer edges. A unification scheme based solely on inclination angle and the presence of a donut-shaped absorbing morphology may thus be an oversimplification (Elvis 2012). It is also not clear how geometrically thick structures (scale heights $H/R \sim
1$) comprised of cold gas (100 K) are supported vertically over long timescales (Krolik & Begelman 1988).
Meanwhile, the community has accumulated observations of variations in the X-ray absorbing column $N_{\rm H}$ in both (optically classified) type Is and IIs, with timescales of variability ranging from hours to years. For instance, Risaliti, Elvis, & Nicastro (2002; hereafter REN02) claim that variations in $N_{\rm H}$ in a sample of X-ray-bright, Compton-thin and moderately Compton-thick type IIs are ubiquitous, with typical variations up to factors of $\sim1.5-3$. Current X-ray missions such as *XMM-Newton*, *Chandra* and *Suzaku* have enabled high precision studies of variations in $N_{\rm H}$ in numerous AGN; major findings include the following:
$\bullet$ Numerous moderately Compton-thick variations ($\Delta$$N_{\rm H} \sim 10^{23-24}$ cm$^{-2}$) on timescales $\la
1-2$ d in NGC 1365 (Risaliti [et al.]{} 2005, 2007, 2009a) and ESO 323–G77 (Miniutti [et al.]{} 2014); moderately Compton-thick variations on timescales from days to months in NGC 7582 (Bianchi [et al.]{} 2009).
$\bullet$ Changes in covering fraction of partial-covering absorbers on timescales from $<$1 d (NGC 4151, Puccetti [et al.]{} 2007; NGC 3516, Turner [et al.]{} 2008; Mkn 766, Risaliti [et al.]{} 2011; SWIFT J2127.4+5654, Sanfrutos [et al.]{} 2013) to months–years in NGC 4151 (de Rosa [et al.]{}2007 and Markowitz [et al.]{} in preparation, using *BeppoSAX* and *Rossi X-ray Timing Explorer* (*RXTE*) data, respectively).
$\bullet$ Time-resolved spectroscopy of full eclipse events (ingress and egress), yielding constraints on clouds’ density profiles for long-duration (3–6 months) eclipses in NGC 3227 (Lamer [et al.]{} 2003) and Cen A (Rivers [et al.]{} 2011b) and for eclipses $\la$1 d in NGC 1365 (Maiolino [et al.]{} 2010) and SWIFT J2127.4+5654 (Sanfrutos [et al.]{} 2013).
These results suggest that the circumnuclear absorbing gas is *clumpy*, with non-homogeneous or clumpy absorbers being invoked to explain time-variable X-ray absorption as far back as *Ariel V* and *Einstein* observations (Barr [et al.]{} 1977; Holt [et al.]{} 1980). With the concept of a homogeneous, axisymmetric absorber thus under scrutiny, the community has been developing torus models incorporating clumpy gas, e.g., Elitzur & Shlosman (2006) and Nenkova [et al.]{} (2008a, 2008b; see Hönig 2013 for a review), although suggestions that the torus should consist of clouds go as far back as, e.g., Krolik & Begelman (1986, 1988). In the most recent models, total line-of-sight absorption for a given source is quantified as a viewing dependent *probability* based on the size and locations of clouds, although typically, clouds are preferentially distributed towards the equatorial plane. The fraction of obscured sources depends on the average values and distributions of such parameters as the average number of clouds lying along a radial path, and the thickness of the cloud distribution. Clouds are possibly supported vertically by, e.g., radiation pressure (Krolik 2007) or disk winds (Elitzur & Shlosman 2006).
Observations so far suggest that clouds are typically on the order of $10^{13-15}$ cm in diameter, with number densities $\sim 10^{8-11}$ cm$^{-3}$. Inferred distances from the black hole, usually based on constraints from X-ray ionization levels and assumption of Keplerian motion, range from light-days and commensurate with clouds in the Broad Line Region (BLR; e.g., for NGC 1365 and SWIFT J2127.4+5654) to several light-months (Rivers [et al.]{} 2011b for Cen A) and commensurate with the IR-emitting torus in that object. We emphasize “commensurate,” as X-ray absorbers lie along the line of sight, but BLR clouds likely contain components out of the line of sight.
Recent IR interferometric observations have spatially resolved distributions of dust down to radii of tenths of pc (e.g., Kishimoto [et al.]{} 2009, 2011; Tristram [et al.]{} 2009; Pott [et al.]{} 2010). In addition, there are suggestions from reverberation mapping of the thermal continuum emission in four Seyferts that warm dust gas extends down to $\sim10-80$ light-days, likely just outside the outer BLR (Suganuma [et al.]{} 2006). As suggested by Netzer & Laor (1993), Elitzur (2007), and Gaskell et al. (2008), the BLR and dusty torus may be part of a common radially extended structure, spanning radii inside and outside the dust sublimation radius $R_{\rm d}$, respectively, since dust embedded in the gas outside $R_{\rm d}$ suppresses optical/UV line emission. Optical obscuration is due to dusty gas, while X-ray obscuration can come from dusty or dust-free gas. X-ray obscuration is thus the only way to probe obscuration inside $R_{\rm d}$.
In order to gain a more complete picture of the geometry of circumnuclear gas in AGN, however, the community needs to gauge the relevance of clumpy-absorber models over a wide range of length scales, including both inside and outside $R_{\rm d}$, and to clarify the links between the distributions of dusty gas, X-ray-absorbing gas, and the BLR. To date, however, observational constraints to limit parameter space in clumpy-torus models has been lacking because there has been no statistical survey so far. One of our goals for this paper is to derive constraints on clumpy-torus models via variable X-ray-absorbing gas, including estimates of the probability that the line of sight to the AGN is intercepted by a cloud.
We use the vast archive of *RXTE* multi-timescale light curve monitoring of AGN, as described in $\S$2. We search for changes in full- or partial-covering $N_{\rm H}$ in Seyferts. We use a combination of light curve hardness ratios and time-resolved spectroscopy to identify and confirm eclipses, which are summarized in $\S$3. We use the observed eclipse durations and the observation sampling patterns to estimate the probability to observe a source undergoing an eclipse in $\S$4. In $\S$5, we infer the eclipsing clouds’ radial locations and physical properties, relate the X-ray-absorbing clouds to other AGN emitting components, and describe the resulting observational constraints for key parameters of clumpy-torus models. The results are summarized in $\S$6. In a separate paper (Nikutta et al., in preparation), we will extend our analysis of the cloud properties based on the X-ray data presented in this paper.
OBSERVATIONS, DATA REDUCTION, AND ECLIPSE IDENTIFICATION
========================================================
Our strategy to identify eclipse events in general follows that used in successful detections by, e.g., Risaliti [et al.]{} (2009b, 2011), Rivers [et al.]{} (2011b), etc. We first extract sub-band light curves for all objects with sufficient X-ray monitoring, and examine hardness ratios to identify possible eclipse events, which manifest themselves via sudden increases in hardness ratio. We then perform time-resolved spectroscopy, binning individual adjacent observations as necessary to achieve adequate signal to noise ratio, to attempt to confirm such events as being due to an increase in $N_{\rm H}$ as opposed to a flattening of the continuum photon index.
*RXTE* has already revealed long-term (months in duration) eclipse events with $\Delta$$N_{\rm H} \sim 10^{23}$ cm$^{-2}$ for four objects: as mentioned above, complete eclipse events were confirmed via time-resolved spectroscopy for NGC 3227 in 2000–1 (Lamer [et al.]{} 2003) and Cen A in 2010–2011 (Rivers [et al.]{} 2011b). As discussed by Rivers [et al.]{} (2011b) and Rothschild [et al.]{} (2011), there is evidence for an absorption event in Cen A in 2003–2004 with a similar value of $\Delta$$N_{\rm H}$ to the 2010–2011 event. Smith [et al.]{}(2001) and Akylas [et al.]{} (2002) presented evidence for a decrease in $N_{\rm H}$ during 1996–1997 in the Compton-thin Sy 2 Mkn 348, suggesting that *RXTE* witnessed the tail end of an absorption event with $\Delta$$N_{\rm H} \ga 14 \times 10^{22}$ cm$^{-2}$.
Target selection from the *RXTE* database
-----------------------------------------
*RXTE* operated from 1995 December until 2012 January. We consider data from its Proportional Counter Array (PCA; Jahoda [et al.]{} 2006), sensitive over 2–60 keV. *RXTE*’s unique attributes – large collecting area for the PCA, rapid slewing and flexible scheduling – made it the first X-ray mission to permit sustained monitoring campaigns, with regularly spaced visits, usually 1–2 ks each, over durations of weeks to years.
*RXTE* visited 153 AGN during the mission, with sustained monitoring (multiple individual observations) spanning durations of $\sim4$ d or longer occurring for 118 of them. *RXTE* monitored AGN for a variety of science pursuits, including, for example, interband correlations to probe accretion disk structure and jet-disk links (e.g., Arévalo [et al.]{} 2008; Breedt [et al.]{} 2009; Chatterjee [et al.]{} 2011), X-ray timing analysis to constrain variability mechanisms in Seyferts (e.g., Markowitz [et al.]{} 2003; McHardy [et al.]{} 2006), and coordinated multi-wavelength campaigns on blazars during giant outbursts to constrain spectral energy distributions (SEDs) and thus models for particle injection/acceleration in jets (e.g., Krawczynski [et al.]{} 2002; Collmar [et al.]{} 2010). The archive thus features a wide range of sampling frequencies and durations from object to object. Typical long-term campaigns consisted of one observation every 2–4 d for durations of months to years (15.4 yr in the longest case, NGC 4051). A few tens of objects were subject to more intensive monitoring consisting of e.g., 1–4 visits per day for durations of weeks.
For this paper, we do not consider sources visited less than four times during the mission; many sources were in fact visited hundreds to more than a thousand times during *RXTE*’s lifetime. We rejected sources whose mean 2–10 keV flux is $\la 8 \times
10^{-12}$ [ergcm$^{-2}$s$^{-1}$]{}; such sources had very large error bars in the sub-band light curves and hardness ratios, and poor constraints from spectral modeling.
We also searched for eclipse events in the 29 blazars that were monitored with *RXTE* and have average 2–10 keV fluxes $>8\times 10^{-12}$ erg cm$^{-2}$ s$^{-1}$. These are sources considered under unification schemes to have jets aligned along the line of sight, with jet emission drowning out emission from the accretion disk and corona. We found no evidence for eclipses from the hardness ratio light curves.[^2] We do however include 3C 273 in our final sample, since its X-ray spectrum is likely a composite of typical Sy 1.0 and blazar spectra (e.g., Soldi [et al.]{} 2008).
As per $\S2.3$ below, we can detect if a source changes from Compton thin to Compton thick. However, we cannot accurately quantify changes in $N_{\rm H}$ where there is already a steady full-covering Compton-thick absorber present. This is due to the energy resolution and bandpass. The presence of Compton reflection in some of the Compton-thick sources observed with *RXTE* (Rivers [et al.]{} 2011a, 2013) combined with the fact that Compton-thick absorption causes a rollover around 10 keV means there is not sufficient “leverage” in the spectrum to fully deconvolve the power law, Compton reflection, and absorption with short exposure times. In such cases, we also cannot detect addition absorption by $10^{22-23}$ cm$^{-2}$. We checked the hardness ratio light curves (see below) of the seven Compton-thick sources monitored with *RXTE*, but there was no significant evidence for such sources becoming Compton thin or unabsorbed; the only variations in hardness ratio were small and consistent with modest variations in the photon index of the coronal power law. We exclude Compton-thick-absorbed sources from our final sample and do not discuss them further.
The final target list consists of 37 type I and 18 Compton-thin type II AGN, listed in Tables \[tab:monit1\] and \[tab:monit2\], respectively. We use the optically classified subtypes from the NASA/IPAC Extragalactic Database as follows: we group subtypes 1.0, 1.2, and 1.5 together into the type I category. We have no Sy 1.8s in our sample. We group subtypes 1.9 (objects where H$\alpha$ is the only broad line detected in non-polarized optical light) and 2 (objects with no broad lines detected in non-polarized optical light) into the type II category. 15 of the 18 type IIs are Sy 2s. In general, it is not entirely clear whether Sy 2s intrinsically lack BLRs or if the BLR in those objects is present but “optically hidden,” manifesting itself only in scattered/polarized optical emission or via Paschen lines in the IR band. However, 9/15 Sy 2s and 2/3 Sy 1.9s show evidence for “optically hidden” BLRs, with references listed in Table \[tab:monit2\]. We adhere to the assumption that BLRs exist in all our objects. Our distinction between type Is and type IIs is thus based on the assumption of relatively increasing levels of obscuration in the optical band, independent of assumptions about system orientation. The average redshift of the type Is is $\langle z \rangle = 0.045$, with all but five objects (3C 273, MR 2251–178, PDS 456, PG 0052+251, and PG 0804+761) having $z<0.100$. For the type IIs, the average redshift is $\langle z \rangle = 0.011$.
The majority of these 55 objects are well studied with most major X-ray missions. Previous publications (e.g., REN02, Patrick et al. 2012) usually find $N_{\rm H} > \sim 5\times 10^{21}$ cm$^{-2}$ for each of the 18 type IIs, with NGC 6251 a possible exception ($N_{\rm H} \la 5 \times 10^{20}$ cm$^{-2}$, Dadina 2007; González-Martín et al. 2009). X-ray obscuration in type Is is generally not as common, and with generally lower columns. For example, 28 of the 37 type Is in our monitored sample overlap with the *Suzaku* sample of Patrick et al. (2012). Those authors model full- or partial-covering neutral or warm absorbers in 21 of the 28 sources, but only 12 have at least one component with $N_{\rm H} > 1
\times 10^{22}$ cm$^{-2}$, with only two of those having neutral absorbers (NGC 4151 and PDS 456).
We define an “object-year” as one target being monitoring for a total of one year including smaller gaps in monitoring due to, e.g., satellite sun-angle constraints or missing individual observations, but excluding lengthy gaps $>75$ per cent[^3] of the full duration, e.g., yearly observing cycles when no observations were scheduled. We estimate totals of 189 and 41 object-years for the type Is and the Compton-thin type IIs, respectively. (These values fall to 169 and 26 object-years when sun-angle gaps are removed.) Consequently, with a total of roughly 230 years of monitoring 55 AGN, and with a wide dynamic range in timescales sampled, the present data set is by far the largest ever available for statistical studies of cloud events in AGN to timescales from days to years.
-------------- ---------------- ---------- ------------------------- --------------- --------------------- -------------------------------------------------
Source Opt. Redshift log($L_{\rm 2-10~keV}$) $D_{\rm min}$ $D_{\rm max}$
name class. $z$ (erg s$^{-1}$) (d) (Tot. gap frac.) Comments
3C 111 BLRG/Sy1 0.0485 43.9 0.41 14.77 yr ($54.3\%$)
3C 120 BLRG/Sy1 0.0330 43.4 0.21 11.14 yr ($32.5\%$)
3C 273 QSO/FSRQ/Sy1 0.1583 45.7 0.37 15.91 yr ($28.2\%$)
3C 382 BLRG/Sy1 0.0579 44.0 0.38 7.59 yr ($99.7\%$)
3C 390.3 BLRG/Sy1 0.0561 43.8 0.66 8.66 yr ($73.6\%$)
Ark 120 Sy1 0.0327 43.4 0.33 5.50 yr ($72.4\%$)
Ark 564 NLSy1 0.0247 42.9 0.21 6.19 yr ($32.6\%$)
Fairall 9 Sy1 0.0470 43.4 0.24 6.32 yr ($16.1\%$) Candidate event, 2001.3
Mkn 335 NLSy1 0.0258 42.7 5.2 0.99 yr ($15.0\%$) $^{\dagger}$
Mkn 766 NLSy1 0.0129 42.5 0.47 10.65 yr ($27.2\%$)
NGC 3783 Sy1 0.0097 42.6 0.37 15.92 yr ($30.1\%$) Double eclipse, 2008.3.
Candidate event, 2008.7
Candidate event, 2011.1
NGC 3998 Sy1/LINER 0.0035 40.8 0.60 0.99 yr ($0\%$) $^{\dagger}$
NGC 4593 Sy1 0.0090 42.3 0.34 10.51 yr ($39.2\%$)
PDS 456 Sy1/QSO 0.1840 44.2 0.20 11.32 yr ($88.9\%$) $^{\dagger}$
PG 0804+761 Sy1 0.1000 43.9 0.91 5.91 yr ($80.2\%$)
PG 1121+343 NLSy1 0.0809 43.4 1.8 0.89 yr ($40.7\%$) $^{\dagger}$
Pic A BLRG/Sy1/LINER 0.0351 43.2 3.3 3.3 d ($0\%$)
PKS 0558–504 RL NLSy1 0.1370 44.3 1.3 14.21 yr ($52.8\%$)
PKS 0921–213 FSRQ/Sy1 0.0520 43.2 0.27 4.6 d ($0\%$) $^{\dagger}$
4U 0241+622 Sy1.2 0.0440 43.6 2.2 31.8 d ($0\%$)
IC 4329a Sy1.2 0.0161 43.2 0.25 11.01 yr ($64.2\%$)
MCG–6-30-15 NLSy1.2 0.0077 42.2 0.20 14.78 yr ($26.1\%$)
Mkn 79 Sy1.2 0.0222 42.8 0.39 11.81 yr ($20.0\%$) Eclipses, 2003.5, 2003.6, & 2009.9
Mkn 509 Sy1.2 0.0344 43.4 0.21 10.24 yr ($71.4\%$) Eclipse, 2005.9
Mkn 590 Sy1.2 0.0264 43.2 5.9 0.80 yr ($0\%$)
NGC 7469 Sy1.2 0.0163 42.7 0.20 13.71 yr ($56.9\%$)
PG 0052+251 Sy1.2 0.1545 44.1 3.0 7.51 yr ($89.2\%$) $^{\dagger}$
MCG–2-58-22 Sy1.5 0.0469 43.6 0.32 160.5 d ($88.7\%$)
Mkn 110 NLSy1.5 0.0353 43.4 0.41 11.81 yr ($35.5\%$)
Mkn 279 Sy1.5 0.0305 43.1 0.26 6.00 yr ($99.2\%$)
MR 2251–178 QSO/Sy1.5 0.0640 44.0 1.7 15.05 yr ($58.\%$) 1996 obsn. during eclipse.
NGC 3227 Sy1.5 0.0039 41.5 0.53 6.92 yr ($36.4\%$) Eclipses 2000–1 (Lamer [et al.]{} 2003)
and 2002.8
NGC 3516 Sy1.5 0.0088 42.3 0.64 14.79 yr ($62.4\%$) Candidate event, 2011.7
NGC 4051 NLSy1.5 0.0023 40.8 0.20 15.69 yr ($0.7\%$)
NGC 4151 Sy1.5 0.0033 42.2 0.21 8.36 yr ($70.8\%$) Var. partial covering $N_{\rm H}\sim$
$10^{23.5}$ cm$^{-2}$ (De Rosa [et al.]{} 2007;
Markowitz in prep.)
NGC 5548 Sy1.5 0.0172 42.9 0.26 15.65 ($26.8\%$)
NGC 7213 RLSy1.5 0.0058 41.7 1.1 3.83 yr ($0\%$)
-------------- ---------------- ---------- ------------------------- --------------- --------------------- -------------------------------------------------
\
2–10 keV luminosities refer to the hard X-ray power-law component, which are corrected for absorption, and are taken from Rivers [et al.]{} (2013). Optical classifications are taken from NED. We collectively refer to type Is as including all Seyfert 1.0s, 1.2s and 1.5s. $^{\dagger}$ denotes that the 10–18 keV band had poor signal-to-noise ratio. $D_{\rm min}$ and $D_{\rm max}$ denote the lengths of the minimum and maximum campaign durations, respectively, where one campaign is defined as a minimum of four observations, with no single gap $>75$ per cent of the duration (see $\S$4), i.e., it does not necessarily mean sustained, regular monitoring for the entire duration. The values in parentheses denote the accumulated fraction of missing time of the single longest campaign due to gaps in monitoring (e.g., sun-angle constraints, or campaigns consisting of only a few observations concentrated into $\sim$days–weeks separated by $\sim$years).
----------------- ------------ ---------- ------------------------- --------------- --------------------- -------------------------------------------
Source Opt. Redshift log($L_{\rm 2-10~keV}$) $D_{\rm min}$ $D_{\rm max}$
name class. $z$ (erg s$^{-1}$) (d) (Tot. gap frac.) Comments
NGC 526a NELG/Sy1.9 0.0192 43.1 0.20 5.9 d ($99.0\%$)
NGC 5506 Sy1.9 0.0062 42.4 0.20 8.39 yr ($38.6\%$) Eclipse, 2000.2; BLR:Pa.(N02)
NGC 7314 Sy1.9 0.0048 41.7 0.30 3.55 yr ($65.0\%$) BLR:Pol.(L04)
Cen A NLRG/Sy2 0.0018 41.6 0.24 15.37 yr ($85.5\%$) Eclipses in 2010–1 and $\sim$2003–4:
Rivers [et al.]{} (2011b),
Rothschild [et al.]{} (2011)
NGC 4258 Sy2/LLAGN 0.0015 40.3 0.34 15.07 yr ($37.2\%$) $^{\dagger}$
ESO 103–G35 Sy2 0.0133 42.4 0.66 0.59 yr ($95.3\%$)
IC 5063 Sy2 0.0113 42.1 150.1 0.81 yr ($99.8\%$) BLR:Pol.(L04)
IRAS 04575–7537 Sy2 0.0181 42.7 5.8 0.61 yr ($81.3\%$)
IRAS 18325–5926 Sy2 0.0202 42.8 0.67 2.07 yr ($98.5\%$) BLR:Pol.(L04)
MCG–5-23-16 Sy2/NELG 0.0085 42.7 0.46 9.63 yr ($99.9\%$) BLR:Pa.(G94)
Mkn 348 Sy2 0.0150 42.5 0.36 1.13 yr ($70.2\%$) Eclipse 1996-7: Akylas [et al.]{} (2002);
BLR:Pol.(M90)
NGC 1052 RLSy2 0.0050 41.2 18.1 4.56 yr ($13.8\%$) $^{\dagger}$; BLR:Pol.(B99)
NGC 2110 Sy2 0.0076 42.2 0.65 2.0 d ($0\%$) BLR:Pa.(R03)
NGC 2992 Sy2 0.0077 42.0 27.9 0.90 yr ($10.3\%$) BLR:Pa.(R03)
NGC 4507 Sy2 0.0118 42.8 1.1 15.2 d ($0\%$) BLR:Pol.(M00)
NGC 6251 LERG/Sy2 0.0247 42.1 7.7 0.99 yr ($0\%$) $^{\dagger}$
NGC 7172 Sy2 0.0087 42.1 5.0 12.2 d ($0\%$)
NGC 7582 Sy2 0.0053 41.3 0.61 1.24 yr ($97.3\%$) BLR:Pa.(R03)
----------------- ------------ ---------- ------------------------- --------------- --------------------- -------------------------------------------
\
Same as Table \[tab:monit1\], but for our sample of type IIs, which includes all Seyfert 1.9s and 2s. “BLR:Pa.” indicates a “hidden” BLR with broad Paschen lines; references are: G94 = Goodrich et al. (1994), N02 = Nagar et al. (2002), R03 = Reunanen et al. (2003). “BLR:Pol.” indicates a “hidden” BLR with broad Balmer lines detected in polarized emission; references are : B99 = Barth et al. (1999), L04 = Lumsden et al. (2004), M90 = Miller & Goodrich (1990), M00 = Moran et al. (2000).
Summary of data reduction and light curve extraction
----------------------------------------------------
We extract light curves and spectra for each observation for each target following well-established data reduction pipelines and standard screening criteria. We use <span style="font-variant:small-caps;">heasoft</span> version 6.11 software. We use PCA background models pca\_bkgd\_cmbrightvle\_e5v20020201.mdl or pca\_bkgd\_cmfaintl7\_eMv20051128.mdl for source fluxes brighter or fainter than $\sim18$ mCrb, respectively. We extract PCA STANDARD-2 events from the top Xenon layer to maximize signal-to-noise. We use data from Proportional Counter Units (PCUs) 0, 1 and 2 prior to 1998 December 23; PCUs 0 and 2 from 1998 December 23 until 2000 May 12; and PCU 2 only after 2000 May 12[^4]. We ignore data taken within 20 min of the spacecraft’s passing through the South Atlantic Anomaly, during periods of high particle flux as measured by the ELECTRON2 parameter, when the spacecraft was pointed within 10$\degr$ of the Earth, or when the source was $>$0.02$\degr$ from the optical axis. Good exposure times per observation ID after screening were usually near 1 ks: 80.2 per cent of the 22,934 observations had good exposure times between 0.5 and 1.5 ks, although a few tens of observations had good exposure times 10–20 ks.
The wide range of sampling patterns (durations, presence of gaps) means that our sensitivity to eclipses of a given duration can vary strongly from one object to the next. We do not split up individual observations. The shortest timescale we consider is 0.20 d, which corresponds to four points separated by three satellite orbits (3 $\times$ 5760 s). Tables \[tab:monit1\] and \[tab:monit2\] list the minimum and maximum durations to which we are sensitive. For example, NGC 4051 was subjected to sustained monitoring with one pointing every $\sim$2–14 d regularly for 15.69 yr. There were also six intensive monitoring campaigns, with the frequencies of individual observations spanning 0.26 to a few days, and with durations spanning 9.6–147 d. The full light curve thus affords us sensitivity to full eclipse events on a virtual continuum of timescales from $D_{\rm min}$ = 0.26 d to $D_{\rm max}$ = 15.69 yr.
We extract continuum light curves, one point averaged over each observation, in the 2–10 keV band and in the sub-bands 2–4, 4–7, and 7–10 keV. We also extract 10–18 keV light curves for the 37 objects with average 10–18 keV flux $\ga 6 \times 10^{-12}$ [ergcm$^{-2}$s$^{-1}$]{}(lower fluxes yield large statistical uncertainties and/or large uncertainties due to background systematics). Errors for each light curve point are obtained by dividing the standard deviation of the $N$ 16 s binned count rate light curve points in that observation by $\sqrt{N}$. We define the hardness ratio $HR1$ as $F_{7-10}/F_{2-4}$. Assuming full-covering absorption, $HR1$ peaks at column densities roughly $1-3 \times 10^{23}$ cm$^{-2}$ while giving us sensitivity down to $\sim$ a few $10^{22}$ cm$^{-2}$, as illustrated in Fig. \[fig:HRplotNH\]. We define $HR2$ as $F_{10-18}/F_{4-7}$, which peaks at full-covering columns of roughly $0.8 - 3 \times 10^{24}$ cm$^{-2}$.[^5]
The model $HR$ values in Fig. \[fig:HRplotNH\] are calculated in <span style="font-variant:small-caps;">xspec</span> assuming neutral gas fully covering a power-law component with $\Gamma$=1.8, an Fe K$\alpha$ emission line with an equivalent width of 100 eV and a Compton reflection component modeled with <span style="font-variant:small-caps;">pexrav</span>. We include $N_{\rm H,Gal}$, assumed to be $3\times 10^{20}$ cm$^{-2}$. Models were calculated at values of $N_{\rm H}$ every 0.1 in the log. Black and red lines denote $HR1$ and $HR2$, respectively. The orange and gray lines denote $HR1$ and $HR2$ when there exists an additional power-law component to represent nuclear power-law emission scattered off diffuse, extended gas; such emission is frequently observed in the $<$5–10 keV spectra of Compton-thick absorbed Seyferts (e.g., Lira et al. 2002). This component is modeled by a power law that is unabsorbed (except by $N_{\rm H,Gal}$). It has the same photon index but a normalization $f = 0.01$ times that of the primary power law. The value of $f$ will vary from one object to the next, but values $\la1 - 5$ per cent are typical for type II Seyferts, e.g., Bianchi & Guainazzi (2007), Awaki [et al.]{} (2008), and Yang [et al.]{} (2009). Modest changes in $f$ do not significantly impact our analysis. For values of $N_{\rm H}$ below $\sim 3 \times10^{23}$ cm$^{-2}$, this component does not strongly affect $HR1$, and $HR2$ is similarly not strongly affected below $\sim
2 \times10^{24}$ cm$^{-2}$. For Compton-thick absorption, $HR1$ returns to values below 2, but $HR2$ remains very high, above $\sim$5, thus breaking the degeneracy in $HR1$.
Limitations and caveats
-----------------------
The PCA’s moderate energy resolution ($E$/$\Delta$$E$ $\sim6$ at 6 keV) means the archive is one of *spectral* monitoring as opposed to simply X-ray photometry. A typical 1–2 ks exposure for a typical flux (1 to a few mCrb) AGN yielded PCA energy spectra covering 3 up to $\sim20$ keV, with 2–10 keV continuum flux constrained to within $\sim2$ per cent, and photon index of the coronal power-law continuum $\Gamma$ constrained to $\pm\sim$0.01–0.02, or $\sim 1$ per cent.
The energy resolution and bandpass of the PCA means that we are sensitive only to full-covering or near-full-covering events. Specifically, for 2–10 keV fluxes less than roughly 3 $\times
10^{-10}$ [ergcm$^{-2}$s$^{-1}$]{}(almost all our sources) and/or for accumulated spectra with less than $\sim10^6$ 2–10 keV counts, we are able to detect an absorption event and rule out spectral pivoting of the power law only if the covering fraction is greater than approximately 80–90 per cent. An exception is NGC 4151 (average $F_{2-10}$ from *RXTE* observations = $1.2\times 10^{-10}$ [ergcm$^{-2}$s$^{-1}$]{}); its X-ray spectrum is frequently modeled with complex absorption using neutral or moderately-ionized partial coverers (e.g., Schurch & Warwick 2002). Time-resolved spectroscopy using both *BeppoSAX* (de Rosa [et al.]{} 2007) and *RXTE* (Markowitz [et al.]{} in preparation), respectively, provides evidence for variable absorption by columns of gas near $1-3 \times
10^{23}$ cm$^{-2}$ and with covering fractions ranging from approximately 30–70 per cent over timescales of months to years. This implies numerous eclipses moving in and out of the line of sight over these timescales. However, NGC 4151 is likely the only type I Seyfert in the sample whose 2–10 keV spectrum is strongly affected by neutral partial covering. To avoid biasing our determination of the number of eclipses for the type I class, we do not count eclipses implied by NGC 4151’s variable partial covering and we limit ourselves to full- or near-full-covering absorption for this study, henceforth assuming the origin of the X-ray continuum to be a point source.
We are not highly sensitive to absorption by highly ionized gas. We estimate that our PCA spectra are sensitive to ionization levels up to log ($\xi$, erg cm s$^{-1}$) $\ga 1-2$ \[$\xi \equiv L_{\rm
ion}/(nr^2)$, where $L_{\rm ion}$ is the luminosity of the ionizing continuum, $n$ is the number density of the gas, and $r$ is the distance from the source of the ionizing continuum to the gas cloud\].
In summary, our study is probing restricted regions of the full parameter space that can be used to quantify circumnuclear absorbing gas in Seyferts. Our findings are complementary to those derived using the archival databases of other X-ray missions. *XMM-Newton*, *Chandra*, and *Suzaku* have higher energy resolution and bandpasses extending to lower energies than the PCA, and can probe events with durations $\la1$ d, (as *XMM-Newton* and *Chandra* observations do not suffer from Earth occultation), lower column densities, partial covering, and/or high-ionization absorption (e.g., Turner [et al.]{} 2008; Risaliti [et al.]{} 2011). However, unlike *RXTE*, these missions generally do not perform sustained monitoring and are not able to detect absorption events longer than $\sim$a few days. The other uniqueness of the *RXTE* archive compared to *XMM-Newton* or *Chandra* is the sensitivity to relatively higher column eclipses thanks to energy coverage $>$10 keV.
Identification of candidate eclipse events
------------------------------------------
In this section, we introduce criteria that we use to identify and confirm eclipse events. In brief, “candidate” events are identified via the light curves of hardness ratios and $\Gamma_{\rm app}$, defined below, but confirmation of increased $N_{\rm H}$ via time-resolved spectroscopy is required to move an event into the “secure” category.
Ideally, a full eclipse event, wherein we can constrain ingress and egress, must consist of (at least) two adjacent observations with elevated values of hardness ratio and $N_{\rm H}$ (see $\S$2.4), plus two observations on either side to define the “baseline” values of hardness ratio and $N_{\rm H}$. In cases where there exist large gaps in monitoring, we can still identify eclipses based on elevated values of hardness ratio and $N_{\rm H}$, but we may lack information on when ingress/egress occurred.
Sudden increases in light curve hardness ratio and/or decreases in overall spectral slope could indicate increased absorption, but such spectral variability can also potentially be caused by variations in the photon index of the coronal power-law component that typically dominates Seyfert X-ray spectra. The overall energy spectra of Seyferts lacking substantial X-ray absorption are frequently observed to flatten as the total 2–10 keV flux lowers (e.g., Papadakis [et al.]{}2002; Sobolewska & Papadakis 2009). Some objects’ long-term 2–10 keV flux behavior is consistent with pivoting of the continuum power law at some energy $>$10 keV (e.g., Taylor [et al.]{} 2003). Other objects are consistent with the “two component” model across the range of fluxes observed; in this model, the power law remains constant in $\Gamma$ but its normalization varies, and the presence of a Compton reflection component with constant absolute normalization causes a flattening in the overall observed spectral slope as power-law flux lowers (e.g., Shih [et al.]{} 2002). In either case, $\Gamma$ is observed to usually be higher than $\sim1.5-1.6$, after accounting for absorption, if present. For example, from a large sample of X-ray-selected type I Seyferts, Mateos [et al.]{} (2010) find a mean photon index of $1.96$ with a standard deviation of 0.27. In black hole X-ray binary systems, thought to also possess a disk and corona surrounding the black hole, $\Gamma$ is also usually observed to not be lower than about 1.5 (e.g., McClintock & Remillard 2003; Done & Gierliński 2005).
We fit the spectrum of each individual observation with a simple power law using <span style="font-variant:small-caps;">xspec</span> version 12.7.1, accounting for absorption *only* by the Galactic column, $N_{\rm H,Gal}$ (Kalberla [et al.]{} 2005). For each spectrum of NGC 4151, Cen A, and NGC 5506, we add systematics of 5, 5, and 3 per cent, respectively. Because we are neglecting the absorption $N_{\rm H}$ in excess of the Galactic column and other commonly observed spectral features such as, Compton reflection and Fe K emission, the power-law index we measure in this way is not a direct measure of the photon index of the power law, but is instead only a general indicator of hard X-ray spectral shape, which we label $\Gamma_{\rm app}$ (“apparent” photon index). A value of $\Gamma_{\rm app}$ lower than 1.5 for objects normally lacking absorption $>10^{22-23}$ cm$^{-2}$, particularly if they do not occur near times of low power-law continuum flux as probed by 2–10 or 10–18 keV flux, very likely indicates an increase in $N_{\rm H}$. Following Mateos [et al.]{} (2010), $\Gamma=1.5$ (1.4) indicates a 1.7$\sigma$ (2.1$\sigma$) deviation from $\langle\Gamma\rangle=1.96$. These values correspond to $HR1 = 0.9-1.2$ (1.0–1.3) and $HR2 =
1.9-3.6$ (2.2–4.7), assuming values of the Compton reflection strength $R$ spanning 0.0–2.0[^6].
Hardness ratios alone cannot confirm an increase in absorption. Time-resolved spectra, in many cases, also lack the statistics to rule out such scenarios, and as discussed below, we sometimes must freeze $\Gamma$ to avoid degeneracy between $\Gamma$ and $N_{\rm H}$. For the purposes of identifying *candidate* eclipse events, we adhere to the assumption that $\Gamma$ intrinsically does not vary to values below 1.4–1.5, and so values of $\Gamma_{\rm app}$ below roughly 1.3 and consequently values of $HR1$ above 1.7 are very likely only due to the presence of significant line-of-sight absorption. We assume that the strength of the Compton reflection hump relative to the power law, also a source of spectral hardening, remains constant over time. For sources that are perpetually X-ray obscured (e.g., the Compton-thin type IIs), we rely on the deviations of $HR1$, $HR2$, and $\Gamma_{\rm app}$ from their mean values to identify potential eclipse events.
Our criteria for a “candidate” eclipse event are thus as follows:
$\bullet$ Criterion 1: trends in $HR1$ (and/or $HR2$ depending on $\Delta$$N_{\rm H}$) that indicate a statistically significant deviation compared to the average spectral state, with a minimum of 2$\sigma$ (standard deviations), and at least a 50 per cent increase above the mean value of $HR1$.
$\bullet$ Criterion 2: at least two consecutive points in a row in the hardness ratio light curve have elevated values. This removes single-point outliers due to statistical fluctuations.
$\bullet$ Criterion 3: trends in $\Gamma_{\rm app}$ that deviate at the $\geq$2$\sigma$ level. ($\Gamma_{\rm app}$ tends to be more statistically noisy than $HR1$, and in most of our candidate events, the deviation in $HR1$ is commonly 1–2$\sigma$ more than the corresponding deviation in $\Gamma_{\rm app}$. Consequently, $HR1$ is our primary selector.) For X-ray unobscured sources, $\Gamma_{\rm app}$ must be lower than 1.3.
To classify an eclipse candidate as “secure,” we impose a fourth criterion:
$\bullet$ Criterion 4: the increase in $N_{\rm H}$ must be confirmed via follow-up time-resolved X-ray spectroscopy in binned spectra.
“Secure” events are divided into “secure A” events, wherein $N_{\rm H}$ and $\Gamma$ are deconvolved in the time-resolved spectra, or “secure B” events, wherein $N_{\rm H}$ has to be determined with an assumed frozen value of $\Gamma$. We define “candidate” events as those satisfying criteria 1–3, but not 4. Such events can be attributed to low signal-to-noise ratio, in terms of either low continuum fluxes or due to the inferred weakness of the event, e.g., values of $N_{\rm H}$ only barely above $1 \times 10^{22}$ cm$^{-2}$. Qualitatively speaking, the candidate events had large uncertainties on $N_{\rm H}$ even when $\Gamma$ is frozen (e.g., errors spanning factors of more than a few) and/or had large uncertainties on $\Gamma$ (e.g., several tenths or more) when modeling no excess absorption.
OVERVIEW OF ECLIPSE RESULTS
===========================
We defer all details on individual objects to Appendix A. There, we provide long-term light curves for sub-band continuum fluxes, hardness ratios $HR1$ and $HR2$, and $\Gamma_{\rm app}$. For brevity, we include light curve plots only for the 10 objects with candidate or secure eclipse events; in the other 45 objects, we find no significant sustained deviations in $HR1$, $HR2$ or $\Gamma_{\rm app}$ at the 2$\sigma$ level or greater. However, we also include plots of selected type II objects with evidence for constant, non-zero absorption. Appendix A contains all the details on the observed deviations in the $HR1$, $HR2$, and/or $\Gamma_{\rm app}$ light curves and the subsequent time-resolved spectroscopy. The reader is referred to Lamer [et al.]{} (2003), Rivers [et al.]{} (2011a), and Akylas [et al.]{} (2002) for NGC 3227/2000–1, Cen A, and Mkn 348, respectively; we present new results of time-resolved spectroscopy for all other events.
In this paper, we confirm a total of 12 “secure (A+B)” X-ray absorption eclipses in eight objects (confirmed with spectral fitting) plus four “candidate” eclipses in three objects, all summarized in Table \[tab:ecl1summ\]. For our 16 secure/candidate events, each with $N_{\rm H} \sim 10^{22-23}$ cm$^{-2}$ (see below), each object is bright enough for us to use the 10–18 keV band to probe the uneclipsed continuum. In Appendix A10, we present flux-flux plots that demonstrate that the 2–4 keV flux is affected independently of the behavior of the 10–18 keV continuum during our secure events and that one can distinguish the spectral variability from a discrete eclipse event from that due to variability in the power law. For candidate events, though, the flux-flux plots are unable to fully separate the two types of spectral variability, and this is tied to ambiguity in modeling the time-resolved spectra. Six of these 12 are complete absorption events, with *RXTE* witnessing both ingress and egress. The events span a range of quality, depending on source brightness, the sampling of the observations, the duration, and $N_{\rm H}$: four events’ column density profiles are well-resolved in time even after binning the observations for time-resolved spectroscopy, allowing constraints on the density profile: Cen A/2010–1, Mkn 348/1996–7, NGC 3227/2000–1, and NGC 3783/2008.3. In contrast, other, more rapid events subtend only two points in the $HR1$ light curve, with only one binned energy spectrum demonstrating increased absorption. Inferred durations range from $<$1 d to a few years, with six secure events’ durations in the $\sim$tens of days range, and three of them $\geq$5 months.
We are sensitive to nearly full-covering absorption by columns in a range of $\sim10^{22-25}$ cm$^{-2}$, but the detected absorption events span only peak column densities of $\sim 4 - 26 \times
10^{22}$ cm$^{-2}$, i.e., we see no evidence from our sample for full-covering absorption by Compton-thick clouds. This is not to say that full-covering Compton-thick eclipse events do not occur in Seyferts in general. In other words, although our survey is sensitive to such events, we do not detect any for our monitored sample.
The average values of $N_{\rm H}$ in log space for the two classes (considering only secure events) are nearly identical: 12 and 10 $\times10^{22}$ cm$^{-2}$. In the cases where the column density profiles can be well accessed by several data points after binning for time-resolved spectroscopy, we see no strong evidence for non-symmetric events (along the direction of motion across the line of sight) such as the “comet” shaped clouds inferred by Maiolino [et al.]{} (2010) in NGC 1365. The $N_{\rm H}$($t$) and $HR1$($t$) profiles of NGC 3783/2008.3, however, are particularly intriguing: they suggest two peaks separated by $\sim$11 d, but with the values in between the peaks not returning to post/pre-eclipse levels. The possibility of a double-cloud absorption event is discussed further in $\S$\[sec:3783double\]. In Table \[tab:ecl1summ\], we list parameters both for the full duration and for each clump separately.
Surprisingly, only three type II objects show cloud events. That is, based on their $HR$ light curves, the majority of all other monitored type IIs display no strong evidence (sustained trends in $HR$ more than $50$ per cent above the mean value) for variations in $N_{\rm
H}$, although several were individually monitored for several years or more in some cases. In $\S$5.6, we will focus on the nine type II objects that have sustained monitoring for $\geq$0.6 yr. We will show that seven of them lack long-term ($>$1 d) variations in $N_{\rm H}$ down to $\Delta$$N_{\rm H} \sim 1-9 \times 10^{22}$ cm$^{-2}$ depending on the signal-to-noise ratio and discuss the applicability of clumpy-absorber models to these objects.
In the case of Cen A, in addition to the two eclipses identified, we present in Appendix A evidence (at $\sim$2.2$\sigma$ confidence) that the baseline level of $N_{\rm H}$ dipped by $\sim14$ per cent and then recovered during the first three months of 2010. The reader should bear in mind, however, that Cen A is a radio galaxy and no BLR has been detected, so it may not be representative of all Seyferts, most of which are radio quiet. This observation and its implications are discussed in $\S$5.6.1.
We caution that “peak $N_{\rm H}$”refers only to what we have measured with time-resolved spectroscopy. Since clouds are not point sources, we can only probe that two-dimensional slice of the cloud that transits the line of sight, and there may exist other lines of sight outside that slice which intersect parts of the cloud with higher columns. Consequently, the intrinsic maximum column density of the cloud could be greater than what we measure if *RXTE* was not monitoring the source when that part of the cloud transited the line of sight.
The durations we measure also refer only to that slice of the cloud intersected by the line of sight. In the case of a spherically-symmetric cloud, observed eclipse durations will be, on average, $\pi/4$=0.79 times the maximum durations that could have been observed. (Consequently, average inferred cloud diameters, estimated in $\S$\[sec:diams\], may be underestimated by this modest factor.) In the case of spherical clouds with maximum column density at the center and radial density profiles similar to those for Cen A/2010–1 and NGC 3227/2000–1, then the corresponding effect on peak $N_{\rm H}$ will likely be less than $\sim20$ per cent.
[lllllll]{} Source & & & & Duration & Peak $N_{\rm H}$ &\
name & Type & Event & Category & (d) & ($10^{22}$ cm$^{-2}$) & Comments\
\
NGC 3783 & Sy1 & 2008.3 & Secure B & 14.4–15.4 & $11.2^{+1.7}_{-1.5}$ & $N_{\rm H}$($t$) resolved.\
& & & & $\sim$9.2 & $\sim$4.6 & $11.2^{+1.7}_{-1.5}$ & $8.6^{+1.5}_{-1.3}$ &\
Mkn 79 & Sy1.2 & 2003.5 & Secure B & 12.0–39.4 & $14.4^{+4.8}_{-4.2}$$^{\ddagger}$ & Ingress only.\
& & 2003.6 & Secure B & 34.5–37.9 & $11.5^{+3.2}_{-2.8}$ &\
& & 2009.9 & Secure B & 19.6–40.0 & $7.6\pm2.2$ &\
Mkn 509 & Sy1.2 & 2005.9 & Secure B & $26-91$ & $8.8\pm1.7$$^{\ddagger}$ & Ingress only. $N_{\rm H}$($t$) resolved.\
MR 2251–178 & Sy1.5/QSO & 1996 & Secure A & 3 – 1641 & $6.6^{+0.8}_{-1.4}$$^{\ddagger}$ & Egress before Jun. 1998\
NGC 3227 & Sy1.5 & 2000–1 & Secure A & 77–94 & 19–26 & $N_{\rm H}$($t$) resolved.\
& & 2002.8 & Secure B & 2.1–6.6 & $13.3^{+2.6}_{-2.2}$ &\
\
Cen A & NLRG/Sy2 & $\sim$2003–4 & Secure A & $356-2036$ & 8$\pm$1$^{\ddagger}$ &\
& & 2010–1 & Secure A & $170.2-184.5$ & 8$\pm$1 & $N_{\rm H}$($t$) resolved.\
NGC 5506 & Sy1.9 & 2000.2 & Secure A & 0.20–0.80 & $4.0\pm1.4$ &\
Mkn 348 & Sy2 & 1996–7 & Secure A & $399-693$ & $18\pm3$$^{\ddagger}$ & Egress only. $N_{\rm H}$($t$) resolved.\
\
Fairall 9 & Sy1 & 2001.3 & Candidate & 5.4–15.0 & $<21$ &\
NGC 3783 & Sy1 & 2008.7 & Candidate & 17–28 & $<4$ &\
& & 2011.2 & Candidate & 4.1–15.8 & $10.9^{+6.0}_{-5.3}$ &\
NGC 3516 & Sy1.5 & 2011.7 & Candidate & $\sim57$ & 4.7 & Possible variation in covering fraction\
& & & & & & of partial covering, moderately\
& & & & & & highly ionized absorber\
\
Summary of the 12 secure and 4 candidate eclipse events detected. We list events in the order of: secure events in type Is, secure events in type IIs, and then candidate events (which are all for type I objects). “$N_{\rm H}$($t$) resolved” means that at least several consecutive binned spectra confirm increased $N_{\rm H}$ levels. The double-dagger ($^{\ddagger}$) indicates that intrinsic peak value of $N_{\rm H}$($t$) may be higher than that observed if it occurred during a gap in monitoring.
PROBABILITIES FOR OBSERVING AN X-RAY ECLIPSE EVENT
==================================================
Ideally, we would like to derive the instantaneous probability $\overline{P_{\rm ecl}}$ of catching a given source while it is undergoing an X-ray eclipse event, and then relate that probability to the source’s optical classification and/or the constant presence/absence of X-ray-absorbing gas along the line of sight. Eclipse events detected in the *RXTE* archive are, however, evidently rare, with only 10 different objects showing events and with only 1–3 events in each of those objects. We will thus focus in this section on the average instantaneous probability to detect absorption due to an eclipse event for a given *class* of objects as opposed to individual objects. However, deriving such a probability is not straightforward, because we must factor in biases resulting from our observation sampling, which yields a sensitivity to eclipse durations that is very heterogeneous as a function of timescale, both for object classes and for individual objects. For sustained monitoring, for instance, we are sensitive to a relatively larger number of short-duration eclipses than to long-duration eclipses. For example, given a hypothetical campaign consisting of one observation daily for 64 d, we can detect a maximum of 16 full eclipses of duration 3 d (4 points), 8 full eclipses of duration 7 d (8 points), etc.
We thus quantify an instantaneous probability *density* $p_{\rm ecl}$ as a function of eclipse duration as follows: we first define for each individual object a “selection function” $SF_{\rm ind}$ to quantify our sensitivity, as a function of eclipse duration, to the maximum total number of eclipses which *could* have been potentially observed, given that object’s sampling with *RXTE*; this procedure is described in $\S4.1$. We then produce summed selection functions $SF_{\rm sum I,II}$ to quantify the average sensitivity to the total number of eclipses for each object class (type I and II), as well as to identify potential biases affecting a given object class as a whole. In $\S4.2$, we quantify the total number of eclipse events $N_{\rm ecl}$ actually observed within each object class as a function of eclipse duration. Then, in $\S4.3$, as we have binned our observed events and the selection functions (maximum possible event number) on to the same grid as a function of eclipse duration, we divide $N_{\rm ecl}$ by $SF_{\rm sum I,II}$ to obtain the instantaneous probability density $p_{\rm ecl}$($t_{\rm i}$). This quantifies the likelihood to witness an eclipse as a function of that eclipse’s duration, defined over a discrete set of timescale bins $t_{\rm i}$. Finally, we integrate $p_{\rm ecl}$($t_{\rm i}$) over all timescale bins to obtain $\overline{P_{\rm ecl}}$, the instantaneous probability of witnessing a source in eclipse for *any* eclipse duration.
As a reminder, $\overline{P_{\rm ecl}}$ and $p_{\rm ecl}$($t_{\rm i}$) do not necessarily denote the likelihood to simply observe a source with non-zero X-ray obscuration; this holds true *only* if the source normally devoid of X-ray-absorbing gas along the line of sight. $\overline{P_{\rm ecl}}$ and $p_{\rm ecl}$($t_{\rm i}$) refer to the likelihood of catching the source in state with a higher-than-usual value of $N_{\rm H}$ at any instant due specifically to a discrete (localized in time) eclipse event by a cloud of gas, one that transits the line of sight with an observed duration between $t_{\rm i}$ and $t_{\rm i+1}$ in the case of $p_{\rm ecl}$($t_{\rm
i}$). In $\S$5.5, we will compare our estimates of $\overline{P_{\rm ecl}}$ for each class to the predictions for a clumpy torus to cause a given source to be observed in eclipse.
Selection functions
-------------------
This section describes how we generate “selection functions” for each individual object $SF_{\rm ind}$($t_{\rm i}$) and sum selection functions for each object class $SF_{\rm sumI,II}$($t_{\rm i}$) to quantify the number of light curve observation segments (“campaigns”) of a given duration as a function of event duration. Here we define a “campaign” as consisting of a minimum of four observations, with no single gap between adjacent observations being greater than 75 per cent of the duration.[^7]
We define 19 time bins $t_{\rm i}$ spanning from 0.20 to 5850 d, with time bins equally spaced by 0.235 in log space; in linear space, the $i$th time bin is defined by $0.20\times$($1.718^{i}$) d $\leq t <
0.20\times$($1.718^{i+1}$) d, with $i = 0 ... 18$. $SF_{\rm
ind}$($t_{\rm i}$) effectively tells us the potential maximum number of eclipse events of duration $t_{\rm D}$ satisfying $t_{\rm i} \leq
t_{\rm D} < t_{\rm i+1}$ that *RXTE* was capable of potentially catching, given the observed sampling for that object.
The $SF_{\rm ind}$($t_{\rm i}$) are constructed as follows: consider a light curve with $J$ total observations (data points), and observation times denoted by $X_{j=1}$ ... $X_{j=J}$. For each time bin $t_{\rm i}$, we start at $j=1$ and we identify the first light curve segment $X_{1} .. X_{q} $ whose duration $t_{\rm D}$ satisfies $t_{\rm i} \leq t_{\rm D} < t_{\rm i+1}$. We require a minimum of four data points, since we defined above an eclipse event relying on a minimum of two consecutive points to form a significant peak in the $HR$ light curves, plus one point before and one point after the putative eclipse to denote the “baseline” levels of $HR$. The goal is to have consistency between detecting real eclipse events and how we determine the maximum possible number of potential eclipse events. Consequently, we disqualify a segment if any gap between two adjacent points is $>75$ per cent of the segment’s duration. If the segment $X_{1} ... X_{q}$ satisfies these criteria, then $SF_{\rm
ind}$($t_{\rm i}$) = $SF_{\rm ind}$($t_{\rm i}$) +1, and we restart counting from point $X_{q+1}$. Otherwise, we restart from $j=2$.
When there existed sustained monitoring with a sampling interval $\Delta$$t$ for a duration $t_{\rm D}$, we are sensitive to eclipse events on a continuous range of timescales from $4\Delta$$t$ up to $t_{\rm D}$, with $SF_{\rm ind}$($t_{\rm i}$) increasing towards shorter timescales. *RXTE* monitored 42 sources continuously for $\sim$1 yr and longer (sun-angle gaps notwithstanding), and one of the most common sampling times for these programmes was 3–4 d. Many objects’ selection functions thus feature peaks in the 8.8–15.2 d time bin, with a sharp drop-off in $SF_{\rm ind}$($t_{\rm i}$) below 8.8 d. In addition, there were multiple intensive-monitoring programmes featuring observations several times daily for durations of days to weeks. These programmes lead to large contributions in $SF_{\rm ind}$($t_{\rm i}$) from $\sim$1 to several days. As an example, we consider the case of Mkn 79, which had monitoring observations every $\sim$10 d for a duration of 309 d (MJD 51610–51919), every $\sim$2 d for 15.6 d (MJD 51754.5–51770.1), every $\sim$2 d for 3205 d (MJD 52720–55925) with $\sim$26 d gaps once a year due to sun-angle constraints, and every $\sim6$ hr for 64.6 d (MJD 53691.4–53756.0). The $F_{2-10}$ light curve and the corresponding selection function are plotted in Figs. \[fig:mkn79flux\] and \[fig:mkn79selfxn\], respectively.
We create summed selection functions for each object class as a function of timescale bin ($SF_{\rm sumI,II}$($t_{\rm i}$)) by summing up the individual selection functions for the 37 type I and 18 type II AGN separately (left- and right-hand panels of Fig. \[fig:selfxnALL\]). We see that the number of campaigns per given timescale is on average 3.9 times greater for the type Is than for the type IIs. For both classes, the most common campaign timescales are $\sim$tens of days.
Due to the heterogeneous sampling in the archive, some objects can contribute as many as 200–300 campaigns to an individual $t_{\rm i}$ bin (usually the 8.8–15.2 d bin) while other objects were observed much more sparsely and yield selection functions containing only a few campaigns to a few timescale bins. To estimate the uncertainty on each binned $SF_{\rm sumI,II}$($t_{\rm i}$) point stemming from the varying contributions of individual source selection functions, we employ a Monte Carlo bootstrap procedure. For each object class (type I or type II) of $N_{\rm S}$ objects, we do the following $m=1000$ times: we select at random one individual selection function from the pool of observed $SF_{\rm ind}$ functions, until a total of $N_{\rm
S}$ individual selection functions was accumulated. We explicitly allow for individual objects to be selected multiple times. We sum those to create a simulated summed selection function $SF^{i}_{\rm
sum}$, where $i \in {1 ... m}$. Once we have $m$ $SF^{i}_{\rm sum}$ functions, we calculate the standard deviation within each bin to yield the relative uncertainties plotted in Fig. \[fig:selfxnALL\]. This procedure yields typical bin uncertainties of $\sim$12 and 25 per cent for type Is and IIs, respectively.
We also plot for each object class the number of different objects that contribute to a given timescale bin (Fig. \[fig:numobjpertimescaleALL\]). These plots can identify those timescales over which biases arise because only a small number of different objects contribute to the relevant eclipse duration bin. For both classes of objects, the number of objects contributing to a given timescale is roughly constant from $\sim$1 to $\sim$300 d, but drops off rapidly above $\sim$300 d in the type IIs. In fact, for the type IIs, timescales above 400 d are probed by only 3–5 objects. Especially in the case of type IIs, we are thus relying on the assumption that these few objects (Cen A, NGC 1052, NGC 4258, NGC 5506, and NGC 7314) are truly representative of the whole class in terms of their variable X-ray absorption properties, since they strongly influence our inferences about variable absorption in type IIs on these long timescales.
Number of observed events per timescale bin
-------------------------------------------
In Fig. \[fig:durbars\], we plot the number of eclipse events as a function of each of their durations, using the same 19 timescale bins as above. In the upper panel for each class, each eclipse event is depicted by a separate symbol/bar. In these panels we plot all eclipse events independent of the data quality of the eclipse (13 secure A+B and 3 candidate events). In the lower panel for each class we plot histograms denoting the number of observed eclipses as a function of timescale bin $N_{\rm ecl}$($t_{\rm i}$). The black solid line is the “best estimate” of $N_{\rm ecl}$($t_{\rm i}$), using only the best-estimate values of the durations and ignoring the candidate events.
We also present estimates of $N_{\rm ecl}$($t_{\rm i}$) that correspond to the maximum and minimum likelihoods of witnessing sources in eclipse, given that an event’s contributions to $N_{\rm
ecl}$($t_{\rm i}$) may shift in $t_{\rm i}$ given the measured uncertainties in the observed duration. For each eclipse event, we thus considered those timescale bins $t_{\rm i}$ consistent with the limits of the observed duration (as identified in Fig. \[fig:durbars\]), and identified which one of those timescale bins $t_{\rm i}$ had the highest corresponding value of $SF^{i}_{\rm
sumI,II}$ (lowest probability associated with one single eclipse event). The orange line denotes the final $N_{\rm ecl}$($t_{\rm i}$) histogram corresponding to the minimum probability densities, accumulated over all secure events within each class. The green solid line in each lower panel, meanwhile, denotes $N_{\rm ecl}$($t_{\rm
i}$) corresponding to the *highest* probability densities, computed in a similar fashion as the orange solid line, but this time including the candidate events as well. For type I AGN, the “typical” observed duration timescale is tens of days, although this may not be surprising given the strong peak of the selection function at tens of days. The case of MR 2251–178 is very exceptional, as we detect clear signs for a secure eclipse event but have only poor constraints on its duration ($3$ d $ < t < $ 4.5 yr). If we ignore this event, we do not have any type I eclipses that are consistent with an event duration longer than 100 d despite the numerous campaigns lasting for several hundreds of days to several years. Even though the shapes of the selection functions for type I and IIs are extremely similar, for type II objects, we do not detect any eclipse events with durations in the range 1–100 d. The three type II eclipses with durations of $>$100 d have more than five times fewer observing campaigns than between 10 and 30 d where the maximum of the summed selection function for type II objects occurs. In other words, the histogram of type I eclipses agrees well with the expected distribution based on the summed selection function of type I objects, while type II eclipses do not appear at the expected duration if only their summed selection function is considered.
Calculating the instantaneous eclipse probability densities $p_{\rm ecl}$($t_{\rm i}$) and the instantaneous eclipse probability $\overline{P_{\rm ecl}}$ {#sec:sect43}
---------------------------------------------------------------------------------------------------------------------------------------------------------
Given the values of $N_{\rm ecl}$($t_{\rm i}$) corresponding to the best estimates of duration, we can now estimate the corresponding function of $p_{\rm ecl}$($t_{\rm i}$) for each class by dividing each bin of $N_{\rm ecl}$($t_{\rm i}$) by $SF_{\rm sumI,II}$($t_{\rm i}$). The resulting function of $p_{\rm ecl}$($t_{\rm i}$) for each class is plotted as the black solid lines in Fig. \[fig:durprobs\]. As a reminder, $p_{\rm ecl}$($t_{\rm i}$) describes the likelihood to catch a source undergoing an eclipse due to one specific kind of cloud: one that results in an eclipse with a duration $t_{\rm i} \leq
D < t_{\rm i+1}$. The errors on individual points of $p_{\rm ecl}$($t_{\rm i}$) take into account the error in the selection function obtained from the bootstrapping method only.
The minimum- and maximum-likelihood $p_{\rm ecl}$($t_{\rm i}$) functions are calculated in a similar way, although we divide by $SF_{\rm sumI,II}$($t_{\rm i}$) $\pm$ $\sigma_{\rm SF}$($t_{\rm
i}$), where $\sigma_{\rm SF}$($t_{\rm i}$) denotes the uncertainties in $SF_{\rm sumI,II}$($t_{\rm i}$) as determined by the Monte Carlo bootstrap method; the minimum and maximum functions are, respectively, the orange and green solid lines in Fig. \[fig:durprobs\]. Because $p_{\rm ecl}$($t_{\rm i}$) is in log space, any bin with zero probability density is plotted as $10^{-6}$. “Typical” values of $p_{\rm ecl}$($t_{\rm i}$) when eclipses are detected are $\sim10^{-(2.8-3.4)}$ for type Is and $\sim10^{-(1.3-2.2)}$ for type IIs. We must caution the reader that most probability density values are on the order of 1/$SF_{\rm
sum}$($t_{\rm i}$). We display 1/$SF_{\rm sum}$($t_{\rm i}$) in Fig. \[fig:durprobs\] via cyan and magenta lines for types I and II, respectively. Consequently, the detection of one eclipse within these duration bins would strongly influence the inferred $p_{\rm
ecl}$($t_{\rm i}$) profile. Nonetheless, the data suggest that eclipses of any duration generally occur more frequently in type IIs.
In type IIs, we do not detect events of durations $\sim$tens of days, for which one event for a given timescale bin would have yielded a probability density of $\sim$ 1/200 – 1/1000. If it were the case that eclipses with durations of $\sim$tens of days occurred in type IIs with the same probability density as in type Is, $\sim10^{-(2.8-3.4)}$, our observations would likely not have detected them. This is because the values of those $p_{\rm ecl}$($t_{\rm i}$) points would lie below the detection threshold for eclipses in type IIs, represented by 1/$SF_{\rm sumII}$($t_{\rm i}$), the magenta line in the lower panel of Fig. \[fig:durprobs\]; recall that there were $\sim$3-4 times fewer campaigns for type IIs as there were for type Is.
In type Is, we did not detect events of durations $\sim$hundreds of days (although such a duration cannot be ruled out in the case of MR 2251–178), for which one event for a given timescale bin would have yielded a probability of $\sim$ 1/200 – 1/1000. If it were the case that eclipses with durations of $\sim$hundreds of days occurred in type Is with the same probability density as in type IIs, then the number of monitoring campaigns for type Is would have clearly allowed us to detect them. If this were the case, then the resulting $p_{\rm
ecl}$($t_{\rm i}$) values for type Is would appear in the upper panel of Fig. \[fig:durprobs\] with values higher than the detection threshold, represented by 1/$SF_{\rm sumI}$($t_{\rm i}$), the cyan lines.
Finally, we sum up the values of $p_{\rm ecl}$($t_{\rm i}$) over all timescale bins to obtain estimates of $\overline{P_{\rm ecl}}$. More specifically, best-estimate, minimum and maximum values of $\overline{P_{\rm ecl}}$ can be obtained by summing the functions of $p_{\rm ecl}$($t_{\rm i}$) denoted by the black, orange, and green histogram lines, respectively, in Fig. \[fig:durprobs\]. However, we must bear in mind that we have a very small number of total events, and many timescale bins contain only $\sim0-1$ events. When summing to obtain the maximum value of $\overline{P_{\rm ecl}}$, we opt to be conservative and take into account the selection function, the reciprocal of which denotes the likelihood of witnessing just one event with duration $t_{\rm D}$ satisfying $t_{\rm i} \leq D < t_{\rm
i+1}$. For each $t_{\rm i}$, we use the maximum of either $1/SF_{\rm
sumI,II}$($t_{\rm i}$) (cyan/magenta lines in Fig. \[fig:durprobs\]) or $p_{\rm ecl}$($t_{\rm i}$) (green line). To reiterate, our uncertainties on $\overline{P_{\rm ecl}}$ are conservative estimates in that they take into account the effect of uncertainties in duration on $N_{\rm ecl}$($t_{\rm i}$), the uncertainties in $SF_{\rm sumI,II}$($t_{\rm i}$) as determined by the Monte Carlo bootstrap method, the effects of the selection function, and the presence of candidate events on the maximum probability. Our final estimate for $\overline{P_{\rm ecl}}$ for type Is is 0.006 with a range of 0.003–0.166 (minimum – maximum probability). For type IIs, $\overline{P_{\rm ecl}}$=0.110, with a range of 0.039–0.571. That is, based on our best estimates, type II AGN have a $\sim18$ times higher chance of showing an eclipse (of any duration). These values are summarized in Table \[tab:finalprobs\], and we return to these values later in $\S$5.
We caution that these probabilities refer only to eclipses by full-covering, neutral or lowly-ionized clouds with columns densities $\ga 10^{22}$ up to $\sim10^{25}$ cm$^{-2}$; when one considers the full range of clouds (larger range of $N_{\rm H}$, partial-covering clouds, wider range of ionization) the resulting probabilities will almost certainly be higher.
---------- -------------------------- -------------------------- ---------------------------------------
Object $\overline{P_{\rm ecl}}$ $\overline{P_{\rm ecl}}$ Limits on
class (Best (Min.–max. $\overline{P_{\rm ecl}}$ for Compton-
estimate) range) thick events
Type Is 0.006 0.003–0.166 $<0.158$
Type IIs 0.110 0.039–0.571 $<0.520$
---------- -------------------------- -------------------------- ---------------------------------------
: Inferred probabilities $\overline{P_{\rm ecl}}$ to witness a source undergoing an eclipse event[]{data-label="tab:finalprobs"}
Ramos Almeida et al. (2011) noted that the IR SEDs of Sy 1.8–1.9 AGN had spectral slopes intermediate between those of Sy 1.0–1.5s and Sy 2s, and noted similarities in SED fitting parameter results between the 1.8–1.9s and the 1.0–1.5s. However, we simply do not have enough events to break up the probability estimates for type IIs into further subclasses, and cannot address this point. If we break up the type Is into Sy 1.0s/1.2/1.5s, tantalizingly, the best-estimate values for $\overline{P_{\rm ecl}}$ increase with subclass: 0.0003, 0.0024, and 0.0036, consistent with the notion that relatively higher levels of obscuration exist in higher numbered subclasses. However, the (conservatively-determined) minimum and maximum values indicate nearly identical ranges (\[0.0003-0.1586\], \[0.0014-0.1597\], and \[0.0013-0.1619\]) because with only $\sim0-1$ events per timescale bin, the maximum values tended towards the integral of $1/SF_{\rm
sum1.0/1.2/1.5}$($t_{\rm i}$). Furthermore, any conclusions would be based on a mere 1–4 secure eclipse events and 1–3 objects per subclass, and so we do not address the subclasses further.
In $\S$3, we noted that we did not detect any Compton-thick obscuration events, but this is not to say with full certainty that such events cannot exist. A conservative upper limit on the probability to observe a Compton-thick event of any duration between 0.2 d and 16 yr is obtained by summing $1/SF_{\rm sumI,II}$($t_{\rm
i}$): $\overline{P_{\rm ecl}} < 0.158$ (type Is) or $< 0.520$ (type IIs).
DISCUSSION
==========
There has been much work so far into variable X-ray absorption in AGN. The accumulated research so far, including this work, has revealed eclipse events spanning a wide range of observed durations, ionization levels, and including full or partial covering.
Most of the short-term events (durations $\la 1-3$ d) have been observed in the prominent case of the Sy 1.8 NGC 1365, where rapid and strong variations in $N_{\rm H}$ are observed in a very large fraction of observations. The clouds are inferred to exist in the BLR and are frequently modeled via variations in covering fraction of partial-covering material (e.g., De Rosa et al. 2007; Turner et al.2008; Risaliti et al. 2011). In the context of clumpy-absorber models, there can exist a variable number of clouds in the line of sight at any given time, with a relatively extended X-ray continuum source located behind them.
*RXTE*, in contrast, has detected full-covering, cold or at most modestly ionized clouds. An emerging picture is that clouds in the BLR and clouds in the torus may be two manifestations of the same radially extended cloud distribution, existing inside and outside, respectively, the dust sublimation region (Elitzur 2007). That is, in the simplest picture, the full-covering clouds detected with *RXTE* are part of the same population of clouds as those causing the short-duration events, just occurring at larger radii, as we will demonstrate in $\S$5.2.
It is potentially interesting that 4 of the 10 sources in our sample with secure or candidate eclipse events have multiple such events (certain objects thus seem to be prone to a higher frequency of events than others). The high frequency of events observed in NGC 1365 may potentially be explained by a favorable geometry/viewing angle. However, NGC 1365 may be a special object; a statistical sample on a large number of AGN may be better suited to quantify eclipse events across all AGN.
Dependence of cloud events on AGN physical parameters
-----------------------------------------------------
Having identified secure and candidate eclipses in 10 of the 55 objects monitored with *RXTE*, we can attempt to explore if certain system parameters govern the presence or lack of eclipsing clouds. For example, the 10 sources’ black hole masses (see $\S$5.2) span log($M_{\rm BH}$) = 6.9–8.8, and these values are not extreme compared to black hole masses for Seyferts/quasi-stellar objects (QSOs) in general or our whole sample.
There is no evidence that having an extreme value of luminosity governs the presence/lack of eclipses: the range of log($L_{2-10}$) and log($L_{\rm Bol}$) where eclipses are confirmed with *RXTE* to occur are 41.93–44.73 and 42.9–45.6, respectively.[^8] Again, these values are not extreme.
In the context of clouds being produced in a disk wind, Elitzur & Ho (2009) estimate that such winds cannot exist when the bolometric luminosity drops below $5 \times 10^{39}$ ($\frac{M_{\rm
BH}}{10^7{\ensuremath{\rm M_\odot}\xspace}}$)$^{2/3}$ erg s$^{-1}$. However, we cannot test this with our sample, as the lowest-luminosity objects monitored with *RXTE* have 2–10 keV luminosities of $2-6\times 10^{40}$ erg s$^{-1}$ (NGC 4258, NGC 3998, and NGC 4051).
Accretion rate relative to Eddington, $L_{\rm
Bol}/L_{\rm Edd}$, is also not an obvious factor; values of $\dot{m}$ for the 10 objects span $\sim0.1$ per cent for Cen A up to $\sim16$ per cent for MR 2251–178, with most other objects’ values in the range of 3–10 per cent. These values are typical for Seyferts in general as well as for our sample.
Can radio loudness be a factor? One of our eight sources with secure eclipses is radio loud, compared to 13/55 (24 per cent) sources in the original sample. If we start with 55 objects, 13 of which are radio loud, and select at random eight of the 55 for a sub-sample, the probability $P(X=k)$ that exactly $k$ out of eight sources in the subsample will be radio loud is given by the hypergeometric distribution $$\label{eq:hgd}
P(X=k) = {K \choose k} {N-K \choose n-k} \Big/ {N \choose n}\ ,$$ where $N=55$ is the population size, $K=13$ is the number of successes in the population (radio-loud sources), and $n=8$ and $k$ are the sample size and the number of successes in the sample, respectively. ${\cdot \choose \cdot}$ is a binomial coefficient. The chance of having exactly $k=0, 1, 2, 3$ or $4$ radio-loud sources in a random sample of eight is $P(X=0, 1, 2, 3,$ or $ 4) =$ (0.10, 0.29, 0.34, 0.20, or 0.07). In other words, it is almost as likely to have exactly 1/8 radio-loud sources in the sub-sample as it is to have 2/8 radio-loud sources. There is thus no significant statistical evidence that our sub-sample of sources with secure eclipses is unusual compared to the original sample in terms of fraction of radio-loud sources.
In summary, we find no evidence for a strong dependence of the presence of eclipse events as a function of the most common AGN parameters. One has to keep in mind that only strong correlations could have been detected with our relatively small number of observed events. That being said, the reader is reminded that with a high dynamic range of sampling for 55 objects totaling 230 object-years of monitoring, this study is by far the largest available data set for testing the environment close to supermassive black holes via eclipse events over timescales longer than a few days.
Locations of X-ray obscuring clouds {#sec:locations}
-----------------------------------
Constraints on the distance from the X-ray continuum source to each cloud $r_{\rm cl}$ can be derived from the cloud’s ionization level, column density, and eclipse duration. Following, e.g., $\S$4 of Lamer [et al.]{} (2003), we assume for simplicity that each cloud has a uniform density and ionization parameter, and that clouds are in Keplerian orbits. The cloud diameter $D_{\rm cl}$ = $N_{\rm}/n_{\rm H} = v_{\rm cl}t_{\rm D}$, where $t_{\rm D}$ is the cloud’s crossing time across the line of sight and $v_{\rm cl}$ is the transverse velocity, equal to $\sqrt{GM_{\rm BH}/r_{\rm cl}}$. Solving for $r_{\rm cl}$ and using the definition of $\xi$, one obtains (see, e.g., Eqn. 3 of Lamer [et al.]{} 2003): $r_{\rm cl} = 4\times10^{16} M_7^{1/5} L_{42}^{2/5} t_{\rm D}^{2/5} N_{\rm H,22}^{-2/5} \xi^{-2/5}$ cm, where $M_7 = M_{\rm BH}/10^6 {\ensuremath{\rm M_\odot}\xspace}$, $L_{42}=L_{\rm ion}/$($10^{42}$ erg s$^{-1}$), $N_{\rm H,22}=N_{\rm H}/$($10^{22}$ cm$^{-2}$), and $t_{\rm D}$ is in units of days.
Table \[tab:ecl1dist\] lists the inferred values of $r_{\rm cl}$, obtained as follows: we use estimates of $M_{\rm BH}$ from reverberation mapping where available; otherwise, estimates are from stellar kinematics or gas dynamics, based on empirical relations between optical luminosity and optical line widths, or from modeling reprocessing in accretion disks. The references are listed in column 4 of Table \[tab:ecl1dist\].
For all objects except Cen A, we estimate $L_{\rm ion}$ as follows: we use the best-fitting unabsorbed power law from Rivers et al. (2013), and extrapolate to the 0.1–13.6 keV range to estimate the luminosity $L_{\rm 0.1-13.6 keV}$. The luminosity below $\sim$0.1 keV, however, is expected to contain significant contributions from the thermal accretion disk continuum emission. Vasudevan & Fabian (2009) estimate that the ratio of the 0.0136–0.1 keV luminosity to the bolometric luminosity, $L_{\rm 0.0136-0.1 keV}/L_{\rm Bol}$, ranges from 0.2 to 0.6 for values of $L_{\rm Bol}/L_{\rm Edd}$ ranging from 0.01 to 0.6, respectively. We take values of $L_{\rm Bol}$ and $L_{\rm Bol}/L_{\rm
Edd}$ from Vasudevan et al. (2010) when available. However, for Fairall 9 the value of $L_{\rm Bol}$ is taken from Vasudevan & Fabian (2009), and for MR 2251–178, we use the 2–10 keV flux from Rivers et al. (2013) and we estimate $L_{\rm Bol}/L_{\rm Edd} = 40$ from Marconi et al. (2004). We then assume for simplicity a linear dependence of log10($L_{\rm 0.0136-0.1 keV}/L_{\rm Bol}$) on log10($L_{\rm Bol}/L_{\rm Edd}$) to estimate $L_{\rm 0.0136-0.1 keV}$, and add that to $L_{\rm 0.1-13.6 keV}$ to obtain the values of $L_{\rm
ion}$ listed in Table \[tab:ecl1dist\]. The broadband SED of Cen A is more like that of blazars than radio-quiet Seyferts, with a broad inverse-Compton hump dominating the emission from the optical band to gamma-rays. We assume $\Gamma=1.84$ (Rivers et al. 2013) over 0.4–13.6 keV, breaking to $\Gamma=1.0$ over 0.0136–0.4 keV (Steinle 2010; Roustazadeh & Böttcher 2011); this yields $L_{\rm ion}$ = $10^{42.3}$ erg s$^{-1}$.
Our constraints on the ionization parameters for our clouds are poor. Given the column densities of our clouds and the energy resolution of the PCA, we can safely rule out values of log($\xi$) above $\sim2$. In that case, the absorber would manifest itself above 2 keV only via discrete lines and edges, as opposed to a broad rollover towards lower energies. We therefore calculate distances assuming log($\xi$) = –1, 0, or +1. Uncertainty on $r_{\rm cl}$ is thus dominated by uncertainty on $\xi$. For brevity, we list in Table \[tab:ecl1dist\] only values assuming log($\xi$)=0. Because $r_{\rm cl} \propto \xi^{-2/5}$, values of $r_{\rm cl}$ assuming e.g., log($\xi$) = –1 or +1 are only factors of 2.5 larger or smaller, respectively. However, for Cen A, we do not consider log($\xi$) = +1. We fit the spectrum of the most absorbed spectral bin used in Rivers [et al.]{} (2011b), model the absorption with an <span style="font-variant:small-caps;">xstar</span> table, and obtain log($\xi$) $\la -0.1$. Similarly, for NGC 3227/2000–1, Lamer [et al.]{} (2003) found log($\xi$) = –0.3–0 from a contemporaneous *XMM-Newton* observation. These two events have among the highest quality energy spectra in the group, and one might speculate that such an ionization level may be representative of the rest of the sample. For the candidate event in NGC 3516, we use the best-fitting ionization from Turner et al. (2008), log($\xi$)=2.19, as explained in Appendix A. Finally, we note that although for a few events, it is difficult to estimate reliable uncertainties on $t_{\rm d}$, a $\pm10$ per cent uncertainty on $t_{\rm d}$ translates into only $\pm4$ per cent uncertainty in $r_{\rm cl}$.
We obtain best-estimate values of $r_{\rm cl}$ that are typically tens to hundreds of light-days from the central engine, and lie in the range $1 - 50 \times 10^{4} R_{\rm g}$ ($0.3 - 140 \times 10^{4}
R_{\rm g}$ accounting for uncertainties), where $R_{\rm g} \equiv
GM_{\rm BH}c^{-1}$. We now compare these inferred radii to the locations of typical emitting components in AGN. In Table \[tab:BLRdust\] we list inferred locations for the origins of optical and near-IR broad emission lines and “hidden” broad lines observed in polarized emission for type IIs, locations of IR-emitting dust determined via either interband correlations or IR interferometry, and locations of Fe K$\alpha$ line-emitting gas determined via either X-ray spectroscopy or variability. For reverberation mapping of emission lines, we use time-lag results when available, otherwise we use full width at half-maximum (FWHM) velocities $v_{\rm FWHM}$, assumed for simplicity Keplerian motion, and estimate radial locations via $r = GM_{\rm BH}/( \frac{3}{4}
v_{\rm FWHM}^2)$.
The results are also plotted in Fig. \[fig:plotrs\] in units of $R_{\rm g}$ for each object. All structures for a given object in Fig. \[fig:plotrs\] are plotted on one dimension for clarity, but they do not necessarily overlap in space (e.g., X-ray clouds must lie along the line of sight, but other structures may lie out of it). The objects are listed in order of increasing $R_{\rm
d}$/$R_{\rm g}$. The black points denote the estimates for $r_{\rm cl}$ assuming log($\xi$)=0, but we denote the minimum/maximum range assuming log($\xi$)= –1 to +1 by the gray areas (again, with –1 to 0 for Cen A and –0.3 to 0 for NGC 3227/2000–1).
The radial distance from the black hole where dust sublimates can be estimated via $R_{\rm d} \sim 0.4 (L_{\rm bol}/10^{45} {\rm erg
s}^{-1})^{1/2} (T_{\rm d}/T_{\rm 1500})^{-2.6}~{\rm pc}$, where $T_{\rm d}$ is the dust sublimation radius, here assumed to be 1500 K (Eqn. 2.1 of Nenkova et al. 2008b). However, the boundary between dusty and dust-free zones is likely highly blurred, because relatively larger grains can survive at higher temperatures. In addition, individual components of dust can sublimate at different radii, e.g, graphite grains sublimate at a slightly higher temperature than silicate grains (Schartmann et al. 2005). We use the above equation as an approximation for the outer boundary of the “dust sublimation zone” (DSZ), i.e., we assume that no dust sublimation occurs outside $R_{\rm d}$. We take the inner boundary of the DSZ to be a factor of $\sim$2–3 smaller than $R_{\rm d}$. For example, Nenkova et al. (2008b) point out that the $V$–$K$ band reverberation lags measured by Minezaki et al. (2004) and Suganuma et al. (2006) are $\sim2-3$ times shorter than the light travel times predicted by the above equation. Those experiments thus may be tracking the innermost, larger grains. For simplicity, we assume that the central engine emits isotropically.[^9] The estimated values of $R_{\rm d}$ are listed in Table B1, with dusty regions represented by the purple areas in Fig. \[fig:plotrs\], and the inferred DSZs represented by the fading purple areas.
We list the radial locations of each cloud in units of $R_{\rm d}$ in Table \[tab:ecl1dist\]. The clouds in 7/8 objects are consistent with residing in the DSZ considering uncertainties on $r_{\rm cl}$. However, this “clustering” may be in part associated with our observational bias to select eclipses with events of $\sim$tens of days, as per our selection function. In Cen A, the clouds are inferred to be consistent with residing entirely in the dusty zone. In contrast, NGC 5506 has the lowest value of $r_{\rm cl}$/$R_{\rm
d}$; those clouds are likely the least dusty of the secure events in our sample.
The IR-emitting structures for Cen A, NGC 3783, and NGC 3227 as mapped by either interferometric observations or optical-to-near-IR reverberation mapping are inferred to exist at radii $r_{\rm IR} \sim
3-20 \times 10^4 R_{\rm g}$. Although we can only make a statement based on three objects, values of $r_{\rm cl}$ are generally consistent with these structures, again supporting the notion that the X-ray-absorbing clouds detected with *RXTE* are thus likely dusty.
For six of the seven objects with known BLRs (MR 2251–178, Mkn 509, Mkn 79, NGC 3783, NGC 3227, and Mkn 348), the BLR clouds are inferred to exist at $r_{\rm BLR} \sim 0.1 - 2 \times 10^4 R_{\rm g}$ from the black hole. We thus discuss our results in the context of the notion put forth by Netzer & Laor (1993) that the outer radius of the BLR may correspond to $R_{\rm d}$ and the inner radius of the dusty torus, supported by the results of Suganuma et al. (2006). In these six objects, one can see from Fig. \[fig:plotrs\] that the radial ranges of $r_{\rm BLR}$ are generally smaller than those for both $r_{\rm
cl}$ and $r_{\rm IR}$. That is, best-estimate values of $r_{\rm cl}$ are generally commensurate with the outer portions of the BLR or exist at radii up to $\sim$15 times that of the outer boundary of the BLR, although we must caution that we are dealing with a very small sample of only six objects.
We can conclude that, at least for these six objects, the X-ray-absorbing clouds are likely more consistent with the dusty torus than with the BLR. Note that this conclusion is relatively robust to our assumed values of log($\xi$); even if log($\xi$)=+2, the X-ray-absorbing clouds would be closer to the black hole by only a factor of 2.5. Our results thus demonstrate that radii commensurate with the outer BLR and with the inner dusty torus are radii where X-ray-absorbing clouds may exist.
The seventh object with measured BLR parameters is NGC 5506, where $r_{\rm BLR} \sim 4-10 \times 10^4 R_{\rm g}$ based on IR emission lines. The inferred radial location of the X-ray cloud is $0.3-4.5\times 10^4 R_{\rm g}$ – roughly commensurate with the innermost BLR or distances slightly smaller than the BLR as traced by IR emission lines, and thus likely non-dusty. The eclipse event in NGC 5506 may thus be analogous to short-term ($t_{\rm D} \la$ 1 d) eclipse events observed in other objects, namely NGC 1365 and Mkn 766 (Risaliti et al. 2005, 2007, 2009b, 2011; Maiolino et al. 2010). The variable absorbers detected in those observations have column densities typically $N_{\rm H} \sim 1-5 \times 10^{23}$ cm$^{-2}$ and can be partial covering, as for Mkn 766. They are inferred to exist $\sim10^{2.5} - 10^4 R_{\rm g}$ from the black hole, and are frequently identified as BLR clouds.
We return to a point regarding the type Is from $\S$4.3, regarding eclipse events longer than $\sim100$ d in duration. Our selection function analysis shows that if such events occurred in type Is with the same frequency as in type IIs, we should have detected them. It is thus possible that type Is are devoid of clouds at these radial distances along the line of sight. Another possibility is that these clouds do exist at these radii in type Is, but intersect our line of sight so rarely \[probabilities $<$ 1/(10–300)\] that 16 years of monitoring 37 type I AGN was insufficient to catch even one event (excluding the possibility that the event in MR 2251–178 falls into this category). In either case, we can estimate the corresponding radial distances. We assume a black hole mass of 10$^{7.7} {\ensuremath{\rm M_\odot}\xspace}$, $N_{\rm H} = 12 \times 10^{22}$ cm$^{-2}$, log($\xi$)=0, and log($L_{\rm ion}$) = 44.3, the non-weighted average of the type Is with secure events. Arbitrarily-chosen durations of 100, 316, 1000, and 3162 d correspond to inferred radial distances of 14, 23, 36, and 57 $\times 10^4$ $R_{\rm g}$.
Finally, we investigate any possible trend between the best-fitting values of peak $N_{\rm H}$ and the inferred values of $r_{\rm cl}$ listed in Table \[tab:ecl1dist\], in units of either light-days, $R_{\rm g}$, or $r_{\rm cl}$/$R_{\rm d}$ (Fig. \[fig:NHvsrad\]). Given the small number of eclipse events, and given that values of $N_{\rm H}$ span only a factor of $\sim$7, we find no robust correlations; inclusion/exclusion of only 1–2 secure or candidate points significantly changes the resulting coefficients and null hypothesis probabilities. Note that the overdensity of eclipse events around 200 light-days (left-hand panel), $8\times10^4 R_{\rm g}$ (middle panel), and $\la 1 R_{\rm d}$ (right-hand panel) is likely not intrinsic to the sources but mainly caused by our selection function. This figure should thus not be used to derive constraints on the radial distribution of clouds.
------------- ---------------- --------------------------------------- ------------------ ------------------------------------------- ---------------- --------------------- ---------------------- -------------------------- ---------------------------
Source log($M_{\rm BH}$) $L_{2-10}$, $L_{\rm ion}$, $L_{\rm Bol}$, log($\xi$, erg $r_{\rm cl}$ $r_{\rm cl}$
name Event (${\ensuremath{\rm M_\odot}\xspace}$) Ref. (erg s$^{-1}$) cm s$^{-1}$) (light-days) ($10^4$ $R_{\rm g}$) $r_{\rm cl}$/$R_{\rm d}$ Notes
NGC 3783 2008.3 7.47 VP06 43.2, 44.1, 44.4 0 $147^{+11}_{-10}$ $8.6^{+0.7}_{-0.6}$ $0.62\pm0.04$
(Full event)
Mkn 79 2003.5 7.72 Pe04 43.3, 44.3, 44.7 0 $229^{+83}_{-79}$ $7.5\pm2.7$ $0.68\pm0.24$
2003.6 0 $290^{+42}_{-35}$ $9.6^{+1.4}_{-1.1}$ $0.86^{+0.13}_{-0.09}$
2009.9 0 $314^{+91}_{-74}$ $10.4^{+3.0}_{-2.5}$ $0.93^{+0.27}_{-0.22}$
Mkn 509 2005.9 8.19 VP06 44.3, 44.9, 45.2 0 $851^{+255}_{-278}$ $9.5^{+2.9}_{-3.1}$ $1.42^{+0.42}_{-0.46}$ $^{\dagger}$
MR 2251–178 1996 8.3 W09 44.7, 45.3, 45.6 0 460–5700 4–49 0.5–6.0 $^{\dagger}$$^{\ddagger}$
NGC 3227 2000–1 6.88 D10 42.5, 43.0, 43.5 –0.3 $82^{+9}_{-8}$ $19\pm2$ $0.97\pm0.11$
0 $62^{+7}_{-6}$ $14\pm1$ $0.74\pm0.05$
2002.8 0 $23\pm7$ $5.3\pm1.6$ $0.28^{+0.08}_{-0.09}$
Cen A $\sim$2003–4 7.78 R11 41.9, 42.3, 42.9 0 $214^{+70}_{-93}$ $6.2^{+2.0}_{-2.6}$ $5.3^{+1.6}_{-2.1}$ $^{\dagger}$
2010–1 0 $101^{+7}_{-6}$ $2.9\pm0.2$ $2.4\pm0.2$
NGC 5506 2000.2 7.94 Pa04 43.0, 43.9, 44.3 0 $62^{+26}_{-24}$ $1.2^{+0.6}_{-0.4}$ $0.26^{+0.11}_{-0.08}$
Mkn 348 1996–7 7.18 WZ07 43.0, 44.1, 44.4 0 $432^{+79}_{-73}$ $50\pm9$ $1.82\pm0.37$ $^{\dagger}$
Fairall 9 2001.3 (cand.) 8.41 VP06 43.4, 44.6, 44.8 0 $>$180 $>$1.2 $>0.5$ $^{*}$
NGC 3783 2008.7 (cand.) 7.47 VP06 43.2, 44.1, 44.4 0 $>$230 $>$13.7 $>1.0$ $^{*}$
2011.2 (cand.) 0 $127^{+71}_{-52}$ $7.4^{+4.2}_{-3.0}$ $0.54^{+0.30}_{-0.22}$
NGC 3516 2011.7 (cand.) 7.50 D10 43.0, 43.7, 44.2 +2.19 34 1.9 $\sim0.6$ $^{**}$
7.63 43.1, 43.8, 44.2 200 8.0 1.0
------------- ---------------- --------------------------------------- ------------------ ------------------------------------------- ---------------- --------------------- ---------------------- -------------------------- ---------------------------
\
Estimates of the distance from the central black hole to each eclipsing cloud $r_{\rm cl}$ following $\S$5.2. Uncertainties listed above do not include uncertainty on the ionization parameter $\xi$. Reasonable ranges for log($\xi$) are –1 to +1 for most objects, which translates into factors of 2.5 larger/smaller (for Cen A, values of log($\xi$) ranging from –1 to 0 are plausible; for NGC 3227/2000–1, Lamer et al. (2003) measured log($\xi$) = –0.3–0). $L_{2-10}$ is the 2–10 keV of the hard X-ray power-law component from the long-term time-averaged *RXTE* spectrum (Rivers [et al.]{} 2013). $L_{\rm ion}$ is the 1–1000 Ryd ionizing luminosity; please see $\S5.2$ for details. $L_{\rm Bol}$ is the bolometric luminosity; please see $\S5.1$ for details. Luminosities are corrected for all (intrinsic and Galactic) absorption. $R_{\rm d}$ denotes the outer boundary of the “dust sublimation zone” (Nenkova et al. 2008b), i.e., dust residing at distances greater than $R_{\rm d}$ likely does not sublimate, while distances smaller than $\sim\frac{1}{2}-\frac{1}{3}R_{\rm d}$ are expected to be dust-free. Average values (last row) were determined in log space. References for ${\ensuremath{\rm M_\odot}\xspace}$ are: D10 = Denney [et al.]{} (2010), Pa04 = Papadakis (2004), Pe04 = Peterson [et al.]{} (2004), R11 = References in Rothschild [et al.]{} (2011), VP06 = Vestergaard & Peterson (2006), W09 = Wang [et al.]{} (2009), and WZ07 = Wang & Zhang (2007).\
$^{\dagger}$Estimates of $r$ may be lower if peak $N_{\rm H}$ was higher than that measured because either the beginning or end of the eclipse was not covered by monitoring data.\
$^{\ddagger}$For MR 2251–178, we have only limits to the duration, but no reliable “best estimate.”\
$^{*}$Upper limits on $N_{\rm H}$ were used to derive lower limits on $r_{\rm cl}$.\
$^{**}$Assumed ionization following Turner et al. (2008); see Appendix A.
A possible double-eclipse event in NGC 3783 {#sec:3783double}
-------------------------------------------
The $N_{\rm H}$($t$) and $HR1$ profiles for the NGC 3783/2008.3 event are highly intriguing, as the two spikes suggest two absorption events separated by only 11 d. The fact that, assuming that $\Gamma$ remains constant during this time, $N_{\rm H}$($t$) remains near $4\times 10^{22}$ cm$^{-2}$ in between the spikes is also intriguing. No such complex $N_{\rm H}$($t$) profile has been observed for any other AGN to date.
As illustrated in Fig. \[fig:3783doubleprofile\], we model the $N_{\rm H}$($t$) profile in several ways, fitting only bins \#5–14 (see Table \[tab:3783TRtable\] for bin definition; refers to those bins with non-zero values of $N_{\rm H}$ plus two bins with upper limits before and after). We first test a single uniform-density sphere (gray line), which yields $\chi^2/dof$ = 40.1/11=3.65 and underestimates the spikes in $N_{\rm H}$ in bins \#9 and \#15.
We also fit the profile assuming two independent eclipsing events. We test a profile from two uniform-density spheres (cyan line), and one from two linear-density spheres profile (red line). For the latter, the density profile is $n$($r$) = $n_{\rm max}$(1–$r/R$), where $n_{\rm
max}$ is the number density at the center and $R$ is the outer radius of the cloud along the transverse axis (red line). Each cloud’s center point in time space is held frozen at the midpoints of bins \#9 or 15. The best-fitting models yield $\chi^2/dof$ = 50.2/10=5.02 and 48.9/10=4.89, respectively, and did not correctly fit bin \#9 nor the $\sim$2–4 bins between the spikes. We also fit each spike with a phenomenological $\beta$-profile of the form $N_{\rm H}$($t$) = $N_{\rm
H, max} \sqrt{1 - r/R_{\rm c}}$ (panel c), where $N_{\rm H, max}$ is the column density along a line of sight through the cloud’s center and $R_{\rm c}$ is the core radius (Dapp & Basu 2009). The best fit yields $\chi^2/dof$ = 39.5/10=3.95 but still underestimates the middle part of the profile.
Furthermore, the probability that two separate eclipse events could independently occur so close together in time is low: NGC 3783 was monitored for 3.27 yr, had no observations for 1.88 yr, and then was monitored for an additional 7.84 yr. We perform Monte Carlo simulations wherein we assume a 1.0-d grid and we randomly place three eclipses in the 3.27 or 7.84-yr campaigns, and empirically estimate the probability that any two of their peaks can occur $<$15 d apart: only 0.24 per cent. Combined with the fact that $N_{\rm H}$($t$) does not return to zero between the two spikes, the likelihood that the profile is comprised of two independent eclipse events seems low. The addition of a *third* independent cloud with $N_{\rm H}
\sim 4 \times 10^{22}$ cm$^{-2}$ transiting the line of sight roughly halfway between the maximum transits of the other two would of course yield a good fit to the observed $N_{\rm H}$($t$) profile, but this is statistically highly unlikely ($\sim5\times 10^{-6}$).
Finally, we model a uniform-density shell (green line in Fig. \[fig:3783doubleprofile\]; subtracting a smaller uniform-density sphere from another) to successfully model fit the middle part and the two spikes. The best fit has $\chi^2/dof$ = 13.2/9=1.47, with the outer and inner shell boundaries taking $7.5^{+0.5}_{-1.0}$ and $4.5\pm0.5$ d, respectively, to transit the line of sight. However, we stress that this fit is purely phenomenological; it is not clear how such a structure could be created or survive for long periods in an AGN environment.
We speculate that the observed profile could be caused by two dense clumps with some sort of connecting structure (a dumbbell shape), with parts aligned along the direction of travel across the line of sight. We discuss a possible physical origin for this behavior in $\S$\[sec:profiles\].
Physical properties of obscuring clouds {#sec:diams}
---------------------------------------
We estimate diameters, number densities and masses for each eclipsing cloud using the equations in $\S$\[sec:locations\] and listed them in Table \[tab:clumpsizemass\]. The average diameter across the sample of secure events is $\sim0.25$ light-days, with an average number density of $\sim3\times10^8$ cm$^{-3}$. Nenkova [et al.]{} (2008b) estimate theoretically that a total of $\sim10^4 - 10^5$ clouds comprises the dusty torus. The mass estimates for individual clouds in Table \[tab:clumpsizemass\] span $\sim10^{-8}-10^{-2} {\ensuremath{\rm M_\odot}\xspace}$ with a mean value in log space of $10^{-5.2} {\ensuremath{\rm M_\odot}\xspace}$, thus suggesting a total mass for the clumpy component of the torus (i.e., excluding any thin accretion disk or intercloud diffuse medium) of $\sim 10^{-4} - 10^{3} {\ensuremath{\rm M_\odot}\xspace}$. If the torus clouds are produced in a disk wind, then this estimate indicates the maximum mass associated with the outflow. Since the value of $\sim10^4 - 10^5$ clouds quoted above likely refers only to IR-emitting clouds located outside $R_{\rm d}$, this mass refers to clouds lying outside $r_{\rm d}$ only, and also ignores contributions from non-clumpy material such as any intercloud medium (Stalevski et al. 2012).
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Source Diam. Diam. Num. dens., $n$ log(mass) $D_{\rm X-src}$
name Event ($10^{14}$ cm) (light-days) ($10^8$ cm$^{-3}$) (${\ensuremath{\rm M_\odot}\xspace}$) ($R_{\rm g}$)
NGC 3783 2008.3 $1.3^{+0.7}_{-0.5}$ $0.048^{+0.029}_{-0.018}$ $8.6^{+7.0}_{-3.9}$ $-6.0^{+0.4}_{-0.5}$ $<47$
Mkn 79 2003.5 $2.4^{+2.6}_{-1.5}$ $0.089^{+0.097}_{-0.057}$ $6.0^{+15.9}_{-4.0}$ $-5.4^{+0.5}_{-0.8}$ $<64$
Mkn 79 2003.6 $3.1^{+1.8}_{-1.2}$ $0.11^{+0.07}_{-0.04}$ $3.7^{+3.9}_{-1.9}$ $-5.3\pm0.5$ $<65$
Mkn 79 2009.9 $2.4^{+2.1}_{-1.3}$ $0.088^{+0.078}_{-0.046}$ $3.2^{+5.5}_{-2.0}$ $-5.7^{+0.4}_{-0.5}$ $<57$
Mkn 509 2005.9 $4.9^{+5.7}_{-3.3}$ $0.18^{+0.21}_{-0.12}$ $1.8^{+4.5}_{-1.1}$ $-5.1^{+0.6}_{-0.9}$ $<46$
MR 2251–178 1996 $0.25-91$ $0.009-3.4$ $0.06-30$ $-7.7$ to $-2.7$ $<310$
NGC 3227 2000–1 $5.8^{+0.3}_{-1.0}$ $0.22^{+0.01}_{-0.04}$ $3.9^{+1.4}_{-0.8}$ $-4.5^{+0.1}_{-0.2}$ $<540$
NGC 3227 2002.8 $0.49^{+0.56}_{-0.31}$ $0.018^{+0.020}_{-0.012}$ $27^{+62}_{-17}$ $-6.9^{+0.6}_{-0.8}$ $<92$
Cen A $\sim$2003–4 $120^{+60}_{-90}$ $4.5^{+2.3}_{-3.4}$ $0.07^{+0.23}_{-0.03}$ $-2.3^{+0.3}_{-0.2}$ $<2000$
Cen A 2010–1 $26.9^{+0.1}_{-10.1}$ $1.00^{+0.01}_{-0.38}$ $0.30^{+0.24}_{-0.04}$ $-3.6^{+0.1}_{-0.5}$ $<300$
NGC 5506 2000.2 $0.12^{+0.13}_{-0.08}$ $4.3^{+5.8}_{-2.9}\times10^{-3}$ $34^{+111}_{-23}$ $-8.6^{+0.5}_{-0.9}$ $<1.9$
Mkn 348 1996–7 $20^{+17}_{-10}$ $0.74^{+0.62}_{-0.36}$ $0.90^{+1.17}_{-0.49}$ $-3.5\pm0.5$ $<1600$
Fairall 9 2001.3 $<4.5$ $<0.17$ $\sim 5-27$ $\sim-6.2$ to $-4.7$ $<12$
NGC 3783 2008.7 $<2.8$ $<0.10$ $\sim 1-5$ $\sim-7.0$ to $-5.9$ $<63$
NGC 3783 2011.2 $0.9^{+1.0}_{-0.6}$ $0.035^{+0.035}_{-0.023}$ $12^{+41}_{-8}$ $-6.4^{+0.4}_{-0.8}$ $<43$
NGC 3516 2011.7 $\sim11$ $\sim0.4$ $\sim 0.4$ $\sim-4.6$ $<230$
3.9 0.25 2.6 –5.2 $<120$
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\
Best-estimate values listed here correspond to best-fitting values of $N_{\rm H}$ and values of duration listed in Table \[tab:ecl1summ\] and an assumed ionization state of log($\xi$)=0. Uncertainties listed here are based on the uncertainties on observed duration, $N_{\rm H}$, and the ranges of $r_{\rm cl}$ and log($\xi$) listed in Table \[tab:ecl1dist\]. $D_{\rm X-src}$ denotes inferred upper limits on the size of the X-ray continuum source, assuming that the cloud completely covers the X-ray continuum source. Average values were determined in log space.
Finally, assuming that each eclipsing cloud fully covers the X-ray continuum source behind it, the upper limits on the diameter of the cloud yield inferred upper limits on the size of the X-ray continuum source, $D_{\rm X-src}$. These limits are listed in Table \[tab:clumpsizemass\] in units of $R_{\rm g}$. Among the secure events, the limits are as low as a few tens of $R_{\rm g}$ for NGC 3783, Mkn 79, and Mkn 509, and even as low as $2R_{\rm g}$ in the case of NGC 5506.
The X-ray continuum in Seyferts is generally thought to originate via inverse Compton scattering of the soft UV photons from the accretion disk by hot electrons in a compact corona, possibly located in the innermost accretion disk or at the base of a jet (e.g., Haardt [et al.]{}1997; Markoff [et al.]{} 2005). X-ray microlensing analysis of distant quasars typically yields upper limits on the size of the X-ray corona of $\sim10R_{\rm g}$ (e.g., Chartas [et al.]{} 2012, Chen [et al.]{} 2012, and references therein). Recent results using X-ray reverberation lags imply sizes of a few to $\sim10$ $R_{\rm g}$ (Reis & Miller 2013). Geometrical limits from absorption variability in NGC 1365 typically yield $D_{\rm X-src} \la 30-60 R_{\rm g}$ (Risaliti [et al.]{} 2007, 2009b). Our estimates of $D_{\rm X-src}$ are consistent with these studies.
### Exploring the range in observed density profiles {#sec:profiles}
We speculated in $\S$\[sec:3783double\] that the unusual density profile of the NGC 3783/2008.3 event could be caused by the cloud having a dumbbell shape. Even more speculatively, perhaps this structure is a cloud that is in the process of getting torn in half. As pointed out by Krolik & Begelman (1988), the self-gravity of a single cloud in the vicinity of a supermassive black hole is not highly effective against tidal shearing, and clouds can get significantly stretched out within an orbit. Resistance to tidal shearing requires that the size of the cloud is limited to $\la D_{\rm shear} = 10^{16} N_{\rm H,23} r_{\rm pc}^3
M_7^{-1}$ cm, where $N_{\rm H,23} = N_{\rm H}/10^{23} {\rm cm}$, $r_{\rm pc}$ is the distance from the black hole to the cloud, and $M_7 = M_{\rm BH}/(10^7 {\ensuremath{\rm M_\odot}\xspace})$ (Elitzur & Shlosman 2006). For NGC 3783/2008.3, $D_{\rm shear} \sim 7\times10^{12}$ cm, a factor of $\sim1/18$ times the inferred cloud diameter $D_{\rm cl}$, so the possibility of tidal shearing seems plausible. In fact, for all secure eclipse events, we obtain values of $D_{\rm cl}/D_{\rm shear}$ spanning 2.3–350, with an average value (in log-space) of 26, suggesting that many of these clouds’ sizes put them at risk of being tidally disrupted or sheared.
The unusual density profile could also be consistent with models wherein torus clouds may be in the form of clumpy winds, as opposed to compact, discrete clouds, originating at the accretion disk. These winds may be magnetohydrodynamic (MHD) driven (e.g., Konigl & Kartje 1994; Fukumura et al. 2010), with local column densities that can be consistent with values measured in this paper. Other models feature IR radiation-driven winds (Dorodnitsyn & Kallman 2012; Dorodnitsyn et al. 2012): UV/X-ray photons are reprocessed into IR thermal emission in the torus, and IR radiation pressure vertically drives dust grains. Moderately Compton-thick flows are possible at distances of $\ga$1 pc from the black hole (if our assumption about Keplerian motion is not valid, then distance estimates for our clouds may be in error). The models generally assume that clouds’ self-gravity is negligible. Similarly, Czerny & Hryniewicz (2011) proposed a turbulent, dusty, disk wind as the origin for the low-ionization region of the BLR: clouds rise from the disk, intense external radiation heats the gas and dust sublimates. In all these models, the torus is highly dynamic and can feature cloud motions perpendicular to the disk (including failed winds), and thus not strictly Keplerian. In any case, the $N_{\rm H}$($t$) profile of the NGC 3783/2008.3 event can thus also be explained via a non-homogeneous mass outflow, featuring two overdensities that crossed the line of sight 11 d apart.
However, the profile of NGC 3783/2008.3 contrasts with three of the other eclipse events that have $N_{\rm H}$($t$) profiles that are also well resolved in time but are symmetric and centrally peaked: NGC 3227/2000–1 and Cen A/2010–1, observed with *RXTE*, and SWIFT J2127.4+5654 (Sanfrutos [et al.]{} 2013). Such profiles pose a challenge to MHD/IR-driven wind models. For example, the IR-driven winds of Dorodnitsyn et al. (2012) suggest that $\Delta$$N_{\rm
H}$/$N_{\rm H}$ is typically $\sim$ a few (less than the factors we observe). In addition, such winds are inferred to originate $\sim1-3$ pc from the black hole, farther than the distances inferred for our observed eclipses.
In summary, the self-gravity of these clouds likely dominates over any tidal shearing/internal turbulence, presenting a challenge to the expectation that clouds are easily tidally sheared. In these cases, one or more of the assumptions that go into the calculations of $D_{\rm cl}$ or $D_{\rm shear}$ may be wrong. Alternately, some other physical process may prevent clouds from shearing, e.g., confinement of these clouds by external forces (e.g., gas pressure from the ambient medium, external magnetic fields; Krolik & Begelman 1988) may be important.
Implications for clumpy-absorber models
---------------------------------------
[<span style="font-variant:small-caps;">Clumpy</span>]{} torus models have found a lot of observational support, particularly from recent IR observations. For example, smooth-density torus models always predict that the silicate features at 9.7 and 18 will be in emission for pole-on viewing and in absorption for edge-on viewing. However, samples of mid-IR spectra of type I Seyferts can exhibit silicate features spanning a range of emission *and* absorption (see Hao et al. 2007 for a sample obtained with *Spitzer*); type IIs generally show only weak absorption but a few can show emission (Sturm et al. 2005; Hao et al. 2007). Nenkova et al. (2008b) and Nikutta et al. (2009) demonstrate that the clumpy-torus models are consistent with these observations and can explain mismatches between the optical classification and the expected behavior of the silicate features. Clumpy-torus models predict nearly isotropic mid-IR continuum emission and anisotropic obscuration, as generally observed (e.g., Lutz et al. 2004; Horst et al. 2006). Finally, clumpy tori have been predicted to host both hot and cooler dust in close proximity to each other (Krolik & Begelman 1988), in contrast to smooth-density tori. Recent IR interferometric observations indeed do affirm such a co-existence of hot (800 K) and cooler ($\sim 200-300$ K) dust components in nearby AGN (Jaffe et al. 2004; Poncelet et al. 2006; Tristram et al. 2007; Raban et al. 2009).
In this subsection, we derive the first *X-ray* constraints on the [<span style="font-variant:small-caps;">Clumpy</span>]{} model parameters, thus obtaining constraints in a manner independent from IR SED fitting. We caution, however, that there may not be an exact correspondence between constraints derived from the two methods, since IR emission arises only from dusty clouds outside the DSZ while our X-ray clouds can be dust free. The clumpy-torus models are generally defined using the following free parameters: is the $V$-band optical depth of single clouds; usually, all clouds are assumed to have identical values of . The radial extent of the dusty torus is characterized by $Y$, the ratio of the outer radial extent of clouds to the dust sublimation radius $R_{\rm d}$. ${\hbox{${\cal N}_{i}$}}(\theta)$ is the average number of clouds along a radial line of sight that is an angle $\theta$ from the equatorial plane. Assuming a Gaussian distribution, ${\hbox{${\cal N}_{i}$}}(\theta,\sigma,{\hbox{${\cal N}_0$}}) = {\hbox{${\cal N}_0$}}\exp(-(\theta/\sigma)^2)$, where is the average number of clouds along a radial line of sight in the equatorial plane (typically 5–15), and $\sigma$ parametrizes the angular width of the cloud distribution. The index $q$ describes the radial dependence of the average number of clouds per unit length, ${\hbox{${\cal N}_{C}$}}(r,q) \propto r^{-q}$. The inclination angle of the system is $i
\equiv 90\degr-\theta$, defined such that $i=0$ ($\theta=90\degr$) denotes a system with its equatorial plane in the plane of the sky.
The predicted shape of the IR SED is sensitive to all of these parameters. Ramos Almeida et al. (2009, 2011; hereafter RA09 and RA11) have successfully fit the mid-IR spectra of $\la20$ AGN with [<span style="font-variant:small-caps;">Clumpy</span>]{} torus models and the <span style="font-variant:small-caps;">BayesClumpy</span> tool (Asensio Ramos & Ramos Almeida 2009). There are only three sources in our eclipse sample that overlap with the samples of RA09 and RA11 (NGC 3227, Cen A, and NGC 5506). In the following we attempt to focus on comparing our derived parameter constraints for the type I/II classes as opposed to for individual objects.
$\bullet$ : we can translate our inferred values of $N_{\rm H}$ for individual clouds into values of ${\hbox{$\tau_{\rm V}$}}$, which we can then compare to those values typically used in the ${\textsc{Clumpy}}$ model fitting. From the “secure” clumps we observe, $\Delta$$N_{\rm H} = 4-26 \times
10^{22}~{\rm cm}^{-2}$, with values spanning very similar ranges for both type Is/IIs. Using the Galactic dust/gas conversion ratio, $N_{\rm H} = A_{\rm V} {\rm (mag)} \times 1.8 \times 10^{21} {\rm
cm}^{-2}$ (Predehl & Schmidt 1995), these values translate to $A_{\rm V} = 22-144$ mag, or ${\hbox{$\tau_{\rm V}$}}= 20-132$. The [<span style="font-variant:small-caps;">Clumpy</span>]{} models use one value of $\tau_{\rm V}$ per object/spectrum to represent the weighted average value across all clouds in that object. In contrast, we probe 1–3 individual clouds per object. Subject to small-number statistics, we can therefore test both the inter-object scatter of values, as well as the scatter from cloud to cloud in one given source. From general theoretical considerations, Nenkova et al. (2008b) suggest that be typically 30–100. RA11, who use the entire [<span style="font-variant:small-caps;">Clumpy</span>]{} database of models in their IR SED fitting, estimate values of 40–140 for type Is and 5–95 for type IIs. Alonso-Herrero et al. (2011), with the same approach but different sources, find = 105–146 (type Is) and 49–130 (type IIs). These IR-derived ranges are fully consistent with our X-ray based values. In contrast to the above studies, we find no evidence for different cloud optical depths between type I and II sources.
Three of our sources are also present in the IR samples of RA11 and Alonso-Herrero et al. (2011). In two of these (Cen A and NGC 3227) our X-ray constraints agree very well with the IR ones. However, our value for NGC 5506 is smaller than that in RA11 by a factor of $\sim2$ and smaller than that of Alonso-Herrero et al. (2011) by $\sim5$. For our sources with multiple eclipsing events, the measured scatter in $N_{\rm H}$ or is very low, always $<$2. It seems then that assuming a single average value of for all clouds in a given [<span style="font-variant:small-caps;">Clumpy</span>]{} model is justified.
$\bullet$ $Y$: the IR SED modeling and interferometry indicate the presence of dusty clouds out to $Y \sim 20-25$ (e.g., RA11), although emission from beyond this radius may be difficult to separate from the surrounding host galaxy emission, especially for emission longer than $\sim 10{\hbox{$\mu{\rm m}$}}$. Our inferred locations of the X-ray-absorbing clouds cover out to $Y \sim 20$, in agreement with the IR observations. It should be noted though that clouds at much larger distances from the black hole might exist, but would only be detectable with even longer sustained X-ray monitoring campaigns of a larger number of targets.
$\bullet$ $\sigma$, $i$ ($\theta$) and : we observe eclipse events in type I objects, and derive non-zero values of $P_{\rm ecl}$. If it is the case that all type Is are oriented relatively face-on, then clumpy-torus models featuring sharp-edged tori are strongly disfavored, while models featuring a gradual decline of cloud number when moving away from the equatorial plane are favored (see left- and right-hand panels of Nenkova et al. 2008b, Fig. 1, respectively). IR SED fitting likewise disfavors sharp-edged tori, which would produce a dichotomy in the SED shapes that is not observed (Nenkova et al. 2008b).
We can derive our constraints on $\sigma$ and assuming that an observed baseline level of absorption $N_{\rm H,base}$ corresponds to an average of clouds along the line of sight (${\hbox{${\cal N}_{i}$}}= 0$ in the case of sources normally lacking X-ray obscuration). We also assume that an eclipse denotes ${\hbox{${\cal N}_{i}$}}\rightarrow {\hbox{${\cal N}_{i}$}}+1$. In the case of Cen A, to repeat the exercise from Rothschild et al. (2011) and Rivers et al. (2011b), $\Delta$$N_{\rm H}$/$N_{\rm H,base} \sim 3$, suggesting ${\hbox{${\cal N}_{i}$}}\sim 2$. From IR interferometry, Burtscher et al. (2010) model a best-fitting value for the torus inclination of $i=63\degr$, or $\theta =
27\degr$. The median value for $\sigma$ modeled by RA11 was $\sigma = 20\degr$; in order to agree with this value of $\sigma$, one needs ${\hbox{${\cal N}_0$}}=12$.
In both cases of NGC 5506 and Mkn 348, $\Delta$$N_{\rm H}$/$N_{\rm
H,base} \sim 2$, suggesting ${\hbox{${\cal N}_{i}$}}\sim 1$. Due to the absence of external information on $i$, we can only derive constraints on the ratio $\theta/\sigma$. If we assume that ${\hbox{${\cal N}_0$}}= 10$, $\theta/\sigma$ = 1.52. That is, if such a system is inclined relatively edge-on ($\theta\rightarrow 0$), as is commonly suspected for type IIs, then $\sigma$ must be rather small, indicating a tightly flattened distribution. Otherwise, the inclination of the system would have to deviate more from an equatorial view. This exercise assumes that all obscuring material along the line of sight is in the form of clouds. If the baseline absorption $N_{\rm H}$ is not due to clouds (see $\S$5.6), then ${\hbox{${\cal N}_{i}$}}\sim 0$, suggesting that the inclination of the system may not be close to edge-on, and/or that $\sigma$ is relatively small in these systems. A similar condition, ${\hbox{${\cal N}_{i}$}}\sim 0$, applies to sources normally lacking X-ray absorption, e.g., the type I objects in our sample.
For a [<span style="font-variant:small-caps;">Clumpy</span>]{} torus the probability of the AGN being obscured – a single dust cloud along the line of sight is sufficient – is given by $$\label{eq:pobsc}
{\hbox{$P_{\rm obsc}$}}({\hbox{${\cal N}_0$}}, {\hbox{$\sigma$}}, i) = 1 - \exp\left\{-{\hbox{${\cal N}_0$}}\cdot \exp\left\{ -\left(\frac{90-i}{\sigma}\right)^{\!\!\!2}\right\}\right\}$$ (e.g., Nenkova et al. 2008b; Nikutta et al. 2009). The argument to the outer exponential is the (negated) number of clouds along the line of sight , thus ${\hbox{$P_{\rm obsc}$}}= 1 - \exp\{-{\hbox{${\cal N}_{i}$}}\}$. We can attempt to constrain the [<span style="font-variant:small-caps;">Clumpy</span>]{} parameters , , and $i$ by identifying ranges of consistent with our X-ray monitoring data. In Fig. \[fig:pobscured\] we plot as a function of 2 of the 3 parameters, with each row of panels sampling the third parameter. The scale is linear (see color bar). In each panel we overplot as contour lines $\overline{P_{\rm ecl}}$, the inferred values of the probability to catch an AGN while being obscured by a torus cloud, as computed in $\S$\[sec:sect43\] (Table \[tab:finalprobs\]). The solid lines correspond to the minimum and maximum values of $\overline{P_{\rm ecl}}$, and dotted lines are best-estimate values. The white lines thus outline, given the data and the selection function, the range of permitted values of for type I objects, and the black lines trace the allowed range of probabilities for type II objects. The dotted lines are the best-estimate values of .
{width="80.00000%"}
These inferred minimal and maximal probabilities outline the parameter ranges permissible by our extensive monitoring programmes. Given the observed (and not observed!) eclipsing events, any torus parameters which lie outside the enclosed ranges are unlikely to be found in real sources. We remind the reader, however, that the probability values $\overline{P_{\rm ecl}}$ we derived in $\S$4 refer only to lowly-ionized/neutral, full-covering clouds with $N_{\rm H} \ga
10^{22}$ up to $\sim10^{25}$ cm$^{-2}$. When eclipses are due to clouds with higher ionizations and/or lower column densities, and partial-covering events, the values will be higher. Since three parameters (, , $i$) affect with some mutual dependence on each other, it is not straightforward to give one-dimensional ranges of excluded parameter values. However, for physically reasonable values some constraints can be given ad hoc. For instance, if is between 5 and 15 (see, e.g., Nenkova et al. 2008b), it is clear from Fig. \[fig:pobscured\] (bottom row) that must be smaller than $\approx 45{\hbox{$^\circ$}}$ (even for type II sources); only at the most pole-on viewing angles, can be slightly higher. Similarly, if, for instance, is secured by external information, for the reasonable range of the permitted viewing angles $i$ are very well-constrained (see middle row of Fig. \[fig:pobscured\]). Not surprisingly, the permissible parameter values in type II sources are on average higher than in type I sources, because *increases* when any of the three (, , or $i$) grows. There is some overlap, however, and is most likely indicative of “intermediate” types of AGN.
Overall, Fig. \[fig:pobscured\] shows that whenever one of the three parameters can be secured by other means, the other two allow only relatively narrow ranges. In practice, external information is sometimes available, for instance on the orientation of the AGN system from radio jets or from narrow-line ionization cones, and can be used to infer the two other parameters. We are not attempting a full Bayesian parameter inference on the torus parameters here. Suffice it to say here that our results as depicted in Fig. \[fig:pobscured\] translate into information on the *priors* $P(\bm\theta)$ of the Bayesian inference problem $P(\bm\theta|D) \propto P(\bm\theta) \cdot P(D|\bm\theta)$. Here $\bm\theta = ({\hbox{${\cal N}_0$}}, {\hbox{$\sigma$}}, i, \ldots)$ is the vector of model parameters to be estimated, $P(D|\bm\theta)$ is the *likelihood* that a set of parameter values generates a model compatible with the data, and $P(\bm\theta|D)$ is the full parameter *posterior* (also called inverse probability). Faced with perfect ignorance of the distribution of parameter values before modeling, uniform priors are often assumed. If there is independent knowledge available on some parameters it is imperative to incorporate this information into the Bayesian priors. For example, if the orientation of an AGN was known to be pole-on, one could assume a narrow Gaussian shape on the prior of $i$. We presume that then the Bayesian inference process would yield parameter intervals for and commensurate with the lower-left panel of Fig. \[fig:pobscured\].
The function transitions quite steeply from very low to very high probabilities. Our derived probability ranges (see Table \[tab:finalprobs\]) typically bracket most of the transition region, and also exclude a large fraction of the parameter space that corresponds to extremely low or extremely high . We presume that a source such as NGC 1365, aligned so that (average number of clouds along the line of sight) is $\approx 1$, would be just at the high end, or slightly above, of the type II region encompassed by the black lines in Fig. \[fig:pobscured\].
Can clumpy-torus models be applied to type II objects?
------------------------------------------------------
As detailed in Appendix A and summarized in $\S$3, *RXTE* provided sustained long-term monitoring for durations $\geq$0.6 yr for nine type II objects. In eight of them, $HR1$ is consistent with remaining constant for durations $\geq$0.6 yr (we include the 2011 monitoring of Mkn 348 here). This constancy comes despite variations of typical factors $\sim2-4$ in 2–10 keV continuum flux. For example, excluding the $\la$1-d event in NGC 5506, there are no strong variations in $HR1$ (sustained trends above 2$\sigma$ and/or $\sim1.5 \times \langle HR1 \rangle$) lasting longer than 1 d for a period of 8.39 yr. Each of these sources routinely show evidence for the presence of absorbing gas with $N_{\rm H} \sim 10^{21-23}$ cm$^{-2}$ in their X-ray spectra, suggesting a baseline level of absorption $N_{\rm H,base}$ that is consistent with being constant. Using $HR1$ and its distribution, and assuming for simplicity that $\Gamma$ remains constant for each object throughout the monitoring, we estimate the maximum value of $\Delta$$N_{\rm H}$ possible without significantly varying $HR1$. Those values are listed in Table \[tab:boringsy2s\], and for the brightest and best-monitored objects is typically $\sim1-4\times10^{22}$ cm$^{-2}$.
REN02 claim typically 20–80 per cent variations in $N_{\rm H}$ in most of their sample of type II objects, using individual observations from multiple X-ray missions spanning $\sim$25 yr. In this paper, we rely on measurements from one instrument only, and so we do not face systematic uncertainties associated with cross-calibration between various X-ray missions or with fitting spectra data over different bandpasses. For our eight type IIs, we can rule out variations above $\sim20-30$ per cent for IRAS 04575–7537 and Mkn 348 (2011 monitoring only), and variations above $\sim50-60$ per cent for NGC 1052, NGC 4258, and NGC 5506.
In the context of the clumpy-torus paradigm, having a non-zero value $N_{\rm H,base}$ requires a non-zero number of clouds along the line of sight at all times, with the observed value of $N_{\rm H,base}$ being the sum of the values of $N_{\rm H}$ for the individual clouds. Assuming that all clouds have the same physical characteristics ($n$, $N_{\rm H}$, etc.), the measured long-term constancy in $N_{\rm H,base}$ would require that the number of clouds along the line of sight ${\hbox{${\cal N}_{i}$}}$ either remain constant for many years or is sufficiently large such that the addition/subtraction of one cloud intersecting or transiting along the line of sight does not change the total measured value of $N_{\rm H}$ by a detectable amount. However, with most clouds modeled with ${\textsc{Clumpy}}$ having $\tau_{\rm V} \sim 30-100$, which corresponds to $N_{\rm H} \sim 6-20 \times 10^{22}$ cm$^{-2}$ assuming the Galactic dust/gas ratio, values of $N_{\rm H,base}$ of $10^{22}-10^{23}$ cm$^{-2}$ imply that ${\hbox{${\cal N}_{i}$}}$ cannot be more than a few.
We can explore if it is feasible to have the observed duration for each source correspond to a period where ${\hbox{${\cal N}_{i}$}}$ remains constant. That is, we assume that all cloud contributing to the total observed value of $N_{\rm H}$ remain in the line of sight for the duration, and we assume for simplicity that every cloud has identical $N_{\rm H}$. We also assume that every such cloud is $D_{\rm cl}$ = 1 light-day in diameter and on Keplerian orbits. Furthermore, we assume that the fastest velocity (closest to the black hole) cloud has just started to transit the line of sight when the long-term monitoring began, and that the cloud begins to leave the line of sight just when the monitoring ends. In the case of NGC 5506 ($M_{\rm BH}=10^{7.94}
{\ensuremath{\rm M_\odot}\xspace}$), an 8.39-year-long transit by such a cloud would require it to be located at least $r_{\rm cl} = G M_{\rm BH} (t_{\rm d}/D_{\rm
cl})^2$ = 39 pc from the black hole. For NGC 1052 ($M_{\rm
BH}=10^{8.19} {\ensuremath{\rm M_\odot}\xspace}$; Woo & Urry 2002), a 4.56-year transit would place it $\geq20$ pc from the black hole. For NGC 4258, ($t_{\rm
d}$=6.81 yr, 2005 March – 2011 December; $M_{\rm BH} = 3.9\times 10^7
{\ensuremath{\rm M_\odot}\xspace}$, Miyoshi et al. 1995; Herrnstein et al. 1999), $r_{\rm cl}
\geq$ 11 pc.[^10] A cloud diameter of 0.1 light-days, also a feasible value given the range listed in Table \[tab:clumpsizemass\], would yield values of $r_{\rm
cl}$ an order of magnitude higher.
The clumpy-torus models typically assume that clouds cannot exist in large numbers out to several tens of $r_{\rm d}$, or very roughly 10, 5 and 0.5 pc for NGC 5506, NGC 1052, and NGC 4258, respectively[^11]. In these objects, it is thus not very likely that the same clouds used in the ${\textsc{Clumpy}}$ models can also be responsible for the non-variable baseline level of absorption.
Furthermore, Lamer et al. (2003) and Rivers et al. (2011b) demonstrated that the transiting clouds responsible for the NGC 3227/2000–1 and Cen A/2010–1 events were non-uniform, with a number density increasing towards the center. If it were the case that these two clouds were representative of all AGN clouds in terms of their density profiles, then any single cloud moving across the line of sight should produce a secular variation in observed $N_{\rm
H}$ and $HR1$. However, we find no evidence for any strong secular trend in any type II object, to the limits indicated by Table \[tab:boringsy2s\]. In addition, a transiting spherical cloud with a constant *column* density profile would require $n$ to decrease as one goes from the cloud edge to the center.
We conclude that it is difficult for clumpy-torus models to satisfactorily explain the constancy of $N_{\rm H,base}$, especially in the cases of the few longest-monitored and X-ray brightest type II objects observed with *RXTE*.
It is thus more likely that $N_{\rm H,base}$ arises in a non-clumpy, highly-homogeneous (to within the above $\Delta$$N_{\rm H}$ limits) medium; such a medium could be located at any radial location along the line of sight, subject to restrictions from the medium’s ionization parameter. That is, the cloud responsible for the $\la1$-d eclipse in NGC 5506 and the material responsible for $N_{\rm H,base} \sim 2\times10^{22}$ cm$^{-2}$, for instance, are likely physically separate entities. One possibility to explain $N_{\rm H,base}$ is a smooth, relatively low-density, dusty intercloud medium in which the relatively higher density clouds are embedded (Stalevski et al. 2012). To be consistent with the approximate range of baseline values of $N_{\rm H}$ typically measured in type IIs’ X-ray spectra ($0.3 - 30 \times 10^{22}$ cm$^{-2}$), the intercloud medium could have values of $\tau_{\rm V} \sim 1.6 - 160$ or $\tau_{10\micron} \sim 0.07 - 7$.[^12] However, we do not see evidence for such a medium in type I AGN.
Another possibility we explore is X-ray-absorbing gas distant from the black hole and associated with the host galaxy, as has been suggested e.g., by Bianchi [et al.]{} (2009) for NGC 7582. For example, dust lanes/patches associated with the host galaxy are considered a candidate for the obscuration of BLR lines in some systems (Malkan et al. 1998).
With the exception of Cen A (see $\S$\[sec:cenasmalldip\]), we rely on Shao et al. (2007) and Driver et al. (2007), who derive the inclination dependence of dust extinction for disk-dominated galaxies. In the optical bands, the extinction is typically $\la1-2$ mag, derived over a range of inclination angles from face-on to nearly edge-on ($i \sim 80-85 \degr$), corresponding to a maximum $N_{\rm H}
\sim 4 \times 10^{21}$ cm$^{-2}$ for a nearly edge-on disk. If the obscuring dust is distributed uniformly throughout the disk of the host galaxy, then such a component may explain relatively low observed values of $N_{\rm H,base}$ in, for instance, NGC 2992 ($N_{\rm H,base}
\sim 4 \times 10^{21}$ cm$^{-2}$) with the disk component of its host galaxy oriented close to edge-on (Jarrett et al. 2003). However, for objects with host disk inclinations far from edge-on (e.g., NGC 4258 and Mkn 348: Hunt et al. 1999; Jarrett et al. 2003) and/or for objects with $N_{\rm H,base} \sim$ a few $\times 10^{22}$ cm$^{-2}$ and higher, it is difficult for such a distribution of dust to be associated with $N_{\rm H,base}$. Of course, we cannot rule out the possibility of having more concentrated regions of gas lying along the line of sight not associated with the large-scale disk structure (such as a Giant Molecular Cloud). Dust-*free* X-ray-absorbing gas in any of these configurations is also a candidate.
A final possible explanation for constancy in $N_{\rm H}$ can be, as mentioned above, the presence of a large number of low-density clouds, with ${\hbox{${\cal N}_{i}$}}$ remaining nearly constant. However, this would require each cloud to have $N_{\rm H} \ll 10^{22}$ cm$^{-2}$. Such values contrast with those used by Nenkova et al. (2008b), RA09, RA11, and the values measured in this paper, but such clouds are not physically implausible. Contributions from such clouds, if they are dusty, to the total IR emission are negligible. However, the sum of many clouds along the line of sight that each have $N_{\rm H} \ll
10^{22}$ cm$^{-2}$, be they dusty or dust-free, could yield appreciable X-ray absorption.
### A small dip in $N_{\rm H,base}$ in Cen A {#sec:cenasmalldip}
Up to this point, our discussion of the baseline level of X-ray obscuration in type IIs has centered on constant (to within our sensitivity limits) values of $N_{\rm H, base}$. However, $N_{\rm H, base}$ for Cen A warrants more attention in light of the small dip in early 2010.
The famous dust lane crossing the host elliptical of Cen A supplies 3–6 mag. of optical extinction (Ebneter & Balick 1983), corresponding to $\sim5-10 \times 10^{21}$ cm$^{-2}$, far short of observed values of $N_{\rm H,base} \sim 1-2 \times 10^{23}$ cm$^{-2}$. It is generally accepted that a more compact and higher density distribution is responsible for observed values of $N_{\rm H}$ in X-ray spectra of Cen A.
One possibility is that we have witnessed one cloud leaving the line of sight followed by another cloud entering the line of sight two months later. Best-fitting values of $N_{\rm H}$($t$) indicate a drop from $21.7\pm0.9$ to $18.6^{+0.9}_{-0.8}$ and returning to $N_{\rm H}$($t$) $ = 21.9 \pm 0.7 \times 10^{22}$ cm$^{-2}$. These values suggest a scenario in which ${\hbox{${\cal N}_{i}$}}$ dropped temporarily from $\sim$ 7 to 6, with each cloud contributing $\Delta$$N_{\rm H} \sim 3 \pm 1 \times
10^{22}$ cm$^{-2}$, or less than half the column inferred to exist for the clouds causing the 2003–2004 and 2010–2011 *increases* in total $N_{\rm H}$. If all these clouds are part of the same clumpy structure and have a common origin, then the assumption of each cloud in a given AGN having exactly uniform values of $N_{\rm H}$ is an oversimplification. $N_{\rm H} = 3 \pm 1 \times 10^{22}$ cm$^{-2}$ is not an implausible value for a cloud, and only a factor of 2–3 less than that found for other clouds detected in our sample.
$N_{\rm H,base}$ in Cen A could also be attributed to non-clumpy, spatially-extended material that is not entirely homogeneous; perhaps this component can be associated with an intercloud medium. Here, $N_{\rm H,base}$ is usually $\sim 21.8 \times
10^{22}$ cm$^{-2}$, but there exists an *underdense* region with $\Delta$$N_{\rm H} = -3 \times 10^{22}$ cm$^{-2}$ that transited the line of sight in 2010. Assuming an arbitrary distance to the black hole of 100–200 light-days (the location of the absorbing clouds as per $\S$5.2), the $\sim$80-d duration of the dip implies that the underdense region is on the order of half a light-day across.
Longer-term variability in $N_{\rm H,base}$ may also exist: Rothschild et al. (2011) measured values of $14-19\times10^{22}$ cm$^{-2}$ during the 1996–2009 *RXTE* observations, excluding the 2003–2004 points. Hopefully, future X-ray monitoring will further resolve the variability in $N_{\rm H, base}$ and allow us to discriminate among the above scenarios.
SUMMARY
=======
The AGN community is in the process of shifting away from quantifying emission and absorption processes in Seyferts by modeling circumnuclear gas via a homogeneous, Compton-thick “donut” morphology. Instead, a new generation of models describe the torus via distributions of numerous individual clouds, usually preferentially distributed towards the equatorial plane. Often, the clouds are embedded in some outflowing wind from the cold, thin accretion disk that feeds the black hole. Observational support for these models so far has come mainly from fitting IR SEDs in small samples of Seyferts. However, a statistical survey of the environment around supermassive black holes has been needed to properly constrain parameter space in the clumpy-absorber models.
We present the first such survey, the longest AGN X-ray monitoring study to date. Our survey quantifies line-of-sight X-ray absorption by clouds that transit the line of sight to the central engine. Our goals are to assess the relevance of clumpy-torus models as a function of optical classification by exploring absorption over a wide range of length scales (both inside and outside the dust sublimation zone), and to explore links between X-ray absorbers, IR-emitting dusty clouds, and the BLR.
We use the vast public archive of *RXTE* observations of AGN. The archive features a wide array of sustained monitoring campaigns that make us sensitive to variability in line of sight absorption over a high dynamic range of timescales spanning from 0.2 d to 16 yr. Our final sample consists of 37 type I and 18 Compton-thin-obscured type II Seyferts and totals 230 “object-years,” the largest ever available for statistical studies of cloud events in AGN on timescales from days to years.
We use hardness ratio light curves to identify potential eclipse events, and attempt to confirm the events with follow-up time-resolved spectroscopy. We are sensitive to full-covering, neutral or lowly-ionized clouds with columns densities $\ga 10^{22}$ up to $\sim10^{25}$ cm$^{-2}$. Our results are thus complementary to those derived with $\sim$1 d long-look observations using missions with *Chandra*, *XMM-Newton*, and *Suzaku*.
Our primary results are as follows:
$\bullet$ We find 12 “secure” X-ray absorption events in eight Seyferts (confirmed with spectral fitting) plus four “candidate” eclipses in three Seyferts. As four eclipse events were published previously (NGC 3227/2000–1, Cen A/2003–4, Cen A/2010–1, and Mkn 348/1996–7), we triple the number of events detected in the *RXTE* archive. The events span a wide range in duration, from $\la1$ d to over a year. We model the eclipsing clouds to have column densities spanning $4-26 \times 10^{22}$ cm$^{-2}$. Importantly, we do not detect any full-covering clouds that are Compton thick, although if such clouds are partial-covering clouds then our experiment would not be highly sensitive to them.
$\bullet$ We derived the probability to catch a type I/II source undergoing an eclipse event that has any duration between 0.2 d and 16 yr, taking into account the inhomogeneous sampling in our X-ray observations. For type Is, it is 0.006 (conservative range: 0.003–0.166); for type IIs, 0.110 (0.039–0.571). Our uncertainties are conservative, as they take into account our selection function, candidate eclipse events in addition to the secure ones, uncertainties in the observed durations, and uncertainties in the contributions of individual objects’ sampling patterns to our total sensitivity function. As a reminder, these values indicate the probability to observe a source undergoing an eclipse event (of any duration $t_{\rm
D}$ between 0.2 d and 16 yr) due only to a cloud passage through the line of sight, and are independent of long-term constant absorption, e.g., associated with gas in the host galaxy. We conservatively estimate the upper limit to observe a Compton-thick eclipse event (with 0.2 d $< t_{\rm D} <$ 16 yr) to be $<$0.158 or $<$0.520 in type I and II objects, respectively. To repeat the caveat from $\S4$, these probabilities refer only to eclipses by full-covering, neutral or lowly-ionized clouds with column densities $\ga 10^{22}$ up to $\sim10^{25}$ cm$^{-2}$; when one considers the full range of clouds (larger range of $N_{\rm H}$, partial-covering clouds, wider range of ionization) the resulting probabilities will almost certainly be higher.
Although subject to low number statistics, our observations indicate differences in the distributions of observed eclipse event durations and probabilities for type I and II objects. The type I objects have $\sim4$ times as many campaigns and twice the number of targets as the type IIs, but despite this, we do not detect eclipse events with durations longer than 100 d (although we cannot rule this out for the poorly-constrained event in MR 2251–178). If it were the case that eclipses with durations of $\sim$hundreds of days occurred in type Is with the same frequency density as in type IIs, then the monitoring campaigns on the type Is should have detected them. This implies that we are “missing” clouds in type Is along the line of sight at radial distances of $\ga$ a few $\times 10^5 R_{\rm g}$. This does not necessarily imply *intrinsic* differences in the cloud distributions of between type Is and IIs, just potential differences along the line of sight.
Despite the generally low probabilities of observing an eclipse, 4/10 objects show secure or candidate eclipse events multiple times. Perhaps the system is close to edge-on, and/or the total number of clouds is very high in these objects. If this is the case, then an object for which an eclipse has been detected has a higher chance of showing an additional eclipse compared to sources with no eclipses so far. In addition, perhaps there exists some dispersion between the averaged derived probabilities of cloud eclipses for a class of objects and the probabilities for individual objects.
$\bullet$ We see no obvious dependence of the likelihood to observe eclipses in a given object on the usual AGN parameters ($M_{\rm BH}$, $L_{\rm Bol}$, radio loudness).
$\bullet$ We estimate the locations of the clouds from the central black hole $r_{\rm cl}$ based on ionization parameter. Best-estimate values of $r_{\rm cl}$ are typically tens to hundreds of light-days, or $1-50 \times 10^4 R_{\rm g}$ ($0.3 - 140 \times 10^{4} R_{\rm g}$ accounting for uncertainties). In 7/8 objects, the clouds are consistent with residing at the estimated location of the dust sublimation zone (see Fig. \[fig:plotrs\]). The eighth object is Cen A, where the clouds are estimated to exist just outside the dust sublimation zone. For the three objects in our sample whose dusty tori have been mapped via either IR interferometry or optical-to-near-IR reverberation mapping (Cen A, NGC 3783, and NGC 3227), the clouds’ radial distances are commensurate with those of the IR-emitting tori. For six of the seven objects with estimates of the locations of optical/UV/IR BLR clouds, the X-ray-absorbing clumps are commensurate with the outer portions of the BLR or radial distances outside the known BLR within factors of a few to 10. Only in one object (NGC 5506) is the eclipsing cloud inferred to be commensurate with the inner BLR. This cloud thus may be akin to those clouds inferred to exist in the BLRs of other objects (namely NGC 1365).
Our results thus confirm the existence of X-ray-absorbing, neutral/low-ionization clouds with $N_{\rm H} \sim10^{22-23}$ cm$^{-2}$ at these indicate radial distances from the black hole. This is not to say that X-ray-absorbing clouds cannot exist at other distances; our selection function analysis indicates that we are biased towards detecting eclipses with durations of $\sim$tens of days. The 12 “secure” clouds have an average diameter of 0.25 light-days, an average number density of $3\times 10^{8}$ cm$^{-3}$, and an average mass of $10^{-5.2} {\ensuremath{\rm M_\odot}\xspace}$. We find no statistically significant difference between the individual cloud properties of type Is and IIs. Assuming that the clouds are 100 per cent full covering, their inferred diameters imply upper limits to the size of the X-ray continuum-emitting region. These limits are as low as $2 R_{\rm g}$ for NGC 5506 and $45-65 R_{\rm g}$ for NGC 3783, Mkn 79, and Mkn 509.
Given their inferred locations and diameters, the X-ray-absorbing clouds are likely too small to obscure the view of the entire BLR in the type Is studied here. However, such clouds could potentially obscure and redden large parts of the BLR and temporarily turn a type I into a type II AGN over timescales of months to years. The clouds would need to be sufficiently dusty, be at least several light-days in diameter, and be located many light-months away from the black hole. Such events have been observed as variations in optical broad line strengths (Goodrich 1989; Tran et al. 1992; Aretxaga et al. 1999). The probability to observe an X-ray eclipse in a type I by either a dusty or dust-free cloud, derived in $\S$4, thus does not serve as a prediction for the probability to observe a type classification change in the optical band. The latter can also occur via changes in geometry or illumination of the BLR (Cohen et al. 1986).
$\bullet$ Three eclipse events (NGC 3783/2008.3, NGC 3227/2000–1, and Cen A/2010–1) have column density profiles $N_{\rm H}$($t$) that are well resolved in time. The event in NGC 3783 at 2008.3 is the best time-resolved $N_{\rm H}$($t$) profile not yet published, and seems to show evidence for a double-peaked profile with peaks separated by 11 d, with $N_{\rm H}$($t$) not returning to baseline levels in the period between the peaks. The profile could be explained by a dumbbell-shaped (along the direction of transit) cloud. One possibility is that we are witnessing an eclipse by a cloud in the process of being tidally disrupted, or may be filamentary in structure. Models incorporating clumpy disk winds (e.g., MHD-driven or IR radiation-driven) as opposed to discrete, compact structures, may also be relevant for explaining the observed profile. Such models characterize the torus as a highly dynamic structure, with uplifts and failed winds in addition to Keplerian motion. In contrast, the density profiles for the clouds in Cen A and NGC 3227 have been modeled by Rivers et al. (2011b) and Lamer et al. (2003), respectively, to be centrally peaked, suggesting that self-gravity likely dominates over tidal shearing or internal turbulence.
$\bullet$ We provide constraints for the parameters used to quantify the cloud distributions in the ${\textsc{Clumpy}}$ models ($\tau_{\rm V}$, $Y$, ${\hbox{${\cal N}_0$}}$, $i$, $\sigma$), which so far have been constrained observationally only via IR SED fitting. As some of our clouds are in the dust sublimation zone, the extent to which our constraints for models that are based on the IR-emitting region is not fully understood. However, our study provides the first ever statistical constraints from *X-ray* observations and constitutes a completely independent way of studying the structure and geometry of the environment around supermassive black holes. The X-ray constraints on these parameters derived in this paper can be incorporated into the priors of future Bayesian IR SED fitting analyses. Our X-ray column densities are equivalent to values of $\tau_{\rm V}$ of 20–132, consistent with typical values used by clumpy-torus theorists (Nenkova et al. 2008b) and constrained so far by IR SED modeling (e.g., RA11). We find clouds out to $Y=r_{\rm
cl}/R_{\rm d} \sim 20$, also consistent with the IR observations. Constraints on $q$ based on our X-ray data will be provided in a separate paper (Nikutta et al., in preparation).
The fact that we observe eclipses in both type Is and type IIs disfavors sharp-edged cloud distributions and supports torus models with as soft, e.g., Gaussian, distribution above/below the equatorial plane. Furthermore, we compare our constraints on the integrated probability to witness a type I or type II object undergoing an eclipse (of any duration) to the predictions $P_{\rm obsc}$ from ${\textsc{Clumpy}}$ to obtain constraints in (${\hbox{${\cal N}_0$}}$, $\sigma$, $i$) parameter space. When there is external information on any one of these parameters (e.g., the inclination of the system), then constraints on the other two parameters can be obtained following Fig. \[fig:pobscured\].
$\bullet$ We find evidence in eight type II Seyferts for a baseline level of X-ray absorption that remains constant (down to $\sim 0.6-9
\times 10^{22}$ cm$^{-2}$) over timescales from 0.6 to 8.4 yr. The clouds we detect in this paper are not able to explain the constant baseline absorbers, since we would have expected many more “negative” cloud events than observed. The constant total amount of X-ray absorption in type IIs can be explained in the context of clumpy-absorber models only if there exists a large number of very low-density ($\ll 10^{22}$ cm$^{-2}$) clouds along the line of sight, with ${\hbox{${\cal N}_0$}}$ remaining roughly constant within each object. Alternatively, there can exist in type IIs nearly homogeneous X-ray-absorbing gas, whose location along the line of sight in most cases is unconstrained. One possibility is that this gas lies on the order of tens to thousands of parsecs away from the black hole, and is associated with X-ray-absorbing matter in the host galaxy. Another possibility is a medium of non-clumpy, relatively homogeneous gas located further in, such as a relatively low-density intercloud medium in which higher density clumps are embedded (Stalevski et al. 2012). In the case of Cen A, in addition to the two eclipses observed, we find evidence that the baseline level of X-ray absorption dipped by $\sim14$ per cent and then recovered in early 2010. This is consistent with the notion that the material describing $N_{\rm H,base}$ in this object is relatively close to the black hole and indeed not perfectly homogeneous, and that an underdense region transited the line of sight.
*In summary,* our findings are consistent with the notion that both type I and II objects contain clouds with roughly similar properties ($N_{\rm H}$, mass, location from the black hole). The probability to observe an eclipse in type IIs is higher than in type Is. However, in addition in type IIs there is frequently a baseline level of X-ray absorption that is not likely due to clouds, but instead due to an additional, almost homogeneous absorber of unconstrained location (intercloud medium or gas associated with the host galaxy). An exception is Cen A, where the observation of a small dip in the baseline level is consistent with an intercloud medium commensurate with the dusty and X-ray tori.
*Future observational and theoretical work:* Much more observational work needs to be performed to further support and properly test the characteristics of the clumpy-torus model and its key parameters. To date, *RXTE* has been unique in its ability to provide multi-timescale X-ray monitoring for such a large number of AGN. The only future mission potentially capable of creating such an X-ray monitoring archive will be extended Röntgen Survey with an Imaging Telescope (eROSITA) onboard the Russian spacecraft *Spektrum Röntgen/Gamma* (Predehl et al. 2011). eROSITA will scan the entire sky several times during its mission.
Ideally, a repeat of the experiment should aim to uncover X-ray eclipse events spanning as wide a dynamic range in duration and inferred radial distance as possible. In particular, we need better sustained monitoring (with minimal gaps in coverage) on timescales $\ga$ a few years, especially in the type II Seyferts. At the other end, *RXTE* conducted a number of “intensive monitoring” campaigns, featuring, e.g., observations four times daily for 1–2 months. These intensive campaigns allowed us to detect the shortest-duration eclipse event in our sample (the $\la$1 d event in NGC 5506) and enabled us to extract a high-quality $N_{\rm H}$($t$) profile for the NGC 3783/2008.3 eclipse event, and thus illustrate the necessity of intensive sampling.
We need more sustained monitoring of Seyferts spanning a wide range of luminosities, including in the low-luminosity regime, to test the suggestion that the disk wind/torus cloud outflow disappears at low luminosities (Elitzur & Ho 2009).
Finally, additional knowledge about the inclination angle of these accreting black hole systems would help us better constrain the $\sigma$ parameter. Such information can come from, e.g., X-ray spectroscopic modeling of relativistically-broadened reflected emission (Fe K$\alpha$ emission lines and soft excess) from the inner accretion disk (e.g., Patrick et al. 2012).
On the theoretical side, the community can benefit from *dynamical* (time-dependent) models that yield the expected durations and frequency of observed eclipse events (depending on clouds’ diameters and distribution, source viewing angle, source luminosity, etc.). Advances in understanding can also come from models that further explore the interaction between the intercloud medium and the clouds embedded in it.
acknowledgements {#acknowledgements .unnumbered}
================
The authors are very grateful to M. Elitzur for helpful comments. The authors also thank the referee for helpful comments. This research has made use of data obtained from the *RXTE* satellite, a NASA space mission. This work has made use of HEASARC online services, supported by NASA/GSFC, and the NASA/IPAC Extragalactic Database, operated by JPL/California Institute of Technology under contract with NASA. The research was supported by NASA Grant NNX11AD07G and funding from the European Community’s Seventh Framework Programme (/FP7/2007-2013/) under grant agreement no. 229517. RN acknowledges support by the ALMA-CONICYT fund, project no. 31110001. The authors thank Wolfgang Steffen for providing images and video in support of the press release associated with this paper; material can be found at http://cass.ucsd.edu/$\sim$rxteagn/clumpytorus/.
Abdo, A.A., et al. 2010, ApJ, 719, 1433 Akylas A., Georgantopoulos I., Griffiths R.G., Papadakis I.E., Mastichiadis A., Warwick R.S., Nandra K., Smith D.A., 2002, MNRAS, 332, L23 Alonso-Herrero A., et al. 2011, ApJ, 736, 82 Anders E., Grevesse N., 1989, Geochim. Cosmochim. Acta, 53, 197 Antonucci R.R.J., Miller J.S., 1985, ApJ, 297, 621 Arévalo P., Uttley P., Kaspi S., Breedt E., Lira P., McHardy I.M., 2008, MNRAS, 389, 1479 Aretxaga I., Joguet B., Kunth D., Melnick J., Terlevich R.J., 1999, ApJ, 519, L123 Asensio Ramos A., Ramos Almeida C., 2009, ApJ, 696, 2075 Awaki H. et al., 2008, PASJ, 60S, 293 Barr P., White N.E., Sanford P.W., Ives J.C., 1977, MNRAS, 181, 43P Barth A., Filippenko A.V., Moran E.C., 1999, ApJ, 515, L61 Beckert T., Dribe T., Hönig S.F., Weigelt G., 2008, A&A, 486, L17 Bianchi S., Guainazzi M., 2007, in Antonelli, L.A., et al., eds, AIP Conf. Ser. Vol. 924, The Multicolored Landscape of Compact Objects and Their Explosive Origins, Am. Inst. Phys., New York, p. 822 Bianchi S., Piconcelli E., Chiaberge M., Bailón E.J., Matt G., Fiore F., 2009, ApJ, 695, 781 Breedt E. et al., 2009, MNRAS, 394, 427 Brightman M., Nandra K., 2011, MNRAS, 413, 1206 Burtscher L., Meisenheimer K., Jaffe W., Tristram K.R.W., Röttgering H.J.A., 2010, PASA, 27, 290 Chartas G., Kochanek C.S., Dai X., Moore D., Mosquera A.M., Blackburne J.A., 2012, ApJ, 757, 137 Chatterjee R. et al., 2011, ApJ, 734, 43 Chen B., Dai X., Kochanek C.S., Chartas G., Blackburne J.A., Morgan C.W., 2012, ApJ, 755, 24 Chiaberge M., Capetti A., Celotti A., 2001, MNRAS, 324, L33 Cohen R.D., Puetter R.C., Rudy R.J., Ake T.B., Foltz C.B., 1986, ApJ, 311, 135 Collmar W. et al., 2010, A&A, 522, A66 Czerny B., Hryniewicz K., 2011, A&A, 525, L8 Dadina M., 2007, A&A, 461, 1209 Dapp W.B., Basu S., 2009, MNRAS, 395, 1092 Denney K.D. et al., 2010, ApJ, 721, 715 de Rosa A., Piro L., Perola G.C., Capalbi M., Cappi M., Grandi P., Maraschi L., Petrucci P.O., 2007, A&A, 463, 903 Done C., Gierliński M., 2005, MNRAS, 364, 208 Dorodnitsyn A., Kallman T., 2012, ApJ, 761, 70 Dorodnitsyn A., Kallman T., Bisnovatyi-Kogan G.S., 2012, ApJ, 747, 8 Driver S.P., Popescu C.C., Tuffs R.J., Liske J., Graham A.W., Allen P.D., de Propris R., 2007, MNRAS, 379, 1022 Ebneter K., Balick B., 1983, PASP, 95, 675 Elitzur M., 2007, in Ho. L.C., Wang, J.-M., eds, ASP Conf. Ser. Vol. 373, The Central Engine of Active Galactic Nuclei, Astron. Soc. Pac., San Francisco, p. 415 Elitzur M., Ho L.C., 2009, ApJ, 701, L91 Elitzur M., Shlosman I., 2006, ApJ, 648, L101 Elvis M., 2012, J. Phys. Conf. Ser., 372, 012032 Evans D.A., Worrall D.M., Hardcastle M.J., Kraft R.P., Birkinshaw M., 2006, ApJ, 642, 96 Evans I.N., Ford H.C., Kriss G.A., Tsvetanov Z., 1994, in Bicknell G.V., Dopita, M.A., Quinn P.J., eds, ASP Conf. Ser. Vol. 54, The First Stromlo Symposium: The Physics of Active Galaxies, Astron. Soc. Pac., San Francisco, p. 3 Fasano G., Vio R., 1988, Bull. Inf. Cent. Données Stellaires, 35, 191 Fukumura K., Kazanas D., Contopoulos I., Behar E., 2010, ApJ, 715, 636 Gallo L.C., Miniutti G., Miller J.M., Brenneman L.W., Fabian A.C., Guainazzi M., Reynolds C.S., 2011, MNRAS, 411, 607 Garcet O. et al., 2007, A&A, 474, 473 Gaskell C.M., Goosmann R.W., Klimek E.S., 2008, Mem. Soc. Astron. Ital., 79, 1090 Gibson R.R., Marshall H.L., Canizares C.R., Lee J.C., 2005, ApJ, 627, 83 Gofford J. et al., 2011, MNRAS, 414, 3307 González-Martín O. et al., 2009, A&A, 506, 1107 Goodrich R.W., 1989, ApJ, 340, 190 Goodrich R.W., Veilleux S., Hill G.J., 1994, ApJ, 422, 521 Haardt F., Maraschi L., Ghisellini G., 1997, ApJ, 432, L95 Hao L., Weedman D.W., Spoon H.W.W., Marshall J.A., Levenson N.A., Elitzur M., Houck J.R., 2007, ApJ, 655, L77 Herrnstein J. et al., 1999, Nature, 400, 539 Holt S.S., Mushotzky R.F., Boldt E.A., Serlemitsos P.J., Becker R.H., Szymkowiak A.E., White N.E., 1980, ApJ, 241, L13 Hönig S.F., 2013, in Packham C., Mason R., Alonso-Herrero A., eds, Proc. of the Torus Workshop 2012, San Antonio, preprint (arXiv:1301.1349) Horst H., Smette A., Gandhi P., Duschl W.J., 2006, A&A, 457, L17 Hunt L.K., Malkan M.A., Rush B., Bicay M.D., Nelson B.O., Stanga R.M., Webb, W., 1999, ApJS, 125, 349 Jaffe W. et al., 2004, Nature, 429, 47 Jahoda K., Markwardt C.B., Radeva Y., Rots A.H., Stark M.J., Swank J.H., Strohmayer T.E., Zhang W., 2006, ApJS, 163, 401 Jarrett T.H., Chester T., Cutri R., Schneider S.E., Huchra J.P., 2003, AJ, 125, 525 Kalberla P.M.W. et al., 2005, A&A, 440, 775 Kaspi S., Netzer H., Chelouche D., George I.M., Nandra K., Turner T.J., 2004, ApJ, 611, 68 Kelly B.C., Bechtold J., 2007, ApJS, 168, 1 Kishimoto M., Hönig S.F., Antonucci R., Barvainis R., Kotani T., Tristram K., Weigelt G., Levin K., 2011, A&A, 527, A121 Kishimoto M., Hönig S.F., Antonucci R., Kotani T., Barvainis R., Tristram K., Weigelt G., 2009, A&A, 507, L57 Konigl A., Kartje J.F., 1994, ApJ, 434, 446 Krawczynski H., Coppi P.S., Aharonian F., 2002, MNRAS, 336, 721 Krolik J., 2007, ApJ, 661, 52 Krolik J., Begelman M.C., 1986, ApJ, 308, L55 Krolik J., Begelman M.C., 1988, ApJ, 329, 702 Lamer G., Uttley P., McHardy I.M., 2003, MNRAS, 342, L41 Landt H., Bentz M.C., Ward M.J., Elvis M., Peterson B.M., Korista K.T., Karovska M., 2008, ApJS, 174, 282 Lasota J.-P., Abramowicz M.A., Chen X., Krolik J., Narayan R., Yi I., 1996, ApJ, 462, L142 Lira P., Ward M.J., Zezas A., Murray S.S., 2002, MNRAS, 333, 709 Lira P., Arévalo P., Uttley P., McHardy I., Breedt E., 2011, MNRAS, 415, 1290 Lohfink A., Reynolds C.S., Mushotzky R.F, Wilms J., 2012, ApJ, 749, L31 Lumsden S.L., Alexander D.M., Hough J.H., 2004, MNRAS, 348, 1451 Lutz D., Maiolino R., Moorwood A.F.M., Netzer H., Wagner S.J., Sturm E., Genzel R., 2002, A&A, 396, 439 Lutz D., Maiolino R., Spoon H.W.W., Moorwood A.F.M., 2004, A&A, 418, 465 Maiolino R., Marconi A., Salvati M., Risaliti G., Severgnini P., Oliva E., La Franca F., Vanzi, L., 2001, A&A, 365, 28 Maiolino R. et al., 2010, A&A, 517, A47 Malkan M.A., Gorjian V., Tam R., 1998, ApJS, 117, 25 Marconi A., Risaliti G., Gilli R., Hunt, L.K., Maiolino R., Salvati M., 2004, MNRAS, 351, 169 Markoff S., Nowak M.A., Wilms J., 2005, ApJ, 635, 1203 Markowitz A. et al., 2003, ApJ, 593, 96 Markowitz A. et al., 2007, ApJ, 665, 209 Markowitz A. et al., 2008, PASJ, 60, S277 Markowitz A., Reeves J.N., George I.M., Braito V., Smith R., Vaughan S., Arévalo P., Tombesi F., 2009, ApJ, 691, 922 Mateos S. et al., 2010, A&A, 510, A35 McClintock J.E., Remillard R.A., 2003, in “Compact Stellar X-ray Sources,” eds. Lewin W.H.G., van der Klis M., Cambridge University Press (astro-ph/0306213) McHardy I.M., Koerding E., Knigge C., Uttley P., Fender R.P. 2006, Nature, 444, 730 Meisenheimer K. et al., 2007, A&A, 471, 453 Miller J.S., Goodrich R.W., 1990, ApJ, 355, 456 Minezaki T., Yoshii Y., Kobayashi Y., Enya K., Suganuma M., Tomita H., Aoki T., Peterson B.A., 2004, ApJ, 600, L35 Miniutti G., et al., 2014, MNRAS, 437, 1776 Miyoshi M., Moran J., Herrnstein J., Greenhill L., Nakai N., Diamond P., Inoue M., 1995, Nature, 373, 127 Moran E.C., Barth A., Kay L.E., Filippenko A.V., 2000, ApJ, 540, L73 Nagar N.M., Oliva E., Marconi A., Maiolino R., 2002, A&A, 391, L21 Nandra K., Le T., George I.M., Edelson R.A., Mushotzky R.F., Peterson B.M., Turner T.J., 2000, ApJ, 544, 734 Nenkova M., Sirocky M., Ivezić Ž., Elitzur M., 2008a, ApJ, 685, 147 Nenkova M., Sirocky M., Nikutta R., Ivezić Ž., Elitzur M., 2008b, ApJ, 685, 160 Netzer H., Laor A., 1993, ApJ, 404, L51 Nikutta R., Elitzur M., Lacy M., 2009, ApJ, 707, 1550 Osterbrock D.E., 1981, ApJ, 249, 462 Papadakis I.E., 2004, MNRAS, 348, 207 Papadakis I.E., Petrucci P.O., Maraschi L., McHardy I.M., Uttley P., Haardt F., 2002, ApJ, 573, 92 Patrick A.R., Reeves J.N., Porquet D., Markowitz A., Braito V., Lobban A.P., 2012, MNRAS, 426, 2522 Perola G.C. et al., 2004, A&A, 421, 491 Peterson B.M. et al., 2004, ApJ, 613, 682 Poncelet A., Perrin G., Sol H., 2006, A&A, 450, 483 Pott J.-U., Malkan M.A., Elitzur M., Ghez A.M., Herbst T.M., Schödel R., Woillez, J., 2010, ApJ, 713, 736 Predehl P., Schmidt J., 1995, A&A, 293, 889 Predehl P. et al., 2011, Proc. SPIE, 8145, 247 Puccetti S., Fiore F., Risaliti G., Capalbi M., Elvis M., Nicastro F., 2007, MNRAS, 377, 607 Raban D., Jaffe W., Röttgering H., Meisenheimer K., Tristram K., 2009, MNRAS, 394, 1325 Ramos Almeida C. et al., 2009, ApJ, 702, 1127 (RA09) Ramos Almeida C. et al., 2011, ApJ, 731, 92 (RA11) Reeves J.N., Turner M., 2000, MNRAS, 316, 234 Reeves J.N., Nandra K., George I.M., Pounds K., Turner T.J., Yaqoob T., 2004, ApJ, 602, 648 Reis R.C., Miller, J.M., 2013, ApJ, 769, L7 Reunanen J., Kotilainen J.K., Prieto M.A., 2003, MNRAS, 343, 192 Risaliti G., Elvis M., Nicastro F., 2002, ApJ, 571, 234 (REN02) Risaliti G., Elvis M., Fabbiano G., Baldi A., Zezas A., 2005, ApJ, 623, L93 Risaliti G., Elvis M., Fabbiano G., Baldi A., Zezas A., Salvati M., 2007, ApJ, 659, L111 Risaliti G. et al., 2009a, MNRAS, 393, L1 Risaliti G. et al., 2009b, ApJ, 696, 160 Risaliti G., Nardini E., Salvati M., Elvis E., Fabbiano G., Maiolino R., Pietrini P., Torricelli-Ciamponi G., 2011, MNRAS, 410, 1027 Rivers E.S., Markowitz A., Rothschild R.E., 2011a, ApJS, 193, 3 Rivers E.S., Markowitz A., Rothschild R.E., 2011b, ApJ, 742, L29 Rivers E.S., Markowitz A., Rothschild R.E., 2013, ApJ, 772, 114 Rothschild R.E., Markowitz A., Rivers E.S., Suchy S., Pottschmidt K., Kadler M., Müller C., Wilms J., 2011, ApJ, 733, 23 Roustazadeh P., Böttcher M., 2011, ApJ, 728, 134 Sanfrutos M., Miniutti G., Agís-González B., Fabian A.C., Miller J.M., Panessa F., Zoghbi A., 2013, MNRAS, 436, 1588 Schartmann M., Meisenheimer K., Camenzind M., Wolf S., Henning T., 2005, A&A, 437, 861 Schurch N., Warwick R.S., 2002, MNRAS, 334, 811 Shao Z., Xiao Q., Shen S., Mo H.J., Xia X., Deng Z., 2007, ApJ, 659, 1159 Shih D.C., Iwasawa K., Fabian A.C., 2002, MNRAS, 333, 687 Shu X.W., Yaqoob T., Wang J.X., 2010, ApJS, 187, 581 Smith D.A., Georgantopoulos I., Warwick R.S., 2001, ApJ, 550, 635 Sobolewska M.A., Papadakis I.E., 2009, MNRAS, 399, 1597 Soldi S. et al., 2008, A&A, 486, 411 Stalevski M., Fritz, J., Baes M., Nakos T., Popović L., 2012, MNRAS, 420, 2756 Steinle H., 2010, PASA, 27, 431 Sturm E. et al., 2005, ApJ, 629, L21 Suganuma M. et al., 2006, ApJ, 639, 46 Sulentic J.W., Bachev R., Marziani P., Negrete C.A., Dultzin D., 2007, ApJ, 666, 757 Taylor R.D., Uttley P., McHardy I.M., 2003, MNRAS, 342, L31 Tran H.D., 1995, ApJ, 440, 565 Tran H.D., Osterbrock D.E., Martel A., 1992, AJ, 104, 2072 Tristram K. et al., 2007, A&A, 474, 837 Tristram K. et al., 2009, A&A, 502, 67 Turner T.J., Reeves, J.N., Kraemer, S.B., Miller L., 2008, A&A, 483, 161 Urry C.M., Padovani P., 1995, PASP, 107, 803 Vasudevan R.V., Fabian A.C., 2009, MNRAS, 392, 1124 Vasudevan R.V., Fabian A.C., Gandhi P., Winter L.M., Mushotzky R.F., 2010, MNRAS, 402, 1081 Veilleux S., Goodrich R.W., Hill G.J., 1997, ApJ, 477, 631 Verner D.A., Ferland G.J., Korista K.T., Yakovlev D.G., 1996, ApJ, 465, 487 Vestergaard M., Peterson B.M., 2006, ApJ, 641, 689 Wang J., Mao Y.F., Wei J.Y., 2009, AJ, 137, 3388 Wang J.-M., Zhang E.-P., 2007, ApJ, 660, 1072 Wilms J., Allen A., McCray R., 2000, ApJ, 542, 914 Woo J.-H., Urry C.M., 2002, ApJ, 579, 530 Yang Y., Wilson A.S., Matt G., Terashima Y., Greenhill, L.J., 2009, ApJ, 691, 131
Details of eclipse identification in individual objects
=======================================================
Flux, hardness ratio, and $\Gamma_{\rm app}$ light curves are shown in Figs. \[fig:mega3783\] – \[fig:mega348\] for NGC 3783, MR 2251–178, Mkn 79, Mkn 509, NGC 3227, Cen A, NGC 5506, and Mkn 348, respectively, the objects with “secure” events, plus one “candidate” event for NGC 3783. Also included are plots of two sources with only “candidate” events, Fairall 9 (Fig. \[fig:megafrl9\]) and NGC 3516 (Fig. \[fig:mega3516\]). Errors on each light curve point are 68 per cent confidence. All other sources’ light curves do not yield any strong statistically significant deviations and we do not include them in this paper for brevity. Yellow shaded areas indicate times of events for which we conducted follow-up time-resolved spectroscopy to confirm events as “secure.” We bin up consecutive individual spectra around the period of each candidate event to achieve sufficient variability/noise in binned spectra. Bin sizes are determined as a trade-off between the need to achieve small errors in $\Gamma$ or $N_{\rm H}$ in binned spectra versus the need to trace out the $N_{\rm H}$ profile if possible. The number of 2–10 keV counts needed to deconvolve $N_{\rm H}$ and $\Gamma$ is typically $\sim$ 80000, with typically $\sim$ 15000 counts needed to achieve reasonable constraints on $N_{\rm H}$ assuming a fixed value of $\Gamma$. All spectra are grouped to a minimum of 25 counts bin$^{-1}$. Due to the gradual evolution of the PCA response over time, response files are generated for each observation separately, using <span style="font-variant:small-caps;">pcarsp</span> version 11.7.1. All spectral fitting is done with <span style="font-variant:small-caps;">isis</span> version 1.6.2-16. Uncertainties on all parameters derived from spectral fits, including $N_{\rm H}$ and $\Gamma_{\rm app}$, are 90 per cent confidence ($\Delta\chi^2 = 2.71$ for one interesting parameter) unless otherwise noted.
We apply the best-fit model from the time-averaged *RXTE* spectrum derived by Rivers [et al.]{} (2011a, 2013). In all fits, we keep the energy centroid and width of the Fe K$\alpha$ line and all parameters associated with the Compton reflection hump (except for normalization, which was tied to that of the incident power law) frozen at their time-averaged values. Fe K$\alpha$ emission line intensities are kept frozen unless there is significant improvement in the fit. In all spectral fits to sources that are normally X-ray unobscured (all are type I objects), we consider two models. Model 1 has model components identical to the time-averaged model: power law with $\Gamma$ and 1 keV normalization $A_1$ kept free, Fe K$\alpha$ emission line, Compton reflection hump modeled with <span style="font-variant:small-caps;">pexrav</span>, and $N_{\rm Gal}$ modeled with <span style="font-variant:small-caps;">phabs</span>. In Model 2, we model $N_{\rm H}$ (absorption by neutral gas only at the source, i.e., in excess of $N_{\rm Gal}$) with <span style="font-variant:small-caps;">zphabs</span>. We keep $\Gamma$ free only in the high signal-to-noise cases where $\Gamma$ and $N_{\rm H}$ can be deconvolved with a minimum of degeneracy (“secure A” events). Otherwise $\Gamma$ is kept frozen at either the time-averaged value or the value derived from unabsorbed spectra surrounding the putative obscuration event (“secure B” events). For sources normally X-ray obscured (all are type II objects), $N_{\rm H}$ as modeled with <span style="font-variant:small-caps;">zphabs</span> is included in all fits.
All analysis in this paper uses the cosmic abundances of Wilms et al. (2000) and the cross sections of Verner [et al.]{} (1996) unless otherwise stated. Many previous publications on the objects discussed herein used cosmic abundances similar to those of Anders & Grevesse (1989), which have C, N, O, Ne, Si, S, and Fe elemental abundances relative to H spanning factors of 1.3–1.9 lower than those in Wilms et al. (2000). For a model consisting of a power law with photon index $\Gamma=1.8$ this difference yields values of $N_{\rm H}$ using the <span style="font-variant:small-caps;">angr</span> abundances a factor of $\sim30-40$ per cent greater than corresponding values of using the <span style="font-variant:small-caps;">wilm</span> abundances for the column densities encountered in our sample.
There is a known issue with the estimation of background errors by the *RXTE* data reduction software: the software models the background counts spectrum based on long blank-sky observations, and then assumes Poissonian errors for the background counts spectrum appropriate for the exposure time of the observation of the target. However, the unmodeled residual variance in the background is on the order of 1–2 per cent (Jahoda et al. 2006), and for spectra with exposure times higher than $\sim10-30$ ks, the Poissonian errors may be overestimates. The errors on the net (background-subtracted) spectrum are thus likely overestimates (Nandra et al. 2000), frequently yielding best fits with values of $\chi^2_{\rm r}$ $\sim$ 0.6–0.8 for many of the time-resolved spectral fits below. Because these errors are overestimates, uncertainties on best-fitting model parameters reported in the tables are conservative.
NGC 3783 (Sy 1)
---------------
The long-term light curves for NGC 3783 are shown in Fig. \[fig:mega3783\]. The $HR1$ light curve shows several sharp deviations upward from the mean of 0.9:
$\bullet$ *2008.3: (“Secure B”)* During a period of intensive monitoring one point every $\sim$6 hrs in early 2008, there are two “spikes” to values of $HR1$ = 1.5–1.6 ($4-5\sigma$ deviations), separated by only 11 d and peaking at MJD $\sim$ 54567.0 and 54577.7. The light curves are plotted in Fig. \[fig:mega3783zoom\_TR\]. $\Gamma_{\rm app}$ falls to 1.2–1.3 during these times (3–4$\sigma$ deviation), although $HR2$ shows only mild deviations, at the $1-2\sigma$ level. Interestingly, $HR1$ and $\Gamma_{\rm app}$ do not return to their mean values in the period between the spikes.
We group data into bins approximately 2 d wide and fit the 3–23 keV PCA spectra, with results plotted in Table \[tab:3783TRtable\]. We first apply Model 1, keeping $R$ frozen at 0.41 and $I_{\rm Fe}$ frozen at $1.5\times 10^{-4}$ ph cm$^{-2}$ s$^{-1}$. This yields acceptable fits ($\chi_{\rm red} \sim 0.6-1.3$), but with $\Gamma$ reaching values as low as 1.3 near the positions of the $HR1$ spikes, as illustrated in Fig. \[fig:mega3783zoom\_TR\].
We then apply Model 2, with $\Gamma$ frozen at 1.73; this value is flatter than that from the time-averaged spectrum, 1.86, but equal to the average of the values before and after the putative eclipse event. Values of $\chi_{\rm red}$ are overall similar to the first model. $N_{\rm H}$($t$) is plotted in the lower panel of Fig. \[fig:mega3783zoom\_TR\]. Choosing slightly different bin sizes does not change the overall $N_{\rm H}$($t$) profile. Best-fitting values of $N_{\rm H}$ to spectra \#9 and \#15 are $11.2^{+1.7}_{-1.5}$ and $8.6^{+1.5}_{-1.3}$ $\times 10^{22}$ cm$^{-2}$, respectively, with values $\sim 3-5 \times 10^{22}$ cm$^{-2}$ for the period in between. Upper limits to $N_{\rm H}$ are obtained for spectra \#1–8 and \#16–23.
We can treat this event as a single event by adopting the maximum values of $N_{\rm H}$ (during spectrum $\#9$) as the peak value of $N_{\rm H}$. We use the $HR1$ light curve to assign a duration to the total event of 14.4–15.4 d. Alternatively, assuming two independent occultation events, we derive full durations of 9.2 and 4.6 d based on a dual linear-density sphere model fit to the column density profile, discussed in $\S$\[sec:3783double\].
We do not include the multiple warm absorbers modeled in previous X-ray spectra obtained with *XMM-Newton* or *Chandra* (e.g., Reeves et al. 2004 and references therein). Considering the three zones of warm absorption modeled by Reeves et al. (2004), the lowly-ionized absorber with log($\xi$)=–0.1 and $N_{\rm H} = 1.1
\times 10^{21}$ cm$^{-2}$ has a negligible effect on the PCA spectrum. The two other warm absorbers, one with log($\xi$)=2.1 and $N_{\rm H} =
1.2 \times 10^{22}$ cm$^{-2}$, the other with log($\xi$)=3 and $N_{\rm
H} = $ cm$^{-2}$, have a modest effect on continuum components above 3 keV. When we include these two zones in the model, $\Gamma$, steepens by $\ga0.05$, but the effect on any values of neutral $N_{\rm H}$ we measure is only at the $10$ per cent level or less. Modeling a simple power-law suffices to quantify the continuum, given the energy resolution and photon noise of the time-resolved spectra, and we assume that these absorbers’ parameters are constant.
$\bullet$ At 2008.7 (“Candidate”), $\Gamma_{\rm app}$ drops below a 2$\sigma$ deviation, reaching 1.2, and $HR1$ and $HR2$ increase to $>$3$\sigma$ deviation for 3/4 consecutive observations; see Fig. \[fig:mega3783zoom2008rej\]. However, this also occurs near a period of low continuum flux in all bands, and time-resolved spectroscopy cannot confirm absorption in excess of the Galactic column, with $N_{\rm H} < 4 \times 10^{22}$ cm$^{-2}$. We tentatively classify this as a “candidate” event of duration 12–28 d.
$\bullet$ *2011.2 (“Candidate”)*: In the $HR1$ light curve (see Fig. \[fig:mega3783zoom2008rej\]), there are two consecutive points at $HR1\sim1.9$, a $\sim7\sigma$ deviation. They occur in early 2011 at MJD 55629 and 55633. $\Gamma_{\rm app}$ falls from an average of 1.6 in the surrounding $\pm$5 observations to $1.20\pm0.03$ and $1.09\pm0.03$.
The source was monitored once every 4 d during this time. We perform time-resolved spectroscopy on data from MJD 55589.85 to 55693.73, grouping every three observations (exposure times 2–3 ks), except for MJD 55629.43–55633.56 (1.8 ks). However, this event also occurs near a period of low continuum flux in all bands, including the 10–18 keV band, yielding large statistical uncertainties. Results from spectral fits for Model 1 (again assuming $R$ frozen at 0.41 and $I_{\rm Fe}$ frozen at $1.5\times
10^{-4}$ ph cm$^{-2}$ s$^{-1}$) yields $\Gamma=1.22\pm0.18$ during the putative eclipse. For Model 2, $N_{\rm H} = 11^{+6}_{-7} \times
10^{22}$ cm$^{-2}$ for that spectrum, assuming $\Gamma$ is frozen at 1.62, the average of the values during the other spectra.
Given the large uncertainty in $\Gamma$ and $N_{\rm H}$, the near-identical and low values of $\chi^2_{\rm red}$ for both models ($\sim0.5$), and the low continuum flux in all wavebands, including the 10–18 keV band (see Appendix A10 and Fig. \[fig:fluxflux\]), this event does not clearly pass criterion 4. We hence choose to be conservative and classify this event as a “candidate” event, rather than a “secure B” event. Given the start/stop times of the monitoring observations, we constrain the duration of the obscuration event to be between 4.1 and 15.8 d.
$\bullet$ There are additional deviations in the long-term $HR1$ and $\Gamma_{\rm app}$ light curves which tend to catch the eye, but these deviations did not fulfill our selection criteria. For example, there are single-point increases in $HR1$ at the $2.5-4\sigma$ deviation level near MJD 51610, 52310 (large uncertainty), 53695 (large uncertainty), 54205, 54253, 54530; $\Gamma_{\rm app}$ was 1.3 or higher for each of these cases. In addition, $\Gamma_{\rm app}$ drops to values $<$1.25 at MJD 53639 and 55873, but again, these are single-point decreases, usually occurring during periods of relatively low 2–10 and/or 10–18 keV flux, and time-resolved spectroscopy cannot confirm excess absorption.
-------------------------- ------ -------------------- --------------- ---------------------- ------------ -------------------- ----------------------------------- ---------------------- ------------
Start–stop Expo
(MJD) (ks) $\chi^2_{\rm red}$ $\Gamma$ $A_1$ $F_{2-10}$ $\chi^2_{\rm red}$ $N_{\rm H}$ ($10^{22}$ cm$^{-2}$) $A_1$ $F_{2-10}$
54550.20–54551.83 (\#1) 5.1 0.94 $1.69\pm0.04$ $15.0^{+1.1}_{-1.0}$ 6.07 0.86 $< 1.4$ $15.7^{+0.3}_{-0.2}$ 6.10
54552.23–54553.93 (\#2) 5.1 0.75 $1.74\pm0.03$ $18.1^{+1.2}_{-1.1}$ 6.82 0.76 $< 0.7$ $17.6^{+0.2}_{-0.3}$ 6.80
54554.12–54555.83 (\#3) 5.2 1.02 $1.73\pm0.04$ $17.3\pm1.1$ 6.64 1.02 $< 0.7$ $17.2^{+0.2}_{-0.3}$ 6.63
54556.09–54557.79 (\#4) 4.9 0.86 $1.75\pm0.04$ $20.8^{+0.3}_{-0.8}$ 7.61 0.92 $< 0.4$ $19.6\pm0.2$ 7.56
54558.18–54559.76 (\#5) 5.1 0.61 $1.74\pm0.03$ $18.6^{+1.3}_{-1.1}$ 6.93 0.64 $< 0.4$ $17.9^{+0.2}_{-0.3}$ 6.90
54560.28–54561.87 (\#6) 5.4 0.97 $1.78\pm0.03$ $22.0^{+1.3}_{-1.2}$ 7.70 1.21 $< 0.2$ $19.8\pm0.2$ 7.61
54562.25–54563.68 (\#7) 5.0 1.05 $1.66\pm0.04$ $14.4^{+1.1}_{-1.0}$ 6.17 0.95 $1.5^{+0.9}_{-0.8}$ $16.4\pm0.4$ 6.06
54564.14–54565.79 (\#8) 5.1 0.88 $1.50\pm0.05$ $ 7.2\pm0.7$ 4.04 0.78 $5.1^{+1.3}_{-1.2}$ $11.9^{+0.4}_{-0.3}$ 3.87
54566.17–54567.49 (\#9) 4.9 0.75 $1.28\pm0.06$ $ 4.3^{+0.5}_{-0.4}$ 3.48 0.70 $11.2^{+1.7}_{-1.5}$ $12.0\pm0.5$ 3.27
54567.75–54569.11 (\#10) 5.0 1.02 $1.53\pm0.04$ $ 9.4^{+0.8}_{-0.7}$ 5.00 0.80 $4.7\pm1.1$ $14.6\pm0.4$ 4.69
54569.43–54570.97 (\#11) 4.2 0.74 $1.57\pm0.04$ $11.4^{+1.0}_{-0.8}$ 5.70 0.66 $3.3\pm1.1$ $16.1\pm0.4$ 5.53
54571.27–54572.79 (\#12) 5.5 1.00 $1.59\pm0.04$ $11.9^{+0.9}_{-0.8}$ 5.70 0.68 $3.4\pm0.9$ $16.1\pm0.4$ 5.51
54573.10–54574.47 (\#13) 4.5 0.94 $1.52\pm0.04$ $ 9.5^{+0.8}_{-0.7}$ 5.13 0.58 $5.3\pm1.1$ $15.2^{+0.5}_{-0.4}$ 4.90
54575.21–54576.53 (\#14) 4.2 1.15 $1.52\pm0.06$ $ 9.1\pm0.8$ 4.93 1.06 $4.9\pm1.2$ $14.5^{+0.5}_{-0.4}$ 4.91
54576.85–54578.00 (\#15) 4.7 0.96 $1.37\pm0.05$ $ 6.0\pm0.6$ 4.15 0.81 $8.6^{+1.5}_{-1.3}$ $13.5\pm0.5$ 3.92
54578.53–54580.18 (\#16) 5.0 0.81 $1.69\pm0.04$ $13.5^{+1.1}_{-1.0}$ 5.52 0.76 $0.9\pm0.6$ $14.3\pm0.4$ 5.45
54580.43–54582.00 (\#17) 4.4 1.06 $1.72\pm0.04$ $16.7^{+1.3}_{-1.1}$ 6.45 1.03 $< 0.9$ $16.5\pm0.3$ 6.40
54582.42–54583.96 (\#18) 4.1 0.78 $1.68\pm0.05$ $13.9^{+1.2}_{-1.1}$ 5.72 0.78 $< 1.0$ $14.8\pm0.4$ 5.69
54584.28–54586.01 (\#19) 5.3 0.74 $1.74\pm0.04$ $17.2^{+1.2}_{-1.1}$ 6.41 0.76 $< 0.4$ $16.5\pm0.2$ 6.39
54586.34–54587.90 (\#20) 5.0 1.17 $1.75\pm0.04$ $19.3^{+1.3}_{-1.2}$ 7.11 1.21 $< 0.4$ $18.3\pm0.3$ 7.07
54588.21–54589.86 (\#21) 4.5 0.97 $1.81\pm0.04$ $22.7^{+1.5}_{-1.4}$ 7.60 1.36 $< 0.2$ $19.4\pm0.3$ 7.48
54590.20–54591.88 (\#22) 4.4 1.34 $1.84\pm0.04$ $26.8^{+1.7}_{-1.3}$ 8.43 2.23 $< 0.1$ $21.4\pm0.3$ 8.24
54592.16–54593.85 (\#23) 5.1 0.97 $1.80\pm0.03$ $26.2^{+1.5}_{-1.3}$ 8.83 1.45 $< 0.1$ $22.6^{+0.3}_{-0.2}$ 8.69
-------------------------- ------ -------------------- --------------- ---------------------- ------------ -------------------- ----------------------------------- ---------------------- ------------
\
$A_1$ is the 1 keV normalization of the power law in units of $10^{-3}$ ph cm$^{-2}$ s$^{-1}$ keV$^{-1}$. $F_{2-10}$ is the observed/absorbed model flux in units of $10^{-11}$ erg cm$^{-2}$ s$^{-1}$. $I_{\rm Fe}$ is kept frozen at $1.5 \times 10^{-4}$ ph cm$^{-2}$ s$^{-1}$. For Model 2, $\Gamma$ is frozen at 1.73. Each spectral fit is performed over the 3–23 keV bandpass and has 45 $dof$.
Mkn 79 (Sy 1.2)
---------------
$\bullet$ *2003.5 (“Secure B”) and 2003.6 (“Secure B”):* The long-term light curves are plotted in Fig. \[fig:mega79\]. As seen there and in Fig. \[fig:mega79zoom03\_TR\], during MJD $\sim$ 52805–52818 (2003.5), $HR1$ increases to values ranging from 0.90 (just above $\langle HR1 \rangle$) to 2.4, with large scatter. The average and peak values of $HR1$ during this period correspond to 2.8$\sigma$ and 5$\sigma$ deviation, respectively. $\langle\Gamma_{\rm
app}\rangle$ during this time is 1.25, with a minimum value of 0.92 (3.2$\sigma$ deviation) at MJD 52816.0. The 10–18 keV continuum does not change as rapidly as the $<$7 keV continuum during this time; $HR2$ roughly doubled, to a $2\sigma$ deviation. No data were taken during MJD 50821–50839 due to a sun-angle gap. By MJD 52840, $HR1$ has returned to its mean value of 0.85.
During MJD $\sim$ 52861–52891 (2003.6), $HR1$ increases again, showing deviations as high as $3-6\sigma$, with an average deviation of 2.2$\sigma$ during this time. During this time, $\Gamma_{\rm app}$ reaches values as low as 1.0–1.1 (2.0–2.8$\sigma$), with large scatter; $\langle\Gamma_{\rm app}\rangle=1.39$.
We bin up data from MJD 52770–52925 into five bins of approximately 15–30 d as listed in Table \[tab:79TRtable\]. This yields only one time bin for each of the two candidate events (spectra 2 and 4), but smaller bin sizes would have yielded poor parameter constraints. We fit 3–23 keV data, keeping $R$ frozen at 0.7 (Rivers [et al.]{} 2013). We first test Model 1, with $N_{\rm H}$ frozen at 0, $\Gamma$ free, and $I_{\rm Fe}$ frozen at $5\times10^{-5}$ ph cm$^{-2}$ s$^{-1}$. Best-fitting values are listed in Table \[tab:79TRtable\]; $\Gamma$ reaches values of 1.18$\pm$0.17 and 1.34$\pm$0.10 in spectra 2 and 4, respectively. We then add a neutral column of gas with <span style="font-variant:small-caps;">zphabs</span> for Model 2, keeping $\Gamma$ frozen at 1.78, the average from spectra 1, 3, and 5. $N_{\rm H}$ is $14.4^{+4.8}_{-4.2}$ and $11.5^{+3.2}_{-2.8} \times
10^{22}$ cm$^{-2}$ for spectra 2 and 4, respectively, with upper limits $\sim 2 \times 10^{22}$ cm$^{-2}$ for spectra 1, 3, and 5. Best-fitting values of $\Gamma$ for Model 1 and $N_{\rm H}$ for Model 2 are plotted in Fig. \[fig:mega79zoom03\_TR\].
Values of $\chi^2_{\rm red}$ are virtually identical between Models 1 and 2, and span 0.46–0.81. That is, the observed spectral flattening *could* potentially be due to a combination of $\Gamma$ reaching extremely low values and spectral pivoting of the power law in or near the 10–18 keV band. However, adhering to our assumption that $\Gamma$ should not dip below 1.5, the occultation interpretation is then preferred in each of these cases.
We assign durations based on the $HR1$ light curve, but these durations are approximate due to large scatter in $HR1$. For the 2003.5 event, we observed ingress, but egress likely occurred during the monitoring gap; limits to the duration are 12.0–39.4 d. We cannot rule out that the possibility that the peak of the event occurred during the gap, in which case peak $N_{\rm H}$ might be higher than that observed. For the (complete) 2003.6 event, we estimate a duration of 34.5–37.9 d.
$\bullet$ *2009.9 (“Secure B”)*: At MJD$\sim55155-55197$, some values of $HR1$ deviate as high as the 1–3$\sigma$ level, but $\langle HR1 \rangle$ corresponds to the $\sim1\sigma$ level. $F_{2-10}$ is about half the average value, but $F_{10-18}$ stays roughly constant, yielding an $HR2$ peak at the $\sim2-2.5\sigma$ level albeit with large errors. We sum all spectra taken from MJD 55155 – 55197, plus all data in the $\sim$40 d periods before and after this period (spectra 6, 7, and 8 in Table \[tab:79TRtable\]). $I_{\rm Fe}$ is kept frozen at $4\times10^{-5}$ ph cm$^{-2}$ s$^{-1}$. For spectrum \#7, we apply Model 2 by freezing $\Gamma$ at 1.84, the average of $\Gamma$ from spectra 6 and 8; this yields $N_{\rm H} = 7.6\pm2.2 \times 10^{22}$ cm$^{-2}$. We estimate a duration of $19.6-40.0$ d from the $HR1$ light curve.
$\bullet$ There are additional deviations in the long-term $HR1$ and/or $\Gamma_{\rm app}$ light curves which tend to catch the eye, but these deviations do not fulfill our selection criteria and are rejected:
At 2007.3, at MJD $54208-54219$, $HR1$ increases to $\sim$1.2, but these are only $\sim1-2\sigma$ deviations. There are only a few points at $1\sigma$ deviation in the $HR2$ light curve as well; the 10–18 and 2–10 keV continua are both $\sim40$ per cent below average during this period. $\langle\Gamma_{\rm app}\rangle$ during this period is 1.37, just under a 1$\sigma$ deviation, and with large scatter. Time-resolved spectroscopy to data summed between MJD $54208$ and $54219$, assuming that $\Gamma$ is frozen at 1.78, yields $N_{\rm H} < 9.2 \times
10^{22}$ cm$^{-2}$.
At 2008.2, MJD$\sim 54535-54590$, and several times near the end of 2008, MJD $\sim54725-54830$, there are deviations in $HR1$ up to 2–5$\sigma$ levels, but with very large scatter and frequently large uncertainties. Roughly a third of the values of $\Gamma_{\rm app}$ during these times are $<$1.3, but there is very large scatter, and average values of $\Gamma_{\rm app}$ are at the $\sim1\sigma$ level of deviation. Both the 10–18 and 2–10 keV continuua were $\sim30-60$ per cent of their average values, and follow-up time-resolved spectroscopy cannot confirm significant increases in $N_{\rm H}$ (upper limits are in the range $3-5 \times
10^{22}$ cm$^{-2}$).
Near 2009.1, at MJD $\sim54856-54876$, a couple of $HR1$ points reach the 2$\sigma$ level, but errors are large, and $\langle HR1 \rangle$ during this time is a $\ga1\sigma$ deviation. All continuum flux light curves experience dips during this time to relatively low levels. Time-resolved spectroscopy cannot confirm any significant increase in $N_{\rm H}$.
Fig. \[fig:mega79zoom89rej\] shows a zoom-in on the 2008.0–2009.3 data with rejected events.
----------------------- ------ -------------------- --------------- ------------- ------------ -------------------- ----------------------------------- --------------------- ------------
Start–stop Expo
(MJD) (ks) $\chi^2_{\rm red}$ $\Gamma$ $A_1$ $F_{2-10}$ $\chi^2_{\rm red}$ $N_{\rm H}$ ($10^{22}$ cm$^{-2}$) $A_1$ $F_{2-10}$
52770.2–52804.3 (\#1) 14.8 0.67 $1.73\pm0.06$ $4.7\pm0.5$ 1.98 0.70 $<$2.0 $5.1\pm0.1$ 1.97
52806.2–52818.2 (\#2) 3.4 0.50 $1.18\pm0.17$ $1.2\pm0.3$ 1.22 0.50 $14.4^{+4.8}_{-4.2}$ $4.9^{+0.6}_{-0.5}$ 1.14
52843.7–52858.3 (\#3) 4.7 0.64 $1.87\pm0.11$ $6.0\pm1.2$ 1.89 0.69 $<$1.4 $5.1\pm0.2$ 1.85
52861.0–52891.3 (\#4) 9.7 0.73 $1.34\pm0.10$ $1.7\pm0.3$ 1.26 0.80 $11.5^{+3.2}_{-2.8}$ $4.7\pm0.3$ 1.18
52895.8–52925.6 (\#5) 11.6 0.72 $1.75\pm0.07$ $4.6\pm0.7$ 1.81 0.74 $<$2.1 $5.0^{+0.2}_{-0.1}$ 1.80
55115.8–55153.8 (\#6) 14.6 0.92 $1.83\pm0.04$ $6.5\pm0.7$ 2.12 0.94 $<1.1$ $6.4\pm0.1$ 2.08
55155.8–55195.1 (\#7) 18.6 0.79 $1.44\pm0.08$ $1.8\pm0.3$ 1.17 1.15 $7.6\pm2.2$ $4.4\pm0.3$ 1.17
55197.9–55235.7 (\#8) 18.0 0.69 $1.84\pm0.05$ $6.2\pm0.4$ 1.99 0.94 $<1.0$ $5.9\pm0.1$ 1.94
----------------------- ------ -------------------- --------------- ------------- ------------ -------------------- ----------------------------------- --------------------- ------------
\
$A_1$ is the 1 keV normalization of the power law in units of $10^{-3}$ ph cm$^{-2}$ s$^{-1}$ keV$^{-1}$. $F_{2-10}$ is the observed/absorbed model flux in units of $10^{-11}$ erg cm$^{-2}$ s$^{-1}$. $I_{\rm Fe}$ is kept frozen at 5 and $4 \times
10^{-5}$ ph cm$^{-2}$ s$^{-1}$ for the 2003 and 2009 spectra, respectively. For Model 2, $\Gamma$ is frozen at 1.78 and 1.84 for the 2003 and 2009 spectra, respectively. Each spectral fit is performed over the 3–23 keV bandpass and has 45 $dof$.
Mkn 509 (Sy 1.2)
----------------
The long-term light curves are plotted in Fig. \[fig:mega509\], with a zoom-in on late 2005 plotted in Fig. \[fig:mega509zoom\_TR\].
*2005.9 (“Secure B”):* The $HR1$ light curve in Fig. \[fig:mega509zoom\_TR\] shows an upturn from the baseline value of $\sim$0.6 to 1.2 over a 30 d period in late 2009, with a $\sim5\sigma$ peak deviation by MJD 53736, after which monitoring was interrupted for 59 d due to sun-angle constraints. $\Gamma_{\rm app}$ drops from about 1.7 to 1.3 during this time, a $\sim4\sigma$ deviation.
During this time, *RXTE* monitored the source once every 3 d. For time-resolved spectroscopy, we grouped the individual spectra into bins of 15 d each. We fit the 3–23 keV spectra keeping $R$ and $I_{\rm Fe}$ frozen at their respective time-averaged values of 0.15 and $7\times 10^{-5}$ ph cm$^{-2}$ s$^{-1}$; results are listed in Table \[tab:509TRtable\]. With Model 1, acceptable fits ($\chi^2_{\rm red}$ = 0.52–0.68) are obtained, although $\Gamma$ flattens to the very low value of $1.37\pm0.06$ during spectrum \#6. We then test Model 2, freezing $\Gamma$ to 1.72, the average of the values in spectra \#1–3 and \#7, and close to the time-averaged value found by Rivers [et al.]{} (2011), 1.75. Acceptable fits spanning a nearly identical range of $\chi^2_{\rm red}$ are obtained, with $N_{\rm H}$ peaking at $8.8\pm1.7 \times 10^{22}$ cm$^{-2}$ during spectrum \#6, and with upper limits during spectra \#1–3 and 7. Best-fitting values of $\Gamma$ for Model 1 and $N_{\rm H}$ for Model 2 are plotted in Fig. \[fig:mega509zoom\_TR\].
Under the assumption that an eclipse occurred, it is difficult to pin down the exact duration because the event ended sometime during the 60 d sun-angle gap, between MJD 53736 and 53795; peak $N_{\rm H}$ could thus be higher than that measured during spectrum $\#6$, $8.8\pm1.7
\times 10^{22}$ cm$^{-2}$. From the $HR1$ light curve, we assign a duration of 26–94 d. If the occulting cloud is symmetric in density along the transverse direction, then the total duration can be $\sim55-90$ d, with peak $N_{\rm H}$ occurring immediately before the sun-angle gap in the former case (i.e., during the times constrained by spectrum $\#5$), or with peak $N_{\rm H}$ slightly higher and occurring $\sim$15 d into the gap in the latter case.
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Start–stop Expo
(MJD) (ks) $\chi^2_{\rm red}$ $\Gamma$ $A_1$ $F_{2-10}$ $\chi^2_{\rm red}$ $N_{\rm H}$ ($10^{22}$ cm$^{-2}$) $A_1$ $F_{2-10}$
53650.31–53662.56 ($\#1$) 6.4 0.67 $1.74^{+0.04}_{-0.05}$ $11.7\pm1.0$ 4.32 0.68 $<0.5$ $11.4\pm0.2$ 4.31
53665.18–53677.10 ($\#2$) 8.0 0.63 $1.72\pm0.04$ $11.1\pm0.9$ 4.22 0.63 $<0.8$ $11.1\pm0.2$ 4.22
53680.11–53692.37 ($\#3$) 9.7 0.52 $1.71^{+0.04}_{-0.03}$ $11.8^{+0.9}_{-0.7}$ 4.55 0.53 $<0.8$ $12.0^{+0.2}_{-0.1}$ 4.55
53695.25–53707.58 ($\#4$) 8.3 0.54 $1.65^{+0.04}_{-0.05}$ $8.6\pm0.7$ 3.68 0.61 $1.1\pm0.8$ $10.0^{+0.4}_{-0.3}$ 3.64
53710.46–53722.19 ($\#5$) 8.2 0.53 $1.52^{+0.04}_{-0.05}$ $6.5^{+0.6}_{-0.5}$ 3.45 0.73 $4.5\pm1.2$ $10.4\pm0.3$ 3.33
53725.91–53736.34 ($\#6$) 7.8 0.67 $1.37\pm0.06$ $3.7\pm0.4$ 2.51 0.80 $8.8\pm1.7$ $8.4^{+0.4}_{-0.3}$ 2.37
53795.10–53806.97 ($\#7$) 7.9 0.68 $1.70\pm0.06$ $11.5^{+1.0}_{-0.8}$ 4.56 0.65 $<0.9$ $12.1\pm0.2$ 4.58
--------------------------- ------ -------------------- ------------------------ ---------------------- ------------ -------------------- ----------------------------------- ---------------------- ------------
\
$A_1$ is the 1 keV normalization of the power law in units of $10^{-3}$ ph cm$^{-2}$ s$^{-1}$ keV$^{-1}$. $F_{2-10}$ is the observed/absorbed model flux in units of $10^{-11}$ erg cm$^{-2}$ s$^{-1}$. $I_{\rm Fe}$ is kept frozen at $7 \times 10^{-5}$ ph cm$^{-2}$ s$^{-1}$. Each spectral fit is performed over the 3–23 keV bandpass and has 45 $dof$.
MR 2251–178 (Sy 1.5/QSO)
------------------------
$\bullet$ *1996 (“Secure A”)* This source was observed by *RXTE* for 3 d in 1996 MJD 50426-9, then subjected to sustained monitoring during 2004 March – 2011 December, as seen in Fig. \[fig:mega2251\]. Values of $HR1$ in 1996 are $\sim1.0-1.1$, a $\sim4\sigma$ deviation from the mean value of 0.68 from the 2004–2011 data. $\Gamma_{\rm app}$ during 1996 is 1.5, a $\sim2-3\sigma$ deviation.
The X-ray spectrum of this source is characterized by absorption from gas spanning a wide range of ionization states; Gofford et al. (2011) quantify five such zones using combined *Suzaku* + *Swift*-Burst Alert Telescope spectra. Furthermore, there is evidence for the column densities of at least some of these zones to vary over timescales of months–years (Kaspi et al. 2004). *RXTE* is not strongly sensitive to the effect of the highest ionization zones of absorption, log($\xi$) $\ga3$, e.g., “Zones 3–5” in Gofford et al. (2011). Any absorbing column we detect in excess of $N_{\rm Gal}$ may potentially be analogous to (or even potentially identified as) an overdense region of “Zone 1” from Gofford et al. (2011; with log($\xi$) $\sim-0.23$ and $N_{\rm H} \sim 5\times 10^{20}$ cm$^{-2}$), or the absorber with log($\xi$) $\sim+0.02$ and $N_{\rm H}
\sim 2.4 \times 10^{21}$ cm$^{-2}$ measured by Gibson et al. (2005) using a 2002 *Chandra*-High-Energy Transmission Grating Spectrometer (HETGS) observation.[^13] Here, we fit the *RXTE* spectra assuming full-covering neutral absorption only.
We sum all the data during the 1996 campaign; the good exposure for the PCA was 90.1 ks. The good exposure time per High-Energy X-ray Timing Experiment (HEXTE) cluster is 20.1/20.5 ks for A/B, with the source detected only out to 25 keV, so we use PCA data only. We fit 3–23 keV data, and assume no reflection component and no high-energy rollover (see Rivers [et al.]{} 2011a). We first apply Model 1, keeping the Fe K$\alpha$ line energy and width $\sigma$ frozen at 6.0 keV and 0.9 keV, respectively. This fit yields $\Gamma=1.52\pm0.01$, but $\chi^2/dof$ is 175.2/42 = 4.2, with poor data/model residuals showing strong curvature up to the $\pm$6 per cent level below 10 keV. We apply Model 2, keeping $\Gamma$ free, and obtain a fit with $\chi^2/dof$ = 95.8/41 and data/model residuals $\la2$ per cent, consistent with PCA calibration. Best-fitting model parameters are $N_{\rm H} = 6.6^{+0.8}_{-1.4} \times 10^{22}$ cm$^{-2}$, $\Gamma=1.73^{+0.02}_{-0.06}$, and $I_{\rm Fe}$ fell to $3.1\pm1.3
\times 10^{-5}$ ph cm$^{-2}$ s$^{-2}$.
Fig. \[fig:mega2251zoom\] shows a zoom-in on the 1996 data; we find no evidence then for strong variations in $HR1$ on timescales $<$3 d. Constraints on the duration of the event from *RXTE* data alone are poor; we can only set a lower limit of 3 d, the length of the 1996 campaign. Peak $N_{\rm H}$ may of course be higher than that observed in 1996. Dadina (2007), fitting two *BeppoSAX* observations in 1998 June and 1998 November, noted total columns consistent with $\sim1-2\times$$N_{\rm H,Gal}$. In the context of an absorption event, this measurement suggests that the event had ended by 1998 June. Reeves & Turner (2000), fitting *ASCA* spectra obtained in 1993 November–December, found $N_{\rm H} < 1 \times 10^{20}$ cm$^{-2}$, implying an upper limit for the duration of the event of $\sim4.5$ yr.
NGC 3227 (Sy 1.5)
-----------------
The long-term light curves are plotted in Fig. \[fig:mega3227\].
$\bullet$ *2000.9–2001.2 (“Secure A”):* Lamer [et al.]{} (2003) confirmed a complete eclipse event (ingress and egress) by a full-covering cloud. They determined the cloud to be mildly ionized via a contemporaneous *XMM-Newton* observation, with log($\xi$, erg cm s$^{-1}$) $\sim -0.3 - 0$. Our $HR1$, $HR2$, and $\Gamma_{\rm app}$ light curves show peak deviations up to the $\ga6-8$, $\ga3-5$, and the $\sim4$$\sigma$ levels, respectively, and $\Gamma_{\rm app}$ gets as low as $\sim0.5$. We refer the reader to Lamer [et al.]{} (2003) for details of time-resolved spectral fitting and fitting the $N_{\rm H}$($t$) profile. Lamer [et al.]{} (2003) found values of peak $N_{\rm H}$ of $19$ and $26 \times 10^{22}$ cm$^{-2}$ based on fits to the $N_{\rm
H}$($t$) profile using a uniform-density sphere and a $\beta$-profile fits, respectively; we adopt the latter value for the rest of this paper. Based on our $HR1$ light curve, we adopt a duration of 77–94 d, similar to values used by Lamer [et al.]{} (2003).
$\bullet$ *2002.8 (“Secure B”):* As seen in Fig. \[fig:mega3227zoom02\_TR\], two consecutive points at MJD 52565.7 and 52567.8 show anomalously high $HR1$ values: $2.54\pm0.15$ (2.9$\sigma$) and $1.92\pm0.10$ (1.8$\sigma$), respectively. $\Gamma_{\rm app} = 1.01\pm0.02$ and $1.12\pm0.02$, $\sim2\sigma$ deviations. $HR2$ shows only $\sim1.5\sigma$ deviations.
We sum spectra obtained during MJD 52541–52581 in groups of two to match the time resolution of the putative event, and fit each 3–23 keV spectrum, keeping $I_{\rm Fe}$ frozen at $6\times10^{-5}$ ph cm$^{-2}$ s$^{-1}$. Fitting spectrum \#7 with Model 1 yields a fit with $\chi^2_{\rm red}=1.34$ and with strong systematic curvature in the data/model residuals $\sim3-7$ keV; $\Gamma$ is $1.13\pm0.08$. Applying Model 1 to the other spectra yields an average value of $\Gamma$ of 1.61. Applying Model 2 to spectrum \#7 with $\Gamma$ frozen at 1.61 fixed data/model residuals; $\chi^2_{\rm red}=0.55$, and $N_{\rm H} = 13.3^{+2.6}_{-2.2} \times 10^{22}$ cm$^{-2}$. Results for all spectra are listed in Table \[tab:3227TRtable02\]. From the start/stop times of the two observations, limits on the duration of the event are 2.1–6.6 d.
Using *XMM-Newton*, Markowitz et al. (2009) modeled two outflowing zones of ionized absorption, both with $N_{\rm H} \sim 1-2 \times 10^{21}$ cm$^{-2}$, and with log($\xi$) $\sim 1.2-1.4$ and $\sim$2.9. Neither are expected to significantly impact the time-resolved PCA spectra. Due to the much higher column and lower ionization, it is likely that the eclipse events detected with *RXTE* are likely distinct from these outflowing warm absorbers.
$\bullet$ Additional events do not meet our selection criteria and are rejected due to being single points only (failing criterion 2). As seen in Fig. \[fig:mega3227zoom2004\], a single point at MJD 53207 (2004 Jul 21) shows a very strong $HR1$ deviation up to $3.0\pm0.29$, a $3.7\sigma$ deviation. $\Gamma_{\rm app} =
0.80\pm0.03$, a 3.2$\sigma$ deviation; $HR2$ shows a $2\sigma$ deviation.
At 2000.3, $HR1$ goes to a 6$\sigma$ deviation. There are four single points in the period 2000.3–2000.4 where $HR2$ goes to a $\geq$2$\sigma$ deviation, but with extremely large errors. Again, however, each of these events fails criterion 2 and is rejected.
The bottom two panels in Fig. \[fig:mega3227zoom02\_TR\] also suggest a significant increase in $N_{\rm H}$ for spectrum \#4, at MJD 52555. Given the fact that the continuum flux was relatively low in all wavebands, the near-identical values of $\chi^2_{\rm red}$ between Models 1 and 2, and especially the fact that only 1 point shows a $>$1$\sigma$ deviation in both the $HR1$ and $\Gamma_{\rm app}$ light curves (thus failing criterion 2), we do not consider this as an event.
-------------------------- ------ -------------------- --------------- ---------------------- ------------ -------------------- ----------------------------------- -------------- ------------
Start–stop Expo
(MJD) (ks) $\chi^2_{\rm red}$ $\Gamma$ $A_1$ $F_{2-10}$ $\chi^2_{\rm red}$ $N_{\rm H}$ ($10^{22}$ cm$^{-2}$) $A_1$ $F_{2-10}$
52541.11–52543.16 (\#1) 1.7 0.58 $1.69\pm0.08$ $9.7^{+1.5}_{-1.3}$ 4.12 0.66 $<0.7$ $8.3\pm0.3$ 4.06
52544.99–52547.03 (\#2) 1.9 0.57 $1.68\pm0.09$ $7.9\pm1.3$ 3.38 0.62 $<0.7$ $6.8\pm0.3$ 3.34
52549.13–52551.92 (\#3) 2.0 0.62 $1.57\pm0.07$ $8.4\pm1.3$ 4.33 0.64 $<1.4$ $9.1\pm0.3$ 4.36
52553.36–52555.07 (\#4) 1.6 0.70 $1.30\pm0.11$ $3.3\pm0.7$ 2.72 0.60 $7\pm3$ $7.1\pm0.6$ 2.56
52556.96–52558.98 (\#5) 1.9 0.33 $1.69\pm0.06$ $12.9^{+1.6}_{-1.4}$ 5.42 0.46 $<0.5$ $11.1\pm0.3$ 5.34
52561.97–52563.04 (\#6) 1.7 1.03 $1.44\pm0.04$ $6.7^{+1.0}_{-0.9}$ 4.23 0.72 $<4.5$ $10.0\pm0.5$ 4.00
52565.74–52567.87 (\#7) 1.8 1.34 $1.13\pm0.08$ $3.0^{+0.5}_{-0.4}$ 3.18 0.55 $13.3^{+2.6}_{-2.2}$ $9.9\pm0.6$ 2.97
52569.63–52573.72 (\#8) 1.8 1.03 $1.56\pm0.08$ $7.4\pm1.1$ 3.85 0.95 $<2.0$ $8.4\pm0.5$ 3.71
52575.90–52578.28 (\#9) 1.9 0.79 $1.59\pm0.10$ $6.2^{+1.2}_{-1.0}$ 3.10 0.80 $<2.0$ $6.4\pm0.3$ 3.11
52579.57–52581.31 (\#10) 1.7 0.95 $1.67\pm0.08$ $9.7^{+1.5}_{-1.3}$ 4.23 0.98 $<0.8$ $8.6\pm0.3$ 4.19
-------------------------- ------ -------------------- --------------- ---------------------- ------------ -------------------- ----------------------------------- -------------- ------------
\
$A_1$ is the 1 keV normalization of the power law in units of $10^{-3}$ ph cm$^{-2}$ s$^{-1}$ keV$^{-1}$. $F_{2-10}$ is the observed/absorbed model flux in units of $10^{-11}$ erg cm$^{-2}$ s$^{-1}$. $I_{\rm Fe}$ is kept frozen $6 \times 10^{-5}$ ph cm$^{-2}$ s$^{-1}$. For Model 2, $\Gamma$ is frozen at 1.61. Each spectral fit is performed over the 3–23 keV bandpass and has 45 $dof$.
Cen A (NLRG)
------------
$\bullet$ *2010–2011 (“Secure A”)*: Our $HR1$, $HR2$, and $\Gamma_{\rm app}$ light curves (Fig. \[fig:megacena\]) each suggest strong spectral flattening (at roughly 3–4$\sigma$ deviation) over a $\sim$6 month period in 2010–2011 (MJD $\sim$ 55430–55620), during a period of monitoring once every 2 d. The means and standard deviations of $HR1$, $HR2$, and $\Gamma_{\rm app}$ in Fig. \[fig:megacena\] were calculated omitting data during this period to avoid biases due to the large number of individual points during the eclipse. A zoom-in on this time period is shown in Fig. \[fig:megacena1011zoom\]. Time-resolved spectroscopy by Rivers [et al.]{} (2011b) successfully deconvolved $\Gamma$ and $N_{\rm H}$ and confirmed a complete eclipse (ingress and egress) with peak $\Delta$$N_{\rm H}$ of $8 \times 10^{22}$ cm$^{-2}$ above the “baseline” level of $\sim 20 \times 10^{22}$ cm$^{-2}$; the reader is referred to that paper for details. With spectra binned to once every 10 d and an event duration of a little over 170 d, this event is likely the best-observed eclipse so far in terms of resolving the $N_{\rm H}$ profile.
The profile is consistent with a cloud whose density was symmetric along the direction of transit (as opposed to a “comet-shaped” cloud, e.g., Maiolino [et al.]{} 2010); to the eye the $HR1$ and $HR2$ light curves may look slightly skewed, but fits to the $N_{\rm H}$ profile were consistent with a symmetric cloud. A centrally-concentrated sphere with a linear density profile was a better fit than a constant-density sphere. From our $HR1$ light curve, we adopt a duration of 170.2–184.5 d.
$\bullet$ *2003–2004 (“Secure A”)*: Similar changes in $HR1$ and $\Gamma_{\rm app}$ suggest an event with roughly similar $\Delta$$N_{\rm H}$ peaking in 2003–2004. However, constraints on the duration of the event and peak $N_{\rm H}$ are poor because *RXTE* observed Cen A during this time via three clusters of observations separated by months–years, as opposed to via sustained monitoring. Spectral fits by Rothschild [et al.]{} (2011; see their fig. 4) reveal $N_{\rm H} = 23 \times 10^{22}$ cm$^{-2}$ during the 2003 March campaign and $24-26 \times 10^{22}$ cm$^{-2}$ during 2004 January and February, compared to $15-18 \times 10^{22}$ cm$^{-2}$ measured during the 2000 January and earlier campaigns and the 2005 August and later campaigns. The spectral fits of Rothschild [et al.]{} (2011) thus indicate an observed $\Delta$$N_{\rm H}$ = $8 \pm 1 \times 10^{22}$ cm$^{-2}$ with a similar baseline $N_{\rm H}$ as the 2010–2011 event. Constraints on the duration of the event from the $HR1$ light curve are 356–2036 d. We also present evidence that the “baseline” level of $N_{\rm H}$ is not constant. There is a small “dip” in the $HR1$ light curve in 2010, shortly after the 2-d monitoring commenced. We sum up individual spectra from the first four months of monitoring in groups of three (6 d; Rivers et al. 2011b binned every five spectra / 10 d). We leave $\Gamma$ free, but hold $I_{\rm Fe}$ frozen at $4.2\times 10^{-4}$ ph cm$^{-2}$ s$^{-1}$. The resulting values of $\Gamma$($t$) and $N_{\rm H}$($t$) are displayed in Fig. \[fig:cenasmalldip\] and listed in Table \[tab:cenasmalldiptable\]. The value of $N_{\rm H}$($t$) reaches a maximum in bin \#2 (covering MJD 55226.7–55232.3), with $N_{\rm H} = 21.7\pm0.9 \times 10^{22}$ cm$^{-2}$, reaches a minimum in bin \#8 (MJD 55264.5–55270.4) with $N_{\rm H} = 18.6^{+0.9}_{-0.8}
\times 10^{22}$ cm$^{-2}$, and returns to a maximum value of $21.9\pm0.7 \times 10^{22}$ cm$^{-2}$ in bin \#21 (MJD 55341.1–55350.5). As the uncertainties on each $N_{\rm H}$ point are 90 per cent confidence, this difference is a $\sim2.2\sigma$ result. This result cannot be due to degeneracy between $\Gamma$ and $N_{\rm H}$: an increase in $\Gamma$ is expected to be associated with an *increase* in $N_{\rm H}$, contrary to what is observed. However, freezing $\Gamma$ at the average value of 1.85 does not significantly change the $N_{\rm H}$($t$) light curve; those values are plotted in gray in Fig. \[fig:cenasmalldip\]. Summing over all spectra, the total value of $\chi^2$/$dof$ is 1067.95/1113, compared to 1037.92/1092 with $\Gamma$ free; this means that allowing $\Gamma$ to be free improves the fits only at 93.3 per cent confidence according to an $F$-test.
-------------------------- ------ -------------------- ----------------------- ------------------------ ------------------------------------ ------------
Start–stop Expo $N_{\rm H}$ $A_1$
(MJD) (ks) $\chi^2_{\rm red}$ ($10^{22}$ cm$^{-2}$) $\Gamma$ (ph cm$^{-2}$ s$^{-1}$ keV$^{-1}$) $F_{2-10}$
55221.53–55225.65 (\#1) 2.4 0.88 $21.7\pm0.9$ 1.82$\pm$0.03 $0.214^{+0.016}_{-0.014}$ 34.62
55227.83–55231.27 (\#2) 2.4 0.97 $21.8\pm0.9$ 1.81$\pm$0.03 $0.214^{+0.016}_{-0.014}$ 35.13
55233.29–55237.65 (\#3) 2.8 1.08 $21.3\pm0.8$ 1.85$\pm$0.03 $0.243^{+0.016}_{-0.014}$ 37.66
55239.84–55243.70 (\#4) 2.2 1.02 $21.2\pm0.9$ 1.88$\pm$0.03 $0.254^{+0.020}_{-0.017}$ 37.43
55247.62–55251.64 (\#5) 3.3 0.97 $18.8^{+0.7}_{-0.6}$ 1.85$\pm$0.02 $0.244^{+0.014}_{-0.012}$ 39.85
55253.90–55257.92 (\#6) 2.5 0.84 $19.2\pm0.9$ 1.87$\pm$0.03 $0.187^{+0.016}_{-0.014}$ 29.29
55259.59–55263.45 (\#7) 2.6 0.70 $20.5^{+1.0}_{-0.9}$ 1.88$\pm$0.03 $0.191^{+0.016}_{-0.014}$ 28.64
55265.56–55269.48 (\#8) 2.6 0.97 $18.6^{+0.9}_{-0.8}$ 1.82$\pm$0.03 $0.190^{+0.014}_{-0.013}$ 32.66
55271.35–55275.12 (\#9) 2.4 0.93 $18.9^{+0.9}_{-0.8}$ 1.86$\pm$0.03 $0.217^{+0.016}_{-0.014}$ 35.00
55277.71–55281.56 (\#10) 2.4 0.75 $19.4^{+0.9}_{-0.8}$ 1.88$\pm$0.03 $0.220^{+0.017}_{-0.015}$ 33.54
55283.86–55287.54 (\#11) 2.2 0.76 $18.9^{+0.9}_{-0.8}$ 1.84$\pm$0.03 $0.209^{+0.016}_{-0.014}$ 34.64
55289.33–55293.49 (\#12) 2.8 1.08 $19.3^{+0.8}_{-0.7}$ $1.83^{+0.03}_{-0.02}$ $0.221^{+0.014}_{-0.013}$ 37.10
55295.29–55299.64 (\#13) 2.8 1.42 $19.2^{+0.9}_{-0.8}$ 1.83$\pm$0.03 $0.184^{+0.014}_{-0.012}$ 31.11
55303.40–55306.94 (\#14) 2.5 1.03 $20.0\pm0.9$ 1.84$\pm$0.03 $0.191^{+0.015}_{-0.013}$ 30.76
55309.87–55313.43 (\#15) 2.6 0.66 $19.5\pm0.8$ 1.83$\pm$0.03 $0.219^{+0.015}_{-0.013}$ 36.18
55315.44–55319.17 (\#16) 3.1 1.18 $20.4\pm0.7$ 1.84$\pm$0.02 $0.271^{+0.015}_{-0.013}$ 42.99
55321.25–55325.45 (\#17) 2.9 0.59 $19.9\pm0.8$ 1.84$\pm$0.02 $0.225^{+0.015}_{-0.013}$ 36.36
55327.28–55331.54 (\#18) 2.5 0.95 $21.0^{+0.9}_{-0.8}$ 1.82$\pm$0.03 $0.217^{+0.015}_{-0.014}$ 35.43
55333.61–55338.38 (\#19) 2.4 1.03 $21.3\pm0.8$ $1.84^{+0.03}_{-0.02}$ $0.272^{+0.018}_{-0.016}$ 42.31
55339.42–55342.86 (\#20) 2.1 1.24 $20.9^{+0.9}_{-0.8}$ 1.85$\pm$0.03 $0.250^{+0.018}_{-0.016}$ 39.02
55345.33–55349.51 (\#21) 2.6 0.91 $21.9\pm0.7$ 1.88$\pm$0.02 $0.303^{+0.019}_{-0.017}$ 43.94
-------------------------- ------ -------------------- ----------------------- ------------------------ ------------------------------------ ------------
\
Results of time-resolved spectroscopy for Cen A to quantify the small “dip” in the baseline level of $N_{\rm H}$ in early 2010 (approx. MJD 55230–55330). $A_1$ is the 1 keV normalization of the power law. $F_{2-10}$ is the observed/absorbed model flux in units of $10^{-11}$ erg cm$^{-2}$ s$^{-1}$. $I_{\rm Fe}$ is frozen at $4.2 \times 10^{-4}$ ph cm$^{-2}$ s$^{-1}$. Each spectral fit is performed using the 3–30 keV bandpass and has 52 $dof$.
NGC 5506 (Sy 1.9)
-----------------
$\bullet$ *2000.2 (“Secure A”)*: The long-term light curves are plotted in Fig. \[fig:mega5506\]. During a period of intensive monitoring in early 2000 (Fig. \[fig:mega5506zoom\_TR\], with four observations per day, two consecutive observations (at MJD 51624.9 and 51625.1) show anomalous increases in $HR1$ from its mean value of 0.88 with a standard deviation of 0.10 to values of 1.41$\pm$0.02 and 1.48$\pm$0.02, deviations of 5.2 and 6.0$\sigma$. $\Gamma_{\rm app}$ drops from 1.6 to 1.3 ($>4\sigma$ deviation).
We perform time-resolved spectroscopy by summing spectra together in groups of two. We fit 3–23 keV data, allowing $N_{\rm, H}$, $\Gamma$, $A_1$ and $I_{\rm Fe}$ to vary from their best-fitting time-averaged values. Uncertainties on $I_{\rm Fe}$ within each time bin are usually over 50 per cent, and so we re-fit with $I_{\rm Fe}$ held at $2.5 \times 10^{-4}$ ph cm$^{-2}$ s$^{-1}$. The results are listed in Table \[tab:5506TRtable\]. The resulting best-fitting parameters for $\Gamma$ and $N_{\rm H}$ are plotted in Fig. \[fig:mega5506zoom\_TR\].
$N_{\rm H}$ increases from an average of $5.4\pm1.0\times 10^{22}$ to $9.4\pm1.1\times 10^{22}$ cm$^{-2}$ during MJD 51624.9–51625.1. For the MJD 51624.9–51625.1 spectrum, freezing $\Gamma$ to 1.93 causes $\chi^2_{\rm red}$ to increase from 1.08 to 1.19; $N_{\rm H}$ goes to a value of $10.7\pm0.6\times 10^{22}$ cm$^{-2}$ (consistent with the value obtained with $\Gamma$ free), but there are systematic data/model residuals around 5 keV, at the $\sim8$ per cent level.
We conclude that a short-duration obscuration event with $\Delta$$N_{\rm H} = 4.0\pm1.4 \times 10^{22}$ cm$^{-2}$ occurred. Given the start and stop times of these observations and the surrounding ones, the duration of the event must be in the range $17.2-69.0$ ks.
--------------------------- ------ -------------------- ----------------------- ------------------------ ---------------------------------------------- ------------
Start–stop Expo $N_{\rm H}$ $A_1$
(MJD) (ks) $\chi^2_{\rm red}$ ($10^{22}$ cm$^{-2}$) $\Gamma$ ($10^{-2}$ ph cm$^{-2}$ s$^{-1}$ keV$^{-1}$) $F_{2-10}$
51623.47–51623.68 ($\#1$) 1.7 0.61 $4.6\pm1.1$ $1.98^{+0.06}_{-0.05}$ $4.1^{+0.6}_{-0.5}$ 9.53
51623.86–51624.14 ($\#2$) 1.6 0.71 $6.2^{+1.2}_{-1.1}$ $2.06^{+0.06}_{-0.05}$ $5.1^{+0.7}_{-0.6}$ 9.91
51624.47–51624.67 ($\#3$) 2.3 0.66 $5.0\pm0.9$ $2.06\pm0.05$ $5.3^{+0.6}_{-0.5}$ 10.73
51624.94–51625.14 ($\#4$) 2.0 1.08 $9.4\pm1.1$ $1.86^{+0.05}_{-0.04}$ $4.0^{+0.5}_{-0.4}$ 9.53
1.19 $10.7\pm0.6$ 1.93 (frozen) $4.7\pm0.1$ 9.45
51625.47–51625.67 ($\#5$) 2.3 0.79 $4.9\pm1.2$ $2.01\pm0.06$ $3.7^{+0.5}_{-0.4}$ 8.09
51625.94–51626.14 ($\#6$) 2.0 0.71 $5.1\pm1.1$ $2.02^{+0.06}_{-0.05}$ $4.2\pm0.6$ 8.84
51626.47–51626.67 ($\#7$) 2.3 0.91 $6.5\pm0.9$ $2.08^{+0.05}_{-0.04}$ $5.9\pm0.6$ 10.73
--------------------------- ------ -------------------- ----------------------- ------------------------ ---------------------------------------------- ------------
\
Results of time-resolved spectroscopy for NGC 5506 in early 2000. $A_1$ is the 1 keV normalization of the power law. $F_{2-10}$ is the observed/absorbed model flux in units of $10^{-11}$ erg cm$^{-2}$ s$^{-1}$. $I_{\rm Fe}$ is frozen at $2.5 \times 10^{-4}$ ph cm$^{-2}$ s$^{-1}$. Each spectral fit is performed over the 3–23 keV bandpass and has 44 $dof$.
Mkn 348 (Sy 2)
--------------
$\bullet$ *1996–1997 (“Secure A”)*: The long-term light curves are plotted in Fig. \[fig:mega348\]. *RXTE* observed this source 25 times over a 16 d period in 1996 May–June, followed by sporadic observations from 1996 December through 1997 July. The source was monitored regularly once every two days during 2011 January – November followed by four times daily monitoring for 30 d in 2011 November–December. Our $HR1$ light curve shows a decrease from 5–8 in 1996 ($\sim$2–3$\sigma$) to 3 by 1997, with values $\sim$1.5–3 in 2011.
Smith [et al.]{} (2001) and Akylas [et al.]{} (2002), fitting spectra from the 1996–1997 campaigns, reported a steady decrease in $N_{\rm H}$ from $\sim$27 to 12 $\times
10^{22}$ cm$^{-2}$; they assumed $\Gamma$ frozen at 1.85 and <span style="font-variant:small-caps;">angr</span> abundances. We divide data from the 1996–1997 campaign into six coarse time slices, to confirm the decrease in $N_{\rm H}$, albeit with a larger time resolution. We assume a simple full-covering neutral absorber and used the best-fitting model from Rivers [et al.]{} (2013); we keep $R$ frozen at 0.3, and $I_{\rm Fe}$ frozen at $3\times 10^{-5}$ ph cm$^{-2}$ s$^{-1}$ (when we allow $I_{\rm
Fe}$ to vary, values are almost always consistent with this value at the 90 per cent level). Assuming <span style="font-variant:small-caps;">angr</span> abundances, we obtain best-fitting values of $N_{\rm H}$($t$) consistent with Akylas [et al.]{} (2002; their fig. 2). Assuming <span style="font-variant:small-caps;">wilm</span> abundances instead, we obtain values of $N_{\rm H}$ an average of 1.42 higher. That is, we measure a decrease from $\sim 32$ to $\sim20$ $\times 10^{22}$ cm$^{-2}$, as listed in Table \[tab:348TRtable\].
We also perform spectral fits to the 2011 data, dividing them into three coarse time slices (MJD 55562–55647, 55684–55888, and 55890–55921), with the divisions corresponding to a sun-angle gap and the start of the intensive monitoring. We apply the same model as above, and we obtain best-fitting values of $N_{\rm H}$ of $\sim 15-17
\times 10^{22}$ cm$^{-2}$ (using <span style="font-variant:small-caps;">wilm</span> abundances), with corresponding values of $\Gamma$ listed in Table \[tab:348TRtable\].
As shown in Table \[tab:348TRtable\], if we assume that the value of $N_{\rm H}$ throughout 1996–1997 was $16.6 \times 10^{22}$ cm$^{-2}$ (the average of the three values obtained for 2011), we obtain fits for spectra \# 1–5 with significantly worse values of $\chi^2_{\rm
red}$, strong curvature in the data/model residuals, and values of $\Gamma$ as low as 1.2.
Spectral fits to an *XMM-Newton* EPIC observation in 2002 July performed by Brightman & Nandra (2011) yield $N_{\rm H} =
13.1^{+0.7}_{-1.3} \times 10^{22}$ cm$^{-2}$ (with $\Gamma=1.68^{+0.07}_{-0.10}$, <span style="font-variant:small-caps;">angr</span> abundances, thus $\sim 18
\times 10^{22}$ cm$^{-2}$ with <span style="font-variant:small-caps;">wilm</span> abundances), consistent with the idea that $N_{\rm H}$ has varied only weakly since 1997 July. Assuming the “baseline” $N_{\rm H}$ level is $\sim
15-18 \times 10^{22}$ cm$^{-2}$ (<span style="font-variant:small-caps;">wilm</span> abundances), the *XMM-Newton* and 2011 *RXTE* data thus support the suggestion by Akylas [et al.]{} (2002) that *RXTE* just caught the tail end of the eclipse in 1997 July.
We assign a minimum duration of 399 d (from the start of the *RXTE* monitoring at MJD 50227 until the point when *RXTE* measured a value of $N_{\rm H}$ consistent with the “baseline” level, after MJD 50626. Using the peak value of $N_{\rm
H}$ from spectrum \#3 and a baseline level of $16.6 \times 10^{22}$ cm$^{-2}$, we adopt $\Delta$$N_{\rm H} = 18\pm3 \times 10^{22}$ cm$^{-2}$.
The Tartarus database of AGN X-ray spectra observed with *ASCA*[^14] indicates absorption by a column $\sim 10 \times 10^{22}$ cm$^{-2}$ for an observation on 1995 August 4 (MJD 49933), suggesting an upper limit to the duration of 693 d. If the event happened to start immediately after the *ASCA* observation, and if it had an $N_{\rm H}$($t$) profile that was symmetric in time, then the event peak would likely have occurred in 1996 July, just after the 1996 *RXTE* observations started. If this is the case then *RXTE* was lucky enough to observe the approximate peak $N_{\rm
H}$ value. We cannot rule out, however, that peak $N_{\rm H}$ occurred before the *RXTE* observations started and thus might be higher than that observed by *RXTE*.
------------------------- ------- ------------------------ ---------------------- ------------------------ --------------------- ------------ ------------------------ ---------------
Start–stop Expo
(MJD) (ks) $\chi^2_{\rm red}/dof$ $N_{\rm H}$ $\Gamma$ $A_1$ $F_{2-10}$ $\chi^2_{\rm red}/dof$ $\Gamma$
50227.05–50231.99 (\#1) 28.8 1.48/46 $32.4^{+2.6}_{-2.5}$ $1.55\pm0.06$ $5.5^{+0.9}_{-0.7}$ 1.27 4.25/47 $1.20\pm0.03$
50233.16–50237.03 (\#2) 79.0 2.62/46 $34.7^{+2.6}_{-2.3}$ $1.59^{+0.06}_{-0.05}$ $4.1^{+0.6}_{-0.5}$ 0.85 6.96/47 $1.19\pm0.02$
50238.44–50243.35 (\#3) 65.8 2.36/46 $35.1^{+2.7}_{-2.5}$ $1.55\pm0.06$ $3.9\pm0.4$ 0.87 6.17/47 $1.15\pm0.02$
50446.44–50528.99 (\#4) 18.2 0.50/46 $34.4^{+6.5}_{-5.5}$ $1.59^{+0.15}_{-0.13}$ $3.6^{+1.5}_{-1.0}$ 0.74 1.05/47 $1.19\pm0.06$
50576.94–50598.06 (\#5) 7.5 0.50/46 $27.1^{+4.0}_{-3.6}$ $1.66^{+0.11}_{-0.10}$ $7.1^{+2.1}_{-1.6}$ 1.46 0.98/47 $1.40\pm0.05$
50626.91–50641.07 (\#6) 27.8 1.02/46 $19.5^{+1.0}_{-0.9}$ $1.71\pm0.03$ $12.2\pm1.0$ 2.67 1.51/47 $1.63\pm0.02$
55562.45–55646.78 (\#7) 38.4 1.31/44 $14.8^{+2.3}_{-2.2}$ $1.69\pm0.08$ $6.5^{+0.8}_{-0.7}$ 1.64 1.32/45 $1.74\pm0.04$
55685.99–55888.42 (\#8) 75.2 0.72/44 $17.7^{+1.6}_{-1.5}$ $1.68\pm0.05$ $7.1\pm0.9$ 1.71 0.73/45 $1.65\pm0.03$
55890.93–55920.54 (\#9) 124.0 1.16/44 $17.2^{+1.1}_{-1.0}$ $1.71\pm0.04$ $8.7^{+0.8}_{-0.7}$ 2.01 1.15/45 $1.69\pm0.02$
------------------------- ------- ------------------------ ---------------------- ------------------------ --------------------- ------------ ------------------------ ---------------
\
Results of time-resolved spectroscopy for Mkn 348; see also Akylas [et al.]{} (2002) for spectral fits to the 1996–1997 data. Column densities, which are in units of $10^{22}$ cm$^{-2}$, refer to total observed columns, i.e., baseline level plus eclipsing cloud. $A_1$ is the 1 keV normalization of the power law in units of $10^{-3}$ ph cm$^{-2}$ s$^{-1}$ keV$^{-1}$. $F_{2-10}$ is the observed/absorbed model flux in units of $10^{-11}$ erg cm$^{-2}$ s$^{-1}$. $I_{\rm Fe}$ is frozen at $3\times 10^{-5}$ ph cm$^{-2}$ s$^{-1}$. Each spectral fit is performed using the 3–23 keV bandpass.
Additional objects with “candidate” events:
-------------------------------------------
*Fairall 9 (Sy 1), 2001.3 (“candidate”)*: The long-term light curve is plotted in Fig. \[fig:megafrl9\].
There is a small spike in $HR1$ over two consecutive points, at MJD 52029 and 52033 ($HR1_{\rm peak} \sim 1.3$ ; $>3\sigma$), plotted in Fig. \[fig:megafrl9zoom\]. However, the uncertainties are very high in each case ($\sim$0.3-0.5). $\Gamma_{\rm app}$ is roughly $1.2-1.4$ with uncertainties 0.06–0.08. We sum the two spectra together for spectral fitting, but the derived constraints are very poor: adopting the parameter values from the time-averaged fit from Rivers [et al.]{}(2013), adding an extra column for neutral absorption using <span style="font-variant:small-caps;">zphabs</span>, and freezing $\Gamma$ to 2.0 yields $N_{\rm H} < 2.1 \times 10^{23}$ cm$^{-2}$. The event satisfies criteria 1–3, but with only an upper limit to $N_{\rm H}$ and with all continuum flux levels rather low, we classify this as a “candidate” event with a duration in the range 4–12 d .
$\bullet$ There are also two single point anomalously high values of $HR1$ at MJD 51257 and 51274 with values of $HR1$ and $\Gamma_{\rm
app}$ similar to the above “candidate” event, but they fail criterion 2. Furthermore, spectral fitting to each spectrum yields $N_{\rm H} <
4 \times 10^{23}$ cm$^{-2}$, and so these events are rejected and not discussed further.
Lohfink et al. (2012) note the trio of low-flux “dips” in the 2–10 and 10–20 keV continuum light curves of Fairall 9 in mid- to late-2000 (MJD $\sim$ 51780, 51830, and 51860). They explore if these sudden decreases in the X-ray continuum flux could be associated with structural changes in the inner accretion disk or with absorption by Compton-thick clouds transiting the line of sight. However, our $HR1$ and $HR2$ light curves do not reveal anything obvious during mid- to late-2000: values are usually consistent with $\langle HR1 \rangle$ or $\langle HR2 \rangle$ and/or have large error bars and large point-to-point scatter.
*NGC 3516 (Sy 1.5), 2011.7, “candidate” event:* This source’s $HR1$ light curve (Fig. \[fig:mega3516\]) shows systematic trends that are visually not obviously correlated with 2–10 or 10–18 keV continuum flux. Of peculiar interest is a “spike” in $HR$ to $\sim$ 2.4 ($\sim4\sigma$) during Fall 2011. $\Gamma_{\rm app}$ is usually 1.49$\pm$0.19 for NGC 3516 but reaches values as low as 0.8–0.9 in Fall 2011.
The X-ray spectrum of this object is commonly characterized by complex absorption, which can be variable on timescales of years, as documented by Turner [et al.]{} (2008). For example, Markowitz [et al.]{} (2008) found the source to be in a Compton-thin absorbed state in 2005. Turner [et al.]{} (2008), fitting *Chandra* HETGS and *XMM-Newton* EPIC and RGS spectra, list four “zones” of absorption in addition to $N_{\rm H,Gal}$. Zone 3 (using their notation) is expected to impact modeling of PCA spectra of NGC 3516. This absorber is a partial-covering, moderately-ionized (log$\xi$ = 2.19) absorber; Turner [et al.]{} (2008) argue that variations in the covering fraction $CF$ on timescales of $\sim$ half a day can explain observed spectral variability. Rivers [et al.]{} (2011a), fitting the time-averaged PCA + HEXTE spectrum, modeled Zone 3 with log($\xi$) frozen at 2.19 and column density $N_{\rm H,WA}$ frozen at $2 \times
10^{23}$ cm$^{-2}$, the best-fitting parameters from Turner [et al.]{} (2008), but leave the covering fraction $CF$ free, obtaining a best-fitting value of $CF=0.55 \pm 0.10$.
For time-resolved spectroscopy, we sum the spectra from 2011 into five bins spanning approximately 80 d each; these spectra typically have exposure times of 11–22 ks and 25–50 $\times 10^3$ counts in the 3–23 keV band. We fit 3–23 keV spectra using the best-fitting model parameters from Rivers [et al.]{} (2011a), keeping log ($\xi$) and $N_{\rm
H,WA}$ frozen at 2.19 for simplicity.
We test three simple models for the spectral variability: a model with $CF$ free, $\Gamma$ frozen at 1.85 (from Rivers [et al.]{} 2011a), and $N_{\rm H,WA}$ frozen at $2 \times 10^{23}$ cm$^{-2}$ (which we call Model PC1); $\Gamma$ frozen at 1.85, $CF$ frozen at 0.75, and $N_{\rm
H,WA}$ free (Model PC2); and $\Gamma$ free, $CF$ frozen at the arbitrary value of 0.75, and $N_{\rm H,WA}$ frozen at $2 \times
10^{23}$ cm$^{-2}$ (Model PC3). $A_1$ and $I_{\rm Fe}$ are free parameters in all fits. The results are summarized in Table A8.
For Model PC1, best-fitting values of $CF$ span $0.34$ to $1.00$, with the highest value occurring for the Fall 2011 spectrum ($\#4$). However, uncertainties on $CF$ are extremely large; we also cannot rule out full-covering for spectrum $\#2$. Values of $\chi^2_{\rm
red}$ between each of the three models were very similar and usually $\la$1. For model PC2, spectrum $\#4$ yields a value of $N_{\rm H,WA}$ about twice that for spectra $\#2$, 3 and 5, but $\chi^2_{\rm red}$ was 1.57, with strong data/model residuals (at the $\sim10-20$ per cent level) below 4–5 keV.
For Model PC3, best-fitting values of $\Gamma$ are usually 1.7 and greater, with $\Gamma=1.55\pm0.04$ for the Fall 2011 spectrum; such values are not implausible based on empirical grounds. Since we cannot rule out flattening of the primary power law as the driver behind the observed variations in $HR1$, we cannot confirm that variations in properties of the ionized absorber occurred. More definitive results can be obtained if this source is subjected to future long-term X-ray spectral monitoring using higher energy resolution and a wider bandpass than the PCA, such as eROSITA or the proposed *Large Observatory For X-ray Timing*.
We thus classify the Fall 2011 event as a “candidate” full-covering event, with a duration of $\sim57$ d based on the points in the $HR1$ light curve above $1\sigma$ deviation. With best-fitting covering fractions of 74, 100, and 79 per cent before, during, and after the event, respectively, in Model PC1, we use a covering fraction of 23.5 per cent and assign $\Delta$$N_{\rm H}$ =(0.235 $\times$ $2\times10^{23}$ cm$^{-2}$) = $4.7\times10^{22}$ cm$^{-2}$.
--------------------------- ------ -------------------- ------------------------ ---------------------- ------------ -------------------- --------------------- --------------
Start–stop Expo
(MJD) (ks) $\chi^2_{\rm red}$ $CF$ $A_1$ $F_{2-10}$ $\chi^2_{\rm red}$ $N_{\rm H,WA}$ $A_1$
(44 $dof$) (45 $dof$)
55561.55–55637.70 ($\#1$) 17.4 0.67 $0.34^{+0.10}_{-0.09}$ $ 6.3\pm1.1$ 4.60 0.76 $8.7^{+1.3}_{-2.0}$ $13.5\pm0.4$
55641.58–55721.90 ($\#2$) 17.7 0.84 $0.84^{+0.16}_{-0.14}$ $12.8\pm1.0$ 2.55 1.02 $35\pm5$ $11.4\pm0.7$
55725.88–55769.57 ($\#3$) 10.8 0.50 $0.74^{+0.24}_{-0.18}$ $10.1\pm1.3$ 2.44 0.49 $28^{+7}_{-5}$ $10.0\pm0.8$
55773.55–55873.27 ($\#4$) 22.2 0.98 $1.00^{+0.00}_{-0.08}$ $14.5^{+0.3}_{-0.4}$ 2.01 1.57 $65^{+2}_{-4}$ $12.3\pm0.3$
55877.52–55925.39 ($\#5$) 11.4 1.08 $0.79^{+0.19}_{-0.15}$ $13.4^{+1.2}_{-1.3}$ 2.92 1.13 $30\pm5$ $12.3\pm0.9$
--------------------------- ------ -------------------- ------------------------ ---------------------- ------------ -------------------- --------------------- --------------
--------------------------- -------------------- --------------- ----------------------
Start–stop
(MJD) $\chi^2_{\rm red}$ $\Gamma$ $A_1$
(45 $dof$)
55561.55–55637.70 ($\#1$) 1.53 $2.12\pm0.03$ $30.3^{+0.9}_{-1.2}$
55641.58–55721.90 ($\#2$) 0.90 $1.78\pm0.04$ $9.5\pm0.7$
55725.88–55769.57 ($\#3$) 0.49 $1.86\pm0.05$ $10.4^{+1.1}_{-1.0}$
55773.55–55873.27 ($\#4$) 1.03 $1.55\pm0.04$ $ 5.0\pm0.4$
55877.52–55925.39 ($\#5$) 1.11 $1.83\pm0.04$ $11.8\pm1.0$
--------------------------- -------------------- --------------- ----------------------
\
Results of time-resolved spectroscopy for the 2011 campaign of NGC 3516. Each model incorporates a partial-covering absorber with log($\xi$)=2.19. Covering fraction $CF$ is free in Model PC1, and frozen at 0.75 in Models PC2 and PC3. $N_{\rm H,WA}$ is free in Model PC2 and frozen at $20 \times 10^{22}$ cm$^{-2}$ in Models PC1 and PC3. $\Gamma$ is free in Model PC3, and frozen at 1.85 in Models PC1 and PC2. $N_{\rm H}$ is listed in units of $10^{22}$ cm$^{-2}$. $A_1$ is the 1 keV normalization of the *covered* power law only in units of $10^{-3}$ ph cm$^{-2}$ s$^{-1}$ keV$^{-1}$. $F_{2-10}$ is the observed/absorbed model flux in units of $10^{-11}$ erg cm$^{-2}$ s$^{-1}$; values for Models PC2 and PC3 were virtually identical to those for model PC1. $I_{\rm
Fe}$ is left free, with best-fitting values and uncertainties typically $\sim6\pm3\times 10^{-5}$ ph cm$^{-2}$ s$^{-1}$ for Models PC1 and PC2 and typically $\sim8\pm4\times 10^{-5}$ ph cm$^{-2}$ s$^{-1}$ for Model PC3.
Summary of eclipse events manifested via flux-flux plots
--------------------------------------------------------
If there is an eclipse by a discrete cloud with $N_{\rm H} \sim
10^{22-23}$ cm$^{-2}$, as implied by the above spectral fits, then the 2–4 keV band should be impacted much more strongly than the 10–18 keV band. The 10–18 keV band probes the uneclipsed portion of the continuum power-law component in this case. In Figure \[fig:fluxflux\], we plot 2–4 keV flux as a function of 10–18 keV flux for each observation during each eclipse event, along with flux points representing non-eclipsed periods (the gray points). We use only the 75 data points before and after the putative eclipse (excluding other putative eclipse events) for clarity, to avoid overcrowding the figure. The green dashed line is a best fit to the gray points, using a linear regression algorithm that accounts for uncertainties in both quantities (Fasano & Vio 1988). The green dot-dashed lines indicate the $\pm 1\sigma$ distribution of the gray points. When there is no eclipse, the 2–4 and 10–18 keV bands are both probing the continuum and track variations in the power-law continuum. During each of the secure events, the flux-flux points track the spectral variability away from the main distribution, especially near the peak of each event. These plots do not directly yield quantitative information about $N_{\rm H}$ or $\Gamma$; those quantities are best obtained via time-resolved spectroscopy. However, these plots confirm that the 2–4 keV band is impacted independently of the behavior of the primary power law. For each of the candidate events, however, the low levels of 10–18 kev continuum flux (as indicated in bottom row of panels) introduce ambiguity in terms of our ability to use time-resolved spectroscopy to rule out flattening of the primary power law as the cause of observed spectral flattening.
Type IIs with $\geq$0.6 years of monitoring
-------------------------------------------
In Figs. \[fig:sy2iras\]–\[fig:sy27314\] we plot the long-term continuum and $HR1$ light curves for six type II objects with at least 0.6 years of sustained monitoring. X-ray spectral observations typically confirm the presence of an X-ray column $\ga10^{22}$ cm$^{-2}$ in each of these objects (e.g., REN02). As hardness ratios are sensitive to changes in either $\Gamma$ or $N_{\rm H}$, the lack of any obvious systematic trends (above 2.0$\sigma$ from $\langle HR1
\rangle$, and/or trends $\sim50$ per cent greater than $\langle HR1 \rangle$) suggests the lack of strong variations in either of these parameters. If we assume that $\Gamma$ is intrinsically constant, then we can use the average and 1$\sigma$ standard deviation of $HR1$ and the best-fitting time-averaged parameters from Rivers et al. (2013) to estimate the corresponding maximum change in $N_{\rm H}$; these estimates of $\Delta$$N_{\rm H}$ are listed in Table \[tab:boringsy2s\].
---------------------- ----------------------- ---------- --------------------------------------------------- --------------------------------------------------- -----------------------
Typical Typical $\Delta$$N_{\rm H}$
Source $\langle HR1 \rangle$ $\sigma$ $N_{\rm H}$ ($10^{22}$ cm$^{-2}$) $\Gamma$ ($10^{22}$ cm$^{-2}$)
IRAS 04575–7537 0.56 0.06 $3.6\pm2.6$ (R13) $2.48\pm0.22$ (R13) $\sim0.6$
Mkn 348 (2011 only) 2.35 0.84 $16.6\pm1.6$ (TW); $\sim11-16$ (R02) $1.69\pm0.06$ (TW) $\sim5$
NGC 1052 1.93 0.87 $13.6\pm5.2$ (R13); $31^{+11}_{-26}$ (B11) $1.71\pm0.29$ (R13); $1.7^{+0.1}_{-0.2}$ (B11) $\sim6$
NGC 2992 0.84 0.17 0.4–1.6 (R02); $0.41\pm0.17$ (B11) $1.78\pm0.18$ (R13); $1.59\pm0.03$ (B11) $\la4$
NGC 4258 1.21 0.61 $8.4\pm3.9$ (R13); $\sim9-14$ (R02) $1.80\pm0.17$ (R13) $\sim5$
NGC 5506$^{\dagger}$ 0.89 0.10 $1.9\pm0.5$ (R11); $\sim2-4$ (R02) $1.93\pm0.03$ (R11); $1.82^{+0.05}_{-0.04}$ (B11) $\sim1$
NGC 6251 0.36 0.49 $\la0.4$ (G09) $2.38\pm0.23$ (R13); $1.67\pm0.06$ (E06) $\la9$
NGC 7314 0.68 0.09 $\sim0.8-1.3$ (R02); $0.60^{+0.01}_{-0.03}$ (B11) $1.99\pm0.10$ (R13); $1.95^{+0.02}_{-0.01}$ (B11) $\sim2$
---------------------- ----------------------- ---------- --------------------------------------------------- --------------------------------------------------- -----------------------
\
Estimates of $\Delta$$N_{\rm H}$ corresponding to the standard deviations $\sigma$ of $HR1$. $^{\dagger}$For NGC 5506, the two observations at MJD 51624.94 and 51625.14 during an eclipse event are excluded. References for typical values of $\Gamma$ and $N_{\rm H}$ are: B11 = Brightman & Nandra (2011); E06 = Evans et al. (2006); G09 = González-Martín et al. (2009); R11 = Rivers et al. (2011a); R13 = Rivers et al. (2013); R02 = Risaliti, Elvis & Nicastro (2002) and references therein; TW = this work (Mkn 348: average of the three values derived from time-resolved fitting).
For NGC 4258, the average value of $HR1$ drops from 1.46 (stand. dev. 0.35) for the 1997–2000 data to 1.07 (stand. dev. 0.68) for the 2005–2011 data. While such a drop is consistent with the average values of $\Gamma$ or $N_{\rm H}$ systematically varying, spectral fitting on data summed over 1997–2000 and over 2005–2011 failed to provide evidence for changes in $\Gamma$ or $N_{\rm H}$, as uncertainties on these parameters were extremely large. Similarly, the average value of $HR1$ for NGC 5506 (Fig. \[fig:mega5506\]) also seems to fall very slightly towards the end of 1999: $\langle HR1 \rangle$ = 1.03 (stand. dev. 0.08) for data before MJD 51300, and 0.85 (stand. dev. 0.06) for data after MJD 51500 (excluding the eclipse event covering the two observations at MJD 51624.94 and 51625.14). Spectral fitting on these two data sets, however, yields values of $N_{\rm H}$ consistent with the average at 90 per cent confidence.
Inferred radial distances of various emission/absorption components in our eight primary sources
================================================================================================
In this appendix, we provide the inferred distances from the central black hole of various emission/absorption components in our eight sources with “secure” eclipse events; Fig. \[fig:plotrs\] is based on these values. In Table B1, we list the X-ray clumps’ minimum and maximum distances as calculated in $\S$5.2: assuming a range of log($\xi$) from –1 to +1 for all events except for the two events in Cen A (log($\xi$) = –1 to 0) and NGC 3227/2000–1 (–0.3 to 0). We also list inferred locations of IR-emitting tori from either interferometry or reverberation mapping, BLR emission lines, and Fe K$\alpha$ line-emitting gas. The reader is reminded that even if distances from the black hole are commensurate, structures may not physically overlap (e.g., emission originating from out of the line of sight versus X-ray absorption along the line of sight).
Source Component Radius (light-days) Radius ($R_{\rm g}$) Ref.
------------- ------------------------------------------------------------------------ ---------------------- -------------------------------------- -----------
NGC 3783 He <span style="font-variant:small-caps;">ii</span> $1.4^{+0.8}_{-0.5}$ $850^{+480}_{-300}$ P04
Si <span style="font-variant:small-caps;">iv</span> $2.0^{+0.9}_{-1.1}$ $1210^{+550}_{-670}$ P04
C <span style="font-variant:small-caps;">iv</span> $3.8^{+1.0}_{-0.9 }$ $2300^{+610}_{-550}$ P04
H$\beta$ $10.2^{+3.3}_{-2.3}$ $6180^{+2000}_{-1390}$ P04
He <span style="font-variant:small-caps;">i</span>$\lambda$2.058$\mu$ $14.7^{+4.7}_{-3.2}$ $8570^{+2740}_{-1870}$ R02
Br$\gamma$ (broad) $21.4^{+1.5}_{-1.3}$ $1.253^{+0.083}_{-0.075}\times 10^4$ R02
Fe K$\alpha$ width, HETGS $66^{+39}_{-20}$ $3.9^{+2.2}_{-1.2} \times 10^4$ S10
IR torus (B08, Model A) 48–76 $2.8-4.4 \times 10^4$ B08
$H$-band-emitting dust $66^{+6}_{-7}$ $3.9\pm0.4\times 10^4$ L11
$K$-band-emitting dust $76^{+11}_{-17}$ $4.4^{+0.6}_{-1.0} \times 10^4$ L11
X-ray clump 55–400 3.2–23 $\times 10^4$ This work
$R_{\rm d}$ 240 $13.9 \times 10^4 $ This work
IR torus (B08, Model B) 250–357 $14.6-20.9 \times 10^4$ B08
Mkn 79 Fe K$\alpha$ width, EPIC $3.0^{+0.7}_{-0.8}$ $990^{+210}_{-240}$ Ga11
H$\beta$ $9.0^{+8.3}_{-7.8}$ $ 3100^{+2800}_{-2600}$ P04
H$\beta$ $16.0^{+6.4}_{-5.8}$ $ 5400^{+2200}_{-2000}$ P04
H$\beta$ $16.1\pm6.6$ $ 5500\pm 2200$ P04
C <span style="font-variant:small-caps;">iv</span> 16.9 5590 K07
Br$\gamma$ 20.5 6750 L08
H$\alpha$ 23.3 7660 L08
H$\beta$ 26.4 8690 L08
He <span style="font-variant:small-caps;">i</span>$\lambda$5876 27.9 9170 L08
O <span style="font-variant:small-caps;">i</span>$\lambda$1.1287$\mu$ 30.7 10090 L08
Pa$\alpha$, Pa$\beta$, Pa$\delta$, Pa$\epsilon$ 29.0–31.5 9530– 10360 L08
O <span style="font-variant:small-caps;">i</span>$\lambda$8446 32.6 10720 L08
He <span style="font-variant:small-caps;">i</span>$\lambda$1.0830$\mu$ 57.1 18800 L08
$R_{\rm d}$ 340 $11.1 \times 10^4 $ This work
X-ray clump 60–1020 $2.0-34 \times 10^4$ This work
Mkn 509 He <span style="font-variant:small-caps;">ii</span> $33.5^{+8.2}_{-7.1}$ $3870^{+950}_{-820}$ P04
He <span style="font-variant:small-caps;">i</span>$\lambda$5876 59 6550 L08
H$\beta$ 68 7570 L08
H$\beta$ $79.6^{+6.1}_{-5.4}$ $9190^{+700}_{-620}$ P04
Pa$\alpha$, Pa$\beta$, Pa$\gamma$, Pa$\delta$, Pa$\epsilon$ 93–120 10330–13340 L08
O <span style="font-variant:small-caps;">i</span>$\lambda$8446 99 11020 L08
H$\alpha$ 100 11180 L08
H$\gamma$ 103 11450 L08
Br$\gamma$ 111 12390 L08
He <span style="font-variant:small-caps;">i</span>$\lambda$1.0830$\mu$ 119 13320 L08
Fe K$\alpha$ line width, HETGS $123^{+254}_{-89}$ $1.4^{+2.8}_{-1.0} \times 10^4$ S10
O <span style="font-variant:small-caps;">i</span>$\lambda$1.1287$\mu$ 126 14010 L08
$R_{\rm d}$ 610 $6.7 \times 10^4 $ This work
X-ray clump 230–2800 2.6–31 $\times 10^4$ This work
MR 2251–178 Fe K$\alpha$ width, *Suzaku* $>$9 $>$750 Go11
H$\beta$ 27 2300 S07
C <span style="font-variant:small-caps;">iv</span> $85^{+15}_{-13}$ $7230^{+1310}_{-1040}$ S07
$R_{\rm d}$ 910 $8.1 \times 10^4 $ This work
X-ray clump 200–2500 1.7–120 $\times 10^4$ This work
Source Component Radius (light-days) Radius ($R_{\rm g}$) Ref.
---------- ------------------------------------------------------------------------ ------------------------ ------------------------------------ -------------------
NGC 3227 He <span style="font-variant:small-caps;">i</span>$\lambda$5876 2.0 4580 L08
H$\beta$ 3.4 7640 L08
Pa$\epsilon$ 3.4 7690 L08
H$\gamma$ 3.7 8330 L08
O <span style="font-variant:small-caps;">i</span>$\lambda$8446 4.0 8980 L08
H$\beta$ $3.8\pm0.8$ $9000\pm1890$ D10
Br$\gamma$ 4.0 9010 L08
H$\alpha$ 4.4 10010 L08
O <span style="font-variant:small-caps;">i</span>$\lambda$1.1287$\mu$ 4.9 11220 L08
He <span style="font-variant:small-caps;">i</span>$\lambda$1.0830$\mu$ 5.7 12860 L08
Pa$\beta$, Pa$\delta$ 6.0 13600 L08
Fe K$\alpha$ width, EPIC $7.2^{+12.7}_{-4.9}$ $1.7^{+3.0}_{-1.2} \times 10^4$ M09
H$\alpha$ $18.9^{+8.7}_{-11.3}$ $4.48^{+2.06}_{-2.68} \times 10^4$ P04
$K$-band-emitting dust $\sim$20 $4.74 \times 10^4$ S06
X-ray clump 6.5–91 $1.5 - 21 \times 10^4$ This work
$R_{\rm d}$ 85 $19.2 \times 10^4 $ This work
Fe K$\alpha$ variability $<$ 700 $< 1.7 \times 10^6$ M09
Cen A $R_{\rm d}$ 42 $1.22 \times 10^4 $ This work
Fe K$\alpha$ width, *Suzaku* $>$ 66 $>1.9 \times 10^4$ Ma07
IR Torus 120–360 3.5–11 $\times 10^4$ Me07
X-ray clump 94–710 2.7–20 $\times 10^4$ This work
NGC 5506 X-ray clump 15–220 $0.3-4.4 \times 10^4 $ This work
Pa$\beta$ $\sim$190 $3.7 \times 10^4 $ N02
Fe K$\alpha$ width, HETGS $220^{+764}_{-127}$ $4.4^{+15.2}_{-2.5} \times 10^4 $ S10 (68$\%$ err.)
$R_{\rm d}$ 240 $4.8 \times 10^4 $ This work
Br$\gamma$ $250^{+40}_{-30}$ $5.0^{+0.7}_{-0.6} \times 10^4 $ N02
Br$\alpha$ $420^{+70}_{-60}$ $8.3^{+1.6}_{-1.2} \times 10^4$ L02
Mkn 348 H$\beta$ $1.19\pm0.03$ 1340$\pm$30 T95
H$\alpha$ $1.47^{+0.10}_{-0.11}$ $1650^{+130}_{-110} $ T95
Br$\gamma$ $\sim$ 16.6 $1.87 \times 10^4 $ V97
Pa$\beta$ $\sim$ 30.4 $3.42 \times 10^4 $ V97
$R_{\rm d}$ 235 $26.8 \times 10^4 $ This work
X-ray Clump 140–1290 $16-150 \times 10^4$ This work
\
For BLR line lags from reverberation mapping (P04, D10), $\tau_{\rm
cent}$ is used if available. For Mkn 79, we use “unflagged” H$\beta$ values only. $R_{\rm d}$ denotes the approximate outer boundary of the DSZ, i.e., dust residing at distances greater than $R_{\rm d}$ likely does not sublimate, while distances smaller than $\sim\frac{1}{2}-\frac{1}{3}R_{\rm d}$ are likely to be dust-free. References for Column (5) are: B08 = Beckert [et al.]{} (2008), D10 = Denney [et al.]{} (2010), Ga11 = Gallo [et al.]{} (2011), Go11 = Gofford et al. (2011), K07 = Kelly & Bechtold (2007), L02 = Lutz [et al.]{} (2002), L08 = Landt [et al.]{} (2008), L11 = Lira [et al.]{} (2011), Ma07 = Markowitz [et al.]{}(2007), M09 = Markowitz [et al.]{} (2009), Me07 = Meisenheimer [et al.]{}(2007), N02 = Nagar [et al.]{} (2002), P04 = Peterson [et al.]{} (2004), R03 = Reunanen [et al.]{} (2003), S06 = Suganuma [et al.]{} (2006), S07 = Sulentic [et al.]{} (2007), S10 = Shu [et al.]{} (2010; 68 per cent uncertainties used), T95 = Tran (1995), V97 = Veilleux [et al.]{} (1997).
[^1]: E-mail: [email protected]
[^2]: This is not surprising, given blazars’ orientation and that lines of sight along the poles have the lowest likelihood to have obscuring clouds in ${\textsc{Clumpy}}$ models ($\S$5.5), and additionally given that jets might destroy clouds or push them aside.
[^3]: Other values, e.g., between 50 and 90 per cent yield virtually identical results for this calculation.
[^4]: PCUs 3 and 4 (and also PCU 1 starting late 1998/early 1999) suffered from repeated discharge problems. PCU 0 lost its propane veto layer following a suspected micrometeroid hit on 2000 May 12.
[^5]: Values of $F_{10-18}/F_{2-4}$ peak at column densities $\sim 0.8-2 \times 10^{23}$ cm$^{-2}$, quite similar to $HR1$, so we use 4–7 keV as our lower energy band for $HR2$.
[^6]: Here, $R$ is defined following the convention of the <span style="font-variant:small-caps;">xspec</span> model <span style="font-variant:small-caps;">pexrav</span>, with a normalization defined relative to that of the illuminating power law and with $R=1$ corresponding to a sky-covering fraction of 2$\pi$ sr as seen from the illuminating source.
[^7]: Other values, e.g., between 50 and 90 per cent yield virtually identical results for this calculation as well; the effects on the summed selection functions are always negligible compared with the uncertainties stemming from the varying contributions of individual source selection functions ($\S$4.1).
[^8]: Values for $L_{\rm Bol}$ for most sources are taken from Vasudevan et al. (2010), except for Fairall 9 (Vasudevan & Fabian 2009), and MR 2251–178 and Cen A ($L_{\rm 2-10}$ from Rivers et al. 2013 and $L_{\rm Bol}/L_{2-10}$ corrections from Marconi et al. 2004 for both objects).
[^9]: This may not be true for Cen A, whose continuum emission is likely mildly beamed. A typical value of Doppler $\delta$ for Cen A is 1.2 (Chiaberge et al. 2001). Using 0.66 as the spectral index (Abdo et al. 2010), the flux will be boosted by $\delta^{2+\alpha} \sim 1.6$, which translates into only a $\sim26$ per cent effect on our estimate of $R_{\rm d}$.
[^10]: Resulting values of $r_{\rm cl }$ for NGC 2992 ($t_{\rm d}$=0.90 yr; $M_{\rm BH} = 5.2\times10^7~{\ensuremath{\rm M_\odot}\xspace}$, Woo & Urry 2002), Mkn 348 ($t_{\rm d}$=0.98 yr, 2011 only), and NGC 7314 ($t_{\rm d}$=1.56 yr, 1999–2000; $M_{\rm BH} = 1.4 \times 10^6$ ${\ensuremath{\rm M_\odot}\xspace}$, Vasudevan et al. 2010) are 0.26, 0.09, and 0.02 pc, respectively.
[^11]: Bolometric luminosities for NGC 5506, NGC 1052 and NGC 4258 from Vasudevan et al. (2010), Woo & Urry (2002), and Lasota et al. (1996), respectively.
[^12]: We employ the optical properties of the composite silicate/graphite grains used in the [<span style="font-variant:small-caps;">Clumpy</span>]{} models, with $\tau_{\rm V} = 23.6\tau_{10\micron}$.
[^13]: “Zone 2” from Gofford et al. (2011), which has log($\xi$) $\sim2.21$, may also potentially have a modest impact on *RXTE* spectra. However, at the column density measured by Gofford et al. (2011), $6\times 10^{21}$ cm$^{-2}$, the effect on the X-ray continuum near 3–5 keV is only at the $\sim4$ per cent level.
[^14]: http://heasarc.gsfc.nasa.gov/W3Browse/asca/tartarus.html
| ArXiv |
---
abstract: 'As the realization of a fully operational quantum computer remains distant, *quantum simulation*, whereby one quantum system is engineered to simulate another, becomes a key goal of great practical importance. Here we report on a variational method exploiting the natural physics of cavity QED architectures to simulate strongly interacting quantum fields. Our scheme is broadly applicable to any architecture involving tunable and strongly nonlinear interactions with light; as an example, we demonstrate that existing cavity devices could simulate models of strongly interacting bosons. The scheme can be extended to simulate systems of entangled multicomponent fields, beyond the reach of existing classical simulation methods.'
author:
- Sean Barrett
- Klemens Hammerer
- Sarah Harrison
- 'Tracy E. Northup'
- 'Tobias J. Osborne'
title: Simulating Quantum Fields with Cavity QED
---
Modelling interacting many-particle systems classically is a challenging yet tractable problem. However, in the quantum regime, it becomes rapidly intractable, owing to the dramatic increase in the number of variables required to describe the system. Feynman [@Feynman1982] realized that an alternate approach would be to exploit quantum mechanics to carry out simulations beyond the reach of classical computers. This idea was the basis of Lloyd’s simulation algorithm [@Lloyd1996], a procedure for a *digital* quantum computer to simulate the dynamics of a strongly interacting quantum system. In contrast, there is also an *analogue* approach to quantum simulation, where the simulator’s Hamiltonian is tailored to match that of the simulated system [@Buluta2009]. The complementary aspects of the analogue and digital methods, reviewed in [@Buluta2009; @Aspuru-Guzik2012; @Bloch2012; @Johanning2009; @Lewenstein2007], have led to a host of recent experiments [@Friedenauer2008; @Gerritsma2010; @Haller2010; @Kim2010; @Lanyon2010; @Islam2011; @Simon2011; @Barreiro2011; @Lanyon2011].
To date, most experimental implementations of quantum simulation algorithms have been focussed on the task of simulating *quantum lattice systems*, with comparatively less attention paid to systems with continuous degrees of freedom. The archetypal example of a quantum system with a continuous degree of freedom is the *quantum field*. Currently, quantum simulations of quantum field theories have relied on discretization of the dynamical degrees of freedom. One body of recent theoretical work is focussed on the analogue simulation of discretized quantum fields, using cold atoms in optical lattices [@Lepori2010; @Bermudez2010; @Cirac2010; @Semiao2011; @Kapit2011] and coupled cavity arrays [@Hartmann2006; @Greentree2006; @Angelakis2007]. Complementing this are proposals for digital quantum simulation on a universal quantum computer of the zero-temperature [@Byrnes2006] and thermal [@Temme2011] dynamics of non-abelian gauge theories and, more recently, a digital quantum simulation [@Jordan2012; @Jordan2011] of scattering processes of a discretized $\lambda\phi^4$ quantum field.
In this paper we report on an *analogue* algorithm to simulate the ground-state physics of a one-dimensional strongly interacting quantum field using the *continuous* output of a cavity-QED apparatus [@Raimond2001; @Miller2005; @Walther2006; @Haroche2006; @Girvin2009]. Our method involves no discretization of the dynamical degrees of freedom; the simulation register is the continuous electromagnetic output mode of the cavity. The variational wave function generated in this way therefore belongs to an extremely expressive class, namely the class of continuous matrix product states, as we will show. We argue that our approach is already realizable with state-of-the-art cavity-QED technology.
{width="1.5\columnwidth"}
We concentrate on simulating quantum fields modelling collections of strongly interacting bosons in one spatial dimension. These systems are compactly described in second quantization using the quantum field annihilation and creation operators $\hat{\psi}(x)$ and $\hat{\psi}^\dag(x)$, which obey the canonical commutation relations $[\hat{\psi}(x), \hat{\psi}^\dag(y)] = \delta(x-y)$. The task is to determine the ground state of a given field-theoretic Hamiltonian $\hat{\mathcal{H}}(\hat\psi,\hat\psi^\dagger)$. The prototypical form of such a Hamiltonian is $$\label{eq:Ham}
\hat{\mathcal{H}} = \int (\hat{T} + \hat{W} + \hat{N})\, dx$$ where $\hat{T} = \frac{d\hat{\psi}^\dag(x)}{dx}\frac{d\hat{\psi}(x)}{dx}$, $\hat{W} = \int w(x-y) \hat{\psi}^\dag(x)\hat{\psi}^\dag(y)\hat{\psi}(y)\hat{\psi}(x)\, dy$, and $\hat{N} = -\mu\hat{\psi}^\dag(x)\hat{\psi}(x)$ describe the kinetic energy, two-particle interactions with potential $w(x-y)$, and the chemical potential, respectively. Our approach provides a quantum variational algorithm for finding the ground states of an arbitrary Hamiltonian that is translation-invariant and consists of finite sums of polynomials of creation/annihilation operators and their derivatives.
The apparatus proposed to simulate the ground-state physics of $\hat{\mathcal{H}}$ is a single-mode cavity coupled to the quantum degrees of freedom of some intracavity medium (Fig. \[fig:dictionary\]); our proposal is not tied to the specific nature of the medium, so long as one or more tunable nonlinear interactions are present that are sufficiently strong at the single-photon level. Below we consider the example of a single trapped atom coupled to the cavity via electronic transitions. The system is described by a Hamiltonian $\hat H_{\mathrm{sys}}(\lambda)$ that depends on a set of controllable parameters $\lambda$, for example, externally applied fields. When the cavity is driven, either directly through one of its mirrors or indirectly through the medium, the intracavity field relaxes to a stationary state, and the cavity emits a steady-state beam of photons in a well-defined mode.
The crucial idea underlying our proposal is to regard the steady-state cavity output as a continuous register recording a *variational* quantum state $|\Psi(\lambda)\rangle$ of a one-dimensional quantum field with control parameters $\lambda$ *as* the variational parameters. This representation is chosen so that the spatial location $x$ of the simulated translation-invariant field is identified with the value of the time-stationary cavity output mode exiting the cavity at time $t = x/s$. The arbitrary scaling parameter $s$ is included in the set of variational parameters $\lambda$. We complete this identification by equating the annihilation operator $\hat{\psi}(x)$ of the simulated quantum field with the field operator $\hat{E}^{+}(t)$ for the positive-frequency electric field of the cavity output mode [^1], via $\hat{\psi}(x) =\hat E^{+}(t)/\sqrt{s}$.
Recall that the variational method proceeds by minimizing the average energy density of the variational state $f(\lambda) = \langle \Psi(\lambda)|\hat{T}+\hat{W}+\hat{N}|\Psi(\lambda)\rangle$ over the variational parameters $\lambda$. A key point in our scheme is that — with the identification of the field operators $\hat E^{+}(t)$ and $\hat{\psi}(x)$ in hand — the value of $f(\lambda)$ can be determined from standard optical measurements on the cavity output field, namely the measurement of *Glauber correlation functions* [@Glauber1963; @Mandel1995], see Fig. \[fig:dictionary\]. This result is easily seen for the Hamiltonian of Eq. . Thanks to the linearity of the expectation value, we can separately measure $\langle \hat{T} \rangle$, $\langle \hat{W}\rangle$, and $\langle \hat{N} \rangle$. The expectation value of the chemical potential term corresponds to a function of the *intensity* of the output beam via $\langle \hat{N} \rangle = -\frac{\mu}{s}\langle\hat E^{-}(t)\hat E^{+}(t)\rangle$. The kinetic energy term $\langle \hat{T} \rangle$ corresponds to the limit $$\begin{aligned}
\langle \hat{T} \rangle = \lim_{\epsilon_1,\epsilon_2\rightarrow 0} \frac{1}{s^3\epsilon_1\epsilon_2}&\left(g^{(1)}(t+\epsilon_1,t+\epsilon_2)-g^{(1)}(t+\epsilon_1,t)\right.\\
&\left.-g^{(1)}(t,t+\epsilon_2)+g^{(1)}(t,t)\right),\end{aligned}$$ where $g^{(1)}(t_1,t_2)=\langle\hat E^{-}(t_1)\hat E^{+}(t_2)\rangle$; this quantity can be estimated by choosing a finite but small value for $\epsilon_1$ and $\epsilon_2$. Note that this procedure does not amount to a simple space discretization because the output is a continuous quantum register. The final term $\langle \hat{W}\rangle$ depends on two-point spatial correlation functions $\langle \hat{\psi}^\dag(x)\hat{\psi}^\dag(y)\hat{\psi}(y)\hat{\psi}(x)\rangle$, which translate to measurements of $g^{(2)}(t_1,t_2)=\langle\hat E^{-}(t_1)\hat E^{-}(t_2)\hat E^{+}(t_2)\hat E^{+}(t_1)\rangle$. The detection schemes to estimate all terms in the showcase Hamiltonian are presented in Fig. \[fig:dictionary\]. From a wider perspective, *any* Glauber correlation function $g^{(n,m)}=\langle\hat E^{-}(t_1)\cdots\hat E^{-}(t_n)\hat E^{+}(t'_m)\cdots\hat E^{+}(t'_1)\rangle$, i.e., any $n+m$-point field correlation function composed of $n$ creation operators $E^{-}$ and $m$ annihilation operators $E^{+}$ [@Glauber1963; @Mandel1995], can be measured with similar, albeit more complex setups, such as in [@Gerber2009; @Koch2011]. Thus, upon identifying $E^{-}(t)$ and $E^+(t)$ with $\psi^\dagger(x)$ and $\psi(x)$, respectively, our scheme admits the measurement of any equivalent energy density $\propto \langle\hat\psi^\dagger(x_1)\cdots\hat\psi^\dagger(x_n)\hat\psi(x'_m)\cdots\hat\psi(x'_1)\rangle$, and therefore ultimately the simulation of arbitrary Hamiltonians $\hat{\mathcal{H}}(\hat\psi,\hat\psi^\dagger)$.
Once $f({\lambda})=\langle \Psi(\lambda)|\hat{\mathcal{H}}|\Psi(\lambda)\rangle$ has been experimentally estimated for a given $\lambda$, the next step is to apply the variational method to minimize $f(\lambda)$. Minimization is carried out by adaptively tuning the parameters $\lambda$ in the system Hamiltonian $\hat{H}_\mathrm{sys}(\lambda)$ and iteratively reducing $f(\lambda)$, for example, using a standard numerical gradient-descent method. Once the optimum choice of ${\lambda}$ is found, the resulting cavity output field is a variational approximation to the ground state of $\hat{\mathcal{H}}$, and relevant observables of the field theory can be directly measured using the detection schemes of Fig. \[fig:dictionary\]. We emphasize that our method applies also to cases where a numerical estimation of $f(\lambda)$ cannot be performed efficiently due to the size and complexity of the system to be simulated, and we suggest that this is exactly the strength of our approach. Moreover, the optimization may be performed experimentally without theoretically calculating the cavity-QED system dynamics; indeed, it is not necessary to accurately characterize $H_{\mathrm{sys}}$ or its relation to the adjustable parameters $\lambda$.
Why should the stationary output of a cavity-QED apparatus be an expressive class capable of capturing the ground-state physics of strongly interacting fields? It is possible to show that such states are of *continuous matrix product state* (cMPS) type, a variational class of quantum field states recently introduced for the classical simulation of both nonrelativistic and relativistic quantum fields [@Verstraete2010; @Osborne2010; @Haegeman2010a; @Haegeman2010b]. These states are a generalisation of *matrix product states* (MPS) [@Fannes1992; @Verstraete2008; @Cirac2009; @Schollwoeck2011], which have enjoyed unparalleled success in the study of strongly correlated phenomena in one dimension in conjunction with the *density matrix renormalization group* (DMRG) [@White1992; @Schollwoeck2005]. It turns out that *all* quantum field states admit a cMPS description, providing a compelling argument for their utility as a variational class [@Completeness]. Crucially, the cMPS formalism turns out to be *identical* to the input-output formalism of cavity QED [@Collett1984]. This identification was anticipated in [@Verstraete2010; @Osborne2010; @Schoen2005; @Schoen2007], and we elucidate it further in the supplemental material. It implies that *quantum field states emerging from a cavity are of cMPS type* and thus fulfill the necessary conditions for being a suitable and expressive class of variational quantum states.
![Two-particle correlations in the Lieb-Liniger model are reproduced in simulations of an ion-trap cavity experiment. This *critical* model exhibits a transition between the *superfluid* regime for $v \approx 0$ and the *Tonks-Girardeau* regime for $v\gg 0$, which is seen in the value of the correlation function at $t = 0$. (a) The Lieb-Liniger ground state is simulated for interaction strengths $v = \{ 0.07\, \text{(red)}, 3.95\, \text{(orange)}, 60.20\, \text{(yellow)}, 625.95\, \text{(green)}\}$, and correlation functions $\langle \hat{\psi}^{\dagger}(0)\hat{\psi}^{\dagger}(x)\hat{\psi}(x)\hat{\psi}(0) \rangle$ calculated as in [@Verstraete2010; @Osborne2010] using 338 variational parameters. (b) Two-photon correlation functions $g^{(2)}{(t)}$ for an experiment with the parameters of [@Stute2012]. Although there are visible differences, with just three variational parameters $\{ g, \Omega, s \}$ the transition in the correlation functions is approximately reproduced. It is worth emphasizing how unusual it is for a variational calculation with only a few parameters to reproduce anything more than the coarsest features of a correlation function, e.g., if mean-field theory is used one does not obtain nontrivial correlators. Strikingly, the transition in (a) is captured even in the presence of realistic decay channels (inset). Note that this transition is analogous to that observed in [@Dubin2010].[]{data-label="fig:pilot_study"}](g2.pdf){width="\columnwidth"}
Even though cavity-QED output states are of cMPS type, can a realistic system in the presence of decoherence reproduce the relevant physics of a strongly interacting quantum field? As a test case, we demonstrate that the paradigmatic cavity-QED system, comprising a single trapped atom coupled to a high-finesse cavity mode, is capable of simulating the ground-state physics of an equally paradigmatic field, namely, the Lieb-Liniger model [@Lieb1963]. This model describes hard-core bosons with a delta-function interaction and is given by Eq. with $w(x-y) = v\delta(x-y)$, where $v$ describes the interaction strength. Our simulator consists of a two-level atom interacting with one cavity mode, described (in a suitable rotating frame) by the on-resonance Jaynes-Cummings Hamiltonian $$\begin{aligned}
\hat H_{sys} = g(\hat\sigma^{+}\hat a + \hat\sigma^{-}\hat a^{\dagger}) + \Omega(\hat\sigma^{+} + \hat\sigma^{-}),
\label{HJC}\end{aligned}$$ where $\hat\sigma^{+}$ is the atomic raising operator and $\hat a$ is the cavity photon annihilation operator, $g$ the atom–cavity coupling, and $\Omega$ the laser drive. The cavity-QED Hamiltonian $\hat H_{sys}$ can be realized in various experimental architectures [@Raimond2001; @Miller2005; @Walther2006; @Haroche2006; @Girvin2009]. Here we choose the example of a trapped calcium ion in an optical cavity, with which tunable photon statistics have previously been demonstrated [@Dubin2010], and we show in the supplemental information (see below) how $g$ and $\Omega$ can function as variational parameters.
In an experiment, to measure the variational energy density $f(\lambda)$, the output beam would be allowed to relax to steady state, the intensity $I$, $g^{(1)}$, and $g^{(2)}$ functions estimated as depicted in Fig. \[fig:dictionary\], and $f(\lambda)$ finally estimated from postprocessing this data. Measurement schemes 2a and 2b of Fig. \[fig:dictionary\] are just the standard laboratory techniques of photon detection and Hanbury-Brown and Twiss interferometry. Measurement scheme 2c represents an interferometer with variable path length that is used to estimate the derivative of the quantum field in the kinetic energy term $\langle \hat{T} \rangle$. Length shifts on the mm scale correspond to ps values of $\epsilon_1$ and $\epsilon_2$ in the estimation of $\langle \hat{T} \rangle$; as these values are six orders of magnitude smaller than the relevant timescales of the experiment, they are considered sufficiently small.
Obviously the test model chosen here is simple enough to admit a *classical* simulation, which we carry out for the experimental parameters of [@Stute2012]. Exploiting a simple gradient-descent method, we find the values of $\lambda = \{ g, \Omega, s \}$ that minimize $f(\lambda)$ for a given value of $v$. This procedure is repeated over a range of $v$ values of interest, and the corresponding optimized values of $\lambda$ are then used to compute quantities of interest, e.g., spatial-correlation functions as shown in Fig. \[fig:pilot\_study\]. Remarkably, we find that just these three variational parameters $\lambda=(g,\Omega, s)$, when varied in the experimentally feasible parameter regime of [@Dubin2010; @Stute2012] in the presence of losses, allow for a quantum simulation of Lieb-Liniger ground-state physics. One expects that the ground-state approximation would improve by increasing the dimension of the auxiliary system and by allowing sufficiently general internal couplings and couplings to the field. In the context of atom-cavity systems, this can be done by making use of the rich level structure of atoms (i.e., Zeeman splittings) and making use of motional degrees of freedom. For sufficiently complex intracavity dynamics a classical simulation will become unfeasible, and at the same time it becomes conceivable that such a simulator will outperform the best classical methods.
The reliability of a quantum simulator can be compromised by decoherence effects, as was recently emphasized in [@Hauke2011]. Our simulation of the ion-cavity system includes both cavity decay (at rate $\kappa$) and decay of the ion due to spontaneous emission (at rate $\gamma$). The cavity decay rate rescales the parameter $s$ linking measurement time and simulated space, and thus it can be considered as a variational parameter itself. We emphasize that cavity decay does not function as a decoherence channel in our scheme but is rather an essential element of the cMPS formalism. In contrast, spontaneous emission sets the limit for the timescale over which coherent dynamics can be observed. For our present example, we can show that the regime of strong cooperativity $\mathcal{C} = g^2/\kappa\gamma \gtrsim 1$ is sufficient to allow a simulation of the Lieb-Liniger model despite detrimental losses. The Lieb-Liniger model exhibits nontrivial variations (Friedel oscillations) of the two-point correlation function on a length scale $\xi=\langle\hat\psi(x)\hat\psi^\dagger(x)\rangle^{-1}$. This length scale in the simulated model translates to a time scale over which the stationary output field (as the simulating register) should exhibit similar nontrivial features. From the previously established scaling rule one finds that the required time scale is $\tau=\xi/s=\langle \hat E^{-}(t)\hat E^{+}(t)\rangle^{-1}$. By means of the cavity input-output relations, we relate the output photon flux to the mean intracavity photon number $\langle\hat E^{-}(t)\hat E^{+}(t)\rangle=\kappa\langle\hat a^\dagger \hat a\rangle$. In the bad cavity limit $\kappa \gg g$ the cavity mode can be adiabatically eliminated. In this case $\langle a^\dagger a\rangle\simeq (g/\kappa)^2$, such that $\tau=\kappa/g^2$. On the other hand, the characteristic decoherence time of the ion is determined by the spontaneous emission rate $2\gamma$. Beyond a time $1/2\gamma$ the second order correlation function $g^{(2)}$ will be trivial. We therefore demand $\tau\lesssim 1/2\gamma$, which is equivalent to the requirement of a large cooperativity $\mathcal{C}\gtrsim 1$. For the exemplary case of the ion-cavity system considered above the cooperativity was indeed $\mathcal{C}\simeq 1.8$, see supplementary material. While equivalent conditions must be determined on a case-to-case basis, we expect that nontrivial quantum simulations in cavity QED will not be possible in the weak-coupling regime. Finally, there are overall losses associated with scattering and absorption in cavity mirrors, detection path optics, and photon-counter efficiency. However, while these losses reduce the efficiency with which photon correlations are detected, they do not otherwise affect the system dynamics.
A natural question is when our scheme would provide a practical advantage over classical computers in the simulation of quantum fields. We expect this to be the case in particular for the simulation of fields with multiple components, or species of particles, with canonical field-annihilation operators $\hat{\psi}_\alpha(x)$, $\alpha = 1, 2, \ldots, N$. This situation arises in at least two settings: firstly, for vector bosons in gauge theories with gauge group $\textsl{SU}(N)$, and secondly, in the nonrelativistic setting of cold atomic gases with multiple species. Variational calculations using cMPS fail in these settings, as the number of variational parameters must scale as $D\sim 2^N$. On the other hand, in a cavity-QED quantum simulation multiple output fields are naturally accessible via polarization or higher order cavity modes, and at the same time large effective Hilbert space dimensions can be achieved, e.g., with trapped ions or atoms. With $N \gtrsim 10$, substantial practical speedups are already expected with respect to the classical cMPS algorithm, which requires a number of operations scaling as $2^{3\times N}$.
The realisation that ground-state cMPS and the field states emerging from a cavity are connected can be exploited to characterise the correlations of the emitted light. Indeed, we obtain a simple criteria to determine when the correlations in the light field are nonclassical: if it turns out that the simulated hamiltonian is quadratic in the field operators; and (b) contains only “ultralocal” terms, i.e., no derivatives in the field operators, then the ground state is a trivial (i.e., gaussian) product state, and there would be no nonclassical correlations in the emitted light.
The output of a cavity-QED apparatus admits a natural interpretation as a variational class of quantum-field states. We have demonstrated that this allows an *analogue* quantum simulation procedure for strongly correlated physics using current technology. This result opens up an entirely new perspective for all cavity-QED systems which exhibit sufficiently strong nonlinearities at the single-photon level. This includes not only optical cavities coupled to atoms, but also superconducting circuits with super-strong coupling to solid state quantum systems [@Bozyigit2010], as well as other nonlinear systems achieving a high optical depth without cavities, such as atomic ensembles exhibiting Rydberg blockade [@Pritchard], or coupled to nanophotonic waveguides [@Vetsch2010; @Goban2012]. Looking further afield, since the input-output and cMPS formalisms generalize in a natural way to fermionic settings [@Sun1999; @Search2002; @Gardiner2004], our simulation procedure might be applicable to cavity-like microelectronic settings involving fermionic degrees of freedom. We hope our work will inspire explorations of these promising directions.
We acknowledge helpful conversations and comments by Jens Eisert, Frank Verstraete, Ignacio Cirac, Bernhard Neukirchen, Jutho Haegeman, Konstantin Friebe, Jukka Kiukas and Reinhard Werner. This work was supported by the Centre for Quantum Engineering and Space-Time Research (QUEST), the ERC grant QFTCMPS, the Austrian Science Fund (FWF) through the SFB FoQuS (FWF Project No. F4003), the European Commission (AQUTE, COQUIT), the Engineering and Physical Sciences Research Council (EPSRC) and through the FET-Open grant MALICIA (265522).
Supplementary Material
======================
Continuous Matrix Product States and Cavity QED
-----------------------------------------------
The matrix product state formalism has recently been generalized to the setting of quantum fields in [@Verstraete2010; @Haegeman2010; @Haegeman2010b] giving rise to *continuous matrix product states* (cMPS). These states refer to one-dimensional bosonic fields with annihilation and creation operators $\hat\psi_\alpha(x)$ and $\hat\psi_\alpha^\dagger(x)$ (such that $[\hat\psi_\alpha(x),\hat\psi_\beta^\dagger(x)]=\delta_{\alpha\beta}\delta(x-y)$), and are defined as $$\begin{aligned}
\label{eq:Psi}
|\Psi\rangle&=\mathrm{tr}_{aux}\{\hat U\}|\Omega\rangle,\end{aligned}$$ where $|\Omega\rangle$ denotes the vacuum state of the quantum field and $$\begin{aligned}
\hat U&=\mathcal{P}\exp\left(\int_{-\infty}^\infty dx \hat H_{\text{cMPS}}(x)\right).\end{aligned}$$ Here $\mathcal{P}$ denotes path ordering, and the (non Hermitian) Hamiltonian is $$\begin{aligned}
\label{eq:HcMPS}
\hat H_{\text{cMPS}}(x)=\hat Q\otimes\mathds{1}+\sum_\alpha \hat R_\alpha\otimes \hat\psi^\dagger_\alpha(x)\end{aligned}$$ with $\hat Q$ and $\hat R_\alpha$ being $D\times D$ matrices acting on an auxiliary system of dimension $D$. $\mathrm{tr}_{aux}$ in Eq. is the trace over this auxiliary system.
As was shown in [@Verstraete2010; @Osborne2010] an equivalent representation of the state $|\Psi\rangle$ as defined in Eq. is given by $$\begin{aligned}
\label{eq:Psi2}
\hat\rho&\propto \lim_{x_0,x_1\rightarrow\infty}\mathrm{tr}_{aux}\left\{\hat U(x_0,x_1)|\Omega\rangle\langle\Omega|\otimes|\psi\rangle\langle\psi|\hat U^\dagger(x_0,x_1)\right\}\end{aligned}$$ where $$\begin{aligned}
\hat U(x_0,x_1)&=\mathcal{P}\exp\left(\int_{x_0}^{x_1} dx \hat H_{\text{cMPS}}(x)\right),\end{aligned}$$ and $|\psi\rangle$ is an arbitrary state of the auxiliary system, which plays the role of a boundary condition at $x_0$. The two states in and are equivalent in the sense that they give rise to identical expectation values for normal- and position-ordered expressions of field operators $\langle\hat\psi^\dagger(y_1)\ldots\hat\psi^\dagger(y_n)\hat\psi(y_{n+1})\ldots \hat\psi(y_m)\rangle$, see [@Verstraete2010; @Osborne2010]. Note that the trace in is a proper partial trace over the auxiliary system (in contrast to the trace in ).
Consider, on the other hand, a cavity with several relevant modes described by annihilation/creation operators $\hat a_\alpha,\,\hat a^\dagger_\beta$ (such that $[\hat a_\alpha,\hat a^\dagger_\beta]=\delta_{\alpha\beta}$) which are coupled to some intracavity medium via a system Hamiltonian $\hat H_{sys}$. Moreover, each cavity is coupled to a continuum of field modes ($[\hat a_\alpha(\omega),\hat a^\dagger_\beta(\bar\omega)]=\delta(\omega-\bar\omega)\delta_{\alpha\beta}$) through one of its end mirrors. (The generalization to the case of double-sided cavities, or ring cavities with several outputs per mode is immediate.) The total Hamiltonian for the cavities, the intracavity medium, and the outside field is $$\begin{aligned}
\hat H_{cQED}(t)&=\hat H_{sys}\otimes\mathds{1}\\
&\quad+i\sum_\alpha \int d\omega\sqrt{\frac{\kappa_\alpha(\omega)}{2\pi}}\left(\hat a_\alpha \otimes \hat a_\alpha^\dagger(\omega)e^{-i\omega t}-\mathrm{h.c.}\right).\end{aligned}$$ This Hamiltonian is written in an interaction picture with respect to the free energy of the continuous fields, and it is taken in a frame rotating at the resonance frequencies of the cavities. In the interaction picture and rotating frame each integral extends over a band width of frequencies around the respective cavity frequencies. In the optical domain the Born-Markov approximation, which assumes $\kappa_\alpha(\omega)=\mathrm{const.}$ in the relevant band width, holds to an excellent degree. In this case it is common to define time-dependent operators $\hat E^+_\alpha(t)=\int d\omega/\sqrt{2\pi}\,\hat a_\alpha(\omega)\exp(-i\omega t)$, which fulfill $[\hat E^+_\alpha(t),\hat E^-_\beta(t')]=\delta(t-t')$. As in the main text, these operators correspond to the electric field components at the cavity output, and are defined such that $\langle\hat E^{-}(t)\hat E^{+}(t) \rangle$ denotes the flux of photons per second. The Hamiltonian then is $$\begin{aligned}
\hat H_{cQED}(t)&=\hat H_{sys}\otimes\mathds{1}+i \sum_\alpha \sqrt{\kappa_\alpha}\left(\hat a_\alpha \otimes \hat E_\alpha^-(t)-\mathrm{h.c.}\right).\end{aligned}$$ If the outside field modes are in the vacuum, standard quantum optical calculations [@Gardiner2004a] show that this Hamiltonian is equivalent to an *effective* non-Hermitian Hamiltonian $$\begin{aligned}
\hat H_{cQED}(t)&=\hat H_{sys}\otimes\mathds{1}-i\sum_\alpha\frac{\kappa_\alpha}{2}\hat a^\dagger_\alpha \hat a_\alpha\otimes\mathds{1}+i \sum_\alpha \sqrt{\kappa_\alpha} \hat a_\alpha \otimes \hat E_\alpha^-(t).\end{aligned}$$ When this is compared to Eq. the identification of the formalism of cMPS and cavity QED is immediate. If we assume that the field and the cavity system is in a state $|\Omega\rangle\otimes|\psi\rangle$ at some initial time $t_0$ the final state of the field outside the cavity at time $t_1$ is given by expression , when we identify $x=s\,t$, and $\hat H_{\text{cMPS}}(x)=-i\hat H_{cQED}(x/s)$, that is $$\begin{aligned}
\label{eq:identification}
\hat \psi(x)&= \frac{1}{\sqrt{s}}\hat E^+_\alpha(t), &
\hat R_\alpha&=\sqrt{\frac{\kappa_\alpha}{s}}\hat a_\alpha, &
\hat Q&= -\frac{i}{s}\hat H_{sys}-\sum_\alpha\frac{\hat R^\dagger_\alpha \hat R_\alpha}{2},\end{aligned}$$ with an arbitrary scaling factor $s$. Therefore, the state of the output modes of a cavity is always a continuous matrix product state. Formally these states are cMPS with an infinite-dimensional auxiliary system $D\rightarrow\infty$. However, due to energy constraints, the dimensions of the cavity system are effectively finite. The relevant dimension of the cavity Hilbert space then sets the dimension $D$ of the auxiliary system in the cMPS formalism.
Quantum Variational Algorithm
-----------------------------
As an exemplary test case we demonstrated that the cavity QED system comprising a single trapped atom strongly coupled to a single high-finesse cavity mode is capable of simulating the ground-state physics of the Lieb-Liniger model. The atom-cavity system is described (in a suitable rotating frame) by the on-resonance Jaynes-Cummings Hamiltonian $$\begin{aligned}
H_{sys} = g(\hat \sigma^{+}\hat a + \hat \sigma^{-}\hat a^{\dagger}) + \Omega(\hat\sigma^{+} + \hat\sigma^{-}),
\label{HJC1}\end{aligned}$$ where $\hat\sigma^{+}$ is the atomic raising operator and $\hat a$ is the cavity-photon annihilation operator, $g$ the atom cavity coupling, and $\Omega$ the laser drive. Photons leak out of the cavity with leakage rate $\kappa$, and it is assumed that, in a real experiment, this output light can be measured by various detection setups.
The Lieb-Liniger model, with Hamiltonian $$\begin{aligned}
{\cal \hat H} &= \int_{-\infty}^\infty (\hat{T}+\hat{W}+\hat{N})dx \\
&= \int_{-\infty}^\infty \Big[ \frac{d \hat \psi^\dagger(x)}{dx} \frac{d \hat \psi(x)}{dx} + v~ \hat \psi^\dagger(x)\hat \psi^\dagger(x) \hat\psi(x) \hat\psi(x)\\
&\hspace{5cm} - \mu \hat \psi^\dagger(x) \hat \psi(x) \Big]
dx,\end{aligned}$$ describes hard-core bosons with contact interaction of strength $v$. We performed variational optimizations for a range of values for $v$. We did this using a simple gradient-descent minimization of the average energy density $f(\lambda) = \langle \Psi(\lambda)|\hat{T}+\hat{W}+\hat{N}|\Psi(\lambda)\rangle$. Using the parameters that minimize $f( \lambda)$ we then calculate other quantities of interest, such as correlation functions for the simulated ground state field. We have focussed on a gradient-descent algorithm for clarity; in practice a more sophisticated optimization procedure using the time-dependent variational principle, or conjugate gradients, could be used.
Our variational parameters $ \lambda = \left( g, \Omega, s \right)$ enter $f(\lambda)$ as follows. The expectation value of the energy density in the Lieb-Liniger model is $$\begin{aligned}
f(\lambda; v,\mu) = \langle \hat{T} \rangle +\langle \hat{W} \rangle + \langle \hat{N} \rangle\end{aligned}$$ where each term may be written in terms of the experimentally observed correlation functions via the correspondence $\hat\psi(x) = \hat E^+(t)/\sqrt{s}$, giving $$\begin{aligned}
\langle \hat{T} \rangle &= \lim_{\epsilon_1,\epsilon_2\rightarrow 0} \frac{1}{s^3\epsilon_1\epsilon_2}\left(g^{(1)}(t+\epsilon_1,t+\epsilon_2)-g^{(1)}(t+\epsilon_1,t)\right.\\
&\left.\quad-g^{(1)}(t,t+\epsilon_2)+g^{(1)}(t,t)\right) \,, \\
\langle \hat{W}\rangle & = \frac{v}{s^2} g^{(2)}(t,t) \,, \\
\langle \hat{N} \rangle & = -\frac{\mu}{s} g^{(1)}(t,t) \,,\end{aligned}$$ where $g^{(1)}(t_1,t_2)=\langle \hat E^{-}(t_1)\hat E^{+}(t_2)\rangle$ and $g^{(2)}(t_1,t_2)=\langle \hat E^{-}(t_1)\hat E^{-}(t_2)\hat E^{+}(t_2)\hat E^{+}(t_1)\rangle$. In an experimental simulation, these correlation functions would be measured directly in the laboratory, and the results would be fed back into a classical computer performing the optimization algorithm. However, for the purposes of our proof-of-principle simulation, we calculate the correlation functions directly by means of the input-output formalism. Making use of the cavity input-output relation $\hat E^{+}(t) = \hat E^+_{\mathrm{(in)}}(t) + \sqrt{\kappa}\hat a(t)$, where $\hat E^+_{\mathrm{(in)}}(t)$ denotes the field impinging on the cavity at time $t$ (which is assumed to be in the vacuum state), the expectation value may be written [@Verstraete2010; @Osborne2010]
$$\begin{aligned}
f(\lambda; v,\mu) = \mathrm{tr} \left\{ \left(\left[\hat Q,\hat R\right]\right)^\dagger \left[\hat Q,\hat R\right] \hat \rho_{\tiny \mbox{ss}} \right\} &+ v~ \mathrm{tr} \left\{ \left(\hat R^\dagger\right)^2\hat R^2 ~\hat\rho_{\tiny \mbox{ss}}\right\}\\
&- \mu~\mathrm{tr} \left\{ \hat R^\dagger \hat R \hat\rho_{\tiny \mbox{ss}}\right\}
\end{aligned}$$
where $\hat R$ and $\hat Q$ are defined in Eq. , and $\hat\rho_{\tiny \mbox{ss}}$ is the unique steady state of the atom-cavity system, satisfying $$\begin{aligned}
\frac{d \hat\rho_{\tiny \mbox{ss}} }{dt} = -i \left[ \hat H_{\tiny\mbox{sys}}, \hat\rho_{\tiny \mbox{ss}}\right] + \kappa \hat a \hat\rho_{\tiny \mbox{ss}} \hat a^\dagger -\frac{\kappa}{2} \left[ \hat a^\dagger \hat a, \hat\rho_{\tiny \mbox{ss}}\right]=0.\end{aligned}$$ Note that above, we write $f(\lambda; v,\mu)$ to highlight the dependence of $f$ on $v$ and $\mu$. Hereafter we set $\mu=1$ and minimize $f(\lambda; v,\mu)$ for a range of different values of $v$. Solutions for other values of $\mu$ can be obtained by means of a scaling transformation, as described in [@Verstraete2010; @Osborne2010].
The general outline of the algorithm to determine our optimum values of the variational parameters $\lambda$, for a given choice of $v$ and $\mu$, is thus:
initialize $\textit{tol}$ (tolerance), and $\epsilon$ (step size) initialize $\lambda$ to an arbitrary value\
set $\lambda^{\prime} = \lambda$\
calculate $\nabla f(\lambda; v,\mu)$\
update experimental parameters as $\lambda \leftarrow \lambda - \epsilon\nabla f(\lambda; v,\mu)$\
calculate $f(\lambda^{\prime}; v,\mu)$ and $f(\lambda; v,\mu)$\
Note that in our simulation, $\nabla f(\lambda; v,\mu)$ is estimated numerically by evaluating $f(\lambda+\Delta_i; v,\mu)$ for small values of $\Delta_i$, while in an experiment, $\nabla f(\lambda; v,\mu)$ is found with the aid of measurements of $\langle \hat{T} \rangle$, $\langle \hat{W}\rangle$, and $\langle \hat{N} \rangle$.
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| ArXiv |
---
abstract: 'Estimation of 3D human pose from monocular image has gained considerable attention, as a key step to several human-centric applications. However, generalizability of human pose estimation models developed using supervision on large-scale in-studio datasets remains questionable, as these models often perform unsatisfactorily on unseen in-the-wild environments. Though weakly-supervised models have been proposed to address this shortcoming, performance of such models relies on availability of paired supervision on some related tasks, such as 2D pose or multi-view image pairs. In contrast, we propose a novel kinematic-structure-preserved unsupervised 3D pose estimation framework[^1], which is not restrained by any paired or unpaired weak supervisions. Our pose estimation framework relies on a minimal set of prior knowledge that defines the underlying kinematic 3D structure, such as skeletal joint connectivity information with bone-length ratios in a fixed canonical scale. The proposed model employs three consecutive differentiable transformations named as forward-kinematics, camera-projection and spatial-map transformation. This design not only acts as a suitable bottleneck stimulating effective pose disentanglement, but also yields interpretable latent pose representations avoiding training of an explicit latent embedding to pose mapper. Furthermore, devoid of unstable adversarial setup, we re-utilize the decoder to formalize an energy-based loss, which enables us to learn from in-the-wild videos, beyond laboratory settings. Comprehensive experiments demonstrate our state-of-the-art unsupervised and weakly-supervised pose estimation performance on both Human3.6M and MPI-INF-3DHP datasets. Qualitative results on unseen environments further establish our superior generalization ability.'
author:
- |
Jogendra Nath Kundu[^2]Siddharth SethRahul M VMugalodi Rakesh\
**R. Venkatesh BabuAnirban Chakraborty\
Indian Institute of Science, Bangalore, India\
[{jogendrak, siddharthseth}@iisc.ac.in, [email protected], [email protected], {venky, anirban}@iisc.ac.in]{}**
bibliography:
- 'ms.bib'
title: |
Kinematic-Structure-Preserved Representation for\
Unsupervised 3D Human Pose Estimation
---
\[tab:char\]
Introduction
============
Building general intelligent systems, capable of understanding the inherent 3D structure and pose of non-rigid humans from monocular RGB images, remains an illusive goal in the vision community. In recent years, researchers aim to solve this problem by leveraging the advances in two key aspects, a) improved architecture design [@newell2016stacked; @chu2017multi] and b) increasing collection of diverse annotated samples to fuel the supervised learning paradigm [@VNect_SIGGRAPH2017]. However, obtaining 3D pose ground-truth for non-rigid human-bodies is a highly inconvenient process. Available motion capture systems, such as body-worn sensors (IMUs) or multi-camera structure-from-motion (SFM), requires careful pre-calibration, and hence usually done in a pre-setup laboratory environment [@ionescu2013human3; @zhang2017martial]. This often restricts diversity in the collected dataset, which in turn hampers generalization of the supervised models trained on such data. For instance, the widely used Human3.6M [@ionescu2013human3] dataset captures 3D pose using 4 fixed cameras (only 4 backgrounds scenes), 11 actors (limited apparel variations), and 17 action categories (limited pose diversity). A model trained on this dataset delivers impressive results when tested on samples from the same dataset, but does not generalize to an unknown deployed environment, thereby yielding non-transferability issue.
To deal with this problem, researchers have started exploring innovative techniques to reduce dependency on annotated real samples. Aiming to enhance appearance diversity on known 3D pose samples (CMU-MoCap), synthetic datasets have been proposed, by compositing a diverse set of human template foregrounds with random backgrounds [@varol2017learning]. However, models trained on such samples do not generalize to a new motion (e.g. a particular dance form), apparel, or environment much different from the training samples, as a result of large domain shift. Following a different direction, several recent works propose weakly-supervised approaches [@zhou2017towards], where they consider access to a large-scale dataset with paired supervision on some related-tasks other than the task in focus (3D pose estimation). Particularly, they access multiple cues for weak supervision, such as, a) paired 2D ground-truth, b) unpaired 3D ground-truth (3D pose without the corresponding image), c) multi-view image pair ([Rhodin et al. [-@rhodin2018unsupervised]]{}), d) camera parameters in a multi-view setup etc. (see Table \[tab:char\] for a detailed analysis).
While accessing such weak paired-supervisions, the general approach is to formalize a self-supervised consistency loop, such as 2D$\rightarrow$3D$\rightarrow$2D [@tung2017adversarial], view-1$\rightarrow$3D$\rightarrow$view-2 [@kocabas2019self], etc. However, the limitations of domain-shift still persists as a result of using annotated data (2D ground-truth or multi-view camera extrinsic). To this end, without accessing such paired samples, [@jakab2019learning] proposed to leverage unpaired samples to model the natural distribution of the expected representations (2D or 3D pose) using adversarial learning. Obtaining such samples, however, requires access to a 2D or 3D pose dataset and hence the learning process is still biased towards the action categories presented in that dataset. One can not expect to have access to any of the above discussed paired or unpaired weak supervisory signals for an unknown deployed environment (e.g. frames of a dance-show where the actor is wearing a rare traditional costume). This motivates us to formalize a fully-unsupervised framework for monocular 3D pose estimation, where the pose representation can be adapted to the deployed environment by accessing only the RGB video frames devoid of dependency on any explicit supervisory signal.
**Our contributions.** We propose a novel unsupervised 3D pose estimation framework, relying on a carefully designed kinematic structure preservation pipeline. Here, we constrain the latent pose embedding, to form interpretable 3D pose representation, thus avoiding the need for an explicit latent to 3D pose mapper. Several recent approaches aim to learn a prior characterizing kinematically plausible 3D human poses using available MoCap datasets ([Kundu et al. ]{}[-@kundu2019bihmp]). In contrast, we plan to utilize minimal kinematic prior information, by adhering to the restrictions to not use any external unpaired supervision. This involves, a) access to the knowledge of hierarchical limb connectivity, b) a vector of allowed bone length ratios, and c) a set of 20 synthetically rendered images with diverse background and pose (a minimal dataset with paired supervision to standardize the model towards the intended 2D or 3D pose conventions). The aforementioned prior information is very minimal in comparison to the pose-conditioned limits formalized by ([Akhter et al. ]{}[-@akhter2015pose]) in terms of both dataset size and parameters associated to define the constraints.
In the absence of multi-view or depth information, we infer 3D structure, directly from the video samples, for the unsupervised 3D pose estimation task. One can easily segment moving objects from a video, in absence of any background (BG) motion. However, this is only applicable to in-studio static camera feeds. Aiming to work on in-the-wild YouTube videos , we formalize separate unsupervised learning schemes for videos with both static and dynamic BG. In absence of background motion, we form pairs of video frames with a rough estimate of the corresponding BG image, following a training scheme to disentangle foreground-apparel and the associated 3D pose. However, in the presence of BG motion, we lack in forming such consistent pairs, and thus devise a novel energy-based loss on the disentangled pose and appearance representations. In summary,
- We formalize a novel collection of three differentiable transformations, which not only acts as a bottleneck stimulating effective pose disentanglement but also yields interpretable latent pose representations avoiding training of an explicit latent-to-pose mapper.
- The proposed energy-based loss, not only enables us to learn from in-the-wild videos, but also improves generalizability of the model as a result of training on diverse scenarios, without ignoring any individual image sample.
- We demonstrate *state-of-the-art* unsupervised and weakly-supervised 3D pose estimation performance on both Human3.6M and MPI-INF-3DHP datasets.
Related Works {#sec:related-works}
=============
**3D human pose estimation.** There is a plethora of fully-supervised 3D pose estimations works [@fang2018learning; @mehta2017monocular; @VNect_SIGGRAPH2017], where the performance is bench-marked on the same dataset, which is used for training. Such approaches do not generalize on minimal domain shifts beyond the laboratory environment. In absence of large-scale diverse outdoor datasets with 3D pose annotations, datasets with 2D pose annotations is used as a weak supervisory signal for transfer learning using various 2D to 3D lifting techniques ([Tung et al. [-@tung2017adversarial]]{}; [Chen et al. [-@chen20173d]]{}; [Ramakrishna et al. [-@ramakrishna2012]]{}). However, these approaches still rely on availability of 2D pose annotations. Avoiding this, ([Kocabas et al. [-@kocabas2019self]]{}; [Rhodin et al. [-@rhodin2018unsupervised]]{}) proposed to use multi-view correspondence acquired by synchronized cameras. But in such approaches ([Rhodin et al. [-@rhodin2018unsupervised]]{}), the latent pose representation remains un-interpretable and abstract, thereby requiring a substantially large amount of 3D supervision to explicitly train a *latent-to-pose mapping* mapper. We avoid training of such explicit mapping, by casting the latent representation, itself as the 3D pose coordinates. This is realized as a result of formalizing the geometry-aware bottleneck.
**Geometry-aware representations.** To capture intrinsic structure of objects, the general approach is to disentangle individual factors of variations, such as appearance, camera viewpoint and other pose related cues, by leveraging inter-instance correspondence. In literature, we find unsupervised land-mark detection techniques [@zhang2018unsupervised], that aim to utilize a relative transformation between a pair of instances of the same object, targeting the 2D pose estimation task. To obtain such pairs, these approaches rely on either of the following two directions, viz. a) frames from a video with an acceptable time-difference [@jakab2018unsupervised], or b) synthetically simulated 2D transformations [@rocco2017convolutional]. However, such techniques fail to capture the 3D structure of the object in the absence of multi-view information. The problem becomes more challenging for deformable 3D skeletal structures as found in diverse human poses. Recently [@jakab2018unsupervised] proposed an unsupervised 2D landmark estimation method to disentangle pose from appearance using a conditional image generation framework. However, the predicted 2D landmarks do not match with the standard human pose key-points, hence are highly un-interpretable with some landmarks even lying on the background. Such outputs can not be used for a consequent task requiring a structurally consistent 2D pose input. Defining structural constraints in 2D is highly ill-posed, considering images as projections of the actual 3D world. Acknowledging this, we plan to estimate 3D pose separately with camera parameters followed by a camera-projection to obtain the 2D landmarks. As a result of this inverse-graphics formalization, we have the liberty to impose structural constraints directly on the 3D skeletal representation, where the bone-length and other kinematic constraints can be imposed seamlessly using consistent rules as compared to the corresponding 2D representation. A careful realization of 3D structural constraints not only helps us to obtain interpretable 2D landmarks but also reduces the inherent uncertainty associated with the process of lifting a monocular 2D images to 3D pose [@chen2019unsupervised], in absence of any additional supervision such as multi-view or depth cues.
{width="1.0\linewidth"}
Approach {#sec:approach}
========
Our aim is to learn a mapping function, that can map an RGB image of human to its 3D pose by accessing minimal kinematic prior information. Motivated by ([Rhodin et al. [-@rhodin2018unsupervised]]{}), we plan to cast it as an unsupervised disentanglement of three different factors , a) foreground (FG) appearance, b) background (BG) appearance, and c) kinematic pose. However, unlike ([Rhodin et al. [-@rhodin2018unsupervised]]{}) in absence of multi-view pairs, we have access to simple monocular video streams of human actions consisting of both static and dynamic BG.
Architecture
------------
As shown in Fig. \[fig:main1\]A, we employ two encoder networks each with a different architecture, $E_P$ and $E_A$ to extract the local-kinematic parameters $v_k$ (see below) and FG-appearance, $f_a$ respectively from a given RGB image. Additionally, $E_P$ also outputs 6 camera parameters, denoted by $c$, to obtain coordinates of the camera-projected 2D landmarks, $p_{2D}$.
One of the major challenges in learning factorized representations [@denton2017unsupervised] is to realize purity among the representations. More concretely, the appearance representation should not embed any pose related information and vice-versa. To achieve this, we enforce a bottleneck on the pose representation by imposing kinematic-structure based constraints (in 3D) followed by an inverse-graphics formalization for 3D to 2D re-projection. This introduces three pre-defined transformations , a) Forward kinematic transformation, $\mathcal{T}_{fk}$ and b) Camera projection transformation $\mathcal{T}_c$, and c) Spatial-map transformation $\mathcal{T}_m$.
### a) Forward kinematic transformation, $\mathcal{T}_{fk}$
Most of the prior 3D pose estimation approaches ([Chen et al. [-@chen2019unsupervised]]{}; [Rhodin et al. [-@rhodin2018unsupervised]]{}) aim to either directly regress joint locations in 3D or depth associated with the available 2D landmarks. Such approaches do not guarantee validity of the kinematic structure, thus requiring additional loss terms in the optimization pipeline to explicitly impose kinematic constraints such as bone-length and limb-connectivity information [@habibie2019wild]. In contrast, we formalize a view-invariant local-kinematic representation of the 3D skeleton based on the knowledge of skeleton joint connectivity. We define a canonical rule (see Fig. \[fig:main1\]B), by fixing the neck and pelvis joint (along z-axis, with pelvis at the origin) and restricting the [trunk to hip-line (line segment connecting the two hip joints) angle]{}, to rotate only about x-axis on the YZ-plane(1-DOF) in the canonical coordinate system $C$ (Cartesian system defined at the pelvis as origin). Our network regresses one pelvis to hip-line angle and 13 unit-vectors (all 3-DOF), which are defined at their respective parent-relative local coordinate systems, $L^{Pa(j)}$, where $Pa(j)$ denotes the parent joint of $j$ in the skeletal kinematic tree. Thus, $v_k\in \mathbb{R}^{40}$ (1+13\*3). These predictions are then passed on to the forward-kinematic transformation to obtain the 3D joint coordinates $p_{3D}$ in $C$, $\mathcal{T}_{fk}:v_k\rightarrow p_{3D}$ where $p_{3D}\in \mathbb{R}^{3J}$, with $J$ being the total number of skeleton joints. First, positions of the 3 root joints, $p_{3D}^{(j)}$ for $j$ as left-hip, right-hip and neck, are obtained using the above defined canonical rule after applying the estimate of the [trunk to hip-line angle]{}, $v_k^{(0)}$. Let $\textit{len}^{(j)}$ store the length of the line-segment (in a fixed canonical unit) connecting a joint $j$ with $Pa(j)$. Then, $p_{3D}^{(j)}$ for rest of the joints is realized using the following recursive equation, $p_{3D}^{(j)} = p_{3D}^{(Pa(j))}+\textit{len}^{(j)}v_k^{(j)}$. See Fig. \[fig:main1\]B (dotted box) for a more clear picture.
### b) Camera-projection transformation, $\mathcal{T}_{c}$
As $p_{3D}$ is designed to be view-invariant, we rely on estimates of the camera extrinsics $c$ (3 angles, each predicted as 2 parameters, the $\sin$ and $\cos$ component), which is used to rotate and translate the camera in the canonical coordinate system $C$, to obtain 2D landmarks of the skeleton (using the rotation and translation matrices, $R_c$ and $T_c$ respectively). Note that, these 2D landmarks are expected to register with the corresponding joint locations in the input image. Thus, the 2D landmarks are obtained as, $p_{2D}^{(j)} = P(R_c*p_{3D}^{(j)}+T_c)$, where $P$ denotes a fixed perspective camera transformation.
### c) Spatial-map transformation, $\mathcal{T}_{m}$
After obtaining coordinates of the 2D landmarks $p_{2D}\in\mathbb{R}^{2J}$, we aim to effectively aggregate it with the spatial appearance-embedding $f_a$. Thus, we devise a transformation procedure $\mathcal{T}_{m}$, to transform the vectorized 2D coordinates into spatial-maps denoted by $f_{2D}\in \mathbb{R}^{H\times W\times \textit{Ch}}$, which are of consistent resolution to $f_a$, $\mathcal{T}_m:p_{2D}\rightarrow f_{2D}$. To effectively encode both joint locations and their connectivity information, we propose to generate two sets of spatial maps namely, a) heat-map, $f_{hm}$ and b) affinity-map, $f_{am}$ (, $f_{2D}:(f_{hm},f_{am})$). Note that, the transformations to obtain these spatial maps must be fully differentiable to allow the disentaglement of pose using the cross-pose image-reconstruction loss, computed at the decoder output (discussed in Sec. [3.3a]{}). Keeping this in mind, we implement a novel computational pipeline by formalizing translated and rotated Gaussians to represent both joint positions ($f_{hm}$) and skeleton-limb connectivity ($f_{am}$). We use a constant variance $\sigma$ along both spatial directions to realize the heat-maps for each joint $j$, as $f_{hm}^{(j)}(u) = \exp(-0.5||u-p_{2d}^{(j)}||^2/\sigma^{2})$, where $u:[u_x,u_y]$ denotes the spatial-index in a $H\times W$ lattice (see Fig. \[fig:main2\]A).
We formalize the following steps to obtain the affinity maps based on the connectivity of joints in the skeletal kinematic tree (see Fig. \[fig:main2\]A). For each limb (line-segment), $l$ with endpoints $p_{2D}^{l(j_1)}$ and $p_{2D}^{l(j_2)}$, we first compute location of its mid-point, $\mu^{(l)}:[\mu_x^{(l)},\mu_y^{(l)}]$ and slope $\theta^{(l)}$. Following this, we perform an affine transformation to obtain, $u^\prime = R_{\theta^{(l)}}*(u-\mu^{(l)})$, where $R_{\theta^{(l)}}$ is the 2D rotation matrix. Let, $\sigma_x^{(l)}$ and $\sigma_y^{(l)}$ denote variance of a Gaussian along both spatial directions representing the limb $l$. We fix $\sigma_y^{(l)}$ from prior knowledge of the limb width. Whereas, $\sigma_x^{(l)}$ is computed as $\alpha*len(l)$ in the 2D euclidean space (see Supplementary). Finally, the affinity map is obtained as, $$\begin{aligned}
f_{am}^{(l)}(u) = \exp(-0.5||u_x^\prime/\sigma_x^{(l)}||^2-0.5||u_y^\prime/\sigma_y^{(l)}||^2) \end{aligned}$$
$\mathcal{T}_{fk}$, $\mathcal{T}_{c}$ and $\mathcal{T}_{m}$ (collectively denoted as $\mathcal{T}_k$) are designed using perfectly differentiable operations, thus allowing back-propagation of gradients from the loss functions defined at the decoder output. As shown in Fig. \[fig:main1\]A, the decoder takes in a tuple of spatial-pose-map representation and appearance ($f_{2D}$ and $f_a$ respectively, concatenated along the channel dimension) to reconstruct an RGB image. To effectively disentangle BG information in $f_a$, we fuse the background image $B_t$ towards the end of decoder architecture, inline with ([Rhodin et al. [-@rhodin2018unsupervised]]{}).
Access to minimal prior knowledge
---------------------------------
One of the key objectives of this work is to solve the unsupervised pose estimation problem with minimal access to prior knowledge whose acquisition often requires manual annotation or a data collection setup, such as CMU-MoCap . Adhering to this, we restrict the proposed framework from accessing any paired or unpaired data samples as shown in Table \[tab:char\]. Here, we list the specific prior information that has been considered in the proposed framework,
- Kinematic skeletal structure (the joint connectivity information) with bone-length ratios in a fixed canonical scale. Note that, [we do not consider access to the kinematic angle limits]{} for the limb joints, as such angles are highly pose dependent particularly for diverse human skeleton structures [@akhter2015pose].
- A set of 20 synthetically rendered SMPL models with diverse 3D poses and FG appearance [@varol2017learning]. We have direct paired supervision loss (denoted by $\mathcal{L}_{prior}$) on these samples to standardize the model towards the intended 2D or 3D pose conventions (see Supplementary).
{width="1.0\linewidth"}
Unsupervised training procedure
-------------------------------
In contrast to [@jakab2018unsupervised], we aim to disentangle foreground (FG) and background (BG) appearances, along with the disentanglement of pose. In a generalized setup, we also aim to learn from in-the-wild YouTube videos in contrast to in-studio datasets, avoiding dataset-bias.
### Separating paired and unpaired samples.
For an efficient disentanglement, we aim to form image tuples of the form $(I_s,I_t, B_t)$. Here, $I_s$ and $I_t$ are video frames, which have identical FG-appearance with a nonidentical *kinematic-pose* (pairs formed between frames beyond a certain time-difference). As each video-clip captures action of an individual in a certain apparel, *FG-appearance* remains identical among frames from the same video. Here, $B_t$ denotes an estimate of BG image without the human subject corresponding to the image $I_t$, which is obtained as the median of pixel intensities across a time-window including the frame $I_t$. However, such an estimate of $B_t$ is possible only for scenarios with no camera movement beyond a certain time window to capture enough background evidence (static background with a moving human subject).
Given an in-the-wild dataset of videos, we classify temporal clips of a certain duration ($>$5 seconds) into two groups based on the amount of BG motion in that clip. This is obtained by measuring the pixel-wise L2 loss among the frames in a clip, considering human action covers only 10-20% of pixels in the full video frame (see Supplementary). Following this, we realize two disjoint datasets denoted by $\mathcal{D}_{p}=\{(I_s^{(i)},I_t^{(i)},B_t^{(i)})\}_{i=1}^{N}$ and $\mathcal{D}_{unp}=\{(I_s^{(k)}, I_t^{(k)})\}_{k=1}^M$ as sets of tuples with extractable BG pair (paired) and un-extractable BG pair (unpaired), respectively.
### a) Training objective for paired samples, $\mathcal{D}_p$
As shown in Fig. \[fig:main1\]A, given a source and target image ($I_s$ and $I_t$), we aim to transfer the pose of $I_t$ ($f_{2D}$) to the FG-appearance extracted from $I_s$ ($f_a$) and background from $B_t$ to reconstruct $\hat{I}_t$. Here, the FG and BG appearance information can not leak through pose representation because of the low dimensional bottleneck $p_{2D}\in\mathbb{R}^{2J}$. Moreover, consecutive predefined matrix and spatial-transformation operations further restrict the framework from leaking appearance information through the pose branch even as low-magnitude signals. Note that, the BG of $I_s$ may not register with the BG of $I_t$, when the person moves in the 3D world (even in a fixed camera scenario) as these images are outputs of an off-the shelf person-detector. As a result of this BG disparity and explicit presence of the clean spatially-registered background $B_t$, $D_I$ catches the BG information directly from $B_t$, thereby forcing $f_a$ to solely model FG-appearance from the apparel-consistent source, $I_s$. Besides this, we also expect to maintain perceptual consistency between $I_t$ and $\hat{I}_t$ through the encoder networks, keeping in mind the later energy-based formalization (next section). Thus, all the network parameters are optimized for the paired samples using the following loss function, $\mathcal{L}_P = |I_t - \hat{I}_t| + \lambda_1|p_{2D}-\hat{p}_{2D}|+\lambda_2|f_a - \hat{f}_a|$. Here, $\hat{p}_{2D} = \mathcal{T}_{k}\circ E_P(\hat{I_t})$ and $\hat{f_a} = E_A(\hat{I_t})$.
### b) Training objective for unpaired samples, $\mathcal{D}_{unp}$
Although, we find a good amount of YouTube videos where human motion (e.g. dance videos) is captured on a tripod mounted static camera, such videos are mostly limited to indoor environments. However, a diverse set of human actions are captured in outdoor settings (e.g. sports related activities), which usually involves camera motion or dynamic BG. Aiming to learn a general pose representation, instead of ignoring the frames from video-clips with dynamic BG, we plan to formalize a novel direction to adapt the parameters of $E_P$ and $E_A$ even for such diverse scenarios.
We hypothesize that the decoder $D_I$ expects the pose and FG-appearance representation in a particular form, satisfying the corresponding input distributions, $P(f_{2D})$ and $P(f_a)$. Here, a reliable estimate of $P(f_{2D})$ and $P(f_a)$ can be achieved solely on samples from $\mathcal{D}_p$ in presence of paired supervision, avoiding *mode-collapse*. More concretely, the parameters of $D_I$ should not be optimized on samples from $\mathcal{D}_{unp}$ (as shown in Fig. \[fig:main2\]B with a lock sign). Following this, one can treat $D_I$ analogous to a *critic*, which outputs a reliable prediction (an image of human with pose from $I_t$, FG-appearance from $I_s$ and BG from $B_t$) only when its inputs $f_{2D}$ and $f_{a}$ satisfy the expected distributions- $P(f_{2D})$ and $P(f_a)$ respectively. We plan to leverage this analogy to effectively use the frozen $D_I$ network as an energy function to realize simultaneous adaptation of $E_P$ and $E_A$ for the unpaired samples from $\mathcal{D}_{unp}$.
We denote $B_r$ to represent a random background image. As shown in Fig. \[fig:main2\]B, here $\tilde{I}_t = D_I(f_{2D}, f_{a}, B_r)$, in absence of access to a paired image to enforce a direct pixel-wise loss. Thus, the parameters of $E_P$ and $E_A$ are optimized for the unpaired samples using the following loss function, $\mathcal{L}_{\textit{\textit{UNP}}} = |p_{2D} - \tilde{p}_{2D}| + \lambda_2|f_a - \tilde{f}_a|$, where $\tilde{p}_{2D}=\mathcal{T}^{-1} \circ\mathcal{T}_k\circ E_P\circ\mathcal{T}(\tilde{I}_t)$ and $\tilde{f}_a=E_A(\tilde{I}_t)$. Here, $\mathcal{T}$ and $\mathcal{T}^{-1}$ represents a differentiable spatial transformation (such as image flip or in-plane rotation) and its inverse, respectively. We employ this to maintain a consistent representation across spatial-transformations. Note that, for the flip-operation of $p_{2D}$, we also exchange the indices of the joints associated with the left side to right and vice-versa.
We train on three different loss functions, viz. $\mathcal{L}_{prior}, \mathcal{L}_{P}$, and $\mathcal{L}_{\textit{UNP}}$ at separate iterations, each with different optimizer. Here, $\mathcal{L}_{prior}$ denotes the supervised loss directly on $p_{3D}$ and $p_{2D}$ for the synthetically rendered images on randomly selected backgrounds, as discussed before.
\[tab:protocol2results\]
\[tab:mpiinf3dhp\]
Experiments
===========
In this section, we describe experimental details followed by a thorough analysis of the framework for bench-marking on two widely used datasets, Human3.6M and MPI-INF-3DHP.
We use Resnet-50 (till *res4f*) with ImageNet-pretrained parameters as the base pose encoder $E_P$, whereas the appearance encoder is designed separately using 10 Convolutions. $E_P$ later divides into two parallel branches of fully-connected layers dedicated for $v_k$ and $c$ respectively. We use $J=17$ for all our experiments as shown in Fig. \[fig:main1\]. The channel-wise aggregation of $f_{am}$ (16-channels) and $f_{hm}$ (17-channels) is passed through two convolutional layers to obtain $f_{2D}$ (128-maps), which is then concatenated with $f_a$ (512-maps) to form the input for $D_I$ (each with 14$\times$14 spatial dimension). Our experiments use different AdaGrad optimizers (learning rate: 0.001) for each individual loss components in alternate training iterations, thereby avoiding any hyper-parameter tuning. We perform several augmentations (color jittering, mirroring, and in-plane rotation) of the 20 synthetic samples, which are used to provide a direct supervised loss at the intermediate pose representations.
**Datasets.** The *base-model* is trained on a mixture of two datasets, Human3.6M and an in-house collection of YouTube videos (also refereed as YTube). In contrast to the in-studio H3.6M dataset, YTube contains human subjects in diverse apparel and BG scenes performing varied forms of motion (usually dance forms such as western, modern, contemporary etc.). Note that all samples from H3.6M contribute to the paired dataset $\mathcal{D}_p$, whereas $\sim$40% samples in YTube contributed to $\mathcal{D}_p$ and rest to $\mathcal{D}_{unp}$ based on the associated BG motion criteria. However, as we do not have ground-truth 3D pose for the samples from YTube (in-the-wild dataset), we use MPI-INF-3DHP (also refereed as 3DHP) to quantitatively benchmark generalization of the proposed pose estimation framework.
### a) Evaluation on Human3.6M.
We evaluate our framework on protocol-II, after performing scaling and rigid alignment of the poses inline with the prior arts ([Chen et al. [-@chen2019unsupervised]]{}; [Rhodin et al. [-@rhodin2018unsupervised]]{}). We train three different variants of the proposed framework a) *Ours(unsup.)*, b) *Ours(semi-sup.)*, and c) *Ours(weakly-sup.)* as reported in Table \[tab:protocol2results\]. After training the *base-model* on the mixed YTube+H3.6M dataset, we finetune it on the static H3.6M dataset by employing $\mathcal{L}_{prior}$ and $\mathcal{L}_{p}$ (without using any multi-view or pose supervision) and denote this model as *Ours(unsup.)*. This model is further trained with full supervision on the 2D pose landmarks simultaneously with $\mathcal{L}_{prior}$ and $\mathcal{L}_{p}$ to obtain *Ours(weakly-sup.)*. Finally, we also train *Ours(unsup.)* with supervision on 5% 3D of the entire trainset simultaneously with $\mathcal{L}_{prior}$ and $\mathcal{L}_{p}$ (to avoid over-fitting) and denote it as *Ours(semi-sup.)*. As shown in Table \[tab:protocol2results\], *Ours(unsup.)* clearly outperforms the prior-art ([Rhodin et al. [-@rhodin2018unsupervised]]{}) with a significant margin (89.4 vs. 98.2) even without leveraging multi-view supervision. Moreover, *Ours(weakly-sup.)* demonstrates state-of-the-art performance against prior weakly supervised approaches.
{width="1.00\linewidth"}
\[fig:viewsyn\]
{width="0.98\linewidth"}
\[fig:qualitative\]
### b) Evaluation on MPI-INF-3DHP.
We aim to realize a higher level of generalization in consequence of leveraging rich kinematic prior information. The proposed framework outputs 3D pose, which is bounded by the kinematic plausibility constraints even for unseen apparel, BG and action categories. This characteristic is clearly observed while evaluating performance of our framework on unseen 3DHP dataset. We take *Ours(weakly-sup.)* model trained on YTube+H3.6M dataset to obtain 3D pose predictions on unseen 3DHP testset (9th row in Table \[tab:mpiinf3dhp\]). We clearly outperform the prior work [@chen2019unsupervised] by a significant margin in a fully-unseen setting (8th and 9th row with -3DHP in Table \[tab:mpiinf3dhp\]). Furthermore, our weakly supervised model (with 100% 2D pose supervision) achieves state-of-the-art performance against prior approaches at equal supervision level.
\[tab:ablations\]
### c) Ablation study.
In the proposed framework, our major contribution is attributed to the design of differentiable transformations and an innovative way to facilitate the usage of unpaired samples even in presence of BG motion. Though effectiveness of camera-projection has been studied in certain prior works [@chen2019unsupervised], use of forward-kinematic transformation $\mathcal{T}_{fk}$ and affinity map in the spatial-map transformation $\mathcal{T}_m$ is employed for the first time in such a learning framework. Therefore, we evaluate importance of both $\mathcal{T}_{fk}$ and $\mathcal{T}_m$ by separately bypassing these modules through neural network transformations. Results in Table \[tab:ablations\] clearly highlight effectiveness of these carefully designed transformations for the unsupervised 3D pose estimation task.
### d) Qualitative results.
Fig. \[fig:viewsyn\] depicts qualitative results derived from *Ours(unsup.)* on in-studio H3.6M and in-the-wild YTube dataset. It highlights effectiveness of unsupervised disentanglement through separation or cross-transfer of apparel, pose, camera-view and BG, for novel image synthesis. Though, our focus is to disentangle 3D pose information, separation of apparel and pose transfer is achieved as a byproduct of the proposed learning framework. In Fig. \[fig:qualitative\] we show results on the 3D pose estimation task obtained from *Ours(weakly-sup.)* model. Though we train our model on H3.6M, 3DHP and YTube datasets, results on LSP dataset [@johnson2010clustered] is obtained without training on the corresponding train-set, in a fully-unseen setting. Reliable pose estimation on such diverse unseen images highlights generalization of the learned representations thereby overcoming the problem of dataset-bias.
Conclusion
==========
We present an unsupervised 3D human pose estimation framework, which relies on a minimal set of prior knowledge regarding the underlying kinematic 3D structure. The proposed local-kinematic model indirectly endorses a kinematic plausibility bound on the predicted poses, thereby limiting the model from delivering implausible pose outcomes. Furthermore, our framework is capable of leveraging knowledge from video frames even in presence of background motion, thus yielding superior generalization to unseen environments. In future, we would like to extend such frameworks for predicting 3D mesh, by characterizing the prior knowledge on human shape, alongside pose and appearance.
[ **Acknowledgements.** This work was supported by a Wipro PhD Fellowship (Jogendra) and in part by DST, Govt. of India (DST/INT/UK/P-179/2017).]{}
[^1]: [<https://sites.google.com/view/ksp-human/>]{}
[^2]: equal contribution
| ArXiv |
---
author:
- |
[**Jian Gao, Linzhi Shen, Fang-Wei Fu** ]{}\
[Chern Institute of Mathematics and LPMC, Nankai University]{}\
[Tianjin, 300071, P. R. China]{}\
title: '[**Skew Generalized Quasi-Cyclic Codes Over Finite Fields**]{}'
---
[ In this work, we study a class of generalized quasi-cyclic (GQC) codes called skew GQC codes. By the factorization theory of ideals, we give the Chinese Remainder Theorem over the skew polynomial ring, which leads to a canonical decomposition of skew GQC codes. We also focus on some characteristics of skew GQC codes in details. For a $1$-generator skew GQC code, we define the parity-check polynomial, determine the dimension and give a lower bound on the minimum Hamming distance. The skew quasi-cyclic (QC) codes are also discussed briefly.]{}
[ Skew cyclic codes; Skew GQC codes; $1$-generator skew GQC codes; Skew QC codes]{}
[**Mathematics Subject Classification (2000)** ]{} 11T71 $\cdot$ 94B05 $\cdot$ 94B15
0.2in [**1 Introduction**]{} Recently, it has been shown that codes over finite rings are a very important class of codes and many types of codes with good parameters could be constructed over rings [@Aydin; @Abualrub2; @Siap2]. Skew polynomial rings are an important class of non-commutative rings. More recently, applications in the construction of algebraic codes have been found [@Abualrub1; @Bhaintwal2; @Boucher1; @Boucher2; @Boucher3], where codes are defined as ideals or modules in the quotient ring of skew polynomial rings. The principle motivation for studying codes in this setting is that polynomials in skew polynomials rings have more factorizations than that in the commutative case. This suggests that it may be possible to find good or new codes in the skew polynomial ring with lager minimum Hamming distance. Some researchers have indeed shown that such codes in skew polynomial rings have resulted in the discovery of many new linear codes with better minimum Hamming distances than any previously known linear codes with same parameters [@Abualrub1; @Boucher1].
Quasi-cyclic (QC) codes over commutative rings constitute a remarkable generalization of cyclic codes [@Aydin; @Bhaintwal1; @Conan; @Ling2; @Siap2]. More recently, many codes were constructed over finite fields which meet the best value of minimum distances with the same length and dimension [@Aydin; @Siap2]. In [@Abualrub1], Abualrub et al. have studied skew QC codes over finite fields as a generalization of classical QC codes. They have introduced the notation of similar polynomials in skew polynomial rings and shown that parity-check polynomials for skew QC codes are unique up to similarity. They also constructed some skew QC codes with minimum Hamming distances greater than previously best known linear codes with the given parameters. In [@Bhaintwal2], Bhaintwal studied skew QC codes over Galois rings. He gave a necessary and sufficient condition for skew cyclic codes over Galois rings to be free, and presented a distance bound for free skew cyclic codes. Futhermore, he also discussed the sufficient condition for 1-generator skew QC codes to be free over Galois rings. A canonical decomposition and the dual codes of skew QC codes were also given.
The notion of generalized quasi-cyclic (GQC) codes over finite fields were introduced by Siap and Kulhan [@Siap1] and some further structural properties of such codes were studied by Esmaeili and Yari [@Esmaeili]. Based on the structural properties of GQC codes, Esmaeili and Yari gave some construction methods of GQC codes and obtained some optimal linear codes over finite fields. In [@Cao1], Cao studied GQC codes of arbitrary length over finite fields. He investigated the structural properties of GQC codes and gave an explicit enumeration of all $1$-generator GQC codes. As a natural generalization, GQC codes over Galois rings were introduced by Cao and structural properties and explicit enumeration of GQC codes were also obtained in [@Cao2]. But the problem of researching skew GQC codes over finite fields has not been considered to the best of our knowledge.
Let $\mathbb{F}_{q}$ be a finite field, where $q=p^m$, $p$ is a prime number and $m$ is a positive integer. The Frobenius automorphism $\theta$ of $\mathbb{F}_{q}$ over $\mathbb{F}_p$ is defined by $\theta (a)=a^p$, $a\in\mathbb{F}_{q}$. The automorphism group of $\mathbb{F}_{q}$ is called the Galois group of $\mathbb{F}_{q}$. It is a cyclic group of order $m$ and is generated by $\theta$. Let $\sigma$ be an automorphism of $\mathbb{F}_{q}$. The *skew polynomial ring* $R=\mathbb{F}_q[x, \sigma]$ is the set of polynomials over $\mathbb{F}_q$, where the addition is defined as the usual addition of polynomials and the multiplication is defined by the following basic rule $$(ax^i)(bx^j)=a\sigma^i(b)x^{i+j},~~a,b\in\mathbb{F}_q.$$ From the definition one can see that $R$ is a non-commutative ring unless $\sigma$ is an identity automorphism.
Let $\mid \sigma\mid$ denote the order of $\sigma$ and assume $\mid \sigma\mid=t$. Then there exists a positive integer $d$ such that $\sigma=\theta^d$ and $m=td$. Clearly, $\sigma$ fixes the subfield $\mathbb{F}_{p^d}$ of $\mathbb{F}_q$. Let $Z(\mathbb{F}_q[x,\sigma])$ denote the center of $R$. For $f, g\in R$, $g$ is called a *right divisor* (resp. *left divisor*) of $f$ if there exists $r\in R$ such that $f=rg$ (resp. $f=gr$). In this case, $f$ is called a *left multiple* (resp. *right multiple*) of $g$. Let the division be defined similarly. Then $\bullet$ If $g, f \in Z(\mathbb{F}_q[x, \sigma])$, then $g\cdot f=f\cdot g$. $\bullet$ Over finite fields, a skew polynomial ring is both a right Euclidean ring and a left Euclidean ring.
Let $f, g \in R$. A polynomial $h$ is called a *greatest common left divisor* (gcld) of $f$ and $g$ if $h$ is a left divisor of $f$ and $g$; and if $u$ is another left divisor of $f$ and $g$, then $u$ is a left divisor of $h$. A polynomial $e$ is called a *least common left multiple* (lclm) of $f$ and $g$ if $e$ is a right multiple of $f$ and $g$; and if $v$ is another right multiple of $f$ and $g$, then $v$ is a right multiple of $e$. The *greatest common right divisor* (gcrd) and *least common right multiple* (lcrm) of polynomials $f$ and $g$ are defined similarly.
The main aim of this paper is to study the structural properties of skew generalized quasi-cyclic (GQC) codes over finite fields. The rest of this paper is organized as follows. In Section 2, we survey some well known results of skew cyclic codes and give the BCH-type bound for skew cyclic codes. By the factorization theory of ideals, we give the Chinese Remainder Theorem in skew polynomial rings. In Section 3, using the Chinese Remainder Theorem, we give a necessary and sufficient condition for a code to be a skew GQC code. And this leads to a canonical decomposition of skew GQC codes. In Section 4, we mainly describe some characteristics of $1$-generator GQC codes including parity-check polynomials, dimensions and the minimum Hamming distance bounds. In Section 5, we discuss a special class of skew GQC codes called skew QC codes.
[**2 Skew cyclic codes** ]{} Let $\sigma$ be an automorphism of the finite field $\mathbb{F}_q$ and $n$ be a positive integer such that the order of $\sigma$ divides $n$. A linear code $C$ of length $n$ over $\mathbb{F}_q$ is called *skew cyclic code* or *$\sigma$-cyclic code* if for any codeword $(c_0, c_1, \ldots, c_{n-1})\in C$, the vector $(\sigma(c_{n-1}), \sigma(c_0), \ldots, \sigma(c_{n-2}))$ is also a codeword in $C$. In polynomial representation, a linear code of length $n$ over $\mathbb{F}_q$ is a skew cyclic code if and only if it is a *left ideal* of the ring $R/(x^n-1)$, where $(x^n-1)$ denotes the *two-sided ideal* generated by $x^n-1$. In general, if $f(x)\in R$ generates a two-sided ideal, then a left ideal of $R/(f(x))$ is a linear code over $\mathbb{F}_q$. Such a linear code will be called a *skew linear code* or a *$\sigma$-linear code*. Let $C$ be a linear code of length $n$ over $\mathbb{F}_q$. The Euclidean dual of $C$ is defined as$$C^\perp=\{v \in \mathbb{F}_q^n|~u\cdot v=0, ~\forall u\in C\}.$$ In this paper, we suppose that the order of $\sigma$ divides $n$ and ${\rm gcd}(n,q)=1$. In the following, we list some well known results of skew cyclic codes in Theorem 2.1.
[**Theorem 2.1** ]{}[@Boucher1; @Boucher2] *Let $C$ be a skew cyclic code ($\sigma$-cyclic code) of length $n$ over $\mathbb{F}_q$ generated by a right divisor $g(x)=\sum_{i=0}^{n-k-1}g_ix^i+x^{n-k}$ of $x^n-1$. Then\
* *(i) The generator matrix of $C$ is given by*
$$\left(
\begin{array}{ccccccc}
g_0 & \cdots & g_{n-k-1}& 1 & 0 & \cdots & 0\\
0 & \sigma(g_0) & \cdots & \sigma(g_{n-k-1}) & 1 & \cdots & 0\\
0 & \ddots & \ddots& & & \ddots& \vdots\\
\vdots & & \ddots &\ddots &\cdots &\ddots &0\\
0& \cdots & 0& \sigma^{k-1}(g_0) & \cdots & \sigma^{k-1}(g_{n-k-1})& 1\\
\end{array}
\right)$$
*and $\mid C\mid=q^{n-{\rm deg}(g(x))}$.\
*
*(ii) Let $x^n-1=h(x)g(x)$ and $h(x)=\sum_{i=0}^{k-1}h_ix^i$. Then $C^\perp$ is also a skew cyclic code of length $n$ generated by $\widetilde{h}(x)=x^{{\rm deg}(h(x))}\varphi(h(x))=1+\sigma(h_{k-1})x+\cdots+\sigma^k(h_0)x^k$, where $\varphi$ is an anti-automorphism of $\sigma$ defined as $\varphi(\sum_{i=0}^ta_ix^t)=\sum_{i=0}^tx^{-i}a_i$, where $ \sum_{i=0}^ta_ix^i\in R$. The generator matrix of $ C^\perp$ is given by* $$\left(
\begin{array}{ccccccc}
1 & \sigma(h_{k-1}) & \cdots& \sigma^k(h_0) & 0 & \cdots & 0\\
0 & 1 & \sigma^2(h_{k-1}) & \cdots & \sigma^{k+1}(h_0) & \cdots & 0\\
0 & 0 & \ddots& & & \ddots& 0\\
\vdots & & \ddots &\ddots &\cdots &\ddots &\vdots\\
0& \cdots & 0& 1 & \sigma^{n-k}(h_{k-1}) & \cdots& \sigma^{n-1}(h_0)\\
\end{array}
\right)$$\
*and $\mid C^\perp\mid=q^k$.*\
*(iii) For $c(x)\in R$, $c(x)\in C$ if and only if $c(x)h(x)=0$ in $R$.\
*
*(iv) $C$ is a cyclic code of length $n$ over $\mathbb{F}_q$ if and only if the generator polynomial $g(x)\in \mathbb{F}_{p^d}[x]/(x^n-1)$.*
The monic polynomials $g(x)$ and $h(x)$ in Theorem 2.1 are called the *generator polynomial* and the *parity-check polynomial* of the skew cyclic code $C$, respectively.
[**Theorem 2.2** ]{} *Let $C$ be a skew cyclic code with the generator polynomial $g(x)$ and the check polynomial $h(x)$. Then a polynomial $f(x)\in R/(x^n-1)$ generates $C$ if and only if there exists a polynomial $p(x)\in R$ such that $f(x)=p(x)g(x)$ where $p(x)$ and $h(x)$ are right coprime.*
*Proof* Let $f(x)\in R/(x^n-1)$ generate $C$. Then there exist polynomials $p(x), q(x) \in R/(x^n-1)$ such that $f(x)=p(x)g(x)$ and $g(x)=q(x)f(x)$ in $R/(x^n-1)$. Therefore $g(x)=q(x)p(x)g(x)$. In $R$, we have $$g(x)=q(x)p(x)g(x)+r(x)(x^n-1)=q(x)p(x)g(x)+r(x)h(x)g(x)$$ for some $r(x)\in R$. It follows that $$(1-q(x)p(x)-r(x)h(x))g(x)=0.$$ Since $R$ is a principal ideal domain, we have $1-q(x)p(x)-r(x)h(x)=0$, which implies that $p(x)$ and $h(x)$ are right coprime. Conversely, suppose $f(x)=p(x)g(x)$ where $p(x)$ and $h(x)$ are right coprime. Then there exist polynomials $u(x), v(x) \in R$ such that $u(x)p(x)+v(x)h(x)=1$. Multiplying on right by $g(x)$ both sides, we have $u(x)p(x)g(x)+v(x)h(x)g(x)=g(x)$, which implies that $u(x)p(x)g(x)=u(x)f(x)g(x)$ in $R/(x^n-1)$. Therefore $g(x)\in (f(x))_l$, where $(f(x))_l$ denotes the left ideal generated by $f(x)$ in $R/(x^n-1)$. It means that $(g(x))_l \subseteq (f(x))_l$. Clearly, $(f(x))_l\subseteq (g(x))_l$, and hence $(f(x))_l=(g(x))_l=C$. $\Box$
Let $\mathbb{F}[Y^{q_0}, \circ]=\{a_0Y+a_1Y^{q_0}+\cdots+a_nY^{q_0^n}\mid~a_0,a_1,\ldots,a_n\in \mathbb{F}_q\}$, where $ q_0=p^d$. For $f=a_0Y+a_1Y^{q_0}+\cdots+a_nY^{q_0^n}$ and $g=b_0Y+b_1Y^{q_0}+\cdots+b_tY^{q_0^t}$, define $f+g$ to be ordinary addition of polynomials and define $f\circ g=f(g)$. Thus, $f\circ g=c_0Y+c_1Y^{q_0}+\cdots+c_{n+t}Y^{q_0^{n+t}}$, where $c_i=\sum_{j+s=i}a_jb_s^{q_0^s}$. It is easy to see that $\mathbb{F}[Y^{q_0}, \circ]$ under addition and composition $\circ$ forms a non-commutative ring called *Ore Polynomial* ring (see [@McDonald]).
Define $$\phi:~ R\rightarrow \mathbb{F}_q[Y^{q_0}, \circ],$$ $$\sum a_ix^i\mapsto \sum a_iY^{q_0^i}.$$
[**Lemma 2.3** ]{}[@McDonald Theorem II.13] *The above mapping $\phi:~R\rightarrow \mathbb{F}[Y^{q_0}, \circ]$ is a ring isomorphism between the skew polynomial ring $R=\mathbb{F}_q[x, \sigma]$ and the Ore Polynomial ring $F_q[Y^{q_0}, \circ]$.* $\Box$
For a skew cyclic code over $\mathbb{F}_q$, it can also be described in terms of the $n$-th root of unity. By the above mapping $\phi$, one can verify that $\phi(x^n-1)=Y^{q_0^n}-Y$. Since $\sigma=\theta^d$, the fixed subfield is $\mathbb{F}_{p^d}=\mathbb{F}_{q_0}$. Let $\mathbb{F}_{q^s}$ be the smallest extension of $\mathbb{F}_q$ containing $\mathbb{F}_{q_0^n}$ as a subfield. Then $\mathbb{F}_{q^s}$ is the splitting field of $\phi(x^n-1)$ over $\mathbb{F}_q$. An element $\alpha \in \mathbb{F}_{q^s}$ is called a *right root* of $f\in R$ if $x-\alpha$ is a right divisor of $f$.
Let the extension of $\sigma$ to an automorphism of $\mathbb{F}_{q^s}$ be also denoted by $\sigma$. For any $\alpha\in \mathbb{F}_{q^s}$ define ${\mathcal N}_{\sigma, i}(\alpha)=\sigma^{i-1}(\alpha)\sigma^{i-2}(\alpha)\cdots \sigma(\alpha)\alpha,~i>0$, with ${\mathcal N}_{\sigma, 0}=1$.
[**Lemma 2.4** ]{}[@Jacobson2 Proposition 1.3.11] *Let $f(x)=\sum_{i=0}^ka_ix^i \in R$. Then\
(i) The remainder $r$ on right division of $f(x)$ by $x-\alpha$ is given by $r=a_0{\mathcal N}_{\sigma, 0}(\alpha)+a_1{\mathcal N}_{\sigma, 1}(\alpha)+\cdots +a_k{\mathcal N}_{\sigma, k}(\alpha)$.\
(ii) Let $\beta\in \mathbb{F}_{q^s}$. Then $(x-\beta)\mid_rf(x)$ if and only if $\sum_{i=0}^ka_i{\mathcal N}_{\sigma, i}(\beta)=0$.* $\Box$
Note that $\sigma(\alpha)=\theta^d(\alpha)=\alpha^{p^d}=\alpha^{q_0}$, and hence ${\mathcal N}_{\sigma, i}(\alpha)=\alpha^{\frac{q_0^i-1}{q_0-1}}$. The following result can also be found in [@Chaussade Lemma 4], here we give another proof by Lemma 2.3.
[**Lemma 2.5** ]{} *Let $f(x)\in R$, and $\mathbb{F}_{q^s}$ be the smallest extension of $\mathbb{F}_q$ in which $\phi(f(x))$ splits. Then a non-zero element $\alpha \in \mathbb{F}_{q^s}$ is a root of $\phi(f(x))$ if and only if $\alpha^{q_0}/\alpha$ is a right root of $f(x)$.*
*Proof* If $\alpha^{q_0}/\alpha$ is a right root of $f(x)$, then $x-\alpha^{q_0}/\alpha$ is a right divisor of $f(x)$. From Lemma 2.3, we have $Y^{q_0}-\alpha^{q_0}/\alpha Y$ is a factor of $\phi (f(x))$. Therefore $\alpha$ is the root of $\phi(f(x))$. Conversely, suppose $\alpha$ is the root of $\phi(f(x))$. Let $f(x)=k(x)(x-\alpha^{q_0}/\alpha)+r$, where $r\in \mathbb{F}_q$. Then $\phi(f(x))=\phi(k(x))\circ \phi(x-\alpha^{q_0}/\alpha)+\phi(r)$. From the discussion above, $\alpha$ is the root of $\phi(x-\alpha^{q_0}/\alpha)$. Therefore $\alpha$ is also the root of $\phi(r)$, i.e., $r\alpha=0$. Since $\alpha$ is a non-zero element in $\mathbb{F}_{q^s}$, we have $r=0$. This implies that $\alpha^{q_0}/\alpha$ is a right root of $f(x)$. $\Box$
Since $\phi(x^n-1)=Y^{q_0^n}-Y$ splits into linear factors in $\mathbb{F}_{q^s}$, it follows from Lemma 2.3 that $x^n-1$ also splits into linear factors in $\mathbb{F}_{q^s}[x, \sigma]$. It is well known that the non-zero roots of $Y^{q_0^n}-Y$ are precisely the elements of $\{1, \gamma, \ldots, \gamma^{q_0^n-2}\}$, where $\gamma$ is a primitive element of $\mathbb{F}_{q_0^n}$. Therefore, by Lemma 2.5, $x-(\gamma^i)^{q_0}/\gamma^i$ is the right factor of the skew polynomial $x^n-1$. It means that there are several different factorizations of the skew polynomial $x^n-1$.
In the following, we give the BCH-type bound for the skew cyclic code over $\mathbb{F}_q$.
[**Theorem 2.6** ]{} *Let $C$ be a skew cyclic code of length $n$ generated by a monic right factor $g(x)$ of the skew polynomial $x^n-1$ in $R$. If $x-\gamma^j$ is a right divisor of $g(x)$ for all $j=b, b+1, \ldots, b+\delta-2$, where $b\geq 0$ and $\delta \geq 1$, then the minimum Hamming distance of $C$ is at least $ \delta$.*
*Proof* Let $c(x)=\sum_{i=0}^{n-1}c_ix^i$ be a codeword of $C$. Then $c(x)$ is a left multiple of $g(x)$, and hence $x-\gamma^j$ is a right divisor of $c(x)$, for all $0\leq j \leq b+\delta-2$. From Lemma 2.4, $x-\gamma^j$ is a right divisor of $c(x)$ if and only if $\sum_{i=0}^{n-1}c_i{\mathcal N}_{\sigma, i}(\gamma^j)=0$, $j=b, b+1, \ldots, b+\delta-2$. Therefore the matrix $$\label{matrix}
\left(
\begin{array}{cccc}
1 & {\mathcal N}_{\sigma, 1}(\gamma^b) & \cdots & {\mathcal N}_{\sigma, n-1}(\gamma^b)\\
1 & {\mathcal N}_{\sigma, 1}(\gamma^{b+1}) & \cdots & {\mathcal N}_{\sigma, n-1}(\gamma^{b+1})\\
\vdots & \vdots & \ddots & \vdots \\
1& {\mathcal N}_{\sigma, 1}(\gamma^{b+\delta-2}) & \cdots & {\mathcal N}_{\sigma, n-1}(\gamma^{b+\delta-2}) \\
\end{array}
\right)$$ is a parity-check matrix. Any $\delta-1$ columns of (\[matrix\]) form a $(\delta-1)\times (\delta-1)$ matrix and denote $D$ as its determinant. Since $D$ is a Vandermonde determinant, $D=0$ if and only if ${\mathcal N}_{\sigma, i}(\gamma)={\mathcal N}_{\sigma, j}(\gamma)$, for $i\neq j$. It is equivalent to $$\gamma^{\frac{q_0^i-q_0^j}{q_0-1}}=\gamma^{\frac{q_0^j(q_0^{i-j}-1)}{q_0-1}}=1.$$ In particular, $\gamma^{q_0^j(q_0^{i-j}-1)}=1$ implies that $(q_0^n-1)\mid q_0^j(q_0^{i-j}-1)$. Since ${\rm gcd}(q_0^n-1, q_0^j)=1$, it follows that $(q_0^n-1)\mid (q_0^{i-j}-1)$. Therefore, there exists a positive integer $l$ such that $i-j=nl$. It means that $\frac{q_0^{nl}-1}{q_0-1}=k(q_0^n-1)$ for some positive integer $k$. Thus $(q_0-1)\mid \frac{q_0^{nl}-1}{q_0^n-1}=\sum_{i=0}^{m-1}q_0^{ni}$. It implies that $\gamma^{q_0-1}\mid \gamma$, which is impossible. This shows that any $\delta-1$ columns are linearly independent, and hence the minimum Hamming distance of $C$ is is at least $ \delta$. $\Box$
[**Example 2.7** ]{} Consider $R=\mathbb{F}_{3^2}[x, \sigma]$, where $\sigma=\theta$ is a Frobenius automorphism of $\mathbb{F}_{3^2}$ over $\mathbb{F}_3$. The polynomial $g(x)=x-\alpha^2$ is a right factor of $x^4-1$, where $\alpha$ is a primitive element of $\mathbb{F}_{3^2}$. Since $\phi(x^4-1)=Y^{81}-Y$, it follows that $\phi(x^4-1)$ splits in $\mathbb{F}_{3^4}$. Let $\xi$ be a primitive element of $\mathbb{F}_{3^4}$. Then $\alpha=\xi^{20}$ and $\phi(g(x))=Y^3-\alpha^2 Y$ has a root $\xi^{20}$. Therefore, by Lemma 2.5, $(\xi^{20})^3/\xi^{20}=\xi^{40}$ is a right root of $g(x)$. Let $C$ be a skew cyclic code of length $4$ generated by $g(x)$ over $\mathbb{F}_{3^2}$. Then $C$ is a code with ${\rm dim}(C)=3$ and $d_H(C)\geq 2$. In fact, $C$ is an optimal $[4,3,2]$ skew cyclic code over $\mathbb{F}_{3^2}$. Also for each $i=1,2,\ldots,40$, $(\xi^i)^3/\xi^i=\xi^{2i}$ is a right root of $x^4-1$, which implies that there are $10$ different factorizations of skew polynomial $x^4-1$ over $\mathbb{F}_{3^2}$.
We now consider the factorization theory of (two-sided) ideals or two-sided elements. An element $a^* \in R$ is called two-sided element if $Ra^*=a^*R$.
[**Theorem 2.8** ]{}[@McDonald Theorem II.12] *If a polynomial $f^*$ generates a two-sided ideal in $R$, then $f^*$ has the form $$(a_0+a_1x^t+\cdots +a_nx^{nt})x^m,$$ where $a_i\in \mathbb{F}_q$ and $t=\mid \sigma\mid$.*
Obvious $\mathbb{F}_{q_0}[x]=\{ b_0+b_1x+\cdots +b_nx^n \mid~b_i\in \mathbb{F}_{q_0}\}$ forms a commutative subring of $R$. Then the center of $R$ is $\mathbb{F}_{q_0}[x]\cap \mathbb{F}_q[x^t]$ where $\mathbb{F}_q[x^t]=\{ a_0+a_1x^t+\cdots +a_nx^{nt}\mid~a_i\in \mathbb{F}_{q_0}\}$, i.e., the center of $R$ is $\mathbb{F}_{q_0}[x^t]=\mathbb{F}_{p^d}[x^t]$ (see [@McDonald]).
If $Ra^*$ is a non-zero (two-sided) maximal ideal in $R$, or equivalently, $a^*\neq 0$ and $R/Ra^*$ is a simple ring, then we call the two-sided element $a^*$ a *two-sided maximal* (t.s.m) element. Let $a, b$ be non-zero elements in $R$. Then $a$ is said to be *left similar* to $b$ ($a\sim_lb$) if and only if $R/Ra\cong R/Rb$. Two elements are left similar if and only if they are right similar.
Let $a$ be a non-zero element in $R$. If $a$ is not a unit in $R$, then $a$ can be written as $a=p_1p_2\cdots p_s$, where $p_1,p_2,\ldots,p_s$ are irreducible. Moreover, if $a=p_1p_2\cdots p_s=p_1'p_2'\cdots p_t'$, where $p_i$ and $p_j'$ are irreducible, then $s=t$ and there exists a permutation $(1',2', \ldots, s')$ of $(1,2,\ldots,s)$ such that $p_i\sim p_{i'}'$ (see [@Jacobson2]).
[**Lemma 2.9** ]{}[@Jacobson2 Theorem 1.2.17$'$, Theorem 1.2.19] *Let $a^*$ ba a non-zero two-sided element in $R$ and $a^*$ not a unit. Then\
(i) $a^*=p_1^*p_2^*\cdots p_m^*$, where $p_i^*, 1\leq i \leq m$, are t.s.m elements and such a factorization is unique up to order and unit multipliers.\
(ii) Let $p_i^*=p_{i,1}p_{i,2}\cdots p_{i,n}$, where $p_{i,j}, 1\leq i \leq m, 1\leq j \leq n$ are irreducible. Then $p_{i,1}, p_{i,2}, \ldots, p_{i,n}$ are all similar.*
[**Example 2.10** ]{} Consider $R=\mathbb{F}_{3^3}[x, \sigma]$, where $\sigma=\theta$ is a Frobenius automorphism of $\mathbb{F}_{3^3}$ over $\mathbb{F}_3$. The fixed field of $\sigma$ is $\mathbb{F}_3$. Let $f(x)=x^6-x^3-2 \in R$. Since $f(x)\in \mathbb{F}_3[x^3]$, $f(x)$ is a two-sided element of $R$. A factorization of $f(x)$ in $R$ is $f(x)=(x^3+1)(x^3-2)$. Clearly, both $x^3+1$ and $x^3-2$ are two-sided elements. Moreover, they must be t.s.m elements because they have the smallest degree for a polynomial of the form $x^t, t\geq 1$, to be a two-sided element.
[**Remark 2.1** ]{} Note that there is an error in Example 4 in [@Bhaintwal2], where the author claimed that the fixed field of $\sigma$ is $\mathbb{F}_9$. But it is well known that $\mathbb{F}_9$ is not a subfield of $\mathbb{F}_{27}$ at all. We have corrected it in Example 2.10.
Suppose $x^n-1$ has a factorization of the form $x^n-1=f_1f_2\cdots f_k$, where $f_1,f_2,\ldots,f_k$ are irreducible polynomials. Since $x^n-1$ is a two-sided element, by Lemma 2.9, it can also be factorized as $x^n-1=f_1^*f_2^*\cdots f_t^*$, where each $f_i^*$ is a t.s.m element and is a product of all polynomials similar to an irreducible factor $f_i$ of $x^n-1$. Since ${\rm gcd}(q, n)=1$, it follows that all factors $f_1^*,f_2^*,\ldots,f_t^*$ are distinct. Also since $(f_i^*)$ is maximal, we can see that $f_i^*$ and $f_j^*$ are coprime for all $i\neq j$. Denote $\widehat{f}_i^*$ as the product of all $f_j^*$ except $f_i^*$, we have the following *Chinese Remainder Theorem* in the skew polynomial ring $\mathbb{F}_q[x, \sigma]$.
[**Theorem 2.11** ]{} *Let $x^n-1=f_1^*f_2^*\cdots f_t^*$ be the unique representation of $x^n-1$ as a product of pairwise coprime t.s.m elements in $R$. Since ${\rm gcd}(\widehat{f}_i^*, f_i^*)=1$, there exist polynomials $b_i, c_i \in R$ such that $b_i\widehat{f}_i^*+c_if_i^*=1$. Let $e_i=b_i\widehat{f}_i^* \in R$. Then\
(i) $e_1, e_2, \ldots, e_t$ are mutually orthogonal in $R$;\
(ii) $e_1+e_2+\cdots +e_t=1$ in $R$;\
(iii) $R_i=(e_i)$ is a two-sided ideal of $R$ and $e_i$ is the identity in $(e_i)$;\
(iv) $R=R_1\bigoplus R_2 \bigoplus \cdots \bigoplus R_t$;\
(v) For each $i=1,2,\ldots,t$, the map $$\psi:~R/(f_i^*)\rightarrow R_i$$ $$g+(f_i^*)\mapsto (g+(x^n-1))e_i$$ is a well-defined isomorphism of rings;\
(vi) $R\cong R/(f_1^*)\bigoplus R/(f_2^*)\bigoplus \cdots \bigoplus R/(f_t^*)$.*
*Proof* (i) Suppose $e_i=0$ for some $i=1,2,\ldots,t$, i.e., $b_i\widehat{f}_i^*\in (x^n-1)$ in $R$. Then $b_i\widehat{f}_i^*\in (f_i^*)$. Thus $1=b_i\widehat{f}_i^*+c_if_i^*\in (f_i^*)$, which is a contradiction. Hence, for each $i=1,2,\ldots,t$, $e_i\neq 0$. Thus we have $b_i\widehat{f}_i^*b_j\widehat{f}_j^* \in (x^n-1)$ for $i\neq j$. This implies that $e_ie_j=0$ in $R$. (ii) We have $b_1\widehat{f}_1^*+\cdots +b_t\widehat{f}_t^*-1 \in (f_i^*)$, for all $i=1,2,\ldots,t$. Therefore $b_1\widehat{f}_1^*+\cdots +b_t\widehat{f}_t^*-1 \in (x^n-1)$. Thus $e_1+\cdots +e_t=1$ in $R$. (iii) Let $Re_i=(e_i)_l$. Then $(e_i)_l\subseteq (\widehat{f}_i^*)$. On the other hand, $\widehat{f}_i^*=\widehat{f}_i^*(b_i\widehat{f}_i^*+c_if_i^*)=\widehat{f}_i^*b_i\widehat{f}_i^*$ in $R$, which implies $(\widehat{f}_i^*)\subseteq (e_i)_l$. Therefore $(e_i)_l=(\widehat{f}_i^*)$. Similarly, one can prove that $e_iR=(e_i)_r=(\widehat{f}_i^*)$, which implies that $(e_i)$ is a two-sided ideal of $R$. Clearly, $e_i$ is the identity in $(e_i)$. (iv) For any $a\in R$, $a$ can be represented as $a=ae_1+ae_2+\cdots +ae_t$. Since $ae_i\in (e_i)$, $R=(e_1)+(e_2)+\cdots +(e_t)$. Assume that $a_1+a_2+\cdots+a_t=0$, where $a_i\in (e_i)$. Multiplying on the left (or on the right) by $e_i$, we obtain that $a_1e_i+a_2e_i+\cdots +a_te_i=a_ie_i=a_i$, for $i=1,2,\ldots,t$. Therefore $R=R_1\bigoplus R_2 \bigoplus \cdots \bigoplus R_t$. (v) Let $g+(f_i^*)=g'+(f_i^*)$, for $g, g'\in R$. Then $g-g'\in (f_i^*)$. But $b_i\widehat{f}_i^* \in (f_j^*)$ for all $i\neq j$. Therefore $(g-g')b_i\widehat{f}_i^*\in (x^n-1)$. Hence, $(g+(x^n-1))e_i=(g'+(x^n-1))e_i$ in $R$, which implies that the map $\psi$ is well-defined. Clearly, $\psi$ is a surjective homomorphism of rings. Let $g+(f_i^*)\in R/(f_i^*)$ statisfy $(g+(x^n-1))e_i=(x^n-1)$. Then $gb_i\widehat{f}_i^* \in (x^n-1)\subseteq (f_i^*)$. Thus $g\in (f_i^*)$, i.e., $g+(f_i^*)=(f_i^*)$. This implies that the kernel of $\varphi$ is zero. (vi) From (iv) and (v), one can deduce this result immediately. $\Box$
[**3 Skew GQC codes**]{} In this section, we investigate the structural properties of skew GQC codes. We give the definition of skew GQC codes first. [**Definition 3.1** ]{} *Let $R=\mathbb{F}_q[x,\sigma]$ be a skew polynomial ring and $m_1, m_2, \ldots,m_l$ positive integers. Let $t$ be a divisor of each $m_i$, where $t$ is the order of $\sigma$ and $i=1,2,\ldots,l$. Denote ${\mathcal R}_i=R/(x^{m_i}-1)$ for $i=1,2,\ldots,l$. Any left $R$-submodule of the $R$-module ${\mathcal R}={\mathcal R}_1\times {\mathcal R}_2\times\cdots\times{\mathcal R}_l$ is called a skew generalized quasi-cyclic (GQC) code over $\mathbb{F}_q$ of block length $(m_1,m_2,\ldots,m_l)$ and length $\sum_{i=1}^lm_i$.*
Let $m_i\geq 1$, $i=1,2,\ldots,l$, and ${\rm gcd}(q, m_i)=1$. Then, by Lemma 2.9, $x^{m_i}-1$ has a unique factorization $x^{m_i}-1=f_{i1}^*f_{i2}^*\cdots f_{ir_i}^*$, where $f_{ij}^*$ , $j=1,2,\ldots,r_i$, are pairwise coprime monic t.s.m elements in $R$. Let $\{ g_1^*, g_2^*, \ldots, g_s^*\}=\{ f_{ij}^* \mid 1\leq i\leq l, 1\leq j \leq r_i\}$. Then we have $$x^{m_i}-1=g_1^{*d_{i1}}g_2^{*d_{i2}}\ldots g_s^{*d_{is}},$$ where $d_{ik}=1$ if $g_k^*=f_{i,j}^*$ for some $1\leq j \leq r_i$ and $d_{i, k}=0$ if ${\rm gcd}(g_k^*, x^{m_i}-1)=1$, for all $1\leq i \leq l$ and $1\leq k \leq s$. Suppose $n_j=\mid \{i\mid f_{i,\lambda}^*=g_j^*, 1\leq \lambda \leq r_i, 1\leq i \leq l, 1\leq j\leq s \}\mid$. Let ${\mathcal M}_j=(R/(g_j^*))^{n_j}$. Then we have
[**Theorem 3.2** ]{} *Let ${\mathcal R}={\mathcal R}_1\times {\mathcal R}_2\times\cdots \times {\mathcal R}_l$, where ${\mathcal R}_i=R/(x^{m_i}-1)$ for all $i=1,2,\ldots,l$. Then there exists an $R$-module isomorphism $\phi$ from ${\mathcal R}$ onto ${\mathcal M}_1\times {\mathcal M}_2 \times \cdots \times {\mathcal M}_s$ such that a linear code $C$ is a skew GQC code of block length $(m_1, m_2, \ldots, m_l)$ and length $\sum_{i=1}^lm_i$ over $\mathbb{F}_q$ if and only if for each $1\leq k \leq s $ there is a unique left $R$-module $M_k$ of ${\mathcal M}_k$ such that $\phi (C)=M_1\times M_2\times\cdots \times M_s$.* *Proof* Denote $$g^*=g_1^*g_2^*\cdots g_s^*,~ \widehat{g}_k^*=\frac{g^*}{g_k^*},$$ $$\widetilde{g}_{i,k}^*=\frac{x^{m_i}-1}{g_k^{*d_{ik}}}, ~i=1,2,\ldots,l,~k=1,2,\ldots,s.$$ Then there exists a polynomial $w_{i,k}^*\in R$ such that $$\widehat{g}_k^*=w_{i,k}^*\widetilde{g}_{i,k}^*,~i=1,2,\ldots,l,~k=1,2,\ldots,s.$$ Since $g_k^*$ and $\widehat{g}_k^*$ are coprime, there exist polynomials $b_k,~s_k\in R$ such that $b_k\widehat{g}_k^*+s_kg_k^*=1$, which implies that $b_kw_{i,k}^*\widetilde{g}_{i,k}^*+c_kg_k^*=1$ in $R$. Let $\varepsilon_{ik}=b_kw_{i,k}^*\widetilde{g}_{i,k}^*+(x^{m_i}-1)=b_k\widehat{g}_k^*+(x^{m_i}-1)\in {\mathcal R}_i$. Then, by Theorem 2.11, we have
(i) $\varepsilon_{ik}=0$ if and only if ${\rm gcd}(g_k^*, x^{m_i}-1)=1,~k=1,2,\ldots,s.$
\[\]
(ii) $\varepsilon_{i1},\varepsilon_{i2},\ldots,\varepsilon_{is}$ are mutually orthogonal in ${\mathcal R}_i$.
\[\]
(iii) $\varepsilon_{i1}+\varepsilon_{i2}+\cdots+\varepsilon_{is}=1$ in ${\mathcal R}_i$.
\[\]
(iv) Let ${\mathcal R}_{ik}=(\varepsilon_{ik})$ be the principle ideal of ${\mathcal R}_i$ generated by $\varepsilon_{ik}$. Then $\varepsilon_{ik}$ is the identity of ${\mathcal R}_{ik}$ and ${\mathcal R}_{ik}=(b_k\widehat{g}_k^*)$. Hence $R_{ik}=\{ 0\}$ if and only if ${\rm gcd}(g_k^*, x^{m_i}-1)=1$.
\[\]
(v) ${\mathcal R}_i=\bigoplus_{k=1}^s{\mathcal R}_{ij}$.
\[\]
(vi) For each $k=1,2,\ldots,s$, the mapping $\phi_{ik}:~{\mathcal R}_{ik}\rightarrow R/(g_k^{*d_{ik}})$, defined by $$\phi_{ik}:~fb_k\widehat{g}_k^*+(x^{m_i}-1)\mapsto f+(g_k^{*d_{ik}}), ~\mbox{where}\; f\in R,$$ is a well defined isomorphism of rings.
\[\]
(vii) ${\mathcal R}_i=R/(x^{m_i}-1)\cong \bigoplus_{j=1}^sR/(g_j^{*d_{ij}})$.
From (vi), we have a well defined $R$-module isomorphism $\Phi_k$ from $b_k\widehat{g}_k^*{\mathcal R}$ onto $R/(g_k^{*d_{ik}})\times \cdots \times R/(g_k^{*d_{ik}})$, which defined by $$\Phi_k:~(\alpha_1, \ldots, \alpha_l)\mapsto (\phi_{1k}(\alpha_1),\ldots,\phi_{lk}(\alpha_l)),~ \mbox{where}\; \alpha_i\in {\mathcal R}_{ik}, i=1,2,\ldots,l.$$ $\Phi_k$ can introduce a natural $R$-module isomorphism $\mu_k$ from $b_k\widehat{g}_k^*{\mathcal R}$ onto ${\mathcal M}_k$.
For any $c=(c_0,c_1,\ldots,c_l)\in {\mathcal R}$, from (v) we deduce $c=(b_1\widehat{g}_1^*c_1+\cdots+b_s\widehat{g}_s^*c_1, \ldots, \\ b_1\widehat{g}_1^*c_l+\cdots+b_s\widehat{g}_s^*c_l)=b_1\widehat{g}_1^*c+\cdots+b_s\widehat{g}^*_sc$, where $b_k\widehat{g}_k^*c\in b_k\widehat{g}_k^*{\mathcal R}_1\times\cdots\times b_k\widehat{g}_k^*{\mathcal R}_l$ for all $k=1,2,\ldots,s$. Hence ${\mathcal R}=b_1\widehat{g}_1^*{\mathcal R}+\cdots+b_s\widehat{g}_s^*{\mathcal R}$. Let $c_1$, $c_2$, $\ldots$, $c_s\in {\mathcal R}$ satisfying $b_1\widehat{g}_1^*c_1+\cdots+b_s\widehat{g}_s^*c_s=0$. Since $(x^{m_i}-1)\mid g^*$ for all $i=1,2,\ldots,l$, it follows that $g^*{\mathcal R}=\{0\}$. Then for each $k=1,2,\ldots,s$, from $b_k\widehat{g}_k^*+s_kg_k^*=1$, $g^*=g_k^*\widehat{g}_k^*$ and $g^*\mid \widehat{g}_\tau^* \widehat{g}_\sigma^*$ for all $1\leq \tau\neq \sigma \leq s$, we deduce $b_k\widehat{g}_k^*c_k=0$. Hence ${\mathcal R}=\bigoplus_{j=1}^sb_j\widehat{g}_j^*{\mathcal R}$.
Define $\phi:~\beta_1+\beta_2+\cdots+\beta_s\mapsto (\mu_1(\beta_1), \mu_2(\beta_2), \ldots, \mu_s(\beta_s))~\mbox{where}\; \beta_k\in b_k\widehat{g}_k^*{\mathcal R}, k=1,2,\ldots,s$. Then $\phi$ is an $R$-module isomorphism from ${\mathcal R}$ onto ${\mathcal M}_1\times\cdots\times{\mathcal M}_s$. For any left $R$-module $M_j$, it is obvious that $M_1\times\cdots\times M_s$ is a left $R$-submodule of ${\mathcal M}_1\times\cdots\times{\mathcal M}_s$. Therefore there is a unique left $R$-submodule $C$ of ${\mathcal R}$ such that $\phi(C)=M_1\times\cdots\times M_s$. $\Box$
Since ${\mathcal M_k}=(R/(g_k^*))^{n_k}=\bigoplus_{i=1}^lR/(g_k^{*d_{ik}})$ is up to an $R$-module isomorphism, Theorem 3.2 can lead to a canonical decomposition of skew GQC codes as follows.
[**Theorem 3.3** ]{} *Let $C$ be a skew GQC code of block length $(m_1, m_2, \ldots, m_l)$ and length $\sum_{i=1}^lm_i$ over $\mathbb{F}_q$. Then $$C=\bigoplus_{i=1}^sC_i$$ where $C_i$, $1\leq i\leq s$, is a linear code of length $l$ over $R/(g_i^{*d_{ik}})$ and each $j$-th, $1\leq j\leq l$, component in $C_i$ is zero if $d_{ji}=0$ and an element of the ring $R/(g_i^*)$ otherwise.* $\Box$
Let $m_1=m_2=\cdots m_l=m$. Then a skew GQC code $C$ is a *skew quasi-cyclic* (QC) code of length $ml$ over $\mathbb{F}_q$. From Theorems 3.2 and 3.3, we have the following result.
[**Corollary 3.4** ]{} *Let $R=\mathbb{F}_q[x, \sigma]$, ${\rm gcd}(m,q)=1$ and $x^m-1=g_1^*g_2^*\cdots g_s^*$, where $g_1^*, g_2^*, \ldots, g_s^*$ are pairwise coprime monic t.s.m elements in $R$. Then we have\
(i) There is an $R$-module isomorphism $\phi$ from ${\mathcal R}=(R/(x^m-1))^l, ~l\geq 1$, onto $(R/(g_1^*))^l\times (R/(g_2^*))^l\times \cdots \times (R/(g_s^*))^l$.\
(ii) $C$ is a skew QC code of length $ml$ over $\mathbb{F}_q$ if and only if there is a left $R$-submodule $M_i$ of $(R/(g_i^*))^l, ~i=1,2,\ldots,s$, such that $\phi(C)=M_1\times M_2\times\cdots \times M_s$.\
(iii) A skew QC code $C$ of length $ml$ can be decomposed as $C=\bigoplus_{i=1}^sC_i$, where each $C_i$ is a linear code of length $l$ over $R/(g_i^*)$, $i=1,2,\ldots,s$.* $\Box$
A skew GQC code $C$ of block length $(m_1, m_2, \ldots, m_l)$ and length $\sum_{i=1}^lm_i$ is called a *$\rho$-generator* over $\mathbb{F}_q$ if $\rho$ is the smallest positive integer for which there are codewords $c_i(x)=(c_{i,1}(x),c_{i,2}(x), \ldots, c_{i,l}(x))$, $1\leq i \leq \rho$, in $C$ such that $C=Rc_1(x)+Rc_2(x)+\cdots +Rc_\rho(x)$.
Assume that the dimension of each $C_i$, $i=1,2,\ldots,s$, is $k_i$, and set ${\mathcal K}={\rm max} \\{\{ k_i \mid 1\leq i\leq s\}}$. Now by generalizing Theorem 3 of [@Esmaeili], we get
[**Theorem 3.5** ]{} *Let $C$ be a $\rho$-generator skew GQC code of block length $(m_1, m_2, \ldots, m_l)$ and length $\sum_{i=1}^lm_i$ over $\mathbb{F}_q$. Let $C=\bigoplus_{i=1}^sC_i$, where each $C_i$, $i=1,2,\ldots,s$, is with dimension $k_i$ and ${\mathcal K}={\rm max} {\{ k_i \mid 1\leq i\leq s\}}$. Then $\rho={\mathcal K}$. In fact, any skew GQC code $C$ with $C=\bigoplus_{i=1}^sC_i$, where each $C_i$, $i=1,2,\ldots,s$, is with dimension $k_i$ satisfying $\rho={\rm max }_{1\leq i\leq s} k_i$, is a $\rho$-generator skew GQC code.*
*Proof* Let $C$ be a $\rho$-generator skew GQC code generated by the elements $c^{(j)}(x)=(c_1^{(j)}(x), c_2^{(j)}, \ldots, c_l^{(j)}(x)) \in {\mathcal R}, ~j=1,2,\ldots,\rho$. Then for each $i=1,2,\ldots,s$, $C_i$ is spanned as a left $R$-module by $\widetilde{c}^{(j)}(x)=(\widetilde{c}_1^{(j)}(x), \widetilde{c}_2^{(j)}(x), \ldots, \widetilde{c}_l^{(j)}(x))$, where $\widetilde{c}_\nu^{(j)}(x)=c_\nu^{(j)}(x)~({\rm mod} g_i^*)$ if $g_i^*$ is a factor of $x^{m_i}-1$ and $\widetilde{c}_\nu^{(j)}(x)=0$ otherwise, $\nu=1,2,\ldots,l$. Hence $k_i\leq \rho$ for each $i$, and so ${\mathcal K}\leq \rho$. On the other hand, since ${\mathcal K}={\rm max}_{1\leq i\leq s} k_i$, there exist $q_i^{(j)}(x)\in R^l$, $1\leq j \leq {\mathcal K}$, such that $q_i^{(j)}(x)$ span $C_i$, $1\leq i \leq s$, as a left $R$-module. Then, by Theorem 3.3, for each $1\leq j \leq {\mathcal K}$, there exists $q^{(j)}(x)\in C$ such that $q_i^{(j)}(x)=q^{(j)}(x)~({\rm mod}~g_i^*)$ and $C$ is generated by $q_i^{(j)}(x)$, $1\leq j \leq{\mathcal K}$. Hence $\rho \leq {\mathcal K}$, which implies that $\rho={\mathcal K}$. $\Box$
If $C$ is a $1$-generator skew GQC code of block length $(m_1, m_2, \ldots, m_l)$ and length $\sum_{i=1}^lm_i$ over $\mathbb{F}_q$, then by Theorem 3.5, each $C_i$, $i=1,2,\ldots,s$, is either trivial or an $[l, 1]$ linear code over $R/(g_i^*)$. Conversely, any linear code $C$ is a $1$-generator GQC code when each $C_i$, $i=1,2,\ldots,s$, is with dimension at most $1$.
[**Example 3.6** ]{} Let $R=\mathbb{F}_{3^2}[x, \sigma]$, where $\sigma$ is the Frobenius automorphism of $\mathbb{F}_{3^2}$ over $\mathbb{F}_3$. Let ${\mathcal R}=R/(x^4-1)\times R/(x^8-1)$ and $C$ be a $2$-generator skew GQC code of block length $(4,8)$ and length $4+8=12$ generated by $c_1(x)=(x^3-x, x^3-\alpha x)$ and $c_2(x)=(x^3, x^3-2\alpha x)$, where $\alpha$ is a $4$-th primitive element in $\mathbb{F}_{3^2}$ over $\mathbb{F}_{3}$. Since $x^4-1=(x^2-1)(x^2-2)$ and $x^8-1=(x^2-1)(x^2-2)(x^2-\alpha)(x^2-\alpha^2)$, by Theorem 3.2, $${\mathcal R}\cong (R/(x^2-1))^2\times (R/(x^2-1))^2\times R/(x^2-\alpha)\times R/(x^2-\alpha^2).$$ Then up to an $R$-module isomorphism $$\begin{array}{ccc}
{\mathcal R} & \cong & (R/(x^2-1), R/(x^2-1))\\
& \bigoplus & (R/(x^2-2), R/(x^2-2))\\
& \bigoplus & (0, R/(x^2-\alpha))\\
& \bigoplus & (0 , R/(x^2-\alpha^2)).\\
\end{array}$$ This implies that the skew GQC code $C$ can be decomposed into $C=\bigoplus_{i=1}^4C_i$, where $\bullet$ $C_1$ is the $[2,2]$ linear code with the basis $(0, (1-\alpha)x)$ and $(x, (1-2\alpha)x)$ over $R/(x^2-1)$; $\bullet$ $C_2$ is the $[2,2]$ linear code with the basis $(x, (2-\alpha)x)$ and $(2x, (2-2\alpha)x)$ over $R/(x^2-2)$; $\bullet$ $C_3$ is the $[2,1]$ linear code with the basis $(0, 2\alpha x)$ over $R/(x^2-\alpha)$; $\bullet$ $C_4$ is the $[2,1]$ linear code with the basis $(0, (\alpha^2-\alpha)x)$ over $R/(x^2-\alpha^2)$. Let $k_i$ be the dimension of $C_i$, $i=1,2,3,4$. Then $${\rm max}\; k_i=2={\rm the ~number ~of ~generators ~of ~}C.$$
[**4 $1$-generator skew GQC codes**]{} In this section, we discuss some structural properties of $1$-generator skew GQC codes over $\mathbb{F}_q$. Let $R=\mathbb{F}_q[x, \sigma]$ and ${\mathcal R}=R/(x^{m_1}-1)\times R/(x^{m_2}-1)\times\cdots\times R/(x^{m_l}-1)$.
[**Definition 4.1** ]{} *Let $C$ be a $1$-generator skew GQC code generated by $c(x)=(c_1(x),\\ c_2(x), \ldots,c_l(x))\in {\mathcal R}$. The the monic polynomial $h(x)$ of minimum degree satisfying $c(x)h(x)=0$ is called the parity-check polynomial of $C$.*
Let $C$ be a $1$-generator skew GQC code of block length $(m_1, m_2, \ldots, m_l)$ and length $\sum_{i=1}^lm_i$ with the generator $(c_1(x), c_2(x),
\ldots, c_l(x))$, $c_i(x)\in {\mathcal R}_i=R/(x^{m_i}-1), ~i=1,2,\ldots,l$. Define a well defined $R$-homomorphism $\varphi_i$ from ${\mathcal R}$ onto ${\mathcal R}_i$ such that $\varphi_i(c_1(x), c_2(x), \ldots, c_l(x))=c_i(x)$. Then $\varphi_i(C)$ is a skew cyclic code of length $m_i$ generated by $c_i(x)$ in ${\mathcal R}_i$. From Theorem 2.2, we have $\varphi_i(C)=(p_i(x)g_i(x))$, where $g_i(x)$ is a right divisor of $x^{m_i}-1$ and $p_i(x)$ and $h_i(x)=(x^{m_i}-1)/g_i(x)$ are right coprime. Therefore, $h_i(x)=(x^{m_i}-1)/g_i(x)=(x^{m_i}-1)/{\rm gcld}(c_i(x), x^{m_i}-1)$ is the parity-check polynomial of $\varphi_i(C)$. It means that $h(x)={\rm lclm}\{h_1(x), h_2(x), \ldots, h_l(x)\}$ is the parity-check polynomial of $C$. Define a map $\psi$ from $R$ to ${\mathcal R}$ such taht $\psi(a(x))=c(x)a(x)$ . This is an $R$-module homomorphism with the kernel $(h(x))_r$, which implies that $C\cong R/(h(x))_r$. Thus ${\rm dim}(C)={\rm deg}(h(x))$.
As stated above, we have the following result. [**Theorem 4.2**]{} *Let $C$ be a $1$-generator skew GQC code of block length $(m_1, m_2, \ldots, m_l)$ and length $\sum_{i=1}^lm_i$ generated by $c(x)=(c_1(x), c_2(x), \ldots, c_l(x))\in {\mathcal R}$. Then the parity-check polynomial of $C$ is $h(x)={\rm lclm}\{h_1(x), h_2(x), \ldots, h_l(x)\}$, where $h_i(x)=(x^{m_i}-1)/{\rm gcld}(c_i(x), x^{m_i}-1)$, $i=1,2,\ldots,l$, and the dimension of $C$ is equal to the degree of $h(x)$.* $\Box$
Let $h_1(x)$ and $h_2(x)$ be the parity-check polynomials of $1$-generator skew GQC codes $C_1$ and $C_2$, respectively. If $C_1=C_2$, then $h_1(x)=h_2(x)$, which implies that ${\rm deg }(h_1(x))={\rm deg}(h_2(x))$. It means that $R/(h_1(x))_r=R/(h_2(x))_r$. Conversely, suppose $h_1(x)$ and $h_2(x)$ are similar. Then we have $R/(h_1(x))_r\cong R/(h_2(x))_r$, which implies that $C_1= C_2$. Then from the discussion above, we have $C_1=C_2$ if and only if $h_1(x)\sim h_2(x)$, i.e., any $1$-generator skew GQC code has a unique parity-check polynomial up to similarity.
[**Theorem 4.3**]{} *Let $C$ be a $1$-generator skew GQC code of block length $(m_1, m_2,\ldots, m_l)$ and length $\sum_{i=1}^lm_i$ generated by $c(x)=(c_1(x), c_2(x), \ldots,c_l(x))\in {\mathcal R}$. Suppose $h_i(x)$ is given as in Theorem 4.2 and $h(x)={\rm lclm}\{h_1(x),h_2(x),\ldots,h_l(x)\}$. Let $\delta_i$ denote the number of consecutive powers of a primitive $m_i$-th root of unity that among the right zeros of $(x^{m_i}-1)/h_i(x)$. Then\
(i) $d_{\rm H}(C)\geq \sum_{{i}\not \in K}(\delta_i+1)$, where $K\subseteq \{1,2,\ldots,l\}$ is a set of maximum size such that ${\rm lclm}_{i\in K}h_i(x)\neq h(x)$.\
(ii) If $h_1(x)=h_2(x)=\cdots =h_l(x)$, then $d_{\rm H}(C)\geq\sum_{i=1}^l(\delta_i+1)$.*
*Proof* Let $a(x)\in C$ be a nonzero codeword. Then there exists a polynomial $f(x)\in R$ such that $a(x)=f(x)c(x)$. Since for each $i=1,2,\ldots,l$, the $i$-th component is zero if and only if $(x^{m_i}-1)\mid f(x)c_i(x)$, i.e., if and only if $h_i(x)\mid f(x)$. Therefore $a(x)=0$ if and only if $h(x)\mid f(x)$. So $a(x)\neq 0$ if and only if $h(x)\nmid f(x)$. This implies that $c(x)\neq 0$ has the most number of zero blocks whenever $h(x)\neq {\rm lclm}_{i\in K}h_i(x)$, where ${\rm lclm}_{i\in K}h_i(x)\mid f(x)$, and $K$ is a maximal subset of $\{1,2,\ldots,l\}$ having this property. Thus, $d_{\rm H}(C)\geq \sum_{i\notin K}d_i$, where $d_i=d_{\rm H}(\varphi_i(C))\geq \delta_i+1$. Clearly, $K=\emptyset$ if and only if $h_1(x)=h_2(x)=\cdots =h_l(x)$. Therefore, from the discussion above, we have if $h_1(x)=h_2(x)=\cdots =h_l(x)$, then $d_{\rm H}(C)=\sum_{i=1}^ld_i\geq \sum_{i=1}^l(\delta_i+1)$. $\Box$
From Theorems 4.2 and 4.3, we have the following corollary immediately.
[**Corollary 4.4**]{} *Let $C$ be a $1$-generator skew QC code of length $ml$ generated by $c(x)=(c_1(x), c_2(x), \ldots, c_l(x))\in (R/(x^m-1))^l$. Suppose $h_i(x)=(x^m-1)/{\rm gcld}(c_i(x), x^m-1)$, $i=1,2,\ldots,l$, and $h(x)={\rm lclm}\{h_1(x),h_2(x),\ldots,h_l(x)\}$. Then\
(i) The dimension of $C$ is the degree of $h(x)$.\
(ii) Let $\delta_i$ denote the number of consecutive powers of a primitive $m_i$-th root of unity that among the right zeros of $(x^{m}-1)/h_i(x)$. Then $d_{\rm H}(C)\geq \sum_{{i}\not \in K}(\delta_i+1)$, where $K\subseteq \{1,2,\ldots,l\}$ is a set of maximum size such that ${\rm lclm}_{i\in K}h_i(x)\neq h(x)$.\
(iii) If $h_1(x)=h_2(x)=\cdots =h_l(x)$, then $\delta_i=\delta$ for each $i=1,2,\ldots,l$ and $d_{\rm H}(C)\geq l(\delta+1)$.* $\Box$
[**Example 4.5**]{} Let $R=\mathbb{F}_{3^2}[x,\sigma]$, where $\sigma$ is the Frobenius automorphism of $\mathbb{F}_{3^2}$ over $\mathbb{F}_3$. The polynomial $g(x)=x-\alpha^2$ is a right divisor of $x^4-1$, where $\alpha$ is a primitive element of $\mathbb{F}_{3^2}$. Consider the $1$-generator GQC code $C$ of block length $(4,8)$ and length $4+8=12$ generated by $c(x)=(g(x), g(x))$. Then, by Theorem 4.3, $h(x)=(x^8-1)/(x-\alpha^2)$ and $d_H(C)\geq 2$. A generator matrix for $C$ is given as follows $$G=\left(
\begin{array}{cccccccccccc}
-\alpha^2 & 1 & 0 & 0 & -\alpha^2 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & -\alpha^6 & 1 & 0 & 0 & -\alpha^6 & 1& 0 & 0 & 0& 0 & 0\\
0 & 0 & -\alpha^2 & 1 & 0 & 0 & -\alpha^2 & 1 & 0 & 0 & 0 & 0\\
1 & 0 & 0 & -\alpha^6 & 0 & 0 & 0 & -\alpha^6 & 1 & 0 & 0 & 0\\
-\alpha^2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & -\alpha^2 & 1 & 0 & 0\\
0 & -\alpha^6 & 1 & 0 & 0 & 0 & 0& 0 & 0 & -\alpha^6& 1 & 0\\
0 & 0 & -\alpha^2 & 1 & 0 & 0& 0 & 0 & 0 & 0 & -\alpha^2 & 1\\
\end{array}
\right).$$ $C$ is an optimal $[12, 8, 4]$ skew GQC code over $\mathbb{F}_{3^2}$ actually.
[**Example 4.6**]{} Let $R=\mathbb{F}_{3^2}[x,\sigma]$, where $\sigma$ is the Frobenius automorphism of $\mathbb{F}_{3^2}$ over $\mathbb{F}_3$. The polynomial $g(x)=x-\alpha^2$ is a right divisor of $x^4-1$, where $\alpha$ is a primitive element of $\mathbb{F}_{3^2}$. Consider the $1$-generator skew QC code $C$ of length $ml=12$ and index $3$ generated by $c(x)=(g(x), g(x), g(x))$ over $\mathbb{F}_{3^2}$. Then $h(x)=h_1(x)=h_2(x)=h_3(x)=(x^4-1)/(x-\alpha^2)$. Thus, by Corollary 4.4, $C$ is a skew QC code of length $12$ and index $3$ with dimension $3$ and the minimum Hamming distance at least $3\times 2=6$. A generator matrix for $C$ is given as follows
$$G=\left(
\begin{array}{cccccccccccc}
-\alpha^2 & 1 & 0 & 0 & -\alpha^2 & 1 & 0 & 0 & -\alpha^2 & 1 & 0 & 0\\
0 & -\alpha^6 & 1 & 0 & 0 & -\alpha^6 & 1& 0 & 0 & -\alpha ^6& 1 & 0\\
0 & 0 & -\alpha^2 & 1 & 0 & 0& -\alpha^2 & 1 & 0 & 0 & -\alpha ^2 & 1\\
\end{array}
\right).$$
From the generator matrix $G$, we see that $C$ is an $[12, 3, 6]$ skew QC code over $\mathbb{F}_{3^2}$.
[**5 Skew QC codes**]{} Skew quasi-cyclic (QC) codes as a special class of skew generalized quasi-cyclic (GQC) codes, have the similar structural properties to skew GQC codes such as Corollary 3.4 and Corollary 4.4. But in this section, we use another view presented in [@Lally] to research skew QC codes over finite fields. The dual codes of skew QC codes are also discussed briefly.
For convenience, we write an element $a\in \mathbb{F}_q^{lm}$ as a $m$- tuple $a=(a_0, a_1, \ldots, a_{m-1})$, where $a_i=(a_{i,0}, a_{i,1}, \ldots, a_{i,
(l-1)}) \in \mathbb{F}_q^l$. Let the map $T_{\sigma, l}$ on $\mathbb{F}_q^{lm}$ be defined as follows
$$T_{\sigma, l}(a_0, a_1, \ldots, a_{m-1})=(\sigma(a_{m-1}), \sigma(a_0), \ldots, \sigma(a_{m-2})),$$ where $\sigma(a_i)=(\sigma(a_{i,0}), \sigma(a_{i,1}), \ldots, \sigma(a_{i,l-1}))$. Define a one-to-one correspondence
$$\eta: \mathbb{F}_q^{lm}\rightarrow {\mathcal R}^l,$$ $$(a_{0,0}, a_{0,1}, \ldots, a_{0,l-1},a_{1,0}, a_{1,1}, \ldots, a_{1,l-1}, \ldots, a_{m-1,0}, a_{m-1,1}, \ldots, a_{m-1,l-1})$$ $$\mapsto a(x)=(a_0(x), a_1(x), \ldots, a_{l-1}(x)),$$ where $a_j(x)=\sum_{i=0}^{m-1}a_{i,j}x^i$ for $j=0,1,\ldots,l-1$. Then a skew QC code $C$ of length $lm$ with index $l$ defined as in Corollary 3.4 is equivalent to a linear code of length $lm$, which is invariant under the map $T_{\sigma,l}$.
Let $v=(v_{0,0}, v_{0,1},\ldots,v_{0,l-1}, v_{1,0}, v_{1,1}, \ldots,v_{1,l-1},\ldots, v_{m-1,0}, v_{m-1,1}, \ldots, v_{m-1,l-1} )\in
\mathbb{F}_q^{ml}$. Let $\{1, \xi, \xi^2, \ldots, \xi^{l-1}\}$ be a basis of $\mathbb{F}_{q^l}$ over $\mathbb{F}_q$. Define an isomorphism between $\mathbb{F}_q^{ml}$ and $\mathbb{F}_{q^l}^m$, for $i=0,1,\ldots,m-1$, associating each $l$-tuple $(v_{i,0}, v_{i,1}, \ldots,v_{i,l-1})$ with the element $v_i\in
\mathbb{F}_{q^l}$ where $v_i=v_{i,0}+v_{i,1}\xi+\cdots+v_{i,l-1}\xi^{l-1}$. Then every element in $\mathbb{F}_q^{ml}$ is a one-to-one correspondence with an element in $\mathbb{F}_{q^l}^m$. The operator $T_{\sigma,l}$ on $(v_{0,0}, v_{0,1},\ldots,v_{0,l-1}, v_{1,0},v_{1,1}, \ldots, v_{1,l-1},\ldots, v_{m-1,0}, v_{m-1,1}, \ldots,v_{m-1,l-1} )\in \mathbb{F}_q^{ml}$ corresponds to the element $(\sigma(v_{m-1}), \sigma(v_0),\ldots, \sigma(v_{m-2}))\in \mathbb{F}_{q^l}^m$ under the above isomorphism. The vector $v\in \mathbb{F}_q^{ml}$ can be associated with the polynomial $v(x)=v_0+v_1x+\cdots+v_{m-1}x^{m-1}\in \widetilde{R}=\mathbb{F}_{q^l}[x, \sigma]$. Clearly, there is an $R/(x^m-1)$-module isomorphism between $\mathbb{F}_q^{ml}$ and $\widetilde{ R}[x]/( x^m-1)$ that is defined by $\phi(v)=v(x)$. It follows that there is a one-to-one correspondence between the left $R/( x^m-1)$-submodule of $\widetilde{ R}/( x^m-1)$ and the skew QC code of length $ml$ with index $l$ over $\mathbb{F}_q$. In addition, a skew QC code of length $ml$ with index $l$ over $\mathbb{F}_q$ can also be regarded as an $R$-submodule of $\widetilde{ R}/( x^m-1)$ because of the equivalence of $\mathbb{F}_q^{ml}$ and $\widetilde{ R}/( x^m-1)$.
Let $C$ be a skew QC code of length $ml$ with index $l$ over $\mathbb{F}_q$, and generated by the elements $v_1(x), v_2(x),\ldots, v_\rho(x)\in \widetilde{R}/( x^m-1)$ as a left $R/(x^m-1)$-submodule of $\widetilde{R}[x]/( x^m-1)$. Then $C=\{a_1(x)v_1(x)+a_2(x)v_2(x)+\cdots+a_\rho(x)v_\rho(x)| a_i(x)\in R/( x^m-1 ), i=1,2,\ldots,\rho\}$. As discussed above, $C$ is also an $R$-submodule of $\widetilde{R}/( x^m-1)$. As an $R$-submodule of $\widetilde{R}/( x^m-1)$, $C$ is generated by the following set $\{v_1(x), xv_1(x), \ldots, x^{m-1}v_1(x),v_2(x), xv_2(x), \ldots,x^{m-1}v_2(x), \ldots,v_\rho(x), xv_\rho(x), \ldots,\\ x^{m-1}v_\rho(x)\}$.
Since $R/( x^m-1)$ is a subring of $\widetilde{R}[x]/( x^m-1)$ and $C$ is a left $R/( x^m-1)$-submodule of $\widetilde{R}/( x^m-1)$, $C$ is in particular a left submodule of an $\widetilde{R}/( x^m-1)$-submodule of $\widetilde{R}/( x^m-1)$, i.e., the skew cyclic code $ \widetilde{C}$ of length $m$ over $\widetilde{R}$. Therefore, $d_H(C)\geq d_H(\widetilde{C})$, where $d_H(C)$ and $d_H(\widetilde {C})$ are the minimum Hamming distance of $C$ and $\widetilde{C}$, respectively. Lally [@Lally Theorem 5] has obtained another lower bound on the minimum Hamming distance of the QC code over finite fields. In the following, we generalized these results to skew QC codes.
[**Theorem 5.1**]{} *Let $C$ be a $\rho$-generator skew QC code of length $ml$ with index $l$ over $\mathbb{F}_q$ and generated by the set $\{v_i(x)=\widetilde{v}_{i,0}+\widetilde{v}_{i,1}x+\cdots +\widetilde{}v_{i,m-1}x^{m-1}, i=1,2,\ldots,\rho\}\subseteq \widetilde{R}/(x^m-1)$. Then $C$ has lower bound on minimum Hamming distance given by $$d_H(C)\geq d_H(\widetilde{C})d_H(B),$$ where $\widetilde{C}$ is a skew cyclic code of length $m$ over $\widetilde{ R}$ with generator polynomial ${\rm gcld}(v_1(x), \\ v_2(x), \ldots, v_\rho (x), x^m-1)$ and $B$ is a skew linear code of length $l$ generated by $\{{\mathcal V}_{i,j}, i=1,2,\ldots, \rho, j=0,1,\ldots, m-1\}\subseteq \mathbb{F}_q^l$ where each ${\mathcal V}_{i,j}$ is the vector corresponding to the coefficients $\widetilde{v}_{i,j} \in \mathbb{F}_{q^l}$ with respect to a $\mathbb{F}_q$-basis $\{1, \xi, \ldots, \xi^{l-1}\}$.* $\Box$
Define the Euclidean inner product of $u, v\in \mathbb{F}_q^{lm}$ by $$u\cdot v=\sum_{i=0}^{m-1}\sum_{j=0}^{l-1}u_{i,j}v_{i,j}.$$ Let $C$ be a skew QC code of length $lm$ with index $l$, $u\in C$ and $v\in C^\perp$. Since $\sigma^m=1$, we have $u\cdot T_{\sigma,l}(v)=\sum_{i=0}^{m-1}u_i\cdot \sigma(v_{i+m-1})=\sum_{i=0}^{m-1}\sigma(\sigma^{m-1}(u_i)\cdot v_{i+m-1})=\sigma(T_{\sigma,l}^{m-1}(u)\cdot v)=\sigma(0)=0$, where $i+m-1$ is taken modulo $m$. Hence $T_{\sigma, l}(v)\in C^\perp$, which implies that the dual code of skew QC code $C$ is also a skew QC code of the same index.
We define a conjugation map $^-$ on $R$ such that $\overline{ax^i}=\sigma^{-i}x^{m-i}$, for $ax^i\in R$. On $R^l$, we define the Hermitian inner product of $a(x)=(a_0(x), a_1(x), \ldots, a_{l-1}(x))$ and $b(x)=(b_0(x), b_1(x), \ldots, b_{l-1}(x))\in R^l$ by $$\langle a(x), b(x)\rangle=\sum_{i=0}^{l-1}a(x)\cdot \overline{b_i(x)}.$$ By generalizing Proposition 3.2 of [@Ling1], we get
[**Proposition 5.2**]{} *Let $u, v \in \mathbb{F}_q^{lm}$ and $u(x)$ and $v(x)$ be their polynomial representations in $R^l$, respectively. Then $T_{\sigma,l}^k(u)\cdot v=0$ for all $0\leq k \leq m-1$ if and only if $\langle u(x), v(x)\rangle=0$.* $\Box$
Let $C$ be a skew QC code of length $lm$ with index $l$ over $\mathbb{F}_q$. Then, by Theorem 5.1, $$C^\perp=\{v(x)\in R^l\mid \langle c(x), v(x)\rangle=0,~\forall c(x)\in C\}.$$ Furthermore, by Corollary 3.4 (iii), we have $C^\perp=\bigoplus_{i=1}^sC_i^\perp$.
In [@Ling3], some results for $\rho$-generator QC codes and their duals over finite fields are given. These results can also be generalized to skew $\rho$-generator QC codes over finite fields. By generalizing Corollary 6.3, Corollary 6.4 in [@Ling3] and Theorem 3.5 in this paper, we get the following result.
[**Theorem 5.3**]{} *Let $C$ be a $\rho$-generator skew QC code of length $lm$ with index $l$ over $\mathbb{F}_q$. Let $C=\bigoplus_{i=1}^sC_i$, where each $C_i$, $i=1,2,\ldots,s$, is with dimension $k_i$. Then\
(i) $C$ is a ${\mathcal K}$-generator skew QC code and $C^\perp$ is an $(l-{\mathcal K}')$-generator skew QC code, where ${\mathcal K}={\rm max}_{1\leq i\leq s} k_i$ and ${\mathcal K}'={\rm min}_{1\leq i\leq s} k_i$.\
(ii) Let $l\geq 2$. If $C^\perp$ is also an $\rho$-generator skew QC code, then ${\rm min}_{1\leq i\leq s} k_i=l-\rho$ and $l\leq 2\rho$.\
(iii) If $C$ is a self-dual $\rho$-generator skew QC code, then $l$ is even and $l\leq 2\rho$.* $\Box$
For a $1$-generator skew QC code of length $lm$ with index $l$ and the canonical decomposition $C=\bigoplus_{i=1}^sC_i$, $C^\perp$ is also a $1$-generator skew QC code if and only if $l=2$ and ${\rm dim}(C_i)=1$ for each $i=1,2,\ldots,s$.
[**6 Conclusion**]{} The structural properties of skew cyclic codes and skew GQC codes over finite fields are studied. Using the factorization theory of ideals, we give the Chinese Remainder Theorem in the skew polynomial ring $\mathbb{F}_q[x, \sigma]$, which leads to a canonical decomposition of skew GQC codes. Moreover, we give some characteristics of $\rho$-generator skew GQC codes. For $1$-generator skew GQC codes, we give their parity-check polynomials and dimensions. A lower bound on the minimum Hamming distance of $1$-generator skew GQC codes is given. These special codes may lead to some good linear codes over finite fields. Finally, skew QC codes are also discussed in details.
In this paper, we restrict on the condition that the order of $\sigma$ divides each $m_i$, $i=1,2,\ldots,l$. If we remove this condition, then the polynomial $x^m-1$ may not be a central element. This implies that the set $R/(x^m-1)$ is not a ring anymore. In this case, the cyclic code in $R/(x^m-1)$ will not be an ideal. It is just a left $R$-submodule, and we call it a *module skew cyclic code*. A GQC code in ${\mathcal R}$ is also a left $R$-submodule of ${\mathcal R}$, and we call it a *module skew GQC code*. Most of our results on skew cyclic codes and skew GQC codes in this paper depend on the fact that $x^m-1$ is a central element of $R$. Since in the module skew case this is not ture anymore, some results stated in this paper cannot be held. Therefore, the structural properties of module skew cyclic and module skew GQC codes are also interesting open problems for further consideration. Another interesting open problem is to find some new or good linear codes over finite fields from skew GQC codes.
[**Acknowledgments**]{} This research is supported by the National Key Basic Research Program of China (973 Program Grant No. 2013CB834204), the National Natural Science Foundation of China (Nos. 61171082, 60872025, 10990011).
[s21]{} Abualrub, T., Aydin, N., Siap, I.: *On the construction of skew quasi-cyclic codes*. IEEE Trans. Inform. Theory. 56, 2081-2090(2010).
Aydin, N.: *Quasi-Cyclic Codes over $\mathbb{Z}_4$ and Some New Binary Codes*. IEEE Trans. Inform. Theory. 48, 2065-2069(2002).
Abualrub, T., Siap, I.: *Cyclic codes over the ring $\mathbb{Z}_2+u\mathbb{Z}_2$ and $\mathbb{Z}_2+u\mathbb{Z}_2+u^2\mathbb{Z}_2$.* Des. Codes and Cryptography. 42, 273-287(2007).
Bhaintwal, M., Wasan, S.: *On quasi-cyclic codes over $\mathbb{Z}_q$*. Appl. Algebra Eng. Commun. Comput. 20, 459-480(2009).
Bhaintwal, M.: *Skew quasi-cyclic codes over Galois rings*. Des. Codes Cryptogr. 62, 85-101(2012).
Boucher, D., Geisemann, W., Ulmer, F.: *Skew-cyclic codes*. Appl. Algebra Eng. Commun. Comput. 18, 379-389(2007).
Boucher, D., Solé, P., Ulmer, F.: *Skew constacyclic codes over Galois rings*. Adv. Math. Commun. 2, 273-292(2008).
Boucher, D., Ulmer, F.: *Coding with skew polynomial rings*. J. Symbolic Comput. 44, 1644-1656(2009).
Cao, Y.: *Structural properties and enumeration of $1$-generator generalized quasi-cyclic codes*. Des. Codes Cryptogr, 60, 67-79(2011)
Cao, Y.: *Generalized quasi-cyclic codes over Galois rings: structural properties and enumeratio*n. Appl. Algebra Eng. Commun. Comput. 22, 219-233(2011)
Chaussade, L., Loidreau, P: *Skew codes of prescribed distance or rank*. Des. Codes and Cryptogr. 50, 267-284(2009)
Conan, J., Séguin, G.: *Structural Properties and Enumeration of Quasi Cyclic Codes*. Appl. Algebra Eng. Commun. Comput. 4, 25-39(1993).
Esmaeili, M., Yari, S.: *Generalized quasi-cyclic codes: structural properties and codes construction*. Appl. Algebra Eng. Commun. Comput. 20, 159-173(2009).
Jacobson, N: *Theory of Rings*. Ann. Math. Sco. New York(1943).
Jacobson, N: *Finite Dimensional Division Algebras over Fields*. Springer, New York(1996).
Lally, K.: *Quasicyclic codes of index $l$ over $\mathbb{F}_q$ viewed as $\mathbb{F}_q[x]$-submodules of $\mathbb{F}_{q^l}[x]/(x^m-1)$.* in Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Lecture Notes in Comput. Sci. 2643, Springer-Verlag, Berlin, Heidelberg, 244-253(2003).
Ling, S., Solé, P.: *On the algebra structure of quasi-cyclic codes I: finite fields.* IEEE Trans. Inform. Theory. 47, 2751-2760(2001).
Ling, S., Solé, P.: *On the algebra structure of quasi-cyclic codes II: chain rings.* Des. Codes and Cryptography. 30 (1), 113-130(2003).
Ling, S., Solé, P.: *On the algebra structure of quasi-cyclic codes III: generator theory.* IEEE Trans. Inform. Theory. 51, 2692-2700(2005).
McDonald, B. R.: *Finite Rings with Identity*. Marcel Dekker, New York(1974).
Siap, I., Kulhan, N.: *The structure of generalized quasi-cyclic codes.* Appl. Math. E-Notes. 5, 24-30(2005).
Siap, I., Abualrub, T., Yildiz, B. : *One generator quasi-cyclic codes over $\mathbb{F}_2+u\mathbb{F}_2$.* J. Frank. Inst. 349, 284-292(2012).
| ArXiv |
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abstract: 'Suppose that $E$ and $E''$ denote real Banach spaces with dimension at least $2$ and that $D\varsubsetneq E$ and $D''\varsubsetneq E''$ are uniform domains with homogeneously dense boundaries. We consider the class of all $\varphi$-FQC (freely $\varphi$-quasiconformal) maps of $D$ onto $D''$ with bilipschitz boundary values. We show that the maps of this class are $\eta$-quasisymmetric. As an application, we show that if $D$ is bounded, then maps of this class satisfy a two sided Hölder condition. Moreover, replacing the class $\varphi$-FQC by the smaller class of $M$-QH maps, we show that $M$-QH maps with bilipschitz boundary values are bilipschitz. Finally, we show that if $f$ is a $\varphi$-FQC map which maps $D$ onto itself with identity boundary values, then there is a constant $C\,,$ depending only on the function $\varphi\,,$ such that for all $x\in D$, the quasihyperbolic distance satisfies $k_D(x,f(x))\leq C$.'
address:
- 'Yaxiang. Li, Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China'
- 'Matti. Vuorinen, Department of Mathematics and Statistics, University of Turku, FIN-20014 Turku, Finland'
- 'Xiantao. Wang, Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China'
author:
- 'Y. Li'
- 'M. Vuorinen'
- 'X. Wang ${}^{~\mathbf{*}}$'
title: Quasiconformal maps with bilipschitz or identity boundary values in Banach spaces
---
Introduction and main results {#intro}
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Many results of classical function theory have their counterparts in the context of quasiconformal maps in the Euclidean $n$-dimensional space $\mathbb{R}^n$. J. Väisälä [@Vai6-0; @Vai6; @Vai5] has developed a theory of quasiconformality in the Banach space case which differs from the finite dimensional theory in many respects because tools such as conformal invariants and measures of sets are no longer available. These classical tools are replaced by fundamental objects from metric space geometry such as curves, their lengths, and approximately length minimizing curves. Väisälä used these notions in the setup of several metric space structures on the same underlying Banach space and developed effective methods based on these basic notions. In addition to the norm metric he considered two hyperbolic type metric structures, the quasihyperbolic metric and the distance ratio metric. The quasihyperbolic metric $k_D$ of a domain $D$ has a key role as quasiconformality is defined in terms of it in the Banach space case. Only recently some basic properties of quasihyperbolic metric have been studied: the convexity of quasihyperbolic balls was studied by R. Klén [@k; @k2], A. Rasila and J. Talponen [@rt; @krt], Väisälä [@Vai6']. Rasila and Talponen also proved the smoothness of quasihyperbolic geodesics in [@rt2] applying now stochastic methods.
Given domains $D, D'$ in Banach spaces $E$ and $E'$, respectively, our basic problem is to study the class of homeomorphisms $f\in QC^L_{\varphi}(D,D')$, where $$\label{intro-eq-1}
QC^L_{\varphi}(D,D') = \{f: \overline{D}\to \overline{D}'\; { \rm homeo}\;\Big| f|_{D} \;{\rm is}\; {\rm a}\;\varphi{\rm -FQC}\;{\rm map}\; {\rm and}\; f|_{\partial D}\; {\rm is}\; L{\rm -bilipschitz}\}\, .$$ For the definition of $\varphi$-FQC and $L$-bilipschitz maps see Section \[sec-2\]. The class $QC^L_{\varphi}(D,D')$ is very wide and many particular cases of interest are obtained by choosing $D, D', \varphi, L$ in a suitable way as we will see below.
Our first result deals with the case when both $D$ and $D'$ are uniform domains. In this case we prove that the class consists of quasisymmetric maps. More precisely, we prove the following theorem.
\[thm1.2\] Let $D\subsetneq E$, $D'\subsetneq E'$ be $c$-uniform domains. If $f\in QC^L_{\varphi}(D,D')$, then $f$ is $\eta$-QS in $\overline{D}$ with $\eta$ depending on $c$, $L$ and $\varphi$ only.
Applying this result to the case of a bounded domain $D$ we obtain the second result. Recall that in the case of ${\mathbb R}^n$ results of this type have been proved by R. Näkki and B. Palka [@np]. For the definitions, see Section \[sec-2\].
\[thm1.3\] Let $D\subsetneq E$, $D'\subsetneq E'$ be $c$-uniform domains. If $f\in QC^L_{\varphi}(D,D')$ and $D$ is bounded, then for all $x,y\in D$, $$\frac{|x-y|^{1/{\alpha}}}{C}\leq|f(x)-f(y)|\leq C|x-y|^{\alpha},$$ where $C\geq 1$ and $\alpha\in(0,1)$ depend on $c$, $L$, $\varphi$ and $\operatorname{diam}(D)$.
Our third result concerns the case when both $D$ and $D'$ are uniform domains and $\varphi(t)=Mt$ for some fixed $M \ge 1\,.$ We also require a density condition of the boundary of a domain. This $(r_1,r_2)$-HD condition will be defined in Section 2.
\[thm1.4\] Let $D\subsetneq E$, $D'\subsetneq E'$ be $c$-uniform domains and the boundary of $D$ be $(r_1,r_2)$-HD. If $f\in QC^L_{\varphi}(D,D')$ with $\varphi(t)=Mt$, then $f$ is $M'$-bilipschitz in $\overline{D}$, where $M'$ depends only on $c$, $r_1$, $r_2$, $L$ and $M$.
Our fourth result deals with the case when $D=D'$, $L=1$ and, moreover, the boundary mapping $f|_{\partial D}:\partial D\to\partial D$ is the identity. This problem has been studied very recently in [@MV; @M2; @VZ]. Originally, the problem was motivated by Teichmüller’s work on plane quasiconformal maps [@K; @T] and then extended to the higher dimensional case by several authors: [@AV], [@M2], [@MV; @VZ]. Our result is as follows.
\[thm1.1\] Let $D\subsetneq E$ be a $c$-uniform domain with $(r_1,r_2)$-HD boundary. If $f$ is a $\varphi$-FQC map which maps $D$ onto itself with identity boundary values, then for all $x\in D$, $$k_D(x, f(x))\leq C,$$ where $C$ is a constant depending on $r_1$, $r_2$, $c$ and $\varphi$ only.
For the case $n=2$, when $D$ is the unit disk, the sharp bound is due to Teichmüller [@K; @T]. For the case of unit ball in $\mathbb{R}^n, n\ge 2,$ nearly sharp results appear in [@MV; @VZ]. In both of these cases one uses the hyperbolic metric in place of the quasihyperbolic metric.
We do not know whether there are sharp results for the Banach spaces, too. For instance, it is an open problem whether Theorem \[thm1.1\] could be refined for the case $D=\mathbb{B}$, the unit ball, to the effect that $C\rightarrow 0$ when $\varphi$ approaches the identity map.
The organization of this paper is as follows. In Section \[sec-4\], we will prove Theorems \[thm1.2\], \[thm1.3\], \[thm1.4\] and \[thm1.1\]. In Section \[sec-2\], some preliminaries are stated.
Preliminaries {#sec-2}
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We adopt mostly the standard notation and terminology from Väisälä [@Vai6-0; @Vai5]. We always use $E$ and $E'$ to denote real Banach spaces with dimension at least $2$. The norm of a vector $z$ in $E$ is written as $|z|$, and for every pair of points $z_1$, $z_2$ in $E$, the distance between them is denoted by $|z_1-z_2|$, the closed line segment with endpoints $z_1$ and $z_2$ by $[z_1,
z_2]$. Moreover, we use $\mathbb{B}(x, r)$ to denote the ball with center $x\in E$ and radius $r$ $(> 0)$, and its boundary and closure are denoted by $\mathbb{S}(x,\; r)$ and $\overline{\mathbb{B}}(x,\; r)$, respectively. In particular, we use $\mathbb{B}$ to denote the unit ball $\mathbb{B}(0,\; 1)$. The one-point extension of $E$ is the Hausdorff space $\dot{E}=E\cup\{\infty\}$, where the neighborhoods of $\infty$ are the complements of closed bounded sets of $E$. The boundary $\partial A$ and the closure $\overline{A}$ of a set $A\subset E$ are taken in $\dot{E}$.
The [*quasihyperbolic length*]{} of a rectifiable arc or a path $\alpha$ in the norm metric in $D$ is the number (cf. [@Geo; @GP; @Vai6-0]):
$$\ell_k(\alpha)=\int_{\alpha}\frac{|dz|}{d_{D}(z)},$$ where $d_D(z)$ denotes the distance from $z$ to the boundary $\partial D$ of $D$.
For each pair of points $z_1$, $z_2$ in $D$, the [*quasihyperbolic distance*]{} $k_D(z_1,z_2)$ between $z_1$ and $z_2$ is defined in the usual way: $$k_D(z_1,z_2)=\inf\ell_k(\alpha),$$ where the infimum is taken over all rectifiable arcs $\alpha$ joining $z_1$ to $z_2$ in $D$.
For each pair of points $z_1$, $z_2$ in $D$, the [*distance ratio metric*]{} $j_D(z_1,z_2)$ between $z_1$ and $z_2$ is defined by $$j_D(z_1,z_2)=\log\Big(1+\frac{|z_1-z_2|}{\min\{d_D(z_1),d_D(z_2)\}}\Big).$$
For all $z_1$, $z_2$ in $D$, we have (cf. [@Vai6-0])
$$\label{eq(0000)} k_{D}(z_1, z_2)\geq
\inf\left\{\log\Big(1+\frac{\ell(\alpha)}{\min\{d_{D}(z_1), d_{D}(z_2)\}}\Big)\right\}\geq j_D(z_1, z_2)$$
$$\geq
\Big|\log \frac{d_{D}(z_2)}{d_{D}(z_1)}\Big|,$$ where the infimum is taken over all rectifiable curves $\alpha$ in $D$ connecting $z_1$ and $z_2$. Moreover, if $|z_1-z_2|\le d_D(z_1)$, we have [@Vai6-0; @Mvo1] $$\label{vu1}
k_D(z_1,z_2)\le \log\Big( 1+ \frac{
|z_1-z_2|}{d_D(z_1)-|z_1-z_2|}\Big).$$
Gehring and Palka [@GP] introduced the quasihyperbolic metric of a domain in $\mathbb{R}^n$ and it has been recently used by many authors in the study of quasiconformal mappings and related questions [@HIMPS; @krt; @rt] etc.
A domain $D$ in $E$ is called $c$-[*uniform*]{} in the norm metric provided there exists a constant $c$ with the property that each pair of points $z_{1},z_{2}$ in $D$ can be joined by a rectifiable arc $\alpha$ in $ D$ satisfying (see [@Martio-80; @Vai; @Vai4])
1. \[wx-4\] ${\displaystyle}\min_{j=1,2}\ell (\alpha [z_j, z])\leq c\,d_{D}(z)
$ for all $z\in \alpha$, and
2. \[wx-5\] $\ell(\alpha)\leq c\,|z_{1}-z_{2}|$,
where $\ell(\alpha)$ denotes the length of $\alpha$ and $\alpha[z_{j},z]$ the part of $\alpha$ between $z_{j}$ and $z$. Moreover, $\alpha$ is said to be a [*uniform arc*]{}.
In [@Vai6], Väisälä characterized uniform domains as follows.
\[pre-lem-1\] For a domain $D$, the following are quantitatively equivalent:
1. $D$ is a $c$-uniform domain;
2. $k_D(z_1,z_2)\leq c'\; j_D(z_1,z_2)$ for all $z_1,z_2\in D$;
3. $k_D(z_1,z_2)\leq c'_1\; j_D(z_1,z_2) +d$ for all $z_1,z_2\in D$.
In the case of domains in $ {\mathbb R}^n \,,$ the equivalence of items (1) and (3) in Theorem D is due to Gehring and Osgood [@Geo] and the equivalence of items (2) and (3) due to Vuorinen [@Mvo1]. Many of the basic properties of this metric may be found in [@Geo; @krt; @rt; @Vai6-0; @Vai6].
In [@Vai5], Väisälä proved the following examples for some special uniform domain.
\(1) Each ball $B\subset E$ is $2$-uniform;
\(2) Every bounded convex domain $G\subset E$ is uniform;
\(3) Half space $H\subset E$ is $c$-uniform for all $c>2$.
Suppose $G\varsubsetneq E\,,$ $G'\varsubsetneq
E'\,,$ and $M \ge 1\,.$ We say that a homeomorphism $f: G\to G'$ is [*$M$-bilipschitz*]{} if $$|x-y|/M \leq |f(x)-f(y)|\leq M\,|x-y|$$ for all $x$, $y\in G$, and [*$M$-QH*]{} if $$k_{G}(x,y)/M\leq k_{G'}(f(x),f(y))\leq M\,k_{G}(x,y)$$ for all $x$, $y\in G$.
Clearly, if $f$ is $M$-bilipschitz or $M$-QH, then also $f^{-1}$ has the same property.
Let $G\not=E$ and $G'\not=E'$ be metric spaces, and let $\varphi:[0,\infty)\to [0,\infty)$ be a growth function, that is, a homeomorphism with $\varphi(t)\geq t$. We say that a homeomorphism $f: G\to G'$ is [*$\varphi$-semisolid*]{} if $$k_{G'}(f(x),f(y))\leq \varphi(k_{G}(x,y))$$ for all $x$, $y\in G$, and [*$\varphi$-solid*]{} if both $f$ and $f^{-1}$ satisfy this condition.
We say that $f$ is [*fully $\varphi$-semisolid*]{} (resp. [*fully $\varphi$-solid*]{}) if $f$ is $\varphi$-semisolid (resp. $\varphi$-solid) on every subdomain of $G$. In particular, when $G=E$, the corresponding subdomains are taken to be proper ones. Fully $\varphi$-solid maps are also called [*freely $\varphi$-quasiconformal maps*]{}, or briefly [*$\varphi$-FQC maps*]{}.
Clearly, if $f$ is freely $\varphi$-quasiconformal, then so is $f^{-1}\,.$
If $E=\mathbb{R}^n=E'$, then $f$ is $FQC$ if and only if $f$ is quasiconformal (cf. [@Vai6-0]). See [@Vai1; @Mvo1] for definitions and properties of $K$-quasiconformal maps, or briefly $K$-QC maps.
Let $X$ be a metric space and $\dot{X}=X\cup \{\infty\}$. By a triple in $X$ we mean an ordered sequence $T=(x,a,b)$ of three distinct points in $X$. The ratio of $T$ is the number $$\rho(T)=\frac{|a-x|}{|b-x|}.$$ If $f: X\to Y$ is an injective map, the image of a triple $T=(x,a,b)$ is the triple $fT=(fx,fa,fb)$.
Let $X$ and $Y$ be two metric spaces, and let $\eta: [0, \infty)\to [0, \infty)$ be a homeomorphism. An embedding $f: X\to Y$ is said to be [*$\eta$-quasisymmetric*]{}, or briefly $\eta$-$QS$, if $\rho(f(T))\leq \eta(\rho(T))$ for each triple $T$ in $X$.
It is known that an embedding $f: X\to Y$ is $\eta$-$QS$ if and only if $\rho(T)\leq t$ implies that $\rho(f(T))\leq \eta(t)$ for each triple $T$ in $X$ and $t\geq 0$ (cf. [@TV]).
A quadruple in $X$ is an ordered sequence $Q=(a,b,c,d)$ of four distinct points in $X$. The cross ratio of $Q$ is defined to be the number $$\tau(Q)=|a,b,c,d|=\frac{|a-b|}{|a-c|}\cdot\frac{|c-d|}{|b-d|}.$$ Observe that the definition is extended in the well known manner to the case where one of the points is $\infty$. For example, $$|a,b,c,\infty|= \frac{|a-b|}{|a-c|}.$$ If $X_0 \subset \dot{X}$ and if $f: X_0\to \dot{Y}$ is an injective map, the image of a quadruple $Q$ in $X_0$ is the quadruple $fQ=(fa,fb,fc,fd)$.
Let $X$ and $Y$ be two metric spaces and let $\eta: [0, \infty)\to [0, \infty)$ be a homeomorphism. An embedding $f: X\to Y$ is said to be [*$\eta$-quasimöbius*]{} (cf. [@Vai2]), or briefly $\eta$-$QM$, if the inequality $\tau(f(Q))\leq \eta(\tau(Q))$ holds for each quadruple $Q$ in $X$.
Observe that if $\infty\in X$ and if $f:X\to Y$ is $\eta$-quasimöbius with $f(\infty)=\infty$, then $f$ is $\eta$-quasisymmetric (see [@Vai5 6.18]). Conversely, the following result holds.
\[pre-lem-2\]$($[@Vai2 Theorem 3.12]$)$ Suppose that $X$ and $Y$ are bounded spaces, that $\lambda>0$, that $z_1,z_2,z_3\in X$, and that $f:X \to Y$ is $\theta$-quasimöbius such that $$|z_i-z_j|\geq \operatorname{diam}(X)/{\lambda}\, \mbox{,}\, |f(z_i)-f(z_j)|\geq\operatorname{diam}(Y)/{\lambda}$$ for $i\neq j$. Then $f$ is $\eta$-quasisymmetric with $\eta=\eta_{\theta,\lambda}$.
Concerning the relation between the class of uniform domains and quasimöbius maps, Väisälä proved the following result.
\[pre-lem-3\] Suppose that $D\varsubsetneq E$ and $D'\varsubsetneq E'$, that $D$ and $D'$ are $c$-uniform domain, and that $f:D\to D'$ is a $\varphi$-FQC map. Then $f$ extends to a homeomorphism $\overline{f}: \overline{D}\to
\overline{D}'$ and $\overline{f}$ is $\theta_1$-QM in $\overline{ D}$.
Finally we introduce the concept of homogeneous density from [@TV].
[([@TV Definition 3.8])]{} A space $X$ is said to be [*homogeneously dense*]{}, abbreviated HD, if there are numbers $r_1$, $r_2$ such that $0<r_1\leq r_2<1$ and such that for each pair of points $a,b\in X$ there is $x\in X$ satisfying the condition $$r_1|b-a|\leq|x-a|\leq r_2|b-a|.$$ We also say that $X$ is $(r_1,r_2)$-HD or simply $r$-HD, where $r=(r_1,r_2)$.
By the definition, obviously, a HD space has no isolated point. And for all $0<r_1\leq r_2<1$, every connected domain is $(r_1,r_2)$-HD, $[0,1]\cup [2,3]$ is $(\frac{1}{6},\frac{1}{4})$-HD (see [@TV]). Particularly, a finite union of connected nondegenerate sets (i.e. the set is not a point) is $(r_1,r_2)$-HD with some constants $0<r_1\leq r_2<1$. For a HD space, Tukia and Väisälä proved the following properties in [@TV].
\[pre-lem-4\][([@TV Lemma 3.9])]{} $(\textit{1})$ Let $X$ be $(r_1,r_2)$-HD and let $m$ be a positive integer. Then $X$ is $(r_1^m,r_2^m)$-HD.
$(\textit{2})$ Let $X$ be $r$-HD and let $f:X\to Y$ be $\eta$-QS. Then $fX$ is $\mu$-HD, where $\mu$ depends only on $\eta$ and $r$.
Moreover, we prove the following property.
\[lem-2-2\] Let $D\subsetneq E $ be a domain with $(r_1,r_2)$-HD boundary and let $x\in D$. Then for all $x_0\in \partial D$ with $|x-x_0|\leq 2d_D(x)$ there exists some point $x_1\in \partial D$ such that $$\label{eq-th-ll}\frac{1}{2}d_D(x)
\leq |x_0-x_1|\leq \big(2+\frac{17}{2r_1}\big) d_D(x).$$
By Lemma \[pre-lem-4\] we may assume that $0<r_1\leq r_2<\frac{1}{3}$. For example, if $r_2\geq \frac{1}{3}$, then there exists a positive integer $m$ such that $r_2^m<\frac{1}{3}$. In fact we can choose $m-1$ to be the integer part of $\log_{r_2}\frac{1}{3}$, and by Lemma \[pre-lem-4\] the $(r_1,r_2)$-HD property of $\partial D$ implies that $\partial D$ is $(r_1^m,r_2^m)$-HD with $r_2^m<\frac{1}{3}$.
For a given $x\in D$, let $x_0\in \partial D$ be such that $|x-x_0|\leq 2d_D(x)\,.$ We divide the proof into three cases.
[*Case I*]{}: $\partial D\subset \overline{\mathbb{\mathbb{B}}}\big(x,\frac{5}{2}d_D(x)\big)$.
Obviously, $D$ is bounded. Let $x_1\in \partial D$ be such that $|x_0-x_1|\geq \frac{1}{3}\operatorname{diam}(D)$. Then $$\frac{2}{3}d_D(x)\leq |x_0-x_1|\leq 5d_D(x),$$ which shows that $x_1$ is the desired point and satisfies .
[*Case II*]{}: $\partial D\cap \Big(\mathbb{B}\big(x, \frac{1}{r_1}d_D(x)\big)\setminus \overline{\mathbb{B}}\big(x,\frac{5}{2}d_D(x)\big)\Big)\neq \emptyset$.
Let $x_2\in \partial D\cap \mathbb{\mathbb{B}}\big(x, \frac{1}{r_1}d_D(x)\big)\setminus \overline{\mathbb{B}}\big(x,\frac{5}{2}d_D(x)\big)$. Then $$|x_0-x_2|\geq |x_2-x|-|x-x_0|\geq \frac{1}{2}d_D(x)$$ and $$|x_0-x_2|\leq |x_0-x|+|x-x_2|\leq \big(\frac{1}{r_1}+2\big)d_D(x).$$
Obviously, $x_2$ is the needed point.
[*Case III*]{}: $\partial D\cap \Big( \mathbb{B}\big(x, \frac{1}{r_1}d_D(x)\big)\setminus \overline{\mathbb{B}}\big(x,\frac{5}{2}d_D(x)\big)\Big)= \emptyset$.
Let $\omega=\partial D \cap (E\setminus \mathbb{B}\big(x, \frac{1}{r_1}d_D(x))\big)$ and $d_1$ denote the distance from $\omega$ to $\mathbb{B}\big(x, \frac{1}{r_1}d_D(x)\big)$, i.e., $d_1=d\Big(\omega, \mathbb{B}\big(x, \frac{1}{r_1}d_D(x)\big)\Big)$. If $d_1=0$, let $x_3\in \omega$ be such that $d(x_3, \mathbb{B}\big(x, \frac{1}{r_1}d_D(x))\big)\leq \frac{1}{2}d_D(x)$. Hence $$(\frac{1}{r_1}-2)d_D(x)\leq |x_0-x_3|\leq |x_0-x|+|x-x_3|\leq (\frac{1}{r_1}+\frac{5}{2})d_D(x).$$ So $x_3$ is the desired point.
On the other hand, if $d_1>0$, let $x_4\in \omega$ be such that $$\label{lem-2-eq1}d(x_4, \mathbb{B}\big(x, \frac{1}{r_1}d_D(x))\big)\leq \frac{3}{2}d_1.$$ We claim that the point $x_4$ satisfies . To see this, we first prove $$\label{eq-th-ss} d_1<\frac{5}{r_1}d_D(x).$$ Suppose on the contrary that $d_1\geq \frac{5}{r_1}d_D(x).$ Then by there exists some point $u\in \partial D$ such that $$\begin{aligned}
|u-x_0|&\geq& r_1|x_0-x_4|\geq r_1(|x_4-x|-|x-x_0|)\\&\geq& r_1\big(\frac{6}{r_1}-2\big)d_D(x)= (6-2r_1)d_D(x)\end{aligned}$$ and $$\begin{aligned}
|u-x_0|&\leq& r_2|x_0-x_4|\leq r_2(|x_0-x|+|x-x_4|)\\&\leq& r_2\big(2+\frac{1}{r_1}\big)d_D(x)+\frac{3r_2}{2}d_1\leq d_1,\end{aligned}$$ which shows that $u\in \partial D\cap \Big(\mathbb{B}\big(x, \frac{1}{r_1}d_D(x)\big)\setminus \overline{\mathbb{B}}\big(x,\frac{5}{2}d_D(x)\big)\Big)$. This is a contradiction. Hence holds.
By , we have $$\big(\frac{1}{r_1}-2\big)d_D(x)\leq|x_1-x_0|\leq \big(2+\frac{1}{r_1}\big)d_D(x)+\frac{3}{2}d_1\leq \big(2+\frac{17}{2r_1}\big)d_D(x).$$ Hence the point $x_4$ has the required properties, and so the proof of the lemma is complete.
The discussions in the case III also follows from [@H Lemma 11.7].
Proofs of Theorems \[thm1.2\], \[thm1.3\], \[thm1.4\] and \[thm1.1\] {#sec-4}
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For convenience, in the following, we always assume that $x$, $y$, $z$, $\ldots$ denote points in $D$ and $x'$, $y'$, $z'$, $\ldots$ the images in $D'$ of $x$, $y$, $z$, $\ldots$ under $f$, respectively. We start with some known results that are necessary for the following proofs.
\[proof-lem-1\] Suppose that $x,y\in D\neq E$ and that either $|x-y|\leq \frac{1}{2}d_D(x)$ or $k_D(x,y)\leq 1$. Then $$\frac{1}{2}\frac{|x-y|}{d_D(x)}\leq k_D(x,y)\leq 2\frac{|x-y|}{d_D(x)}.$$
\[proof-lem-2\] Suppose that $X$ is connected, that $f:X\to Y$ is $\eta$-quasisymmetric, and that $A\subset X$ is bounded. Then $f|_A$ satisfies a two-sided Hölder condition $$|x-y|^{{1}/{\alpha}}/M\leq |fx-fy|\leq M|x-y|^{\alpha}\;\;\;\;\; for \; x,y \in A,$$ where $\alpha=\alpha(\eta)\leq1$ and $M=M(\eta, d(A),d(fA))\geq 1.$
Since $f:\partial D\to \partial D'$ is $L$-bilipschitz, we know that the boundedness of $D$ (resp. $D'$) implies the boundedness of $D'$ (resp. $D$). In fact, suppose on the contrary that $D$ is bounded and $D'$ is unbounded. Then let $w_1', w_2'\in \partial D'$ such that $|w_1'-w_2'|\geq 4L \operatorname{diam}(D)$. Then we have $$\operatorname{diam}(D)\geq \frac{1}{2}|w_1-w_2|\geq \frac{1}{2L}|w_1'-w_2'|\geq 2\operatorname{diam}(D),$$ which is a contradiction.
If $D$ is unbounded, then $\infty\in \partial D$, by auxiliary inversions we normalize the situation such that $f(\infty)=\infty.$ Hence by Lemma \[pre-lem-3\], $f$ is $\eta$-QS in $\overline{D}$ with $\eta$ depending on $c$, $L$ and $\varphi$.
In the following, we assume that $D$ is bounded. Then $$\label{thm-1-4}\frac{1}{4L}\operatorname{diam}(D)\leq\operatorname{diam}(D')\leq 4L\operatorname{diam}(D).$$ Let $z_1, z_2\in \partial D$ be such that $|z_1-z_2|\geq \frac{1}{2}\operatorname{diam}(D)$ and let $z_3'\in \partial D' $ be such that $$\min\{|z_1'-z_3'|,|z_2'-z_3'|\}\geq \frac{1}{6}\operatorname{diam}(D').$$ Then by , we have $$|z_1'-z_2'|\geq \frac{1}{L}|z_1-z_2|\geq \frac{1}{2}\operatorname{diam}(D)\geq \frac{1}{8L^2}\operatorname{diam}(D')$$ and $$\min\{|z_1-z_3|,|z_2-z_3|\}\geq\frac{1}{L}\min\{|z_1'-z_3'|,|z_2'-z_3'|\}\geq\frac{1}{24L^2}\operatorname{diam}(D),$$ which, in combination with Lemma \[pre-lem-2\] and Lemma \[pre-lem-3\], shows that $f$ is $\eta$-QS in $\overline{D}$ with $\eta$ depending on $c$, $L$ and $\varphi$.
The proof of Theorem \[thm1.3\] easily follows from Theorem \[thm1.2\] and Lemma \[proof-lem-2\].
In the remaining part of this paper, we always assume that $D$ and $D'$ are $c$-uniform subdomains in $E$ and $E'$, respectively, that the boundary of $D$ is $(r_1,r_2)$-homogeneously dense, that $f: D\to D'$ is a $\varphi$-FQC map, and that $f$ extends to a homeomorphism $\overline{f}:
\overline{D}\to \overline{D'}$ such that $\overline{f}:\partial
D\to \partial D'$ is $L$-bilipschitz.
We first show that the following lemma holds.
\[lem-1\] There is a constant $M_1=M_1(c,L,\varphi,r_1,r_2)$ such that for given $x\in D$ the following hold:\
$(1)$ For $x_0\in \partial D$ with $|x-x_0|\leq 2d_D(x)$, we have $$|x_0'-x'|\leq
M_1d_{D}(x).$$ $(2)$ For all $x_1\in \partial D$, we have $$\label{eq-lem-ls}\frac{1}{2(2L+M_1)}|x_1-x|\leq|x'_1-x'|\leq 2(2L+M_1)|x_1-x|.$$
We first prove $(1)$.
For a fixed $x\in D$, let $x_0\in \partial D$ be such that $|x-x_0|\leq 2d_D(x)$. Let $x_2$ be the intersection point of $\mathbb{S}(x, \frac{1}{2}d_{D}(x))$ with $[x_0, x]$. Then by we have $$k_{D}(x_2,x)\leq
\log\Big(1+\frac{|x-x_2|}{d_{D}(x)-|x-x_2|}\Big)=\log 2,$$ which implies that
$$\log\frac{|x'_2-x'|}{|x'_2-x'_0|}\leq k_{D'}(x'_2,x')\leq \varphi(k_{D}(x_2,x))=\varphi(\log 2).$$ Hence $$\label{lem-1-0}|x'_2-x'|\leq e^{\varphi(\log 2)} |x'_2-x'_0|,$$ and so $$\label{lem-1-1}|x'-x'_0|\leq|x'-x'_2|+|x'_2-x'_0|
\leq(e^{\varphi(\log 2)}+1)|x'_2-x'_0|.$$
Since $\partial D$ is $(r_1,r_2)$-HD, we see from Lemma \[lem-2-2\] that there must exist some point $x_3\in \partial D$ such that $$\label{lem-1-16}\frac{1}{2}d_D(x)\leq |x_3-x_0|\leq \big(2+\frac{17}{2r_1}\big)d_D(x).$$ Hence
$$\label{lem-1-6'}|x-x_3|\leq |x-x_0|+|x_0-x_3|\leq
\big(4+\frac{17}{2r_1}\big)d_D(x)$$
and $$\label{lem-1-6}\frac{1}{2L}d_{D}(x)\leq
\frac{1}{L}|x_3-x_0|\leq |x'_3-x'_0|\leq L|x_3-x_0|\leq L\big(2+\frac{17}{2r_1}\big)d_{D}(x).$$ By Lemma \[pre-lem-3\] we see that $f^{-1}$ is $\theta$-quasimöbius in $\overline{D}$, where $\theta=\theta(c,\varphi)$. It follows from (\[lem-1-0\]), (\[lem-1-16\]), (\[lem-1-6’\]) and (\[lem-1-6\]) that $$\begin{aligned}
\frac{1}{6\big(4+\frac{17}{2r_1}\big)}&\leq&\frac{|x_3-x_0|}{|x_2-x_0|}\cdot\frac{|x_2-x|}{|x-x_3|}\leq
\theta
\Big(\frac{|x'_3-x'_0|}{|x'_2-x'_0|}\cdot\frac{|x'_2-x'|}{|x'-x'_3|}\Big)\\&\leq&
\theta\Big(\frac{L\big(2+\frac{17}{2r_1}\big)e^{\varphi(\log 2)}d_D(x)}{|x'-x'_3|}\Big) ,\end{aligned}$$ which, together with (\[lem-1-1\]), shows that $$\begin{aligned}
|x'-x'_0|&\leq& |x'-x'_3|+|x'_3-x'_0|\leq (\lambda L\big(2+\frac{17}{2r_1}\big)e^{\varphi(\log 2)}+1)d_D(x)\\&\leq & 2\lambda L\big(2+\frac{17}{2r_1}\big)e^{\varphi(\log 2)}d_D(x),\end{aligned}$$ where $\lambda={1}/{\theta^{-1}(\frac{1}{6\big(4+\frac{17}{2r_1}\big)})}$. Thus the proof of $(1)$ is complete by taking $M_1=2\lambda L\big(2+\frac{17}{2r_1}\big)e^{\varphi(\log 2)}$.
Now we are going to prove $(2)$.
We first observe that $f:\partial D\to \partial D'$ is $\eta$-QS with $\eta(t)=L^2t$. Hence Lemma \[pre-lem-4\] shows that $\partial D'$ is $(\lambda_1,\lambda_2)$-HD with $\lambda_1, \lambda_2$ depending only on $L$, $r_1$ and $r_2.$ Since $f^{-1}$ is also a $\varphi$-FQC map, it is easily seen that we only need to prove the right hand side of . For $x\in D$, we let $y_1\in
\partial D$ be such that $$\label{lem-2-13}|x-y_1|\leq 2d_D(x).$$ Then it follows from Lemma \[lem-1\] $(1)$ that $$\label{lem-2-7}|x'-y'_1|\leq M_1d_D(x)\leq M_1|x-y_1|.$$
For $x_1\in \partial D$, on one hand, if $|y_1-x_1|\leq 2 |x-y_1|$, then by , $$\begin{aligned}
|x'-x'_1|&\leq&
|x'-y'_1|+|y'_1-x'_1|\leq M_1|x-y_1|+L|y_1-x_1|\\ \nonumber &\leq&(2L+M_1)|x-y_1|\leq 2(2L+M_1)d_D(x)\\
\nonumber &\leq& 2(2L+M_1)|x-x_1|.\end{aligned}$$
On the other hand, if $|y_1-x_1|>2|x- y_1|$, then we have $$|x-x_1|>|y_1-x_1|-|x-y_1|>\frac{1}{2}|y_1-x_1|,$$ which, together with (\[lem-2-7\]), shows that $$\begin{aligned}
|x'-x'_1|&\leq&
|x'-y'_1|+|y'_1-x'_1|\leq M_1|x-y_1|+L|y_1-x_1|\\ \nonumber &\leq& 2M_1d_D(x)+2L|x-x_1|\leq 2(L+M_1)|x-x_1|.\end{aligned}$$ Hence the proof of is complete.
Supposing that $f \in QC^L_{\varphi}(D,D')$ is $M$-QH, we show that $f$ is $M'$-bilipschitz from $\overline{D}$ to $\overline{D}'$. Lemma \[pre-lem-4\] yields that $\partial D'$ is $(\lambda_1,\lambda_2)$-HD with $\lambda_1, \lambda_2$ depending only on $L$, $r_1$ and $r_2$. Then by Lemma \[lem-1\] and the fact that $``f^{-1}$ is also $M$-QH and a $M$-QH map is a $\varphi$-FQC map with $\varphi(t)=Mt$" we know that it suffices to show that for all $z_1,z_2\in D$, the following holds: $$\label{thm-1-2}|z_1'-z_2'|\leq M'|z_1-z_2|.$$
Fix $z_1,z_2\in D\,.$ Without loss of generality, we may assume that $$\max\{d_D(z_1),d_D(z_2)\}=d_D(z_1).$$
Consider first the case $|z_1-z_2|\leq \frac{1}{2M}d_D(z_1)\,.$ Then by Lemma \[proof-lem-1\], $$k_{D'}(z_1',z_2')\leq M k_D(z_1,z_2)\leq 2M \frac{|z_1-z_2|}{d_D(z_1)}\leq 1,$$ which shows that $$\frac{1}{2}\frac{|z_1'-z_2'|}{d_{D'}(z_1')}\leq k_D(z_1',z_2')\leq M k_D(z_1,z_2)\leq 2M\frac{|z_1-z_2|}{d_D(z_1)}.$$ Hence Lemma \[lem-1\] shows that $$\label{thm-1-2-proof1}|z_1'-z_2'|\leq 4M\frac{|z_1-z_2|}{d_D(z_1)}d_{D'}(z_1')\leq 4MM_1|z_1-z_2|.$$
Next we consider the case $|z_1-z_2|> \frac{1}{2M}d_D(z_1)$. We let $z\in \partial D$ be such that $|z_1-z|\leq 2 d_D(z_1)$. If $|z_1-z|\leq \frac{1}{2}|z_2-z|$, then $$|z_1-z_2|\geq |z_2-z|-|z_1-z|\geq \frac{1}{2}|z_2-z|,$$ and so Lemma \[lem-1\] yields $$\label{thm-1-2-proof2}|z_1'-z_2'|\leq |z_1'-z'|+|z_2'-z'|\leq M_1 d_D(z_1)+ 2(2L+M_1)|z_2-z|$$$$\begin{aligned}
\leq 2(MM_1+4L+2M_1)|z_1-z_2|.\end{aligned}$$ On the other hand, if $|z_1-z|\geq \frac{1}{2}|z_2-z|$, then by Lemma \[lem-1\] we have $$\label{thm-1-2-proof3}|z_1'-z_2'|\leq |z_1'-z'|+|z_2'-z'|\leq M_1 d_D(z_1)+ 2(2L+M_1)|z_2-z|$$$$\begin{aligned}
\leq M_1 d_D(z_1)+ 4(2L+M_1)|z_1-z|\leq 2M(9M_1+16L)|z_1-z_2|.\end{aligned}$$ By taking $M'= 2M(9M_1+16L)$ we see from , and that holds. Hence the proof of Theorem \[thm1.4\] is complete.
\[remark\]
1. In Theorem \[thm1.4\], the hypothesis “$f$ is FQC" alone does not imply the conclusion “$f$ is bilipschitz". As an example, we consider the radial power map $f_{\alpha}: \mathbb{B}\to \mathbb{B}$ with $f_{\alpha}(x)=|x|^{\alpha-1}x$ and $\alpha\geq 1$. By [@Vai6-0 6.5] we see that $f_{\alpha}$ is a FQC map and $f_{\alpha}|_{\partial \mathbb{B}}$ is the identity on the boundary, but $f_{\alpha}$ is not bilipschitz (see [@Vai5 6.8]).
2. If the boundary of $D$ is not HD, then “$f$ being QH" does not always imply that “$f$ is bilipschitz". We still consider the radial power map $f_{\alpha}: E\setminus\{0\}\to E\setminus\{0\}$ with $f_{\alpha}(x)=|x|^{\alpha-1}x$ and $\alpha\geq 1$. On one hand, the domain $E\setminus\{0\}$ has only two boundary components: $\{0\}$ and $\{\infty\}$, and so the boundary is not HD. On the other hand, $f$ is $\alpha$-QH (see [@Vai5 5.21]) and it is the identity on the boundary. But it is not bilipschitz.
Given $x\in D=D'$, let $z'\in \partial D'$ be such that $d_{D'}(x')\geq \frac{1}{2}|x'-z'|$. Then Lemma \[lem-1\] yields $$d_{D'}(x')\geq \frac{1}{4(2L+M_1)}|x-z|\geq \frac{1}{4(2L+M_1)}d_D(x).$$
Let $z_1\in \partial D$ be such that $|x-z_1|\leq 2 d_D(x)$. Then it follows from Lemma \[lem-1\] that $$|x-x'|\leq |x-z_1|+|x'-z_1|\leq (2+M_1)d_D(x).$$
Hence by Lemma \[pre-lem-1\] we see that $$k_D(x,x')\leq c'\log\Big(1+\frac{|x-x'|}{\min\{d_D(x),d_D(x')\}}\Big)\leq c'\log\big(1+4(2+M_1)(2L+M_1)\big).$$
[**Acknowledgement.**]{} This research was finished when the first author was an academic visitor in Turku University and the first author was supported by the Academy of Finland grant of Matti Vuorinen with the Project number 2600066611. She thanks Department of Mathematics in Turku University for hospitality.
[99]{}
, An extremal displacement mapping in n-space. *Complex analysis Joensuu 1978* (Proc. Colloq., Univ. Joensuu, Joensuu, 1978), pp. 1–9, Lecture Notes in Math., 747, Springer, Berlin, 1979.
, Uniform domains and the quasi-hyperbolic metric, *J. Analyse Math.,* [**36**]{} (1979), 50–74.
, Quasiconformally homogeneous domains, *J. Analyse Math.,* [**30**]{} (1976), 172–199.
, [Isometries of some hyperbolic-type path metrics, and the hyperbolic medial axis.]{} In the tradition of Ahlfors-Bers. IV, 63–74, Contemp. Math., 432, Amer. Math. Soc., Providence, RI, 2007.
, Lectures on analysis on metric spaces. Universitext. Springer-Verlag, New York, 2001. x+140 pp.
, Local convexity properties of quasihyperbolic balls in punctured space. *J. Math. Anal. Appl.* [**342**]{} (2008), no. 1, 192–201.
, *Close-to-convexity of quasihyperbolic and j-metric balls.* *Ann. Acad. Sci. Fenn. Math.* [**35**]{} (2010), no. 2, 493–501.
Quasihyperbolic geometry in euclidean and Banach spaces, Proc. ICM2010 Satellite Conf. International Workshop on Harmonic and Quasiconformal Mappings (HMQ2010), eds. D. Minda, S. Ponnusamy, N. Shanmugalingam, J. Analysis [**18**]{} (2011),261–278, arXiv:1104.3745v1 \[math.CV\].
[K]{} [J. Krzyż]{}, On an extremal problem of F. W. Gehring, *Bull. Acad. Polon. Sci. S¨¦r. Sci. Math. Astronom. Phys.*, [**16**]{} (1968), 99–101.
, On quasiconformal maps with identity boundary values, *Trans. Amer. Math. Soc.*, [**363**]{} (2011), no. 4, 2467–2479.
, The Teichmüller problem for mean distortion, *Ann. Acad. Sci. Fenn. Ser. A I Math.*, [**34**]{} (2009), 233–247.
, Definitions of uniform domains, *Ann. Acad. Sci. Fenn. Ser. A I Math.,* [**5**]{} (1980), 197–205.
, Lipschitz conditions and quasiconformal mappings. *Indiana Univ. Math. J.* [**31**]{} (1982), no. 3, 377–401.
Convexity properties of quasihyperbolic balls on Banach spaces, Ann. Acad. Sci. Fenn. [**37**]{} (2012), 215-228, arXiv: 1007.3197v1 \[math. CV\].
On Quasihyperbolic Geodesics in Banach Spaces, arXiv:1301.0900 \[math.CV\]
, Ein Verschiebungssatz der quasikonformen Abbildung. (German) *Deutsche Math.,* [**7**]{} (1944), 336–343.
, Quasisymmetric embeddings of metric spaces, *Ann. Acad. Sci. Fenn. Ser. A I Math.,* [**5**]{} (1980), 97-114.
, Lectures on n-dimensional quasiconformal mappings, Springer-Verlag, 1971.
, Quasimöbius maps, *J. Analyse Math.,* [**44**]{} (1985), 218–234.
, Uniform domains, *Tohoku Math. J.,* [**40**]{} (1988), 101–118.
, Free quasiconformality in Banach spaces. I, *Ann. Acad. Sci. Fenn. Ser. A I Math.,* [**15**]{} (1990), 355-379.
, Free quasiconformality in Banach spaces. II, *Ann. Acad. Sci. Fenn. Ser. A I Math.,* [**16**]{} (1991), 255-310.
, Relatively and inner uniform domains, *Conformal Geom. Dyn.,* [**2**]{} (1998), 56–88.
, The free quasiworld, Freely quasiconformal and related maps in Banach spaces, *Banach Center Publ.,* [**48**]{} (1999), 55–118. , Quasihyperbolic geometry of planar domains. *Ann. Acad. Sci. Fenn. Math.* [**34**]{} (2009), no. 2, 447–473.
, Conformal invariants and quasiregular mappings, *J. Analyse Math.,* [**45**]{} (1985), 69–115.
, Distortion of quasiconformal mappings with identity boundary values, *arXiv: 1203.0427v1\[math.CV\],* [**2**]{} Mar (2012).
| ArXiv |
---
abstract: 'Massive multiple-input multiple-output (MIMO) systems achieve high sum spectral efficiency by offering an order of magnitude increase in multiplexing gains. In time division duplexing systems, however, the reuse of uplink training pilots among cells results in additional channel estimation error, which causes downlink inter-cell interference, even when large numbers of antennas are employed. Handling this interference with conventional network MIMO techniques is challenging due to the large channel dimensionality. Further, the implementation of large antenna precoding/combining matrices is associated with high hardware complexity and power consumption. In this paper, we propose multi-layer precoding to enable efficient and low complexity operation in full-dimensional massive MIMO, where a large number of antennas is used in two dimensions. In multi-layer precoding, the precoding matrix of each base station is written as a product of a number of precoding matrices, each one called a layer. Multi-layer precoding (i) leverages the directional characteristics of large-scale MIMO channels to manage inter-cell interference with low channel knowledge requirements, and (ii) allows for an efficient implementation using low-complexity hybrid analog/digital architectures. We present a specific multi-layer precoding design for full-dimensional massive MIMO systems. The performance of this precoding design is analyzed and the per-user achievable rate is characterized for general channel models. The asymptotic optimality of the proposed multi-layer precoding design is then proved for some special yet important channels. Numerical simulations verify the analytical results and illustrate the potential gains of multi-layer precoding compared to traditional pilot-contaminated massive MIMO solutions.'
author:
- 'Ahmed Alkhateeb, Geert Leus, and Robert W. Heath, Jr. [^1] [^2][^3]'
title: ' Multi-Layer Precoding: A Potential Solution for Full-Dimensional Massive MIMO Systems'
---
Introduction {#sec:Intro}
============
Massive MIMO promises significant spectral efficiency gains for cellular systems. Scaling up the number of antennas, however, faces several challenges that prevent the corresponding scaling of the gains [@Larsson2014; @Rusek2013; @HeathJr2016; @Truong2013]. First, the training and feedback of the large channels has high overhead in frequency division duplexing (FDD) systems. To overcome that, channel reciprocity in conjunction with time division duplexing (TDD) systems is used [@Marzetta2010; @Bjoernson2016]. Reusing the uplink training pilots among cells, however, causes channel estimation errors which in turn lead to downlink inter-cell interference, especially for cell-edge users [@Marzetta2010]. Managing this inter-cell interference using traditional network MIMO techniques requires high coordination overhead, which could limit the overall system performance [@Lozano2013]. Another challenge with the large number of antennas lies in the hardware implementation [@HeathJr2016; @Singh2009]. Traditional MIMO precoding techniques normally assumes complete baseband processing, which requires dedicating an RF chain per antenna. This may lead to high cost and power consumption in massive MIMO systems [@HeathJr2016]. Therefore, developing precoding techniques that can overcome the challenges of inter-cell interference and complete baseband processing is of great interest.
Prior Work
----------
Inter-cell interference is a critical problem for general MIMO systems. Typical solutions for managing this interference require some sort of collaboration between the base stations (BSs) [@Gesbert2010]. The overhead of this cooperation, though, can limit the system performance [@Lozano2013]. When the number of antennas grows to infinity, the performance of the network becomes limited by pilot contamination [@Marzetta2010], which is one form of inter-cell interference. Pilot contamination happens because of the channel estimation errors that result from reusing the uplink training pilots among users in TDD massive MIMO systems. Several solutions have been proposed to manage inter-cell interference in massive MIMO systems [@Huh2012; @Jose2011; @Ashikhmin2012; @Yin2013]. In [@Huh2012; @Jose2011], multi-cell zero-forcing and MMSE MIMO precoding strategies were developed to cancel or reduce the inter-cell interference. The solutions in [@Huh2012; @Jose2011], however, require global channel knowledge at every BS, which makes them feasible only for small numbers of antennas [@Lu2014]. Pilot contamination precoding was proposed in [@Ashikhmin2012] to overcome the pilot contamination problem, relying on the channel covariance knowledge. The technique in [@Ashikhmin2012], though, requires sharing the transmitted messages between all BSs, which is difficult to achieve in practice. In [@Yin2013], the directional characteristics of large-dimensional channels were leveraged to improve the uplink channel training in TDD systems. This solution, however, requires fully-digital hardware and does not leverage the higher degrees of freedom provided in full-dimensional massive MIMO systems.
Precoding approaches that divide the processing between two stages have been developed in [@ElAyach2014; @Alkhateeb2014b; @Bogale2014; @Liang2014; @Adhikary2013] for mmWave and massive MIMO systems. Motivated by the high cost and power consumption of the RF, [@ElAyach2014] developed hybrid analog/digital precoding algorithms for mmWave systems. Hybrid precoding divides the precoding between RF and baseband domains, and requires a much smaller number of RF chains compared to the number of antennas. For multi-user systems [@Alkhateeb2014b] proposed a two-stage hybrid precoding design where the first precoding matrix is designed to maximize the signal power for each user and the second matrix is designed to manage the multi-user interference. Similar solutions were also developed for massive MIMO systems [@Bogale2014; @Liang2014], with the general objective of maximizing the system sum-rate. In [@Adhikary2013], a two-stage joint spatial division and multiplexing (JSDM) precoding scheme was developed to reduce the channel training overhead in FDD massive MIMO systems. In JSDM, the base station (BS) divides the mobile stations (MSs) into groups of approximately similar covariance eigenspaces, and designs a pre-beamforming matrix based on the large channel statistics. The interference between the users of each group is then managed using another precoding matrix given the effective reduced-dimension channels. The work in [@ElAyach2014; @Alkhateeb2014b; @Bogale2014; @Liang2014; @Adhikary2013], however, did not consider out-of-cell interference, which ultimately limits the performance of massive MIMO systems.
Contribution
------------
In this paper, we introduce a general framework, called multi-layer precoding, that (i) coordinates inter-cell interference in full-dimensional massive MIMO systems leveraging large channel characteristics and (ii) allows for efficient implementations using hybrid analog/digital architectures. Note that most of the literature on full-dimensional MIMO systems did not assume massive MIMO [@Nam2013; @Kim2014a; @Seifi2014a], and the two systems were studied independently using different tools and theories. In this paper, we refer to full-dimensional massive MIMO as a two-dimensional MIMO system, which adopts large numbers of antennas in the two dimensions. The main contributions of our work are summarized as follows.
- Designing a specific multi-layer precoding solution for full-dimensional massive MIMO systems. The proposed precoding strategy decouples the precoding matrix of each BS as a multiplication of three precoding matrices, called layers. The three precoding layers are designed to avoid inter-cell interference, maximize effective signal power, and manage intra-cell multi-user interference, with low channel training overhead.
- Analyzing the performance of the proposed multi-layer precoding design. First, the per-user achievable rate using multi-layer precoding is derived for a general channel model. Then, asymptotic optimality results for the achievable rates with multi-layer precoding are derived for two special channel models: the one-ring and the single-path models. Lower bounds on the achievable rates for the cell-edge users are also characterized under the one-ring channel model.
The developed multi-layer precoding solutions are also evaluated by numerical simulations. Results show the multi-layer precoding can approach the single-user rate, which is free of inter-cell and intra-cell interference, in some special cases. Further, results illustrate that significant rate and coverage gains can be obtained by multi-layer precoding compared to conventional conjugate beamforming and zero-forcing massive MIMO solutions.
We use the following notation throughout this paper: $\bA$ is a matrix, $\ba$ is a vector, $a$ is a scalar, and $\cA$ is a set. $|\bA|$ is the determinant of $\bA$, $\|\bA \|_F$ is its Frobenius norm, whereas $\bA^T$, $\bA^H$, $\bA^*$, $\bA^{-1}$, ${\ensuremath{\bA^{\dagger}}}$ are its transpose, Hermitian (conjugate transpose), conjugate, inverse, and pseudo-inverse respectively. $[\bA]_{r,:}$ and $[\bA]_{:,c}$ are the $r$th row and $c$th column of the matrix $\bA$, respectively. $\mathrm{diag}(\ba)$ is a diagonal matrix with the entries of $\ba$ on its diagonal. $\bI$ is the identity matrix and $\mathbf{1}_{N}$ is the $N$-dimensional all-ones vector. $\bA \otimes \bB$ is the Kronecker product of $\bA$ and $\bB$, and $\bA \circ \bB$ is their Khatri-Rao product. $\cN(\bm,\bR)$ is a complex Gaussian random vector with mean $\bm$ and covariance $\bR$. $\bbE\left[\cdot\right]$ is used to denote expectation.
System and Channel Models {#sec:Model}
=========================
In this section, we present the full-dimensional massive MIMO system and channel models adopted in the paper.
System Model {#sec:SysModel}
------------
![A full-dimensional MIMO cellular model where each BS has a 2D antenna array and serves $K$ users.[]{data-label="fig:Model"}](FDMIMO_Cellular.pdf){width=".95\columnwidth"}
Consider a cellular system model consisting of $B$ cells with one BS and $K$ MS’s in each cell, as shown in . Each BS is equipped with a two-dimensional (2D) antenna array of $N$ elements, $N=N_\mathrm{V}$ (vertical antennas) $\times N_\mathrm{H}$ (horizontal antennas), and each MS has a single antenna. We assume that all BSs and MSs are synchronized and operate a TDD protocol with universal frequency reuse. In the downlink, each BS $b=1, 2, ..., B$, applies an $N \times K$ precoder $\bF_b$ to transmit a symbol for each user, with a power constraint $\|\left[\bF_b\right]_{:,k}\|^2=1$, $k=1, 2, ..., K$. Uplink and downlink channels are assumed to be reciprocal. If $\bh_{b c k}$ denotes the $N \times 1$ uplink channel from user $k$ in cell $c$ to BS $b$, then the received signal by this user in the downlink can be written as $$y_{c k}=\sum_{b =1}^{B} \bh_{b c k}^* \bF_{b} \bs_{b} + n_{c k},
\label{eq:Received}$$ where $\bs_{b}=\left[s_{b,1}, ..., s_{b,K}\right]^T$ is the $K \times 1$ vector of transmitted symbols from BS $b$, such that $\bbE \left[\bs_{b} \bs_{b}^*\right]=\frac{P}{K} \bI$, with $P$ representing the average total transmitted power, and $n_{c k}\sim \cN (0, \sigma^2 )$ is the Gaussian noise at user $k$ in cell $c$. It is useful to expand as $$y_{c k}=\underbrace{\bh_{c c k}^* \left[\bF_{c}\right]_{:,k} s_{c,k}}_\text{Desired signal}+\underbrace{ \sum_{m \neq k} \bh_{c c k}^* \left[\bF_{c}\right]_{:,m} s_{c,m}}_\text{Intra-cell interference}+ \underbrace{\sum_{b \neq c} \bh_{b c k}^* \bF_{b} \bs_{b}}_\text{Inter-cell interference} + n_{c k},$$ to illustrate the different components of the received signal.
Channel Model {#sec:ChModel}
-------------
We consider a full-dimensional MIMO configuration where 2D antenna arrays are deployed at the BS’s. Consequently, the channels from the BS’s to each user have a 3D structure. Extensive efforts are currently given to 3D channel measurements, modeling, and standardization [@Kammoun2014; @Zhong2013]. One candidate is the Kronecker product correlation model, which provides a reasonable approximation of 3D covariance matrices[@Ying2014]. In this model, the covariance of the 3D channel $\bh_{b c k}$, defined as $\bR_{b c k}= \bbE\left[\bh_{b c k} \bh_{b c k}^*\right]$, is approximated by $$\bR_{b c k}= \bR_{b c k}^\mathrm{A} \otimes \bR_{b c k}^\mathrm{E},$$ where $\bR_{b c k}^\mathrm{A}$ and $\bR_{b c k}^\mathrm{E}$ represent the covariance matrices in the azimuth and elevation directions. If $\bR_{b c k}^\mathrm{A}= \bU_{b c k}^\mathrm{A} {\boldsymbol{\Lambda}_{b c k}^{\mathrm{A}}} {\bU_{b c k}^\mathrm{A}}^*$ and $\bR_{b c k}^\mathrm{E}= \bU_{b c k}^\mathrm{E} {\boldsymbol{\Lambda}_{b c k}^{\mathrm{E}}} {\bU_{b c k}^\mathrm{E}}^*$ are the eigenvalue decompositions of $\bR_{b c k}^\mathrm{A}$ and $\bR_{b c k}^\mathrm{E}$, then using Karhunen-Loeve representation, the channel $\bh_{b c k}$ can be expressed as $$\label{eq:channel}
\bh_{b c k}= \left[\bU_{b c k}^\mathrm{A} {\boldsymbol{\Lambda}_{b c k}^{\mathrm{A}}}^{\frac{1}{2}} \otimes \bU_{b c k}^\mathrm{E} {\boldsymbol{\Lambda}_{b c k}^{\mathrm{E}}}^{\frac{1}{2}}\right] \bw_{b c k},$$ where $\bw_{b c k} \sim \cN(\boldsymbol{0}, \bI)$ is a $\mathrm{rank}\left(\bR_{b c k}^\mathrm{A}\right) \mathrm{rank}\left(\bR_{b c k}^\mathrm{E}\right) \times 1$ vector, with $\mathrm{rank}(\bA)$ representing the rank of the matrix $\bA$. Without loss of generality, and to simplify the notation, we assume that all the users have the same ranks for the azimuth and elevation covariance matrices, which are denoted as $r_\mathrm{A}$ and $r_\mathrm{E}$.
Multi-Layer Precoding: The General Concept {#sec:Concept}
==========================================
In this section, we briefly introduce the motivation and general concept of multi-layer precoding. Given the system model in [[Section]{} \[sec:Model\]]{}, the signal-to-interference-plus-noise ratio (SINR) at user $k$ in cell $c$ is $$\mathrm{SINR}_{c k}=\frac{\frac{P}{K} \left|\bh_{c c k}^* \left[\bF_c\right]_{:,k}\right|^2}{\frac{P}{K}
\displaystyle{\sum_{m \neq k}} |\bh_{c c k}^* \left[\bF_c\right]_{:,m}|^2
+ \frac{P}{K} \displaystyle{\sum_{b \neq c}} \|\bh_{b c k}^* \bF_{b}\|^2 + \sigma^2},$$ where $\left|\bh_{c c k}^* \left[\bF_c\right]_{:,k}\right|^2$ is the desired signal power, ${\sum_{m \neq k}} |\bh_{c c k}^* \left[\bF_c\right]_{:,m}|^2$ is the intra-cell multi-user interference, and ${\sum_{b \neq c}} \|\bh_{b c k}^* \bF_{b}\|^2$ is the inter-cell interference. Designing one precoding matrix per BS to manage all these kinds of signals by, for example, maximizing the system sum-rate is non-trivial. This normally leads to a non-convex problem whose closed-form solution is unknown [@Gesbert2010]. Also, coordinating inter-cell interference between BS’s typically results in high cooperation overhead that makes the value of this cooperation limited [@Lozano2013]. Another challenge lies in the entire baseband implementation of these precoding matrices, which may yield high cost and power consumption in massive MIMO systems [@HeathJr2016].
Our objective is to design the precoding matrices, $\bF_{b}$, $b=1, 2, ..., B$, such that (i) they manage the inter-cell and intra-cell interference with low requirements on the channel knowledge, and (ii) they can be implemented using low-complexity hybrid analog/digital architectures [@Alkhateeb2014b], i.e., with a small number of RF chains. Next, we present the main idea of multi-layer precoding, a potential solution to achieve these objectives.
Inspired by prior work on multi-user hybrid precoding [@Alkhateeb2014b] and joint spatial division multiplexing [@Adhikary2013], and leveraging the directional characteristics of large-scale MIMO channels [@Yin2013], we propose to design the precoding matrix $\bF_{c}$ as a product of a number of precoding matrices (layers). In this paper, we will consider a 3-layer precoding matrix $$\bF_{c}= \bF_{c}^{(1)} \bF_{c}^{(2)} \bF_{c}^{(3)},$$ where these precoding layers are designed according to the following criteria.
- **One precoding objective per layer:** Each layer is designed to achieve only one precoding objective, e.g., maximizing desired signal power, minimizing inter-cell interference, or minimizing multi-user interference. This simplifies the precoding design problem and divides it into easier and/or convex sub-problems. Further, this decouples the required channel knowledge for each layer.
- **Successive dimensionality reduction:** Each layer is designed such that the effective channel, including this layer, has smaller dimensions compared to the original channel. This reduces the channel training overhead of every precoding layer compared to the previous one. Further, this makes a successive reduction in the dimensions of the precoding matrices, which eases implementing them using hybrid analog/digital architectures [@HeathJr2016; @ElAyach2014; @Alkhateeb2014b; @Han2015] with small number of RF chains.
- **Different channel statistics:** These precoding objectives are distributed over the precoding layers such that $\bF_{c}^{(1)}$ requires slower time-varying channel state information compared with $\bF_{c}^{(2)}$, which in turn requires slower channel state information compared with $\bF_{c}^{(3)}$. Given the successive dimensionality reduction criteria, this means that the first precoding layer, which needs to be designed based on the large channel matrix, requires very large-scale channel statistics and needs to be updated every very long period of time. Similarly, the second and third precoding layers, which are designed based on the effective channels that have less dimensions, need to be updated more frequently.
In the next sections, we will present a specific multi-layer precoding design for full-dimensional massive MIMO systems, and show how it enables leveraging the large-scale MIMO channel characteristics to manage different kinds of interference with limited channel knowledge. We will also show how the multiplicative and successive reduced dimension structure of multi-layer precoding allows for efficient implementations using hybrid analog/digital architectures.
Proposed Multi-Layer Precoding Design {#sec:Algorithm}
=====================================
In this section, we present a multi-layer precoding algorithm for the full-dimensional massive MIMO system and channel models described in [[Section]{} \[sec:Model\]]{}. Following the proposed multi-layer precoding criteria explained in [[Section]{} \[sec:Concept\]]{}, we propose to design the $N_\mathrm{V} N_\mathrm{H} \times K$ precoding matrix $\bF_b$ of cell $b$, $b=1,...,B$ as $$\begin{aligned}
\label{eq:prec_layer}
\bF_b&=\bF_b^{(1)} \bF_b^{(2)} \bF_b^{(3)}, \end{aligned}$$ where the first precoding layer $\bF^{(1)}_b$ is dedicated to avoid the out-of-cell interference, the second precoding layer $\bF_b^{(2)}$ is designed to maximize the effective signal power, and the third layer $\bF_b^{(3)}$ is responsible for canceling the intra-cell multi-user interference. Writing the received signal at user $k$ in cell $c$ in terms of the multi-layer precoding in , we get $$\label{eq:Received_0}
y_{c k}=\underbrace{\bh_{c c k}^* \bF^{(1)}_{c} \bF^{(2)}_{c} \bF^{(3)}_{c}\bs_{c}}_\text{received signal from serving BS}+ \underbrace{\sum_{b \neq c} \bh_{b c k}^* \bF^{(1)}_{b} \bF^{(2)}_{b} \bF^{(3)}_{b} \bs_{b}}_\text{received signal from other BSs} + n_{c k}.$$
Next, we explain in detail the proposed design of each precoding layer as well as the required channel knowledge.
First Layer: Inter-Cell Interference Management
-----------------------------------------------
We will design the first precoding layer ${\bF_b}^{(1)}$ to avoid the inter-cell interference, i.e., to cancel the second term of . Exploiting the Kronecker structure of the channel model in , we propose to construct the first layer as $$\label{eq:First_layer}
\bF^{(1)}_b={\bF^\mathrm{A}_b}^{(1)} \otimes {\bF^\mathrm{E}_b}^{(1)}.$$
Adopting the channel model in with $\overline{\bw}_{b c k}=\left({\boldsymbol{\Lambda}_{b c k}^{\mathrm{A}}}^{\frac{1}{2}} \otimes {\boldsymbol{\Lambda}_{b c k}^{\mathrm{E}}}^{\frac{1}{2}}\right) \bw_{b c k}$ and employing the Kronecker precoding structure in , the second term of the received signal $y_{c k}$ in can be expanded as $$\sum_{b \neq c} \bh_{b c k}^* \bF^{(1)}_{b} \bF^{(2)}_{b} \bF^{(3)}_{b} \bs_{b}= \sum_{b \neq c} \overline{\bw}_{b c k}^* \left(\bU_{b c k}^{\mathrm{A}^*} {\bF_b^\mathrm{A}}^{(1)} \otimes \bU_{b c k}^{\mathrm{E}^*} {\bF_b^\mathrm{E}}^{(1)} \right) \bF^{(2)}_{b} \bF^{(3)}_{b} \bs_{b}.$$
Avoiding the inter-cell interference for the users at cell $c$ can then be satisfied if $\bF_b^{(1)}, b \neq c$ is designed such that $ \bU_{b c k}^{\mathrm{E}^*} {\bF_b^\mathrm{E}}^{(1)}=\boldsymbol{0}, \forall k$. Equivalently, for any cell $c$ to avoid making interference on the other cell users, it designs its precoder ${\bF_c^\mathrm{E}}^{(1)}$ to be in the null-space of the elevation covariance matrices of all the channels connecting BS $c$ and the other cell users, i.e., to be in $\mathrm{Null}\left(\sum_{b \neq c} \sum_{k\in \cK_b} \bR_{c b k}^\mathrm{E}\right)$ with $\cK_b$ denoting the subset of $K$ scheduled users in cell $b$. Note that with large numbers of vertical antennas and for several channel models, this elevation inter-cell interference covariance will not have full rank and may actually have just a small overlap with the desired users’ channels, as will be shown in [[Section]{} \[sec:Perf\]]{}.
Thanks to the directional structure of large-scale MIMO channels, we further note that with a large number of vertical antennas, $N_\mathrm{V}$, the null-space $\mathrm{Null}\left(\sum_{b \neq c} \sum_{k\in \cK_b} \bR_{c b k}^\mathrm{E}\right)$ of different scheduled users $\cK_b$ will have a large overlap. This means that designing ${\bF_c^\mathrm{E}}^{(1)}$ based on the interference covariance subspace averaged over different scheduled users may be sufficient. Leveraging this intuition relaxes the required channel knowledge to design the first precoding layer. Hence, we define the average interference covariance matrix for BS $c$ as $$\bR_{c}^{\mathrm{I}}=\sum_{b \neq c} \bbE_{\cK_b} \left[\bR^{E}_{c b k}\right].
\label{eq:Interference}$$
In this paper, we manage the inter-cell interference in the elevation space, and therefore, set ${\bF^\mathrm{A}_b}^{(1)}=\bI_{N_\mathrm{H}}$. Let $\left[\bU_{c}^\mathrm{I} \ \bU_{c}^\mathrm{NI}\right] \boldsymbol{\Lambda}_c \left[\bU_{c}^\mathrm{I} \bU_{c}^\mathrm{NI}\right]^*$ represent the eigen-decomposition of $\bR_{c}^{\mathrm{I}}$ with the $N_\mathrm{v}\times r_\mathrm{I}$ matrix $\bU_{c}^\mathrm{I}$ and $N_\mathrm{v}\times r_\mathrm{NI}$ matrix $\bU_{c}^\mathrm{NI}$ corresponding to the non-zero and zero eigenvalues, respectively. Then, we design the first precoding layer $\bF_c^{(1)}$ to be in the null-space of the average interference covariance matrix by setting $$\label{eq:first_layer2}
\bF_c^{(1)}=\bI_{N_\mathrm{H}} \otimes \bU_{c}^\mathrm{NI},$$ which is an $N_\mathrm{V} N_\mathrm{H} \times \rN N_\mathrm{H}$ matrix.
Given the design of the first precoding layer in , and defining the $\rN \times \rE$ effective elevation eigen matrix $\overline{\bU}_{c c k}^\mathrm{E}={\bUN_{c}}^* \bU^\mathrm{E}_{c c k}$, the received signal at user $k$ of cell $c$ in becomes $$y_{c k} = \overline{\bw}_{c c k}^* \left(\bU_{c c k}^{\mathrm{A}^*} \otimes \overline{\bU}_{c c k}^{\mathrm{E}^*}\right) \bF^{(2)}_{c} \bF^{(3)}_{c} \bs_{c}+ n_{c k}.
\label{eq:Received_2}$$ Note that the first precoding layer in acts as a spatial filter that entirely eliminates the inter-cell interference in the elevation domain. This filter, however, may have a negative impact on the desired signal power for the served users at cell $c$ if they share the same elevation subspace with the out-of-cell users. Therefore, this first layer precoding design is particularly useful for systems with low-rank elevation subspaces. It is worth mentioning here that recent measurements of 3D channels show that elevation eigenspaces may have low ranks at both low-frequency and millimeter wave systems [@3GPP_LTE; @Akdeniz2014; @Hur2016]. Relaxations of the precoding design in are proposed in [[Section]{} \[sec:Discuss\]]{} to compromise between inter-cell interference avoidance and desired signal power degradation.
**Required channel knowledge:** The design of the first precoding layer in requires only the knowledge of the interference covariance matrix *averaged* over different scheduled users. It depends therefore on very large time-scale channel statistics, which means that this precoding layer needs to be updated every very long period of time. This makes its acquisition overhead relatively small from an overall system perspective. In fact, this is a key advantage of the decoupled multi-layer precoding structure that allows dedicating one layer for canceling the out-of-cell interference based on large time-scale channel statistics while leaving the other layers to do other functions based on different time scales. This can not be done by typical precoding approaches that rely on one precoding matrix to manage different precoding objectives, as this precoding matrix will likely need to be updated based on the fastest channel statistics.
Second Layer: Desired Signal Beamforming
----------------------------------------
The second precoding layer $\bF_c^{(2)}$ is designed to focus the transmitted power on the served users’ effective subspaces, i.e., on the user channels’ subspaces including the effect of the first precoding layer. If we define the matrix consisting of the effective eigenvectors of user $k$ in cell $c$ as $\overline{\bU}_{c c k}=\left(\bU_{c c k}^{\mathrm{A}} \otimes \overline{\bU}_{c c k}^{\mathrm{E}}\right)$, then we design the second precoding layer $\bF_c^{(2)}$ as a large-scale conjugate beamforming matrix, i.e., we set $$\label{eq:second_layer}
\bF_c^{(2)}=\left[\overline{\bU}_{c c 1}, ..., \overline{\bU}_{c c K}\right],$$ which has $N_\mathrm{H} \rN \times K \rA \rE$ dimensions. Given the second precoding layer design, and defining $\bG_{c,(k,r)}=\overline{\bU}_{c c k}^* \overline{\bU}_{c c r}$, the received signal by user $k$ in cell $c$ can be written as $$y_{c k} = \overline{\bw}_{c c k}^* \left[\bG_{c,(k,1)}, ..., \bG_{c,(k,K)}\right] \bF_{c}^{(3)} \bs_c + n_{c k}.
\label{eq:Received_2x}$$ The main objectives of this precoding layer can be summarized as follows. First, the effective channel vectors, including the first and second precoding layers, will have reduced dimensions compared to the original channels, especially when large numbers of antennas are employed. This reduces the overhead associated with training the effective channels, which is particularly important for FDD systems [@Adhikary2013; @Alkhateeb2014b]. Second, this precoding layer supports the multiplicative structure of multi-layer precoding with successive dimensionality reduction, which simplifies its implementation using hybrid analog/digital architectures, as will be briefly discussed in [[Section]{} \[sec:Discuss\]]{}.
**Required channel knowledge:** The design of the second precoding layer requires only the knowledge of the effective eigenvector matrices $\overline{\bU}_{c c k}, k=1, ..., K$, which depends on the large-scale channel statistics. It is worth noting that during the uplink training of the matrices $\overline{\bU}_{c c k}$, the first precoding layer works as spatial filtering for the other cell interference. Hence, this reduces (and ideally eliminates) the channel estimation error due to pilot reuse among cells, and consequently leads to a pilot decontamination effect.
Third Layer: Multi-User Interference Management
-----------------------------------------------
The third precoding layer $\bF_c^{(3)}$ is designed to manage the multi-user interference based on the effective channels, i.e., including the effect of the first and second precoding layers. If we define the effective channel of user $k$ in cell $c$ as $\overline{\bh}_{c k}=\left[\bG_{c,(k,1)}, ..., \bG_{c,(k,K)}\right]^* \overline{\bw}_{c c k}$, and let $\overline{\bH}_c=[\overline{\bh}_{c 1}, ..., \overline{\bh}_{c K}]$, then we construct the third precoding layer $\bF_c^{(3)}$ as a zero-forcing matrix $$\label{eq:third_layer}
\bF_c^{(3)}=\overline{\bH}_c\left(\overline{\bH}_c^* \overline{\bH}_c\right)^{-1} \bUpsilon_{c},$$ where $\bUpsilon_{c}$ is a diagonal power normalization matrix that ensures satisfying the precoding power constraint $\|\left[\bF_b\right]_{:,k}\|^2=1$. Note that this zero-forcing design requires $N_\mathrm{H} \rN \geq K \rA \rE$, which is satisfied with high probability in massive MIMO systems, especially with sparse and low-rank channels. Given the design of the precoding matrix $\bF_c^{(3)}$, the received signal at user $k$ in cell $c$ can be expressed as $$y_{c k} = \left[\bUpsilon_{c}\right]_{k,k} s_{c,k} + n_{c k}.
\label{eq:Received_3}$$
**Required channel knowledge:** The design of the third precoding layer relies on the instantaneous effective channel knowledge. Thanks to the first and second precoding layers, these effective channels should have much smaller dimensions compared to the original channels in massive MIMO systems, which reduces the required training overhead.
Performance Analysis {#sec:Perf}
====================
The proposed multi-layer precoding design in [[Section]{} \[sec:Algorithm\]]{} eliminates inter-cell interference as well as multi-user intra-cell interference, assuming that every BS $b$ has the knolwedge of its users’ effective channels and channel covariance $\overline{\bH}_b$, $\left\{\bR_{cck}\right\}$ and the averaged inter-cell interference covariance in the elevation dimension $\bR_c^\mathrm{I}$. This interference cancellation, however, may have a penalty on the desired signal power which is implicitly captured by the power normalization factor $\left[\bUpsilon_c\right]_{k,k}$ in . In this section, we will first characterize the achievable rate by the proposed multi-layer precoding design for a general channel model in Lemma \[lem:Ach\_Rate\]. Then, we will show that this precoding design can achieve optimal performance for some special yet important channel models in Section \[subsec:OneRing\] and [[Section]{} \[subsec:Physical\]]{}.
\[lem:Ach\_Rate\] Consider the system and channel models in [[Section]{} \[sec:Model\]]{} and the multi-layer precoding design in [[Section]{} \[sec:Algorithm\]]{}. The achievable rate by user $k$ in cell $c$ is given by $$\label{eq:Ach_Rate_1}
R_{ck}=\log_2\left(1+ \frac{\mathsf{SNR}}{\left( \bW^*_c {\bF_{c}^{(2)}}^* {\bF_{c}^{(2)}} {\bF_{c}^{(2)}}^* {\bF_{c}^{(2)}} \bW_c \right)_{k,k}^{-1}}\right),$$ where $\bW_c=\bI_{K} \circ \left[\overline{\bw}_{c c 1}, ..., \overline{\bw}_{c c K}\right]$ and $\mathsf{SNR}=\frac{P}{K \sigma^2}$.
See Appendix \[app:Ach\_Rate\]
Note that the achievable rate in is upper bounded by the single-user rate—the rate when the user is solely served in the network—which is given by $\overline{R}_{ck}=\log_2\left(1+\mathsf{SNR} \left\|\overline{\bw}_{c c k}\right\|^2\right)$. Therefore, Lemma \[lem:Ach\_Rate\] indicates that the proposed multi-layer precoding can achieve optimal performance if ${\bF_c^{(2)}}^*{\bF_c^{(2)}}=\bI$. To achieve that, it is sufficient to satisfy the following two conditions.
(i) [$\bG_{c,(k,m)}=\boldsymbol{0}, \forall m\neq k$, a condition that captures the impact of multi-user interference cancellation on the desired signal power.]{}
(ii) [$\bG_{c,(k,k)}=\left({\bU_{c c k}^{\mathrm{A}}}^* \otimes {{\bU}_{c c k}^{\mathrm{E}}}^*\right){\bF_c^{(1)}} {\bF_c^{(1)}}^*\left({\bU_{c c k}^{\mathrm{A}}} \otimes {\bU}_{c c k}^{\mathrm{E}}\right)=\bI, \forall k$, a condition that captures the possible impact of the inter-cell interference avoidance on the desired signal power.]{}
Next, we characterize the performance of multi-layer precoding for two special yet important channel models, namely, the one-ring and single-path channel models.
Performance with One-Ring Channel Models {#subsec:OneRing}
----------------------------------------
Motivated by its analytical tractability and meaningful geometrical interpretation, we will consider the one-ring channel model in this subsection [@Shiu2000; @Petrus2002; @Abdi2002; @Zhang2007]. This will enable us to draw useful insights into the performance of multi-layer precoding, which can then be extended to more general channel models. Note that due to its tractability, one-ring channel models have also been adopted in prior massive MIMO work [@Bjoernson2014; @Adhikary2013; @Yin2013; @Shen2016].
![An illustration of the one-ring channel model in the azimuth direction. The BS that, has a UPA in the y-z plane, serves a mobile user in the x-y plane at distance $d_{c k}$. The user is surrounded by scatterers on a ring of radius $r_{c k}$, and its channel experiences an azimuth angular spread $\Delta_\rm{A}$.[]{data-label="fig:OneRing"}](OneRingV2.pdf){width=".45\columnwidth"}
The one-ring channel model describes the case when a BS is elevated away from scatterers and is communicating with a mobile user that is surrounded by a ring of scatterers. Consider a BS at height $H_\mathrm{BS}$ employing an $N_V \times N_H$ UPA, and serving a mobile user at a distance $d_{ck}$ with azimuth and elevation angles $\phi_{ck}, \theta_{ck}$, as depicted in . If the mobile is surrounded by scatterers on a ring of radius $r_{ck}$ in the azimuth dimension, then the azimuth angular spread $\Delta_\mathrm{A}$ can be approximated as $\Delta_\mathrm{A}=\arctan{\left(\frac{r_{ck}}{d_{ck}}\right)}$. Further, assuming for simplicity that the received power is uniformly distributed over the ring, then the correlation between any two antenna elements with orders $n_1, n_2$ in the horizontal direction is given by $$\label{eq:A_Cov}
\left[\bR^{A}_{cck}\right]_{n_1,n_2}=\frac{1}{2 \Delta_\mathrm{A}}\int_{-\Delta_\mathrm{A}}^{\Delta_\mathrm{A}} {e^{- j \frac{2 \pi}{\lambda} d (n_2-n_1) \sin(\phi_{ck}+\alpha)\sin(\theta_{ck})} d\alpha}.$$
The elevation correlation matrix can be similarly defined for the user $k$, in terms of its elevation angular spread $\Delta_\mathrm{E}$.
In the next theorem, we characterize the achievable rate for an arbitrary user $k$ in cell $c$ under the one-ring channel model.
\[th:CellCenter\] Consider the full-dimensional cellular system model in [[Section]{} \[sec:SysModel\]]{} with cells of radius $r_\mathrm{cell}$, and the channel model in [[Section]{} \[sec:ChModel\]]{} with the one-ring correlation matrices in . Let $\phi_{c k}, \theta_{c k}$ denote the azimuth and elevation angles of user $k$ at cell $c$, and let $\Delta_\mathrm{A}, \Delta_\mathrm{E}$ represent the azimuth and elevation angular spread. Define the maximum distance with no blockage on the desired signal power as $d_\mathrm{max}=H_\mathrm{BS} \tan\left(\arctan\left(\frac{r_\mathrm{cell}}{H_\mathrm{BS}}\right)- 2 \Delta_\mathrm{E}\right)$. If $\left|\phi_{c k}-\phi_{c m}\right| \geq 2 \Delta_\rm{A}$ or $\left|\theta_{c k}-\theta_{c m}\right| \geq 2 \Delta_\rm{E}$, $\forall m \neq k$, and $d_{c k} \leq d_\rm{max}$, then the achievable rate of user $k$ at cell $c$, when applying the multi-layer precoding algorithm in [[Section]{} \[sec:Algorithm\]]{}, satisfies $$\label{eq:Rate_InCell}
\lim_{N_\rm{V}, N_\rm{H} \rightarrow \infty} R_{c k}=\overline{R}_{c k}=\log_2\left(1+\mathsf{SNR} \left\|\overline{\bw}_{c c k}\right\|^2 \right).$$
See Appendix \[app:CellCenter\]
Theorem \[th:CellCenter\] indicates that the achievable rate with multi-layer precoding converges to the optimal single-user rate for the users that are not at the cell edge ($r_\rm{cell}-d_\mathrm{max}$ away from cell edge), provided that they maintain either an azimuth or elevation separation by double the angular spread. For example, consider a cellular system with cell radius $100$m and BS antenna height $50$m, if the elevation angular spread equals $\Delta_\rm{E}=3^\circ$, then all the users within $\sim 80$m distance from the BS achieve optimal rate. It is worth noting here that these rates do not experience any pilot contamination or multi-user interference impact and can, therefore, grow with the antenna numbers or transmit power without any bound on the maximum values that they can reach.
The angular separation between the users in Theorem \[th:CellCenter\] can be achieved via user scheduling techniques or other network optimization tools. In fact, even without user scheduling, this angular separation is achieved with high probability as will be illustrated by simulations in [[Section]{} \[sec:Results\]]{} under reasonable system and channel assumptions. Further, for sparse channels with finite number of paths, it can be shown that this angular separation is not required to achieve the optimal rate. Studying these topics are interesting future extensions.
In the following theorem, we derive a lower bound on the achievable rate with multi-layer precoding for the cell-edge users.
\[th:CellEdge\] Consider the system and channel models described in Theorem \[th:CellCenter\]. If $\left|\phi_{c k}-\phi_{c m}\right| \geq 2 \Delta_\rm{A}$ or $\left|\theta_{c k}-\theta_{c m}\right| \geq 2 \Delta_\rm{E}$, $\forall m \neq k$, and $d_\rm{max} \leq d_{c k} \leq r_\rm{cell}$, then the achievable rate of user $k$ at cell $c$, when applying the multi-layer precoding algorithm in [[Section]{} \[sec:Algorithm\]]{}, satisfies $$\label{eq:Rate_CellEdge}
\lim_{N_\rm{V}, N_\rm{H} \rightarrow \infty} R_{c k} \geq \log_2\left(1+\mathsf{SNR} \left\|\overline{\bw}_{c c k}\right\|^2 \sigma^2_\rm{min}\left(\overline{\bU}_{c c k}^\rm{E}\right)\right),$$ where $\sigma_\rm{min}\left(\bA\right)$ denotes the minimum singular value of the matrix $\bA$.
See Appendix \[app:CellEdge\]
Theorem \[th:CellEdge\] indicates that cell edge users experience some degradation in their $\mathsf{SNRs}$ as a cost for the perfect inter-cell interference avoidance. In [[Section]{} \[sec:Discuss\]]{}, we will discuss some solutions that make compromises between the degradation of the desired signal power and the management of the inter-cell interference for cell-edge users, under the multi-layer precoding framework.
Performance with Single-Path Channel Models {#subsec:Physical}
-------------------------------------------
Rank-1 channel models describe the cases where the signal propagation through the channel is dominated by one line-of-sight (LOS) or non-LOS (NLOS) path. This is particularly relevant to systems with sparse channels, such as mmWave systems [@Bai2014; @Rappaport2013a; @Hur2016]. A special case of rank-1 channel models is the single-path channels. Consider a user $k$ at cell $c$ with a single path channel, defined by its azimuth and elevation angles $\phi_{c k}, \theta_{c k}$. Then, the channel vector can be expressed as $$\label{eq:Rank1_CH}
\bh_{c c k}=\rho^{\frac{1}{2}}_{cc k} \ \beta_{c k} \ \ba_\rm{A}\left(\phi_{c k}, \theta_{c k}\right) \otimes \ba_\rm{E}\left(\phi_{c k}, \theta_{c k}\right),$$ where $\ba_\rm{A}\left(\phi_{c k}, \theta_{c k}\right)$ and $\ba_\rm{E}\left(\phi_{c k}, \theta_{c k}\right)$ are the azimuth and elevation array response vectors, $\beta_{c k}$ is the complex path gain, and $\rho_{c c k}$ is its path loss.
In the next corollary, we characterize the achievable rate of the proposed multi-layer precoding design for single-path channels.
\[cor:Rank1\] Consider the full-dimensional cellular system model in [[Section]{} \[sec:SysModel\]]{}, and the single-path channel model in . When applying the multi-layer precoding algorithm in [[Section]{} \[sec:Algorithm\]]{}, the achievable rate of user $k$ at cell $c$ satisfies $$\lim_{N_\rm{V}, N_\rm{H} \rightarrow \infty} R_{c k}= \overline{R}_{c k} = \log_2\left(1+ \mathsf{SNR} \left\|\bh_{ c c k}\right\|^2 \right).$$
The proof is similar to that of Theorem \[th:CellCenter\], and is omitted due to space limitations.
Corollary \[cor:Rank1\] indicates that the proposed multi-layer precoding design can achieve an optimal performance for single-path channels, making it a promising solution for mmWave and low channel rank massive MIMO systems. This will also be verified by numerical simulations in [[Section]{} \[sec:Results\]]{}.
Discussion and Extensions {#sec:Discuss}
=========================
While we proposed and analyzed a specific multi-layer precoding design in this paper, there are many possible extensions as well as important topics that need further investigations. In this section, we briefly discuss some of these points, leaving their extensive study for future work.
Multi-Layer Precoding with Augmented Vertical Dimensions {#subsec:Aug}
--------------------------------------------------------
As explained in [[Section]{} \[sec:Algorithm\]]{}, the proposed multi-layer precoding design attempts to perfectly avoid the inter-cell interference by forcing its transmission to be in the elevation null-space of the interference. While this guarantees optimal performance for cell-interior users and decontaminates the pilots for all the cell users, it may also block some of the desired signal power at the cell-edge. In this section, we propose a modified design for the first precoding layer $\bF_c^{(1)}$ that compromises between the inter-cell interference avoidance and the desired signal degradation. The main idea of the proposed design, that we call multi-layer precoding with augmented vertical dimensions, is to simply extend the null-space of the inter-cell interference via exploiting the structure of large channels. This is summarized as follows. Leveraging Lemma 2 in [@Yin2013], the rank of the one-ring correlation matrix can be related to its angular range $\left[\theta_\rm{min}, \theta_\rm{max}\right]$ as $$\text{rank}\left(\bR\right)=\frac{N D}{\lambda} \left(\cos(\theta_\rm{min})-\cos(\theta_\rm{max})\right) \ \text{as} \ N \rightarrow \infty.$$ Applying this lemma to the elevation inter-cell interference subspace, setting $\theta_\rm{min}=\pi/2$, BS $c$ can estimate its maximum interference elevation angle, denoted $\theta_c^\rm{I}$, as $$\theta_c^\rm{I}=\arccos\left(-\frac{\text{rank}\left(\bR_c^\rm{I}\right) \lambda}{N_\rm{V} D}\right).$$
Extending the null space of the interference can then be done by virtually reducing the inter-cell interference subspace. Let $\delta_\rm{E}$ denote the angular range of the extended subspace. The modified inter-cell interference covariance can then be calculated as $$\left[\overline{\bR}_c^\rm{I}\right]_{n_1,n_2}=\frac{1}{\theta_c^\rm{I}-\delta_\rm{E}-\pi/2} \int_{\frac{\pi}{2}}^{\theta_c^\rm{I}-\delta_\rm{E}} e^{j k D (n_2-n_1) \cos(\alpha)} d\alpha.$$
Finally, if $\left[\overline{\bU}_{c}^\mathrm{I} \ \overline{\bU}_{c}^\mathrm{NI}\right] \overline{\boldsymbol{\Lambda}}_c \left[\overline{\bU}_{c}^\mathrm{I} \overline{\bU}_{c}^\mathrm{NI}\right]^*$ represents the eigen-decomposition of $\overline{\bR}_c^\rm{I}$, with $\overline{\bU}_{c}^\mathrm{I}$ and $\overline{\bU}_{c}^\mathrm{NI}$ correspond to the non-zero and zero eigenvalues, then the modified first precoding layer can be constructed as $$\bF_{c}^{(1)}= \bI \otimes \overline{\bU}_c^\rm{NI}.$$
Note that under this multi-layer precoding design, only cell edge users will experience inter-cell interference and pilot contamination while optimal performance is still guaranteed for cell-interior users. This yields an advantage for multi-layer precoding over conventional massive MIMO precoding schemes, which will also be illustrated by numerical simulations in [[Section]{} \[sec:Results\]]{}.
TDD and FDD Operation with Multi-Layer Precoding
------------------------------------------------
While we focused on TDD systems in this paper, the fact that multi-layer precoding relies on channel covariance knowledge makes it attractive for FDD operation as well. In FDD systems, the adjacent cells will cooperate to construct the elevation inter-cell interference subspace, which is needed to build the first precoding layer. Since this channel knowledge is of very large-scale statistics and this precoding layer needs to be updated every long time period, this cooperation overhead can be reasonably low. Given the first layer spatial filtering, every BS can estimate its users covariance knowledge free of inter-cell interference. Thanks to the multiplicative structure of the multi-layer precoding and its successive dimensionality reduction, only the third precoding layer requires the instantaneous knowledge of the effective channel, which has much smaller dimensions. It is worth noting here that other FDD massive MIMO precoding schemes, such as JSDM [@Adhikary2013] with its user grouping functions, can be easily integrated into the proposed multi-layer precoding framework for full-dimensional massive MIMO cellular systems.
In TDD systems, the required channel knowledge for the three stages can be done through uplink training on different time scales. One important note is that the second precoding layer (and its channel training) may not be needed in TDD systems with fully-digital transceivers, as the instantaneous channels can be easily trained in the uplink with a small number of pilots. This precoding layer, however, is important if multi-layer precoding is implemented using hybrid architectures, as will be shown in the following subsection.
Multi-Layer Precoding using Hybrid Architectures
------------------------------------------------
![The figure shows a hybrid analog/digital architecture, at which baseband precoding, RF precoding, and antenna downtilt beamforming can be utilized to implement the multi-layer precoding algorithm.[]{data-label="fig:Beam"}](Hybrid_MLP.pdf){width=".5\columnwidth"}
Thanks to the multiplicative structure and the specific multi-layer precoding design in [[Section]{} \[sec:Algorithm\]]{}, we note that each precoding layer has less dimensions compared to the prior layers. This allows the multi-layer precoding matrices to be implemented using hybrid analog/digital architectures[@HeathJr2016; @ElAyach2014; @Alkhateeb2014b; @Han2015], which reduces the required number of RF chains. In this section, we briefly highlight one possible idea for the hybrid analog/digital implementation, leaving its optimization and extensive investigation for future work.
Considering the three-stage multi-layer precoding design in [[Section]{} \[sec:Algorithm\]]{}, we propose to implement the first and second layers in the RF domain and perform the third layer precoding at baseband, as depicted in . Given the successive dimensional reductions, the required number of RF chains is expected to be much less than the number of antennas, especially in sparse and low-rank channels. As the first precoding layer focuses on avoiding the inter-cell interference in the elevation direction, we can implement it using downtilt directional antenna patterns. We assume that each antenna port has a directional pattern and electrically adjusted downtilt angle [@Kammoun2014; @Seifi2014a]. For example, the 3GPP antenna port elevation gain $G^\mathrm{E}\left(\theta\right)$ is defined as [@Kammoun2014] $$G^\mathrm{E}(\theta)=G^\mathrm{E}_\mathrm{max}-\min\left\{12 \left(\frac{\theta-\theta_\mathrm{tilt}}{\theta_\mathrm{3dB}}\right)^2, \mathrm{SL}\right\},$$ where $\theta_\mathrm{tilt}$ is the downtilt angle, and SL is the sidelobe level. Therefore, one way to approximate ${\bF_c^{\mathrm{E}}}^{(1)}$ is to adjust the downtilt angle $\theta_\mathrm{tilt}$ to minimize the leakage transmission outside the interference null-space $\bU_c^\mathrm{NI}$.
Once $\bF_c^{(1)}$ is implemented, the second precoding layer $\bF_c^{(2)}$ can be designed similar to [@Alkhateeb2014b], i.e., each column of $\bF_c^{(2)}$ can be approximated by a beamsteering vector taken from a codebook that captures the analog hardware constraints. Finally, the third precoding layer $\bF_c^{(3)}$ is implemented in the baseband to manage the multi-user interference based on the effective channels.
Simulation Results {#sec:Results}
==================
In this section, we evaluate the performance of the proposed multi-layer precoding algorithm using numerical simulations. We also draw insights into the impact of the different system and channel parameters.
We consider a single-tier 7-cell cellular system model as depicted in , and calculate the performance for the cell in the center. Unless otherwise mentioned, every BS is assumed to a have a UPA, oriented in the y-z plane, at a height $H_\rm{BS}=35$m, and serving users at cell radius $r_\rm{cell}=100$m. Users are randomly and uniformly dropped in the cells, and every cell randomly schedules $K=20$ users to be served at the same time and frequency slot. The BS transmit power is $P=5$ dB and the receiver noise figure is $7$ dB. The system operates at a carrier frequency $4$ GHz, with a bandwidth $10$ MHz, and a path loss exponent $3.5$. Two channel models are assumed, namely, the single-path and the one-ring channel models.
The BSs in the adopted system apply the multi-layer precoding algorithm in [[Section]{} \[sec:Algorithm\]]{}. The required channel knowledge is perfectly obtained from the geometry of the network, i.e., no actual channel estimation is applied. We assume a universal pilot reuse, i.e., all the cells randomly assign the same $K$ orthogonal pilots to its users. The channels of the users sharing the same pilots are added at every BS, which simulates the interference of the other cells’ users in the channel estimation phase. In more detail, the channel estimation and multi-layer precoding are done as follows. First, the averaged interference covariance matrix $\bR_b^\mathrm{I}$ of every BS $b$ is constructed by averaging the elevation interference covariance over $40$ realizations of scheduled users, each has $20$ users/cell. Using this knowledge, the first-layer precoders are obtained according to . Then, the effective channel covariance matrices $\overline{\bU}_{c c k}$ are calculated by applying the first precoding layer on the sum of the channel covariance matrices of the users that share the same pilots from the different cells. Note that the first-layer precoders act as spatial filters that reduce (and ideally eleminate) the contributions of the other cells in the sum of the channel covariance. The second-layer precoders are then obtained following . The effective channels are similarly calculated, applying the first and second precoding layers, from which the third-layer precoders are constructed using . For the other precoding schemes we compare with, the channels are similarly constructed using the geometry and by adding the co-pilot user channels. Next, we present the simulation results for the two adopted channel models.
Results with Single-Path Channels
---------------------------------
In this section, we adopt a single-path model for the user channels as described in . The azimuth and elevation angles are geometrically determined based on users’ locations relative to the BSs, and the complex path gains $\beta_{c k} \sim \mathcal{CN} \left(0,1\right)$.
\[fig:Imact\_Red\]
**Optimality with large antennas:** In , we compare the per-user achievable rate of multi-layer precoding with the single-user rate and the rate with conventional conjugate beamforming. The BSs are assumed to employ UPAs that have $N_\rm{H}=30$ horizontal antennas and different numbers of vertical antennas. First, we note that the per-user achievable rate with multi-layer precoding approaches the optimal single-user rate as the number of antennas grow large. This verifies the asymptotic optimality result of multi-layer precoding given in Corollary \[cor:Rank1\]. Note that the single-user rate is the rate if only this user is served in the network, i.e., with no inter-cell or multi-user intra-cell interference. In the figure, we also plot the achievable rate with conventional conjugate beamforming. This assumes that channels are estimated using uplink training and then conjugate beamforming is applied in the downlink data transmission [@Marzetta2010]. As a function of the path-loss $\rho_{b c k}$ in , the conjugate beamforming rate is theoretically bounded from above by [@Marzetta2010] $$\overline{R}_{c k}^{CB}= \log_2\left(1+\mathsf{SNR} \frac{\rho_{c c k}^2}{\sum_{b \neq c} \rho_{b c k}^2}\right),$$ which limits its rate from growing with the number of antennas beyond this value. Interestingly, the multi-layer precoding rate does not have a limit on its rate and can grow with the number of antennas and transmit power without a theoretical limit. The intuition behind that lies in the inter-cell interference avoidance using averaged channel covariance knowledge in multi-layer precoding. This works as a spatial filtering that avoids uplink channel estimation errors due to pilot reuse among cells and cancels inter-cell interference in the downlink data transmission. Therefore, the multi-layer precoding rate is free of the pilot-contamination impact. Note that while the asymptotic optimality of multi-layer precoding is realized at large antenna numbers, shows it can still achieve gain over conventional massive MIMO beamforming schemes at much lower number of antennas.
**Impact of antenna heights and cell radii:**
![The rate coverage gain of the proposed multi-layer precoding algorithm over conventional conjugate beamforming and zero-forcing is illustrated. This rate coverage is also shown to be close to the single-user case. The BSs are assumed to employ $120 \times 30$ UPAs at heights $H_\rm{BS}=35$m, the cell radius is $r_\rm{cell}=100$m, and the users have single-path channels.[]{data-label="fig:Cov_Rank1"}](Cov_R1_N120x30_R100_H35_U20.pdf){width=".7\columnwidth"}
In , we evaluate the impact of the BS antenna height and cell radius on the achievable rates. This figure adopts the same system and channel assumptions as in . In (a), the achievable rates for multi-layer precoding, single-user, and conjugate beamforming are compared for different antenna heights, assuming cells of radius $200$m. The figure shows that multi-layer precoding approaches single-user rates at higher antenna heights. This is intuitive because forcing the transmission to become in the elevation null-space of the interference may have less impact on the desired signal blockage if higher antennas are employed. Note that the convergence to the single-user rate is expected to happen at lower antenna heights when large arrays are deployed. These achievable rates are again compared in (b), but for different cell radii. This figure illustrates that a higher cell radius generally leads to less rate because of the higher path loss. Further, the difference between single-user and multi-layer precoding rates increases at higher cell radii. In fact, this is similar to the degradation with smaller antenna heights, i.e., due to the impact of the inter-cell interference avoidance on the desired signal power. For reasonable antenna heights and cell radii, however, the multi-layer precoding still achieves good gain over conventional conjugate beamforming.
**Rate coverage:** To evaluate the rate coverage of multi-layer precoding, we plot . The same setup of is adopted again with cells of radius $100$m, and BSs with $120 \times 30$ UPAs at heights $35$m. First, the figure shows that multi-layer precoding achieves very close coverage to the single-user case, especially for users not at the cell edge. For example, $\sim 60\%$ of the multi-layer precoding users get the same rate of the single-user case. At the cell edge, some degradation is experienced due to the first precoding layer that filters out-of-cell interference and affects the desired signal power. This loss, though, is expected to decrease as more antennas are employed. The figure also shows significant rate coverage gain over conventional conjugate beamforming and zero-forcing precoding solutions.
![The rate coverage gain of the proposed multi-layer precoding algorithms over conventional conjugate beamforming and zero-forcing is illustrated. This rate coverage is also shown to be close to the single-user case. Further, the modified algorithm with augmented vertical dimensions can overcome the cell-edge blockage. The BSs are assumed to employ $100 \times 40$ UPAs at heights $H_\rm{BS}=35$m, the cell radius is $r_\rm{cell}=100$m. The users have one-ring channel models of azimuth and elevation angular spread $\Delta_\rm{A}=5^\circ, \Delta_\rm{E}=3^\circ$[]{data-label="fig:Cov_OneRing_80"}](Cov_M100x40_R100_H35_U20F.pdf){width=".63\columnwidth"}
![The rate coverage gain of the proposed multi-layer precoding algorithms over conventional conjugate beamforming and zero-forcing is illustrated. This rate coverage is also shown to be close to the single-user case. Further, the modified algorithm with augmented vertical dimensions can overcome the cell-edge blockage. The BSs are assumed to employ $140 \times 40$ UPAs at heights $H_\rm{BS}=35$m, the cell radius is $r_\rm{cell}=100$m. The users have one-ring channel models of azimuth and elevation angular spread $\Delta_\rm{A}=5^\circ, \Delta_\rm{E}=3^\circ$.[]{data-label="fig:Cov_OneRing_140"}](Cov_M140x40_R100_H35_U20F.pdf){width=".63\columnwidth"}
Results with One-Ring Channels
------------------------------
In this section, we adopt a one-ring model for the user channels as described in . The azimuth and elevation angles are geometrically determined based on users’ locations relative to the BSs, and the angular spread is set to $\Delta_\rm{A}=5^\circ, \Delta_\rm{E}=3^\circ$. Every BS randomly selects $K=20$ users to be served, i.e., no scheduling is done to guarantee the angular separation condition in Theorem \[th:CellCenter\] and Theorem \[th:CellEdge\].
![The rate coverage gain of the proposed multi-layer precoding algorithms over conventional single-cell conjugate beamforming and multi-cell MMSE precoding. This rate coverage is also shown to be close to the single-user case. The BSs are assumed to be at heights $H_\rm{BS}=35$m, the cell radius is $r_\rm{cell}=100$m. The users have one-ring channel models of azimuth and elevation angular spread $\Delta_\rm{A}=5^\circ, \Delta_\rm{E}=3^\circ$.[]{data-label="fig:Comp_MMSE"}](Comparisons_M80x20_U5_H35_R100_MMSE.pdf){width=".7\columnwidth"}
**Rate coverage:** In -, we compare the rate coverage of multi-layer precoding, single-user, and conventional conjugate beamforming, for different antenna sizes. We also plot the rate coverage of the multi-layer precoding with augmented vertical dimensions described in [[Section]{} \[subsec:Aug\]]{}, assuming an extended angle $\delta_\rm{E}=2 \Delta_\rm{E}$. This choice makes the maximum no-blockage distance $d_\rm{max}$, defined in Theorem \[th:CellCenter\], to be equal to the cell radius. Optimization of this parameter deserves more study in future extensions. considers the system model in [[Section]{} \[sec:Model\]]{} with $100 \times 40$ BS UPAs and one-ring channel model. First, the figure shows that multi-layer precoding achieves close coverage to the single-user case at the cell center. For the cell edge, though, multi-layer precoding users experience high blockage, which results from the elevation inter-cell interference avoidance. This can be improved when augmenting vertical subspaces as described in [[Section]{} \[subsec:Aug\]]{}. Different than the multi-layer precoding case, the small degradation at the cell-edge users is due to inter-cell interference, not signal blockage. Further, it is important to note that the cell-center users still achieve the asymptotic optimal rate with the modified algorithm in [[Section]{} \[subsec:Aug\]]{}, i.e., no inter-cell interference or pilot contamination impact exist. The same behavior is shown again in , when larger array sizes are employed. In this case, though, the cell-edge blockage with multi-layer precoding is less as a better separation between the desired cell and the other cells’ users can be achieved. In the two figures, multi-layer precoding with augmented vertical subspaces is shown to have a good coverage gain over conventional massive MIMO precoding solutions.
In , we consider the same system and channel models as in , but with $80 \times 20$ UPAs and $K=5$ users to reduce the computational complexity. compares the rate coverage of the proposed augmented dimension based multi-layer precoding with the single-user rate and the single-cell conjugate beamforming. The figure also plots the rate coverage of the multi-cell MMSE precoding in [@Jose2011] that manages the inter-cell interference. As shown in the figure, multi-layer precoding achieves a close performance to single-user rate and good gain over single-cell precoding. also illustrates that multi-layer precoding achieves a reasonable gain over multi-cell MMSE precoding despite the requirement of less channel knowledge.
![The achievable rates of the proposed multi-layer precoding algorithms are compared to the single-user rate and the rate with conventional conjugate beamforming, for different distances from cell center. The BSs are assumed to employ $120 \times 40$ UPAs at heights $H_\rm{BS}=35$m and the cell radius is $r_\rm{cell}=100$m. The users have one-ring channel models of azimuth and elevation angular spread $\Delta_\rm{A}=5^\circ, \Delta_\rm{E}=3^\circ$.[]{data-label="fig:Dist"}](Distance_M120x40_R100_H35_U20.pdf){width=".7\columnwidth"}
**Rates at the cell-interior and cell-edge:** To illustrate the achievable rates for cell-interior and cell-edge users, we plot the achievable rates of multi-layer precoding, single-user, and conventional conjugate beamforming in . The rates are plotted versus the user distance to the BS, normalized by the cell radius $r_\rm{cell}=100$m. The figure confirms the asymptotic optimal performance of multi-layer precoding at the cell-interior, given in Theorem \[th:CellCenter\]. At the cell edge, users experience some blockage that can be fixed with the augmented vertical dimension modification in [[Section]{} \[subsec:Aug\]]{}. Compared to the conventional conjugate beamforming performance, the multi-layer precoding with augmented vertical dimensions has better performance, even at the cell edge.
Conclusion {#sec:Conclusion}
==========
In this paper, we proposed a general precoding framework for full-dimensional massive MIMO systems, called multi-layer precoding. We developed a specific design for multi-layer precoding that efficiently manages different kinds of interference, leveraging the large channel characteristics. Using analytical derivations and numerical simulations, we showed that multi-layer precoding can guarantee asymptotically optimal performance for the cell-interior users under the one-ring channel models and for all the users under single-path channels. For the cell-edge users, we proposed a modified multi-layer precoding design that compromises between desired signal power maximization and inter-cell interference avoidance. Results indicated that multi-layer precoding can achieve close performance, in terms of rate and coverage, to the single-user case. Further, results showed that multi-layer precoding achieves clear gains over conventional massive MIMO precoding techniques. For future work, it would be interesting to investigate and optimize the implementation of multi-layer precoding using hybrid analog/digital architectures. It is also important to develop techniques for the channel training and estimation under hybrid architecture hardware constraints.
{#app:Ach_Rate}
To prove the achievable rate in , it is sufficient to prove that the power normalization factor $\left[\bUpsilon\right]_{k,k}$ that satisfies the multi-layer precoding power constraint $\left\|\left[\bF_c^{(1)} \bF_c^{(2)} \bF_c^{(3)}\right]_{:,k}\right\|^2=1$ is given by $\left[\bUpsilon\right]_{k,k}=\sqrt{\left(\left( \bW^*_c {\bF_{c}^{(2)}}^* {\bF_{c}^{(2)}} {\bF_{c}^{(2)}}^* {\bF_{c}^{(2)}} \bW_c \right)_{k,k}^{-1}\right)^{-1}}$. Using this values of $\left[\bUpsilon\right]_{k,k}$, the multi-layer precoding power constraint can be written as $$\begin{aligned}
\left\|\left[\bF_c^{(1)} \bF_c^{(2)} \bF_c^{(3)}\right]_{:,k}\right\|^2 & = \left[\bUpsilon\right]_{:,k}^* \bF_c^{(3)} \bF_c^{(2)} \bF_c^{(1)}\bF_c^{(1)} \bF_c^{(2)} \bF_c^{(3)} \left[\bUpsilon\right]_{:,k} \\
&\stackrel{(a)}{=} \left[\bUpsilon\right]_{:,k}^* \bF_c^{(3)} \bF_c^{(2)} \bF_c^{(2)} \bF_c^{(3)} \left[\bUpsilon\right]_{:,k} \\
&\stackrel{}{=} \left[\bUpsilon\right]_{:,k}^* \left(\overline{\bH}_c^* \overline{\bH}_c\right)^{-1} \overline{\bH}_c^* \bF_c^{(2)} \bF_c^{(2)} \overline{\bH}_c \left(\overline{\bH}_c^* \overline{\bH}_c\right)^{-1} \left[\bUpsilon\right]_{:,k} \\
&\stackrel{(b)}{=} \left[\bUpsilon\right]_{:,k}^* \left(\bW_c^* {\bF_c^{(2)}}^* \bF_c^{(2)} {\bF_c^{(2)}}^* \bF_c^{(2)} \bW_c \right)^{-1} \bW_c^* {\bF_c^{(2)}}^* \bF_c^{(2)} \bF_c^{(2)} \bF_c^{(2)} {\bF_c^{(2)}}^* \bF_c^{(2)} \bW_c \nonumber \\
& \hspace{10pt} \times \left(\bW_c^*{\bF_c^{(2)}}^* \bF_c^{(2)} {\bF_c^{(2)}}^* \bF_c^{(2)} \bW_c\right)^{-1} \left[\bUpsilon\right]_{:,k} \\
&\stackrel{}{=} \left[\bUpsilon\right]_{:,k}^* \left(\bW_c^* {\bF_c^{(2)}}^* \bF_c^{(2)} {\bF_c^{(2)}}^* \bF_c^{(2)} \bW_c \right)^{-1} \bW_c^* {\bF_c^{(2)}}^* \bF_c^{(2)} {\bF_c^{(2)}}^* \bF_c^{(2)} \bW_c \nonumber \\
& \hspace{10pt} \times \left(\bW_c {\bF_c^{(2)}}^* \bF_c^{(2)} \bW_c^*\right)^{-1} \bW_c^* {\bF_c^{(2)}}^* \bF_c^{(2)} {\bF_c^{(2)}}^* \bF_c^{(2)} \bW_c \nonumber \\
& \hspace{10pt} \times \left(\bW_c^*{\bF_c^{(2)}}^* \bF_c^{(2)} {\bF_c^{(2)}}^* \bF_c^{(2)} \bW_c\right)^{-1} \left[\bUpsilon\right]_{:,k} \\
&\stackrel{}{=} \left[\bUpsilon\right]_{:,k}^* \left(\bW_c {\bF_c^{(2)}}^* \bF_c^{(2)} \bW_c^*\right)^{-1} \left[\bUpsilon\right]_{:,k} \\
& \stackrel{(c)}{=} 1,
\end{aligned}$$ where (a) follows by noting that $\bF_c^{(1)}$ has a semi-unitary structure. The effective channel matrix $\overline{\bH}_c=[\overline{h}_{c 1}, ..., \overline{h}_{c K}]$ with $\overline{h}_{c k}=\left[\bG_{c,(k,1)}, ..., \bG_{c,(k,K)}\right]^* \overline{\bw}_{c c k}, k=1, ..., K$ can also be written as $\overline{\bH}_c={\bF_c^{(2)}}^* \bF_c^{(2)} \bW_c$ with $\bW_c=\bI_{K} \circ \left[\overline{\bw}_{c c 1}, ..., \overline{\bw}_{c c K}\right]$, which leads to (b). Finally, (c) follows by substituting with $\left[\bUpsilon\right]_{k,k}=\sqrt{\left(\left( \bW^*_c {\bF_{c}^{(2)}}^* {\bF_{c}^{(2)}} {\bF_{c}^{(2)}}^* {\bF_{c}^{(2)}} \bW_c \right)_{k,k}^{-1}\right)^{-1}}$.
{#app:CellCenter}
Considering the system and channel models in [[Section]{} \[sec:Model\]]{} and applying the multi-layer precoding algorithm in [[Section]{} \[sec:Algorithm\]]{}, the achievable rate by user $k$ at cell $c$ is given by Lemma \[lem:Ach\_Rate\] $$R_{ck}=\log_2\left(1+ \frac{\mathsf{SNR}}{\left( \bW^*_c {\bF_{c}^{(2)}}^* {\bF_{c}^{(2)}} {\bF_{c}^{(2)}}^* {\bF_{c}^{(2)}} \bW_c \right)_{k,k}^{-1}}\right).$$
If $\bG_{c,(k,m)}=\boldsymbol{0}, \forall m\neq k$ and $\bG_{c,(k,k)}=\bI$, then by noting that the matrix $\bW_c$ has a block diagonal structure and using the matrix inversion lemma [@Zhang2006], we get $\left( \bW^*_c {\bF_{c}^{(2)}}^* {\bF_{c}^{(2)}} {\bF_{c}^{(2)}}^* {\bF_{c}^{(2)}} \bW_c \right)_{k,k}^{-1}=\left\|\overline{\bw}_{c c k}\right\|^{-2}$. Therefore, to complete the proof, it is sufficient to prove that (i) $\lim_{N_\rm{V}, N_\rm{H}\rightarrow \infty} \bG_{c,(k,m)}=\boldsymbol{0}, \forall m\neq k$ and (ii) $ \lim_{N_\rm{V}, N_\rm{H}\rightarrow \infty} \bG_{c,(k,k)}=\bI$. To do that, we will first present the following useful lemma, which is a modified version of Lemma 3 in [@Yin2013].
\[lem:NullRange\] Consider a user $k$ at cell $c$ with an azimuth angle $\phi_{c k}$. Adopt the one-ring channel model in with an azimuth angular spread $\Delta_\rm{A}$ and correlation matrix $\bR_{c c k}^\rm{A}$. Define the unit-norm azimuth array response vector associated with an azimuth angle $\phi_m$ and elevation angle $\theta_m$ as $\bu_m=\frac{\ba(\phi_m, \theta_m)}{\sqrt{(N_\rm{H})}}$, where $\ba(\phi_m,\theta_m)=\left[1, ..., e^{j k D (N_\rm{H}-1) \sin(\theta_x) \sin(\phi_x)}\right]$. If the angle $\phi_x \notin \left[\phi_{c k}-\Delta_\rm{A}, \phi_{c k}+\Delta_\rm{A}\right]$, then $$\bu_x \in \text{Null}\left(\bR_{c c k}^\rm{A}\right), \hspace{10pt} \text{as} \hspace{10pt} N_\rm{H} \rightarrow \infty.$$
First, note that $\left[\bR_{c c k}^\rm{A}\right]_{n_1,n_2}$ in , can also be written as $$\left[\bR_{c c k}^\rm{A}\right]_{n_1,n_2}= \frac{1}{ 2 \Delta_\rm{A}} \int_{-\Delta_\rm{A}}^{\Delta_\rm{A}} \left[\ba\left(\phi_{ck}+\alpha, \theta_{c k} \right) \ba^*\left(\phi_{ck}+\alpha, \theta_{c k} \right)\right]_{n_1,n_2} d \alpha$$ Then, we have $$\begin{aligned}
\bu_m^* \bR \bu_m &= \frac{1}{ 2 \Delta_\rm{A} N_\mathrm{H}} \int_{-\Delta_\rm{A}}^{\Delta_\rm{A}} \ba^*(\phi_m,\theta_m) \ba\left(\phi_{ck}+\alpha, \theta_{c k} \right) \ba^*\left(\phi_{ck}+\alpha, \theta_{c k} \right) \ba(\phi_m,\theta_m) d \alpha \\
& = \frac{1}{ 2 \Delta_\rm{A}} \int_{-\Delta_\rm{A}}^{\Delta_\rm{A}}
\frac{1}{ N_\mathrm{H}} \left| \ba^*(\phi_m,\theta_m) \ba\left(\phi_{ck}+\alpha, \theta_{c k} \right) \right|^2 d \alpha.
\end{aligned}$$ Using Lemma 1 in [@ElAyach2012a], we reach $$\lim_{{N_H}\rightarrow \infty} \bu_m^* \bR \bu_m = 0, \hspace{10pt} \forall \phi_m \notin \left[\phi_{c k}-\Delta_\rm{A}, \phi_{c k}+\Delta_\rm{A}\right].$$
Now, to prove that $\bG_{c,(k,m)}=\overline{\bU}_{c c k}^* \overline{\bU}_{c c r}={{\bU}^{\rm{A}^*}_{c c k}} {\bU}_{c c r}^\rm{A} \otimes {\overline{\bU}^{\rm{E}^*}_{c c k}} \overline{\bU}_{c c r}^\rm{E}=\boldsymbol{0}$, we need to prove that either ${\bU}^{\rm{A}^*}_{c c k} {\bU}_{c c r}^\rm{A}=\boldsymbol{0}$ or ${\overline{\bU}^{\rm{E}^*}_{c c k}} \overline{\bU}_{c c r}^\rm{E}=\boldsymbol{0}$. If $\left|\phi_{c k}-\phi_{c m}\right| \geq 2 \Delta_\rm{A}$, then the columns of $\bU_{c c m}^\rm{A} \in \text{Span}\left\{\frac{\ba(\phi_m)}{\sqrt{(N_\rm{H})}}\left| \phi_{m} \in \left[\phi_{c m}-\Delta_\rm{A}, \phi_{c m}+\Delta_\rm{A}\right]\right. \right\} \subseteq \text{Null}\left(\bR_{c c k}^\rm{A}\right)$ as $N_\mathrm{H} \rightarrow \infty$, which follows from Lemma \[lem:NullRange\]. This leads to $\lim_{N_\rm{H} \rightarrow \infty} {\bU_{c c k}^\rm{A}}^* \bU_{c c m}^\rm{A} = \boldsymbol{0}$. Similarly, if $\left|\theta_{c k}-\theta_{c m}\right| \geq 2 \Delta_\rm{E}$, then $\lim_{N_\rm{V} \rightarrow \infty} \bU_{c c k}^{\rm{E}^*} \bU_{c c m}^\rm{E} = \boldsymbol{0}$. Further, since $d \leq d_\mathrm{max}$, we have $\left|\theta_{c k}-\theta_{I}\right| \geq 2 \Delta_\rm{E}$, for any elevation angle $\theta_I$ of another cell user. This implies that $\bU_{c c k}^\rm{E} \in \text{Range} \left\{\bU^\rm{NI}_c\right\}$ as $N_\rm{V} \rightarrow \infty$ by Lemma \[lem:NullRange\], and $\exists \bA_{c k}$ such that $\bU_{c c k}^\rm{E}=\bU_c^\rm{NI} \bA_{c k}$. For the $\bU_{c c m}$, it can be generally expressed as $\bU_{c c m}=\bU_c^\rm{NI} \bA_{c m} + \bU_c^\rm{I} \bB_{c m}$ for some matrices $\bA_{c m}, \bB_{c m}$ of proper dimensions. As $\lim_{N_\rm{V} \rightarrow \infty} \bU_{c c k}^{\rm{E}^*} \bU_{c c m}^\rm{E} = \boldsymbol{0}$, we have $\lim_{N_\rm{V} \rightarrow \infty} \bA_{c k}^* \bA_{c m} = \boldsymbol{0}$. Then, $\overline{\bU}_{c c k}^{\rm{E}^*} \overline{\bU}_{c c m}^\rm{E}=\bA_{c k}^* \bA_{c m}=\boldsymbol{0}$ as $N_\mathrm{V} \rightarrow \infty$. This completes the proof of the first condition, $\bG_{c,(k,m)}=\boldsymbol{0}$ if $\left|\phi_{c k}-\phi_{c m}\right| \geq 2 \Delta_\rm{A}$ or $\left|\theta_{c k}-\theta_{c m}\right| \geq 2 \Delta_\rm{E}$, $\forall m \neq k$.
To prove that $\lim_{N_\rm{V}, N_\rm{H}\rightarrow \infty} \bG_{c,(k,k)}=\bI$, we need to show that $\overline{\bU}_{c c k}^{\rm{E}^*} \overline{\bU}_{c c k}^\rm{E}=\bI$. Since $\bU_{c c k}^\rm{E}$ can be written as $\bU_{c c k}^\rm{E}=\bU_c^\rm{NI} \bA_{c k}$ when $N_\rm{V} \rightarrow \infty$, then we have $\bU_{c c k}^{\rm{E}^*} \bU_{c c k}^\rm{E}= \bA_{c k}^* \bA_{c k} = \bI$. This results in $\overline{\bU}_{c c k}^{\rm{E}^*} \overline{\bU}_{c c k}^\rm{E}=\bA_{c k}^* \bA_{c k} = \bI$ as $N_\rm{V} \rightarrow \infty$, which completes the proof.
{#app:CellEdge}
Similar to the proof of Theorem \[th:CellCenter\], if $\left|\phi_{c k}-\phi_{c m}\right| \geq 2 \Delta_\rm{A}$ or $\left|\theta_{c k}-\theta_{c m}\right| \geq 2 \Delta_\rm{E}$, $\forall m \neq k$, then $\lim_{N_\rm{V}, N_\rm{H} \rightarrow \infty} \bG_{c,(k,m)}=\boldsymbol{0}$. Using the matrix inversion lemma and leveraging the block diagonal structure of $\bW_c$, we get $\left( \bW^*_c {\bF_{c}^{(2)}}^* {\bF_{c}^{(2)}} {\bF_{c}^{(2)}}^* {\bF_{c}^{(2)}} \bW_c \right)_{k,k}^{-1} = \left(\overline{\bw}^*_{c c k} \bG_{c (k,k)} \overline{\bw}_{c c k}\right)^{-1}$. Note that since $d>d_\rm{max}$, $\bU_{c c k}$ is not guaranteed to be in $\text{Range}\left(\bUN_c\right)$, and $\overline{\bU}_{c c k}^{\rm{E}^*} \overline{\bU}_{c c k}^\rm{E} \neq \bI$ in general. The achievable rate of user $k$ at cell $c$ can therefore be written as $$\begin{aligned}
\lim_{N_\rm{V}, N_\rm{H} \rightarrow \infty} R_{c k} &=\log_2\left(1+ \mathsf{SNR} \ \overline{\bw}^*_{c c k} \bG_{c (k,k)} \overline{\bw}_{c c k} \right) \\
& \stackrel{(a)}{\geq} \log_2\left(1+ \mathsf{SNR} \ \left|\overline{\bw}_{c c k} \right\|^2 \sigma^2_\rm{min}\left(\bI \otimes \overline{\bU}_{c c k}^\rm{E}\right) \right) \\
& \stackrel{(b)}{=} \log_2\left(1+ \mathsf{SNR} \ \left|\overline{\bw}_{c c k} \right\|^2 \sigma^2_\rm{min}\left(\overline{\bU}_{c c k}^\rm{E}\right) \right),\end{aligned}$$ where (a) follows by applying the Rayleigh-Ritz theorem [@Lutkepohl1997], and (b) results from the properties of the Kronecker product.
[10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[ l@\#1 =l@\#1 \#2]{}]{}
A. Alkhateeb, G. Leus, and R. W. Heath, “Multi-layer precoding for full-dimensional massive [MIMO]{} systems,” in *Asilomar Conference on Signals, Systems and Computers*, Nov 2014, pp. 815–819.
E. Larsson, O. Edfors, F. Tufvesson, and T. Marzetta, “Massive [MIMO]{} for next generation wireless systems,” *IEEE Communications Magazine*, vol. 52, no. 2, pp. 186–195, Feb. 2014.
F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, T. L. Marzetta, O. Edfors, and F. Tufvesson, “Scaling up [MIMO]{}: Opportunities and challenges with very large arrays,” *IEEE Signal Processing Magazine*, vol. 30, no. 1, pp. 40–60, Jan 2013.
R. W. Heath, N. González-Prelcic, S. Rangan, W. Roh, and A. M. Sayeed, “An overview of signal processing techniques for millimeter wave [MIMO]{} systems,” *IEEE Journal of Selected Topics in Signal Processing*, vol. 10, no. 3, pp. 436–453, April 2016.
K. T. Truong and R. W. Heath, “Effects of channel aging in massive [MIMO]{} systems,” *Journal of Communications and Networks*, vol. 15, no. 4, pp. 338–351, Aug 2013.
T. Marzetta, “Noncooperative cellular wireless with unlimited numbers of base station antennas,” *IEEE Transactions on Wireless Communications*, vol. 9, no. 11, pp. 3590–3600, November 2010.
E. Björnson, E. G. Larsson, and T. L. Marzetta, “Massive [MIMO]{}: Ten myths and one critical question,” *IEEE Communications Magazine*, vol. 54, no. 2, pp. 114–123, February 2016.
A. Lozano, R. W. Heath, and J. G. Andrews, “Fundamental limits of cooperation,” *IEEE Transactions on Information Theory*, vol. 59, no. 9, pp. 5213–5226, Sept 2013.
J. Singh, S. Ponnuru, and U. Madhow, “Multi-gigabit communication: the [ADC]{} bottleneck,” in *Proc. of IEEE International Conference on Ultra-Wideband (ICUWB)*, Vancouver, BC, Sept. 2009, pp. 22–27.
D. Gesbert, S. Hanly, H. Huang, S. S. Shitz, O. Simeone, and W. Yu, “Multi-cell [MIMO]{} cooperative networks: [A]{} new look at interference,” *IEEE Journal on Selected Areas in Communications*, vol. 28, no. 9, pp. 1380–1408, December 2010.
H. Huh, A. M. Tulino, and G. Caire, “Network [MIMO]{} with linear zero-forcing beamforming: Large system analysis, impact of channel estimation, and reduced-complexity scheduling,” *IEEE Transactions on Information Theory*, vol. 58, no. 5, pp. 2911–2934, May 2012.
J. Jose, A. Ashikhmin, T. L. Marzetta, and S. Vishwanath, “Pilot contamination and precoding in multi-cell [TDD]{} systems,” *IEEE Transactions on Wireless Communications*, vol. 10, no. 8, pp. 2640–2651, August 2011.
A. Ashikhmin and T. Marzetta, “Pilot contamination precoding in multi-cell large scale antenna systems,” in *IEEE International Symposium on Information Theory Proceedings (ISIT)*, July 2012, pp. 1137–1141.
H. Yin, D. Gesbert, M. Filippou, and Y. Liu, “A coordinated approach to channel estimation in large-scale multiple-antenna systems,” *IEEE Journal on Selected Areas in Communications*, vol. 31, no. 2, pp. 264–273, February 2013.
L. Lu, G. Y. Li, A. L. Swindlehurst, A. Ashikhmin, and R. Zhang, “An overview of massive [MIMO]{}: Benefits and challenges,” *IEEE Journal of Selected Topics in Signal Processing*, vol. 8, no. 5, pp. 742–758, Oct 2014.
O. El Ayach, S. Rajagopal, S. Abu-Surra, Z. Pi, and R. Heath, “Spatially sparse precoding in millimeter wave [MIMO]{} systems,” *IEEE Transactions on Wireless Communications*, vol. 13, no. 3, pp. 1499–1513, Mar. 2014.
A. Alkhateeb, G. Leus, and R. Heath, “Limited feedback hybrid precoding for multi-user millimeter wave systems,” *IEEE Transactions on Wireless Communications*, vol. 14, no. 11, pp. 6481–6494, Nov. 2015.
T. Bogale and L. B. Le, “Beamforming for multiuser massive [MIMO]{} systems: Digital versus hybrid analog-digital,” in *Proc. of IEEE Global Communications Conference (GLOBECOM)*, Dec. 2014, pp. 4066–4071.
L. Liang, W. Xu, and X. Dong, “Low-complexity hybrid precoding in massive multiuser [MIMO]{} systems,” *IEEE Wireless Communications Letters*, vol. 3, no. 6, pp. 653–656, Dec 2014.
A. Adhikary, J. Nam, J.-Y. Ahn, and G. Caire, “Joint spatial division and multiplexing: The large-scale array regime,” *IEEE Transactions on Information Theory*, vol. 59, no. 10, pp. 6441–6463, Oct. 2013.
Y. H. Nam, B. L. Ng, K. Sayana, Y. Li, J. Zhang, Y. Kim, and J. Lee, “Full-dimension mimo [(FD-MIMO)]{} for next generation cellular technology,” *IEEE Communications Magazine*, vol. 51, no. 6, pp. 172–179, June 2013.
Y. Kim, H. Ji, J. Lee, Y. H. Nam, B. L. Ng, I. Tzanidis, Y. Li, and J. Zhang, “Full dimension mimo [(FD-MIMO)]{}: the next evolution of [MIMO]{} in [LTE]{} systems,” *IEEE Wireless Communications*, vol. 21, no. 2, pp. 26–33, April 2014.
N. Seifi, J. Zhang, R. Heath, T. Svensson, and M. Coldrey, “Coordinated [3D]{} beamforming for interference management in cellular networks,” *IEEE Transactions on Wireless Communications*, vol. 13, no. 10, pp. 5396–5410, Oct 2014.
A. Kammoun, H. Khanfir, Z. Altman, M. Debbah, and M. Kamoun, “Preliminary results on [3D]{} channel modeling: From theory to standardization,” *IEEE Journal on Selected Areas in Communications*, vol. 32, no. 6, pp. 1219–1229, June 2014.
Z. Zhong, X. Yin, X. Li, and X. Li, “Extension of [ITU]{} [IMT]{}-advanced channel models for elevation domains and line-of-sight scenarios,” in *Proc. of IEEE Vehicular Technology Conference*, Sept 2013, pp. 1–5.
D. Ying, F. W. Vook, T. A. Thomas, D. J. Love, and A. Ghosh, “Kronecker product correlation model and limited feedback codebook design in a [3D]{} channel model,” in *Proc. of IEEE International Conference on Communications (ICC)*, June 2014, pp. 5865–5870.
S. Han, C.-L. I, Z. Xu, and C. Rowell, “Large-scale antenna systems with hybrid analog and digital beamforming for millimeter wave [5G]{},” *IEEE Communications Magazine*, vol. 53, no. 1, pp. 186–194, Jan. 2015.
G. T. 36.873, “Study on [3D]{} channel model for lte,” Sep. 2014.
M. Akdeniz, Y. Liu, M. Samimi, S. Sun, S. Rangan, T. Rappaport, and E. Erkip, “Millimeter wave channel modeling and cellular capacity evaluation,” *IEEE Journal on Sel. Areas in Communications*, vol. 32, no. 6, pp. 1164–1179, June 2014.
S. Hur, S. Baek, B. Kim, Y. Chang, A. F. Molisch, T. S. Rappaport, K. Haneda, and J. Park, “Proposal on millimeter-wave channel modeling for [5G]{} cellular system,” *IEEE Journal of Selected Topics in Signal Processing*, vol. 10, no. 3, pp. 454–469, Apr. 2016.
D.-S. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn, “Fading correlation and its effect on the capacity of multielement antenna systems,” *IEEE Transactions on Communications*, vol. 48, no. 3, pp. 502–513, Mar 2000.
P. Petrus, J. H. Reed, and T. S. Rappaport, “Geometrical-based statistical macrocell channel model for mobile environments,” *IEEE Transactions on Communications*, vol. 50, no. 3, pp. 495–502, Mar 2002.
A. Abdi and M. Kaveh, “A space-time correlation model for multielement antenna systems in mobile fading channels,” *IEEE Journal on Selected Areas in Communications*, vol. 20, no. 3, pp. 550–560, Apr 2002.
M. Zhang, P. J. Smith, and M. Shafi, “An extended one-ring [MIMO]{} channel model,” *IEEE Transactions on Wireless Communications*, vol. 6, no. 8, pp. 2759–2764, August 2007.
E. Björnson, J. Hoydis, M. Kountouris, and M. Debbah, “Massive [MIMO]{} systems with non-ideal hardware: [Energy]{} efficiency, estimation, and capacity limits,” *IEEE Transactions on Information Theory*, vol. 60, no. 11, pp. 7112–7139, Nov 2014.
J. C. Shen, J. Zhang, E. Alsusa, and K. B. Letaief, “Compressed [CSI]{} acquisition in [FDD]{} massive [MIMO]{}: How much training is needed?” *IEEE Transactions on Wireless Communications*, vol. 15, no. 6, pp. 4145–4156, June 2016.
T. Bai, A. Alkhateeb, and R. Heath, “Coverage and capacity of millimeter-wave cellular networks,” *IEEE Communications Magazine*, vol. 52, no. 9, pp. 70–77, Sept. 2014.
T. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. Wong, J. Schulz, M. Samimi, and F. Gutierrez, “Millimeter wave mobile communications for [5G]{} cellular: It will work!” *IEEE Access*, vol. 1, pp. 335–349, May 2013.
F. Zhang, *The Schur complement and its applications*.1em plus 0.5em minus 0.4emSpringer Science & Business Media, 2006, vol. 4.
O. El Ayach, R. W. Heath Jr, S. Abu-Surra, S. Rajagopal, and P. Zhouyue, “The capacity optimality of beam steering in large millimeter wave [MIMO]{} systems,” in *Proc. of IEEE International Workshop on Signal Processing Advances in Wireless Communications (SPAWC)*, Cesme, Turkey, June 2012, pp. 100–104.
H. Lutkepohl, “Handbook of matrices.” *Computational Statistics and Data Analysis*, vol. 2, no. 25, p. 243, 1997.
[^1]: Ahmed Alkhateeb and Robert W. Heath Jr. are with The University of Texas at Austin (Email: aalkhateeb, [email protected]). Geert Leus is with Delft University of Technology (Email: [email protected]).
[^2]: This material is based upon work supported in part by the National Science Foundation under Grant No. NSF-CCF-1319556.
[^3]: A shorter version of this paper was presented at Asilomar Conference on Signals, Systems, and Computers, Nov., 2014 [@Alkhateeb2014c].
| ArXiv |
---
abstract: |
The evolution of number density, size and intrinsic colour is determined for a volume-limited sample of visually classified early-type galaxies selected from the HST/ACS images of the GOODS North and South fields (version 2). The sample comprises $457$ galaxies over $320$ arcmin$^2$ with stellar masses above $3\cdot
10^{10}$in the redshift range 0.4$<$z$<$1.2. Our data allow a simultaneous study of number density, intrinsic colour distribution and size. We find that the most massive systems (${\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}}3\cdot
10^{11}M_\odot$) do not show any appreciable change in comoving number density or size in our data. Furthermore, when including the results from 2dFGRS, we find that the number density of massive early-type galaxies is consistent with no evolution between z=1.2 and 0, i.e. over an epoch spanning more than half of the current age of the Universe. Massive galaxies show very homogeneous [*intrinsic*]{} colour distributions, featuring red cores with small scatter. The distribution of half-light radii – when compared to z$\sim$0 and z$>$1 samples – is compatible with the predictions of semi-analytic models relating size evolution to the amount of dissipation during major mergers. However, in a more speculative fashion, the observations can also be interpreted as weak or even no evolution in comoving number density [*and size*]{} between 0.4$<$z$<$1.2, thus pushing major mergers of the most massive galaxies towards lower redshifts.
author:
- |
Ignacio Ferreras$^{1}$[^1], Thorsten Lisker$^2$, Anna Pasquali$^3$, Sadegh Khochfar$^4$, Sugata Kaviraj$^{1,5}$\
$^1$ Mullard Space Science Laboratory, Unversity College London, Holmbury St Mary, Dorking, Surrey RH5 6NT\
$^2$ Astronomisches Rechen-Institut, Zentrum für Astronomie, Universität Heidelberg, Mönchhofstr. 12-14, D-69120 Heidelberg, Germany\
$^3$ Max-Planck-Institut für Astronomie, Koenigstuhl 17, D-69117 Heidelberg, Germany\
$^4$ Max-Planck-Institut für Extraterrestrische Physik, Giessenbachstrasse, D-85748 Garching, Germany\
$^5$ Astrophysics subdepartment, The Denys Wilkinson Building, Keble Road, Oxford OX1 3RH
date: 'January 20, 2009: To be published in MNRAS'
title: 'On the formation of massive galaxies: A simultaneous study of number density, size and intrinsic colour evolution in GOODS'
---
\[firstpage\]
galaxies: evolution — galaxies: formation — galaxies: luminosity function, mass function — galaxies: high redshift
Introduction {#sec:intro}
============
During the past decades the field of extragalactic astrophysics has undergone an impressive development, from simple models that were compared with small, relatively nearby samples to current surveys extending over millions of Mpc$^3$ at redshifts beyond z$\sim$1 along with numerical models that can probe cosmological volumes with the aid of large supercomputers. However, in the same period of time, our knowledge of the ’baryon physics’ relating the dark and luminous matter components has progressed much slower, mainly due to the highly non-linear processes that complicate any ab initio approach to this complex problem.
The evolution of the most massive galaxies constitutes one of the best constraints one can impose on the modelling of galaxy formation. Within the current paradigm of galaxy growth in a $\Lambda$CDM cosmology, massive galaxies evolve from subsequent mergers of smaller structures. The most massive galaxies are early-type in morphology and are dominated by old stellar populations, with a tight mass-metallicity relation and abundance ratios suggesting a quick build-up of the stellar component [see e.g. @ren06]. On the other hand, semi-analytic models of galaxy formation predict a more extended assembly history (if not star formation) from major mergers. By carefully adjusting these models, it has been possible to generate realizations that are compatible with the observed stellar populations in these galaxies [e.g. @kav06; @deluc06; @bow06]
In this paper we study the redshift evolution of a sample of the most massive early-type galaxies from the catalogue of @egds09, which were visually selected from the [*HST*]{}/ACS images of the GOODS North (HDFN) and South (CDFS) fields [@giav04]. Our data set complements recent work exploring the issue of size and stellar mass evolution [e.g. @Bun05; @McIn05; @fran06; @fon06; @Borch06; @brown07; @Truj07; @vdk08]. The coverage (320 arcmin$^2$), depth ($1\sigma$ surface brightness limit per pixel of $24.7$ AB mag/arcsec$^2$ in the $i$ band) and high-resolution (FWHM$\sim 0.12$ arcsec) of these images allow us to perform a consistent analysis of the redshift evolution of the comoving number density, size and intrinsic colour of these galaxies.
{width="5in"}
The sample {#sec:sample}
==========
The [*HST*]{}/ACS images of the GOODS North and South fields (v2.0) were used to perform a visual classification of spheroidal galaxies. This is a continuation of @fer05 – that was restricted to the CDFS field. However, notice that our sample does [*not*]{} apply the selection based on the Kormendy relation, i.e. the only constraint in this sample is visual classification. The analysis of the complete sample is presented in @egds09. Over the $320$ arcmin$^2$ field of view of the North and South GOODS/ACS fields, the total sample comprises $910$ galaxies down to $i_{\rm
AB}=24$ mag (of which 533/377 are in HDFN/CDFS). The available photometric data – both space and ground-based – were combined with spectroscopic or photometric redshifts in order to determine the stellar mass content. Spectroscopic redshifts are available for 66% of the galaxies used in this paper. The photometric redshifts have an estimated accuracy of $\Delta (z)/(1+z)\sim 0.002\pm 0.09$ [@egds09]. Stellar masses are obtained by convolving the synthetic populations of @bc03 with a grid of exponentially decaying star formation histories [see appendix B of @egds09 for details]. A @chab03 Initial Mass Function is assumed. Even though the intrinsic properties of a stellar population (i.e. its age and metallicity distribution) cannot be accurately constrained with broadband photometry, the stellar mass content can be reliably determined to within $0.2-0.3$ dex provided the adopted IMF gives an accurate representation of the true initial mass function [see e.g. @fsb08].
The sizes are computed using a non-parametric approach that measures the total flux within an ellipse with semimajor axis $a_{\rm
TOT}<1.5a_{\rm Petro}$. The eccentricity of the ellipse is computed from the second order moments of the surface brightness distribution. The half-light radius is defined as R$_{50}\equiv\sqrt{a_{50}\times
b_{50}}$, where $a_{50}$ and $b_{50}$ are respectively the semimajor and semiminor axes of the ellipse that engulfs 50% of the total flux. Those values need to be corrected for the loss of flux caused by the use of an aperture [see e.g. @gra05]. We used a synthetic catalogue of galaxies with Sersic profiles and the same noise and sampling properties as the original GOODS/ACS images to build fitting functions for the corrections in flux and size. The corrections depend mostly on R$_{50}$ and, to second order, on the Sersic index. Most of this correction is related to the ratio between the size of the object and the size of the Point Spread Function of the observations. The dependence with Sersic index (or in general surface brightness slope) is milder and for this correction the concentration [as defined in @ber00] was used as a proxy.
We compared our photometry with the GOODS-MUSIC data [@graz06] in the CDFS. Our sample has 351 galaxies in common with that catalogue, and the difference between our total+corrected $i$-band magnitudes and the total magnitudes from GOODS-MUSIC is $\Delta i\equiv i_{\rm ours}- i_{\rm MUSIC}=-0.17\pm 0.16$ mag. This discrepancy is mostly due to our corrections of the total flux. A bootstrap method using synthetic images show that our corrections are accurate with respect to the true total flux to within 0.05 mag, and to within 9% in half-light radius [see appendix A of @egds09]. Our estimates of size were also compared with the GALFIT-based parametric approach of @gems on the GEMS survey. Out of 133 galaxies in common, the median of the difference defined as $($R$_{50}^{\rm ours}-$R$_{50}^{\rm GEMS})/$R$_{50}^{\rm ours}$ is $-0.01 \pm 0.16$ (the error bar is defined as the semi-interquartile range).
{width="5in"}
We focus here on a volume-limited sample comprising early-type galaxies with stellar mass $M_s{\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}}3\times 10^{10}$. This sample is binned according to fixed steps in comoving volume (a standard $\Lambda$CDM cosmology with $\Omega_m=0.3$ and $h=0.7$ is used throughout). The complete sample of 910 galaxies from @egds09 is shown in figure \[fig:sample\]. Solid (open) circles represent early-type galaxies whose colours are compatible with an older (younger) stellar population. This simple age criterion is based on a comparison of the observed optical and NIR colours with the predictions from a set of templates with exponentially decaying star formation histories, all beginning at redshift z$_{\rm F}=3$, with solar metallicity. The “old” population is compatible with formation timescales $\tau{\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}1$ Gyr [see @egds09 for details]. The black dots in the figure correspond to the sample of $457$ galaxies used in this paper.
{width="5in"}
We further subdivide this sample into three mass bins, starting at $\log ({\rm M}_s/$$)=10.5$ with a width $\Delta\log ({\rm
M}_s/$$)=0.5$ dex. For comparison, the characteristic stellar mass from the mass function of the GOODS-MUSIC sample is shown as a dashed line [@fon06], although we warn that the GOODS-MUSIC masses are calculated using a @salp55 IMF, which will give a systematic 0.25 dex overestimate in $\log$M$_s$ with respect to our choice of IMF. Our sample is safely away from the limit imposed by the cut in apparent magnitude ($i_{AB}\leq 24$). The curved solid lines give that limit for two extreme star formation histories, corresponding to the “old” and “young” populations as defined above. Notice that within our sample of massive early-type galaxies there are NO galaxies whose colours are compatible with young stellar populations (i.e. open circles).
The evolution of massive galaxies {#sec:evol}
=================================
The redshift evolution of the comoving number density is shown in figure \[fig:logn\] (black dots). The ($1\sigma$) error bars include both Poisson noise as well as the effect of a 0.3 dex uncertainty in the stellar mass estimates. These uncertainties are computed using a Monte Carlo run of 10,000 realizations. The figure includes data from GOODS-MUSIC [@fon06], COMBO17 [@Bell04] and Pal/DEEP2 [@con07]. At z=0 we show an estimate from the segregated 2dFGRS luminosity function [@crot05]. We take their Schechter fits for early-type galaxies within an environment with a mean density defined by a contrast – measured inside radius $8h^{-1}$ Mpc – in the range $\delta_8=-0.43\cdots+0.32$ (black open circles). In order to illustrate possible systematic effects in 2dFGRS, we also include the result for their full volume sample as grey open circles. The 2dFGRS data are originally given as luminosity functions in the rest-frame $b_J$. We took a range of stellar populations typical of early-type galaxies in order to translate those luminosities into stellar masses. The error bars shown for the 2dFGRS data represent the uncertainty caused by this translation from light into mass over a wide range of stellar populations (with typical M/L($b_J$) in the range $7\cdots 12$ / ). The black solid lines show semi-analytic model (SAM) predictions from @ks06a. Their SAM follows the merging history of dark matter halos generated by the Extended Press-Schechter formalism down to a mass resolution of $M_{\rm min}=5 \times 10^9$ , and follows the baryonic physics within these halos using recipes laid out in @kb05 [and references therein]. The grey dashed lines are the predictions from the Millennium simulation @deluc06. This model is extracted from their web-based database[^2], and is not segregated with respect to galaxy morphology. This explains the excess number density in the low-mass bin (bottom panel). In the two higher mass bins most of the galaxies have an early-type morphology. The predictions of the Millennium simulation are in agreement with the middle bin – i.e. masses between $10^{11}$ and $3\cdot 10^{11}$M$_\odot$. However, for the most massive bin, the sharp decrease in density with redshift of the models is in remarkable disagreement with the observations. In contrast, @ks06a predict a nearly constant density at the highest mass bin out to z$<$1.
The main reason for this discrepancy is that AGN feedback in the Millennium simulation prohibits the growth of massive galaxies by gas cooling and subsequent star formation in order to reproduce the right colour-bimodality and the luminosity function at z=0. As shown in @ks08 the existence of a characteristic mass scale for the shut-off of star formation will lead to dry merging being the main mechanism for the growth of massive galaxies. In that respect the evolution of the number density of massive galaxies in the Millennium simulation is mainly driven by mergers. The difference between that model and @ks06a is probably due to the different merger rates in their models. The Millennium simulation predicts a lower major merger rate compared to @ks06a almost by a factor 10 (Hopkins et al., in preparation).
{width="5in"}
Figure \[fig:Re\] shows the redshift evolution of the half-light radius. Our methodology follows a non-parametric approach avoiding the degeneracies intrinsic to profile fitting. Nevertheless, we compared our size estimates with those using a parametric approach like GALFIT [@gems] and there is good agreement (see §\[sec:sample\]). Our data (black dots) are compared with @Truj07 [grey triangles] and with a z$\sim$0 measurement from the SDSS [@Shen03 taking their early-type sample]. The error bars give the RMS scatter of the size distribution within each mass and redshift bin. The lines correspond to the models of @ks06b. These models associate size evolution to the amount of dissipation encountered during major mergers along the merging history of an early-type galaxy. The points at high redshift (z$>$1.2) correspond to [*individual*]{} measurements from the literature (see caption for details). In all the comparisons shown in this paper with work from the literature, we have checked that the initial mass functions used are similar, so that stellar masses are compared consistently. All results quoted either use a @chab03 IMF or functions very close to it in terms of the total mass expected per luminosity unit, which – for early-type systems – mainly reduces to the shape of the low-mass end of the IMF. Other functions used in the quoted data were @kro93 [@kro01] or @bg03. Only for the GOODS-MUSIC data [@fon06] the Salpeter IMF (1955) was used, which will always give a systematic overestimate of $\sim 0.25$ dex in stellar mass with respect to the previous choices given its (unphysical) extrapolation of the same power law down to the low stellar mass cutoff [see e.g. @bc03]. A single-law Salpeter IMF is an unlikely choice for the stellar populations in early-type galaxies as shown by comparisons of photometry with kinematics [@cap06] or with gravitational lensing [@fsb08].
Similarly to the density evolution, we also apply a simple power law fit only to our data points: R$_e\propto (1+z)^\beta$. The solid lines give those best fits, and the power law index is given in each panel. Taking into account all data points betweeen z$=$0 and z$\sim$2.5 one sees a clear trend of decreasing size with redshift for all three mass bins. However our data suggest milder size evolution for the most massive early-type galaxies between z$=$1.2 and z$=$0.4, corresponding to a 4 Gyr interval of cosmic time.
The depth and high spatial resolution of the ACS images also allow us to probe in detail the [*intrinsic*]{} colour distribution of the galaxies (i.e. the colour distribution within each galaxy). We follow the approach described in @fer05 which, in a nutshell, registers the images in the two bands considered for a given colour, degrades them by the Point Spread Function of the other passband, and perfoms an optimal Voronoi tessellation in order to achieve a S/N per bin around $10$ while preserving spatial resolution. The final binned data is used to fit a linear relation between colour and $\log
(R/R_e)$ from which we determine the slope and the scatter about the best fit (using a biweight estimator). Figure \[fig:clr\] shows the observer-frame V$-$i colour gradient ([*bottom*]{}) and scatter ([*top*]{}) as a function of stellar mass ([*left*]{}) and half-light radius ([*right*]{}). The black dots correspond to binned data in stellar mass, showing the average and RMS value within each bin. Notice the significant trend with increasing stellar mass towards redder cores (i.e. more negative colour gradients) and small scatter. The colour gradient is in most cases nearly flat, and only for the lowest mass bin do we find significantly large gradients. For comparison, we also show as small grey dots a continuation of the original sample from @egds09 towards lower stellar masses. Blue cores (positive colour gradients) dominate in spheroidal galaxies below $10^{10}$. The homogeneous intrinsic colour distribution thereby suggests no significant star formation and a fast rearranging process of the stellar populations if mergers take place during the observed redshift range. Notice this sample only targets objects visually classified as early-type galaxies. The early phases of major merging are therefore excluded from our sample. Nevertheless, the number density at the massive end (upper panel of figure \[fig:logn\]) does not change significantly between z=0 and z$\sim$1, already suggesting that major merger events must be rare over those redshifts.
Discussion and Conclusions {#sec:discussion}
==========================
Using a volume-limited sample of massive spheroidal galaxies from the v2.0 ACS/HST images of the GOODS North and South fields we have consistently estimated the number density, size and intrinsic colour distribution over the redshift range 0.4$<$z$<$1.2. In combination with other samples we find a significant difference in the redshift evolution according to stellar mass, in agreement with recent work based on other samples or different selection criteria [see e.g. @Bun05; @McIn05; @fran06; @fon06; @Borch06; @brown07; @Truj07; @vdk08]. The most massive galaxies – which impose the most stringent constraints on models of galaxy formation – keep a constant comoving number density between z$\sim$1 and 0 (i.e. over half of the current age of the Universe) but present a significant size evolution, roughly a factor 2 increase between z=1 and 0. Note, however that within our sample, there is no significant size evolution over the redshift range z=0.4$\cdots$1.2. It is by extending the analysis to higher redshifts that the size evolution shows up at the most massive bin [e.g. @vdk08; @bui08]. When velocity dispersion is added to the analysis, a significant difference is found in the $\sigma$-R$_e$ distribution between z=0 and z=1, suggesting an important change in the dynamics of these galaxies [@vdwel08].
Some of the semianalytic models of massive galaxy evolution [@ks06a; @ks06b] are in good agreement with these observations. These models follow the standard paradigm of early-type galaxy growth through major mergers, with the ansatz that size evolution is related to the amount of dissipation during major mergers. The decreasing evolution in the comoving number density at high masses is explained within the models by a balance between the ’sink’ (loss due to mergers of massive galaxies generating more massive galaxies) and ’source’ terms (gain from mergers at lower mass) over the redshifts considered. One could argue that the sink terms would generate a population of extremely massive galaxies (above a few $10^{12}$), possibly the central galaxies within massive groups or clusters. However, this population – with predicted comoving number densities below $10^{-6}$Mpc$^{-3}$ – are very hard to study with current surveys. Furthermore, environment effects in these systems will complicate the analysis of size evolution [e.g. @ko08].
It is important to note that the lack of evolution in the number density relates to the bright end of the luminosity function. @fab07 found a significant change in the number density of [*red*]{} galaxies with redshift. However, they also emphasize that this change does not refer to the most luminous galaxies. If we include all mass bins in our sample, we do find a significant decrease in the number density with redshift, as the lower mass bins – which contribute the most in numbers – do have a rather steep decrease in density (see figure \[fig:logn\]). This difference suggests that the (various) mechanisms playing a role in the transition from blue cloud to red sequence must be strongly dependent on the stellar mass of the galaxies involved.
In a more speculative fashion, our data are also suggestive of weak or even [*no evolution*]{} in the number density of the most massive early-type galaxies over a redshift range 0.4$<$z$<$1.2. This would imply a negligible role of major mergers at the most massive end for z$>$0.4, thereby pushing this stage of galaxy formation towards lower redshifts [@ks08]. Another speculative scenario for the evolution of massive spheroidal galaxies would involve negligible major mergers at these redshifts and a significant amount of minor mergers which will ’puff up’ the galaxy. Minor mergers are considered the cause of recent star formation observed in NUV studies of early-type galaxies [@kav07]. Larger surveys of Luminous Red Galaxies are needed to confirm or disprove this important issue.
Baldry, I. K. & Glazebrook, K., 2003, ApJ, 593, 258
Bell, E. F. [[et al. ]{}]{} 2004, ApJ, 608, 752
Bershady, M. A., Jangren, A. & Conselice, C. J., 2000, AJ, 119, 2645
Borch, A., [[et al. ]{}]{}, 2006, A& A, 453, 869
Bower, R. G, Benson, A. J., Malbon, R., Helly, J. C., Frenk, C. S., Baugh, C. M., Cole, S. & Lacey, C. G. 2006, MNRAS, 370, 645
Brown, M. J. I., Dey, A., Jannuzi, B. T., Brand, K., Benson, A. J., Brodwin, M., Croton, D. J. & Eisenhardt, P. R., 2007, ApJ, 654, 858
Bruzual, G. & Charlot, S., 2003, MNRAS, 344, 1000
Buitrago, F., Trujillo, I., Conselice, C. J., Bouwens, R. J., Dickinson, M. & Yan, H., 2008, arXiv:0807.4141
Bundy, K., Ellis, R. S. & Conselice, C. J., 2005, ApJ, 625, 621
Cappellari, M., [[et al. ]{}]{}, 2006, MNRAS, 366, 1126
Chabrier, G., 2003, PASP, 115, 763
Cimatti, A., [[et al. ]{}]{}, 2008, A& A, 482, 21
Conselice, C. J., [[et al. ]{}]{}, 2007, MNRAS, 381, 962
Croton, D. J., [[et al. ]{}]{}2005, MNRAS, 356, 1155
Damjanov, I., [[et al. ]{}]{}2008, arXiv:0807:1744
De Lucia, G., Springel, V., White, S. D. M., Croton, D. & Kauffmann, G.2006, MNRAS, 366, 499
Faber, S. M., [[et al. ]{}]{}, 2007, ApJ, 665, 265
Ferreras, I., Lisker, T., Carollo, C. M., Lilly, S. J. & Mobasher, B. 2005, ApJ, 635, 243
Ferreras, I., Saha, P. & Burles, S. 2008, MNRAS, 383, 857
Ferreras, I., Lisker, T., Pasquali, A. & Kaviraj, S. 2009, MNRAS submitted, arXiv:0901.2123
Fontana, A., [[et al. ]{}]{} 2006, A& A, 459, 745
Franceschini, A., [[et al. ]{}]{}, 2006, A& A, 453, 397
Giavalisco, M., [[et al. ]{}]{}, 2004, ApJ, 600, L93
Graham, A. W., Driver, S. P., Petrosian, V., Conselice, C. J., Bershady, M. A., Crawford, S. M. & Goto, T. 2005, AJ, 130, 1535
Grazian, A. [[et al. ]{}]{}, 2006, A& A, 449, 951.
Häussler, B. [[et al. ]{}]{}, 2007, ApJS, 172, 615
Kaviraj, S., Devriendt, J. E. G., Ferreras, I., Yi, S. K. & Silk, J. 2006, arXiv:astro-ph/0602347
Kaviraj, S., Peirani, S., Khochfar, S., Silk, J. & Kay, S., 2007, MNRAS, in press, arXiv:0711.1493
Khochfar, S. & Burkert, A. 2005, MNRAS, 359, 1379
Khochfar, S. & Ostriker, J. P. 2008, ApJ, 680, 54
Khochfar, S. & Silk, J. 2006a, MNRAS, 370, 902
Khochfar, S. & Silk, J. 2006b, ApJ, 648, L21
Khochfar, S. & Silk, J. 2008, arXiv:0809.1734
Kroupa, P., Tout, C. A. & Gilmore, G., 1993, MNRAS, 262, 545
Kroupa, P., 2001, MNRAS, 322, 231
McIntosh, D., [[et al. ]{}]{}, 2005, ApJ, 632, 191
Renzini, A. 2006, ARA& A, 44, 141
Salpeter, E. E., 1955, ApJ, 121, 161
Shen, S., Mo, H. J., White, S. D. M., Blanton, M. R., Kauffmann, G., Voges, W., Brinkmann, J. & Csabai, I. 2003, MNRAS, 343, 978
Toft, S., 2007, ApJ, 671, 285
Trujillo, I., Conselice, C. J., Bundy, K., Cooper, M. C., Eisenhardt, P., & Ellis, R. S. 2007, MNRAS, 382, 109
van der Wel, A., Holden, B. P., Zirm, A. W., Franx, M., Rettura, A., Illingworth, G. D. & Ford, H. C., 2008, arXiv:0808.0077
van Dokkum, P. G., [[et al. ]{}]{}2008, ApJ, 677, L5
Zirm, A. W., [[et al. ]{}]{}2007, ApJ, 656, 66
[^1]: E-mail: [email protected]
[^2]: http://www.mpa-garching.mpg.de/millennium
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*Int. J. Bifurcation and Chaos* **13**[, 3147-3233, (2003). Tutorial and Review paper.]{}
**TOWARD A THEORY OF CHAOS**
A. Sengupta\
Department of Mechanical Engineering\
Indian Institute of Technology. Kanpur\
E-Mail: [email protected]
**ABSTRACT**
[This paper formulates a new approach to the study of chaos in discrete dynamical systems based on the notions of inverse ill-posed problems, set-valued mappings, generalized and multivalued inverses, graphical convergence of a net of functions in an extended multifunction space [@Sengupta2000], and the topological theory of convergence. Order, chaos, and complexity are described as distinct components of this unified mathematical structure that can be viewed as an application of the theory of convergence in topological spaces to increasingly nonlinear mappings, with the boundary between order and complexity in the topology of graphical convergence being the region in $\textrm{Multi}(X)$ that is susceptible to chaos. The paper uses results from the discretized spectral approximation in neutron transport theory [@Sengupta1988; @Sengupta1995] and concludes that the numerically exact results obtained by this approximation of the Case singular eigenfunction solution is due to the graphical convergence of the Poisson and conjugate Poisson kernels to the Dirac delta and the principal value multifunctions respectively. In $\textrm{Multi}(X)$, the continuous spectrum is shown to reduce to a point spectrum, and we introduce a notion of]{} *latent chaotic states* [to interpret superposition over generalized eigenfunctions. Along with these latent states, spectral theory of nonlinear operators is used to conclude that nature supports complexity to attain efficiently a multiplicity of states that otherwise would remain unavailable to it. ]{}
*Keywords:* [chaos, complexity, ill-posed problems, graphical convergence, topology, multifunctions.]{}
**Prologue**
**1.**
@Peitgen1992
**2.**
Mitchell Feigenbaum’s *Foreword* (pp 1-7) in @Peitgen1992
**3.** [^1]
Opening address of Heitor Gurgulino de Souza, Rector United Nations University, Tokyo @Grebogi1997
**4.** [^2]
@Gallagher1999
**5.**
@Goldenfeld1999
**6.**
@Gleick1987
**7.**
@Waldrop1992
**8.**
@Brown1996
**9.**
@Falconer1990
**10.**
@Robinson1999
**1. Introduction**
The purpose of this paper is to present an unified, self-contained mathematical structure and physical understanding of the nature of chaos in a discrete dynamical system and to suggest a plausible explanation of *why* natural systems tend to be chaotic. The somewhat extensive quotations with which we begin above, bear testimony to both the increasingly significant — and perhaps all-pervasive — role of nonlinearity in the world today as also our imperfect state of understanding of its manifestations. The list of papers at both the UN Conference [@Grebogi1997] and in *Science* [@Gallagher1999] is noteworthy if only to justify the observation of @Gleick1987 that “chaos seems to be everywhere”. Even as everybody appears to be finding chaos and complexity in all likely and unlikely places, and possibly because of it, it is necessary that we have a clear mathematically-physical understanding of these notions that are supposedly reshaping our view of nature. This paper is an attempt to contribute to this goal. To make this account essentially self-contained we include here, as far as this is practicable, the basics of the background material needed to understand the paper in the form of *Tutorials* and an extended *Appendix.*
The paradigm of chaos of the kneading of the dough is considered to provide an intuitive basis of the mathematics of chaos [@Peitgen1992], and one of our fundamental objectives here is to recount the mathematical framework of this process in terms of the theory of ill-posed problems arising from non-injectivity [@Sengupta1997], *maximal ill-posedness,* and *graphical convergence* of functions [@Sengupta2000]. A natural mathematical formulation of the kneading of the dough in the form of *stretch-cut-and-paste* and *stretch-cut-and-fold* operations is in the ill-posed problem arising from the increasing non-injectivity of the function $f$ modeling the kneading operation.
***Begin Tutorial1: Functions and Multifunctions***
A *relation,* or *correspondence,* between two sets $X$ and $Y$, written $\mathscr{M}\!:X\qquad Y$, is basically a rule that associates subsets of $X$ to subsets of $Y$; this is often expressed as $(A,B)\in\mathscr{M}$ where $A\subset X$ and $B\subset Y$ and $(A,B)$ is an ordered pair of sets. The domain $$\mathcal{D}(\mathscr{M})\overset{\textrm{def}}=\{ A\subset X\!:(\!\exists Z\in\mathscr{M})(\pi_{X}(Z)=A)\}$$ and range $$\mathcal{R}(\mathscr{M})\overset{\textrm{def}}=\{ B\subset Y\!:(\!\exists Z\in\mathscr{M})(\pi_{Y}(Z)=B)\}$$
of $\mathscr{M}$ are respectively the sets of $X$ which under $\mathscr{M}$ corresponds to sets in $Y$; here $\pi_{X}$ and $\pi_{Y}$ are the projections of $Z$ on $X$ and $Y$ respectively. Equivalently, $\mathcal{D}(\mathcal{M})=\{ x\in X\!:\mathscr{M}(x)\neq\emptyset\}$ and $\mathcal{R}(\mathscr{M})=\bigcup_{x\in\mathcal{D}(\mathcal{M})}\mathscr{M}(x)$. The *inverse* $\mathscr M^{-}$ of $\mathscr{M}$ is the relation $$\mathscr M^{-}=\{(B,A)\!:(A,B)\!\in\mathscr{M}\}$$ so that $\mathscr M^{-}$ assigns $A$ to $B$ iff $\mathscr{M}$ assigns $B$ to $A$. In general, a relation may assign many elements in its range to a single element from its domain; of especial significance are *functional relations* $f$[^3] that can assigns only a unique element in $\mathcal{R}(f)$ to any element in $\mathcal{D}(f)$. Fig. \[Fig: functions\] illustrates the distinction between arbitrary and functional relations $\mathscr{M}$ and $f$. This difference between functions (or maps) and multifunctions is basic to our developments and should be fully understood. Functions can again be classified as injections (or $1:1$) and surjections (or onto). $f\!:X\rightarrow Y$ is said to be *injective* (or *one-to-one*) if $x_{1}\neq x_{2}\Rightarrow f(x_{1})\neq f(x_{2})$ for all $x_{1},x_{2}\in X$, while it is *surjective* (or *onto*) if $Y=f(X)$. $f$ is *bijective* if it is both $1:1$ and onto.
Associated with a function $f\!:X\rightarrow Y$ is its inverse $f^{-1}\!:Y\supseteq\mathcal{R}(f)\rightarrow X$ that exists on $\mathcal{R}(f)$ iff $f$ is injective. Thus when $f$ is bijective, $f^{-1}(y):=\{ x\in X\!:y=f(x)\}$ exists for every $y\in Y$; infact $f$ is bijective iff $f^{-1}(\{ y\})$ is a singleton for each $y\in Y$. Non-injective functions are not at all rare; if anything, they are very common even for linear maps and it would be perhaps safe to conjecture that they are overwhelmingly predominant in the nonlinear world of nature. Thus for example, the simple linear homogeneous differential equation with constant coefficients of order $n>1$ has $n$ linearly independent solutions so that the operator $D^{n}$ of $D^{n}(y)=0$ has a $n$-dimensional null space. Inverses of non-injective, and in general non-bijective, functions will be denoted by $f^{-}$. If $f$ is not injective then
$$A\subset f^{-}f(A)\overset{\textrm{def}}=\textrm{sat}(A)$$ where $\textrm{sat}(A)$ is the *saturation of* $A\subseteq X$ *induced by* $f$; if $f$ is not surjective then $${\textstyle ff^{-}(B):=B\bigcap f(X)\subseteq B.}$$ If $A=\textrm{sat}(A)$, then $A$ is said to be *saturated,* and $B\subseteq\mathcal{R}(f)$ whenever $ff^{-}(B)=B$. Thus for non-injective $f$, $f^{-}f$ is not an identity on $X$ just as $ff^{-}$ is not **$\mathbf{1}_{Y}$** if $f$ is not surjective. However the set of relations $$ff^{-}f=f,\qquad f^{-}ff^{-}=f^{-}\label{Eqn: f_inv_f}$$ that is always true will be of basic significance in this work. Following are some equivalent statements on the injectivity and surjectivity of functions $f\!:X\rightarrow Y$.
(Injec) $f$ is $1:1$ $\Leftrightarrow$ there is a function $f_{\textrm{L }}\!:Y\rightarrow X$ called the left inverse of $f$, such that $f_{\textrm{L}}f=\mathbf{1}_{X}$ $\Leftrightarrow$ $A=f^{-}f(A)$ for all subsets $A$ of $X$$\Leftrightarrow$$f(\bigcap A_{i})=\bigcap f(A_{i})$.
(Surjec) $f$ is onto $\Leftrightarrow$ there is a function $f_{\textrm{R }}\!:Y\rightarrow X$ called the right inverse of $f$, such that $ff_{\textrm{R}}=\mathbf{1}_{Y}$ $\Leftrightarrow$ $B=ff^{-}(B)$ for all subsets $B$ of $Y$.
As we are primarily concerned with non-injectivity of functions, saturated sets generated by equivalence classes of $f$ will play a significant role in our discussions. A relation $\mathscr{E}\mathcal{E}$ on a set $X$ is said to be an *equivalence relation* if it is[^4]
(ER1) Reflexive: $(\forall x\in X)(x\mathcal{E}x)$.
(ER2) Symmetric: $(\forall x,y\in X)(x\mathcal{E}y\Longrightarrow y\mathcal{E}x)$.
(ER3) Transitive: $(\forall x,y,z\in X)(x\mathcal{E}y\wedge y\mathcal{E}z\Longrightarrow x\mathcal{E}z)$.
Equivalence relations group together unequal elements $x_{1}\neq x_{2}$ of a set as equivalent according to the requirements of the relation. This is expressed as $x_{1}\sim x_{2}\textrm{ }(\textrm{mod }\mathcal{E})$ and will be represented here by the shorthand notation $x_{1}\sim_{\mathcal{E}}x_{2}$, or even simply as $x_{1}\sim x_{2}$ if the specification of $\mathcal{E}$ is not essential. Thus for a noninjective map if $f(x_{1})=f(x_{2})$ for $x_{1}\neq x_{2}$, then $x_{1}$ and $x_{2}$ can be considered to be equivalent to each other since they map onto the same point under $f$; thus $x_{1}\sim_{f}x_{2}\Leftrightarrow f(x_{1})=f(x_{2})$ defines the equivalence relation $\sim_{f}$ induced by the map $f$. Given an equivalence relation $\sim$ on a set $X$ and an element $x\in X$ the subset $$[x]\overset{\textrm{def}}=\{ y\in X\!:y\sim x\}$$ is called the *equivalence class of $x$;* thus $x\sim y\Leftrightarrow[x]=[y]$*.* In particular, equivalence classes generated by $f\!:X\rightarrow Y$, $[x]_{f}=\{ x_{\alpha}\in X\!:f(x_{\alpha})=f(x)\}$, will be a cornerstone of our analysis of chaos generated by the iterates of non-injective maps, and the equivalence relation $\sim_{f}:=\{(x,y)\!:f(x)=f(y)\}$ generated by $f$ is uniquely defined by the partition that $f$ induces on $X$. Of course as $x\sim x$, $x\in[x]$. It is a simple matter to see that any two equivalence classes are either disjoint or equal so that the equivalence classes generated by an equivalence relation on $X$ form a disjoint cover of $X.$ The *quotient set of $X$ under $\sim$,* denoted by $X/\sim\;:=\{[x]\!:x\in X\}$, has the equivalence classes $[x]$ as its elements; thus $[x]$ plays a dual role either as subsets of $X$ or as elements of $X/\sim$. The rule $x\mapsto[x]$ defines a surjective function $Q\!:X\rightarrow X/\sim$ known as the *quotient map.*
**Example 1.1.** Let $$S^{1}=\{(x,y)\in\mathbb{R}^{2})\!:x^{2}+y^{2}=1\}$$
be the unit circle in $\mathbb{R}^{2}$. Consider $X=[0,1]$ as a subspace of $\mathbb{R}$, define a map $$q\!:X\rightarrow S^{1},\qquad s\longmapsto(\cos2\pi s,\sin2\pi s),\,\, s\in X,$$
from $\mathbb{R}$ to $\mathbb{R}^{2}$, and let $\sim$ be the equivalence relation on $X$ $$s\sim t\Longleftrightarrow(s=t)\vee(s=0,t=1)\vee(s=1,t=0).$$ If we bend $X$ around till its ends touch, the resulting circle represents the quotient set $Y=X/\sim$ whose points are equivalent under $\sim$ as follows $$[0]=\{0,1\}=[1],\qquad[s]=\{ s\}\,\textrm{for all }s\in(0,1).$$
Thus $q$ is bijective for $s\in(0,1)$ but two-to-one for the special values $s=0\textrm{ and }1$, so that for $s,t\in X$,$$s\sim t\Longleftrightarrow q(s)=q(t).$$ This yields a bijection $h\!:X/\sim\:\rightarrow S^{1}$ such that $$q=h\circ Q$$
defines the quotient map $Q\!:X\rightarrow X/\sim$ by $h([s])=q(s)$ for all $s\in[0,1]$. The situation is illustrated by the commutative diagram of Fig. \[Fig: quotient\] that appears as an integral component in a different and more general context in Sec. 2. It is to be noted that commutativity of the diagram implies that if a given equivalence relation $\sim$ on $X$ is completely determined by $q$ that associates the partitioning equivalence classes in $X$ to unique points in $S^{1}$, then $\sim$ is identical to the equivalence relation that is induced by $Q$ on $X$. Note that a larger size of the equivalence classes can be obtained by considering $X=\mathbb{R}_{+}$ for which $s\sim t\Leftrightarrow|s-t|\in\mathbb{Z}_{+}$.$\qquad\blacksquare$
***End Tutorial1***
One of the central concepts that we consider and employ in this work is the inverse $f^{-}$ of a nonlinear, non-injective, function $f$; here the equivalence classes $[x]_{f}=f^{-}f(x)$ of $x\in X$ are the saturated subsets of $X$ that partition $X$. While a detailed treatment of this question in the form of the non-linear ill-posed problem and its solution is given in Sec. 2 [@Sengupta1997], it is sufficient to point out here from Figs. \[Fig: functions\](c) and \[Fig: functions\](d), that the inverse of a noninjective function is not a function but a multifunction while the inverse of a multifunction is a noninjective function. Hence one has the general result that$$\begin{aligned}
f\textrm{ is a non injective function} & \Longleftrightarrow & f^{-}\textrm{ is a multifunction}.\label{Eqn: func-multi}\\
f\textrm{ is a multifunction} & \Longleftrightarrow & f^{-}\textrm{ is a non injective function}\nonumber \end{aligned}$$
The inverse of a multifunction $\mathscr{M}\!:X\qquad Y$ is a generalization of the corresponding notion for a function $f\!:X\rightarrow Y$ such that $$\mathscr M^{-}(y)\overset{\textrm{def}}=\{ x\in X\!:y\in\mathscr{M}(x)\}$$
leads to $${\textstyle \mathscr M^{-}(B)=\{ x\in X\!:\mathscr{M}(x)\bigcap B\neq\emptyset\}}$$
for any $B\subseteq Y$, while a more restricted inverse that we shall not be concerned with is given as $\mathscr M^{+}(B)=\{ x\in X\!:\mathscr{M}(x)\subseteq B\}$. Obviously, $\mathscr M^{+}(B)\subseteq\mathscr M^{-}(B)$. A multifunction is injective if $x_{1}\neq x_{2}\Rightarrow\mathscr{M}(x_{1})\bigcap\mathscr{M}(x_{2})=\emptyset$, and in common with functions it is true that $$\begin{aligned}
\mathscr{M}\left(\bigcup_{\alpha\in\mathbb{{D}}}A_{\alpha}\right)= & \bigcup_{\alpha\in\mathbb{{D}}}\mathscr{M}(A_{\alpha})\\
\mathscr{M}\left(\bigcap_{\alpha\in\mathbb{{D}}}A_{\alpha}\right)\subseteq & \bigcap_{\alpha\in\mathbb{{D}}}\mathscr{M}(A_{\alpha})\end{aligned}$$
and where $\mathbb{D}$ is an index set. The following illustrates the difference between the two inverses of $\mathscr{M}$. Let $X$ be a set that is partitioned into two disjoint $\mathscr{M}$-invariant subsets $X_{1}$ and $X_{2}$. If $x\in X_{1}$ (or $x\in X_{2}$) then $\mathscr{M}(x)$ represents that part of $X_{1}$ (or of $X_{2}$ ) that is realized immediately after one application of $\mathscr{M}$, while $\mathscr M^{-}(x)$ denotes the possible precursors of $x$ in $X_{1}$ (or of $X_{2}$) and $\mathscr M^{+}(B)$ is that subset of $X$ whose image lies in $B$ for any subset $B\subset X$.
In this work the multifunctions we are explicitly concerned with arise as the inverses of non-injective functions.
The second major component of our theory is the *graphical convergence of a net of functions to a multifunction.* In Tutorial2 below, we replace for the sake of simplicity and without loss of generality, the net (which is basically a sequence where the index set is not necessarily the positive integers; thus every sequence is a net but the family[^5] indexed, for example, by $\mathbb{Z}$, the set of *all* integers, is a net and not a sequence) with a sequence and provide the necessary background and motivation for the concept of graphical convergence.
***Begin Tutorial2: Convergence of Functions***
This Tutorial reviews the inadequacy of the usual notions of convergence of functions either to limit functions or to distributions and suggests the motivation and need for introduction of the notion of graphical convergence of functions to multifunctions. Here, we follow closely the exposition of @Korevaar1968, and use the notation $(f_{k})_{k=1}^{\infty}$ to denote real or complex valued functions on a bounded or unbounded interval $J$.
A sequence of piecewise continuous functions $(f_{k})_{k=1}^{\infty}$ is said to converge to the function $f$, notation $f_{k}\rightarrow f$, on a bounded or unbounded interval $J$[^6]
\(1) *Pointwise* if$$f_{k}(x)\longrightarrow f(x)\qquad\textrm{for all }x\in J,$$
that is: Given any arbitrary real number $\varepsilon>0$ there exists a $K\in\mathbb{N}$ that may depend on $x$, such that $|f_{k}(x)-f(x)|<\varepsilon$ for all $k\geq K$.
\(2) *Uniformly* if $$\sup_{x\in J}|f(x)-f_{k}(x)|\longrightarrow0\qquad\textrm{as }k\longrightarrow\infty,$$
that is: Given any arbitrary real number $\varepsilon>0$ there exists a $K\in\mathbb{N}$, such that $\sup_{x\in J}|f_{k}(x)-f(x)|<\varepsilon$ for all $k\geq K$.
\(3) *In the mean of order $p\geq1$* if $|f(x)-f_{k}(x)|^{p}$ is integrable over $J$ for each $k$ $$\int_{J}|f(x)-f_{k}(x)|^{p}\longrightarrow0\qquad\textrm{as }k\rightarrow\infty.$$
For $p=1$, this is the simple case of *convergence in the mean.*
\(4) *In the mean $m$-integrally* if it is possible to select indefinite integrals $$f_{k}^{(-m)}(x)=\pi_{k}(x)+\int_{c}^{x}dx_{1}\int_{c}^{x_{1}}dx_{2}\cdots\int_{c}^{x_{m-1}}dx_{m}f_{k}(x_{m})$$
and
$$f^{(-m)}(x)=\pi(x)+\int_{c}^{x}dx_{1}\int_{c}^{x_{1}}dx_{2}\cdots\int_{c}^{x_{m-1}}dx_{m}f(x_{m})$$
such that for some arbitrary real $p\geq1$, $$\int_{J}|f^{(-m)}-f_{k}^{(-m)}|^{p}\longrightarrow0\qquad\textrm{as }k\rightarrow\infty.$$
where the polynomials $\pi_{k}(x)$ and $\pi(x)$ are of degree $<m$, and $c$ is a constant to be chosen appropriately.
\(5) *Relative to test functions $\varphi$* if $f\varphi$ and $f_{k}\varphi$ are integrable over $J$ and $$\int_{J}(f_{k}-f)\varphi\longrightarrow0\qquad\textrm{for every }\varphi\in\mathcal{C}_{0}^{\infty}(J)\textrm{ as }k\longrightarrow\infty,$$
where $\mathcal{C}_{0}^{\infty}(J)$ is the class of infinitely differentiable continuous functions that vanish throughout some neighbourhood of each of the end points of $J$. For an unbounded $J$, a function is said to vanish in some neighbourhood of $+\infty$ if it vanishes on some ray $(r,\infty)$.
While pointwise convergence does not imply any other type of convergence, uniform convergence on a bounded interval implies all the other convergences.
It is to be observed that apart from pointwise and uniform convergences, all the other modes listed above represent some sort of an averaged contribution of the entire interval $J$ and are therefore not of much use when pointwise behaviour of the limit $f$ is necessary. Thus while limits in the mean are not unique, oscillating functions are tamed by $m$-integral convergence for adequately large values of $m$, and convergence relative to test functions, as we see below, can be essentially reduced to $m$-integral convergence. On the contrary, our graphical convergence — which may be considered as a pointwise biconvergence with respect to both the direct and inverse images of $f$ just as usual pointwise convergence is with respect to its direct image only — allows a sequence (in fact, a net) of functions to converge to an arbitrary relation, unhindered by external influences such as the effects of integrations and test functions. To see how this can indeed matter, consider the following
**Example 1.2.** Let $f_{k}(x)=\sin kx,\, k=1,2,\cdots$ and let $J$ be any bounded interval of the real line. Then $1$-integrally we have$$f_{k}^{(-1)}(x)=-\frac{1}{k}\cos kx=-\frac{1}{k}+\int_{0}^{x}\sin kx_{1}dx_{1},$$
which obviously converges to $0$ uniformly (and therefore in the mean) as $k\rightarrow\infty$. And herein lies the point: even though we cannot conclude about the exact nature of $\sin kx$ as $k$ increases indefinitely (except that its oscillations become more and more pronounced), we may very definitely state that $\lim_{k\rightarrow\infty}(\cos kx)/k=0$ uniformly. Hence from$$f_{k}^{(-1)}(x)\longrightarrow0=0+\int_{0}^{x}\lim_{k\rightarrow\infty}\sin kx_{1}dx_{1}$$ it follows that $$\lim_{k\rightarrow\infty}\sin kx=0\label{Eqn: intsin}$$ $1$-integrally.
Continuing with the same sequence of functions, we now examine its test-functional convergence with respect to $\varphi\in\mathcal{C}_{0}^{1}(-\infty,\infty)$ that vanishes for all $x\notin(\alpha,\beta)$. Integrating by parts, $$\begin{aligned}
{\displaystyle {\displaystyle \int_{-\infty}^{\infty}f_{k}\varphi}}= & {\displaystyle \int_{\alpha}^{\beta}\varphi(x_{1})\sin kx_{1}dx_{1}}\\
= & -\frac{1}{k}\left[\varphi(x_{1})\cos kx_{1}\right]_{\alpha}^{\beta}-\frac{1}{k}\int_{\alpha}^{\beta}\varphi^{\prime}(x_{1})\cos kx_{1}dx_{1}\end{aligned}$$
The first integrated term is $0$ due to the conditions on $\varphi$ while the second also vanishes because $\varphi\in\mathcal{C}_{0}^{1}(-\infty,\infty)$. Hence $$\int_{-\infty}^{\infty}f_{k}\varphi\longrightarrow0=\int_{\alpha}^{\beta}\lim_{k\rightarrow\infty}\varphi(x_{1})\sin ksdx_{1}$$ for all $\varphi$, and leading to the conclusion that $$\lim_{k\rightarrow\infty}\sin kx=0\label{Eqn: testsin}$$ test-functionally.$\qquad\blacksquare$
This example illustrates the fact that if $\textrm{Supp}(\varphi)=[\alpha,\beta]\subseteq J$[^7], integrating by parts sufficiently large number of times so as to wipe out the pathological behaviour of $(f_{k})$ gives $$\begin{aligned}
\int_{J}f_{k}\varphi= & \int_{\alpha}^{\beta}f_{k}\varphi\\
= & \int_{\alpha}^{\beta}f_{k}^{(-1)}\varphi^{\prime}=\cdots=(-1)^{m}\int_{\alpha}^{\beta}f_{k}^{(-m)}\varphi^{m}\end{aligned}$$
where $f_{k}^{(-m)}(x)=\pi_{k}(x)+\int_{c}^{x}dx_{1}\int_{c}^{x_{1}}dx_{2}\cdots\int_{c}^{x_{m-1}}dx_{m}f_{k}(x_{m})$ is an $m$-times arbitrary indefinite integral of $f_{k}$. If now it is true that $\int_{\alpha}^{\beta}f_{k}^{(-m)}\rightarrow\int_{\alpha}^{\beta}f^{(-m)}$, then it must also be true that $f_{k}^{(-m)}\varphi^{(m)}$ converges in the mean to $f^{(-m)}\varphi^{(m)}$ so that $$\int_{\alpha}^{\beta}f_{k}\varphi=(-1)^{m}\int_{\alpha}^{\beta}f_{k}^{(-m)}\varphi^{(m)}\longrightarrow(-1)^{m}\int_{\alpha}^{\beta}f^{(-m)}\varphi^{(m)}=\int_{\alpha}^{\beta}f\varphi.$$ In fact the converse also holds leading to the following Equivalences between $m$-convergence in the mean and convergence with respect to test-functions, [@Korevaar1968].
**Type 1 Equivalence.** If $f$ and $(f_{k})$ are functions on $J$ that are integrable on every interior subinterval, then the following are equivalent statements.
\(a) For every interior subinterval $I$ of $J$ there is an integer $m_{I}\geq0$, and hence a smallest integer $m\geq0$, such that certain indefinite integrals $f_{k}^{(-m)}$ of the functions $f_{k}$ converge in the mean on $I$ to an indefinite integral $f^{(-m)}$; thus $\int_{I}|f_{k}^{(-m)}-f^{(-m)}|\rightarrow0.$
\(b) $\int_{J}(f_{k}-f)\varphi\rightarrow0$ for every $\varphi\in\mathcal{C}_{0}^{\infty}(J)$.
A significant generalization of this Equivalence is obtained by dropping the restriction that the limit object $f$ be a function. The need for this generalization arises because metric function spaces are known not to be complete: Consider the sequence of functions (Fig. \[Fig: FuncSpace\](a)) $$\begin{aligned}
f_{k}(x)= & \left\{ \begin{array}{lcl}
0 & \textrm{} & \textrm{if }a\leq x\leq0\\
kx & \textrm{} & \textrm{if }0\leq x\leq1/k\\
1 & \textrm{} & \textrm{if }1/k\leq x\leq b\end{array}\right.\label{Eqn: Lp[a,b]}\end{aligned}$$
which is not Cauchy in the uniform metric $\rho(f_{j},f_{k})=\sup_{a\leq x\leq b}|f_{j}(x)-f_{k}(x)|$ but is Cauchy in the mean $\rho(f_{j},f_{k})=\int_{a}^{b}|f_{j}(x)-f_{k}(x)|dx$, or even pointwise. However in either case, $(f_{k})$ cannot converge in the respective metrics to a *continuous function* and the limit is a discontinuous unit step function $$\Theta(x)=\left\{ \begin{array}{lcl}
0 & & \textrm{if }a\leq x\leq0\\
1 & & \textrm{if }0<x\leq b\end{array}\right.$$ with graph $([a,0],0)\bigcup((0,b],1)$, which is also integrable on $[a,b]$. Thus even if the limit of the sequence of continuous functions is not continuous, both the limit and the members of the sequence are integrable functions. This Riemann integration is not sufficiently general, however, and this type of integrability needs to be replaced by a much weaker condition resulting in the larger class of the Lebesgue integrable complete space of functions $L[a,b]$.[^8]
The functions in Fig \[Fig: FuncSpace\](b1), $$\delta_{k}(x)=\left\{ \begin{array}{ccl}
k & & \textrm{if }0<x<1/k\\
0 & & x\in[a,b]-(0,1/k),\end{array}\right.$$
can be associated with the arbitrary indefinite integrals $$\Theta_{k}(x)\overset{\textrm{def}}=\delta_{k}^{(-1)}(x)=\left\{ \begin{array}{lcl}
0 & & a\leq x\leq0\\
kx & & 0<x<1/k\\
1 & & 1/k\leq x\leq b\end{array}\right.$$
of Fig. \[Fig: FuncSpace\](b2), which, as noted above, converge in the mean to the unit step function $\Theta(x)$; hence $\int_{-\infty}^{\infty}\delta_{k}\varphi\equiv\int_{\alpha}^{\beta}\delta_{k}\varphi=-\int_{\alpha}^{\beta}\delta_{k}^{(-1)}\varphi^{\prime}\rightarrow-\int_{0}^{\beta}\varphi^{\prime}(x)dx=\varphi(0)$. But there can be no *functional relation $\delta(x)$* for which $\int_{\alpha}^{\beta}\delta(x)\varphi(x)dx=\varphi(0)$ for *all* $\varphi\in C_{0}^{1}[\alpha,\beta]$, so that unlike in the case in Type 1 Equivalence, the limit in the mean $\Theta(x)$ of the indefinite integrals $\delta_{k}^{(-1)}(x)$ *cannot be expressed as the indefinite integral $\delta^{(-1)}(x)$ of some function $\delta(x)$ on any interval containing the origin.* This leads to the second more general type of equivalence
**Type 2 Equivalence.** If $(f_{k})$ are functions on $J$ that are integrable on every interior subinterval, then the following are equivalent statements.
\(a) For every interior subinterval $I$ of $J$ there is an integer $m_{I}\geq0$, and hence a smallest integer $m\geq0$, such that certain indefinite integrals $f_{k}^{(-m)}$ of the functions $f_{k}$ converge in the mean on $I$ to an integrable function $\Theta$ which, unlike in Type 1 Equivalence, need not itself be an indefinite integral of some function $f$.
\(b) $c_{k}(\varphi)=\int_{J}f_{k}\varphi\rightarrow c(\varphi)$ for every $\varphi\in\mathcal{C}_{0}^{\infty}(J)$.
Since we are now given that $\int_{I}f_{k}^{(-m)}(x)dx\rightarrow\int_{I}\Psi(x)dx$, it must also be true that $f_{k}^{(-m)}\varphi^{(m)}$ converges in the mean to $\Psi\varphi^{(m)}$ whence $$\int_{J}f_{k}\varphi=(-1)^{m}\int_{I}f_{k}^{(-m)}\varphi^{(m)}\longrightarrow(-1)^{m}\int_{I}\Psi\varphi^{(m)}\left(\neq(-1)^{m}\int_{I}f^{(-m)}\varphi^{(m)}\right).$$
The natural question that arises at this stage is then: What is the nature of the relation (not function any more) $\Psi(x)$? For this it is now stipulated, despite the non-equality in the equation above, that as in the mean $m$-integral convergence of $(f_{k})$ to a *function* $f$, $$\Theta(x):=\lim_{k\rightarrow\infty}\delta_{k}^{(-1)}(x)\overset{\textrm{def}}=\int_{-\infty}^{x}\delta(x^{\prime})dx^{\prime}\label{Eqn: delta1}$$
*defines* the non-functional relation (“generalized function”) $\delta(x)$ integrally as a solution of the integral equation (\[Eqn: delta1\]) of the first kind; hence formally[^9] $$\delta(x)=\frac{d\Theta}{dx}\label{Eqn: delta2}$$
***End Tutorial2***
The above tells us that the “delta function” is not a function but its indefinite integral is the piecewise continuous *function* $\Theta$ obtained as the mean (or pointwise) limit of a sequence of non-differentiable functions with the integral of $d\Theta_{k}(x)/dx$ being preserved for all $k\in\mathbb{Z}_{+}$. What then is the delta (and not its integral)? The answer to this question is contained in our multifunctional extension $\textrm{Multi}(X,Y)$ of the function space $\textrm{Map}(X,Y)$ considered in Sec. 3. Our treatment of ill-posed problems is used to obtain an understanding and interpretation of the numerical results of the discretized spectral approximation in neutron transport theory [@Sengupta1988; @Sengupta1995]. The main conclusions are the following: In a one-dimensional discrete system that is governed by the iterates of a nonlinear map, the dynamics is chaotic if and only if the system evolves to a state of *maximal ill-posedness.* The analysis is based on the non-injectivity, and hence ill-posedness, of the map; this may be viewed as a mathematical formulation of the *stretch-and-fold* and *stretch-cut-and-paste* kneading operations of the dough that are well-established artifacts in the theory of chaos and the concept of maximal ill-posedness helps in obtaining a *physical understanding* of the nature of chaos. We do this through the fundamental concept of the *graphical convergence* of a sequence (generally a net) of functions [@Sengupta2000] that is allowed to converge graphically, when the conditions are right, to a set-valued map or multifunction. Since ill-posed problems naturally lead to multifunctional inverses through functional generalized inverses [@Sengupta1997], it is natural to seek solutions of ill-posed problems in multifunctional space $\textrm{Multi}(X,Y)$ rather than in spaces of functions $\textrm{Map}(X,Y)$; here $\textrm{Multi}(X,Y)$ is an extension of $\textrm{Map}(X,Y)$ that is generally larger than the smallest dense extension $\textrm{Multi}_{\mid}(X,Y)$.
Feedback and iteration are natural processes by which nature evolves itself. Thus almost every process of evolution is a self-correction process by which the system proceeds from the present to the future through a controlled mechanism of input and evaluation of the past. Evolution laws are inherently nonlinear and complex; here *complexity* is to be understood as the natural manifestation of the nonlinear laws that govern the evolution of the system.
This work presents a mathematical description of complexity based on [@Sengupta1997] and [@Sengupta2000] and is organized as follows. In Sec. 1, we follow [@Sengupta1997] to give an overview of ill-posed problems and their solution that forms the foundation of our approach. Secs. 2 to 4 apply these ideas by defining a chaotic dynamical system as a *maximally ill-posed problem;* by doing this we are able to overcome the limitations of the three Devaney characterizations of chaos [@Devaney1989] that apply to the specific case of iteration of transformations in a metric space, and the resulting graphical convergence of functions to multifunctions is the basic tool of our approach. Sec. 5 analyzes graphical convergence in $\textrm{Multi}(X)$ for the discretized spectral approximation of neutron transport theory, which suggests a natural link between ill-posed problems and spectral theory of non-linear operators. This seems to offer an answer to the question of *why* a natural system should increase its complexity, and eventually tend toward chaoticity, by becoming increasingly nonlinear.
**2. Ill-Posed Problem and its solution**
This section based on @Sengupta1997 presents a formulation and solution of ill-posed problems arising out of the non-injectivity of a function $f\!:X\rightarrow Y$ between topological spaces $X$ and $Y$. A workable knowledge of this approach is necessary as our theory of chaos leading to the characterization of chaotic systems as being a *maximally ill-posed* state of a dynamical system is a direct application of these ideas and can be taken to constitute a mathematical representation of the familiar *stretch-cut-and paste* and *stretch-and-fold* paradigms of chaos. The problem of finding an $x\in X$ for a given $y\in Y$ from the functional relation $f(x)=y$ is an inverse problem that is *ill-posed* (or, the equation $f(x)=y$ is ill-posed) if any one or more of the following conditions are satisfied.
(IP1) $f$ *is not injective.* This *non-uniqueness* problem of the solution for a given $y$ is the single most significant criterion of ill-posedness used in this work.
(IP2) *$f$ is not surjective.* For a $y\in Y$, this is the *existence* problem of the given equation.
(IP3) When $f$ *is bijective,* the inverse *$f^{-1}$* is not continuous, which means that small changes in $y$ may lead to large changes in $x$.
A problem $f(x)=y$ for which a solution exists, is unique, and small changes in data $y$ lead to only small changes in the solution $x$ is said to be *well-posed* or *properly posed.* This means that $f(x)=y$ is well-posed if $f$ is bijective and the inverse $f^{-1}\!:Y\rightarrow X$ is continuous; otherwise the equation is *ill-posed* or *improperly posed.* It is to be noted that the three criteria are not, in general, independent of each other. Thus if $f$ represents a bijective, bounded linear operator between Banach spaces $X$ and $Y$, then the inverse mapping theorem guarantees that the inverse $f^{-1}$ is continuous. Hence ill-posedness depends not only on the algebraic structures of $X$, $Y$, $f$ but also on the topologies of $X$ and $Y$.
**Example 2.1.** As a non-trivial example of an inverse problem, consider the heat equation$$\frac{\partial\theta(x,t)}{\partial t}=c^{2}\frac{\partial^{2}\theta(x,t)}{\partial x^{2}}$$
for the temperature distribution $\theta(x,t)$ of a one-dimensional homogeneous rod of length $L$ satisfying the initial condition $\theta(x,0)=\theta_{0}(x),\textrm{ }0\leq x\leq L$, and boundary conditions $\theta(0,t)=0=\theta(L,t),\,0\leq t\leq T$, having the Fourier sine-series solution $$\theta(x,t)=\sum_{n=1}^{\infty}A_{n}\sin\left(\frac{n\pi}{L}x\right)e^{-\lambda_{n}^{2}t}\label{Eqn: heat1}$$
where $\lambda_{n}=(c\pi/a)n$ and $$A_{n}=\frac{2}{L}\int_{0}^{a}\theta_{0}(x^{\prime})\sin\left(\frac{n\pi}{L}x^{\prime}\right)dx^{\prime}$$ are the Fourier expansion coefficients. While the direct problem evaluates $\theta(x,t)$ from the differential equation and initial temperature distribution $\theta_{0}(x)$, the inverse problem calculates $\theta_{0}(x)$ from the integral equation $$\theta_{T}(x)=\frac{2}{L}\int_{0}^{a}k(x,x^{\prime})\theta_{0}(x^{\prime})dx^{\prime},\qquad0\leq x\leq L,$$
when this final temperature $\theta_{T}$ is known, and $$k(x,x^{\prime})=\sum_{n=1}^{\infty}\sin\left(\frac{n\pi}{L}x\right)\sin\left(\frac{n\pi}{L}x^{\prime}\right)e^{-\lambda_{n}^{2}T}$$ is the kernel of the integral equation. In terms of the final temperature the distribution becomes $$\theta_{T}(x)=\sum_{n=1}^{\infty}B_{n}\sin\left(\frac{n\pi}{L}x\right)e^{-\lambda_{n}^{2}(t-T)}\label{Eqn: heat2}$$
with Fourier coefficients $$B_{n}=\frac{2}{L}\int_{0}^{a}\theta_{T}(x^{\prime})\sin\left(\frac{n\pi}{L}x^{\prime}\right)dx^{\prime}.$$
In $L^{2}[0,a]$, Eqs. (\[Eqn: heat1\]) and (\[Eqn: heat2\]) at $t=T$ and $t=0$ yield respectively $$\Vert\theta_{T}(x)\Vert^{2}=\frac{L}{2}\sum_{n=1}^{\infty}A_{n}^{2}e^{-2\lambda_{n}^{2}T}\leq e^{-2\lambda_{1}^{2}T}\Vert\theta_{0}\Vert^{2}\label{Eqn: heat3}$$ $$\Vert\theta_{0}\Vert^{2}=\frac{L}{2}\sum_{n=1}^{\infty}B_{n}^{2}e^{2\lambda_{n}^{2}T}.\label{Eqn: heat4}$$
The last two equations differ from each other in the significant respect that whereas Eq. (\[Eqn: heat3\]) shows that the direct problem is well-posed according to (IP3), Eq. (\[Eqn: heat4\]) means that in the absence of similar bounds the inverse problem is ill-posed.[^10]$\qquad\blacksquare$
**Example 2.2.** Consider the ****Volterra integral equation of the first kind $$y(x)=\int_{a}^{x}r(x^{\prime})dx^{\prime}=Kr$$
where $y,r\in C[a,b]$ and $K\!:C[0,1]\rightarrow C[0,1]$ is the corresponding integral operator. Since the differential operator $D=d/dx$ under the sup-norm $\Vert r\Vert=\sup_{0\leq x\leq1}|r(x)|$ is unbounded, the inverse problem $r=Dy$ for a differentiable function $y$ on $[a,b]$ is ill-posed, see Example 6.1. However, $y=Kr$ becomes well-posed if $y$ is considered to be in $C^{1}[0,1]$ with norm $\Vert y\Vert=\sup_{0\leq x\leq1}|Dy|$. This illustrates the importance of the topologies of $X$ and $Y$ in determining the ill-posed nature of the problem when this is due to (IP3).$\qquad\blacksquare$
Ill-posed problems in nonlinear mathematics of type (IP1) arising from the non-injectivity of $f$ can be considered to be a generalization of non-uniqueness of solutions of linear equations as, for example, in eigenvalue problems or in the solution of a system of linear algebraic equations with a larger number of unknowns than the number of equations. In both cases, for a given $y\in Y$, the solution set of the equation $f(x)=y$ is given by $$f^{-}(y)=[x]_{f}=\{ x^{\prime}\in X:f(x^{\prime})=f(x)=y\}.$$
A significant point of difference between linear and nonlinear problems is that unlike the special importance of 0 in linear mathematics, there are no preferred elements in nonlinear problems; this leads to a shift of emphasis from the null space of linear problems to equivalence classes for nonlinear equations. To motivate the role of equivalence classes, let us consider the null spaces in the following linear problems.
\(a) Let $f:\mathbb{R}^{2}\rightarrow\mathbb{R}$ be defined by $f(x,y)=x+y$, $(x,y)\in\mathbb{R}^{2}$. The null space of $f$ is generated by the equation $y=-x$ on the $x$-$y$ plane, and the graph of $f$ is the plane passing through the lines $\rho=x$ and $\rho=y.$ For each $\rho\in\textrm{R}$ the equivalence classes $f^{-}(\rho)=\{(x,y)\in\textrm{R}^{2}\!:x+y=\rho\}$ are lines on the graph parallel to the null set.
\(b) For a linear operator $A\!:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$, $m<n$, satisfying (1) and (2), the problem $Ax=y$ reduces $A$ to echelon form with rank $r$ less than $\min\{ m,n\}$, when the given equations are consistent. The solution however, produces a generalized inverse leading to a set-valued inverse $A^{-}$ of $A$ for which the inverse images of $y\in\mathcal{R}(A)$ are multivalued because of the non-trivial null space of $A$ introduced by assumption (1). Specifically, a null-space of dimension $n-r$ is generated by the free variables $\{ x_{j}\}_{j=r+1}^{n}$ which are arbitrary: this is illposedness of type (1). In addition, $m-r$ rows of the row reduced echelon form of $A$ have all 0 entries that introduces restrictions on $m-r$ coordinates $\{ y_{i}\}_{i=r+1}^{m}$ of $y$ which are now related to $\{ y_{i}\}_{i=1}^{r}$: this illustrates illposedness of type (2). Inverse ill-posed problems therefore generate multivalued solutions through a generalized inverse of the mapping.
\(c) The eigenvalue problem $$\left(\frac{d^{2}}{dx^{2}}+\lambda^{2}\right)y=0\qquad y(0)=0=y(1)$$
has the following equivalence class of 0 $$[0]_{D^{2}}=\{\sin(\pi mx)\}_{m=0}^{\infty},\qquad D^{2}=\left(d^{2}/dx^{2}+\lambda^{2}\right),$$
as its eigenfunctions corresponding to the eigenvalues $\lambda_{m}=\pi m$.
Ill-posed problems are primarily of interest to us explicitly as noninjective maps $f$, that is under the condition of (IP1). The two other conditions (IP2) and (IP3) are not as significant and play only an implicit role in the theory. In its application to iterative systems, the degree of non-injectivity of $f$ defined as the number of its injective branches, increases with iteration of the map. A necessary (but not sufficient) condition for chaos to occur is the increasing non-injectivity of $f$ that is expressed descriptively in the chaos literature as *stretch-and-fold* or *stretch-cut-and-paste* operations. This increasing noninjectivity that we discuss in the following sections, is what causes a dynamical system to tend toward chaoticity. Ill-posedness arising from non-surjectivity of (injective) $f$ in the form of *regularization* [@Tikhonov1977] **has received wide attention in the literature of ill-posed problems; this however is not of much significance in our work.
***Begin Tutorial3: Generalized Inverse***
In this Tutorial, we take a quick look at the equation $a(x)=y$, where $a\!:X\rightarrow Y$ is a linear map that need not be either one-one or onto. Specifically, we will take $X$ and $Y$ to be the Euclidean spaces $\mathbb{R}^{n}$ and $\mathbb{R}^{m}$ so that $a$ has a matrix representation $A\in\mathbb{R}^{m\times n}$ where $\mathbb{R}^{m\times n}$ is the collection of $m\times n$ matrices with real entries. The inverse $A^{-1}$ exists and is unique iff $m=n$ and $\textrm{rank}(A)=n$; this is the situation depicted in Fig. \[Fig: functions\](a). If $A$ is neither one-one or onto, then we need to consider the multifunction $A^{-}$, a functional choice of which is known as the *generalized inverse* $G$ of $A$. A good introductory text for generalized inverses is @Campbell1979Figure \[Fig: MP\_Inverse\](a) introduces the following definition of the *Moore-Penrose* generalized inverse $G_{\textrm{MP}}$.
**Definition 2.1.** ***Moore-Penrose Inverse.*** *If $a\!:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ is a linear transformation with matrix representation $A\in\mathbb{R}^{m\times n}$ then the* Moore-Penrose inverse $G_{\textrm{MP}}\in\mathbb{R}^{n\times m}$ of $A$ *(we will use the same notation* $G_{\textrm{MP}}\!:\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$ *for the inverse of the map $a$) is the noninjective map defined in terms of the row and column spaces of $A$,* $\textrm{row}(A)=\mathcal{R}(A^{\textrm{T}})$, $\textrm{col}(A)=\mathcal{R}(A)$*, as* $$G_{\textrm{MP}}(y)\overset{\textrm{def}}=\left\{ \begin{array}{lcl}
(a|_{\textrm{row}(A)})^{-1}(y), & & \textrm{if }y\in\textrm{col}(A)\\
0 & & \textrm{if }y\in\mathcal{N}(A^{\textrm{T}}).\end{array}\right.\qquad\square\label{Eqn: Def: Moore-Penrose}$$
Note that the restriction $a|_{\textrm{row}(A)}$ of $a$ to $\mathcal{R}(A^{\textrm{T}})$ is bijective so that the inverse $(a|_{\textrm{row}(A)})^{-1}$ is well-defined. The role of the transpose matrix appears naturally, and the $G_{\textrm{MP}}$ of Eq. (\[Eqn: Def: Moore-Penrose\]) is the unique matrix that satisfies the conditions
$$\begin{array}{c}
AG_{\textrm{MP}}A=A,\quad G_{\textrm{MP}}AG_{\textrm{MP}}=G_{\textrm{MP}},\\
(G_{\textrm{MP}}A)^{\textrm{T}}=G_{\textrm{MP}}A,\quad(AG_{\textrm{MP}})^{\textrm{T}}=AG_{\textrm{MP}}\end{array}\label{Eqn: MPInverse}$$
that follow immediately from the definition (\[Eqn: Def: Moore-Penrose\]); hence $G_{\textrm{MP}}A$ and $AG_{\textrm{MP}}$ are orthogonal projections[^11] onto the subspaces $\mathcal{R}(A^{\textrm{T}})=\mathcal{R}(G_{\textrm{MP}})$ and $\mathcal{R}(A)$ respectively. Recall that the range space $\mathcal{R}(A^{\textrm{T}})$ of $A^{\textrm{T}}$ is the same as the *row space* $\textrm{row}(A)$ of $A$, and $\mathcal{R}(A)$ is also known as the *column space* of $A$, $\textrm{col}(A)$.
**Example 2.3.** For $a\!:\mathbb{R}^{5}\rightarrow\mathbb{R}^{4}$, let $$A=\left(\begin{array}{rrrrr}
1 & -3 & 2 & 1 & 2\\
3 & -9 & 10 & 2 & 9\\
2 & -6 & 4 & 2 & 4\\
2 & -6 & 8 & 1 & 7\end{array}\right)$$
By reducing the augmented matrix $\left(A|y\right)$ to the row-reduced echelon form, it can be verified that the null and range spaces of $A$ are $3$- and $2$-dimensional respectively. A basis for the null space of $A^{\textrm{T}}$ and of the row and column space of $A$ obtained from the echelon form are respectively $$\left(\begin{array}{r}
-2\\
0\\
1\\
0\end{array}\right),\textrm{ }\left(\begin{array}{r}
1\\
-1\\
0\\
1\end{array}\right);\quad\textrm{and }\left(\begin{array}{r}
1\\
-3\\
0\\
3/2\\
1/2\end{array}\right),\textrm{ }\left(\begin{array}{r}
0\\
0\\
1\\
-1/4\\
3/4\end{array}\right);\textrm{ }\left(\begin{array}{r}
1\\
0\\
2\\
-1\end{array}\right),\textrm{ }\left(\begin{array}{r}
0\\
1\\
0\\
1\end{array}\right).$$
According to its definition Eq. (\[Eqn: Def: Moore-Penrose\]), the Moore-Penrose inverse maps the middle two of the above set to $(0,0,0,0,0)^{\textrm{T}}$, and the $A$-image of the first two (which are respectively $(19,70,38,51)^{\textrm{T}}$ and $(70,275,140,205)^{\textrm{T}}$ lying, as they must, in the span of the last two), to the span of $(1,-3,2,1,2)^{\textrm{T}}$ and $(3,-9,10,2,9)^{\textrm{T}}$ because $a$ restricted to this subspace of $\mathbb{R}^{5}$ is bijective. Hence $$G_{\textrm{MP}}\left(A\left(\begin{array}{r}
1\\
-3\\
0\\
3/2\\
1/2\end{array}\right)\textrm{ }A\left(\begin{array}{r}
0\\
0\\
1\\
-1/4\\
3/4\end{array}\right)\begin{array}{rr}
-2 & 1\\
0 & -1\\
1 & 0\\
0 & 1\end{array}\right)=\left(\begin{array}{rrrr}
1 & 0 & 0 & 0\\
-3 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
3/2 & -1/4 & 0 & 0\\
1/2 & 3/4 & 0 & 0\end{array}\right).$$
The second matrix on the left is invertible as its rank is $4$. This gives $${\displaystyle G_{\textrm{MP}}=\left(\begin{array}{rrrr}
9/275 & -1/275 & 18/275 & -2/55\\
-27/275 & 3/275 & -54/275 & 6/55\\
-10/143 & 6/143 & -20/143 & 16/143\\
238/3575 & -57/3575 & 476/3575 & -59/715\\
-129/3575 & 106/3575 & -258/3575 & 47/715\end{array}\right)}\label{Eqn: MPEx5}$$
as the Moore-Penrose inverse of $A$ that readily verifies all the four conditions of Eqs. (\[Eqn: MPInverse\]). The basic point here is that, as in the case of a bijective map, $G_{\textrm{MP}}A$ and $AG_{\textrm{MP}}$ are identities on the row and column spaces of $A$ that define its rank. For later use — when we return to this example for a simpler inverse $G$ — given below are the orthonormal bases of the four fundamental subspaces with respect to which $G_{\textrm{MP}}$ is a representation of the generalized inverse of $A$; these calculations were done by MATLAB. The basis for
\(a) the column space of $A$ consists of the first $2$ columns of the eigenvectors of $AA^{\textrm{T}}$: $$\begin{array}{c}
(-1633/2585,-363/892,\textrm{ }3317/6387,\textrm{ }363/892)^{\textrm{T}}\\
(-929/1435,\textrm{ }709/1319,\textrm{ }346/6299,-709/1319)^{\textrm{T}}\end{array}$$
\(b) the null space of $A^{\textrm{T}}$ consists of the last $2$ columns of the eigenvectors of $AA^{\textrm{T}}$:$$\begin{array}{c}
(-3185/8306,\textrm{ }293/2493,-3185/4153,\textrm{ }1777/3547)^{\textrm{T}}\\
(323/1732,\textrm{ }533/731,\textrm{ }323/866,\textrm{ }1037/1911)^{\textrm{T}}\end{array}$$
\(c) the row space of $A$ consists of the first $2$ columns of the eigenvectors of $A^{\textrm{T}}A$: $$\begin{array}{c}
(421/13823,\textrm{ }44/14895,-569/918,-659/2526,\textrm{ }1036/1401)\\
(661/690,\textrm{ }412/1775,\textrm{ }59/2960,-1523/10221,-303/3974)\end{array}$$
\(d) the null space of $A$ consists of the last $3$ columns of the of $A^{\textrm{T}}A$:$$\begin{array}{c}
(-571/15469,-369/776,\textrm{ }149/25344,-291/350,-389/1365)\\
(-281/1313,\textrm{ }956/1489,\textrm{ }875/1706,-1279/2847,\textrm{ }409/1473)\\
(292/1579,-876/1579,\textrm{ }203/342,\textrm{ }621/4814,\textrm{ }1157/2152)\end{array}$$
The matrices $Q_{1}$ and $Q_{2}$ with these eigenvectors $(x_{i})$ satisfying $\Vert x_{i}\Vert=1$ and $(x_{i},x_{j})=0$ for $i\neq j$ as their columns are *orthogonal matrices* with the simple inverse criterion $Q^{-1}=Q^{\textrm{T}}$.$\qquad\blacksquare$
***End Tutorial3***
The basic issue in the solution of the inverse ill-posed problem is its reduction to an well-posed one when restricted to suitable subspaces of the domain and range of $A$. Considerations of geometry leading to their decomposition into orthogonal subspaces is only an additional feature that is not central to the problem: recall from Eq. (\[Eqn: f\_inv\_f\]) that any function $f$ must necessarily satisfy the more general set-theoretic relations $ff^{-}f=f$ and $f^{-}ff^{-}=f^{-}$ of Eq. (\[Eqn: MPInverse\]) for the multiinverse $f^{-}$ of $f\!:X\rightarrow Y$. The second distinguishing feature of the MP-inverse is that it is defined, by a suitable extension, on all of $Y$ and not just on $f(X)$ which is perhaps more natural. The availability of orthogonality in inner-product spaces allows this extension to be made in an almost normal fashion. As we shall see below the additional geometric restriction of Eq. (\[Eqn: MPInverse\]) is not essential to the solution process, and infact, only results in a less canonical form of the inverse.
***Begin Tutorial4: Topological Spaces***
This Tutorial is meant to familiarize the reader with the basic principles of a topological space. A topological space $(X,\mathcal{U})$ is a set $X$ with a class[^12] $\mathcal{U}$ of distinguished subsets, called *open sets of $X$,* that satisfy
(T1) The empty set $\emptyset$ and the whole $X$ belong to $\mathcal{U}$
(T2) Finite intersections of members of $\mathcal{U}$ belong to $\mathcal{U}$
(T3) Arbitrary unions of members of $\mathcal{U}$ belong to $\mathcal{U}$.
**Example 2.4.** (1) The smallest topology possible on a set $X$ is its *indiscrete topology* when the only open sets are $\emptyset$ and $X$; the largest is the *discrete topology* where every subset of $X$ is open (and hence also closed).
\(2) In a metric space $(X,d)$, let $B_{\varepsilon}(x,d)=\{ y\in X\!:d(x,y)<\varepsilon\}$ be an open ball at $x$. Any subset $U$ of $X$ such that for each $x\in U$ there is a $d$-ball $B_{\varepsilon}(x,d)\subseteq U$ in $U$, is said to be an open set of $(X,d)$. The collection of all these sets is the topology induced by $d$. The topological space $(X,\mathcal{U})$ is then said to be *associated with (induced by)* $(X,d)$.
\(3) If $\sim$ is an equivalence relation on a set $X$, the set of all saturated sets $[x]_{\sim}=\{ y\in X\!:y\sim x\}$ is a topology on $X;$ this topology is called the *topology of saturated sets.*
We argue in Sec. 4.2 that this constitutes the defining topology of a chaotic system.
\(4) For any subset $A$ of the set $X$, the $A$-*inclusion topology on $X$* consists of $\emptyset$ and every superset of $A$, while the $A$-*exclusion topology on* $X$ consists of all subsets of $X-A$. Thus $A$ is open in the inclusion topology and closed in the exclusion, and in general every open set of one is closed in the other.
The special cases of the *$a$-inclusion* and *$a$-exclusion* topologies for $A=\{ a\}$ are defined in a similar fashion.
\(5) The *cofinite* and *cocountable topologies* in which the open sets of an infinite (resp. uncountable) set $X$ are respectively the complements of finite and countable subsets, are examples of topologies with some unusual properties that are covered in Appendix A1. If $X$ is itself finite (respectively, countable), then its cofinite (respectively, cocountable) topology is the discrete topology consisting of all its subsets. It is therefore useful to adopt the convention, unless stated to the contrary, that cofinite and cocountable spaces are respectively infinite and uncountable.$\qquad\blacksquare$
In the space $(X,\mathcal{U})$, a *neighbourhood of a point* $x\in X$ is a nonempty subset $N$ of $X$ that contains an open set $U$ containing $x$; thus $N\subseteq X$ is a neighbourhood of $x$ iff $$x\in U\subseteq N\label{Eqn: Def: nbd1}$$ for some $U\in\mathcal{U}$. The largest open set that can be used here is $\textrm{Int}(N)$ (where, by definition, $\textrm{Int}(A)$ is the largest open set that is contained in $A$) so that the above neighbourhood criterion for a subset $N$ of $X$ can be expressed in the equivalent form $$N\subseteq X\textrm{ is a }\mathcal{U}-\textrm{neighbourhood of }x\textrm{ iff }x\in\textrm{Int}_{\mathcal{U}}(N)\label{Eqn: Def: nbd2}$$ implying that a subset of $(X,\mathcal{U})$ is a neighbourhood of all its interior points, so that $N\in\mathcal{N}_{x}\Rightarrow N\in\mathcal{N}_{y}$ for all $y\in\textrm{Int}(N)$. The collection of all neighbourhoods of $x$ $$\mathcal{N}_{x}\overset{\textrm{def}}=\{ N\subseteq X\!:x\in U\subseteq N\textrm{ for some }U\in\mathcal{U}\}\label{Eqn: Def: nbd system}$$ **is the *neighbourhood system* at $x$, and the subcollection $U$ of the topology used in this equation constitutes a *neighbourhood* (*local*) *base* or *basic neighbourhood system, at* $x$, see Def. A1.1 of Appendix A1. The properties
(N1) $x$ belongs to every member $N$ of *$\mathcal{N}_{x}$,*
(N2) The intersection of any two neighbourhoods of *$x$* is another neighbourhood of $x$: $N,M\in\mathcal{N}_{x}\Rightarrow N\bigcap M\in\mathcal{N}_{x}$,
(N3) Every superset of **any neighbourhood of $x$ is a neighbourhood of $x$: $(M\in\mathcal{N}_{x})\wedge(M\subseteq N)\Rightarrow N\in\mathcal{N}_{x}$.
that characterize *$\mathcal{N}_{x}$* completely are a direct consequence of the definition (\[Eqn: Def: nbd1\]), (\[Eqn: Def: nbd2\]) that may also be stated as
(N0) Any neighbourhood $N\in\mathcal{N}_{x}$ contains another neighbourhood $U$ of $x$ that is a *neighbourhood of each of its point*s: $((\forall N\in\mathcal{N}_{x})(\exists U\in\mathcal{N}_{x})(U\subseteq N))\!:(\forall y\in U\Rightarrow U\in\mathcal{N}_{y})$.
Property (N0) infact serves as the defining characteristic of an open set, and *$U$* can be identified with the largest open set $\textrm{Int}(N)$ contained in $N$; hence *a set $G$ in a topological space is open iff it is a neighbourhood of each of its points.* Accordingly if *$\mathcal{N}_{x}$* is a given class of subsets of $X$ associated with each $x\in X$ satisfying $(\textrm{N}1)-(\textrm{N}3)$, then (N0) defines the special class of neighbourhoods $G$ $$\mathcal{U}=\{ G\in\mathcal{N}_{x}\!:x\in B\subseteq G\textrm{ for all }x\in G\textrm{ and a basic nbd }B\in\mathcal{N}_{x}\}\label{Eqn: nbd-topology}$$ as the unique topology on $X$ that contains a basic neighbourhood of each of its points, for which the neighbourhood system at $x$ **coincides exactly with the assigned collection** *$\mathcal{N}_{x}$*; compare Def A1.1.** Neighbourhoods in topological spaces are a generalization of the familiar notion of distances of metric spaces that quantifies “closeness” of points of $X$.
A *neighbourhood of a nonempty subset* $A$ of $X$ that will be needed later on is defined in a similar manner: $N$ is a neighbourhood of $A$ iff $A\subseteq\textrm{Int}(N)$, that is $A\subseteq U\subseteq N$; thus the neighbourhood system at $A$ is given by $\mathcal{N}_{A}=\bigcap_{a\in A}\mathcal{N}_{a}:=\{ G\subseteq X\!:G\in\mathcal{N}_{a}\textrm{ for every }a\in A\}$ is the class of common neighbourhoods of each point of $A$.
Some examples of neighbourhood systems at a point $x$ in $X$ are the following:
\(1) In an indiscrete space $(X,\mathcal{U})$, $X$ is the only neighbourhood of every point of the space; in a discrete space any set containing $x$ is a neighbourhood of the point.
\(2) In an infinite cofinite (or uncountable cocountable) space, every neighbourhood of a point is an open neighbourhood of that point.
\(3) In the topology of saturated sets under the equivalence relation $\sim$, the neighbourhood system at $x$ consists of all supersets of the equivalence class $[x]_{\sim}$.
\(4) Let $x\in X$. In the $x$-inclusion topology, $\mathcal{N}_{x}$ consists of all the non-empty open sets of $X$ which are the supersets of $\{ x\}$. For a point $y\neq x$ of $X$, $\mathcal{N}_{y}$ are the supersets of $\{ x,y\}$.
For any given class $_{\textrm{T}}\mathcal{S}$ of subsets of $X$, a unique topology $\mathcal{U}(_{\textrm{T}}\mathcal{S})$ can always be constructed on $X$ by taking all *finite* *intersections* $_{\textrm{T}}\mathcal{S}_{\wedge}$ of members of $\mathcal{S}$ followed by *arbitrary* *unions* $_{\textrm{T}}\mathcal{S}_{\wedge\vee}$ of these finite intersections. $\mathcal{U}(_{\textrm{T}}\mathcal{S}):=\,_{\textrm{T}}\mathcal{S}_{\wedge\vee}$ is the smallest topology on $X$ that contains $_{\textrm{T}}\mathcal{S}$ and is said to be *generated by* $_{\textrm{T}}\mathcal{S}$. For a given topology $\mathcal{U}$ on $X$ satisfying $\mathcal{U}=\mathcal{U}(_{\textrm{T}}\mathcal{S})$, $_{\textrm{T}}\mathcal{S}$ is a *subbasis,* and $_{\textrm{T}}\mathcal{S}_{\wedge}:=\,_{\textrm{T}}\mathcal{B}$ a *basis, for the topology* $\mathcal{U}$; for more on topological basis, see Appendix A1. The topology generated by a subbase essentially builds not from the collection $_{\textrm{T}}\mathcal{S}$ itself but from the finite intersections $_{\textrm{T}}\mathcal{S}_{\wedge}$ of its subsets; in comparison the base generates a topology directly from a collection $_{\textrm{T}}\mathcal{S}$ of subsets by forming their unions. Thus whereas *any* class of subsets can be used as a subbasis, a given collection must meet certain qualifications to pass the test of a base for a topology: these and related topics are covered in Appendix A1. Different subbases, therefore, can be used to generate different topologies on the same set $X$ as the following examples for the case of $X=\mathbb{R}$ demonstrates; here $(a,b)$, $[a,b)$, $(a,b]$ and $[a,b]$, for $a\leq b\in\mathbb{R}$, are the usual open-closed intervals in $\mathbb{R}$[^13]. The subbases $_{\textrm{T}}\mathcal{S}_{1}=\{(a,\infty),(-\infty,b)\}$, $_{\textrm{T}}\mathcal{S}_{2}=\{[a,\infty),(-\infty,b)\}$, $_{\textrm{T}}\mathcal{S}_{3}=\{(a,\infty),(-\infty,b]\}$ and $_{\textrm{T}}\mathcal{S}_{4}=\{[a,\infty),(-\infty,b]\}$ give the respective bases $_{\textrm{T}}\mathcal{B}_{1}=\{(a,b)\}$, $_{\textrm{T}}\mathcal{B}_{2}=\{[a,b)\}$, $_{\textrm{T}}\mathcal{B}_{3}=\{(a,b]\}$ and $_{\textrm{T}}\mathcal{B}_{4}=\{[a,b]\}$, $a\leq b\in\mathbb{R}$, leading to the *standard (usual*)*, lower limit (Sorgenfrey*)*, upper limit,* and *discrete* (take $a=b$) topologies on $\mathbb{R}$. Bases of the type $(a,\infty)$ and $(-\infty,b)$ provide the *right* and *left ray* topologies on $\mathbb{R}$.
*This feasibility of generating different topologies on a set can be of great practical significance because open sets determine convergence characteristics of nets and continuity characteristics of functions, thereby making it possible for nature to play around with the structure of its working space in its kitchen to its best possible advantage.*[^14] **
Here are a few essential concepts and terminology for topological spaces.
**Definition 2.2.** ***Boundary, Closure, Interior*.** *The* *boundary of $A$ in $X$* *is the set of points $x\in X$ such that every neighbourhood $N$ of $x$ intersects both $A$ and $X-A$:* $${\textstyle \textrm{Bdy}(A)\overset{\textrm{def}}=\{ x\in X\!:(\forall N\in\mathcal{N}_{x})((N\bigcap A\neq\emptyset)\wedge(N\bigcap(X-A)\neq\emptyset))\}}\label{Eqn: Def: Boundary}$$ *where $\mathcal{N}_{x}$ is the neighbourhood system of Eq. (\[Eqn: Def: nbd system\]) at $x$.*
*The* *closure of $A$* *is the set of all points $x\in X$ such that each neighbourhood of $x$ contains at least one point of $A$* ***that may be $\boldmath{x}$ itself****. Thus the set* $${\textstyle \textrm{Cl}(A)\overset{\textrm{def}}=\{ x\in X\!:(\forall N\in\mathcal{N}_{x})\textrm{ }(N\bigcap A\neq\emptyset)\}}\label{Eqn: Def: Closure}$$ *of all points in $X$ adherent* *to* *$A$ is given by* *is the union* ***of $A$ with its boundary.*
*The* *interior of $A$* $$\textrm{Int}(A)\overset{\textrm{def}}=\{ x\in X\!:(\exists N\in\mathcal{N}_{x})\textrm{ }(N\subseteq A)\}\label{Eqn: Def: Interior}$$ *consisting of those points of $X$ that are in $A$ but not in its boundary,* $\textrm{Int}(A)=A-\textrm{Bdy}(A)$*, is the largest open subset of $X$ that is contained in $A$. Hence it follows that* $\textrm{Int}(\textrm{Bdy}(A))=\emptyset$, *the boundary of $A$ is the intersection of the closures of $A$ and $X-A$,* *and a subset $N$ of $X$ is a neighbourhood of $x$ iff* $x\in\textrm{Int}(N)$*.$\qquad\square$*
The three subsets $\textrm{Int}(A)$, $\textrm{Bdy}(A)$ and *exterior* of $A$ defined as $\textrm{Ext}(A):=\textrm{Int}(X-A)=X-\textrm{Cl}(A)$, are pairwise disjoint and have the full space $X$ as their union.
**Definition 2.3.** ***Derived and Isolated sets.*** *Let $A$ be a subset of $X$. A point $x\in X$ (which may or may not be a point of $A$) is a* *cluster point of* $A$ *if every neighbourhood $N\in\mathcal{N}_{x}$ contains atleast one point of $A$* ***different from*** *$\mathbf{x}$. The* *derived set of $A$* $${\textstyle \textrm{Der}(A)\overset{\textrm{def}}=\{ x\in X\!:(\forall N\in\mathcal{N}_{x})\textrm{ }(N\bigcap(A-\{ x\})\neq\emptyset)\}}\label{Eqn: Def: Derived}$$ *is the set of all cluster points of $A$. The complement of* $\textrm{Der}(A)$ in $A$ $$\textrm{Iso}(A)\overset{\textrm{def}}=A-\textrm{Der}(A)=\textrm{Cl}(A)-\textrm{Der}(A)\label{Eqn: Def: Isolated}$$ *are the* *isolated* *points* *of* $A$ *to which no proper sequence in $A$ converges, that is there exists a neighbourhood of any such point that contains no other point of $A$ so that* *the only sequence that converges to* $a\in\textrm{Iso}(A)$ *is the constant sequence $(a,a,a,\cdots)$.$\qquad\square$*
Clearly,
$$\begin{array}{ccl}
{\textstyle \textrm{Cl}(A)} & = & A\bigcup\textrm{Der}(A)=A\bigcup\textrm{Bdy}(A)\\
& = & \textrm{Iso}(A)\bigcup\textrm{Der}(A)=\textrm{Int}(A)\bigcup\textrm{Bdy}(A)\end{array}$$
with the last two being disjoint unions, and $A$ is closed iff $A$ contains all its cluster points, $\textrm{Der}(A)\subseteq A$, iff $A$ contains its closure. Hence $$\begin{gathered}
A=\textrm{Cl}(A)\Longleftrightarrow\textrm{Cl}(A)=\{ x\in A\!:((\exists N\in\mathcal{N}_{x})(N\subseteq A))\vee((\forall N\in\mathcal{N}_{x})(N\bigcap(X-A)\neq\emptyset))\}\end{gathered}$$ Comparison of Eqs. (\[Eqn: Def: Boundary\]) and (\[Eqn: Def: Derived\]) also makes it clear that $\textrm{Bdy}(A)\subseteq\textrm{Der}(A)$. The special case of $A=\textrm{Iso}(A)$ with $\textrm{Der}(A)\subseteq X-A$ is important enough to deserve a special mention:
**Definition 2.4.** ***Donor set.*** *A proper, nonempty subset $A$ of $X$ such that* $\textrm{Iso}(A)=A$ *with* $\textrm{Der}(A)\subseteq X-A$ *will be called* *self-isolated* *or* *donor.* *Thus sequences eventually in a donor set converges only in its complement; this is the opposite of the characteristic of a closed set where all converging sequences eventually in the set must necessarily converge in it. A closed-donor set with a closed neighbour has no derived or boundary sets, and will be said to be* *isolated in $X$.*$\qquad\square$
**Example 2.5.** In an isolated set sequences converge, if they have to, simultaneously in the complement (because it is donor) and in it (because it is closed). Convergent sequences in such a set can only be constant sequences. Physically, if we consider adherents to be contributions made by the dynamics of the corresponding sequences, then an isolated set is secluded from its neighbour in the sense that it neither receives any contributions from its surroundings, nor does it give away any. In this light and terminology, a closed set is a *selfish* set (recall that a set $A$ is closed in $X$ iff every convergent net of $X$ that is eventually in $A$ converges in $A$; conversely a set is open in $X$ iff the only nets that converge in $A$ are eventually in it), **whereas a set with a derived set that intersects itself and its complement may be considered to be *neutral.* Appendix A3 shows the various possibilities for the derived set and boundary of a subset $A$ of $X$.$\qquad\blacksquare$
Some useful properties of these concepts for a subset $A$ of a topological space $X$ are the following.
\(a) $\textrm{Bdy}_{X}(X)=\emptyset$,
\(b) $\textrm{Bdy}(A)=\textrm{Cl}(A)\bigcap\textrm{Cl}(X-A)$,
\(c) $\textrm{Int}(A)=X-\textrm{Cl}(X-A)=A-\textrm{Bdy}(A)=\textrm{Cl}(A)-\textrm{Bdy}(A)$,
\(d) $\textrm{Int}(A)\bigcap\textrm{Bdy}(A)=\emptyset$,
\(e) $X=\textrm{Int}(A)\bigcup\textrm{Bdy}(A)\bigcup\textrm{Int}(X-A)$,
\(f) $${\textstyle \textrm{Int}(A)=\bigcup\{ G\subseteq X\!:G\textrm{ is an open set of }X\textrm{ contained in }A\}}\label{Eqn: interior}$$
\(g) $${\textstyle \textrm{Cl}(A)=\bigcap\{ F\subseteq X\!:F\textrm{ is a closed set of }X\textrm{ containing }A\}}\label{Eqn: closure}$$
A straightforward consequence of property (b) is that the boundary of any subset $A$ of a topological space $X$ is closed in $X$; this significant result may also be demonstrated as follows. If $x\in X$ is not in the boundary of $A$ there is some neighbourhood $N$ of $x$ that does not intersect both $A$ and $X-A$. For each point $y\in N$, $N$ is a neighbourhood of that point that does not meet $A$ and $X-A$ simultaneously so that $N$ is contained wholly in $X-\textrm{Bdy}(A)$. We may now take $N$ to be open without any loss of generality implying thereby that $X-\textrm{Bdy}(A)$ is an open set of $X$ from which it follows that $\textrm{Bdy}(A)$ is closed in $X$.
Further material on topological spaces relevant to our work can be found in Appendix A3.
***End Tutorial4***
Working in a general topological space, we now recall the solution of an ill-posed problem $f(x)=y$ [@Sengupta1997] that leads to a multifunctional inverse $f^{-}$ through the generalized inverse $G$. Let $f\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ be a (nonlinear) function between two topological space $(X,\mathcal{U})$ and $(Y,\mathcal{V})$ that is neither one-one or onto. Since $f$ is not one-one, $X$ can be partitioned into disjoint equivalence classes with respect to the equivalence relation $x_{1}\sim x_{2}\Leftrightarrow f(x_{1})=f(x_{2})$. Picking a representative member from each of the classes (this is possible by the Axiom of Choice; see the following Tutorial) produces a *basic set* $X_{\textrm{B}}$ of $X$; it is basic as it corresponds to the row space in the linear matrix example which is all that is needed for taking an inverse. $X_{\textrm{B}}$ is the counterpart of the quotient set $X/\sim$ of Sec. 1, with the important difference that whereas the points of the quotient set are the equivalence classes of $X$, $X_{\textrm{B}}$ *is a subset of* $X$ with each of the classes contributing a point to $X_{\textrm{B}}$. It then follows that $f_{\textrm{B}}\!:X_{\textrm{B}}\rightarrow f(X)$ is the bijective restriction $a|_{\textrm{row}(A)}$ that reduces the original ill-posed problem to a well-posed one with $X_{\textrm{B}}$ and $f(X)$ corresponding respectively to the row and column spaces of $A$, and $f_{\textrm{B}}^{-1}\!:f(X)\rightarrow X_{\textrm{B}}$ is the basic inverse from which the multiinverse $f^{-}$ is obtained through $G$, which in turn corresponds to the Moore-Penrose inverse $G_{\textrm{MP}}$. The topological considerations (obviously not for inner product spaces that applies to the Moore-Penrose inverse) needed to complete the solution are discussed below and in Appendix A1.
***Begin Tutorial5: Axiom of Choice and Zorn’s Lemma***
Since some of our basic arguments depend on it, this Tutorial contains a short description of the Axiom of Choice that has been described as “one of the most important, and at the same time one of the most controversial, principles of mathematics”. What this axiom states is this: For any set $X$ there exists a function $f_{\textrm{C}}\!:\mathcal{P}_{0}(X)\rightarrow X$ such that $f_{\textrm{C}}(A_{\alpha})\in A_{\alpha}$ for every non-empty subset $A_{\alpha}$ of $X$; here $\mathcal{P}_{0}(X)$ is the class of all subsets of $X$ except $\emptyset$. Thus, if $X=\{ x_{1},x_{2},x_{3}\}$ is a three element set, a possible choice function is given by
$$\begin{array}{c}
f_{\textrm{C}}(\{ x_{1},x_{2},x_{3}\})=x_{3},\quad f_{\textrm{C}}(\{ x_{1},x_{2}\})=x_{1},\quad f_{\textrm{C}}(\{ x_{2},x_{3}\})=x_{3},\quad f_{\textrm{C}}(\{ x_{3},x_{1}\})=x_{3},\\
f_{\textrm{C}}(\{ x_{1}\})=x_{1},\quad f_{\textrm{C}}(\{ x_{2}\})=x_{2},\quad f_{\textrm{C}}(\{ x_{3}\})=x_{3}.\end{array}$$
It must be appreciated that the axiom is only an existence result that asserts *every set* to have a choice function, even when nobody knows how to construct one in a specific case. Thus, for example, how does one pick out the isolated irrationals $\sqrt{2}$ or $\pi$ from the uncountable reals? There is no doubt that they do exist, for we can construct a right angled triangle with sides of length $1$ or a circle of radius $1$. The axiom tells us that these choices are possible even though we do not know how exactly to do it; all that can be stated with confidence is that we can actually pick up rationals arbitrarily close to these irrationals.
The axiom of choice is essentially meaningful when $X$ is infinite as illustrated in the last two examples. This is so because even when $X$ is denumerable, it would be physically impossible to make an infinite number of selections either all at a time or sequentially: the Axiom of Choice nevertheless tells us that this is possible. The real strength and utility of the Axiom however is when $X$ and some or all of its subsets are uncountable as in the case of the choice of the *single element* $\pi$ from the reals. To see this more closely in the context of maps that we are concerned with, let $f\!:X\rightarrow Y$ be a non-injective, onto map. To construct a functional right inverse $f_{r}\!:Y\rightarrow X$ of $f$, we must choose, for each $y\in Y$ one *representative* element $x_{\textrm{rep}}$ from the set $f^{-}(y)$ and define $f_{r}(y)$ to be that element according to $f\circ f_{r}(y)=f(x_{\textrm{rep}})=y$. If there is no preferred or natural way to make this choice, the axiom of choice allows us to make an arbitrary selection from the infinitely many that may be possible from $f^{-}(y)$. When a natural choice is indeed available, as for example in the case of the initial value problem $y^{\prime}(x)=x;\, y(0)=\alpha_{0}$ on $[0,a]$, the definite solution $\alpha_{0}+x^{2}/2$ may be selected from the infinitely many $\int_{0}^{x}x^{\prime}dx^{\prime}=\alpha+x^{2}/2,\textrm{ }0\leq x\leq a$ that are permissible, and the axiom of choice sanctions this selection. In addition, each $y\in Y$ gives rise to the family of solution sets $A_{y}=\{ f^{-}(y)\!:y\in Y\}$ and the real power of the axiom is its assertion that it is possible to make a choice $f_{\textrm{C}}(A_{y})\in A_{y}$ on every $A_{y}$ simultaneously; this permits the choice **on every $A_{y}$ of the collection to be made at the same time.
***Pause Tutorial5***
Figure \[Fig: GenInv\] shows our [@Sengupta1997] formulation and solution of the inverse ill-posed problem $f(x)=y$. In sub-diagram $X-X_{\textrm{B}}-f(X)$, the surjection $p\!:X\rightarrow X_{\textrm{B}}$ is the counterpart of the quotient map $Q$ of Fig. \[Fig: quotient\] that is known in the present context as the *identification* of $X$ with $X_{\mathrm{B}}$ (as it *identifies* each saturated subset of $X$ with its representative point in $X_{\textrm{B}}$), with the space $(X_{\textrm{B}},\textrm{FT}\{\mathcal{U};p\})$ carrying the *identification topology* $\textrm{FT}\{\mathcal{U};p\}$ being known as an *identification space.* By sub-diagram $Y-X_{\textrm{B}}-f(X)$, the image $f(X)$ of $f$ gets the *subspace topology*[^15] $\textrm{IT}\{ j;\mathcal{V}\}$ from $(Y,\mathcal{V})$ by the inclusion $j\!:f(X)\rightarrow Y$ when its open sets are generated as, and only as, $j^{-1}(V)=V\bigcap f(X)$ for $V\in\mathcal{V}$. Furthermore if the bijection $f_{\textrm{B}}$ connecting $X_{\textrm{B}}$ and $f(X)$ (which therefore acts as a $1:1$ correspondence between their points, implying that these sets are set-theoretically identical except for their names) is image continuous, then by Theorem A2.1 of Appendix 2, so is the *association* $q=f_{\textrm{B}}\circ p\!:X\rightarrow f(X)$ that associates saturated sets of $X$ with elements of $f(X)$; this makes $f(X)$ look like an identification space of $X$ by assigning to it the topology $\textrm{FT}\{\mathcal{U};q\}$. On the other hand if $f_{\textrm{B}}$ happens to be preimage continuous, then $X_{\textrm{B}}$ acquires, by Theorem A2.2, the initial topology $\textrm{IT}\{ e;\mathcal{V}\}$ by the *embedding* $e\!:X_{\textrm{B}}\rightarrow Y$ that embeds $X_{\textrm{B}}$ into $Y$ through $j\circ f_{\textrm{B}}$, making it look like a subspace of $Y$[^16]. In this dual situation, $f_{\textrm{B}}$ has the highly interesting topological property of being simultaneously image and preimage continuous when the open sets of $X_{\textrm{B}}$ and $f(X)$ — which are simply the $f_{\textrm{B}}^{-1}$-images of the open sets of $f(X)$ which, in turn, are the $f_{\textrm{B}}$-images of these saturated open sets — can be considered to have been generated by $f_{\textrm{B}}$, and are respectively the smallest and largest collection of subsets of $X$ and $Y$ that makes $f_{\textrm{B}}$ *ini*(tial-fi)*nal continuous* [@Sengupta1997]*.* A bijective ininal function such as $f_{\textrm{B}}$ is known as a *homeomorphism* and ininality for functions that are neither $1:1$ **nor onto is a generalization of homeomorphism for bijections; refer Eqs. (\[Eqn: INI\]) and (\[Eqn: HOM\]) for a set-theoretic formulation of this distinction. A homeomorphism $f\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ renders the homeomorphic spaces $(X,\mathcal{U})$ and $(Y,\mathcal{V})$ topologically indistinguishable which may be considered to be identical in as far as their topological properties are concerned.
**Remark.** It may be of some interest here to speculate on the significance of *ininality* in our work. Physically, a map $f\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ between two spaces can be taken to represent an interaction between them and the algebraic and topological characters of $f$ determine the nature of this interaction. A simple bijection merely sets up a correspondence, that is an interaction, between every member of $X$ with some member $Y$, whereas a continuous map establishes the correspondence among the special category of “open” sets. Open sets, as we see in Appendix A1, are the basic ingredients in the theory of convergence of sequences, nets and filters, and the characterization of open sets in terms of convergence, namely that *a set $G$ in $X$ is open in it if every net or sequence that converges in $X$ to a point in $G$ is eventually in $G$*, see Appendix A1, may be interpreted to mean that such sets represent groupings of elements that require membership of the group before permitting an element to belong it; an open set unlike its complement the closed or *selfish* set, however, does not forbid a net that has been eventually in it to settle down in its selfish neighbour, who nonetheless will never allow such a situation to develop in its own territory. An ininal map forces these well-defined and definite groups in $(X,\mathcal{U})$ and $(Y,\mathcal{V})$ to interact with each other through $f$; this is not possible with simple continuity as there may be open sets in $X$ that are not derived from those of $Y$ and non-open sets in $Y$ whose inverse images are open in $X$. *It is our hypothesis that the driving force behind the evolution of a system represented by the input-output relation $f(x)=y$ is the attainment of the ininal triple state $(X,f,Y)$ for the system.* A preliminary analysis of this hypothesis is to be found in Sec. 4.2.
For ininality of the interaction, it is therefore necessary to have$$\begin{aligned}
\textrm{FT}\{\mathcal{U};f_{<}\} & = & \textrm{IT}\{ j;\mathcal{V}\}\label{Eqn: ininal}\\
\textrm{IT}\{\,_{<}f;\mathcal{V}\} & = & \textrm{FT}\{\mathcal{U};p\}\};\nonumber \end{aligned}$$
in what follows we will refer to the injective and surjective restrictions of $f$ by their generic topological symbols of embedding $e$ and association $q$ respectively. What are the topological characteristics of $f$ in order that the requirements of Eq. (\[Eqn: ininal\]) be met? From Appendix A1, it should be clear by superposing the two parts of Fig. \[Fig: Initial-Final\] over each other that given $q\!:(X,\mathcal{U})\rightarrow(f(X),\textrm{FT}\{\mathcal{U};q\})$ in the first of these equations, $\textrm{IT}\{ j;\mathcal{V}\}$ will equal $\textrm{FT}\{\mathcal{U};q\}$ iff $j$ is an ininal open inclusion and $Y$ receives $\textrm{FT}\{\mathcal{U};f\}$. In a similar manner, preimage continuity of $e$ requires $p$ to be open ininal and $f$ to be preimage continuous if the second of Eq. (\[Eqn: ininal\]) is to be satisfied. Thus under the restrictions imposed by Eq. (\[Eqn: ininal\]), the interaction $f$ between $X$ and $Y$ must be such as to give $X$ the smallest possible topology of $f$-saturated sets and $Y$ the largest possible topology of images of all these sets: $f$, under these conditions, is an ininal transformation. Observe that a direct application of parts (b) of Theorems A2.1 and A2.2 to Fig. \[Fig: GenInv\] implies that Eq. (\[Eqn: ininal\]) is satisfied iff $f_{\textrm{B}}$ is ininal, that is iff it is a homeomorphism. Ininality of $f$ is simply a reflection of this as it is neither $1:1$ nor onto.
The $f$- and $p$-images of each saturated set of $X$ are singletons in $Y$ (these saturated sets in $X$ arose, in the first place, as $f^{-}(\{ y\})$ for $y\in Y$) and in $X_{\textrm{B}}$ respectively. This permits the embedding $e=j\circ f_{\textrm{B}}$ to give $X_{\textrm{B}}$ the character of a virtual subspace of $Y$ just as $i$ makes $f(X)$ a real subspace. Hence the inverse images $p^{-}(x_{r})=f^{-}(e(x_{r}))$ with $x_{r}\in X_{\textrm{B}}$, and $q^{-}(y)=f^{-}(i(y))$ with $y=f_{\textrm{B}}(x_{r})\in f(X)$ are the same, and are just the corresponding $f^{-}$ images via the injections $e$ and $i$ respectively. $G$, a left inverse of $e$, is a generalized inverse of $f$. $G$ is a generalized inverse because the two set-theoretic defining requirements of $fGf=f$ and $GfG=G$ for the generalized inverse are satisfied, as Fig. \[Fig: GenInv\] shows, in the following forms $$jf_{\textrm{B}}Gf=f\qquad Gjf_{\textrm{B}}G=G.$$ In fact the commutativity embodied in these equalities is self evident from the fact that $e=if_{\textrm{B}}$ is a left inverse of $G$, that is $eG=\bold1_{Y}$. On putting back $X_{\textrm{B}}$ into $X$ by identifying each point of $X_{\textrm{B }}$ with the set it came from yields the required set-valued inverse $f^{-}$, and $G$ may be viewed as a functional selection of the multiinverse $f^{-}$.
An *injective branch* of a function $f$ in this work refers to the restrictions $f_{\mathrm{B}}$ and its associated inverse $f_{\mathrm{B}}^{-1}$.
The following example of an inverse ill-posed problem will be useful in fixing the notations introduced above. Let $f$ on $[0,1]$ be the function of \[Fig: gen-inv\].
Then $f(x)=y$ is well-posed for $[0,1/4)$, and ill-posed in [\[]{}1/4,1[\]]{}. There are two injective branches of $f$ in $\{[1/4,3/8)$$\bigcup$ $(5/8,1]\}$, and $f$ is constant ill-posed in $[3/8,5/8]$. Hence the basic component $f_{\textrm{B}}$ of $f$ can be taken to be $f_{\textrm{B}}(x)=2x$ for $x\in[0,3/8)$ having the inverse $f_{\textrm{B}}^{-1}(y)=x/2$ with $y\in[0,3/4]$. The generalized inverse is obtained by taking $[0,3/4]$ as a subspace of $[0,1]$, while the multiinverse $f^{-}$ follows by associating with every point of the basic domain $[0,1]_{\textrm{B}}=[0,3/8]$, the respective equivalent points $[3/8]_{f}=[3/8,5/8]$ and $[x]_{f}=\{ x,7/4-3x\}\textrm{ for }x\in[1/4,3/8)$. Thus the inverses $G$ and $f^{-}$ of $f$ are[^17]
$$G(y)=\left\{ \begin{array}{ccl}
y/2, & & y\in[0,3/4]\\
0, & & y\in(3/4,1]\end{array}\right.,\quad f^{-}(y)=\left\{ \begin{array}{ccl}
y/2, & & y\in[0,1/2)\\
\{ y/2,7/4-3y/2\}, & & y\in[1/2,3/4)\\
{}[3/8,5/8], & & y=3/4\\
0, & & y\in(3/4,1],\end{array}\right.$$
which shows that $f^{-}$ is multivalued. In order to avoid cumbersome notations, an injective branch of $f$ will always refer to a representative basic branch $f_{\textrm{B}}$, and its “inverse” will mean either $f_{\textrm{B}}^{-1}$ or $G$.
**Example 2.3, Revisited.** The row reduced echelon form of the augmented matrix $(A|b)$ of Example 2.3 is
$${\displaystyle (A|b)\longrightarrow\left(\begin{array}{rrrrrcl}
1 & -3 & 0 & 3/2 & 1/2 & & 5b_{1}/2-b_{2}/2\\
0 & 0 & 1 & -1/4 & 3/4 & & -3b_{1}/4+b_{2}/4\\
0 & 0 & 0 & 0 & 0 & & -2b_{1}+b_{3}\\
0 & 0 & 0 & 0 & 0 & & b_{1}-b_{2}+b_{4}\end{array}\right)}\label{Eqn: RowReduce}$$
The multifunctional solution $x=A^{-}b$, with $b$ any element of $Y=\mathbb{R}^{4}$ not necessarily in the of image of $a$, is$$x=A^{-}b=Gb+x_{2}\left(\begin{array}{c}
3\\
1\\
0\\
0\\
0\end{array}\right)+x_{4}\left(\begin{array}{r}
-3/2\\
0\\
1/4\\
1\\
0\end{array}\right)+x_{5}\left(\begin{array}{r}
-1/2\\
0\\
-3/4\\
0\\
1\end{array}\right),$$
with its multifunctional character arising from the arbitrariness of the coefficients $x_{2}$, $x_{4},$ and $x_{5}$. The generalized inverse $$G=\left(\begin{array}{rrrr}
5/2 & -1/2 & 0 & 0\\
0 & 0 & 0 & 0\\
-3/4 & 1/4 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\end{array}\right)\!:Y\rightarrow X_{\textrm{B}}\label{Eqn: GenInvEx5}$$ is the unique matrix representation of the functional inverse $a_{\textrm{B}}^{-1}\!:a(\mathbb{R}^{5})\rightarrow X_{\textrm{B}}$ extended to $Y$ defined according to[^18] $$g(b)=\left\{ \begin{array}{ccl}
a_{\textrm{B}}^{-1}(b), & & \textrm{ if }b\in\mathcal{R}(a)\\
0, & & \textrm{ if }b\in Y-\mathcal{R}(a),\end{array}\right.\label{Eqn: Def: GenInv}$$ that bears comparison with the basic inverse $$A_{\textrm{B}}^{-1}(b^{*})=\left(\begin{array}{rrrr}
5/2 & -1/2 & 0 & 0\\
0 & 0 & 0 & 0\\
-3/4 & 1/4 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\end{array}\right)\left(\begin{array}{c}
b_{1}\\
b_{2}\\
2b_{1}\\
b_{2}-b_{1}\end{array}\right)\!:a(\mathbb{R}^{5})\rightarrow X_{\textrm{B}}$$ between the $2$-dimensional column and row spaces of $A$ which is responsible for the particular solution of $Ax=b$. Thus $G$ is simply $A_{\textrm{B}}^{-1}$ acting on its domain $a(X)$ considered a subspace of $Y$, suitably extended to the whole of $Y$. That it is indeed a generalized inverse is readily seen through the matrix multiplications $GAG$ and $AGA$ that can be verified to reproduce $G$ and $A$ respectively. Comparison of Eqs. (\[Eqn: Def: Moore-Penrose\]) and (\[Eqn: Def: GenInv\]) shows that the Moore-Penrose inverse differs from ours through the geometrical constraints imposed in its definition, Eqs. (\[Eqn: MPInverse\]). Of course, this results in a more complex inverse (\[Eqn: MPEx5\]) as compared to our very simple (\[Eqn: GenInvEx5\]); nevertheless it is true that both the inverses satisfy $$\begin{aligned}
E((E(G_{\textrm{MP}}))^{\textrm{T}}) & = & \left(\begin{array}{ccccc}
1 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0\end{array}\right)\\
\\ & = & E((E(G))^{\textrm{T}})\end{aligned}$$
where $E(A)$ is the row-reduced echelon form of $A$. The canonical simplicity of Eq. (\[Eqn: GenInvEx5\]) as compared to Eq. (\[Eqn: MPEx5\]) is a general feature that suggests a more natural choice of bases by the map $a$ than the orthogonal set imposed by Moore and Penrose. This is to be expected since the MP inverse, governed by Eq. (\[Eqn: MPInverse\]), is a subset of our less restricted inverse described by only the first two of (\[Eqn: MPInverse\]); more specifically the difference is made clear in Fig. \[Fig: MP\_Inverse\](a) which shows that for any $b\notin\mathcal{R}(A)$, only $G_{\textrm{MP}}(b_{\bot})=0$ as compared to $G(b)=0$. This seems to imply that introducing extraneous topological considerations into the purely set theoretic inversion process may not be a recommended way of inverting, and the simple bases comprising the row and null spaces of $A$ and $A^{\textrm{T}}$ — that are mutually orthogonal just as those of the Moore-Penrose — are a better choice for the particular problem $Ax=b$ than the general orthonormal bases that the MP inverse introduces. These “good” bases, with respect to which the generalized inverse $G$ has a considerably simpler representation, are obtained in a straight forward manner from the row-reduced forms of $A$ and $A^{\textrm{T}}$. These bases are
\(a) The column space of $A$ is spanned by the columns $(1,\textrm{ }3,\textrm{ }2,\textrm{ }2)^{\textrm{T}}$ and $(1,\textrm{ }5,\textrm{ }2,\textrm{ }4)^{\textrm{T}}$ of $A$ that correspond to the basic columns containing the leading $1$’s in the row-reduced form of $A$,
\(b) The null space of $A^{\textrm{T}}$ is spanned by the solutions $(-2,\textrm{ }0,\textrm{ }1,\textrm{ }0)^{\textrm{T}}$ and $(1,-1,\textrm{ }0,\textrm{ }1)^{\textrm{T}}$ of the equation $A^{\textrm{T}}b=0$,
\(c) The row space of $A$ is spanned by the rows $(1,-3,\textrm{ }2,\textrm{ }1,\textrm{ }2)$ and $(3,-9,\textrm{ }10,\textrm{ }2,\textrm{ }9)$ of $A$ corresponding to the non-zero rows in the row-reduced form of $A$,
\(d) The null space of $A$ is spanned by the solutions $(3,\textrm{ }1,\textrm{ }0,\textrm{ }0,\textrm{ }0)$, $(-6,\textrm{ }0,\textrm{ }1,\textrm{ }4,\textrm{ }0)$, and $(-2,\textrm{ }0,-3,\textrm{ }0,\textrm{ }4)$ of the equation $Ax=0$.$\qquad\blacksquare$
The main differences between the natural “good” bases and the MP-bases that are responsible for the difference in form of the inverses, is that the later have the additional restrictions of being orthogonal to each other (recall the orthogonality property of the $Q$-matrices), and the more severe one of basis vectors mapping onto basis vectors according to $Ax_{i}=\sigma_{i}b_{i}$, $i=1,\cdots,r$, where the $\{ x_{i}\}_{i=1}^{n}$ and $\{ b_{j}\}_{j=1}^{m}$ are the eigenvectors of $A^{\textrm{T}}A$ and $AA^{\textrm{T}}$ respectively and $(\sigma_{i})_{i=1}^{r}$ are the positive square roots of the non-zero eigenvalues of $A^{\textrm{T}}A$ (or of $AA^{\textrm{T}}$), with $r$ denoting the dimension of the row or column space. This is considered as a serious restriction as the linear combination of the basis $\{ b_{j}\}$ that $Ax_{i}$ should otherwise have been equal to, allows a greater flexibility in the matrix representation of the inverse that shows up in the structure of $G$. These are, in fact, quite general considerations in the matrix representation of linear operators; thus the basis that diagonalizes an $n\times n$ matrix (when this is possible) is not the standard “diagonal” orthonormal basis of $\mathbb{R}^{n}$, but a problem-dependent, less canonical, basis consisting of the $n$ eigenvectors of the matrix. The $0$-rows of the inverse of Eq. (\[Eqn: GenInvEx5\]) result from the $3$-dimensional null-space variables $x_{2}$, $x_{4}$, and $x_{5}$, while the $0$-columns come from the $2$-dimensional image-space dependency of $b_{3}$, $b_{4}$ on $b_{1}$ and $b_{2}$, that is from the last two zero rows of the reduced echelon form (\[Eqn: RowReduce\]) of the augmented matrix.
We will return to this theme of the generation of a most appropriate problem-dependent topology for a given space in the more general context of chaos in Sec. 4.2.
In concluding this introduction to generalized inverses we note that the inverse $G$ of $f$ comes very close to being a right inverse: thus even though $AG\not\neq\bold1_{2}$ its row-reduced form $$\left(\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\end{array}\right)$$
is to be compared with the corresponding less satisfactory $$\left(\begin{array}{cccr}
1 & 0 & 2 & -1\\
0 & 1 & 0 & 1\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\end{array}\right)$$
representation of $AG_{\textrm{MP}}$.
**3. Multifunctional extension of function spaces**
The previous section has considered the solution of ill-posed problems as multifunctions and has shown how this solution may be constructed. Here we introduce the multifunction space $\textrm{Multi}_{\mid}(X)$ as the first step toward obtaining a smallest dense extension $\textrm{Multi}(X)$ of the function space $\textrm{Map}(X)$. $\textrm{Multi}_{\mid}(X)$ is basic to our theory of chaos [@Sengupta2000] in the sense that a chaotic state of a system can be fully described by such an indeterminate multifunctional state. In fact, multifunctions also enter in a natural way in describing the spectrum of nonlinear functions that we consider in Section 6; this is required to complete the construction of the smallest extension $\textrm{Multi}(X)$ of the function space $\textrm{Map}(X)$. The main tool in obtaining the space $\textrm{Multi}_{\mid}(X)$ from $\textrm{Map}(X)$ is a generalization of the technique of pointwise convergence of continuous functions to (discontinuous) functions. In the analysis below, we consider nets instead of sequences as the spaces concerned, like the topology of pointwise convergence, may not be first countable, Appendix A1.
***3.1. Graphical convergence of a net of functions***
Let $(X,\mathcal{U})$ and $(Y,\mathcal{V})$ be Hausdorff spaces and $(f_{\alpha})_{\alpha\in\mathbb{D}}:X\rightarrow Y$ be a net of piecewise continuous functions, not necessarily with the same domain or range, and suppose that for each $\alpha\in\mathbb{D}$ there is a finite set $I_{\alpha}=\{1,2,\cdots P_{\alpha}\}$ such that $f_{\alpha}^{-}$ has $P_{\alpha}$ functional branches possibly with different domains; obviously $I_{\alpha}$ is a singleton iff $f$ is a injective. For each $\alpha\in\mathbb{D}$, define functions $(g_{\alpha i})_{i\in I_{\alpha}}\!:Y\rightarrow X$ such that $$f_{\alpha}g_{\alpha i}f_{\alpha}=f_{\alpha i}^{I}\qquad i=1,2,\cdots P_{\alpha,}$$
where $f_{\alpha i}^{I}$ is a basic injective branch of $f_{\alpha}$ on some subset of its domain: $g_{\alpha i}f_{\alpha i}^{I}=1_{X}$ on $\mathcal{D}(f_{\alpha i}^{I})$, $f_{\alpha i}^{I}g_{\alpha i}=1_{Y}$ on $\mathcal{D}(g_{\alpha i})$ for each $i\in I_{\alpha}$. The use of nets and filters is dictated by the fact that we do not assume $X$ and $Y$ to be first countable. In the application to the theory of dynamical systems that follows, $X$ and $Y$ are compact subsets of $\mathbb{R}$ when the use of sequences suffice.
In terms of the residual and cofinal subsets $\textrm{Res}(\mathbb{D})$ and $\textrm{Cof}(\mathbb{D})$ of a directed set $\mathbb{D}$ (Def. A1.7), with $x$ and $y$ in the equations below being taken to belong to the required domains, define subsets $\mathcal{D}_{-}$ of $X$ and $\mathcal{R}_{-}$ of $Y$ as $$\mathcal{D}_{-}=\{ x\in X\!:((f_{\nu}(x))_{\nu\in\mathbb{D}}\textrm{ converges in }(Y,\mathcal{V}))\}\label{Eqn: D-}$$ $$\mathcal{R}_{-}=\{ y\in Y\!:\textrm{ }(\exists i\in I_{\nu})((g_{\nu i}(y))_{\nu\in\mathbb{D}}\textrm{ converges in }(X,\mathcal{U}))\}\label{Eqn: R-}$$
Thus:
$\mathcal{D}_{-}$ is the set of points of $X$ on which the values of a given net of functions $(f_{\alpha})_{\alpha\in\mathbb{D}}$ converge pointwise in $Y$. Explicitly, this is the subset of $X$ on which subnets[^19] in $\textrm{Map}(X,Y)$ combine to form a net of functions that converge pointwise to a limit function $F:\mathcal{D}_{-}\rightarrow Y$.
$\mathcal{R}_{-}$ is the set of points of $Y$ on which the values of the nets in $X$ generated by the injective branches of $(f_{\alpha})_{\alpha\in\mathbb{D}}$ converge pointwise in $Y$. Explicitly, this is the subset of $Y$ on which subnets of injective branches of $(f_{\alpha})_{\alpha\in\mathbb{D}}$ in $\textrm{Map}(Y,X)$ combine to form a net of functions that converge pointwise to a family of limit functions $G:\mathcal{R}_{-}\rightarrow X$. Depending on the nature of $(f_{\alpha})_{\alpha\in\mathbb{D}}$, there may be more than one $\mathcal{R}_{-}$ with a corresponding family of limit functions on each of them. To simplify the notation, we will usually let $G:\mathcal{R}_{-}\rightarrow X$ denote all the limit functions on all the sets $\mathcal{R}_{-}$.
If we consider cofinal rather than residual subsets of $\mathbb{D}$ then corresponding $\mathbb{D}_{+}$ and $\mathbb{R}_{+}$ can be expressed as $$\mathcal{D}_{+}=\{ x\in X\!:((f_{\nu}(x))_{\nu\in\textrm{Cof}(\mathbb{D})}\textrm{ converges in }(Y,\mathcal{V}))\}\label{Eqn: D+}$$ $$\mathcal{R}_{+}=\{ y\in Y\!:(\exists i\in I_{\nu})((g_{\nu i}(y))_{\nu\in\textrm{Cof}(\mathbb{D})}\textrm{ converges in }(X,\mathcal{U}))\}.\label{Eqn: R+}$$
It is to be noted that the conditions $\mathcal{D}_{+}=\mathcal{D}_{-}$ and $\mathcal{R}_{+}=\mathcal{R}_{-}$ are necessary and sufficient for the Kuratowski convergence to exist. Since $\mathcal{D}_{+}$ and $\mathcal{R}_{+}$ differ from $\mathcal{D}_{-}$ and $\mathcal{R}_{-}$ only in having cofinal subsets of $D$ replaced by residual ones, and since residual sets are also cofinal, it follows that $\mathcal{D}_{-}\subseteq\mathcal{D}_{+}$ and $\mathcal{R}_{-}\subseteq\mathcal{R}_{+}$. The sets $\mathcal{D}_{-}$ and $\mathcal{R}_{-}$ serve for the convergence of a net of functions just as $\mathcal{D}_{+}$ and $\mathcal{R}_{+}$ are for the convergence of subnets of the nets (*adherence*). The later sets are needed when subsequences are to be considered as sequences in their own right as, for example, in dynamical systems theory in the case of $\omega$-limit sets.
As an illustration of these definitions, consider the sequence of injective functions on the interval $[0,1]$ $f_{n}(x)=2^{n}x$, for $x\in\left[0,1/2^{n}\right],\textrm{ }n=0,1,2\cdots$. Then $\mathbb{D}_{0.2}$ is the set $\{0,1,2\}$ and only $\mathbb{D}_{0}$ is eventual in $\mathbb{D}$. Hence $\mathcal{D}_{-}$ is the single point set {0}. On the other hand $\mathbb{D}_{y}$ is eventual in $\mathbb{D}$ for all $y$ and $\mathcal{R}_{-}$ is $[0,1]$.
**Definition** **[<span style="font-variant:small-caps;">3.1</span>]{}***[<span style="font-variant:small-caps;">.</span>]{}* ***Graphical Convergence of a net of functions.*** *A net of functions $(f_{\alpha})_{\alpha\in D}\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ is said to* *converge graphically* *if either $\mathcal{D}_{-}\neq\emptyset$ or $\mathcal{R}_{-}\neq\emptyset$; in this case let $F\!:\mathcal{D}_{-}\rightarrow Y$ and $G:\mathcal{R}_{-}\rightarrow X$ be the entire collection of limit functions. Because of the assumed Hausdorffness of $X$ and $Y$, these limits are well defined.*
*The graph of the* *graphical limit* $\mathscr{M}$ *of the net* $(f_{\alpha})\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ ***denoted by* ***$f_{\alpha}\overset{\mathbf{G}}\longrightarrow\mathscr{M}$, is the subset of $\mathcal{D}_{-}\times\mathcal{R}_{-}$that is the union of the graphs of the function $F$ and the multifunction $G^{-}$ $$\mathbf{G}_{\mathscr{M}}=\mathbf{G}_{F}\bigcup\mathbf{G}_{G^{-}}$$*
*where $$\mathbf{G}_{G^{-}}=\{(x,y)\in X\times Y\!:(y,x)\in\mathbf{G}_{G}\subseteq Y\times X\}.\qquad\square$$*
***Begin Tutorial6: Graphical Convergence***
The following two examples are basic to the understanding of the graphical convergence of functions to multifunctions and were the examples that motivated our search of an acceptable technique that did not require vertical portions of limit relations to disappear simply because they were non-functions: the disturbing question that needed an answer was how not to mathematically sacrifice these extremely significant physical components of the limiting correspondences. Furthermore, it appears to be quite plausible to expect a physical interaction between two spaces $X$ and $Y$ to be a consequence of both the direct interaction represented by $f\!:X\rightarrow Y$ and also the inverse interaction $f^{-}\!:Y\rightarrow X$, and our formulation of pointwise biconvergence is a formalization of this idea. Thus the basic examples (1) and (2) below produce multifunctions instead of discontinuous functions that would be obtained by the usual pointwise limit.
**Example 3.1.** (1)
$$f_{n}(x)=\left\{ \begin{array}{lc}
0 & -1\leq x\leq0\\
nx & 0\leq x\leq1/n\\
1 & 1/n\leq x\leq1\end{array}\right.:\quad[-1,1]\rightarrow[0,1]$$ $$g_{n}(y)=y/n:\quad[0,1]\rightarrow[0,1/n]$$
Then$$F(x)=\left\{ \begin{array}{cc}
0 & -1\leq x\leq0\\
1 & 0<x\leq1\end{array}\right.\qquad\mathrm{on}\qquad\mathcal{D}_{-}=\mathcal{D}_{+}=[-1,0]\bigcup(0,1]$$
$$G(y)=0\quad\mathrm{on}\quad\mathcal{R}_{-}=[0,1]=\mathcal{R}_{+}.$$
The graphical limit is $([-1,0],0)\bigcup(0,[0,1])\bigcup((0,1],1)$.
\(2) $f_{n}(x)=nx$ for $x\in[0,1/n]$ gives $g_{n}(y)=y/n:[0,1]\rightarrow[0,1/n].$ Then $$F(x)=0\quad\mathrm{on}\quad\mathcal{D}_{-}=\{0\}=\mathcal{D}_{+},\qquad G(y)=0\quad\mathrm{on}\quad\mathcal{R}_{-}=[0,1]=\mathcal{R}_{+}.$$
The graphical limit is $(0,[0,1])$.$\qquad\blacksquare$
[1.4]{} In these examples that we consider to be the prototypes of graphical convergence of functions to multifunctions, $G(y)=0$ on $\mathcal{R}_{-}$ because $g_{n}(y)\rightarrow0$ for all $y\in\mathcal{R}_{-}$. Compare the graphical multifunctional limits with the corresponding usual pointwise functional limits characterized by discontinuity at $x=0$. Two more examples from @Sengupta2000 that illustrate this new convergence principle tailored specifically to capture one-to-many relations are shown in Fig. \[Fig: Example2\_1\] which also provides an example in Fig. \[Fig: Example2\_1\](c) of a function whose iterates do not converge graphically because in this case both the sets $\mathcal{D}_{-}$ and $\mathcal{R}_{-}$are empty. The power of graphical convergence in capturing multifunctional limits is further demonstrated by the example of the sequence $(\sin n\pi x)_{n=1}^{\infty}$ that converges to $0$ both $1$-integrally and test-functionally, Eqs. (\[Eqn: intsin\]) and (\[Eqn: testsin\]).
[$\quad$(b) $F(x)=1$ on $\mathcal{D}_{-}=\{0\}$ and $G(y)=0$ on $\mathcal{R}_{-}=\{1\}$. Also $F(x)=-1/2,\textrm{ }0,\textrm{ }1,\textrm{ }3/2$ respectively on $\mathcal{D}_{+}=(0,3],\textrm{ }\{2\},\textrm{ }\{0\},\textrm{ }(0,2)$ and $G(y)=0,\textrm{ }0,\textrm{ }2,\textrm{ }3$ respectively on $\mathcal{R}_{+}=(-1/2,1],\textrm{ }[1,3/2),\textrm{ }[0,3/2),\textrm{ }[-1/2,0)$. ]{}
[$\quad$(c) For $f(x)=-0.05+x-x^{2}$, no graphical limit as $\mathcal{D}_{-}=\emptyset=\mathcal{R}_{-}$.]{}
[$\quad$(d) For $f(x)=0.7+x-x^{2}$, $F(x)=\alpha$ on $\mathcal{D}_{-}=[a,c]$, $G_{1}(y)=a$ and $G_{2}(y)=c$ on $\mathcal{R}_{-}=(-\infty,\alpha]$. Notice how the two fixed points and their equivalent images define the converged limit rectangular multi. As in example (1) one has $\mathcal{D}_{-}=\mathcal{D}_{+}$; also $\mathcal{R}_{-}=\mathcal{R}_{+}$.]{}
It is necessary to understand how the concepts of *eventually in* and *frequently in* of Appendix A2 apply in examples (a) and (b) of Fig. [\[Fig: Example2\_1\].]{} In these two examples we have two subsequences one each for the even indices and the other for the odd. For a point-to-point functional relation, this would mean that the sequence frequents the adherence set $\textrm{adh}(x)$ of the sequence $(x_{n})$ but does not converge anywhere as it is not eventually in every neighbourhood of any point. For a multifunctional limit however it is possible, as demonstrated by these examples, for the subsequences to be eventually in every neighbourhood of certain *subsets* common to the eventual limiting sets of the subsequences; this intersection of the subsequential limits is now *defined to be the limit of the original sequence.* A similar situation obtains, for example, in the solution of simultaneous equations: The solution of the equation $a_{11}x_{1}+a_{12}x_{2}=b_{1}$ for one of the variables $x_{2}$ say with $a_{12}\neq0$, is the set represented by the straight line $x_{2}=m_{1}x_{1}+c_{1}$ for all $x_{1}$ in its domain, while for a different set of constants $a_{21}$, $a_{22}$ and $b_{2}$ the solution is the entirely different set $x_{2}=m_{2}x_{1}+c_{2}$, under the assumption that $m_{1}\neq m_{2}$ and $c_{1}\neq c_{2}$. Thus even though the individual equations (subsequences) of the simultaneous set of equations (sequence) may have distinct solutions (limits), the solution of the equations is their common point of intersection.
Considered as sets in $X\times Y$, the discussion of convergence of a sequence of graphs $f_{n}\!:X\rightarrow Y$ would be incomplete without a mention of the convergence of a sequence of sets under the Hausdorff metric that is so basic in the study of fractals. In this case, one talks about the convergence of a sequence of compact subsets of the metric space $\mathbb{R}^{n}$ so that the sequences, as also the limit points that are the fractals, are compact subsets of $\mathbb{R}^{n}$. Let $\mathcal{K}$ denote the collection of all nonempty compact subsets of $\mathbb{R}^{n}$. Then the *Hausdorff metric* $d_{\textrm{H}}$ between two sets on $\mathcal{K}$ is defined to be $$d_{\textrm{H}}(E,F)=\max\{\delta(E,F),\delta(F,E)\}\qquad E,F\in\mathcal{K},$$
where $$\delta(E,F)=\max_{x\in E}\textrm{ }\min_{y\in F}\Vert\mathbf{x-y}\Vert_{2}$$
is $\delta(E,F)$ is the non-symmetric $2$-norm in $\mathbb{R}^{n}$. The power and utility of the Hausdorff distance is best understood in terms of the dilations $E+\varepsilon:=\bigcup_{x\in E}D_{\varepsilon}(x)$ of a subset $E$ of $\mathbb{R}^{n}$ by $\varepsilon$ where $D_{\varepsilon}(x)$ is a closed ball of radius $\varepsilon$ at $x$; physically a dilation of $E$ by $\varepsilon$ is a closed $\varepsilon$-neighbourhood of $E$. Then a fundamental property of $d_{\textrm{H}}$ is that $d_{\textrm{H}}(E,F)\leq\varepsilon$ iff both $E\subseteq F+\varepsilon$ and $F\subseteq E+\varepsilon$ hold simultaneously which leads [@Falconer1990] to the interesting consequence that
*If $(F_{n})_{n=1}^{\infty}$ and $F$ are nonempty compact sets, then $\lim_{n\rightarrow\infty}F_{n}=F$ in the Hausdorff metric iff $F_{n}\subseteq F+\varepsilon$ and $F\subseteq F_{n}+\varepsilon$ eventually. Furthermore if $(F_{n})_{n=1}^{\infty}$ is a decreasing sequence of elements of a filter-base in $\mathbb{R}^{n}$, then the nonempty and compact limit set $F$ is given by $$\lim_{n\rightarrow\infty}F_{n}=F=\bigcap_{n=1}^{\infty}F_{n}.$$* Note that since $\mathbb{R}^{n}$ is Hausdorff, the assumed compactness of $F_{n}$ ensures that they are also closed in **$\mathbb{R}^{n}$; $F$, therefore, is just the adherent set of the filter-base. In the deterministic algorithm for the generation of fractals by the so-called iterated function system (IFS) approach, $F_{n}$ is the inverse image by the $n^{\textrm{th}}\textrm{ }$ iterate of a non-injective function $f$ having a finite number of injective branches and converging graphically to a multifunction. Under the conditions stated above, the Hausdorff metric ensures convergence of any class of compact subsets in $\mathbb{R}^{n}$. It appears eminently plausible that our multifunctional graphical convergence on $\textrm{Map}(\mathbb{R}^{n})$ implies Hausdorff convergence on $\mathbb{R}^{n}$: in fact pointwise biconvergence involves simultaneous convergence of image and preimage nets on $Y$ and $X$ respectively. Thus confining ourselves to the simpler case of pointwise convergence, if $(f_{\alpha})_{\alpha\in\mathbb{D}}$ is a net of functions in $\textrm{Map}(X,Y)$, then the following theorem expresses the link between convergence in $\textrm{Map}(X,Y)$ and in $Y$.
**Theorem 3.1.** *A net of functions* $(f_{\alpha})_{\alpha\in\mathbb{D}}$ *converges to a function* $f$ *in* $(\textrm{Map}(X,Y),\mathcal{T})$ *in the topology of pointwise convergence iff* $(f_{\alpha})$ *converges pointwise to $f$ in the sense that $f_{\alpha}(x)\rightarrow f(x)$ in $Y$ for every $x$ in $X$.$\qquad\square$*
**Proof.** *Necessity.* First consider $f_{\alpha}\rightarrow f$ in $(\textrm{Map}(X,Y),\mathcal{T})$. For an open neighbourhood $V$ of $f(x)$ in $Y$ with $x\in X$, let $B(x;V)$ be a local neighbourhood of $f$ in $(\textrm{Map}(X,Y),\mathcal{T})$, see Eq. (\[Eqn: point\]) in Appendix A1. By assumption of convergence, $(f_{\alpha})$ must eventually be in $B(x;V)$ implying that $f_{\alpha}(x)$ is eventually in $V$. Hence $f_{\alpha}(x)\rightarrow f(x)$ in $Y$.
*Sufficiency.* Conversely, if $f_{\alpha}(x)\rightarrow f(x)$ in $Y$ for every $x\in X$, then for a *finite* collection of points $(x_{i})_{i=1}^{I}$ of $X$ ($X$ may itself be uncountable) and corresponding open sets $(V_{i})_{i=1}^{I}$ in $Y$ with $f(x_{i})\in V_{i}$, let $B((x_{i})_{i=1}^{I};(V_{i})_{i=1}^{I})$ be an open neighbourhood of $f$. From the assumed pointwise convergence $f_{\alpha}(x_{i})\rightarrow f(x_{i})$ in $Y$ for $i=1.2.\cdots.I$, it follows that $(f_{\alpha}(x_{i}))$ is eventually in $V_{i}$ for every $(x_{i})_{i=1}^{I}$. Because $\mathbb{D}$ is a directed set, the existence of a residual applicable globally for all $i=1,2,\cdots,I$ is assured leading to the conclusion that $f_{\alpha}(x_{i})\in V_{i}$ eventually for every $i=1,2,\cdots,I$. Hence $f_{\alpha}\in B((x_{i})_{i=1}^{I};(V_{i})_{i=1}^{I})$ eventually; this completes the demonstration that $f_{\alpha}\rightarrow f$ in $(\textrm{Map}(X,Y),\mathcal{T})$, and thus of the proof.$\qquad\blacksquare$
***End Tutorial6***
***3.2. The Extension*** **Multi$_{\mid}$(*X,Y*)** ***of*** **Map(*X,Y*)**
In this Section we show how the topological treatment of pointwise convergence of functions to functions given in Example A1.1 of Appendix 1 can be generalized to generate the boundary $\textrm{Multi}_{\mid}(X,Y)$ between $\textrm{Map}(X,Y)$ and $\textrm{Multi}(X,Y)$; here $X$ and $Y$ are Hausdorff spaces and $\textrm{Map}(X,Y)$ and $\textrm{Multi}(X,Y)$ are respectively the sets of all functional and non-functional relations between $X$ and $Y$. The generalization we seek defines neighbourhoods of $f\in\textrm{Map}(X,Y)$ to consist of those functional relations in $\textrm{Multi}(X,Y)$ whose images at any point $x\in X$ lies not only arbitrarily close to $f(x)$ (this generates the usual topology of pointwise convergence $\mathcal{T}_{Y}$ of Example A1.1) but whose inverse images at $y=f(x)\in Y$ contain points arbitrarily close to $x$. Thus the graph of $f$ must not only lie close enough to $f(x)$ at $x$ in $V$, but must additionally be such that $f^{-}(y)$ has at least branch in $U$ about $x$; thus $f$ is constrained to cling to $f$ as the number of points on the graph of $f$ increases with convergence and, unlike in the situation of simple pointwise convergence, no gaps in the graph of the limit object is permitted not only, as in Example A1.1 on the domain of $f$, but simultaneously on it range too. We call the resulting generated topology the *topology of pointwise biconvergence on* $\textrm{Map}(X,Y)$, to be denoted by $\mathcal{T}$. Thus for any given integer $I\geq1$, the generalization of Eq. (\[Eqn: point\]) gives for $i=1,2,\cdots,I$, the open sets of $(\textrm{Map}(X,Y),\mathcal{T})$ to be $$\begin{gathered}
B((x_{i}),(V_{i});(y_{i}),(U_{i}))=\{ g\in\mathrm{Map}(X,Y)\!:\\
(g(x_{i})\in V_{i})\wedge(g^{-}(y_{i})\bigcap U_{i}\neq\emptyset)\textrm{ },i=1,2,\cdots,I\},\label{Eqn: func_bi}\end{gathered}$$
where $(x_{i})_{i=1}^{I},(V_{i})_{i=1}^{I}$ are as in that example, $(y_{i})_{i=1}^{I}\in Y$, and the corresponding open sets $(U_{i})_{i=1}^{I}$ in $X$ are chosen arbitrarily[^20]. A local base at $f$, for $(x_{i},y_{i})\in\mathbf{G}_{f}$, is the set of functions of (\[Eqn: func\_bi\]) with $y_{i}=f(x_{i})$ and the collection of all local bases $$B_{\alpha}=B((x_{i})_{i=1}^{I_{\alpha}},(V_{i})_{i=1}^{I_{\alpha}};(y_{i})_{i=1}^{I_{\alpha}},(U_{i})_{i=1}^{I_{\alpha}}),\label{Eqn: local_base}$$ for every choice of $\alpha\in\mathbb{D}$, is a base $_{\textrm{T}}\mathcal{B}$ of $(\textrm{Map}(X,Y),\mathcal{T})$. Here the directed set $\mathbb{D}$ is used as an indexing tool because, as pointed out in Example A1.1, the topology of pointwise convergence is not first countable.
In a manner similar to Eq. (\[Eqn: func\_bi\]), the open sets of $(\mathrm{Multi}(X,Y),\widehat{\mathcal{T}})$, where $\textrm{Multi}(X,Y)$ are multifunctions with only countably many values in $Y$ for every point of $X$ (so that we exclude continuous regions from our discussion except for the “vertical lines” of $\textrm{Multi}_{\mid}(X,Y)$), can be defined by $$\begin{gathered}
\widehat{B}((x_{i}),(V_{i});(y_{i}),(U_{i}))=\{\mathscr{G}\in\mathrm{Multi}(X,Y)\!:(\mathscr{G}(x_{i})\bigcap V_{i}\neq\emptyset)\wedge(\mathscr G^{-}(y_{i})\bigcap U_{i}\neq\emptyset)\}\label{Eqn: multi_bi}\end{gathered}$$
where $$\mathscr G^{-}(y)=\{ x\in X\!:y\in\mathscr{G}(x)\}.$$
and $(x_{i})_{i=1}^{I}\in\mathcal{D}(\mathscr{G}),(V_{i})_{i=1}^{I};(y_{i})_{i=1}^{I}\in\mathcal{R}(\mathscr{G}),(U_{i})_{i=1}^{I}$ are chosen as in the above. The topology $\widehat{\mathcal{T}}$ of $\textrm{Multi}(X,Y)$ is generated by the collection of all local bases $\widehat{B_{\alpha}}$ for every choice of $\alpha\in\mathbb{D}$, and it is not difficult to see from Eqs. (\[Eqn: func\_bi\]) and (\[Eqn: multi\_bi\]), that the restriction **** of $\widehat{\mathcal{T}}$ to $\textrm{Map}(X,Y)$ is just $\mathcal{T}$.
Henceforth $\widehat{\mathcal{T}}$ and $\mathcal{T}$ will be denoted by the same symbol $\mathcal{T}$, and convergence in the topology of pointwise biconvergence in $(\textrm{Multi}(X,Y),\mathcal{T})$ will be denoted by $\rightrightarrows$, with the notation being derived from Theorem 3.1.
**Definition 3.2.** ***Functionization of a multifunction.*** *A net of functions* $(f_{\alpha})_{\alpha\in\mathbb{D}}$ *in* $\textrm{Map}(X,Y)$ *converges in* $(\textrm{Multi}(X,Y),\mathcal{T})$, $f_{\alpha}\rightrightarrows\mathscr{M}$, *if it biconverges pointwise in* $(\textrm{Map}(X,Y),\mathcal{T}^{*})$. *Such a net of functions will be said to be a* *functionization of* $\mathscr{M}$*.$\qquad\square$*
**Theorem 3.2.** *Let $(f_{\alpha})_{\alpha\in\mathbb{D}}$ be a net of functions in $\textrm{Map}(X,Y)$. Then $$f_{\alpha}\overset{\mathbf{G}}\longrightarrow\mathscr{M}\Longleftrightarrow f_{\alpha}\rightrightarrows\mathcal{M}.\qquad\square$$* **Proof.** If $(f_{\alpha})$ converges graphically to $\mathscr{M}$ then either $\mathcal{D}_{-}$ or $\mathcal{R}_{-}$ is nonempty; let us assume both of them to be so. Then the sequence of functions $(f_{\alpha})$ converges pointwise to a function $F$ on $\mathcal{D}_{-}$ and to functions $G$ on $\mathcal{R}_{-}$, and the local basic neighbourhoods of $F$ and $G$ generate the topology of pointwise biconvergence.
Conversely, for pointwise biconvergence on $X$ and $Y$, $\mathcal{R}_{-}$ and $\mathcal{D}_{-}$ must be non-empty.$\qquad\blacksquare$
Observe that the boundary of $\textrm{Map}(X,Y)$ in the topology of pointwise biconvergence is a “line parallel to the $Y$-axis”. We denote this closure of $\textrm{Map}(X,Y)$ as
**Definition 3.3.** $\textrm{Multi}_{\mid}((X,Y),\mathcal{T})=\mathrm{Cl}(\mathrm{Map}((X,Y),\mathcal{T})).$$\qquad\square$
The sense in which $\textrm{Multi}_{\mid}(X,Y)$ is the smallest closed topological extension of $M=\textrm{Map}(X,Y)$ is the following, refer Thm. A1.4 and its proof. Let $(M,\mathcal{T}_{0})$ be a topological space and suppose that$${\textstyle \widehat{M}=M\bigcup\{\widehat{m}\}}$$
is obtained by adjoining an extra point to $M$; here $M=\textrm{Map}(X,Y)$ and $\widehat{m}\in\textrm{Cl}(M)$ is the multifunctional limit in $\widehat{M}=\textrm{Multi}_{\mid}(X,Y)$. Treat all open sets of $M$ generated by local bases of the type (\[Eqn: local\_base\]) with finite intersection property as a filter-base $_{\textrm{F}}\mathcal{B}$ on $X$ that induces a filter $\mathcal{F}$ on $M$ (by forming supersets of all elements of $_{\textrm{F}}\mathcal{B}$; see Appendix A1) and thereby the filter-base $${\textstyle \widehat{_{\textrm{F}}\mathcal{B}}=\{\widehat{B}=B\bigcup\{\widehat{m}\}\!:B\in\,_{\textrm{F}}\mathcal{B}\}}$$
on $\widehat{M}$; this filter-base at $m$ can also be obtained independently from Eq. (\[Eqn: multi\_bi\]). Obviously $\widehat{_{\textrm{F}}\mathcal{B}}$ is an extension of $_{\textrm{F}}\mathcal{B}$ on $\widehat{M}$ and $_{\textrm{F}}\mathcal{B}$ is the filter induced on $M$ by $\widehat{_{\textrm{F}}\mathcal{B}}$. We may also consider the filter-base to be a topological base on $M$ that defines a coarser topology $\mathcal{T}$ on $M$ (through all unions of members of $_{\textrm{F}}\mathcal{B}$) and hence the topology$${\textstyle \widehat{\mathcal{T}}=\{\widehat{G}=G\bigcup\{\widehat{m}\}\!:G\in\mathcal{T}\}}$$
on $\widehat{M}$ to be the topology associated with $\widehat{\mathcal{F}}$. A finer topology on $\widehat{M}$ may be obtained by adding to $\widehat{\mathcal{T}}$ all the discarded elements of $\mathcal{T}_{0}$ that do not satisfy FIP. It is clear that $\widehat{m}$ is on the boundary of $M$ because every neighbourhood of $\widehat{m}$ intersects $M$ by construction; thus $(M,\mathcal{T})$ is dense in $(\widehat{M,}\widehat{\mathcal{T}})$ which is the required topological extension of $(M,\mathcal{T}).$
In the present case, a filter-base at $f\in\mathrm{Map}(X,Y)$ is the neighbourhood system $_{\textrm{F}}\mathcal{B}_{f}$ at $f$ given by decreasing sequences of neighbourhoods $(V_{k})$ and $(U_{k})$ of $f(x)$ and $x$ respectively, and the filter $\widehat{\mathcal{F}}$ is the neighbourhood filter $\mathcal{N}_{f}\bigcup G$ where $G\in$$\textrm{Multi}_{\mid}(X,Y)$. We shall present an alternate, and perhaps more intuitively appealing, description of graphical convergence based on the adherence set of a filter on Sec. 4.1.
As more serious examples of the graphical convergence of a net of functions to multifunction than those considered above, Fig. \[Fig: tent4\] shows the first four iterates of the tent map
$$t(x)=\left\{ \begin{array}{lc}
2x & 0\leq x<1/2\\
2(1-x) & 1/2\leq x\leq1\end{array}\right.\qquad\begin{array}{c}
(t^{1}=t).\end{array}$$
defined on $[0.1]$ and the sine map $f_{n}=|\sin(2^{n-1}\pi x)|,\; n=1,\cdots,4$ with domain $[0,1]$.
These examples illustrate the important generalization that *periodic points may be replaced by the more general equivalence classes* where a sequence of functions converges graphically; this generalization based on the ill-posed interpretation of dynamical systems is significant for non-iterative systems as in second example above. The equivalence classes of the tent map for its two fixed points $0$ and $2/3$ generated by the first 4 iterates are $$[0]_{4}=\left\{ 0,\frac{1}{8},\frac{1}{4},\frac{3}{8},\frac{1}{2},\frac{5}{8},\frac{3}{4},\frac{7}{8},1\right\}$$ $$\left[\frac{2}{3}\right]_{4}=\left\{ c,\frac{1}{8}\mp c,\frac{1}{4}\mp c,\frac{3}{8}\mp c,\frac{1}{2}\mp c,\frac{5}{8}\mp c,\frac{3}{4}\mp c,\frac{7}{8}\mp c,1-c\right\}$$
where $c=1/24$. If the moduli of the slopes of the graphs passing through these equivalent fixed points are greater than 1 then the graphs converge to multifunctions and when these slopes are less than 1 the corresponding graphs converge to constant functions. It is to be noted that the number of equivalent fixed points in a class increases with the number of iterations $k$ as $2^{k-1}+1;$ this *increase in the degree of ill-posedness is typical of discrete chaotic systems and can be regarded as a paradigm of chaos generated by* *the convergence of a family of functions.*
The $m^{\textrm{th}}$ iterate $t^{m}$ of the tent map has $2^{m}$ fixed points corresponding to the $2^{m}$ injective branches of $t^{m}$
$$x_{mj}=\left\{ \begin{array}{ll}
{\displaystyle \frac{j-1}{2^{m}-1}}, & j=1,3,\cdots,(2^{m}-1),\\
{\displaystyle \frac{j}{2^{m}+1}}, & j=2,4,\cdots,2^{m},\end{array}\right.t^{m}(x_{mj})=x_{mj},\textrm{ }j=1,2,\cdots,2^{m}.$$
Let $X_{m}$ be the collection of these $2^{m}$ fixed points (thus $X_{1}=\{0,2/3\}$), and denote by $[X_{m}]$ the set of the equivalent points, one coming from each of the injective branches, for each of the fixed points: thus
$$\begin{array}{crcl}
\mathcal{D}_{-}= & [X_{1}] & = & \{[0],[2/3]\}\\
& [X_{2}] & = & \{[0],[2/5],[2/3],[4/5]\}\end{array}$$
and $\mathcal{D}_{+}=\bigcap_{m=1}^{\infty}[X_{m}]$ is a nonempty countable set dense in $X$ at each of which the graphs of the sequence $(t^{m})$ converge to a multifunction. New sets $[X_{n}]$ will be formed by subsequences of the higher iterates $t^{n}$ for $m=in$ with $i=1,2,\cdots$ where these subsequences remain fixed. For example, the fixed points $2/5$ and $4/5$ produced respectively by the second and fourth injective branches of $t^{2}$, are also fixed for the seventh and thirteenth branches of $t^{4}.$ For the shift map $2x\;\textrm{mod}(1)$ on $[0,1]$, $\mathcal{D}_{-}=\{[0],[1]\}$ where $[0]=\bigcap_{m=1}^{\infty}\{(i-1)/2^{m}\!:i=1,2,\cdots,2^{m}\}$ and $[1]=\bigcap_{m=1}^{\infty}\{ i/2^{m}\!:i=1,2,\cdots,2^{m}\}$.
It is useful to compare the graphical convergence of $(\sin(\pi nx))_{n=1}^{\infty}$ to $[0,1]$ at $0$ and to $0$ at $1$ with the usual integral and test-functional convergences to $0$; note that the point $1/2$, for example, belongs to $\mathcal{D}_{+}$and not to $\mathcal{D}_{-}=\{0,1\}$ because it is frequented by even $n$ only. However for the subsequence $(f_{2^{m-1}})_{m\in\mathbb{Z}_{+}}$, $1/2$ is in $\mathcal{D}_{-}$ because if the graph of $f_{2^{m-1}}$ passes through $(1/2,0)$ for some $m$, then so do the graphs for all higher values . Therefore $[0]=\bigcap_{m=1}^{\infty}\{ i/2^{m-1}\!:i=0,1,\cdots,2^{m-1}\}$ is the equivalence class of $(f_{2^{m-1}})_{m=1}^{\infty}$ and this sequence converges to $[-1,1]$ on this set. Thus our extension $\textrm{Multi}(X)$ is distinct from the distributional extension of function spaces with respect to test functions, and is able to correctly generate the pathological behaviour of the limits that are so crucially vital in producing chaos.
**4. Discrete chaotic systems are maximally ill-posed**
The above ideas apply to the development of a criterion for chaos in discrete dynamical systems that is based on the limiting behaviour of the graphs of a sequence of functions $(f_{n})$ on $X,$ rather than on the values that the sequence generates as is customary. For the development of the maximality of ill-posedness criterion of chaos, we need to refresh ourselves with the following preliminaries.
***Resume Tutorial5: Axiom of Choice and Zorn’s Lemma***
Let us recall from the first part of this Tutorial that for nonempty subsets $(A_{\alpha})_{\alpha\in\mathbb{D}}$ of a nonempty set $X$, the Axiom of Choice ensures the existence of a set $A$ such that $A\bigcap A_{\alpha}$ consists of a single element for every $\alpha$. The choice axiom has far reaching consequences and a few equivalent statements, one of which the Zorn’s lemma that will be used immediately in the following, is the topic of this resumed Tutorial. The beauty of the Axiom, and of its equivalents, is that they assert the existence of mathematical objects that, in general, cannot be demonstrated and it is often believed that Zorn’s lemma is one of the most powerful tools that a mathematician has available to him that is “almost indispensable in many parts of modern pure mathematics” with significant applications in nearly all branches of contemporary mathematics. This “lemma” talks about maximal (as distinct from “maximum”) elements of a partially ordered set, a set in which some notion of $x_{1}$ “preceding” $x_{2}$ for two elements of the set has been defined.
A relation $\preceq$ on a set $X$ is said to be a *partial order* (or simply an *order*) if it is (compare with the properties (ER1)–(ER3) of an equivalence relation, Tutorial1)
(OR1) Reflexive, that is $(\forall x\in X)(x\preceq x)$.
(OR2) Antisymmetric: $(\forall x,y\in X)(x\preceq y\wedge y\preceq x\Longrightarrow x=y)$.
(OR3) Transitive, that is $(\forall x,y,z\in X)(x\preceq y\wedge y\preceq z\Longrightarrow x\preceq z)$. Any notion of order on a set $X$ in the sense of one element of $X$ preceding another should possess at least this property.
The **relation is a *preorder* $\precsim$ if it is only reflexive and transitive, that is if only (OR1) and (OR3) are true. If the hypothesis of (OR2) is also satisfied by a preorder, then this $\precsim$ induces an equivalence relation $\sim$ on $X$ according to $(x\precsim y)\wedge(y\precsim x)\Leftrightarrow x\sim y$ that evidently is actually a partial order iff $x\sim y\Leftrightarrow x=y$. For any element $[x]\in X/\sim$ of the induced quotient space, let $\leq$ denote the generated order in $X/\sim$ so that $$x\precsim y\Longleftrightarrow[x]\leq[y];$$ then $\leq$ is a partial order on $X/\sim$. If every two element of $X$ are *comparable*, in the sense that either $x_{1}\preceq x_{2}$ or $x_{2}\preceq x_{1}$ for all $x_{1},x_{2}\in X$, then $X$ is said to be a *totally ordered set* or a *chain.* A totally ordered subset $(C,\preceq)$ of a partially ordered set $(X,\preceq)$ with the ordering induced from $X$, is known as a *chain in $X$* if $$C=\{ x\in X\!:(\forall c\in X)(c\preceq x\vee x\preceq c)\}.\label{Eqn: chain}$$ The most important class of chains that we are concerned with in this work is that on the subsets $\mathcal{P}(X)$ of a set $(X,\subseteq)$ under the inclusion order; Eq. (\[Eqn: chain\]), as we shall see in what follows, defines a family of chains of nested subsets in $\mathcal{P}(X)$. Thus while the relation $\precsim$ in $\mathbb{Z}$ defined by $n_{1}\precsim n_{2}\Leftrightarrow\mid n_{1}\mid\,\leq\,\mid n_{2}\mid$ with $n_{1},n_{2}\in\mathbb{Z}$ preorders $\mathbb{Z}$, it is not a partial order because although $-n\precsim n\textrm{ and }n\precsim-n$ for any $n\in\mathbb{Z}$, it is does not follow that $-n=n$. A common example of partial order on a set of sets, for example on the power set $\mathcal{P}(X)$ of a set $X$ (see footnote \[Foot: notation\]), is the inclusion relation $\subseteq$: the ordered set $\mathcal{X}=(\mathcal{P}(\{ x,y,z\}),\subseteq)$ is partially ordered but not totally ordered because, for example, $\{ x,y\}\not\subseteq\{ y,x\}$, or $\{ x\}$ is not comparable to $\{ y\}$ unless $x=y$; however $C=\{\{\emptyset,\{ x\},\{ x,y\}\}$ does represent one of the many possible chains of $\mathcal{X}$. Another useful example of partial order is the following: Let $X$ and $(Y,\leq)$ be sets with $\leq$ ordering $Y$, and consider $f,g\in\textrm{Map}(X,Y)$ with $\mathcal{D}(f),\mathcal{D}(g)\subseteq X$. Then
$$\begin{aligned}
(\mathcal{D}(f)\subseteq\mathcal{D}(g))(f=g|_{\mathcal{D}(f)}) & \Longleftrightarrow & f\preceq g\nonumber \\
(\mathcal{D}(f)=\mathcal{D}(g))(\mathcal{R}(f)\subseteq\mathcal{R}(g)) & \Longleftrightarrow & f\preceq g\label{Eqn: FunctionOrder}\\
(\forall x\in\mathcal{D}(f)=\mathcal{D}(g))\textrm{ }(f(x)\leq g(x)) & \Longleftrightarrow & f\preceq g\nonumber \end{aligned}$$
define partial orders on $\textrm{Map}(X,Y)$. In the last case, the order is not total because any two functions whose graphs cross at some point in their common domain cannot be ordered by the given relation, while in the first any $f$ whose graph does not coincide with that of $g$ on the common domain is not comparable to it by this relation.
Let $(X,\preceq)$ be a partially ordered set and let $A$ be a subset of $X$. An element $a_{+}\in(A,\preceq)$ is said to be a *maximal* element of $A$ with respect to $\preceq$ if $$(\forall a\in(A,\preceq))(a_{+}\preceq a)\Longrightarrow\textrm{ }a=a_{+},\label{Eqn: maximal}$$ that is iff there is no $a\in A$ with $a\neq a_{+}$ and $a\succ a_{+}$[^21]. Expressed otherwise, this implies that an element $a_{+}$ of a subset $A\subseteq(X,\preceq)$ is maximal in $(A,\preceq)$ iff it is true that $$(a\preceq a_{+}\in A)\textrm{ }(\textrm{for every }a\in(A,\preceq)\textrm{ comparable to }a_{+});\label{Eqn: maximal1}$$ thus $a_{+}$ in $A$ is a maximal element of $A$ iff it is strictly greater than every *other comparable* element of $A$. This of course does not mean that each element $a$ of $A$ satisfies $a\preceq a_{+}$ because every pair of elements of a partially ordered set need not be comparable: in a totally ordered set there can be at most one maximal element. In comparison, an element $a_{\infty}$ of a subset $A\subseteq(X,\preceq)$ is *the* unique *maximum* (*largest, greatest, last*) element of $A$ iff $$(a\preceq a_{\infty}\in A)\textrm{ }(\textrm{for every }a\in(A,\preceq)),\label{Eqn: maximum}$$
implying that $a_{\infty}$ is *the* element of $A$ that is strictly larger than every other element of $A$. As in the case of the maximal, although this also does not require all elements of $A$ to be comparable to each other, it does require $a_{\infty}$ to be larger than every element of $A$. The dual concepts of minimal and minimum can be similarly defined by essentially reversing the roles of $a$ and $b$ in relational expressions like $a\preceq b$.
The last concept needed to formalize Zorn’s lemma is that of an upper bound: For a subset $(A,\preceq)$ of a partially ordered set $(X,\preceq)$, an element $u$ of $X$ is an *upper bound of* $A$ *in* $X$ iff $$(a\preceq u\in(X,\preceq))\textrm{ }(\textrm{for every }a\in(A,\preceq))\label{Eqn: upper bound}$$ which requires the upper bound $u$ to be larger than all members of $A$, with the corresponding lower bounds of $A$ being defined in a similar manner. Of course, it is again not necessary that the elements of $A$ be comparable to each other, and it should be clear from Eqs. (\[Eqn: maximum\]) and (\[Eqn: upper bound\]) that when an upper bound of a set is in the set itself, then it is the maximum element of the set. If the upper (lower) bounds of a subset $(A,\preceq)$ of a set $(X,\preceq)$ has a least (greatest) element, then this smallest upper bound (largest lower bound) is called *the* *least upper bound* (*greatest lower* *bound*) or *supremum* (*infimum*) *of $A$ in $X$*. Combining Eqs. (\[Eqn: maximum\]) and (\[Eqn: upper bound\]) then yields $$\begin{array}{rcl}
{\displaystyle \sup_{X}A} & = & \{ a_{\leftarrow}\in\Omega_{A}\!:a_{\leftarrow}\preceq u\textrm{ }\forall\textrm{ }u\in(\Omega_{A},\preceq)\}\\
{\displaystyle \inf_{X}A} & = & \{_{\rightarrow}a\in\Lambda_{A}\!:l\preceq\,_{\rightarrow}a\textrm{ }\forall\textrm{ }l\in(\Lambda_{A},\preceq)\}\end{array}\label{Eqn: supinf1}$$ where **$\Omega_{A}=\{\textrm{ }u\in X\!:(\forall\textrm{ }a\in A)(a\preceq u)\}$ **and **$\Lambda_{A}=\{ l\in X\!:(\forall\textrm{ }a\in A)(l\preceq a)\}$ **are the sets of all upper and lower bounds of $A$ in $X$*.* Equation (\[Eqn: supinf1\]) may be expressed in the equivalent but more transparent form as $$\begin{array}{c}
{\displaystyle a_{\leftarrow}={\displaystyle \sup_{X}A}\Longleftrightarrow(a\in A\Rightarrow a\preceq a_{\leftarrow})\wedge(a_{0}\prec a_{\leftarrow}\Rightarrow a_{0}\prec b\preceq a_{\leftarrow}\textrm{ for some }b\in A)}\\
_{\rightarrow}a={\displaystyle \inf_{X}A}\Longleftrightarrow(a\in A\Rightarrow\,_{\rightarrow}a\preceq a)\wedge(_{\rightarrow}a\prec a_{1}\Rightarrow\,_{\rightarrow}a\preceq b\prec a_{1}\textrm{ for some }b\in A)\end{array}\label{Eqn: supinf2}$$ to imply that *$a_{\leftarrow}$* ($_{\rightarrow}a$) is *the* upper (lower) bound of $A$ in $X$ which precedes (succeeds) every other upper (lower) bound of $A$ **in $X$. Notice that uniqueness in the definitions above is a direct consequence of the uniqueness of greatest and least elements of a set. **It must be noted that whereas maximal and maximum are properties of the particular subset and have nothing to do with anything outside it, upper and lower bounds of a set are defined only with respect to a superset that may contain it.
The following example, beside being useful in Zorn’s lemma, is also of great significance in fixing some of the basic ideas needed in our future arguments involving classes of sets ordered by the inclusion relation.
**Example 4.1.** Let $\mathcal{X}=\mathcal{P}(\{ a,b,c\})$ be ordered by the inclusion relation $\subseteq$. The subset $\mathcal{A}=\mathcal{P}(\{ a,b,c\})-\{ a,b,c\}$ has three maximals $\{ a,b\}$, $\{ b,c\}$ and $\{ c,a\}$ but no maximum as there is no $A_{\infty}\in\mathcal{A}$ satisfying $A\preceq A_{\infty}$ for every $A\in\mathcal{A}$, while $\mathcal{P}(\{ a,b,c\})-\emptyset$ the three minimals $\{ a\}$, $\{ b\}$, and $\{ c\}$ but no minimum. This shows that a subset of a partially ordered set may have many maximals (minimals) without possessing a maximum (minimum), but a subset has a maximum (minimum) iff this is its unique maximal (minimal). If $\mathcal{A}=\{\{ a,b\},\{ a,c\}\}$, then every subset of the intersection of the elements of $\mathcal{A}$, namely $\{ a\}$ and $\emptyset$, are lower bounds of $\mathcal{A}$, and all supersets in $\mathcal{X}$ of the union of its elements — which in this case is just $\{ a,b,c\}$ — are its upper bounds. Notice that while the maximal (minimal) and maximum (minimum) are elements of $\mathcal{A}$, upper and lower bounds need not be contained in their sets. In this class $(\mathcal{X},\subseteq)$ of subsets of a set $X$, $X_{+}$ is a maximal element of $\mathcal{X}$ iff $X_{+}$ is not contained in any other subset of $X$, while $X_{\infty}$ is a maximum of $\mathcal{X}$ iff $X_{\infty}$ contains every other subset of $X$.
Let $\mathcal{A}:=\{ A_{\alpha}\in\mathcal{X}\}_{\alpha\in\mathbb{D}}$ be a nonempty subclass of $(\mathcal{X},\subseteq)$, and suppose that both $\bigcup A_{\alpha}$ and $\bigcap A_{\alpha}$ are elements of $\mathcal{X}$. Since each $A_{\alpha}$ is $\subseteq$-less than $\bigcup A_{\alpha}$, it follows that $\bigcup A_{\alpha}$ is an upper bound of $\mathcal{A}$; this is also be the smallest of all such bounds because if $U$ is any other upper bound then every $A_{\alpha}$ must precede $U$ by Eq. (\[Eqn: upper bound\]) and therefore so must $\bigcup A_{\alpha}$ (because the union of a class of subsets of a set is the smallest that contain each member of the class: $A_{\alpha}\subseteq U\Rightarrow\bigcup A_{\alpha}\subseteq U$ for subsets $(A_{\alpha})$ and $U$ of $X$). Analogously, since $\bigcap A_{\alpha}$ is $\subseteq$-less than each $A_{\alpha}$ it is a lower bound of $\mathcal{A}$; that it is the greatest of all the lower bounds $L$ in $\mathcal{X}$ follows because the intersection of a class of subsets is the largest that is contained in each of the subsets: $L\subseteq A_{\alpha}\Rightarrow L\subseteq\bigcap A_{\alpha}$ for subsets $L$ and $(A_{\alpha})$ of $X$. Hence the supremum and infimum of $\mathcal{A}$ in $(\mathcal{X},\subseteq)$ given by $$A_{\leftarrow}=\sup_{(\mathcal{X},\subseteq)}\mathcal{A}=\bigcup_{A\in\mathcal{A}}A\qquad\textrm{and}\qquad_{\rightarrow}A=\inf_{(\mathcal{X},\subseteq)}\mathcal{A}=\bigcap_{A\in\mathcal{A}}A\label{Eqn: supinf3}$$ are both elements of $(\mathcal{X},\subseteq)$. Intuitively, an upper (respectively, lower) bound of $\mathcal{A}$ in $\mathcal{X}$ is any subset of $\mathcal{X}$ that contains (respectively, is contained in) every member of $\mathcal{A}$.$\qquad\blacksquare$
The statement of Zorn’s lemma and its proof can now be completed in three stages as follows. For Theorem 4.1 below that constitutes the most significant technical first stage, let $g$ be a function on $(X,\preceq)$ that assigns to every $x\in X$ an *immediate successor* $y\in X$ such that $${\textstyle \mathscr{M}(x)=\{\textrm{ }y\succ x\!:\not\exists\textrm{ }x_{*}\in X\textrm{ satisfying }x\prec x_{*}\prec y\}}$$ are all the successors of $x$ in $X$ with no element of $X$ lying strictly between $x$ and $y$. Select a representative of $\mathscr{M}(x)$ by a choice function $f_{\textrm{C}}$ such that $$g(x)=f_{\textrm{C}}(\mathscr{M}(x))\in\mathscr{M}(x)$$ is an immediate successor of $x$ chosen from the many possible in the set $\mathscr{M}(x)$. The basic idea in the proof of the first of the three-parts is to express the existence of a maximal element of a partially ordered set $X$ in terms of the existence of a fixed point in the set, which follows as a contradiction of the assumed hypothesis that every point in $X$ has an immediate successor. Our basic application of immediate successors in the following will be to classes $\mathcal{X}\subseteq(\mathcal{P}(X),\subseteq)$ of subsets of a set $X$ ordered by inclusion. In this case for any $A\in\mathcal{X}$, the function $g$ can be taken to be the superset $${\textstyle g(A)=A\bigcup f_{\textrm{C}}(\mathscr{G}(A)),\quad\textrm{where }\mathscr{G}(A)=\{ x\in X-A\!:A\bigcup\{ x\}\in\mathcal{X}\}}\label{Eqn: FilterTower}$$ of $A$. Repeated application of $g$ to $A$ then generates a principal filter, and hence an associated sequence, based at $A$.
**Theorem 4.1.** *Let $(X,\preceq)$ be a partially ordered set that satisfies*
(ST1) *There is a smallest element $x_{0}$ of $X$ which has no immediate predecessor in $X$.*
(ST2) *If $C\subseteq X$ is a totally ordered subset in $X$, then $c_{*}=\sup_{X}C$ is in $X$.*
*Then there exists a maximal element $x_{+}$ of $X$ which has no immediate successor in $X$.*$\qquad\square$
**Proof.** Let $T\subseteq(X,\preceq)$ be a subset of $X$. If the conclusion of the theorem is false then the alternative
(ST3) *Every element $x\in T$ has an immediate successor $g(x)$ in $T$*[^22]
leads, as shown below, to a contradiction that can be resolved only by the conclusion of the theorem. A subset $T$ of $(X,\preceq)$ satisfying conditions (ST1)$-$(ST3) is sometimes known as an $g$*-tower* or an $g$*-sequence:* an obvious example of a tower is $(X,\preceq)$ itself. If $${\textstyle _{\rightarrow}T=\bigcap\{ T\in\mathcal{T}\!:T\textrm{ is an }x_{0}-\textrm{tower}\}}$$ is the $(\mathcal{P}(X),\subseteq)$-infimum of the class $\mathcal{T}$ of all sequential towers of $(X,\preceq)$, we show that this smallest sequential **tower is infact a *sequential totally ordered chain* in $(X,\preceq)$ built from $x_{0}$ by the $g$-function. Let the subset $$C_{\textrm{T}}=\{ c\in X\!:(\forall t\in\,_{\rightarrow}T)(t\preceq c\vee c\preceq t)\}\subseteq X\label{Eqn: tower-chain}$$ of $X$ be an $g$-chain in $_{\rightarrow}T$ in the sense that (cf. Eq. (\[Eqn: chain\])) it is that subset of $X$ each of whose elements is comparable with some element of $_{\rightarrow}T$. The conditions (ST1)$-$(ST3) for $C_{\textrm{T}}$ can be verified as follows to demonstrate that $C_{\textrm{T}}$ is an $g$-tower.
\(1) $x_{0}\in C_{\textrm{T}}$, because it is less than each $x\in\,_{\rightarrow}T$.
\(2) Let $c_{\leftarrow}=\sup_{X}C_{\textrm{T}}$ be the supremum of the chain $C_{\textrm{T}}$ in $X$ so that by (ST2), $c_{\leftarrow}\in X$. Let $t\in\,_{\rightarrow}T$. If there is *some* $c\in C_{\textrm{T}}$ such that $t\preceq c$, then surely $t\preceq c_{\leftarrow}$. Else, $c\preceq t$ for *every* $c\in C_{\textrm{T}}$ shows that $c_{\leftarrow}\preceq t$ because $c_{\leftarrow}$ is the smallest of all the upper bounds $t$ of $C_{\textrm{T}}$. Therefore $c_{\leftarrow}\in C_{\textrm{T}}$.
\(3) In order to show that $g(c)\in C$ whenever $c\in C$ it needs to verified that for all $t\in\,_{\rightarrow}T$, either $t\preceq c\Rightarrow t\preceq g(c)$ or $c\preceq t\Rightarrow g(c)\preceq t$. As the former is clearly obvious, we investigate the later as follows; note that $g(t)\in\,_{\rightarrow}T$ by (ST3). The first step is to show that the subset $$C_{g}=\{ t\in\,_{\rightarrow}T\!:(\forall c\in C_{\textrm{T}})(t\preceq c\vee g(c)\preceq t)\}\label{Eqn: chain_g}$$ of $_{\rightarrow}T$, which is a chain in $X$ (observe the inverse roles of $t$ and $c$ here as compared to that in Eq. (\[Eqn: tower-chain\])), is a tower: Let $t_{\leftarrow}$ be the supremum of $C_{g}$ and take $c\in C$. If there is *some* $t\in C_{g}$ for which $g(c)\preceq t$, then clearly $g(c)\preceq t_{\leftarrow}$. Else, $t\preceq x$ for *each* $t\in C_{g}$ shows that $t_{\leftarrow}\preceq c$ because $t_{\leftarrow}$ is the smallest of all the upper bounds $c$ of $C_{g}$. Hence $t_{\leftarrow}\in C_{g}$.
Property (ST3) for $C_{g}$ follows from a small yet significant modification of the above arguments in which the immediate successors $g(t)$ of $t\in C_{g}$ formally replaces the supremum $t_{\leftarrow}$ of $C_{g}$. Thus given a $c\in C$, if there is *some* $t\in C_{g}$ for which $g(c)\preceq t$ then $g(c)\prec g(t)$; this combined with $(c=t)\Rightarrow(g(c)=g(t))$ yields $g(c)\preceq g(t)$. On the other hand, $t\prec c$ for *every* $t\in C_{g}$ requires $g(t)\preceq c$ as otherwise $(t\prec c)\Rightarrow(c\prec g(t))$ would, from the resulting consequence $t\prec c\prec g(t)$, contradict the assumed hypothesis that $g(t)$ is the immediate successor of $t$. Hence, $C_{g}$ is a $g$-tower in $X$.
To complete the proof that $g(c)\in C_{\textrm{T}}$, and thereby the argument that $C_{\textrm{T}}$ is a tower, we first note that as $_{\rightarrow}T$ is the smallest tower and $C_{g}$ is built from it, $C_{g}=\,_{\rightarrow}T$ must infact be $_{\rightarrow}T$ itself. From Eq. (\[Eqn: chain\_g\]) therefore, for every $t\in\,_{\rightarrow}T$ either $t\preceq g(c)$ or $g(c)\preceq t$, so that $g(c)\in C_{\textrm{T}}$ whenever $c\in C_{\textrm{T}}$. This concludes the proof that $C_{\textrm{T}}$ is actually the tower $_{\rightarrow}T$ in $X$.
From (ST2), the implication of the chain $C_{\textrm{T}}$ $$C_{\textrm{T}}=\,_{\rightarrow}T=C_{g}\label{Eqn: ChainedTower}$$ being the minimal tower $_{\rightarrow}T$ is that the supremum $t_{\leftarrow}$ of the totally ordered $_{\rightarrow}T$ *in its own tower* (as distinct from in the tower $X$: recall that $_{\rightarrow}T$ is a subset of $X$) must be contained in itself, that is $$\sup_{C_{\textrm{T}}}(C_{\textrm{T}})=t_{\leftarrow}\in\,_{\rightarrow}T\subseteq X.\label{Eqn: sup chain}$$ This however leads to the contradiction from (ST3) that $g(t_{\leftarrow})$ be an element of $_{\rightarrow}T$, unless of course
$$g(t_{\leftarrow})=t_{\leftarrow},\label{Eqn: fixed point}$$
which because of (\[Eqn: ChainedTower\]) may also be expressed equivalently as $g(c_{\leftarrow})=c_{\leftarrow}\in C_{\textrm{T}}$. As the sequential totally ordered set $_{\rightarrow}T$ is a subset of $X$, Eq. (48) implies that $t_{\leftarrow}$ is a maximal element of $X$ which allows (ST3) to be replaced by the remarkable inverse criterion that
$(\textrm{ST}3^{\prime})$ If $x\in X$ and $w$ precedes $x,$ $w\prec x$, then $w\in X$
that is obviously false for a general tower $T$. In fact, it follows directly from Eq. (\[Eqn: maximal\]) that under $(\textrm{ST}3^{\prime})$ *any $x_{+}\in X$ is a maximal element of $X$ iff it is a fixed point of $g$* as given by Eq. (\[Eqn: fixed point\]). This proves the theorem and also demonstrates how, starting from a minimum element of a partially ordered set $X$, (ST3) can be used to generate inductively a totally ordered sequential subset of $X$ *leading to a maximal $x_{+}=c_{\leftarrow}\in(X,\preceq)$ that is a fixed point of the generating function $g$* *whenever the supremum* $t_{\leftarrow}$ *of the chain $_{\rightarrow}T$ is in* $X$.$\qquad\blacksquare$
**Remarks.** The proof of this theorem, despite its apparent length and technically involved character, carries the highly significant underlying message that
[0.1in]{} *Any inductive sequential $g$-construction of an infinite chained tower* $C_{\textrm{T}}$ *starting with a smallest element $x_{0}\in(X,\preceq)$ such that a supremum $c_{\leftarrow}$ of the $g$-generated sequential chain* $C_{\textrm{T}}$ *in its own tower is contained in itself, must necessarily terminate with a fixed point relation of the type* (\[Eqn: fixed point\]) *with respect to the supremum. Note from Eqs. (\[Eqn: sup chain\]) and (\[Eqn: fixed point\]) that the role of* (ST2) *applied to a fully ordered tower is the identification of the maximal of the tower — which depends only the tower and has nothing to do with anything outside it — with its supremum that depends both on the tower and its complement.*
Thus although purely set-theoretic in nature, the filter-base associated with a sequentially totally ordered set may be interpreted to lead to the usual notions of adherence and convergence of filters and thereby of a generated topology for $(X,\preceq)$, see Appendix A1 and Example A1.3. This very significant apparent inter-relation between topologies, filters and orderings will form the basis of our approach to the condition of maximal ill-posedness for chaos.
In the second stage of the three stage programme leading to Zorn’s lemma, the tower Theorem 4.1 and the comments of the preceding paragraph are applied at one higher level to a very special class of the power set of a set, the class of all the chains of a partially ordered set, to directly lead to the physically significant
**Theorem 4.2.** **Hausdorff Maximal Principle.** *Every partially ordered set $(X,\preceq)$ has a maximal totally ordered subset*.[^23]$\qquad\square$
**Proof.** Here the base level is $$\mathcal{X}=\{ C\in\mathcal{P}(X)\!:C\textrm{ is a chain in }(X,\preceq)\}\subseteq\mathcal{P}(X)\label{Eqn: Hausdorff}$$ be the set of all the totally ordered subsets of $(X,\preceq)$. Since $\mathcal{X}$ is a collection of (sub)sets of $X$, we order it by the inclusion relation on $\mathcal{X}$ and use the tower Theorem to demonstrate that $(\mathcal{X},\subseteq)$ has a maximal element $C_{\leftarrow}$, which by the definition of $\mathcal{X}$, is the required maximal chain in $(X,\preceq)$.
Let $\mathcal{C}$ be a chain in $\mathcal{X}$ of the chains in $(X,\preceq)$. In order to apply the tower Theorem to $(\mathcal{X},\subseteq)$ we need to verify hypothesis (ST2) that the smallest $$C_{*}=\sup_{\mathcal{X}}\mathcal{C}=\bigcup_{C\in\mathcal{C}}C\label{Eqn: HausdorffChain}$$ of the possible upper bounds of $\mathcal{C}$ (see Eq. (\[Eqn: supinf3\])) is a chain of $(X,\preceq)$. Indeed, if $x_{1},x_{2}\in X$ are two points of $C_{\textrm{sup}}$ with $x_{1}\in C_{1}$ and $x_{2}\in C_{2}$, then from the $\subseteq$-comparability of $C_{1}$ and $C_{2}$ we may choose $x_{1},x_{2}\in C_{1}\supseteq C_{2}$, say. Thus $x_{1}$ and $x_{2}$ are $\preceq$-comparable as $C_{1}$ is a chain in $(X,\preceq)$; $C_{*}\in\mathcal{X}$ is therefore a chain in $(X,\preceq)$ which establishes that the supremum of a chain of $(\mathcal{X},\subseteq)$ is a chain in $(X,\preceq)$.
The tower Theorem 4.1 can now applied to $(\mathcal{X},\subseteq)$ with $C_{0}$ as its smallest element to construct a $g$-sequentially towered fully ordered subset of $\mathcal{X}$ consisting of chains in $X$ $$\mathcal{C}_{\textrm{T}}=\{ C_{i}\in\mathcal{P}(X)\!:C_{i}\subseteq C_{j}\textrm{ for }i\leq j\in\mathbb{N}\}=\,_{\rightarrow}\mathcal{T}\subseteq\mathcal{P}(X)$$ of $(\mathcal{X},\subseteq)$ — consisting of the common elements of all $g$-sequential towers $\mathcal{T}\in\mathfrak{T}$ of $(\mathcal{X},\subseteq)$ — that infact is a principal filter base of chained subsets of $(X,\preceq)$ at $C_{0}$. The supremum (chain in $X$) $C_{\leftarrow}$ of $\mathcal{C}_{\textrm{T}}$ in $\mathcal{C}_{\textrm{T}}$ must now satisfy, by Thm. 4.1, the fixed point $g$-chain of $X$ $$\sup_{\mathcal{C}_{\textrm{T}}}(\mathcal{C}_{\textrm{T}})=C_{\leftarrow}=g(C_{\leftarrow})\in\mathcal{C}_{\textrm{T}}\subseteq\mathcal{P}(X),$$ where the chain $g(C)=C\bigcup f_{\textrm{C}}(\mathscr{G}(C)-C)$ with $\mathscr{G}(C)=\{ x\in X-C\!:C\bigcup\{ x\}\in\mathcal{X}\}$, is an immediate successor of $C$ obtained by choosing one point $x=f_{\textrm{C}}(\mathscr{G}(C)-C)$ from the many possible in $\mathscr{G}(C)-C$ such that the resulting $g(C)=C\bigcup\{ x\}$ is a strict successor of the chain $C$ with no others lying between it and $C$. Note that $C_{\leftarrow}\in(\mathcal{X},\subseteq)$ is only one of the many maximal fully ordered subsets possible in $(X,\preceq)$.$\qquad\blacksquare$
With the assurance of the existence of a maximal chain $C_{\leftarrow}$ among all fully ordered subsets of a partially ordered set $(X,\preceq)$, the arguments are completed by returning to the basic level of $X$.
**Theorem 4.3. Zorn’s Lemma.** *Let $(X,\preceq)$ be a partially ordered set such that every totally ordered subset of $X$ has an upper bound in $X$. Then $X$ has at least one maximal element with respect to its order.$\qquad\square$*
**Proof.** The proof of this final part is a mere application of the Hausdorff Maximal Principle on the existence of a maximal chain $C_{\leftarrow}$ in $X$ to the hypothesis of this theorem that $C_{\leftarrow}$ has an upper bound $u$ in $X$ that quickly leads to the identification of this bound as a maximal element $x_{+}$ of $X$. Indeed, if there is an element $v\in X$ that is comparable to $u$ and $v\not\preceq u$, then $v$ cannot be in $C_{\leftarrow}$ as it is necessary for every $x\in C_{\leftarrow}$ to satisfy $x\preceq u$. Clearly then $C_{\leftarrow}\bigcup\{ v\}$ is a chain in $(X,\preceq)$ bigger than $C_{\leftarrow}$ which contradicts the assumed maximality of $C_{\leftarrow}$ among the chains of $X$.$\qquad\blacksquare$
The sequence of steps leading to Zorn’s Lemma, and thence to the maximal of a partially ordered set, is summarised in Fig. \[Fig: Zorn\].
[(a) The one-level higher subset $\mathcal{X}=\{ C\in\mathcal{P}(X)\!:C\textrm{ is a chain in }(X,\preceq)\}$ of $\mathcal{P}(X)$ consisting of all the totally ordered subsets of $(X,\preceq)$, ]{}
[(b) The smallest common $g$-sequential totally ordered towered chain $\mathcal{C}_{\textrm{T}}=\{ C_{i}\in\mathcal{P}(X)\!:C_{i}\subseteq C_{j}\textrm{ for }i\leq j\}\subseteq\mathcal{P}(X)$ of all sequential $g$-towers of $\mathcal{X}$ by Thm. 4.1, which infact is a principal filter base of totally ordered subsets of $(X,\preceq)$ at the smallest element $C_{0}$. ]{}
[(c) Apply Hausdorff Maximal Principle to $(\mathcal{X},\subseteq)$ to get the subset $\sup_{\mathcal{C}_{\textrm{T}}}(\mathcal{C}_{\textrm{T}})=C_{\leftarrow}=g(C_{\leftarrow})\in\mathcal{C}_{\textrm{T}}\subseteq\mathcal{P}(X)$ of $(X,\preceq)$ as the supremum of $(\mathcal{X},\subseteq)$ in $\mathcal{C}_{\textrm{T}}$. The identification of this supremum as a maximal element of $(\mathcal{X},\subseteq)$ is a consequence of (ST2) and Eqs. (\[Eqn: sup chain\]), (\[Eqn: fixed point\]) that actually puts the supremum into $\mathcal{X}$ itself. ]{}
[By returning to the original level $(X,\preceq)$ ]{}
[(d) Zorn’s Lemma finally yields the required maximal element $u\in X$ as an upper bound of the maximal totally ordered subset $(C_{\leftarrow},\preceq)$ of $(X,\preceq)$. ]{}
[The dashed segment denotes the higher Hausdorff $(\mathcal{X},\subseteq)$ level leading to the base $(X,\preceq)$ Zorn level. ]{}
The three examples below of the application of Zorn’s Lemma clearly reflect the increasing complexity of the problem considered, with the maximals a point, a subset, and a set of subsets of $X$, so that these are elements of $X$, $\mathcal{P}(X)$, and $\mathcal{P}^{2}(X)$ respectively.
**Example 4.2.** (1) Let $X=(\{ a,b,c\},\preceq)$ be a three-point base-level ground set ordered lexicographically, that is $a\prec b\prec c$. A chain $\mathcal{C}$ of the partially ordered Hausdorff-level set $\mathcal{X}$ consisting of subsets of $X$ given by Eq. (\[Eqn: Hausdorff\]) is, for example, $\{\{ a\},\{ a,b\}\}$ and the six $g$-sequential chained towers
$$\begin{array}{c}
\mathcal{C}_{1}=\{\emptyset,\{ a\},\{ a,b\},\{ a,b,c\}\},\qquad\mathcal{C}_{2}=\{\emptyset,\{ a\},\{ a,c\},\{ a,b,c\}\}\\
\mathcal{C}_{3}=\{\emptyset,\{ b\},\{ a,b\},\{ a,b,c\}\},\qquad\mathcal{C}_{4}=\{\emptyset,\{ b\},\{ b,c\},\{ a,b,c\}\}\\
\mathcal{C}_{5}=\{\emptyset,\{ c\},\{ a,c\},\{ a,b,c\}\},\qquad\mathcal{C}_{6}=\{\emptyset,\{ c\},\{ b,c\},\{ a,b,c\}\}\end{array}$$
built from the smallest element $\emptyset$ corresponding to the six distinct ways of reaching $\{ a,b,c\}$ from $\emptyset$ along the sides of the cube marked on the figure with solid lines, all belong to $\mathcal{X}$; see Fig. \[Fig: order\](b). An example of a tower in $(\mathcal{X},\subseteq)$ which is not a chain is $$\mathcal{T}=\{\emptyset,\{ a\},\{ b\},\{ c\},\{ a,b\},\{ a,c\},\{ b,c\},\{ a,b,c\}\}.$$ Hence the common infimum towered chained subset $$\mathcal{C}_{\textrm{T}}=\{\emptyset,\{ a,b,c\}\}=\,_{\rightarrow}\mathcal{T}\subseteq\mathcal{P}(X)$$ of $\mathcal{X}$, with $$\sup_{\mathcal{C}_{\textrm{T}}}(\mathcal{C}_{\textrm{T}})=C_{\leftarrow}=\{ a,b,c\}=g(C_{\leftarrow})\in\mathcal{C}_{\textrm{T}}\subseteq\mathcal{P}(X)$$ the only maximal element of $\mathcal{P}(X)$. Zorn’s Lemma now assures the existence of a maximal element of $c\in X$. Observe how the maximal element of $(X,\preceq)$ is obtained by going one level higher to $\mathcal{X}$ at the Hausdorff stage and returning to the base level $X$ at Zorn, see Fig. \[Fig: Zorn\] for a schematic summary of this sequence of steps.
\(2) *Basis of a vector space.* A linearly independent set of vectors in a vector space $X$ that spans the space is known as the Hamel basis of $X$. To prove the existence of a Hamel basis in a vector space, Zorn’s lemma is invoked as follows.
The ground base level of the linearly independent subsets of $X$ $$\mathcal{X}=\{\{ x_{i_{j}}\}_{j=1}^{J}\in\mathcal{P}(X)\!:\textrm{Span}(\{ x_{i_{j}}\}_{j=1}^{J})=0\Rightarrow(\alpha_{j})_{j=1}^{J}=0\,\forall J\geq1\}\subseteq\mathcal{P}(X)),$$ with $\textrm{Span}(\{ x_{i_{j}}\}_{j=1}^{J}):=\sum_{j=1}^{J}\alpha_{j}x_{i_{j}}$, is such that no $x\in\mathcal{X}$ can be expressed as a linear combination of the elements of $\mathcal{X}-\{ x\}$. $\mathcal{X}$ clearly has a smallest element, say $\{ x_{i_{1}}\}$, for some non-zero $x_{i_{1}}\in X$. Let the higher Hausdorff level $$\mathfrak{X}=\{\mathcal{C}\in\mathcal{P}^{2}(X)\!:\mathcal{C}\textrm{ is a chain in }(\mathcal{X},\subseteq)\}\subseteq\mathcal{P}^{2}(X)$$ collection of the chains $$\mathcal{C}_{i_{K}}=\{\{ x_{i_{1}}\},\{ x_{i_{1}},x_{i_{2}}\},\cdots,\{ x_{i_{1}},x_{i_{2}},\cdots,x_{i_{K}}\}\}\textrm{ }\in\mathcal{P}^{2}(X)$$ of $\mathcal{X}$ comprising linearly independent subsets of $X$ be $g$-built from the smallest $\{ x_{i_{1}}\}$. Any chain $\mathfrak{C}$ of $\mathfrak{X}$ is bounded above by the union $\mathcal{C}_{*}=\sup_{\mathfrak{X}}\mathfrak{C}=\bigcup_{\mathcal{C}\in\mathfrak{C}}\mathcal{C}$ which is a chain in $\mathcal{X}$ containing $\{ x_{i_{1}}\}$, thereby verifying (ST2) for $\mathfrak{X}$. Application of the tower theorem to $\mathfrak{X}$ implies that the chain $$\mathfrak{C}_{\textrm{T}}=\{\mathcal{C}_{i_{1}},\mathcal{C}_{i_{2}},\cdots,\mathcal{C}_{i_{n}},\cdots\}=\,_{\rightarrow}\mathfrak{T}\subseteq\mathcal{P}^{2}(X)$$ in $\mathfrak{X}$ of chains of $\mathcal{X}$ is a $g$-sequential fully ordered towered subset of $(\mathfrak{X},\subseteq)$ consisting of the common elements of all $g$-sequential towers of $(\mathfrak{X},\subseteq)$, that infact is a *chained* *principal ultrafilter on $(\mathcal{P}(X),\subseteq)$ generated by the filter-base $\{\{\{ x_{i_{1}}\}\}\}$* *at $\{ x_{i_{1}}\}$*, where $$\mathfrak{T}=\{\mathcal{C}_{i_{1}},\mathcal{C}_{i_{2}},\cdots,\mathcal{C}_{j_{n}},\mathcal{C}_{j_{n+1}},\cdots\}$$ for some $n\in\mathbb{N}$ is an example of non-chained $g$-tower whenever $(\mathcal{C}_{j_{k}})_{k=n}^{\infty}$ is neither contained in nor contains any member of the $(\mathcal{C}_{i_{k}})_{k=1}^{\infty}$ chain. Hausdorff’s chain theorem now yields the fixed-point $g$-chain $\mathcal{C}_{\leftarrow}\,\in\mathfrak{X}$ of $\mathcal{X}$ $$\sup_{\mathfrak{C}_{\textrm{T}}}(\mathfrak{C}_{\textrm{T}})=\mathcal{C}_{\leftarrow}=\{\{ x_{i_{1}}\},\{ x_{i_{1}},x_{i_{2}}\},\{ x_{i_{1}},x_{i_{2}},x_{i_{3}}\},\cdots\}=g(\mathcal{C}_{\leftarrow})\in\mathfrak{C}_{\textrm{T}}\subseteq\mathcal{P}^{2}(X)$$ as a maximal *totally ordered* *principal filter on $X$ that is generated by the filter-base $\{\{ x_{i_{1}}\}\}$* *at $x_{i_{1}}$*, whose supremum $B=\{ x_{i_{1}},x_{i_{2}},\cdots\}\in\mathcal{P}(X)$ is, by Zorn’s lemma, a maximal element of the base level $\mathcal{X}$. This maximal linearly independent subset of $X$ is the required Hamel basis for $X$: Indeed, if the span of $B$ is not the whole of $X$, then $\textrm{Span}(B)\bigcup x$, with $x\notin\textrm{Span}(B)$ would, by definition, be a linearly independent set of $X$ strictly larger than $B$, contradicting the assumed maximality of the later. It needs to be understood that since the infinite basis cannot be classified as being linearly independent, we have here an important example of the supremum of the maximal chained set not belonging to the set even though this criterion was explicitly used in the construction process according to (ST2) and (ST3).
Compared to this purely algebraic concept of basis in a vector space, is the Schauder basis in a normed space which combines topological structure with the linear in the form of convergence: If a normed vector space contains a sequence $(e_{i})_{i\in\mathbb{Z}_{+}}$ with the property that for every $x\in X$ there is an unique sequence of scalars $(\alpha_{i})_{i\in\mathbb{Z}_{+}}$ such that the remainder $\parallel x-(\alpha_{1}e_{1}+\alpha_{2}e_{2}+\cdots+\alpha_{I}e_{I})\parallel$ approaches $0$ as $I\rightarrow\infty$, then the collection $(e_{i})$ is known as a Schauder basis for $X$.
\(3) *Ultrafilter.* Let $X$ be a set. The set $${\textstyle _{\textrm{F}}\mathcal{S}=\{ S_{\alpha}\in\mathcal{P}(X)\!:S_{\alpha}\bigcap S_{\beta}\neq\emptyset,\textrm{ }\forall\alpha\neq\beta\}\subseteq\mathcal{P}(X)}$$ of all nonempty subsets of $X$ with finite intersection property is known as a *filter subbase on* $X$ and $_{\textrm{F}}\mathcal{B}=\{ B\subseteq X\!:B=\bigcap_{i\in I\subset\mathbb{D}}S_{i}\}$, for $I\subset\mathbb{D}$ a finite subset of a directed set $\mathbb{D}$, is a *filter-base on $X$* *associated with the subbase* $_{\textrm{F}}\mathcal{S}$; cf. Appendix A1. Then the *filter generated by* $_{\textrm{F}}\mathcal{S}$ consisting of every superset of the finite intersections $B\in\,_{\textrm{F}}\mathcal{B}$ of sets of $_{\textrm{F}}\mathcal{S}$ is the smallest filter that contain the subbase $_{\textrm{F}}\mathcal{S}$ and base $_{\textrm{F}}\mathcal{B}$. For notational simplicity, we will denote the subbase $_{\textrm{F}}\mathcal{S}$ in the rest of this example simply by $\mathcal{S}$.
Consider the base-level ground set of all filter subbases on $X$ $$\mathfrak{S}=\{\mathcal{S}\in\mathcal{P}^{2}(X)\!:\bigcap_{\emptyset\neq\mathcal{R}\subseteq\mathcal{S}}\mathcal{R}\neq\emptyset\textrm{ for every finite subset of }\mathcal{S}\}\subseteq\mathcal{P}^{2}(X),$$ ordered by inclusion in the sense that $\mathcal{S}_{\alpha}\subseteq\mathcal{S}_{\beta}\textrm{ for all }\alpha\preceq\beta\in\mathbb{D}$, and let the higher Hausdorff-level $$\widetilde{\mathfrak{X}}=\{\mathfrak{C}\in\mathcal{P}^{3}(X)\!:\mathfrak{C}\textrm{ is a chain in }(\mathfrak{S},\subseteq)\}\subseteq\mathcal{P}^{3}(X)$$ comprising the collection of the totally ordered chains $$\mathfrak{C}_{\kappa}=\{\{ S_{\alpha}\},\{ S_{\alpha},S_{\beta}\},\cdots,\{ S_{\alpha},S_{\beta},\cdots,S_{\kappa}\}\}\in\mathcal{P}^{3}(X)$$ of $\mathfrak{S}$ be $g$-built from the smallest $\{ S_{\alpha}\}$ then an *ultrafilter* on $X$ is a maximal member $\mathcal{S}_{+}$ of $(\mathfrak{S},\subseteq)$ in the usual sense that any subbase $\mathcal{S}$ on $X$ must necessarily be contained in $\mathcal{S}_{+}$ so that $\mathcal{S}_{+}\subseteq\mathcal{S}\Rightarrow\mathcal{S}=\mathcal{S}_{+}$ for any $\mathcal{S}\subseteq\mathcal{P}(X)$ with FIP. The tower theorem now implies that the element $$\widetilde{\mathfrak{C}_{\textrm{T}}}=\{\mathfrak{C}_{\alpha},\mathfrak{C}_{\beta},\cdots,\mathfrak{C}_{\nu},\cdots\}=\,\widetilde{_{\rightarrow}\mathfrak{T}}\subseteq\mathcal{P}^{3}(X)$$ of $\mathcal{P}^{4}(X)$, which is a chain in $\widetilde{\mathfrak{X}}$ of the chains of $\mathfrak{S}$, is a $g$-sequential fully ordered towered subset of the common elements of all sequential towers of $(\widetilde{\mathfrak{X}},\subseteq)$ and a *chained* *principal ultrafilter on $(\mathcal{P}^{2}(X),\subseteq)$ generated by the filter-base $\{\{\{ S_{\alpha}\}\}\}$* *at* $\{ S_{\alpha}\}$; here $$\widetilde{\mathfrak{T}}=\{\mathfrak{C}_{\alpha},\mathfrak{C}_{\beta},\cdots,\mathfrak{C}_{\sigma},\mathfrak{C}_{\varsigma},\cdots\},$$ is an obvious example of non-chained $g$-tower whenever $(\mathfrak{C}_{\sigma})$ is neither contained in, nor contains, any member of the $\mathfrak{C}_{\alpha}$-chain. Hausdorff’s chain theorem now yields the fixed-point $\widetilde{\mathfrak{C}_{\leftarrow}}\,\in\widetilde{\mathfrak{X}}$ $$\sup_{\widetilde{\mathfrak{C}_{\textrm{T}}}}(\widetilde{\mathfrak{C}_{\textrm{T}}})=\widetilde{\mathfrak{C}_{\leftarrow}}=\{\{ S_{\alpha}\},\{ S_{\alpha},S_{\beta}\},\{ S_{\alpha},S_{\beta},S_{\gamma}\},\cdots\}=g(\widetilde{\mathfrak{C}_{\leftarrow}})\in\widetilde{\mathfrak{C}_{\textrm{T}}}\subseteq\mathcal{P}^{3}(X)$$ as a maximal *totally ordered* $g$-chained towered subset of $X$ that is, by Zorn’s lemma, a maximal element of the base level subset $\mathfrak{S}$ of $\mathcal{P}^{2}(X)$. $\widetilde{\mathfrak{C}_{\leftarrow}}$ is a *chained principal ultrafilter on* $(\mathcal{P}(X),\subseteq)$ *generated by the filter-base $\{\{ S_{\alpha}\}\}$* *at $S_{\alpha}$*, while $\mathcal{S}_{+}=\{ S_{\alpha},S_{\beta},S_{\gamma},\cdots\}\in\mathcal{P}^{2}(X)$ is an (non-principal) *ultrafilter on* $X$ — characterized by the property that any collection of subsets on $X$ with FIP (that is any filter subbase on $X$) must be contained the maximal set $\mathcal{S}_{+}$ having FIP — that is not a principal filter unless $\mathcal{S}_{\alpha}$ is a singleton set $\{ x_{\alpha}\}$. $\qquad\blacksquare$
[1.4]{} What emerges from these application of Zorn’s Lemma is the remarkable fact that *infinities (the dot-dot-dots) can be formally introduced as “limiting cases” of finite systems in a purely set-theoretic context* *without the need for topologies, metrics or convergences.* The significance of this observation will become clear from our discussions on filters and topology leading to Sec. 4.2 below. Also, the observation on the successive iterates of the power sets $\mathcal{P}(X)$ in the examples above was to suggest their anticipated role in the complex evolution of a dynamical system that is expected to play a significant part in our future interpretation and understanding of this adaptive and self-organizing phenomenon of nature.
***End Tutorial5***
From the examples in Tutorial5, it should be clear that the sequential steps summarized in Fig. \[Fig: Zorn\] are involved in an application of Zorn’s lemma to show that a partially ordered set has a maximal element with respect to its order. Thus for a partially ordered set $(X,\preceq)$, form the set $\mathcal{X}$ of all chains $C$ in $X$. If $C_{+}$ is a maximal chain of $X$ obtained by the Hausdorff Maximal Principle from the chain $\mathcal{C}$ of all chains of $X$, then its supremum $u$ is a maximal element of $(X,\preceq)$. This sequence is now applied, paralleling Example 4.2(1), to the set of arbitrary relations $\textrm{Multi}(X)$ on an infinite set $X$ in order to formulate our definition of chaos that follows.
Let $f$ be a *noninjective map* in $\textrm{Multi}(X)$ and $P(f)$ the number of injective branches of $f$. Denote by $$F=\{ f\in\textrm{Multi}(X)\!:f\textrm{ is a noninjective function on }X\}\subseteq\textrm{Multi}(X)$$ the resulting basic collection of noninjective functions in $\textrm{Multi}(X)$.
\(i) For every $\alpha$ in some directed set $\mathbb{D}$, let $F$ have the extension property $$(\forall f_{\alpha}\in F)(\exists f_{\beta}\in F)\!:P(f_{\alpha})\leq P(f_{\beta})$$
\(ii) Let a partial order $\preceq$ on $\textrm{Multi}(X)$ be defined, for $f_{\alpha},f_{\beta}\in\textrm{Map}(X)\subseteq\textrm{Multi}(X)$ by $$P(f_{\alpha})\leq P(f_{\beta})\Longleftrightarrow f_{\alpha}\preceq f_{\beta},\label{Eqn: chaos1}$$
with $P(f):=1$ for the smallest $f$, define a partially ordered subset $(F,\preceq)$ of $\textrm{Multi}(X)$. This is actually a preorder on $\textrm{Multi}(X)$ in which functions with the same number of injective branches are equivalent to each other.
\(iii) Let $$C_{\nu}=\{ f_{\alpha}\in\textrm{Multi}(X)\!:f_{\alpha}\preceq f_{\nu}\}\in\mathcal{P}(F),\qquad\nu\in\mathbb{D},$$ be the $g$-chains of non-injective functions of $\textrm{Multi}(X)$ and $$\mathcal{X}=\{ C\in\mathcal{P}(F)\!:C\textrm{ is a chain in }(F,\preceq)\}\subseteq\mathcal{P}(F)$$ denote the corresponding Hausdorff level of the chains of $F$, with $$\mathcal{C}_{\textrm{T}}=\{ C_{\alpha},C_{\beta},\cdots,C_{\nu},\cdots\}=\,_{\rightarrow}\mathcal{T}\subseteq\mathcal{P}(F)$$ being a $g$-sequential chain in $\mathcal{X}$ . **By Hausdorff Maximal Principle, there is a maximal fixed-point $g$-towered chain $C_{\leftarrow}\in\mathcal{X}$ of $F$ $$\sup_{\mathcal{C}_{\textrm{T}}}(\mathcal{C}_{\textrm{T}})=C_{\leftarrow}=\{ f_{\alpha},f_{\beta},f_{\gamma},\cdots\}=g(C_{\leftarrow})\in\mathcal{C}_{\textrm{T}}\subseteq\mathcal{P}(F).$$
Zorn’s Lemma now applied to this maximal chain yields its supremum as the maximal element of $C_{\leftarrow}$, and thereby of $F$. It needs to be appreciated, as in the case of the algebraic Hamel basis, that the existence of this maximal non-functional element was obtained purely set theoretically as the “limit” of a net of functions with increasing non-linearity, without resorting to any topological arguments. Because it is not a function, this supremum does not belong to the functional $g$-towered chain having it as a fixed point, and this maximal chain does not possess a largest, or even a maximal, element, although it does have a supremum.[^24] The supremum is a contribution of the inverse functional relations $(f_{\alpha}^{-})$ in the following sense. From Eq. (\[Eqn: func-multi\]), the net of increasingly non-injective functions of Eq. (\[Eqn: chaos1\]) implies a corresponding net of increasingly multivalued functions ordered inversely by the inverse relation $f_{\alpha}\preceq f_{\beta}\Leftrightarrow f_{\beta}^{-}\preceq f_{\alpha}^{-}$. Thus the inverse relations which are as much an integral part of graphical convergence as are the direct relations, have a smallest element belonging to the multifunctional class. Clearly, this smallest element as the required supremum of the increasingly non-injective tower of functions defined by Eq. (\[Eqn: chaos1\]), serves to complete the significance of the tower by capping it with a “boundary” element that can be taken to bridge the classes of functional and non-functional relations on $X$.
We are now ready to define a *maximally ill-posed problem $f(x)=y$* for *$x,y\in X$* in terms of a *maximally non-injective map $f$* as follows.
**Definition 4.1.** ***Chaotic map.*** *Let $A$ be a non-empty closed set of a compact Hausdorff space $X.$ A function* $f\in\textrm{Multi}(X)$ **(*equivalently the sequence of functions $(f_{i})$*)** *is* *maximally non-injective* *or* *chaotic on* **$A$** *with respect to the order relation* **(\[Eqn: chaos1\])** *if*
*(a) for any $f_{i}$ on $A$ there exists an $f_{j}$ on $A$ satisfying $f_{i}\preceq f_{j}$ for every $j>i\in\mathbb{N}$.*
*(b) the set $\mathcal{D}_{+}$ consists of a countable collection of isolated singletons.$\qquad\square$*
**Definition 4.2.** ***Maximally ill-posed problem.*** *[<span style="font-variant:small-caps;">L</span>]{}et $A$ be a non-empty closed set of a compact Hausdorff space $X$ and let $f$ be a functional relation in* $\textrm{Multi}(X)$*. The problem $f(x)=y$ is* *maximally ill-posed at* **$y$** *if $f$ is chaotic on $A$*.$\qquad\square$
As an example of the application of these definitions, on the dense set $\mathcal{D}_{+}$, the tent map satisfies both the conditions of sensitive dependence on initial conditions and topological transitivity [@Devaney1989] and is also maximally non-injective; the tent map is therefore chaotic on $\mathcal{D}_{+}.$ In contrast, the examples of Secs. 1 and 2 are not chaotic as the maps are not topologically transitive, although the Liapunov exponents, as in the case of the tent map, are positive. Here the $(f_{n})$ are identified with the iterates of $f,$ and the “fixed point” as one through which graphs of all the functions on residual index subsets pass. When the set of points $\mathcal{D}_{+}$ is dense in $[0,1]$ and both $\mathcal{D}_{+}$ and $[0,1]-\mathcal{D}_{+}=[0,1]-\bigcup_{i=0}^{\infty}f^{-i}(\textrm{Per}(f))$ (where $\textrm{Per}(f)$ denotes the set of periodic points of $f$) are totally disconnected, it is expected that at any point on this complement the behaviour of the limit will be similar to that on $\mathcal{D}_{+}$: these points are special as they tie up the iterates on $\textrm{Per}(f)$ to yield the multifunctions. Therefore in any neighbourhood $U$ of a $\mathcal{D}_{+}$-point, there is an $x_{0}$ at which the *forward orbit $\{ f^{i}(x_{0})\}_{i\geq0}$ is chaotic* in the sense that
\(a) the sequence neither diverges nor does it converge in the image space of $f$ to a periodic orbit of any period, and
\(b) the Liapunov exponent given by
$$\begin{aligned}
\lambda(x_{0}) & = & \lim_{n\rightarrow\infty}\ln\left|\frac{df^{n}(x_{0})}{dx}\right|^{1/n}\\
& = & {\displaystyle \lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=0}^{n-1}\ln\left|\frac{df(x_{i})}{dx}\right|,\, x_{i}=f^{i}(x_{0}),}\end{aligned}$$
which is a measure of the average slope of an orbit at $x_{0}$ or equivalently of the average loss of information of the position of a point after one iteration, is positive. Thus *an orbit with positive Liapunov exponent is chaotic if it is not* *asymptotic* (that is neither convergent nor adherent, having no convergent suborbit in the sense of Appendix A1) *to an unstable periodic orbit* *or to any other limit set on which the dynamics is simple.* A basic example of a chaotic orbit is that of an irrational in $[0,1]$ under the shift map and that of the chaotic set its closure, the full unit interval.
Let $f\in\textrm{Map}((X,\mathcal{U}))$ and suppose that $A=\{ f^{j}(x_{0})\}_{j\in\mathbb{N}}$ is a sequential set corresponding to the orbit $\textrm{Orb}(x_{0})=(f^{j}(x_{0}))_{j\in\mathbb{N}}$, and let $f_{\mathbb{R}_{i}}(x_{0})=\bigcup_{j\geq i}f^{j}(x_{0})$ be the $i$-residual of the sequence $(f^{j}(x_{0}))_{j\in\mathbb{N}}$, with $_{\textrm{F}}\mathcal{B}_{x_{0}}=\{ f_{\mathbb{R}_{i}}(x_{0})\!:\textrm{Res}(\mathbb{N})\rightarrow X\textrm{ for all }i\in\mathbb{N}\}$ being the decreasingly nested filter-base associated with $\textrm{Orb}(x_{0})$. The so-called *$\omega$-limit set of* $x_{0}$ given by $$\begin{array}{ccl}
\omega(x_{0}) & \overset{\textrm{def}}= & \{ x\in X\!:(\exists n_{k}\in\mathbb{N})(n_{k}\rightarrow\infty)\textrm{ }(f^{n_{k}}(x_{0})\rightarrow x)\}\\
& = & \{ x\in X\!:(\forall N\in\mathcal{N}_{x})(\forall f_{\mathbb{R}_{i}}\in\,_{\textrm{F}}\mathcal{B}_{x_{0}})\textrm{ }(f_{\mathbb{R}_{i}}(x_{0})\bigcap N\neq\emptyset)\}\end{array}\label{Eqn: Def: omega(x)}$$ is simply the adherence set $\textrm{adh}(f^{j}(x_{0}))$ of the sequence $(f^{j}(x_{0}))_{j\in\mathbb{N}}$, see Eq. (\[Eqn: net adh\]); hence Def. A1.11 of the filter-base associated with a sequence and Eqs. (\[Eqn: adh net2\]), (\[Eqn: adh filter\]), (\[Eqn: filter adh\*\]) and (\[Eqn: net-fil\]) allow us to express $\omega(x_{0})$ more meaningfully as $$\omega(x_{0})=\bigcap_{i\in\mathbb{N}}\textrm{Cl}(f_{\mathbb{R}_{i}}(x_{0})).\label{Eqn: adh_omega_x}$$ It is clear from the second of Eqs. (\[Eqn: Def: omega(x)\]) that for a continuous $f$ and any $x\in X$, $x\in\omega(x_{0})$ implies $f(x)\in\omega(x_{0})$ so that the entire orbit of $x$ lies in $\omega(x_{0})$ whenever $x$ does implying that the $\omega$-limit set is positively invariant; it is also closed because the adherent set is a closed set according to Theorem A1.3. Hence $x_{0}\in\omega(x_{0})\Rightarrow A\subseteq\omega(x_{0})$ reduces the $\omega$-limit set to the closure of $A$ without any isolated points, $A\subseteq\textrm{Der}(A)$. In terms of Eq. (\[Eqn: PrinFil\_Cl(A)\]) involving principal filters, Eq. (\[Eqn: adh\_omega\_x\]) in this case may be expressed in the more transparent form $\omega(x_{0})=\bigcap\textrm{Cl}(\,_{\textrm{F}}\mathcal{P}(\{ f^{j}(x_{0})\}_{j=0}^{\infty}))$ where the principal filter $_{\textrm{F}}\mathcal{P}(\{ f^{j}(x_{0})\}_{j=0}^{\infty})$ at $A$ consists of all supersets of $A=\{ f^{j}(x_{0})\}_{j=0}^{\infty}$, and $\omega(x_{0})$ represents the adherence set of the principal filter at $A$, see the discussion following Theorem A1.3. If $A$ represents a chaotic orbit under this condition, then $\omega(x_{0})$ is sometimes known as a *chaotic set* [@Alligood1997]; thus the chaotic orbit infinitely often visits every member of its chaotic set[^25] which is simply the $\omega$-limit set of a chaotic orbit that is itself contained in its own limit set. Clearly the chaotic set if positive invariant, and from Thm. A1.3 and its corollary it is also compact. Furthermore, if all (sub)sequences emanating from points $x_{0}$ in some neighbourhood of the set converge to it, then $\omega(x_{0})$ is called a *chaotic attractor,* see @Alligood1997. As common examples of chaotic sets that are not attractors mention may be made of the tent map with a peak value larger than $1$ at $0.5$, and the logistic map with $\lambda\geq4$ again with a peak value at $0.5$ exceeding $1$.
[Figure \[Fig: logcob357\],]{} [contd: Multifunctional and cobweb plots of $\lambda_{*}x(1-x)$ where $\lambda_{*}=3.5699456$]{}
It is important that the difference in the dynamical behaviour of the system on $\mathcal{D}_{+}$ and its complement be appreciated. At any fixed point $x$ of $f^{i}$ in $\mathcal{D}_{+}$ (or at its equivalent images in $[x]$) the dynamics eventually gets attached to the (equivalent) fixed point, and the sequence of iterates converges graphically in $\textrm{Multi}(X)$ to $x$ (or its equivalent points).
[Figure \[Fig: logcob357\],]{} [contd: Mulnctional and cobweb plots of $3.57x(1-x)$. ]{}
When $x\notin\mathcal{D}_{+}$, however, the orbit $A=\{ f^{i}(x)\}_{i\in\mathbb{N}}$ is chaotic in the sense that $(f^{i}(x))$ is not asymptotically periodic and not being attached to any particular point they wander about in the closed chaotic set $\omega(x)=\textrm{Der}(A)$ containing $A$ such that for any given point in the set, some subsequence of the chaotic orbit gets arbitrarily close to it. Such sequences do not converge anywhere but only frequent every point of $\textrm{Der}(A)$. Thus although in the limit of progressively larger iterations there is complete uncertainty of the outcome of an experiment conducted at either of these two categories of initial points, whereas on $\mathcal{D}_{+}$ this is due to a random choice from a multifunctional set of equally probable outputs as dictated by the specific conditions under which the experiment was conducted at that instant, on its complement the uncertainty is due to the chaotic behaviour of the functional iterates themselves. Nevertheless it must be clearly understood *that this later behaviour is* *entirely due to the multifunctional limits at the $\mathcal{D}_{+}$ points which completely determine the behaviour of the system on its complement.* As an explicit illustration of this situation, recall that for the shift map $2x\textrm{ mod}(1)$ the $\mathcal{D}_{+}$ points are the rationals on $[0,1]$, and any irrational is represented by a non-terminating and non-repeating decimal so that almost all decimals in $[0,1]$ in any base contain all possible sequences of any number of digits. For the logistic map, the situation is more complex, however. Here the onset of chaos marking the end of the period doubling sequence at $\lambda_{*}=3.5699456$ is signaled by the disappearance of all stable fixed points, Fig. \[Fig: logcob357\](c), with Fig. \[Fig: logcob357\](a) being a demonstration of the stable limits for $\lambda=3.569$ that show up as convergence of the iterates to constant valued functions (rather than as constant valued inverse functions) at stable fixed points, shown more emphatically in Fig\[Fig: log357\](a). What actually happens at $\lambda_{*}$ is shown in Fig. \[Fig: attractor\](a) in the next subsection: the almost vertical lines produced at a large, but finite, iterations $i$ (the multifunctions are generated only in the limiting sense of $i\rightarrow\infty$ and represent a boundary between functional and non-functional relations on a set), decrease in magnitude with increasing iterations until they reduce to points. This gives rise to a (totally disconnected) Cantor set on the $y$-axis in contrast with the connected intervals that the multifunctional limits at $\lambda>\lambda_{*}$ of Figs. \[Fig: attractor\](b)–(d) produce. By our characterization Definition 4.1 of chaos therefore, $\lambda x(1-x)$ is chaotic for the values of $\lambda>\lambda_{*}$ that are shown in Fig. \[Fig: attractor\]. We return to this case in the following subsection.
**
[Figure]{} [\[Fig: log357\], contd: Isolated fixed points of logistic map. The sequence of points generated by the iterates of the map are marked on the $y$-axis of (a)–(c) in]{} *italics*[. The singletons $\{ x\}$ are $\omega$-limit sets of the respective fixed point $x$ and is generated by the constant sequence $(x,x,\cdots)$. Whereas in (a) this is the limit of every point in $(0,1)$, in the other cases these fixed points are isolated in the sense of Def. 2.3. The isolated points, however, give rise to sequences that converge to more than one point in the form of limit cycles as shown in figures (b)–(d). ]{}
As an example of chaos *in a noniterative system*, we investigate the following question: While maximality of non-injectiveness produced by an increasing number of injective branches is necessary for a family of functions to be chaotic, is this also sufficient for the system to be chaotic? This is an important question especially in the context of a non-iterative family of functions where fixed points are of no longer relevant.
Consider the sequence of functions $|\sin(\pi nx)|_{n=1}^{\infty}.$ The graphs of the subsequence $|\sin(2^{n-1}\pi x)|$ and of the sequence $(t^{n}(x))$ on [\[]{}0,1[\]]{} are qualitatively similar in that they both contain $2^{n-1}$ of their functional graphs each on a base of $1/2^{n-1}.$ Thus both $|\sin(2^{n-1}\pi x)|_{n=1}^{\infty}$ and $(t^{n}(x))_{n=1}^{\infty}$ converge graphically to the multifunction [\[]{}0,1[\]]{} on the same set of points equivalent to 0. This is sufficient for us to conclude that $|\sin(2^{n-1}\pi x)|_{n=1}^{\infty}$, and hence $|\sin(\pi nx)|_{n=1}^{\infty}$, is chaotic on the infinite equivalent set [\[]{}0[\]]{}. While Fig. \[Fig: tent4\] was a comparison of the first four iterates of the tent and absolute sine maps, Fig. [\[Fig: tent17\]]{} following shows the “converged” graphical limits for after 17 iterations.
***4.1. The chaotic attractor***
One of the most fascinating characteristics of chaos in dynamical systems is the appearance of attractors the dynamics on which are chaotic. **For a subset $A$ of a topological space $(X,\mathcal{U})$ such that $\mathcal{R}(f(A))$ is contained in $A$ — in this section, unless otherwise stated to the contrary, $f(A)$ *will* *denote the* *graph and not the range (image)* *of* $f$ — which ensures that the iteration process can be carried out in $A$, let $$\begin{array}{ccl}
{\displaystyle f_{\mathbb{R}_{i}}(A)} & = & {\displaystyle \bigcup_{j\geq i\in\mathbb{N}}f^{j}(A)}\\
& = & {\displaystyle \bigcup_{j\geq i\in\mathbb{N}}\left(\bigcup_{x\in A}f^{j}(x)\right)}\end{array}\label{Eqn: absorbing set}$$ generate the filter-base $_{\textrm{F}}\mathcal{B}$ with $A_{i}:=f_{\mathbb{R}_{i}}(A)\in\,_{\textrm{F}}\mathcal{B}$ being decreasingly nested, $A_{i+1}\subseteq A_{i}$ for all $i\in\mathbb{N}$, in accordance with Def. A1.1. The existence of a maximal chain with a corresponding maximal element as asssured by the Hausdorff Maximal Principle and Zorn’s Lemma respectively implies a nonempty core of $_{\textrm{F}}\mathcal{B}$. As in Sec. 3 following Def. 3.3, we now identify the filterbase with the neighbourhood base at $f^{\infty}$ which allows us to define $$\begin{array}{ccl}
{\displaystyle \textrm{Atr}(A_{1})} & \overset{\textrm{def}}= & \textrm{adh}(\,_{\textrm{F}}\mathcal{B})\\
& = & {\displaystyle \bigcap_{A_{i}\in\,_{\textrm{F}}\mathcal{B}}\textrm{Cl}(A_{i})}\end{array}\label{Eqn: attractor_adherence}$$ as the attractor of the set $A_{1}$, where the last equality follows from Eqs.(\[Eqn: Def: omega(A)\]) and (\[Eqn: Def: Closure\]) and the closure is with respect to the topology induced by the neighbourhood filter base $_{\textrm{F}}\mathcal{B}$. Clearly the attractor as defined here is the graphical limit of the sequence of functions $(f^{i})_{i\in\mathbb{N}}$ which may be verified by reference to Def. A1.8, Thm. A1.3 and the proofs of Thms. A1.4 and A1.5, together with the directed set Eq. (\[Eqn: DirectedIndexed\]) with direction (\[Eqn: DirectionIndexed\]). The *basin of attraction* of the attractor is $A_{1}$ because the graphical limit $(\mathcal{D}_{+},F(\mathcal{D}_{+}))\bigcup(G(\mathcal{R}_{+}),\mathcal{R}_{+})$ of Def. 3.1 may be obtained, as indicated above, by a proper choice of sequences associated with $\mathcal{A}$. Note that in the context of iterations of functions, the graphical limit $(\mathcal{D}_{+},y_{0})$ of the sequence $(f^{n}(x))$ denotes a stable fixed point $x_{*}$ with image $x_{*}=f(x_{*})=y_{0}$ to which iterations starting at any point $x\in\mathcal{D}_{+}$ converge. The graphical limits $(x_{i0},\mathcal{R}_{+})$ are generated with respect to the class $\{ x_{i*}\}$ of points satisfying $f(x_{i0})=x_{i*}$, $i=0,1,2,\cdots$ equivalent to unstable fixed point $x_{*}:=x_{0*}$ to which inverse iterations starting at any initial point in $\mathcal{R}_{+}$ must converge. Even though only $x_{*}$ is inverse stable, an equivalent class of graphically converged limit multis is produced at every member of the class $x_{i*}\in[x_{*}]$, resulting in the far-reaching consequence *that every member of the class is as significant as the parent fixed point $x_{*}$ from which they were born in determining the dynamics of the evolving system.* The point to remember about infinite intersections of a collection of sets having finite intersection property, as in Eq. (\[Eqn: attractor\_adherence\]), is that this may very well be empty; recall, however, that in a compact space this is guaranteed not to be so. In the general case, if $\textrm{core}(\mathcal{A})\neq\emptyset$ then $\mathcal{A}$ is the principal filter at this core, and $\textrm{Atr}(A_{1})$ by Eqs. (\[Eqn: attractor\_adherence\]) and (\[Eqn: PrinFil\_Cl(A)\]) is the closure of this core, which in this case of the topology being induced by the filterbase, is just the core itself. $A_{1}$ by its very definition, is a positively invariant set as any sequence of graphs converging to **$\textrm{Atr}(A_{1})$ must be eventually in $A_{1}$: the entire sequence therefore lies in $A_{1}$. Clearly, from Thm. A3.1 and its corollary, the attractor is a positively invariant compact set. A typical attractor is illustrated by the derived sets in the second column of Fig. \[Fig: DerSets\] which also illustrates that the set of functional relations are open in $\textrm{Multi}(X)$; specifically functional-nonfunctional correspondences are neutral-selfish related as in Fig. \[Fig: DerSets\], 3-2, with the attracting graphical limit of Eq. (\[Eqn: attractor\_adherence\]) forming the boundary of (finitely)many-to-one functions and the one-to-(finitely)many multifunctions.
Equation (\[Eqn: attractor\_adherence\]) is to be compared with the *image definition of an attractor* [@Stuart1996] where $f(A)$ denotes the range and not the graph of $f$. Then Eq. (\[Eqn: attractor\_adherence\]) can be used to define a sequence of points $x_{k}\in A_{n_{k}}$ and hence the subset $$\begin{aligned}
\omega(A) & \overset{\textrm{def}}= & \{ x\in X\!:(\exists n_{k}\in\mathbb{N})(n_{k}\rightarrow\infty)(\exists x_{k}\in A_{n_{k}})\textrm{ }(f^{n_{k}}(x_{k})\rightarrow x)\}\nonumber \\
& = & \{ x\in X\!:(\forall N\in\mathcal{N}_{x})(\forall A_{i}\in\mathcal{A})(N\bigcap A_{i}\neq\emptyset)\}\label{Eqn: Def: omega(A)}\end{aligned}$$
as the corresponding attractor of $A$ that satisfies an equation formally similar to (\[Eqn: attractor\_adherence\]) with the difference that the filter-base $\mathcal{A}$ is now in terms of the image $f(A)$ of $A$, which allows the adherence expression to take the particularly simple form $$\omega(A)=\bigcap_{i\in\mathbb{N}}\textrm{Cl}(f^{i}(A)).\label{Eqn: omega(A)_intersect}$$ The complimentary subset excluded from this definition of $\omega(A)$, as compared to $\textrm{Atr}(A_{1})$, that is required to complete the formalism is given by Eq. (\[Eqn: basin\]) below. Observe that the equation for $\omega(A)$ is essentially Eq. (\[Eqn: adh net1\]), even though we prefer to use the alternate form of Eq. (\[Eqn: adh net2\]) as this brings out more clearly the frequenting nature of the sequence. The basin of attraction $$\begin{array}{ccl}
B_{f}(A) & = & \{ x\in A\!:\omega(x)\subseteq\textrm{Atr}(A)\}\\
& = & \{ x\in A\!:(\exists n_{k}\in\mathbb{N})(n_{k}\rightarrow\infty)\textrm{ }(f^{n_{k}}(x)\rightarrow x^{*}\in\omega(A)\textrm{ })\end{array}\label{Eqn: basin}$$ of the attractor is the smallest subset of $X$ in which sequences generated by $f$ must eventually lie in order to adhere at $\omega(A)$. Comparison of Eqs. (\[Eqn: Attractor\_R+\]) with (\[Eqn: R+\]) and (\[Eqn: basin\]) with (\[Eqn: D+\]) show that $\omega(A)$ can be identified with the subset $\mathcal{R}_{+}$ on the $y$-axis on which the multifunctional limits $G\!:\mathcal{R}_{+}\rightarrow X$ of graphical convergence are generated, with its basin of attraction being contained in the $\mathcal{D}_{+}$ associated with the injective branch of $f$ that generates $\mathcal{R}_{+}$. In summary it may be concluded that since definitions (\[Eqn: Def: omega(A)\]) and (\[Eqn: basin\]) involve both the domain and range of $f$, a description of the attractor in terms of the graph of $f$, like that of Eq. (\[Eqn: attractor\_adherence\]), is more pertinent and meaningful as it combines the requirements of both these equations. Thus, for example, as $\omega(A)$ is not the function $G(\mathcal{R}_{+})$, this attractor does not include the equivalence class of inverse stable points that may be associated with $x_{*}$, see for example Fig. \[Fig: omega\].
From Eq. (\[Eqn: Def: omega(A)\]), we may make the particularly simple choice of $(x_{k})$ to satisfy $f^{n_{k}}(x_{-k})=x$ so that $x_{-k}=f_{\textrm{B}}^{-n_{k}}(x)$, where $x_{-k}\in[x_{-k}]:=f^{-n_{k}}(x)$ is the element of the equivalence class of the inverse image of $x$ corresponding to the injective branch $f_{\textrm{B}}$. This choice is of special interest to us as it is the class that generates the $G$-function on $\mathcal{R}_{+}$ in graphical convergence. This allows us to express $\omega(A)$ as $$\omega(A)=\{ x\in X\!:(\exists n_{k}\in\mathbb{N})(n_{k}\rightarrow\infty)(f_{\textrm{B}}^{-n_{k}}(x)=x_{-k}\textrm{ converges in }(X,\mathcal{U}))\};\label{Eqn: Attractor_R+}$$ note that the $x_{-k}$ of this equation and the $x_{k}$ of Eq. (\[Eqn: Def: omega(A)\]) are, in general, quite different points.
A simple illustrative example of the construction of $\omega(A)$ for the positive injective branch of the homeomorphism $(4x^{2}-1)/3$, $-1\leq x\leq1$, is shown in Fig. \[Fig: omega\], where the arrow-heads denote the converging sequences $f^{n_{i}}(x_{i})\rightarrow x$ and $f^{n_{i}-m}(x_{i})\rightarrow x_{-m}$ which proves invariance of $\omega(A)$ for a homeomorphic $f$; here continuity of the function and its inverse is explicitly required for invariance. Positive invariance of a subset $A$ of $X$ implies that for any $n\in\mathbb{N}$ and $x\in A$, $f^{n}(x)=y_{n}\in A$, while negative invariance assures that for any $y\in A$, $f^{-n}(y)=x_{-n}\in A$. Invariance of $A$ in both the forward and backward directions therefore means that for any $y\in A$ and $n\in\mathbb{N}$, there exists a $x\in A$ such that $f^{n}(x)=y$. In interpreting this figure, it may be useful to recall from Def. 4.1 that an increasing number of injective branches of $f$ is a necessary, but not sufficient, condition for the occurrence of chaos; thus in Figs. \[Fig: log357\](a) and \[Fig: omega\], increasing noninjectivity of $f$ leads to constant valued limit functions over a connected $\mathcal{D}_{+}$ in a manner similar to that associated with the classical Gibb’s phenomenon in the theory of Fourier series.
Graphical convergence of an increasingly nonlinear family of functions implied by its increasing non-injectivity may now be combined with the requirements of an attractor to lead to the concept of a chaotic attractor to be that on which the dynamics is chaotic in the sense of Defs. 4.1. and 4.2. Hence
**Definition 4.3.** ***Chaotic Attractor.*** *Let $A$ be a positively invariant subset of $X$. The attractor* $\textrm{Atr}(A)$ *is chaotic on $A$ if there is sensitive dependence on initial conditions for* all *$x\in A$. The sensitive dependence manifests itself as multifunctional graphical limits for all $x\in\mathcal{D}_{+}$ and as chaotic orbits when* $x\not\in\mathcal{D}_{+}$*.*$\qquad\square$
[$$f_{\textrm{f}}(x)=\left\{ \begin{array}{ccl}
2(1+x)/3, & & 0\leq x<1/2\\
2(1-x), & & 1/2\leq x\leq1\end{array}\right.$$ ]{}
The picture of chaotic attractors that emerge from the foregoing discussions and our characterization of chaos of Def. 4.1 is that it it is a subset of $X$ that is simultaneously “spiked” multifunctional on the $y$-axis and consists of a dense collection of singleton domains of attraction on the $x$-axis. This is illustrated in Figure \[Fig: attractor\] which shows some typical chaotic attractors. The first four diagrams (a)$-$(d) are for the logistic map with (b)$-$(d) showing the 4-, 2- and 1-piece attractors for $\lambda=3.575,\textrm{ }3.66,\textrm{ and }3.8$ respectively that are in qualitative agreement with the standard bifurcation diagram reproduced in (e). Figs. (b)$-$(d) have the advantage of clearly demonstrating how the attractors are formed by considering the graphically converged limit as the object of study unlike in Fig. (e) which shows the values of the 501-1001th iterates of $x_{0}=1/2$ as a function of $\lambda$. The difference in Figs. (a) and (b) for a change of $\lambda$ from [$\lambda>\lambda_{*}=3.5699456$]{} to 3.575 is significant as $\lambda=\lambda_{*}$ marks the boundary between the nonchaotic region for $\lambda<\lambda_{*}$ and the chaotic for $\lambda>\lambda_{*}$ (this is to be understood as being suitably modified by the appearance of the nonchaotic windows for some specific intervals in $\lambda>\lambda_{*}$). At $\lambda_{*}$ the generated fractal Cantor set $\Lambda$ is an attractor as it attracts almost every initial point $x_{0}$ so that the successive images $x^{n}=f^{n}(x_{0})$ converge toward the Cantor set $\Lambda$. In Fig. (f) the chaotic attractors for the piecewise continuous function on $[0,1]$
$$f_{\textrm{f}}(x)=\left\{ \begin{array}{ccc}
2(1+x)/3, & & 0\leq x<1/2\\
2(1-x), & & 1/2\leq x\leq1,\end{array}\right.$$
is $[0,1]$ where the dotted lines represent odd iterates and the full lines even iterates of $f$; here the attractor disappears if the function is reflected about the $x$-axis.
[Figure]{} [\[Fig: attractor\]]{}[, contd.]{} [Chaotic attractors for $\lambda=3.66$ and $\lambda=3.8$.]{}
***4.2. Why Chaos? A Preliminary Inquiry***
The question as to why a natural system should evolve chaotically is both interesting and relevant, and this section attempts to advance a plausible answer to this inquiry that is based on the connection between topology and convergence contained in the Corollary to Theorem A1.5. Open sets are groupings of elements that govern convergence of nets and filters, because the required property of being either eventually of frequently in (open) neighbourhoods of a point determines the eventual behaviour of the net; recall in this connection the unusual convergence characteristics in cofinite and cocountable spaces. Conversely for a given convergence characteristic of a class of nets, it is possible to infer the topology of the space that is responsible for this convergence, and it is this point of view that we adopt here to investigate the question of this subsection: recall that our Definitions 4.1 and 4.2 were based on purely algebraic set-theoretic arguments on ordered sets, just as the role of the choice of an appropriate problem-dependent basis was highlighted at the end of Sec. 2.
[Figure]{} [\[Fig: attractor\], contd. Bifurcation diagram and attractors for $f_{\textrm{f}}(x)$.]{}
Chaos as manifest in its attractors is a direct consequence of the increasing nonlinearity of the map with increasing iteration; we reemphasize that this is only a necessary condition so that the increasing nonlinearities of Figs. \[Fig: log357\] and \[Fig: omega\] eventually lead to stable states and not to chaotic instability. Under the right conditions as enunciated following Fig. \[Fig: Zorn\], chaos appears to be the natural outcome of the difference in the behaviour of a function $f$ and its inverse $f^{-}$ under their successive applications. Thus $f=ff^{-}f$ allows $f$ to take advantage of its multi inverse to generate all possible equivalence classes that is available to it, a feature not accessible to $f^{-}=f^{-}ff^{-}$. As we have seen in the foregoing, equivalence classes of fixed points, stable and unstable, are of defining significance in determining the ultimate behaviour of an evolving dynamical system and as the eventual (as also frequent) character of a filter or net in a set is dictated by open neighbourhoods of points of the set, *it is postulated that chaoticity on a set $X$ leads to a reformulation of the open sets of $X$ to equivalence classes generated by the evolving map $f$,* see Example 2.4(3). Such a redefinition of open sets of equivalence classes allow the evolving system to temporally access an ever increasing number of states even though the equivalent fixed points are not fixed under iterations of $f$ except for the parent of the class, and can be considered to be the governing criterion for the cooperative or collective behaviour of the system. The predominance of the role of $f^{-}$ in $f=ff^{-}f$ in generating the equivalence classes (that is exploiting the many-to-one character) of $f$ is reflected as limit multis for $f$ (that is constant $f^{-}$ on $\mathcal{R}_{+}$) in $f^{-}=f^{-}ff^{-}$; this interpretation of the dynamics of chaos is meaningful as graphical convergence leading to chaos is a result of pointwise biconvergence of the sequence of iterates of the functions generated by $f$. But as $f$ is a noninjective function *on* $X$ *possessing the property of increasing nonlinearity in the form of increasing noninjectivity with iteration,* various cycles of disjoint equivalence classes are generated under iteration, see for example Fig. \[Fig: tent4\](a) for the tent map. A reference to Fig. \[Fig: GenInv\] shows that the basic set $X_{\textrm{B}}$, for a finite number $n$ of iterations of $f$, contains the parent of each of these open equivalent sets in the domain of $f$, with the topology on $X_{\textrm{B}}$ being the corresponding $p$-images of these disjoint saturated open sets of the domain. In the limit of infinite iterations of $f$ leading to the multifunction $\mathcal{M}$ (this is the $f^{\infty}$ of Sec. 4.1), the generated open sets constitute a basis for a topology on $\mathcal{D}(f)$ and the basis for the topology of $\mathcal{R}(f)$ are the corresponding $\mathcal{M}$-images of these equivalent classes. *It is our contention that the motive force behind evolution toward a chaos, as defined by Def. 4.1, is the drive toward a state of the dynamical system that supports ininality of the limit multi* $\mathcal{M}$*;* see Appendix A2 with the discussions on Fig. \[Fig: GenInv\] and Eq. (\[Eqn: ininal\]) in Sec. 2. In the limit of infinite iterations therefore, the open sets of the range $\mathcal{R}(f)\subseteq X$ are the multi images that graphical convergence generates at each of these inverse-stable fixed points. $X$ therefore has two topologies imposed on it by the dynamics of $f$: the first of equivalence classes generated by the limit multi $\mathcal{M}$ in the domain of $f$ and the second as $\mathcal{M}$-images of these classes in the range of $f$. Quite clearly these two topologies need not be the same; their intersection therefore can be defined to be the *chaotic topology* *on* $X$ *associated with the chaotic map* $f$ on $X$. Neighbourhoods of points in this topology cannot be arbitrarily small as they consist of all members of the equivalence class to which any element belongs; hence a sequence converging to any of these elements necessarily converges to all of them, and the eventual objective of chaotic dynamics is to generate a topology in $X$ with respect to which elements of the set can be grouped together in as large equivalence classes as possible in the sense that if a net converges simultaneously to points $x\neq y\in X$ then $x\sim y$: $x$ is of course equivalent to itself while $x,y,z$ are equivalent to each other iff they are simultaneously in every open set in which the net may eventually belong. This hall-mark of chaos can be appreciated in terms of a necessary obliteration of any separation property that the space might have originally possessed, see property (H3) in Appendix A3. We reemphasize that a set in this chaotic context is required to act in a dual capacity depending on whether it carries the initial or final topology under $\mathcal{M}$.
This preliminary inquiry into the nature of chaos is concluded in the final section of this work.
**5. Graphical convergence works**
We present in this section some real evidence in support of our hypothesis of graphical convergence of functions in $\textrm{Multi}(X,Y)$. The example is taken from neutron transport theory, and concerns the discretized spectral approximation [@Sengupta1988; @Sengupta1995] of Case’s singular eigenfunction solution of the monoenergetic neutron transport equation, [@Case1967]. The neutron transport equation is a linear form of the Boltzmann equation that is obtained as follows. Consider the neutron-moderator system as a mixture of two species of gases each of which satisfies a Boltzmann equation of the type$$\begin{gathered}
\left(\frac{\partial}{\partial t}+v_{i}.\nabla\right)f_{i}(r,v,t)=\\
=\int dv^{\prime}\int dv_{1}\int dv_{1}^{\prime}\sum_{j}W_{ij}(v_{i}\rightarrow v^{\prime};v_{1}\rightarrow v_{1}^{\prime})\{ f_{i}(r,v^{\prime},t)f_{j}(r,v_{1}^{\prime},t)--f_{i}(r,v,t)f_{j}(r,v_{1},t)\})\end{gathered}$$
where$$W_{ij}(v_{i}\rightarrow v^{\prime};v_{1}\rightarrow v_{1}^{\prime})=\mid v-v_{1}\mid\sigma_{ij}(v-v^{\prime},v_{1}-v_{1}^{\prime})$$
$\sigma_{ij}$ being the cross-section of interaction between species $i$ and $j$. Denote neutrons by subscript 1 and the background moderator with which the neutrons interact by 2, and make the assumptions that
\(i) The neutron density $f_{1}$ is much less compared with that of the moderator $f_{2}$ so that the terms $f_{1}f_{1}$ and $f_{1}f_{2}$ may be neglected in the neutron and moderator equations respectively.
\(ii) The moderator distribution $f_{2}$ is not affected by the neutrons. This decouples the neutron and moderator equations and leads to an equilibrium Maxwellian $f_{\textrm{M}}$ for the moderator while the neutrons are described by the linear equation $$\begin{gathered}
\left(\frac{\partial}{\partial t}+v.\nabla\right)f(r,v,t)=\\
=\int dv^{\prime}\int dv_{1}\int dv_{1}^{\prime}W_{12}(v\rightarrow v^{\prime};v_{1}\rightarrow v_{1}^{\prime})\{ f(r,v^{\prime},t)f_{\textrm{M}}(v_{1}^{\prime})--f(r,v,t)f_{\textrm{M}}(v_{1})\})\end{gathered}$$ This is now put in the standard form of the neutron transport equation [@Williams1967]$$\begin{gathered}
\left(\frac{1}{v}\frac{\partial}{\partial t}+\Omega.v+\mathcal{S}(E)\right)\Phi(r,E,\widehat{\Omega},t)=\int d\Omega^{\prime}\int dE^{\prime}\mathcal{S}(r,E^{\prime}\rightarrow E;\widehat{\Omega}^{\prime}\cdot\widehat{\Omega})\textrm{ }\Phi(r,E^{\prime},\widehat{\Omega}^{\prime},t).\end{gathered}$$ where $E=mv^{2}/2$ is the energy and $\widehat{\Omega}$ the direction of motion of the neutrons. The steady state, monoenergetic form of this equation is Eq. (\[Eqn: NeutronTransport\]) $$\mu\frac{\partial\Phi(x,\mu)}{\partial x}+\Phi(x,\mu)=\frac{c}{2}\int_{-1}^{1}\Phi(x,\mu^{\prime})d\mu^{\prime},\qquad0<c<1,\,-1\leq\mu\leq1$$
and its singular eigenfunction solution for $x\in(-\infty,\infty)$ is given by Eq. (\[Eqn: CaseSolution\_FR\]) $$\begin{gathered}
\Phi(x,\mu)=a(\nu_{0})e^{-x/\nu_{0}}\phi(\mu,\nu_{0})+a(-\nu_{0})e^{x/\nu_{0}}\phi(-\nu_{0},\mu)+\int_{-1}^{1}a(\nu)e^{-x/\nu}\phi(\mu,\nu)d\nu;\end{gathered}$$
see Appendix A4 for an introductory review of Case’s solution of the one-speed neutron transport equation.
--------------- --------------- ------------------------ ------------------------------ ----------------------------------
$\mathcal{R}=X$ $\textrm{Cl}(\mathcal{R})=X$ $\textrm{Cl}(\mathcal{R})\neq X$
Not injective $\cdots$ $P\sigma(\mathscr{L})$ $P\sigma(\mathscr{L})$ $P\sigma(\mathscr{L})$
Not contiuous $C\sigma(\mathscr{L})$ $C\sigma(\mathscr{L})$ $R\sigma(\mathscr{L})$
Continuous $\rho(\mathscr{L})$ $\rho(\mathscr{L})$ $R\sigma(\mathscr{L})$
--------------- --------------- ------------------------ ------------------------------ ----------------------------------
: [\[Table: spectrum\]Spectrum of linear operator $\mathscr{L}\in\textrm{Map}(X)$. Here $\mathscr L_{\lambda}:=\mathscr{L}-\lambda$ satisfies the equation $\mathscr L_{\lambda}(x)=0$, with the resolvent set $\rho(\mathscr{L})$ of $\mathscr{L}$ consisting of all those complex numbers $\lambda$ for which $\mathscr L_{\lambda}^{-1}$]{} [exists as a continuous operator with dense domain. Any value of $\lambda$ for which this is not true is in the spectrum $\sigma(\mathscr{L})$ of $\mathscr{L}$, that is further subdivided into three disjoint components of the point, continuous, and residual spectra according to the criteria shown in the table. ]{}
The term “eigenfunction” is motivated by the following considerations. Consider the eigenvalue equation $$(\mu-\nu)\mathscr F_{\nu}(\mu)=0,\qquad\mu\in V(\mu),\textrm{ }\nu\in\mathbb{R}\label{Eqn: eigen}$$
in the space of multifunctions $\textrm{Multi}(V(\mu),(-\infty,\infty))$, where $\mu$ is in either of the intervals $[-1,1]$ or $[0,1]$ depending on whether the given boundary conditions for Eq. (\[Eqn: NeutronTransport\]) is full-range or half range. If we are looking only for functional solutions of Eq. (\[Eqn: eigen\]), then the unique function $\mathcal{F}$ that satisfies this equation for all possible $\mu\in V(\mu)$ and $\nu\in\mathbb{R}-V(\mu)$ is $\mathcal{F}_{\nu}(\mu)=0$ which means, according to Table \[Table: spectrum\], that the point spectrum of $\mu$ is empty and $(\mu-\nu)^{-1}$ exists for all $\nu$. When $\nu\in V(\mu)$, however, this inverse is not continuous and we show below that in $\textrm{Map}(V(\mu),0)$, $\nu\in V(\mu)$ belongs to the continuous spectrum of $\mu$. This distinction between the nature of the inverses depending on the relative values of $\mu$ and $\nu$ suggests a wider “non-function” space in which to look for the solutions of operator equations, and in keeping with the philosophy embodied in Fig. \[Fig: GenInv\] of treating inverse problems in the space of multifunctions, we consider all $\mathscr F_{\nu}\in\textrm{Multi}(V(\mu),\mathbb{R}))$ satisfying Eq. (\[Eqn: eigen\]) to be eigenfunctions of $\mu$ for the corresponding eigenvalue $\nu$, leading to the following multifunctional solution of (\[Eqn: eigen\])$$\begin{aligned}
\mathscr F_{\nu}(\mu) & = & \left\{ \begin{array}{ccl}
(V(\mu),0), & & \textrm{if }\nu\notin V(\mu)\\
(V(\mu)-\nu,0)\bigcup(\nu,\mathbb{R})), & & \textrm{if }\nu\in V(\mu),\end{array}\right.\end{aligned}$$
where $V(\mu)-\nu$ is used as a shorthand for the interval $V(\mu)$ with $\nu$ deleted. Rewriting the eigenvalue equation (\[Eqn: eigen\]) as $\mu_{\nu}(\mathscr F_{\nu}(\mu))=0$ and comparing this with Fig. \[Fig: GenInv\], allows us to draw the correspondences $$\begin{aligned}
f & \Longleftrightarrow & \mu_{\nu}\nonumber \\
X\textrm{ and }Y & \Longleftrightarrow & \{\mathscr F_{\nu}\in\textrm{Multi}(V(\mu),\mathbb{R})\!:\mathscr F_{\nu}\in\mathcal{D}(\mu_{\nu})\}\nonumber \\
f(X) & \Longleftrightarrow & \{0\!:0\in Y\}\label{Eqn: GenInv_Spectrum}\\
X_{\textrm{B}} & \Longleftrightarrow & \{0\!:0\in X\}\nonumber \\
f^{-} & \Longleftrightarrow & \mu_{\nu}^{-}.\nonumber \end{aligned}$$ Thus a multifunction in $X$ is equivalent to $0$ in $X_{\textrm{B}}$ under the linear map $\mu_{\nu}$, and we show below that this multifunction is infact the Dirac delta “function” $\delta_{\nu}(\mu)$, usually written as $\delta(\mu-\nu)$. This suggests that in $\textrm{Multi}(V(\mu),\mathbb{R})$*, every $\nu\in V(\mu)$ is in the point spectrum of $\mu$*, so that *discontinuous functions that are pointwise limits of functions in function space can be replaced by graphically converged multifunctions in the space of multifunctions*. Completing the equivalence class of $0$ in Fig. \[Fig: GenInv\], gives the multifunctional solution of Eq. (\[Eqn: eigen\]).
From a comparison of the definition of ill-posedness (Sec. 2) and the spectrum (Table \[Table: spectrum\]), it is clear that $\mathscr L_{\lambda}(x)=y$ is ill-posed iff
\(1) $\mathscr L_{\lambda}$ not injective $\Leftrightarrow$ $\lambda\in P\sigma(\mathscr L_{\lambda})$, which corresponds to the first row of Table \[Table: spectrum\].
\(2) $\mathscr L_{\lambda}$ not surjective $\Leftrightarrow$ the values of $\lambda$ correspond to the second and third columns of Table \[Table: spectrum\].
\(3) $\mathscr L_{\lambda}$ is bijective but not open $\Leftrightarrow$ $\lambda\textrm{ is either in }C\sigma(\mathscr L_{\lambda})\textrm{ or }R\sigma(\mathscr L_{\lambda})$ corresponding to the second row of Table \[Table: spectrum\].
We verify in the three steps below that $X=L_{1}[-1,1]$ of integrable functions, $\nu\in V(\mu)=[-1,1]$ belongs to the continuous spectrum of $\mu$.
\(a) *$\mathcal{R}(\mu_{\nu})$ is dense, but not equal to $L_{1}$*. The set of functions $g(\mu)\in L_{1}$ such that $\mu_{\nu}^{-1}g\in L_{1}$ cannot be the whole of $L_{1}$. Thus, for example, the piecewise constant function $g=\textrm{const}\neq0$ on $\mid\mu-\nu\mid\leq\delta>0$ and $0$ otherwise is in $L_{1}$ but not in *$\mathcal{R}(\mu_{\nu})$* as $\mu_{\nu}^{-1}g\not\in L_{1}$. Nevertheless for any $g\in L_{1}$, we may choose the sequence of functions $$g_{n}(\mu)=\left\{ \begin{array}{ccl}
0, & & \textrm{if }\mid\mu-\nu\mid\leq1/n\\
g(\mu), & & \textrm{otherwise}\end{array}\right.$$
in $\mathcal{R}(\mu_{\nu})$ to be eventually in every neighbourhood of $g$ in the sense that $\lim_{n\rightarrow\infty}\int_{-1}^{1}\mid g-g_{n}\mid=0$.
\(b) *The inverse $(\mu-\nu)^{-1}$ exists but is not continuous.* The inverse exists because, as noted earlier, $0$ is the only functional solution of Eq. (\[Eqn: eigen\]). Nevertheless although the net of functions $$\delta_{\nu\varepsilon}(\mu)=\frac{1}{\tan^{-1}(1+\nu)/\varepsilon+\tan^{-1}(1-\nu)/\varepsilon}\left(\frac{\varepsilon}{(\mu-\nu)^{2}+\varepsilon^{2}}\right),\qquad\varepsilon>0$$
is in the domain of $\mu_{\nu}$ because $\int_{-1}^{1}\delta_{\nu\varepsilon}(\mu)d\mu=1$ for all $\varepsilon>0$, $$\lim_{\varepsilon\rightarrow0}\int_{-1}^{1}\mid\mu-\nu\mid\delta_{\nu\varepsilon}(\mu)d\mu=0$$
implying that $(\mu-\nu)^{-1}$ is unbounded.
Taken together, (a) and (b) show that functional solutions of Eq. (\[Eqn: eigen\]) lead to state 2-2 in Table \[Table: spectrum\]; hence $\nu\in[-1,1]=C\sigma(\mu)$.
\(c) The two integral constraints in (b) also mean that $\nu\in C\sigma(\mu)$ is a *generalized eigenvalue* of $\mu$ which justifies calling the graphical limit $\delta_{\nu\varepsilon}(\mu)\overset{\mathbf{G}}\rightarrow\delta_{\nu}(\mu)$ a *generalized,* or singular, *eigenfunction*, see Fig. \[Fig: Poison\] which clearly indicates the convergence of the net of functions[^26].
From the fact that the solution Eq. (\[Eqn: CaseSolution\_FR\]) of the transport equation contains an integral involving the multifunction $\phi(\mu,\nu)$, we may draw an interesting physical interpretation. As the multi appears *every where* on $V(\mu)$ (that is there are no chaotic orbits but only the multifunctions that produce them), we have here a situation typical of *maximal ill-posedness* characteristic of chaos: note that both the functions comprising $\phi_{\varepsilon}(\mu,\nu)$ are non-injective. As the solution (\[Eqn: CaseSolution\_FR\]) involves an integral over all $\nu\in V(\mu)$, the singular eigenfunctions — that collectively may be regarded as representing a *chaotic substate of* the system represented by the solution of the neutron transport equation — combine with the functional components $\phi(\pm\nu_{0},\mu)$ to produce the well-defined, non-chaotic, experimental end result of the neutron flux $\Phi(x,\mu)$.
The solution (\[Eqn: CaseSolution\_FR\]) is obtained by assuming $\Phi(x,\mu)=e^{-x/\nu}\phi(\mu,\nu)$ to get the equation for $\phi(\mu,\nu)$ to be $(\mu-\nu)\phi(\mu,\nu)=-c\nu/2$ with the normalization $\int_{-1}^{1}\phi(\mu,\nu)=1$. As $\mu_{\nu}^{-1}$ is not invertible in $\textrm{Multi}(V(\mu),\mathbb{R})$ and $\mu_{\nu\textrm{B}}\!:X_{\textrm{B}}\rightarrow f(X)$ does not exist, the alternate approach of regularization was adopted in [@Sengupta1988; @Sengupta1995] to rewrite $\mu_{\nu}\phi(\mu,\nu)=-c\nu/2$ as $\mu_{\nu\varepsilon}\phi_{\varepsilon}(\mu,\nu)=-c\nu/2$ with $\mu_{\nu\varepsilon}:=\mu-(\nu+i\varepsilon)$ being a net of bijective functions for $\varepsilon>0$; this is a consequence of the fact that for the multiplication operator every nonreal $\lambda$ belongs to the resolvent set of the operator. The family of solutions of the later equation is given by [@Sengupta1988; @Sengupta1995] $$\phi_{\varepsilon}(\nu,\mu)=\frac{c\nu}{2}\frac{\nu-\mu}{(\mu-\nu)^{2}+\varepsilon^{2}}+\frac{\lambda_{\varepsilon}(\nu)}{\pi_{\varepsilon}}\frac{\varepsilon}{(\mu-\nu)^{2}+\varepsilon^{2}}\label{Eqn: phieps}$$
where the required normalization $\int_{-1}^{1}\phi_{\varepsilon}(\nu,\mu)=1$ gives
$$\begin{array}{ccl}
{\displaystyle \lambda_{\varepsilon}(\nu)} & = & {\displaystyle \frac{\pi_{\varepsilon}}{\tan^{-1}(1+\nu)/\varepsilon+\tan^{-1}(1-\nu)/\varepsilon}\left(1-\frac{c\nu}{4}\ln\frac{(1+\nu)^{2}+\varepsilon^{2}}{(1-\nu)^{2}+\varepsilon^{2}}\right)}\\
& \overset{\varepsilon\rightarrow0}\longrightarrow & \pi\lambda(\nu)\end{array}$$
with $$\pi_{\varepsilon}=\varepsilon\int_{-1}^{1}\frac{d\mu}{\mu^{2}+\varepsilon^{2}}=2\tan^{-1}\left(\frac{1}{\varepsilon}\right)\overset{\varepsilon\rightarrow0}\longrightarrow\pi.$$
These discretized equations should be compared with the corresponding exact ones of Appendix A4. We shall see that the net of functions (\[Eqn: phieps\]) converges graphically to the multifunction Eq. (\[Eqn: singular\_eigen\]) as $\varepsilon\rightarrow0$.
In the discretized spectral approximation., the singular eigenfunction $\phi(\mu,\nu)$ is replaced by $\phi_{\varepsilon}(\mu,\nu)$, $\varepsilon\rightarrow0$, with the integral in $\nu$ being replaced by an appropriate sum. The solution Eq. (\[Eqn: CaseSolution\_HR\]) of the physically interesting half-space $x\geq0$ problem then reduces to [@Sengupta1988; @Sengupta1995] $$\Phi_{\varepsilon}(x,\mu)=a(\nu_{0})e^{-x/\nu_{0}}\phi(\mu,\nu_{0})+\sum_{i=1}^{N}a(\nu_{i})e^{-x/\nu_{i}}\phi_{\varepsilon}(\mu,\nu_{i})\qquad\mu\in[0,1]\label{Eqn: DiscSpect_HR}$$
where the nodes $\{\nu_{i}\}_{i=1}^{N}$ are chosen suitably. This discretized spectral approximation to Case’s solution has given surprisingly accurate numerical results for a set of properly chosen nodes when compared with exact calculations. Because of its involved nature [@Case1967], the exact calculations are basically numerical which leads to nonlinear integral equations as part of the solution procedure. To appreciate the enormous complexity of the exact treatment of the half-space problem, we recall that the complete set of eigenfunctions $\{\phi(\mu,\nu_{0}),\{\phi(\mu,\nu)\}_{\nu\in[0,1]}\}$ are orthogonal with respect to the half-range weight function $W(\mu)$ of half-range theory, Eq. (\[Eqn: W(mu)\]), that is expressed only in terms of solution of the nonlinear integral equation Eq. (\[Eqn: Omega(-mu)\]). The solution of a half-space problem then evaluates the coefficients $\{ a(\nu_{0}),a(\nu)_{\nu\in[0,1]}\}$ from the appropriate half range (that is $0\leq\mu\leq1$) orthogonality integrals satisfied by the eigenfunctions $\{\phi(\mu,\nu_{0}),\{\phi(\mu,\nu)\}_{\nu\in[0,1]}\}$ with respect to the weight $W(\mu)$, see Appendix A4 for the necessary details of the half-space problem in neutron transport theory.
As may be appreciated from this brief introduction, solutions to half-space problems are not simple and actual numerical computations must rely a great deal on tabulated values of the $X$-function. Self-consistent calculations of sample benchmark problems performed by the discretized spectral approximation in a full-range adaption of the half-range problem described below that generate all necessary data, independent of numerical tables, with the quadrature nodes $\{\nu_{i}\}_{i=1}^{N}$ taken at the zeros Legendre polynomials show that the full range formulation of this approximation [@Sengupta1988; @Sengupta1995] can give very accurate results not only of integrated quantities like the flux $\Phi$ and leakage of particles out of the half space, but of also basic ‘"raw‘" data like the extrapolated end point $$z_{0}=\frac{c\nu_{0}}{4}\int_{0}^{1}\frac{\nu}{N(\nu)}\left(1+\frac{c\nu^{2}}{1-\nu^{2}}\right)\ln\left(\frac{\nu_{0}+\nu}{\nu_{0}-\nu}\right)d\nu\label{Eqn: extrapolated}$$
and of the $X$-function itself. Given the involved nature of the exact theory, it is our contention that the remarkable accuracy of these basic data, some of which is reproduced in Table \[Table: extrapolated\], is due to the graphical convergence of the net of functions $$\phi_{\varepsilon}(\mu,\nu)\overset{\mathbf{G}}\longrightarrow\phi(\mu,\nu)$$
shown in Fig. \[Fig: Case\]; here $\varepsilon=1/\pi N$ so that $\varepsilon\rightarrow0$ as $N\rightarrow\infty$. By this convergence, the delta function and principal values in $[-1,1]$ are the multifunctions $([-1,0),0)\bigcup(0,[0,\infty)\bigcup((0,1],0)$ and $\{1/x\}_{x\in[-1,0)}\bigcup(0,(-\infty,\infty))\bigcup\{1/x\}_{x\in(0,1]}$ respectively.
Tables \[Table: extrapolated\] and \[Table: X-function\], taken from @Sengupta1988 and @Sengupta1995, show respectively the extrapolated end point and $X$-function by the full-range adaption of the discretized spectral approximation for two different half range problems denoted as Problems A and B defined as
$$\begin{aligned}
Problem\textrm{ }A\quad & \textrm{Equation}\!:\textrm{ }{\textstyle {\mu\Phi_{x}+\Phi=(c/2)\int_{-1}^{1}\Phi(x,\mu^{\prime})d\mu^{\prime},\; x\geq0}}\\
& \textrm{Boundary condition}:\textrm{ }\Phi(0,\mu)=0,\;\mu\geq0\\
& \textrm{Asymptotic condition}:\textrm{ }\Phi\rightarrow e^{-x/\nu_{0}}\phi(\mu,\nu_{0}),\; x\rightarrow\infty.\\
Problem\textrm{ }B\quad & \textrm{Equation}\!:\textrm{ }{\textstyle {\mu\Phi_{x}+\Phi=(c/2)\int_{-1}^{1}\Phi(x,\mu^{\prime})d\mu^{\prime},\; x\geq0}}\\
& \textrm{Boundary condition}:\textrm{ }\Phi(0,\mu)=1,\;\mu\geq0\\
& \textrm{Asymptotic condition}:\textrm{ }\Phi\rightarrow0,\; x\rightarrow\infty.\end{aligned}$$
The full $-1\leq\mu\leq1$ range form of the half $0\leq\mu\leq1$ range discretized spectral approximation replaces the exact integral boundary condition at $x=0$ by a suitable quadrature sum over the values of $\nu$ taken at the zeros of Legendre polynomials; thus the condition at $x=0$ can be expressed as $$\psi(\mu)=a(\nu_{0})\phi(\mu,\nu_{0})+\sum_{i=1}^{N}a(\nu_{i})\phi_{\varepsilon}(\mu,\nu_{i}),\qquad\mu\in[0,1],\label{Eqn: BC}$$
where $\psi(\mu)=\Phi(0,\mu)$ is the specified incoming radiation incident on the boundary from the left, and the half-range coefficients $a(\nu_{0})$, $\{ a(\nu)\}_{\nu\in[0,1]}$ are to be evaluated using the $W$-function of Appendix 4. We now exploit the relative simplicity of the full-range calculations by replacing Eq. (\[Eqn: BC\]) by Eq. (\[Eqn: HRFR\_Discrete\]) following, where the coefficients $\{ b(\nu_{i})\}_{i=0}^{N}$ are used to distinguish the full-range coefficients from the half-range ones. The significance of this change lies in the overwhelming simplicity of the full-range weight function $\mu$ as compared to the half-range function $W(\mu)$, and the resulting simplicity of the orthogonality relations that follow, see Appendix A4. The basic data of $z_{0}$ and $X(-\nu)$ are then completely generated self-consistently [@Sengupta1988; @Sengupta1995] by the discretized spectral approximation from the full-range adaption $$\sum_{i=0}^{N}b_{i}\phi_{\varepsilon}(\mu,\nu_{i})=\psi_{+}(\mu)+\psi_{-}(\mu),\qquad\mu\in[-1,1],\textrm{ }\nu_{i}\geq0\label{Eqn: HRFR_Discrete}$$
of the discretized boundary condition Eq. (\[Eqn: BC\]), where $\psi_{+}(\mu)$ is by definition the incident flux $\psi(\mu)$ for $\mu\in[0,1]$ and $0$ if $\mu\in[-1,0]$, while $$\psi_{-}(\mu)=\left\{ \begin{array}{ccl}
{\displaystyle \sum_{i=0}^{N}b_{i}^{-}\phi_{\varepsilon}(\mu,\nu_{i})} & & \textrm{if }\mu\in[-1,0],\textrm{ }\nu_{i}\geq0\textrm{ }\\
0 & & \textrm{if }\mu\in[0,1]\end{array}\right.$$ is the the emergent angular distribution out of the medium. Equation (\[Eqn: HRFR\_Discrete\]) corresponds to the full-range $\mu\in[-1,1],\textrm{ }\nu_{i}\geq0$ form $$b(\nu_{0})\phi(\mu,\nu_{0})+\int_{0}^{1}b(\nu)\phi(\mu,\nu)d\nu=\psi_{+}(\mu)+\left(b^{-}(\nu_{0})\phi(\mu,\nu_{0})+\int_{0}^{1}b^{-}(\nu)\phi(\mu,\nu)d\nu\right)\label{Eqn: HRFR}$$ of boundary condition (\[Eqn: BC\_HR\]) with the first and second terms on the right having the same interpretation as for Eq. (\[Eqn: HRFR\_Discrete\]). This full-range simulation merely states that the solution (\[Eqn: CaseSolution\_HR\]) of Eq. (\[Eqn: NeutronTransport\]) holds for all $\mu\in[-1,1]$, $x\geq0$, although it was obtained, unlike in the regular full-range case, from the given radiation $\psi(\mu)$ incident on the boundary at $x=0$ over only half the interval $\mu\in[0,1]$. To obtain the simulated full-range coefficients $\{ b_{i}\}$ and $\{ b_{i}^{-}\}$ of the half-range problem, we observe that there are effectively only half the number of coefficients as compared to a normal full-range problem because $\nu$ is now only over half the full interval. This allows us to generate two sets of equations from (\[Eqn: HRFR\]) by integrating with respect to $\mu\in[-1,1]$ with $\nu$ in the half intervals $[-1,0]$ and $[0,1]$ to obtain the two sets of coefficients $b^{-}$ and $b$ respectively. Accordingly we get from Eq. (\[Eqn: HRFR\_Discrete\]) with $\textrm{ }j=0,1,\cdots,N$ the sets of equations
$$\begin{array}{c}
{\displaystyle {\displaystyle (\psi,\phi_{j-})_{\mu}^{(+)}=-\sum_{i=0}^{N}b_{i}^{-}(\phi_{i+},\phi_{j-})_{\mu}^{(-)}}}\\
b_{j}={\displaystyle \left((\psi,\phi_{j+})_{\mu}^{(+)}+\sum_{i=0}^{N}b_{i}^{-}(\phi_{i+},\phi_{j+})_{\mu}^{(-)}\right)}\end{array}\label{Eqn: FRBC1}$$
where $(\phi_{j\pm})_{j=1}^{N}$ represents $(\phi_{\varepsilon}(\mu,\pm\nu_{j}))_{j=1}^{N}$, $\phi_{0\pm}=\phi(\mu,\pm\nu_{0})$, the $(+)$ $(-)$ superscripts are used to denote the integrations with respect to $\mu\in[0,1]$ and $\mu\in[-1,0]$ respectively, and $(f,g)_{\mu}$ denotes the usual inner product in $[-1,1]$ with respect to the full range weight $\mu$. While the first set of $N+1$ equations give $b_{i}^{-}$, the second set produces the required $b_{j}$ from these ‘"negative‘" coefficients. By equating these calculated $b_{i}$ with the exact half-range expressions for $a(\nu)$ with respect to $W(\mu)$ as outlined in Appendix A4, it is possible to find numerical values of $z_{0}$ and $X(-\nu)$. Thus from the second of Eq. (\[Eqn: Constant\_Coeff\]), $\{ X(-\nu_{i})\}_{i=1}^{N}$ is obtained with $b_{i\textrm{B}}\textrm{ }=a_{i\textrm{B}}$, $i=1,\cdots,N$, which is then substituted in the second of Eq. (\[Eqn: Milne\_Coeff\]) with $X(-\nu_{0})$ obtained from $a_{\textrm{A}}(\nu_{0})$ according to Appendix A4, to compare the respective $a_{i\textrm{A}}$ with the calculated $b_{i\textrm{A}}$ from (\[Eqn: FRBC1\]). Finally the full-range coefficients of Problem A can be used to obtain the $X(-\nu)$ values from the second of Eqs. (\[Eqn: Milne\_Coeff\]) and compared with the exact tabulated values as in Table \[Table: X-function\]. The tabulated values of $cz_{0}$ from Eq. (\[Eqn: extrapolated\]) show a consistent deviation from our calculations of Problem A according to $a_{\textrm{A}}(\nu_{0})=-\exp(-2z_{0}/\nu_{0})$. Since the $X(-\nu)$ values of Problem A in Table \[Table: X-function\] also need the same $b_{0\textrm{A}}$ as input that was used in obtaining $z_{0}$, it is reasonable to conclude that the ‘"exact‘" numerical integration of $z_{0}$ is inaccurate to the extent displayed in Table \[Table: extrapolated\].
----- --------- --------- --------- --------
$N=2$ $N=6$ $N=10$ Exact
0.2 0.78478 0.78478 0.78478 0.7851
0.4 0.72996 0.72996 0.72996 0.7305
0.6 0.71535 0.71536 0.71536 0.7155
0.8 0.71124 0.71124 0.71124 0.7113
0.9 0.71060 0.71060 0.71061 0.7106
----- --------- --------- --------- --------
: \[Table: extrapolated\][Extrapolated end-point $z_{0}$.]{}
-- -- ----------- ----------- ----------- ----------
$\nu_{i}$ Problem A Problem B Exact
0.2133 0.8873091 0.8873091 0.887308
0.7887 0.5826001 0.5826001 0.582500
0.0338 1.3370163 1.3370163 1.337015
0.1694 1.0999831 1.0999831 1.099983
0.3807 0.8792321 0.8792321 0.879232
0.6193 0.7215240 0.7215240 0.721524
0.8306 0.6239109 0.6239109 0.623911
0.9662 0.5743556 0.5743556 0.574355
0.0130 1.5971784 1.5971784 1.597163
0.0674 1.4245314 1.4245314 1.424532
0.1603 1.2289940 1.2289940 1.228995
0.2833 1.0513750 1.0513750 1.051376
0.4255 0.9058140 0.9058410 0.905842
0.5744 0.7934295 0.7934295 0.793430
0.7167 0.7102823 0.7102823 0.710283
0.8397 0.6516836 0.6516836 0.651683
0.9325 0.6136514 0.6136514 0.613653
0.9870 0.5933988 0.5933988 0.593399
-- -- ----------- ----------- ----------- ----------
: [\[Table: X-function\]$X(-\nu)$ by the full range method.]{}
From these numerical experiments and Fig. \[Fig: Case\] we may conclude that the continuous spectrum $[-1,1]$ of the position operator $\mu$ acts as the $\mathcal{D}_{+}$ points in generating the multifunctional Case singular eigenfunction $\phi(\mu,\nu)$. Its rational approximation $\phi_{\varepsilon}(\mu,\nu)$ in the context of the simple simulated full-range computations of the complex half-range exact theory of Appendix A4, clearly demonstrates the utility of graphical convergence of sequence of functions to multifunction. The totality of the multifunctions $\phi(\mu,\nu)$ for all $\nu$ in Fig. \[Fig: Case\](c) and (d) endows the problem with the character of maximal ill-posedness that is characteristic of chaos. This chaotic signature of the transport equation is however latent as the experimental output $\Phi(x,\mu)$ is well-behaved and regular. This important example shows how nature can use hidden and complex chaotic substates to generate order through a process of superposition.
**6. Does Nature support complexity?**
The question of this section is basic in the light of the theory of chaos presented above as it may be reformulated to the inquiry of what makes nature support chaoticity in the form of increasing non-injectivity of an input-output system. It is the purpose of this Section to exploit the connection between spectral theory and the dynamics of chaos that has been presented in the previous section. Since linear operators on finite dimensional spaces do not possess continuous or residual spectra, spectral theory on infinite dimensional spaces essentially involves limiting behaviour to infinite dimensions of the familiar matrix eigenvalue-eigenvector problem. As always this means extensions, dense embeddings and completions of the finite dimensional problem that show up as generalized eigenvalues and eigenvectors. In its usual form, the goal of nonlinear spectral theory consists [@Appel2000] in the study of $T_{\lambda}^{-1}$ for nonlinear operators $T_{\lambda}$ that satisfy more general continuity conditions, like differentiability and Lipschitz continuity, than simple boundedness that is enough for linear operators. The following generalization of the concept of the spectrum of a linear operator to the nonlinear case is suggestive. For a nonlinear map, $\lambda$ need not appear only in a multiplying role, so that an eigenvalue equation can be written more generally as a fixed-point equation $$f(\lambda;x)=x$$ with a fixed point corresponding to the eigenfunction of a linear operator and an “eigenvalue” being the value of $\lambda$ for which this fixed point appears. The correspondence of the residual and continuous parts of the spectrum are, however, less trivial than for the point spectrum. This is seen from the following two examples, [@Roman1975]. Let $Ae_{k}=\lambda_{k}e_{k},\textrm{ }k=1,2,\cdots$ be an eigenvalue equation with $e_{j}$ being the $j^{\textrm{th}}$ unit vector. Then $(A-\lambda)e_{k}:=(\lambda_{k}-\lambda)e_{k}=0$ iff $\lambda=\lambda_{k}$ so that $\{\lambda_{k}\}_{k=1}^{\infty}\in P\sigma(A)$ are the only eigenvalues of $A$. Consider now $(\lambda_{k})_{k=1}^{\infty}$ to be a sequence of real numbers that tends to a finite $\lambda^{*}$; for example let $A$ be a diagonal matrix having $1/k$ as its diagonal entries. Then $\lambda^{*}$ belongs to the continuous spectrum of $A$ because $(A-\lambda^{*})e_{k}=(\lambda_{k}-\lambda^{*})e_{k}$ with $\lambda_{k}\rightarrow\lambda^{*}$ implies that $(A-\lambda^{*})^{-1}$ is an unbounded linear operator and $\lambda^{*}$ a generalized eigenvalue of $A$. In the second example $Ae_{k}=e_{k+1}/(k+1)$, it is not difficult to verify that: (a) The point spectrum of $A$ is empty, (b) The range of $A$ is not dense because it does not contain $e_{1}$, and (c) $A^{-1}$ is unbounded because $Ae_{k}\rightarrow0$. Thus the generalized eigenvalue $\lambda^{*}=0$ in this case belongs to the residual spectrum of $A$. In either case, $\lim_{j\rightarrow\infty}e_{j}$ is the corresponding generalized eigenvector that enlarges the trivial null space $\mathcal{N}(\mathscr L_{\lambda^{*}})$ of the generalized eigenvalue $\lambda^{*}$. In fact in these two and the Dirac delta example of Sec. 5 of continuous and residual spectra, the generalized eigenfunctions arise as the limits of a sequence of functions whose images under the respective $\mathcal{L}_{\lambda}$ converge to $0$; recall the definition of footnote \[Foot: gen\_eigen\]. This observation generalizes to the dense extension $\textrm{Multi}_{|}(X,Y)$ of $\textrm{Map}(X,Y)$ as follows. If $x\in\mathcal{D}_{+}$ is not a fixed point of $f(\lambda;x)=x$, but there is some $n\in\mathbb{N}$ such that $f^{n}(\lambda;x)=x$, then the limit $n\rightarrow\infty$ generates a multifunction at $x$ as was the case with the delta function in the previous section and the various other examples that we have seen so far in the earlier sections.
One of the main goals of investigations on the spectrum of nonlinear operators is to find a set in the complex plane that has the usual desirable properties of the spectrum of a linear operator, @Appel2000. In this case, the focus has been to find a suitable class of operators $\mathcal{C}(X)$ with $T\in\mathcal{C}(X)$, such that the resolvent set is expressed as$$\rho(T)=\{\lambda\in\mathbb{C}\!:(T_{\lambda}\textrm{ is }1:1)(\textrm{Cl}(\mathcal{R}(T_{\lambda})=X)\textrm{ and }(T_{\lambda}^{-1}\in\mathcal{C}(X)\textrm{ on }\mathcal{R}(T_{\lambda}))\}$$
with the spectrum $\sigma(T)$ being defined as the complement of this set. Among the classes $\mathcal{C}(X)$ that have been considered, beside spaces of continuous functions $C(X)$, are linear boundedness $B(X)$, Frechet differentiability $C^{1}(X)$, Lipschitz continuity $\textrm{Lip}(X)$, and Granas quasiboundedness $Q(x)$, where $\textrm{Lip}(X)$ specifically takes into account the nonlinearity of $T$ to define $$\Vert T\Vert_{\textrm{Lip}}=\sup_{x\neq y}\frac{\Vert T(x)-T(y)\Vert}{\Vert x-y\Vert},\qquad|T|_{\textrm{lip}}=\inf_{x\neq y}\frac{\Vert T(x)-T(y)\Vert}{\Vert x-y\Vert}\label{Eqn: LipNorm}$$
that are plainly generalizations of the corresponding norms of linear operators. Plots of $f_{\lambda}^{-}(y)=\{ x\in\mathcal{D}(f-\lambda)\!:(f-\lambda)x=y\}$ for the functions $f\!:\mathbb{R}\rightarrow\mathbb{R}$$$\begin{array}{rcl}
f_{\lambda\textrm{a}}(x) & = & \left\{ \begin{array}{cll}
-1-\lambda x, & & x<-1\\
(1-\lambda)x, & & -1\leq x\leq1\\
1-\lambda x, & & 1<x,\end{array}\right.\\
\\f_{\lambda\textrm{b}}(x) & = & \left\{ \begin{array}{cll}
-\lambda x, & & x<1\\
(1-\lambda x)-1, & & 1\leq x\leq2\\
1-\lambda x, & & 2<x\end{array}\right.\\
\\f_{\lambda\textrm{c}}(x) & = & \left\{ \begin{array}{cll}
-\lambda x & & x<1\\
\sqrt{x-1}-\lambda x & & 1\leq x,\end{array}\right.\\
\\f_{\lambda\textrm{d}}(x) & = & \left\{ \begin{array}{cll}
(x-1)^{2}+1-\lambda x & & 1\leq x\leq1\\
(1-\lambda)x & & \textrm{otherwise}\end{array}\right.\\
\\f_{\lambda\textrm{e}}(x) & = & \tan^{-1}(x)-\lambda x,\\
\\f_{\lambda\textrm{f}}(x) & = & \left\{ \begin{array}{cll}
1-2\sqrt{-x}-\lambda x, & & x<-1\\
(1-\lambda)x, & & -1\leq x\leq1\\
2\sqrt{x}-1-\lambda x, & & 1<x\end{array}\right.\end{array}$$ taken from @Appel2000 are shown in Fig. \[Fig: Appel\]. It is easy to verify that the Lipschitz and linear upper and lower bounds of these maps are as in Table \[Table: Appel\_bnds\].
The point spectrum defined by $$P\sigma(f)=\{\lambda\in\mathbb{C}\!:(f-\lambda)x=0\textrm{ for some }x\neq0\}$$
is the simplest to calculate. Because of the special role played by the zero element $0$ in generating the point spectrum in the linear case, the bounds $m\Vert x\Vert\leq\Vert\mathscr{L}x\Vert\leq M\Vert x\Vert$ together with $\mathscr{L}x=\lambda x$ imply $\textrm{Cl}(P\sigma(\mathscr{L}))=[\Vert\mathscr{L}\Vert_{\textrm{b}},\Vert\mathscr{L}\Vert_{\textrm{B}}]$ — where the subscripts denote the lower and upper bounds in Eq. (\[Eqn: LipNorm\]) and which is sometimes taken to be a descriptor of the point spectrum of a nonlinear operator — as can be seen in Table \[Table: Appel\_spectra\] and verified from Fig. \[Fig: Appel\]. The remainder of the spectrum, as the complement of the resolvent set, is more difficult to find. Here the convenient characterization of the resolvent of a continuous linear operator as the set of all sufficiently large $\lambda$ that satisfy $|\lambda|>M$ is of little significance as, unlike for a linear operator, the non-existence of an inverse is not just due the set $\{ f^{-1}(0)\}$ which happens to be the only way a linear map can fail to be injective. Thus the map defined piecewise as $\alpha+2(1-\alpha)x$ for $0\leq x<1/2$ and $2(1-x)$ for $1/2\leq x\leq1$, with $0<\alpha<1$, is not invertible on its range although $\{ f^{-}(0)\}=1$. Comparing Fig. \[Fig: Appel\] and Table \[Table: Appel\_bnds\], it is seen that in cases (b), (c) and (d), the intervals $[|f|_{\textrm{b}},\Vert f\Vert_{\textrm{B}}]$ are subsets of the $\lambda$-values for which the respective maps are not injective; this is to be compared with (a), (e) and (f) where the two sets are the same. Thus the linear bounds are not good indicators of the uniqueness properties of solution of nonlinear equations for which the Lipschitzian bounds are seen to be more appropriate.
[|c||c|c|c|c|]{} Function& $|f|_{\textrm{b}}$& $\Vert f\Vert_{\textrm{B}}$& $|f|_{\textrm{lip}}$& $\Vert f\Vert_{\textrm{Lip}}$[\
]{} $f_{\textrm{a}}$& $0$& $1$& $0$& $1$[\
]{} $f_{\textrm{b}}$& $0$& $1/2$& $0$& $1$[\
]{} $f_{\textrm{c}}$& $0$& $1/2$& $0$& $\infty$[\
]{} $f_{\textrm{d}}$& $2(\sqrt{2}-1)$& $\infty$& $0$& $2$[\
]{}$f_{\textrm{e}}$& $0$& $1$& $0$& $1$[\
]{} $f_{\textrm{f}}$& $0$& $1$& $0$& $1$[\
]{}
[|c||c|c|]{} Functions& $\sigma_{\textrm{Lip}}(f)$& $P\sigma(f)$[\
]{} $f_{\textrm{a}}$& $[0,1]$& $(0,1]$[\
]{} $f_{\textrm{b}}$& $[0,1]$& $[0,1/2]$[\
]{} $f_{\textrm{c}}$& $[0,\infty)$& $[0,1/2]$[\
]{} $f_{\textrm{d}}$& $[0,2]$& $[2(\sqrt{2}-1),1]$[\
]{} $f_{\textrm{e}}$& $[0,1]$& $(0,1)$[\
]{} $f_{\textrm{f}}$& $[0,1]$& $(0,1)$[\
]{}
In view of the above, we may draw the following conclusions. If we choose to work in the space of multifunctions $\textrm{Multi}(X,\mathcal{T})$, with $\mathcal{T}$ the topology of pointwise biconvergence, when all functional relations are (multi) invertible on their ranges, we may make the following definition for the net of functions $f(\lambda;x)$ satisfying $f(\lambda;x)=x$.
**Definition 6.1.** *Let* $f(\lambda;\cdot)\in\textrm{Multi}(X,\mathcal{T})$ *be a function. The resolvent set of $f$ is given by* $$\rho(f)=\{\lambda\!:(f(\lambda;\cdot)^{-1}\in\textrm{Map}(X,\mathcal{T}))\wedge(\textrm{Cl}(\mathcal{R}(f(\lambda;\cdot))=X)\},$$
*and any $\lambda$ not in $\rho$ is in the spectrum of $f$.$\qquad\square$*
Thus apart from multifunctions, $\lambda\in\sigma(f)$ also generates functions on the boundary of functional and non-functional relations in $\textrm{Multi}(X,\mathcal{T})$. While it is possible to classify the spectrum into point, continuous and residual subsets, as in the linear case, it is more meaningful for nonlinear operators to consider $\lambda$ as being either in the *boundary spectrum* $\textrm{Bdy}(\sigma(f))$ or in the *interior spectrum* $\textrm{Int}(\sigma(f))$, depending on whether or not the multifunction $f(\lambda;\cdot)^{-}$ arises as the graphical limit of a net of functions in either $\rho(f)$ or $R\sigma(f)$. This is suggested by the spectra arising from the second row of Table \[Table: spectrum\] (injective $\mathcal{L}_{\lambda}$ and discontinuous $\mathcal{L}_{\lambda}^{-1}$) that lies sandwiched in the $\lambda$-plane between the two components arising from the first and third rows, see @Naylor1971 Sec. 6.6, for example. According to this simple scheme, the spectral set is a closed set with its boundary and interior belonging to $\textrm{Bdy}(\sigma(f))$ and $\textrm{Int}(\sigma(f))$ respectively. Table \[Table: Appel\_multi\] shows this division for the examples in Fig. \[Fig: Appel\]. Because $0$ is no more significant than any other point in the domain of a nonlinear map in inducing non-injectivity, the division of the spectrum into the traditional sets would be as shown in Table \[Table: Appel\_multi\]; compare also with the conventional linear point spectrum of Table \[Table: Appel\_spectra\]. In this nonlinear classification, the point spectrum consists of any $\lambda$ for which the inverse $f(\lambda;\cdot)^{-}$ is set-valued, irrespective of whether this is produced at $0$ or not, while the continuous and residual spectra together comprise the boundary spectrum. Thus a $\lambda$ can be both in the point and the continuous or residual spectra which need not be disjoint. The continuous and residual spectra are included in the boundary spectrum which may also contain parts of the point spectrum.
Function $\textrm{Int}(\sigma(f))$ $\textrm{Bdy}(\sigma(f))$ $P\sigma(f)$ $C\sigma(f)$ $R\sigma(f)$
------------------ --------------------------- --------------------------- -------------- -------------- --------------
$f_{\textrm{a}}$ $(0,1)$ $\{0,1\}$ $[0,1]$ $\{1\}$ $\{0\}$
$f_{\textrm{b}}$ $(0,1)$ $\{0,1\}$ $[0,1]$ $\{1\}$ $\{0\}$
$f_{\textrm{c}}$ $(0,\infty)$ $\{0\}$ $[0,\infty)$ $\{0\}$ $\emptyset$
$f_{\textrm{d}}$ $(0,2)$ $\{0,2\}$ $(0,2)$ $\{0,2\}$ $\emptyset$
$f_{\textrm{e}}$ $(0,1)$ $\{0,1\}$ $(0,1)$ $\{1\}$ $\{0\}$
$f_{\textrm{f}}$ $(0,1)$ $\{0,1\}$ $(0,1)$ $\{0,1\}$ $\emptyset$
: \[Table: Appel\_multi\][Nonlinear spectra of functions of Fig. \[Fig: Appel\]. Compare the present point spectra with the usual linear spectra of Table \[Table: Appel\_spectra\].]{}
**Example 6.1.** To see how these concepts apply to linear mappings, consider the equation $(D-\lambda)y(x)=r(x)$ where $D=d/dx$ is the differential operator on $L^{2}[0,\infty)$, and let $\lambda$ be real. For $\lambda\neq0$, the unique solution of this equation in $L^{2}[0,\infty)$, is
$$\begin{aligned}
y(x)= & \left\{ \begin{array}{ll}
{\displaystyle e^{\lambda x}\left(y(0)+\int_{0}^{x}e^{-\lambda x^{\prime}}r(x^{\prime})dx^{\prime}\right)}, & \lambda<0\\
{\displaystyle e^{\lambda x}\left(y(0)-\int_{x}^{\infty}e^{-\lambda x^{\prime}}r(x^{\prime})dx^{\prime}\right),} & \lambda>0\end{array}\right.\end{aligned}$$
showing that for $\lambda>0$ the inverse is functional so that $\lambda\in(0,\infty)$ belongs to the resolvent of $D$. However, when $\lambda<0$, apart from the $y=0$ solution (since we are dealing a with linear problem, only $r=0$ is to be considered), $e^{\lambda x}$ is also in $L^{2}[0,\infty)$ so that all such $\lambda$ are in the point spectrum of $D$. For $\lambda=0$ and $r\neq0$, the two solutions are not necessarily equal unless $\int_{0}^{\infty}r(x)=0$, so that the range $\mathcal{R}(D-I)$ is a subspace of $L^{2}[0,\infty)$. To complete the problem, it is possible to show [@Naylor1971] that $0\in C\sigma(D)$, see Ex. 2.2; hence the continuous spectrum forms at the boundary of the functional solution for the resolvent-$\lambda$ and the multifunctional solution for the point spectrum. With a slight variation of problem to $y(0)=0$, all $\lambda<0$ are in the resolvent set, while $\lambda>0$ the inverse is bounded but must satisfy $y(0)=\int_{0}^{\infty}e^{-\lambda x}r(x)dx=0$ so that $\textrm{Cl}(\mathcal{R}(D-\lambda))\neq L^{2}[0,\infty)$. Hence $\lambda>0$ belong to the residual spectrum. The decomposition of the complex $\lambda$-plane for these and some other linear spectral problems taken from @Naylor1971 is shown in Fig. \[Fig: spectrum\]. In all cases, the spectrum due to the second row of Table \[Table: spectrum\] acts as a boundary between that arising from the first and third rows, which justifies our division of the spectrum for a nonlinear operator into the interior and boundary components. Compare Example 2.2.$\qquad\blacksquare$
From the basic representation of the resolvent operator $(\mathbf{1}-f)^{-1}$ $$\mathbf{1}+f+f^{2}+\cdots+f^{i}+\cdots$$
in $\textrm{Multi}(X)$, if the iterates of $f$ converge to a multifunction for some $\lambda$, then that $\lambda$ must be in the spectrum of $f$, which means that the control parameter of a chaotic dynamical system is in its spectrum. Of course, the series can sum to a multi even otherwise: take $f_{\lambda}(x)$ to be identically $x$ with $\lambda=1$, for example, to get $1\in P\sigma(f)$. A comparison of Tables \[Table: spectrum\] and \[Table: Appel\_spectra\] reveal that in case (d), for example, $0$ and $2$ belong to the Lipschtiz spectrum because although $f_{\textrm{d}}^{-1}$ is not Lipschitz continuous, $\Vert f\Vert_{\textrm{Lip}}=2$. It should also be noted that the boundary between the functional resolvent and multifunctional spectral set is formed by the graphical convergence of a net of resolvent functions while the multifunctions in the interior of the spectral set evolve graphically independent of the functions in the resolvent. The chaotic states forming the boundary of the functional and multifunctional subsets of $\textrm{Multi}(X)$ marks the transition from the less efficient functional state to the more efficient multifunctional one.
These arguments also suggest the following. The countably many outputs arising from the non-injectivity of $f(\lambda;\cdot)$ corresponding a given input can be interpreted to define *complexity because* *in a nonlinear system each of these possibilities constitute a experimental result in itself that may not be combined in any definite predetermined manner.* This is in sharp contrast to linear systems where a linear combination, governed by the initial conditions, always generate a unique end result; recall also the combination offered by the singular generalized eigenfunctions of neutron transport theory. This multiplicity of possibilities that have no definite combinatorial property is the basis of the diversity of nature, and is possibly responsible for Feigenbaum’s “historical prejudice”, [@Feigenbaum1992], see Prelude, 2. Thus *order* represented by the functional resolvent passes over to *complexity* of the countably multifunctional interior spectrum via the uncountably multifunctional boundary that is a prerequisite for *chaos.* We may now strengthen our hypothesis offered at the end of the previous section in terms of the examples of Figs. \[Fig: Appel\] and \[Fig: spectrum\], that nature uses chaoticity as an intermediate step to the attainment of states that would otherwise be inaccessible to it. Well-posedness of a system is an extremely inefficient way of expressing a multitude of possibilities as this requires a different input for every possible output. Nature chooses to express its myriad manifestations through the multifunctional route leading either to averaging as in the delta function case or to a countable set of well-defined states, as in the examples of Fig. \[Fig: Appel\] corresponding to the interior spectrum. Of course it is no distraction that the multifunctional states arise respectively from $f_{\lambda}$ and $f_{\lambda}^{-}$ in these examples as $f$ is a function on $X$ that is under the influence of both $f$ and its inverse. The functional resolvent is, for all practical purposes, only a tool in this structure of nature.
The equation $f(x)=y$ is typically an input-output system in which the inverse images at a functional value $y_{0}$ represents a set of input parameters leading to the same experimental output $y_{0};$ this is stability characterized by a complete insensitivity of the output to changes in input. On the other hand, a continuous multifunction at $x_{0}$ is a signal for a hypersensitivity to input because the output, which is a definite experimental quantity, is a choice from the possibly infinite set $\{ f(x_{0})\}$ made by a choice function which represents the experiment at that particular point in time. Since there will always be finite differences in the experimental parameters when an experiment is repeated, the choice function (that is the experimental output) will select a point from $\{ f(x_{0})\}$ that is representative of that experiment and which need not bear any definite relation to the previous values; this is instability and signals sensitivity to initial conditions. Such a state is of high entropy as the number of available states $f_{\textrm{C}}(\{ f(x_{0})\})$ — where $f_{\textrm{C}}$ is the choice function — is larger than a functional state represented by the singleton $\{ f(x_{0})\}.$
**Epilogue**
@Gleick1987
**Appendix**
This Appendix gives a brief overview of some aspects of topology that are necessary for a proper understanding of the concepts introduced in this work.
**A1. Convergence in Topological Spaces: Sequence, Net and Filter.**
In the theory of convergence in topological spaces, *countability* plays an important role. To understand the significance of this concept, some preliminaries are needed.
The notion of a basis, or base, is a familiar one in analysis: a base is a subcollection of a set which may be used to construct, in a specified manner, any element of the set. This simplifies the statement of a problem since a smaller number of elements of the base can be used to generate the larger class of every element of the set. This philosophy finds application in topological spaces as follows.
Among the three properties $(\textrm{N}1)-(\textrm{N}3)$ of the neighbourhood system $\mathcal{N}_{x}$ of Tutorial4, (N1) and (N2) are basic in the sense that the resulting subcollection of $\mathcal{N}_{x}$ can be used to generate the full system by applying $(\textrm{N}3)$; this *basic neighbourhood* *system*, or *neighbourhood (local) bas*e $\mathcal{B}_{x}$ *at* $x$, is characterized by
(NB1) $x$ belongs to each member $B$ of $\mathcal{B}_{x}$*.*
(NB2) The intersection of any two members of **$\mathcal{B}_{x}$ **contains another member of $\mathcal{B}_{x}$: $B_{1},B_{2}\in\mathcal{B}_{x}\Rightarrow(\exists B\in\mathcal{B}_{x}\!:B\subseteq B_{1}\bigcap B_{2})$. **
Formally, compare Eq. (\[Eqn: nbd-topology\]),
**Definition A1.1.** *A neighbourhood (local) base* $\mathcal{B}_{x}$ *at $x$ in a topological space $(X,\mathcal{U})$ is a subcollection of the neighbourhood system $\mathcal{N}_{x}$ having the property that each $N\in\mathcal{N}_{x}$ contains some member of* $\mathcal{B}_{x}$*.* *Thus* $$\mathcal{B}_{x}\overset{\textrm{def}}=\{ B\in\mathcal{N}_{x}\!:x\in B\subseteq N\textrm{ for each }N\in\mathcal{N}_{x}\}\label{Eqn: TBx}$$ *determines the full neighbourhood system* $$\mathcal{N}_{x}=\{ N\subseteq X\!:x\in B\subseteq N\textrm{ for some }B\textrm{ }\in\,\mathcal{B}_{x}\}\label{Eqn: TBx_nbd}$$
*reciprocally as all supersets of the basic elements.$\qquad\square$*
The entire neighbourhood system $\mathcal{N}_{x}$, which is recovered from the base by forming all supersets of the basic neighbourhoods, **is trivially a local base at $x$; non-trivial examples are given below.
The second example of a base, consisting as usual of a subcollection of a given collection, is the topological base $_{\textrm{T}}\mathcal{B}$ that allows the specification of the topology on a set $X$ in terms of a smaller collection of open sets.
**Definition A1.2.** *A base* $_{\textrm{T}}\mathcal{B}$ *in a topological space $(X,\mathcal{U})$ is a subcollection of the topology $\mathcal{U}$ having the property that each $U\in\mathcal{U}$ contains some member of* $_{\textrm{T}}\mathcal{B}$*.* *Thus* $$_{\textrm{T}}\mathcal{B}\overset{\textrm{def}}=\{ B\in\mathcal{U}\!:B\subseteq U\textrm{ for each }U\in\mathcal{U}\}\label{Eqn: TB}$$ *determines reciprocally the topology $\mathcal{U}$ as* $$\mathcal{U}=\left\{ U\subseteq X\!:U=\bigcup_{B\in\,\!_{\textrm{T}}\mathcal{B}\,}B\right\} \qquad\square\label{Eqn: TB_topo}$$
This means that the topology on $X$ can be reconstructed form the base by taking all possible unions of members of the base, and a collection of subsets of a set $X$ is a topological base iff Eq. (\[Eqn: TB\_topo\]) of arbitrary unions of elements of $_{\textrm{T}}\mathcal{B}$ generates a topology on $X$. This topology, which is the coarsest (that is the smallest) that contains $_{\textrm{T}}\mathcal{B}$, is obviously closed under finite intersections. Since the open set $\textrm{Int}(N)$ is a neighbourhood of $x$ whenever $N$ is, Eq. (\[Eqn: TBx\_nbd\]) and the definition Eq. (\[Eqn: Def: nbd system\]) of $\mathcal{N}_{x}$ implies that *the open neighbourhood system of any point in a topological space is an example of a neighbourhood base at that point,* an observation that has often led, together with Eq. (\[Eqn: TB\]), to the use of the term “neighbourhood” as a synonym for “non-empty open set”. The distinction between the two however is significant as neighbourhoods need not necessarily be open sets; thus while not necessary, it is clearly sufficient for the local basic sets $B$ to be open in Eqs. (\[Eqn: TBx\]) and (\[Eqn: TBx\_nbd\]). If Eq. (\[Eqn: TBx\_nbd\]) holds for every $x\in N$, then the resulting $\mathcal{N}_{x}$ reduces to the topology induced by the open basic neighbourhood system $\mathcal{B}_{x}$ as given by Eq. (\[Eqn: nbd-topology\]).
In order to check if a collection of subsets $_{\textrm{T}}\mathcal{B}$ of $X$ qualifies to be a basis, it is not necessary to verify properties $(\textrm{T}1)-(\textrm{T}3)$ of Tutorial4 for the class (\[Eqn: TB\_topo\]) generated by it because of the properties (TB1) and (TB2) below whose strong affinity to (NB1) and (NB2) is formalized in Theorem A1.1.
**Theorem A1.1.** *A collection* $_{\textrm{T}}\mathcal{B}$ *of subsets of $X$ is a* *topological basis on* $X$ *iff*
(TB1) *$X=\bigcup_{B\in\,_{\textrm{T}}\mathcal{B}}B$. Thus each $x\in X$ must belong to some* $B\in\,_{\textrm{T}}\mathcal{B}$ *which implies the existence of a* *local base* *at each point* *$x\in X$.*
(TB2) *The intersection of any two members $B_{1}$ and $B_{2}$ of* $_{\textrm{T}}\mathcal{B}$ *with $x\in B_{1}\bigcap B_{2}$* ***contains another member of* $_{\textrm{T}}\mathcal{B}$: $(B_{1},B_{2}\in\,_{\textrm{T}}\mathcal{B})\wedge(x\in B_{1}\bigcap B_{2})\Rightarrow(\exists B\in\,_{\textrm{T}}\mathcal{B}\!:x\in B\subseteq B_{1}\bigcap B_{2})$.$\qquad\square$
This theorem, together with Eq. (\[Eqn: TB\_topo\]) ensures that a given collection of subsets of a set $X$ satisfying (TB1) and (TB2) induces *some* topology on $X$; compared to this is the result that *any* collection of subsets of a set $X$ is a *subbasis* for some topology on $X$. If $X$, however, already has a topology $\mathcal{U}$ imposed on it, then Eq. **(\[Eqn: TB\]) must also be satisfied in order that the topology generated by $_{\textrm{T}}\mathcal{B}$ is indeed $\mathcal{U}$. The next theorem connects the two types of bases of Defs. A1.1 and A1.2 by asserting that although a local base of a space need not consist of open sets and a topological base need not have any reference to a point of $X$, any subcollection of the base containing a point is a local base at that point.
**Theorem A1.2.** *A collection of open sets* $_{\textrm{T}}\mathcal{B}$ *is a base for a topological space $(X,\mathcal{U})$ iff for each $x\in X$, the subcollection* $$\mathcal{B}_{x}=\{ B\in\mathcal{U}\!:x\in B\in\!\,_{\textrm{T}}\mathcal{B}\}\label{Eqn: base_local base}$$ *of basic sets containing $x$ is a local base at* $x$.$\qquad\square$
**Proof.** *Necessity.* Let $_{\textrm{T}}\mathcal{B}$ be a base of *$(X,\mathcal{U})$* and $N$ be a neighbourhood of $x$, so that $x\in U\subseteq N$ for some open set $U=\bigcup_{B\in\!\,_{\textrm{T}}\mathcal{B}}B$ and basic open sets $B$. Hence $x\in B\subseteq N$ shows, from Eq. (\[Eqn: TBx\]), that $B\in\mathcal{B}_{x}$ is a local basic set at $x$.
*Sufficiency.* If $U$ is an open set of $X$ containing $x$, then the definition of local base Eq. (\[Eqn: TBx\]) requires $x\in B_{x}\subseteq U$ for some subcollection of basic sets $B_{x}$ in $\mathcal{B}_{x}$; hence $U=\bigcup_{x\in U}B_{x}$. By Eq. (\[Eqn: TB\_topo\]) therefore, $_{\textrm{T}}\mathcal{B}$ is a topological base for $X$.$\qquad\blacksquare$
Because the basic sets are open, (TB2) of Theorem A1.1 leads to the following physically appealing paraphrase of Thm. A1.2.
**Corollary.** *A collection* $_{\textrm{T}}\mathcal{B}$ *of open sets of* $(X,\mathcal{U})$ *is a topological base that generates* $\mathcal{U}$ *iff for each open set $U$ of $X$ and each $x\in U$ there is an open set* $B\in\!\,_{\textrm{T}}\mathcal{B}$ *such that $x\in B\subseteq U$*; *that is iff* $$x\in U\in\mathcal{U}\Longrightarrow(\exists B\in\,_{\textrm{T}}\mathcal{B}\!:x\in B\subseteq U).\qquad\square$$ ****Example A1.1.** Some examples of local bases in $\mathbb{R}$ are intervals of the type $(x-\varepsilon,x+\varepsilon)$, $[x-\varepsilon,x+\varepsilon]$ for real $\varepsilon$, $(x-q,x+q)$ for rational $q$, $(x-1/n,x+1/n)$ for $n\in\mathbb{Z}_{+}$, while for a metrizable space with the topology induced by a metric $d$, each of the following is a local base at $x\in X$: $B_{\varepsilon}(x;d):=\{ y\in X:d(x,y)<\varepsilon\}$ and $D_{\varepsilon}(x;d):=\{ y\in X:d(x,y)\leq\varepsilon\}$ for $\varepsilon>0$, $B_{q}(x;d)$ for $\mathbb{Q}\ni q>0$ and $B_{1/n}(x;d)$ for $n\in\mathbb{Z}_{+}$. In $\mathbb{R}^{2}$, two neighbourhood bases at any $x\in\mathbb{R}^{2}$ are the disks centered at $x$ and the set of all squares at $x$ with sides parallel to the axes. Although these bases have no elements in common, they are nevertheless equivalent in the sense that they both generate the same (usual) topology in $\mathbb{R}^{2}$. Of course, the entire neighbourhood system at any point of a topological space is itself a (less useful) local base at that point. By Theorem A1.2, $B_{\varepsilon}(x;d)$, $D_{\varepsilon}(x;d)$, $\varepsilon>0$, $B_{q}(x;d)$, $\mathbb{Q}\ni q>0$ and $B_{1/n}(x;d)$, $n\in\mathbb{Z}_{+}$, for all $x\in X$ are examples of bases in a metrizable space with topology induced by a metric $d$.$\qquad\square$
In terms of local bases and bases, it is now possible to formulate the notions of first and second countability as follows.
**Definition A1.3.** *A topological space is* *first countable* *if each $x\in X$ has some countable neighbourhood base, and is* *second countable* *if it has a countable base.* $\qquad\square$
Every metrizable space $(X,d)$ is first countable as both $\{ B(x,q)\}_{\mathbb{Q}\ni q>0}$ and $\{ B(x,1/n)\}_{n\in\mathbb{Z}_{+}}$ are examples of countable neighbourhood bases at any $x\in(X,d)$; hence $\mathbb{R}^{n}$ is first countable. It should be clear that although every second countable space is first countable, *only a countable first countable space can be second countable*, and a common example of a uncountable first countable space that is also second countable is provided by $\mathbb{R}^{n}$. Metrizable spaces need not be second countable: any uncountable set having the discrete topology is as an example.
**Example A1.2.** The following is an important example of a space that is not first countable as it is needed for our pointwise biconvergence of Section 3. Let $\textrm{Map}(X,Y)$ be the set of all functions between the uncountable spaces $(X,\mathcal{U})$ and $(Y,\mathcal{V})$. Given any integer $I\geq1$, and any *finite* collection of points $(x_{i})_{i=1}^{I}$ of $X$ and of open sets $(V_{i})_{i=1}^{I}$ in $Y$, let $$B((x_{i})_{i=1}^{I};(V_{i})_{i=1}^{I})=\{ g\in\textrm{Map}(X,Y)\!:(g(x_{i})\in V_{i})(i=1,2,\cdots,I)\}\label{Eqn: point}$$
be the functions in $\textrm{Map}(X,Y)$ whose graphs pass through each of the sets $(V_{i})_{i=1}^{I}$ at $(x_{i})_{i=1}^{I}$, and let $_{\textrm{T}}\mathcal{B}$ be the collection of all such subsets of $\textrm{Map}(X,Y)$ for every choice of $I$, $(x_{i})_{i=1}^{I}$, and $(V_{i})_{i=1}^{I}$. The existence of a unique topology $\mathcal{T}$ — the *topology of pointwise convergence* on $\textrm{Map}(X,Y)$ — that is generated by the open sets $B$ of the collection $_{\textrm{T}}\mathcal{B}$ now follows because
(TB1) is satisfied: For any $f\in\textrm{Map}(X,Y)$ there must be some $x\in X$ and a corresponding $V\subseteq Y$ such that $f(x)\in V$, and
(TB2) is satisfied because $$B((s_{i})_{i=1}^{I};(V_{i})_{i=1}^{I})\bigcap B((t_{j})_{j=1}^{J};(W_{j})_{j=1}^{J})=B((s_{i})_{i=1}^{I},(t_{j})_{j=1}^{J};(V_{i})_{i=1}^{I},(W_{j})_{j=1}^{J})$$
implies that a function simultaneously belonging to the two open sets on the left must pass through each of the points defining the open set on the right.
We now demonstrate that $(\textrm{Map}(X,Y),\mathcal{T})$ is not first countable by verifying that it is not possible to have a countable local base at any $f\in\textrm{Map}(X,Y)$. If this is not indeed true, let $B_{f}^{I}((x_{i})_{i=1}^{I};(V_{i})_{i=1}^{I})=\{ g\in\textrm{Map}(X,Y)\!:(g(x_{i})\in V_{i})_{i=1}^{I}\}$, which denotes those members of $_{\textrm{T}}\mathcal{B}$ that contain $f$ with $V_{i}$ an open neighbourhood of $f(x_{i})$ in $Y$, be a countable local base at $f$, see Thm. A1.2. Since $X$ is uncountable, it is now possible to choose some $x^{*}\in X$ different from any of the $(x_{i})_{i=1}^{I}\textrm{ }$ (for example, let $x^{*}\in\mathbb{R}$ be an irrational for rational $(x_{i})_{i}^{I}\textrm{ }$), and let $f(x^{*})\in V^{*}$ where $V^{*}$ is an open neighbourhood of $f(x^{*})$. Then $B(x^{*};V^{*})$ is an open set in $\textrm{Map}(X,Y)$ containing $f$; hence from the definition of the local base, Eq. (\[Eqn: TBx\]), or equivalently from the Corollary to Theorem A1.2, there exists some (countable) $I\in\mathbb{N}$ such that $f\in B^{I}\subseteq B(x^{*};V^{*})$. However, $$\begin{array}{ccc}
f^{*}(x) & = & \begin{cases}
y_{i}\in V_{i}, & \textrm{if }x=x_{i},\textrm{ and }1\leq i\leq I\\
y^{*}\in V^{*} & \textrm{if }x=x^{*}\\
\textrm{arbitrary}, & \textrm{otherwise}\end{cases}\end{array}$$
is a simple example of a function on $X$ that is in $B^{I}$ (as it is immaterial as to what values the function takes at points other than those defining $B^{I}$), but not in $B(x^{*};V^{*})$. From this it follows that *a sufficient condition for the topology of pointwise convergence to be first countable is that $X$ be countable.*$\qquad\blacksquare$
Even though it is not first countable, $(\textrm{Map}(X,Y),\mathcal{T})$ is a Hausdorff space when $Y$ is Hausdorff. Indeed, if $f,g\in(\textrm{Map}(X,Y),\mathcal{T})$ with $f\neq g$, then $f(x)\neq g(x)$ for some $x\in X$. But then as $Y$ is Hausdorff, it is possible to choose disjoint open intervals $V_{f}$ and $V_{g}$ at $f(x)$ and $g(x)$ respectively.
With this background on first and second countability, it is now possible to go back to the question of nets, filters and sequences. Technically, a sequence on a set $X$ is a map $x\!:\mathbb{N}\rightarrow X$ from the set of natural numbers to $X$; instead of denoting this is in the usual functional manner of $x(i)\textrm{ with }i\in\mathbb{N}$, it is the standard practice to use the notation $(x_{i})_{i\in\mathbb{N}}$ for the terms of a sequence. However, if the space $(X,\mathcal{U})$ is not first countable (and as seen above this is not a rare situation), it is not difficult to realize that sequences are inadequate to describe convergence in $X$ simply because it can have only countably many values whereas the space may require uncountably many neighbourhoods to completely define the neighbourhood system at a point. The resulting uncountable generalizations of a sequence in the form of *nets* and *filters* is achieved through a corresponding generalization of the index set $\mathbb{N}$ to the directed set $\mathbb{D}$.
**Definition A1.4.** *A* *directed set* *$\mathbb{D}$ is a preordered set for which the order $\preceq$, known as a* *direction of* $\mathbb{D}$, *satisfies*
\(a) *$\alpha\in\mathbb{D}$ $\Rightarrow$ $\alpha\preceq\alpha$* (that is $\preceq$ is reflexive)*.*
\(b) **$\alpha,\beta,\gamma\in\mathbb{D}\textrm{ such that }(\alpha\preceq\beta\wedge\beta\preceq\gamma)$ $\Rightarrow$ $\alpha\preceq\gamma$ (that is $\preceq$ is transitive).
\(c) $\alpha,\beta\in\mathbb{D}$ $\Rightarrow$ $\exists\gamma\in\mathbb{D}\textrm{ such that }(\alpha\preceq\gamma)\wedge(\beta\preceq\gamma)$*.$\qquad\square$*
While the first two properties are obvious enough and constitutes the preordering of $\mathbb{{D}}$, the third which replaces antisymmetry, ensures that for any finite number of elements of the directed set (recall that a preordered set need not be fully ordered), there is always a successor. Examples of directed sets can be both straight forward, as any totally ordered set like $\mathbb{N}$, $\mathbb{R}$, $\mathbb{Q}$, or $\mathbb{Z}$ and all subsets of a set $X$ under the superset or subset relation (that is $(\mathcal{P}(X),\supseteq)$ or $(\mathcal{P}(X),\subseteq)$ that are directed by their usual ordering, and not quite so obvious as the following examples which are significantly useful in dealing with convergence questions in topological spaces, amply illustrate.
The neighbourhood system $$_{\mathbb{D}}N=\{ N\!:N\in\mathcal{N}_{x}\}$$ at a point $x\in X$, directed by the reverse inclusion direction $\preceq$ defined as $$M\preceq N\Longleftrightarrow N\subseteq M\qquad\textrm{for }M,N\in\mathcal{N}_{x},\label{Eqn: Direction1}$$
is a fundamental example of a *natural direction of $\mathcal{N}_{x}$*. In fact while reflexivity and transitivity are clearly obvious, (c) follows because for any $M,N\in\mathcal{N}_{x}$, $M\preceq M\bigcap N$ and $N\preceq M\bigcap N$. Of course, this direction is not a total ordering on $\mathcal{N}_{x}$. A more naturally useful directed set in convergence theory is $$_{\mathbb{D}}N_{t}=\{(N,t)\!:(N\in\mathcal{N}_{x})(t\in N)\}\label{Eqn: Directed}$$
under its *natural direction* $$(M,s)\preceq(N,t)\Longleftrightarrow N\subseteq M\qquad\textrm{for }M,N\in\mathcal{N}_{x};\label{Eqn: Direction2}$$ **$_{\mathbb{D}}N_{t}$ is more useful than $_{\mathbb{D}}N$ because, unlike the later, $_{\mathbb{D}}N_{t}$ does not require a simultaneous choice of points from every $N\in\mathcal{N}_{x}$ that implicitly involves a simultaneous application of the Axiom of Choice; see Examples A1.2(2) and (3) below. The general indexed variation
$$_{\mathbb{D}}N_{\beta}=\{(N,\beta)\!:(N\in\mathcal{N}_{x})(\beta\in\mathbb{D})(x_{\beta}\in N)\}\label{Eqn: DirectedIndexed}$$
of Eq. (\[Eqn: Directed\]), with natural direction $$(M,\alpha)\leq(N,\beta)\Longleftrightarrow(\alpha\preceq\beta)\wedge(N\subseteq M),\label{Eqn: DirectionIndexed}$$
often proves useful in applications as will be clear from the proofs of Theorems A1.3 and A1.4.
**Definition A1.5.** ***Net.*** *Let $X$ be any set and $\mathbb{D}$ a directed set. A net $\chi\!:\mathbb{D}\rightarrow X$* *in $X$* *is a function* *on the directed set $\mathbb{D}$ with values in $X$.$\qquad\square$*
A net, to be denoted as $\chi(\alpha)$, $\alpha\in\mathbb{D}$, is therefore a function indexed by a directed set. We adopt the convention of denoting nets in the manner of functions and do not use the sequential notation $\chi_{\alpha}$ that can also be found in the literature. Thus, while every sequence is a special type of net, $\chi:\!\mathbb{Z}\rightarrow X$ is an example of a net that is not a sequence.
Convergence of sequences and nets are described most conveniently in terms of the notions of being *eventually in* and *frequently in* every neighbourhood of points. We describe these concepts in terms of nets which apply to sequences with obvious modifications.
**Definition A1.6.** *A net* $\chi\!:\mathbb{D}\rightarrow X$ *is said to be*
\(a) *Eventually in* *a subset $A$* *of* *$X$ if its tail is eventually in $A$*: *$(\exists\beta\in\mathbb{D})\!:(\forall\gamma\succeq\beta)(\chi(\gamma)\in A).$*
\(b) *Frequently in* *a subset $A$* *of* *$X$ if for any index $\beta\in\mathbb{D}$, there is a successor index $\gamma\in\mathbb{D}$ such that $\chi(\gamma)$* is in $A$: *$(\forall\beta\in\mathbb{D})(\exists\gamma\succeq\beta)\!:(\chi(\gamma)\in A).\qquad\square$*
It is not difficult to appreciate that
\(i) A net eventually in a subset is also frequently in it but not conversely,
\(ii) A net eventually (respectively, frequently) in a subset cannot be frequently (respectively, eventually) in its complement.
With these notions of eventually in and frequently in, convergence characteristics of a net may be expressed as follows.
**Definition A1.7.** *A net* *$\chi\!:\mathbb{D}\rightarrow X$ converges to $x\in X$ if it is eventually in every neighbourhood of $x$, that is* $$(\forall N\in\mathcal{N}_{x})(\exists\mu\in\mathbb{D})(\chi(\nu\succeq\mu)\in N).$$ *The point $x$ is known as the* *limit* *of $\chi$ and the collection of all limits of a net is the* *limit set* $$\textrm{lim}(\chi)=\{ x\in X\!:(\forall N\in\mathcal{N}_{x})(\exists\mathbb{R}_{\beta}\in\textrm{Res}(\mathbb{D}))(\chi(\mathbb{R}_{\beta})\subseteq N)\}\label{Eqn: lim net}$$ *of $\chi$, with the set of* *residuals* $\textrm{Res}(\mathbb{D})$ *in $\mathbb{D}$ given by* $$\textrm{Res}(\mathbb{D})=\{\mathbb{R}_{\alpha}\in\mathcal{P}(\mathbb{D})\!:\mathbb{R}_{\alpha}=\{\beta\in\mathbb{D}\textrm{ for all }\beta\succeq\alpha\in\mathbb{D}\}\}.\label{Eqn: residual}$$
*The net* *adheres at* *$x\in X$*[^27] *if it is frequently in every neighbourhood of $x$, that is* $$((\forall N\in\mathcal{N}_{x})(\forall\mu\in\mathbb{D}))((\exists\nu\succeq\mu)\!:\chi(\nu)\in N).$$ *The point $x$ is known as the* *adherent* *of $\chi$ and the collection of all adherents of $\chi$ is the* *adherent set of the net, which* *may be expressed in terms of the* *cofinal subset* *of $\mathbb{D}$* $$\textrm{Cof}(\mathbb{D})=\{\mathbb{C}_{\alpha}\in\mathcal{P}(\mathbb{D})\!:\mathbb{C}_{\alpha}=\{\beta\in\mathbb{D}\textrm{ for some }\beta\succeq\alpha\in\mathbb{D}\}\}\label{Eqn: cofinal}$$ (thus $\mathbb{D}_{\alpha}$ is cofinal in $\mathbb{D}$ iff it intersects every residual in $\mathbb{D}$), *as* $$\textrm{adh}(\chi)=\{ x\in X\!:(\forall N\in\mathcal{N}_{x})(\exists\mathbb{C}_{\beta}\in\textrm{Cof}(\mathbb{D}))(\chi(\mathbb{C}_{\beta})\subseteq N)\}.\label{Eqn: adh net1}$$
*This recognizes, in keeping with the limit set, each subnet of a net to be a net in its own right, and is equivalent to* $${\textstyle \textrm{adh}(\chi)=\{ x\in X\!:(\forall N\in\mathcal{N}_{x})(\forall\mathbb{R}_{\alpha}\in\textrm{Res}(\mathbb{D}))(\chi(\mathbb{R}_{\alpha})\bigcap N\neq\emptyset)\}.\qquad\square}\label{Eqn: adh net2}$$
Intuitively, a sequence is eventually in a set $A$ if it is always in it after a finite number of terms (of course, the concept of a *finite number of terms* is unavailable for nets; in this case the situation may be described by saying that a net is eventually in $A$ if its *tail is in* $A$) and it is frequently in $A$ if it always returns to $A$ to leave it again. It can be shown that a net is eventually (resp. frequently) in a set iff it is not frequently (resp.eventually) in its complement.
The following examples illustrate graphically the role of a proper choice of the index set $\mathbb{D}$ in the description of convergence.
**Example A1.3.** (1) Let $\gamma\in\mathbb{D}$. The eventually constant net $\chi(\delta)=x$ for $\delta\succeq\gamma$ converges to $x$.
\(2) Let $\mathcal{N}_{x}$ be a neighbourhood system at a point $x$ in $X$ and suppose that the net $(\chi(N))_{N\in\mathcal{N}_{x}}$ is defined by $$\chi(M)\overset{\textrm{def}}=s\in M;\label{Eqn: Def: Net1}$$ here the directed index set $_{\mathbb{D}}N$ is ordered by the natural direction (\[Eqn: Direction1\]) of $\mathcal{N}_{x}$. Then $\chi(N)\rightarrow x$ because given any $x$-neighbourhood $M\in\!\:_{\mathbb{D}}N$, it follows from $$M\preceq N\in\,{}_{\mathbb{D}}N\Longrightarrow\chi(N)=t\in N\subseteq M\label{Eqn: DirectedNet1}$$
that a point in any subset of $M$ is also in $M$; $\chi(N)$ is therefore eventually in every neighbourhood of $x$.
\(3) This slightly more general form of the previous example provides a link between the complimentary concepts of nets and filters that is considered below. For a point $x\in X$, and $M,N\in\mathcal{N}_{x}$ with the corresponding directed set $_{\mathbb{D}}M_{s}$ of Eq. (\[Eqn: Directed\]) ordered by its natural order (\[Eqn: Direction2\]), the net $$\chi(M,s)\overset{\textrm{def}}=s\label{Eqn: Def: Net2}$$ converges to $x$ because, as in the previous example, for any given $(M,s)\in\:\!_{\mathbb{D}}N_{s}$, it follows from $$(M,s)\preceq(N,t)\in\!\:_{\mathbb{D}}M_{s}\Longrightarrow\chi(N,t)=t\in N\subseteq M\label{Eqn: DirectedNet2}$$ that $\chi(N,t)$ is eventually in every neighbourhood $M$ of $x$. The significance of the directed set $_{\mathbb{D}}N_{t}$ of Eq. (\[Eqn: Directed\]), as compared to $_{\mathbb{D}}N$, is evident from the net that it induces *without using the Axiom of Choice*: For a subset $A$ of $X$, the net $\chi(N,t)=t\in A$ indexed by the directed set $${\textstyle _{\mathbb{D}}N_{t}=\{(N,t)\!:(N\in\mathcal{N}_{x})(t\in N\bigcap A)\}}\label{Eqn: Closure_Directed}$$ under the direction of Eq. (\[Eqn: Direction2\]), converges to $x\in X$ with all such $x$ defining the closure $\textrm{Cl}(A)$ of $A$. Furthermore taking the directed set to be $${\textstyle _{\mathbb{D}}N_{t}=\{(N,t)\!:(N\in\mathcal{N}_{x})(t\in N\bigcap A-\{ x\})\}}\label{Eqn: Der_Directed}$$ which, unlike Eq. (\[Eqn: Closure\_Directed\]), excludes the point $x$ that may or may not be in the subset $A$ of $X$, induces the net $\chi(N,t)=t\in A-\{ x\}$ converging to $x\in X$, with the set of all such $x$ yielding the derived set $\textrm{Der}(A)$ of $A$. In contrast, Eq. (\[Eqn: Closure\_Directed\]) also includes the isolated points $t=x$ of $A$ so as to generate its closure. Observe how neighbourhoods of a point, which define convergence of nets and filters in a topological space $X$, double up here as index sets to yield a self-consistent tool for the description of convergence.
As compared with sequences where, the index set is restricted to positive integers, the considerable freedom in the choice of directed sets as is abundantly borne out by the two preceding examples, is not without its associated drawbacks. Thus as a trade-off, the wide range of choice of the directed sets may imply that induction methods, so common in the analysis of sequences, need no longer apply to arbitrary nets.
\(4) The non-convergent nets (actually these are sequences)
\(a) $(1,-1,1,-1,\cdots)$ adheres at $1$ and $-1$ and
\(b) $\begin{array}{ccl}
x_{n} & = & {\displaystyle \left\{ \begin{array}{lcl}
n & & \textrm{if }n\textrm{ is odd}\\
1-1/(1+n) & & \textrm{if }n\textrm{ is even}\end{array},\right.}\end{array}$ adheres at $1$ for its even terms, but is unbounded in the odd terms.$\qquad\blacksquare$
A converging sequence or net is also adhering but, as examples (4) show, the converse is false. Nevertheless it is true, as again is evident from examples (4), that in a first countable space where sequences suffice, a sequence $(x_{n})$ adheres at $x$ iff some subsequence $(x_{n_{m}})_{m\in\mathbb{N}}$ of $(x_{n})$ converges to $x$. If the space is not first countable this has a corresponding equivalent formulation for nets with subnets replacing subsequences as follows.
Let $(\chi(\alpha))_{\alpha\in\mathbb{D}}$ be a net. A *subnet* of $\chi(\alpha)$ is the net $\zeta(\beta)=\chi(\sigma(\beta))$, $\beta\in\mathbb{E}$, where $\sigma\!:(\mathbb{E},\leq)\rightarrow(\mathbb{D},\preceq)$ is a function that captures the essence of the subsequential mapping $n\mapsto n_{m}$ in $\mathbb{N}$ by satisfying
(SN1) $\sigma$ is an increasing order-preserving function: it respects the order of $\mathbb{E}$: $\sigma(\beta)\preceq\sigma(\beta^{\prime})$ for every $\beta\leq\beta^{\prime}\in\mathbb{E}$, and
(SN2) For every $\alpha\in\mathbb{D}$ there exists a $\beta\in\mathbb{E}$ such that $\alpha\preceq\sigma(\beta)$.
These generalize the essential properties of a subsequence in the sense that (1) Even though the index sets $\mathbb{D}$ and $\mathbb{E}$ may be different, it is necessary that the values of $\mathbb{E}$ be contained in $\mathbb{D}$, and (2) There are arbitrarily large $\alpha\in\mathbb{D}$ such that $\chi(\alpha=\sigma(\beta))$ is a value of the subnet $\zeta(\beta)$ for some $\beta\in\mathbb{E}$. Recalling the first of the order relations Eq. (\[Eqn: FunctionOrder\]) on $\textrm{Map}(X,Y)$, we will denote a subnet $\zeta$ of $\chi$ by $\zeta\preceq\chi$.
We now consider the concept of filter on a set $X$ that is very useful in visualizing the behaviour of sequences and nets, and in fact filters constitute an alternate way of looking at convergence questions in topological spaces. A filter $\mathcal{F}$ on a set $X$ is a collection of *nonempty* subsets of $X$ satisfying properties $(\textrm{F}1)-(\textrm{F}3)$ below that are simply those of a neighbourhood system $\mathcal{N}_{x}$ without specification of the reference point $x$.
(F1) The empty set $\emptyset$ does not belong to $\mathcal{F}$,
(F2) The intersection of any two members of a filter is another member of the filter: $F_{1},F_{2}\in\mathcal{F}\Rightarrow F_{1}\bigcap F_{2}\in\mathcal{F}$,
(F3) Every superset of **a member of a filter belongs to the filter: $(F\in\mathcal{F})\wedge(F\subseteq G)\Rightarrow G\in\mathcal{F}$; in particular $X\in\mathcal{F}$.
**Example A1.4.** (1) The *indiscrete filter* is the smallest filter on $X$.
\(2) The neighbourhood system $\mathcal{N}_{x}$ is the important *neighbourhood filter at $x$ on $X$,* and any local base at $x$ is also a filter-base for $\mathcal{N}_{x}$. In general for any subset $A$ of $X$, $\{ N\subseteq X\!:A\subseteq\textrm{Int}(N)\}$ is a filter on $X$ at $A$.
\(3) All subsets of $X$ containing a point $x\in X$ is the *principal filter* $_{\textrm{F}}\mathcal{P}(x)$ *on $X$ at $x$.* More generally, if $\mathcal{F}$ consists of all supersets of a *nonempty* subset $A$ of $X$, then $\mathcal{F}$ is the *principal filter* $_{\textrm{F}}\mathcal{P}(A)=\{ N\subseteq X\!:A\subseteq\textrm{Int}(N)\}$ *at $A$. By adjoining the empty set to this filter give the $p$-inclusion and $A$-inclusion topologies on $X$ respectively.* The single element sets $\{\{ x\}\}$ and $\{ A\}$ are particularly simple examples of filter-bases that generate the principal filters at $x$ and $A$.
\(4) For an uncountable (resp. infinite) set $X$, all cocountable (resp. cofinite) subsets of $X$ constitute the *cocountable* (resp. *cofinite* or *Frechet*) filter on $X$. Again, adding to these filters the empty set gives the respective topologies.$\qquad\blacksquare$
Like the topological and local bases $_{\textrm{T}}\mathcal{B}$ and $\mathcal{B}_{x}$ respectively, a subclass of $\mathcal{F}$ may be used to define a filter-base $_{\textrm{F}}\mathcal{B}$ that in turn generate $\mathcal{F}$ on $X$, just as it is possible to define the concepts of limit and adherence sets for a filter to parallel those for nets that follow straightforwardly from Def. A1.7, taken with Def. A1.11.
**Definition A1.8.** *Let $(X,\mathcal{T})$ be a topological space and $\mathcal{F}$ a filter on $X$. Then*$$\textrm{lim}(\mathcal{F})=\{ x\in X\!:(\forall N\in\mathcal{N}_{x})(\exists F\in\mathcal{F})(F\subseteq N)\}\label{Eqn: lim filter}$$ and $${\textstyle \textrm{adh}(\mathcal{F})=\{ x\in X\!:(\forall N\in\mathcal{N}_{x})(\forall F\in\mathcal{F})(F\bigcap N\neq\emptyset)\}}\label{Eqn: adh filter}$$
*are respectively the sets of* *limit points* *and* *adherent* *points* *of $\mathcal{F}$*[^28]*.$\qquad\square$*
A comparison of Eqs. (\[Eqn: lim net\]) and (\[Eqn: adh net2\]) with Eqs. (\[Eqn: lim filter\]) and (\[Eqn: adh filter\]) respectively demonstrate their formal similarity; this inter-relation between filters and nets will be made precise in Definitions A1.10 and A1.11 below. It should be clear from the preceding two equations that $$\textrm{lim}(\mathcal{F})\subseteq\textrm{adh}(\mathcal{F}),\label{Eqn: lim/adh(fil)}$$ with a similar result $$\textrm{lim}(\chi)\subseteq\textrm{adh}(\chi)\label{Eqn: lim/adh(net)}$$ holding for nets because of the duality between nets and filters as displayed by Defs. A1.9 and A1.10 below, with the equality in Eqs. (\[Eqn: lim/adh(fil)\]) and (\[Eqn: lim/adh(net)\]) being true (but not characterizing) for ultrafilters and ultranets respectively, see Example 4.2(3) for an account of this notion . It should be clear from the equations of Definition A1.8 that $$\textrm{adh}(\mathcal{F})=\{ x\in X\!:(\exists\textrm{ a finer filter }\mathcal{G}\supseteq\mathcal{F}\textrm{ on }X)\textrm{ }(\mathcal{G}\rightarrow x)\}\label{Eqn: filter adh}$$ consists of all the points of $X$ to which some finer filter $\mathcal{G}$ (in the sense that $\mathcal{F}\subseteq\mathcal{G}$ implies every element of $\mathcal{F}$ is also in $\mathcal{G}$) converges in $X$; thus $${\textstyle \textrm{adh}(\mathcal{F})=\bigcup\lim(\mathcal{G}\!:\mathcal{G}\supseteq\mathcal{F}),}$$ which corresponds to the net-result of Theorem A1.5 below, that a net *$\chi$* adheres at *$x$* iff there is some subnet of *$\chi$* that converges to *$x$* in *$X$*. Thus if $\zeta\preceq\chi$ is a subnet of $\chi$ and $\mathcal{F}\subseteq\mathcal{G}$ is a filter coarser than $\mathcal{G}$ then $$\begin{aligned}
\lim(\chi)\subseteq\lim(\zeta) & & \lim(\mathcal{F})\subseteq\lim(\mathcal{G})\\
\textrm{adh}(\zeta)\subseteq\textrm{adh}(\chi) & & \textrm{adh}(\mathcal{G})\subseteq\textrm{adh}(\mathcal{F});\end{aligned}$$
a filter $\mathcal{G}$ finer than a given filter $\mathcal{F}$ corresponds to a subnet $\zeta$ of a given net $\chi$. The implication of this correspondence should be clear from the association between nets and filters contained in Definitions A1.10 and A1.11.
A filter-base in $X$ is a *nonempty* family $(B_{\alpha})_{\alpha\in\mathbb{D}}=\!\,_{\textrm{F}}\mathcal{B}$ of subsets of $X$ characterized by
(FB1) There are no empty sets in the collection $_{\textrm{F}}\mathcal{B}$: $(\forall\alpha\in\mathbb{D})(B_{\alpha}\neq\emptyset)$
(FB2) The intersection of any two members of **$_{\textrm{F}}\mathcal{B}$ **contains another member of $_{\textrm{F}}\mathcal{B}$: $B_{\alpha},B_{\beta}\in\,_{\textrm{F}}\mathcal{B}\Rightarrow(\exists B\in\,_{\textrm{F}}\mathcal{B}\!:B\subseteq B_{\alpha}\bigcap B_{\beta})$;
hence any class of subsets of $X$ that does not contain the empty set and is closed under finite intersections is a base for a unique filter on $X$; compare the properties (NB1) and (NB2) of a local basis given at the beginning of this Appendix. Similar to Def. A1.1 for the local base, it is possible to define
**Definition A1.9.** *A filter-base* $_{\textrm{F}}\mathcal{B}$ *in a set $X$ is a subcollection of the filter* $\mathcal{F}$ *on $X$ having the property that each $F\in\mathcal{F}$ contains some member of* $_{\textrm{F}}\mathcal{B}$*.* *Thus* $$_{\textrm{F}}\mathcal{B}\overset{\textrm{def}}=\{ B\in\mathcal{F}\!:B\subseteq F\textrm{ for each }F\in\mathcal{F}\}\label{Eqn: FB}$$ *determines the filter* $$\mathcal{F}=\{ F\subseteq X\!:B\subseteq F\textrm{ for some }B\textrm{ }\in\!\,_{\textrm{F}}\mathcal{B}\}\label{Eqn: filter_base}$$
*reciprocally as all supersets of the basic elements.$\qquad\square$*
This is the smallest filter on $X$ that contains $_{\textrm{F}}\mathcal{B}$ and is said to be *the filter generated by its filter-base* $_{\textrm{F}}\mathcal{B}$; alternatively $_{\textrm{F}}\mathcal{B}$ is the filter-base of $\mathcal{F}$. The entire neighbourhood system $\mathcal{N}_{x}$, the local base $\mathcal{B}_{x}$, $\mathcal{N}_{x}\bigcap A$ for $x\in\textrm{Cl}(A)$, and the set of all residuals of a directed set $\mathbb{D}$ are among the most useful examples of filter-bases on $X$, $A$ and $\mathbb{D}$ respectively. Of course, every filter is trivially a filter-base of itself, and *the singletons $\{\{ x\}\}$, $\{ A\}$ are filter-bases that generate the principal filters $_{\textrm{F}}\mathcal{P}(x)$ and $_{\textrm{F}}\mathcal{P}(A)$ at $x$, and $A$ respectively*.
Paralleling the case of topological subbase $_{\textrm{T}}\mathcal{S}$, a filter subbase $_{\textrm{F}}\mathcal{S}$ can be defined on $X$ to be any collection of subsets of $X$ *with the finite intersection property* (as compared with $_{\textrm{T}}\mathcal{S}$ where no such condition was necessary, this represents the fundamental point of departure between topology and filter) and it is not difficult to deduce that the filter generated by **$_{\textrm{F}}\mathcal{S}$ on $X$ is obtained by taking all finite intersections $_{\textrm{F}}\mathcal{S}_{\wedge}$ of members of $_{\textrm{F}}\mathcal{S}$ followed by their supersets $_{\textrm{F}}\mathcal{S}_{\Sigma\wedge}$. $\mathcal{F}(_{\textrm{F}}\mathcal{S}):=\,_{\textrm{F}}\mathcal{S}_{\Sigma\wedge}$ is the smallest filter on $X$ that contains $_{\textrm{F}}\mathcal{S}$ and is the filter *generated by* $_{\textrm{F}}\mathcal{S}$.
Equation (\[Eqn: adh filter\]) can be put in the more useful and transparent form given by
**Theorem A1.3.** *For a filter $\mathcal{F}$ in a space $(X,\mathcal{T})$* $$\begin{aligned}
{\displaystyle \textrm{adh}(\mathcal{F})} & = & {\displaystyle \bigcap_{F\in\mathcal{F}}\textrm{Cl}(F)}\label{Eqn: filter adh*}\\
& = & {\displaystyle \bigcap_{B\in\,_{\textrm{F}}\mathcal{B}}\textrm{Cl}(B)},\nonumber \end{aligned}$$
*and dually* $\textrm{adh}(\chi)$, *are closed set*s.$\qquad\square$
**Proof.** Follows immediately from the definitions for the closure of a set Eq. (\[Eqn: Def: Closure\]) and the adherence of a filter Eq. (\[Eqn: adh filter\]). As always, it is a matter of convenience in using the basic filters **$_{\textrm{F}}\mathcal{B}$** instead of $\mathcal{F}$ to generate the adherence set.$\qquad\blacksquare$
It is infact true that the limit sets $\lim(\mathcal{F})$ and $\lim(\chi)$ are also closed set of $X$; the arguments involving ultrafilters are omitted.
Similar to the notion of the adherence set of a filter is its *core —* a concept that unlike the adherence, is purely set-theoretic being the infimum of the filter and is not linked with any topological structure of the underlying (infinite) set $X$ — defined as $${\displaystyle \textrm{core }(\mathcal{F})=\bigcap_{F\in\mathcal{F}}F.}\label{Eqn: core}$$ From Theorem A1.3 and the fact that the closure of a set $A$ is the smallest closed set that contains $A$, see Eq. (\[Eqn: closure\]) at the end of Tutorial4, it is clear that in terms of filters$$\begin{aligned}
A & = & \textrm{core}(\,_{\textrm{F}}\mathcal{P}(A))\nonumber \\
\textrm{Cl}(A) & = & \textrm{adh}(\,_{\textrm{F}}\mathcal{P}(A))\label{Eqn: PrinFil_Cl(A)}\\
& = & \textrm{core}(\textrm{Cl}(\,_{\textrm{F}}\mathcal{P}(A)))\nonumber \end{aligned}$$ where $_{\textrm{F}}\mathcal{P}(A)$ is the principal filter at $A$; thus *the core and adherence sets of the principal filter at $A$ are equal respectively to $A$ and* $\textrm{Cl}(A)$ *—* a classic example of equality in the general relation $\textrm{Cl}(\bigcap A_{\alpha})\subseteq\bigcap\textrm{Cl}(A_{\alpha})$ — but both are empty, for example, in the case of an infinitely decreasing family of rationals centered at any irrational (leading to a principal filter-base of rationals at the chosen irrational). This is an important example demonstrating that *the infinite intersection of a non-empty family of (closed) sets with the finite intersection property may be empty,* *a situation that cannot arise on a finite set or an infinite compact set*. Filters on $X$ with an empty core are said to be *free,* and are *fixed* otherwise: notice that by its very definition filters cannot be free on a finite set, and a free filter represents an additional feature that may arise in passing from finite to infinite sets. Clearly $(\textrm{adh}(\mathcal{F})=\emptyset)\Rightarrow(\textrm{core}(\mathcal{F})=\emptyset)$, but as the important example of the rational space in the reals illustrate, the converse need not be true. Another example of a free filter of the same type is provided by the filter-base $\{[a,\infty)\!:a\in\mathbb{R}\}$ in $\mathbb{R}$. Both these examples illustrate the important property that *a filter is free iff it contains the cofinite filter,* and the cofinite filter is the smallest possible free filter on an infinite set. The free cofinite filter, as these examples illustrate, may be typically generated as follows. Let $A$ be a subset of $X$, $x\in\textrm{Bdy}_{X-A}(A)$, and consider the directed set Eq. (\[Eqn: Closure\_Directed\]) to generate the corresponding net in $A$ given by $\chi(N\in\mathcal{N}_{x},t)=t\in A$. Quite clearly, the core of any Frechet filter based on this net must be empty as the point $x$ does not lie in $A$. In general, the intersection is empty because if it were not so then the complement of the intersection — which is an element of the filter — would be infinite in contravention of the hypothesis that the filter is Frechet. It should be clear that every filter finer than a free filter is also free, and any filter coarser than a fixed filter is fixed.
Nets and filters are complimentary concepts and one may switch from one to the other as follows.
**Definition A1.10.** *Let $\mathcal{F}$ be a filter on $X$ and let $_{\mathbb{D}}F_{x}=\{(F,x)\!:(F\in\mathcal{F})(x\in F)\}$ be a directed set with its natural direction $(F,x)\preceq(G,y)\Rightarrow(G\subseteq F)$. The net $\chi_{\mathcal{F}}\,\!:\,_{\mathbb{D}}F_{x}\rightarrow X$ defined by* $$\chi_{\mathcal{F}}(F,x)=x$$ *is said to be* *associated with* *the filter* *$\mathcal{F}$, see Eq. (\[Eqn: DirectedNet2\]).$\qquad\square$*
**Definition A1.11.** *Let $\chi\!:\mathbb{D}\rightarrow X$ be a net and $\mathbb{R}_{\alpha}=\{\beta\in\mathbb{D}\!:\beta\succeq\alpha\in\mathbb{D}\}$ a residual in $\mathbb{D}$. Then* $$_{\textrm{F}}\mathcal{B}_{\chi}\overset{\textrm{def}}=\{\chi(\mathbb{R}_{\alpha})\!:\textrm{Res}(\mathbb{D})\rightarrow X\textrm{ for all }\alpha\in\mathbb{D}\}$$ *is the* *filter-base associated with* *$\chi$, and the corresponding filter $\mathcal{F}_{\chi}$ obtained by taking all supersets of the elements of* $_{\textrm{F}}\mathcal{B}_{\chi}$ *is the* *filter* *associated with* *$\chi$.$\qquad\square$*
$_{\textrm{F}}\mathcal{B}_{\chi}$ is a filter-base in $X$ because $\chi(\bigcap\mathbb{R}_{\alpha})\subseteq\bigcap\chi(\mathbb{R}_{\alpha})$, that holds for any functional relation, proves (FB2). It is not difficult to verify that
\(i) $\chi$ is eventually in $A\Longrightarrow A\in\mathcal{F}_{\chi}$, and
\(ii) $\chi$ is frequently in $A\Longrightarrow(\forall\mathbb{R}_{\alpha}\in\textrm{Res}(\mathbb{D}))(A\bigcap\chi(\mathbb{R}_{\alpha})\neq\emptyset)$ $\Longrightarrow A\bigcap\mathcal{F}_{\chi}\neq\emptyset$ .
Limits and adherences are obviously preserved in switching between nets (respectively, filters) and the filters (respectively, nets) that they generate: $$\begin{aligned}
\lim(\chi)=\lim(\mathcal{F}_{\chi}), & & \textrm{adh}(\chi)=\textrm{adh}(\mathcal{F}_{\chi})\label{Eqn: net-fil}\\
\lim(\mathcal{F})=\lim(\chi_{\mathcal{F}}), & & \textrm{adh}(\mathcal{F})=\textrm{adh}(\chi_{\mathcal{F}}).\label{Eqn: fil-net}\end{aligned}$$
The proofs of the two parts of Eq. (\[Eqn: net-fil\]), for example, go respectively as follows. $x\in\lim(\chi)\Leftrightarrow\chi\textrm{ is eventually in }\mathcal{N}_{x}\Leftrightarrow(\forall N\in\mathcal{N}_{x})(\exists F\in\mathcal{F}_{\chi})\textrm{ such that }(F\subseteq N)\Leftrightarrow x\in\lim(\mathcal{F}_{\chi})$, and $x\in\textrm{adh}(\chi)\Leftrightarrow\chi\textrm{ is frequently in }\mathcal{N}_{x}\Leftrightarrow(\forall N\in\mathcal{N}_{x})(\forall F\in\mathcal{F}_{\chi})\textrm{ }(N\bigcap F\neq\emptyset)\Leftrightarrow x\in\textrm{adh}(\mathcal{F}_{\chi})$; here $F$ is a superset of $\chi(\mathbb{R}_{\alpha})$.
Some examples of convergence of filters are
\(1) Any filter on an indiscrete space $X$ converges to every point of $X$.
\(2) Any filter on a space that coincides with its topology (minus the empty set, of course) converges to every point of the space.
\(3) For each $x\in X$, the neighbourhood filter $\mathcal{N}_{x}$ converges to $x$; this is the smallest filter on $X$ that converges to $x$.
\(4) The *indiscrete* filter $\mathcal{F}=\{ X\}$ converges to no point in the space $(X,\{\emptyset,A,X-A,X\})$, but converges to every point of $X-A$ if $X$ has the topology $\{\emptyset,A,X\}$ because the only neighbourhood of any point in $X-A$ is $X$ which is contained in the filter.
One of the most significant consequences of convergence theory of sequences and nets, as shown by the two theorems and the corollary following, is that this can be used to describe the topology of a set. The proofs of the theorems also illustrate the close inter-relationship between nets and filters.
**Theorem A1.4.** *For a subset $A$ of a topological space $X$,* $$\textrm{Cl}(A)=\{ x\in X\!:(\exists\textrm{ a net }\chi\textrm{ in }A)\textrm{ }(\chi\rightarrow x)\}.\qquad\square\label{Eqn: net closure}$$ **Proof.** *Necessity.* For **$x\in\textrm{Cl}(A)$, construct a **net **$\chi\rightarrow x$ in *$A$* as **follows. Let $\mathcal{B}_{x}$ be a topological local base at $x$, which by definition is the collection of all open sets of $X$ containing $x$. For each $\beta\in\mathbb{D}$, the sets $$N_{\beta}=\bigcap_{\alpha\preceq\beta}\{ B_{\alpha}\!:B_{\alpha}\in\mathcal{B}_{x}\}$$ form a nested decreasing local neighbourhood filter base at $x$. With respect to the directed set $_{\mathbb{D}}N_{\beta}=\{(N_{\beta},\beta)\!:(\beta\in\mathbb{D})(x_{\beta}\in N_{\beta})\}$ of Eq. (\[Eqn: DirectedIndexed\]), define the desired net in $A$ by $${\textstyle \chi(N_{\beta},\beta)=x_{\beta}\in N_{\beta}\bigcap A}$$ where the family of nonempty decreasing subsets $N_{\beta}\bigcap A$ of $X$ constitute the filter-base in $A$ as required by the directed set $_{\mathbb{D}}N_{\beta}$. It now follows from Eq. (\[Eqn: DirectionIndexed\]) and the arguments in Example A1.3(3) that $x_{\beta}\rightarrow x$; compare the directed set of Eq. (\[Eqn: Closure\_Directed\]) for a more compact, yet essentially identical, argument. Carefully observe the dual roles of $\mathcal{N}_{x}$ as a neighbourhood filter base at $x$.
*Sufficiency.* Let $\chi$ be a net in $A$ that converges to $x\in X$. For any $N_{\alpha}\in\mathcal{N}_{x}$, there is a $\mathbb{R}_{\alpha}\in\textrm{Res}(\mathbb{D})$ of Eq. (\[Eqn: residual\]) such that $\chi(\mathbb{R}_{\alpha})\subseteq N_{\alpha}$. Hence the point $\chi(\alpha)=x_{\alpha}$ of $A$ belongs to $N_{\alpha}$ so that $A\bigcap N_{\alpha}\neq\emptyset$ which means, from Eq. (\[Eqn: Def: Closure\]), that $x\in\textrm{Cl}(A)$.$\qquad\blacksquare$
**Corollary.** Together with Eqs. (\[Eqn: Def: Closure\]) and (\[Eqn: Def: Derived\]), is follows that $$\textrm{Der}(A)=\{ x\in X\!:(\exists\textrm{ a net }\zeta\textrm{ in }A-\{ x\})(\zeta\rightarrow x)\}\qquad\square\label{Eqn: net derived}$$
The filter forms of Eqs. (\[Eqn: net closure\]) and (\[Eqn: net derived\]) $$\begin{aligned}
\textrm{Cl}(A) & = & \{ x\in X\!:(\exists\textrm{ a filter }\mathcal{F}\textrm{ on }X)(A\in\mathcal{F})(\mathcal{F}\rightarrow x)\}\label{Eqn: filter cls_der}\\
\textrm{Der}(A) & = & \{ x\in X\!:(\exists\textrm{ a filter }\mathcal{F}\textrm{ on }X)(A-\{ x\}\in\mathcal{F})(\mathcal{F}\rightarrow x)\}\nonumber \end{aligned}$$ then follows from Eq. (\[Eqn: Def: LimFilter\]) and the finite intersection property (F2) of $\mathcal{F}$ so that every neighbourhood of $x$ must intersect $A$ (respectively $A-\{ x\}$) in Eq. (\[Eqn: filter cls\_der\]) to produce the converging net needed in the proof of Theorem A1.3.
We end this discussion of convergence in topological spaces with a proof of the following theorem which demonstrates the relationship that “eventually in” and “frequently in” bears with each other; Eq. (\[Eqn: net adh\]) below is the net-counterpart of the filter equation (\[Eqn: filter adh\]).
**Theorem A1.5.** *If $\chi$ is a net in a topological space $X$, then* $x\in\textrm{adh}(\chi)$ *iff some subnet $\zeta(\beta)=\chi(\sigma(\beta))$ of $\chi(\alpha)$, with $\alpha\in\mathbb{D}$ and $\beta\in\mathbb{E}$ , converges in $X$ to $x$; thus* $$\textrm{adh}(\chi)=\{ x\in X\!:(\exists\textrm{ a subnet }\zeta\preceq\chi\textrm{ in }X)(\zeta\rightarrow x)\}.\qquad\square\label{Eqn: net adh}$$ ****Proof.** *Necessity.* Let $x\in\textrm{adh}(\chi)$. Define a subnet function $\sigma\!:\,_{\mathbb{D}}N_{\alpha}\rightarrow\mathbb{D}$ by $\sigma(N_{\alpha},\alpha)=\alpha$ where $_{\mathbb{D}}N_{\alpha}$ is the directed set of Eq. (\[Eqn: DirectedIndexed\]): (SN1) and (SN2) are quite evidently satisfied according to Eq. (\[Eqn: DirectionIndexed\]). Proceeding as in the proof of the preceding theorem it follows that $x_{\beta}=\chi(\sigma(N_{\alpha},\alpha))=\zeta(N_{\alpha},\alpha)\rightarrow x$ is the required converging subnet that exists from Eq. (\[Eqn: adh net1\]) and the fact that $\chi(\mathbb{R}_{\alpha})\bigcap N_{\alpha}\neq\emptyset$ for every $N_{\alpha}\in\mathcal{N}_{x}$, by hypothesis.
*Sufficiency.* Assume now that $\chi$ has a subnet $\zeta(N_{\alpha},\alpha)$ that converges to $x$. If $\chi$ does not adhere at $x$, there is a neighbourhood $N_{\alpha}$ of $x$ not frequented by it, in which case $\chi$ must be eventually in $X-N_{\alpha}$. Then $\zeta(N_{\alpha},\alpha)$ is also eventually in $X-N_{\alpha}$ so that $\zeta$ cannot be eventually in $N_{\alpha}$, a contradiction of the hypothesis that $\zeta(N_{\alpha},\alpha)\rightarrow x$.[^29]$\qquad\blacksquare$
Eqs. (\[Eqn: net closure\]) and (\[Eqn: net adh\]) imply that the closure of a subset $A$ of $X$ is the class of $X$-adherences of all the (sub)nets of $X$ that are eventually in $A$. This includes both the constant nets yielding the isolated points of $A$ and the non-constant nets leading to the cluster points of $A$, and implies the following physically useful relationship between convergence and topology that can be used as defining criteria for open and closed sets having a more appealing physical significance than the original definitions of these terms. Clearly, the term “net” is justifiably used here to include the subnets too.
The following corollary of Theorem A1.5 summarizes the basic topological properties of sets in terms of nets (respectively, filters).
**Corollary.** Let $A$ be a subset of a topological space $X$. Then
\(1) $A$ is closed in $X$ iff every convergent net of $X$ that is eventually in $A$ actually converges to a point in $A$ (respectively, iff the adhering points of each filter-base on $A$ all belong to $A$). Thus no $X$-convergent net in a closed subset may converge to a point outside it.
\(2) $A$ is open in $X$ iff every convergent net of $X$ that converges to a point in $A$ is eventually in $A$. Thus no $X$-convergent net outside an open subset may converge to a point in the set.
\(3) $A$ is closed-and-open (clopen) in $X$ iff every convergent net of $X$ that converges in $A$ is eventually in $A$ and conversely.
\(4) $x\in\textrm{Der}(A)$ iff some net (respectively, filter-base) in $A-\{ x\}$ converges to $x$; this clearly eliminates the isolated points of $A$ and $x\in\textrm{Cl}(A)$ iff some net (respectively, filter-base) in $A$ converges to $x$.$\qquad\square$
**Remark.** The differences in these characterizations should be fully appreciated: If we consider the cluster points $\textrm{Der}(A)$ of a net $\chi$ in $A$ as the *resource generated by* $\chi$, then a closed subset of $X$ can be considered to be *selfish* as its keeps all its resource to itself: $\textrm{Der}(A)\cap A=\textrm{Der}(A)$. The opposite of this is a *donor* set that donates all its generated resources to its neighbour: $\textrm{Der}(A)\cap X-A=\textrm{Der}(A)$, while for a *neutral* set, both $\textrm{Der}(A)\cap A\neq\emptyset$ and $\textrm{Der}(A)\cap X-A\neq\emptyset$ implying that the convergence resources generated in $A$ and $X-A$ can be deposited only in the respective sets. The clopen sets (see diagram 2-2 of Fig. \[Fig: DerSets\]) are of some special interest as they are boundary less so that no net-resources can be generated in this case as any such limit are required to be simultaneously in the set and its complement.
**Example A1.1, Continued.** This continuation ****of Example A1.2 illustrates how sequential convergence is inadequate in spaces that are not first countable like the uncountable set with cocountable topology. In this topology, a sequence can converge to a point $x$ in the space iff it has only a finite number of distinct terms, and is therefore eventually constant. Indeed, let the complement $$G\overset{\textrm{def}}=X-F,\qquad F=\{ x_{i}\!:x_{i}\neq x,\textrm{ }i\in\mathbb{N}\}$$ of the countably closed sequential set $F$ be an open neighbourhood of $x\in X$. Because a sequence $(x_{i})_{i\in\mathbb{N}}$ in $X$ converges to a point $x\in X$ iff it is eventually in *every* neighbourhood (including $G$) of $x$, the sequence represented by the set $F$ cannot converge to $x$ unless it is of the uncountable type[^30]$$(x_{0},x_{1},\cdots,x_{I},x_{I+1},x_{I+1},\cdots)\label{Eqn: cocount}$$ with only a finite number $I$ of distinct terms actually belonging to the closed sequential set $F=X-G$, and $x_{I+1}=x$. Note that as we are concerned only with the eventual behaviour of the sequence, we may discard all distinct terms from $G$ by considering them to be in $F$, and retain only the constant sequence $(x,x,\cdots)$ in $G$. In comparison with the cofinite case that was considered in Sec. 4, the entire countably infinite sequence can now lie outside a neighbourhood of $x$ thereby enforcing the eventual constancy of the sequence. This leads to a generalization of our earlier cofinite result in the sense that a cocountable filter on a cocountable space converges to every point in the space.
It is now straightforward to verify that for a point $x_{0}$ in an uncountable cocountable space $X$
\(a) Even though no sequence in the open set $G=X-\{ x_{0}\}$ can converge to $x_{0}$, yet $x_{0}\in\textrm{Cl}(G)$ since the intersection of any (uncountable) open neighbourhood $U$ of $x_{0}$ with $G$, being an uncountable set, is not empty.
\(b) By corollary (1) of Theorem A1.5, the uncountable open set $G=X-\{ x_{0}\}$ is also closed in $X$ because if any sequence $(x_{1},x_{2},\cdots)$ in $G$ converges to some $x\in X$, then $x$ must be in $G$ as the sequence must be eventually constant in order for it to converge. But this is a contradiction as $G$ cannot be closed since it is not countable.[^31] By the same reckoning, although $\{ x_{0}\}$ is not an open set because its complement is not countable, nevertheless it follows from Eq. (\[Eqn: cocount\]) that should any sequence converge to the only point $x_{0}$ of this set, then it must eventually be in $\{ x_{0}\}$ so by corollary (2) of the same theorem, $\{ x_{0}\}$ becomes an open set.
\(c) The identity map $\mathbf{1}\!:X\rightarrow X_{\textrm{d}}$, where $X_{\textrm{d}}$ is $X$ with discrete topology, is not continuous because the inverse image of any singleton of $X_{\textrm{d}}$ is not open in $X$. Yet if a sequence converges in $X$ to $x$, then its image $(\mathbf{1}(x))=(x)$ must actually converge to $x$ in $X_{\textrm{d}}$ because a sequence converges in a discrete space, as in the cofinite or cocountable spaces, iff it is eventually constant; this is so because each element of a discrete space being clopen is boundaryless.
This pathological behaviour of sequences in a non Hausdorff, non first countable space does not arise if the discrete indexing set of sequences is replaced by a continuous, uncountable directed set like $\mathbb{R}$ for example, leading to nets in place of sequences. In this case the net can be in an open set without having to be constant-valued in order to converge to a point in it as the open set can be defined as the complement of a closed countable part of the uncountable net. The careful reader could not have failed to notice that the burden of the above arguments, as also of that in the example following Theorem 4.6, is to formalize the fact that since *a closed set is already defined as a countable (respectively finite) set,* the closure operation cannot add further points to it from its complement, and any sequence that converges in an open set in these topologies must necessarily be eventually constant at its point of convergence, a restriction that no longer applies to a net. The cocountable topology thus has the very interesting property of filtering out a countable part from an uncountable set, as for example the rationals in $\mathbb{R}$.$\qquad\blacksquare$
This example serves to illustrate the hard truth that in a space that is not first countable, the simplicity of sequences is not enough to describe its topological character, and in fact “sequential convergence will be able to describe only those topologies in which the number of (basic) neighbourhoods around each point is no greater than the number of terms in the sequences”, @Willard1970. It is important to appreciate the significance of this interplay of convergence of sequences and nets (and of continuity of functions of Appendix A1) and the topology of the underlying spaces.
A comparison of the defining properties (T1), (T2), (T3) of topology $\mathcal{T}$ with (F1), (F2), (F3) of that of the filter $\mathcal{F}$, shows that a filter is very close to a topology with the main difference being with regard to the empty set which must always be in $\mathcal{T}$ but never in $\mathcal{F}$. Addition of the empty set to a filter yields a topology, but removal of the empty set from a topology need not produce the corresponding filter as the topology may contain nonintersecting sets.
The distinction between the topological and filter-bases should be carefully noted. Thus
\(a) While the topological base may contain the empty set, a filter-base cannot.
\(b) From a given topology, form a common base by dropping all basic open sets that do not intersect. Then a (coarser) topology can be generated from this base by taking all unions, and a filter by taking all supersets according to Eq. (\[Eqn: filter\_base\]). For any given filter this expression may be used to extract a subclass $_{\textrm{F}}\mathcal{B}$ as a base for $\mathcal{F}$.
**A2. Initial and Final topology**
The commutative diagram of Fig. \[Fig: GenInv\] contains four sub-diagrams $X-X_{\textrm{B}}-f(X)$, $Y-X_{\textrm{B}}-f(X)$, $X-X_{\textrm{B}}-Y$ and $X-f(X)-Y$. Of these, the first two are especially significant as they can be used to conveniently define the topologies on $X_{\textrm{B}}$ and $f(X)$ from those of $X$ and $Y$, so that $f_{\textrm{B}}$, $f_{\textrm{B}}^{-1}$ and $G$ have some desirable continuity properties; we recall that a function $f\!:X\rightarrow Y$ is continuous if inverse images of open sets of $Y$ are open in $X$. This simple notion of continuity needs refinement in order that topologies on $X_{\textrm{B}}$ and $f(X)$ be unambiguously defined from those of $X$ and $Y$, a requirement that leads to the concepts of the so-called *final* and *initial topologies.* To appreciate the significance of these new constructs, note that if $f\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ is a continuous function, there may be open sets in $X$ that are not inverse images of open — or for that matter of any — subset of $Y$, just as it is possible for non-open subsets of $Y$ to contribute to $\mathcal{U}$. When the triple $\{\mathcal{U},f,\mathcal{V}\}$ are tuned in such a manner that these are impossible, the topologies so generated on $X$ and $Y$ are the initial and final topologies respectively; they are the smallest (coarsest) and largest (finest) topologies on $X$ and $Y$ that make $f\!:X\rightarrow Y$ continuous. It should be clear that every image and preimage continuous function is continuous, but the converse is not true.
Let $\textrm{sat}(U):=f^{-}f(U)\subseteq X$ be the saturation of an open set $U$ of $X$ and $\textrm{comp}(V):=ff^{-}(V)=V\bigcap f(X)\in Y$ be the component of an open set $V$ of $Y$ on the range $f(X)$ of $f$. Let $\mathcal{U}_{\textrm{sat}}$, $\mathcal{V}_{\textrm{comp}}$ denote respectively the saturations $U_{\textrm{sat}}=\{\textrm{sat}(U)\!:U\in\mathcal{U}\}$ of the open sets of $X$ and the components $V_{\textrm{comp}}=\{\textrm{comp}(V)\!:V\in\mathcal{V}\}$ of the open sets of $Y$ whenever these are also open in $X$ and $Y$ respectively. Plainly, $\mathcal{U}_{\textrm{sat}}\subseteq\mathcal{U}$ and $\mathcal{V}_{\textrm{comp}}\subseteq\mathcal{V}$.
**Definition A2.1.** *For a function* $e\!:X\rightarrow(Y,\mathcal{V})$, *the* *preimage* *or* *initial topology of* $X$ *based on (generated by)* **$e$** *and $\mathcal{V}$* *is* $$\textrm{IT}\{ e;\mathcal{V}\}\overset{\textrm{def}}=\{ U\subseteq X\!:U=e^{-}(V)\textrm{ if }V\in\mathcal{V}_{\textrm{comp}}\},\label{Eqn: IT}$$
*while for $q\!:(X,\mathcal{U})\rightarrow Y$, the* *image* *or* *final topology of* $Y$ *based on (generated by) $\mathcal{U}$ and* **$q$** *is* $$\textrm{FT}\{\mathcal{U};q\}\overset{\textrm{def}}=\{ V\subseteq Y\!:q^{-}(V)=U\textrm{ if }U\in\mathcal{U}_{\textrm{sat}}\}.\qquad\square\label{Eqn: FT'}$$
Thus, the topology of $(X,\textrm{IT}\{ e;\mathcal{V}\})$ consists of, and only of, the $e$-saturations of all the open sets of $e(X)$, while the open sets of $(Y,\textrm{FT}\{\mathcal{U};q\})$ are the $q$-images *in* $Y$ (and not just in $q(X)$) of all the $q$-saturated open sets of $X$.[^32] The need for defining (\[Eqn: IT\]) in terms of $\mathcal{V}_{\textrm{comp}}$ rather than $\mathcal{V}$ will become clear in the following. The subspace topology $\textrm{IT}\{ i;\mathcal{U}\}$ of a subset $A\subseteq(X,\mathcal{U})$ is a basic example of the initial topology **by the inclusion map $i\!:X\supseteq A\rightarrow(X,\mathcal{U})$, and we take its generalization $e\!:(A,\textrm{IT}\{ e;\mathcal{V}\})\rightarrow(Y,\mathcal{V})$ that embeds a subset $A$ of $X$ into $Y$ as the prototype of a preimage continuous map. Clearly the topology of $Y$ may also contain open sets not in $e(X)$, and any subset in $Y-e(X)$ may be added to the topology of $Y$ without altering the preimage topology of $X$: *open sets of $Y$ not in $e(X)$ may be neglected in obtaining the preimage topology* as $e^{-}(Y-e(X))=\emptyset$. The final topology on a quotient set by the quotient map $Q\!:(X,\mathcal{U})\rightarrow X/\sim$, which is just the collection of $Q$-images of the $Q$-saturated open sets of $X$, known as the *quotient topology of $X/\sim$,* is the basic example of the image topology and the resulting space $(X/\sim,\textrm{FT}\{\mathcal{U};Q\})$ is called the *quotient space.* We take the generalization $q\!:(X,\mathcal{U})\rightarrow(Y,\textrm{FT}\{\mathcal{U};q\})$ of $Q$ as the prototype of a image continuous function.
The following results are specifically useful in dealing with initial and final topologies; compare the corresponding results for open maps given later.
**Theorem A2.1.** *Let $(X,\mathcal{U})$ and $(Y_{1},\mathcal{V}_{1})$ be topological spaces and let $X_{1}$ be a set. If $f\!:X_{1}\rightarrow(Y_{1},\mathcal{V}_{1})$, $q\!:(X,\mathcal{U})\rightarrow X_{1}$, and $h=f\circ q\!:(X,\mathcal{U})\rightarrow(Y_{1},\mathcal{V}_{1})$ are functions with the topology $\mathcal{U}_{1}$ of $X_{1}$ given by* $\textrm{FT}\{\mathcal{U};q\}$, *then*
\(a) *$f$ is continuous iff $h$ is continuous.*
\(b) *$f$ is image continuous iff* $\mathcal{V}_{1}=\textrm{FT}\{\mathcal{U};h\}$.$\qquad\square$
**Theorem A2.2.** *Let $(Y,\mathcal{V})$ and $(X_{1},\mathcal{U}_{1})$ be topological spaces and let $Y_{1}$ be a set. If $f\!:(X_{1},\mathcal{U}_{1})\rightarrow Y_{1}$, $e\!:Y_{1}\rightarrow(Y,\mathcal{V})$ and $g=e\circ f\!:(X_{1},\mathcal{U}_{1})\rightarrow(Y,\mathcal{V})$ are function with the topology $\mathcal{V}_{1}$ of $Y_{1}$ given by* $\textrm{IT}\{ e;\mathcal{V}\}$*, then*
\(a) *$f$ is continuous iff $g$ is continuous.*
\(b) *$f$ is preimage continuous iff* $\mathcal{U}_{1}=\textrm{IT}\{ g;\mathcal{V}\}$.$\qquad\square$
As we need the second part of these theorems in our applications, their proofs are indicated below. The special significance of the first parts is that they ensure the converse of the usual result that the composition of two continuous functions is continuous, namely that one of the components of a composition is continuous whenever the composition is so.
**Proof of Theorem A2.1.** If $f$ be image continuous, $\mathcal{V}_{1}=\{ V_{1}\subseteq Y_{1}\!:f^{-}(V_{1})\in\mathcal{U}_{1}\}$ and $\mathcal{U}_{1}=\{ U_{1}\subseteq X_{1}\!:q^{-}(U_{1})\in\mathcal{U}\}$ are the final topologies of $Y_{1}$ and $X_{1}$ based on the topologies of $X_{1}$ and $X$ respectively. Then $\mathcal{V}_{1}=\{ V_{1}\subseteq Y_{1}\!:q^{-}f^{-}(V_{1})\in\mathcal{U}\}$ shows that $h$ is image continuous.
Conversely, when $h$ is image continuous, $\mathcal{V}_{1}=\{ V_{1}\subseteq Y_{1}\!:h^{-}(V_{1})\}\in\mathcal{U}\}=\{ V_{1}\subseteq Y_{1}\!:q^{-}f^{-}(V_{1})\}\in\mathcal{U}\}$, with $\mathcal{U}_{1}=\{ U_{1}\subseteq X_{1}\!:q^{-}(U_{1})\in\mathcal{U}\}$, proves $f^{-}(V_{1})$ to be open in $X_{1}$ and thereby $f$ to be image continuous.
**Proof of Theorem A2.2.** If $f$ be preimage continuous, $\mathcal{V}_{1}=\{ V_{1}\subseteq Y_{1}\!:V_{1}=e^{-}(V)\textrm{ if }V\in\mathcal{V}\}$ and $\mathcal{U}_{1}=\{ U_{1}\subseteq X_{1}\!:U_{1}=f^{-}(V_{1})\textrm{ if }V_{1}\in\mathcal{V}_{1}\}$ are the initial topologies of $Y_{1}$ and $X_{1}$ respectively. Hence from $\mathcal{U}_{1}=\{ U_{1}\subseteq X_{1}\!:U_{1}=f^{-}e^{-}(V)\textrm{ if }V\in\mathcal{V}\}$ it follows that $g$ is preimage continuous.
Conversely, when $g$ is preimage continuous, $\mathcal{U}_{1}=\{ U_{1}\subseteq X_{1}\!:U_{1}=g^{-}(V)\textrm{ if }V\in\mathcal{V}\textrm{ }\}=\{ U_{1}\subseteq X_{1}\!:U_{1}=f^{-}e^{-}(V)\textrm{ if }V\in\mathcal{V}\}$ and $\mathcal{V}_{1}=\{ V_{1}\subseteq Y_{1}\!:V_{1}=e^{-}(V)\textrm{ if }V\in\mathcal{V}\}$ shows that $f$ is preimage continuous.$\qquad\blacksquare$
Since both Eqs. (\[Eqn: IT\]) and (\[Eqn: FT’\]) are in terms of inverse images (the first of which constitutes a direct, and the second an inverse, problem) the image $f(U)=\textrm{comp}(V)$ for $V\in\mathcal{V}$ is of interest as it indicates the relationship of the openness of $f$ with its continuity. This, and other related concepts are examined below, where the range space $f(X)$ is always taken to be a subspace of $Y$. Openness of a function *$f\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$* is the “inverse” of continuity, when images of open sets of $X$ are required to be open in $Y$; such a function is said to be *open.* Following are two of the important properties of open functions.
\(1) *If $f\!:(X,\mathcal{U})\rightarrow(Y,f(\mathcal{U}))$ is an open function, then so is* $f_{<}\!:(X,\mathcal{U})\rightarrow(f(X),\textrm{IT}\{ i;f(\mathcal{U})\})$*. The converse is true if $f(X)$ is an open set of $Y$; thus openness of* $f_{<}\!:(X,\mathcal{U})\rightarrow(f(X),f_{<}(\mathcal{U}))$ *implies tha*t *of $f\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ whenever $f(X)$ is open in $Y$ such that $f_{<}(U)\in\mathcal{V}$ for $U\in\mathcal{U}$.* The truth of this last assertion follows easily from the fact that if $f_{<}(U)$ is an open set of $f(X)\subset Y$, then necessarily $f_{<}(U)=V\bigcap f(X)$ for some $V\in\mathcal{V}$, and the intersection of two open sets of $Y$ is again an open set of $Y$.
\(2) *If $f\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ and $g\!:(Y,\mathcal{V})\rightarrow(Z,\mathcal{W})$ are open functions then $g\circ f\!:(X,\mathcal{U})\rightarrow(Z,\mathcal{W})$* *is also open.* It follows that the condition in (1) on $f(X)$ can be replaced by the requirement that the inclusion $i\!:(f(X),\textrm{IT}\{ i;\mathcal{V}\})\rightarrow(Y,\mathcal{V})$ be an open map. This interchange of $f(X)$ with its inclusion $i\!:f(X)\rightarrow Y$ into $Y$ is a basic result that finds application in many situations.
Collected below are some useful properties of the initial and final topologies that we need in this work.
***Initial Topology.*** In Fig. \[Fig: Initial-Final\](b), consider $Y_{1}=h(X_{1})$, $e\rightarrow i$ and $f\rightarrow h_{<}\!:X_{1}\rightarrow(h(X_{1}),\textrm{IT}\{ i;\mathcal{V}\})$. From $h^{-}(B)=h^{-}(B\bigcap h(X_{1}))$ for any $B\subseteq Y$, it follows that for an open set $V$ of $Y$, $h^{-}(V_{\textrm{comp}})=h^{-}(V)$ is an open set of $X_{1}$ which, if the topology of $X_{1}$ is $\textrm{IT}\{ h;\mathcal{V}\}$, are the only open sets of $X_{1}$. Because $V_{\textrm{comp}}$ is an open set of $h(X_{1})$ in its subspace topology, this implies that *the preimage topologies* $\textrm{IT}\{ h;\mathcal{V}\}$ and $\textrm{IT}\{ h_{<};\textrm{IT}\{ i;\mathcal{V}\}\}$ *of $X_{1}$ generated by $h$ and* $h_{<}$ *are the same.* Thus the preimage topology of $X_{1}$ is not affected if $Y$ is replaced by the subspace $h(X_{1})$, the part $Y-h(X_{1})$ contributing nothing to $\textrm{IT}\{ h;\mathcal{V}\}$.
*A preimage continuous function* $e\!:X\rightarrow(Y,\mathcal{V})$ *is not necessarily an open function.* Indeed, if $U=e^{-}(V)\in\textrm{IT}\{ e;\mathcal{V}\}$, it is almost trivial to verify along the lines of the restriction of open maps to its range, that $e(U)=ee^{-}(V)=e(X)\bigcap V$, $V\in\mathcal{V}$, is open in $Y$ (implying that $e$ is an open map) iff $e(X)$ is an open subset of $Y$ (because finite intersections of open sets are open). A special case of this is the important consequence that *the restriction* $e_{<}\!:(X,\textrm{IT}\{ e;\mathcal{V}\})\rightarrow(e(X),\textrm{IT}\{ i;\mathcal{V}\})$ *of* $e\!:(X,\textrm{IT}\{ h;\mathcal{V}\})\rightarrow(Y,\mathcal{V})$ *to its range is an open map.* Even though a preimage continuous map need not be open, it is true that *an injective, continuous and open map $f\!:X\rightarrow(Y,\mathcal{V})$ is preimage continuous.* Indeed, from its injectivity and continuity, inverse images of all open subsets of $Y$ are saturated-open in $X$, and openness of $f$ ensures that these are the only open sets of $X$ the condition of injectivity being required to exclude non-saturated sets from the preimage topology. It is therefore possible to rewrite Eq. (\[Eqn: IT\]) as
$$U\in\textrm{IT}\{ e;\mathcal{V}\}\Longleftrightarrow e(U)=V\textrm{ if }V\in\mathcal{V}_{\textrm{comp}},\label{Eqn: IT'}$$
and to compare it with the following criterion for an *injective, open-continuous* *map* $f\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ that necessarily satisfies $\textrm{sat}(A)=A$ for all $A\subseteq X$ $$U\in\mathcal{U}\Longleftrightarrow(\{\{ f(U)\}_{U\in\mathcal{U}}=\mathcal{V}_{\textrm{comp}})\wedge(f^{-1}(V)|_{V\in\mathcal{V}}\in\mathcal{U}).\label{Eqn: OCINJ}$$
***Final Topology.*** Since it is necessarily produced on the range $\mathcal{R}(q)$ of $q$, the final topology is often considered in terms of a surjection. This however is not necessary as, much in the spirit of the initial topology, $Y-q(X)\neq\emptyset$ inherits the discrete topology without altering anything, thereby allowing condition (\[Eqn: FT’\]) to be restated in the following more transparent form $$V\in\textrm{FT}\{\mathcal{U};q\}\Longleftrightarrow V=q(U)\textrm{ if }U\in\mathcal{U}_{\textrm{sat}},\label{Eqn: FT}$$
and to compare it with the following criterion for a *surjective, open-continuous* *map* $f\!:(X,\mathcal{U})\rightarrow(Y,\mathcal{V})$ that necessarily satisfies $_{f}B=B$ for all $B\subseteq Y$ $$V\in\mathcal{V}\Longleftrightarrow(\mathcal{U}_{\textrm{sat}}=\{ f^{-}(V)\}_{V\in\mathcal{V}})\wedge(f(U)|_{U\in\mathcal{U}}\in\mathcal{V}).\label{Eqn: OCSUR}$$
As may be anticipated from Fig. \[Fig: Initial-Final\], the final topology does not behave as well for subspaces as the initial topology does. This is so because in Fig. \[Fig: Initial-Final\](a) the two image continuous functions $h$ and $q$ are connected by a preimage continuous inclusion $f$, whereas in Fig. \[Fig: Initial-Final\](b) all the three functions are preimage continuous. Thus quite like open functions, although image continuity of $h\!:(X,\mathcal{U})\rightarrow(Y_{1},\textrm{FT}\{\mathcal{U};h\})$ implies that of $h_{<}\!:(X,\mathcal{U})\rightarrow(h(X),\textrm{IT}\{ i;\textrm{FT}\{\mathcal{U};h\}))$ for a subspace $h(X)$ of $Y_{1}$, the converse need not be true unless — entirely like open functions again — either $h(X)$ is an open set of $Y_{1}$ or $i\!:(h(X),\textrm{IT}\{ i;\textrm{FT}\{\mathcal{U};h\}))\rightarrow(X,\textrm{FT}\{\mathcal{U};h\})$ is an open map. Since an open preimage continuous map is image continuous, this makes $i\!:h(X)\rightarrow Y_{1}$ an ininal function and hence all the three legs of the commutative diagram image continuous.
Like preimage continuity, *an image continuous function $q\!:(X,\mathcal{U})\rightarrow Y$ need not be open.* However, although *the restriction of an image continuous function to the saturated open sets of its domain is an open function*, $q$ is unrestrictedly open iff the saturation of every open set of $X$ is also open in $X$. Infact it can be verified without much effort that a continuous, open surjection is image continuous.
Combining Eqs. (\[Eqn: IT’\]) and (\[Eqn: FT\]) gives the following criterion for ininality $$U\textrm{ and }V\in\textrm{IFT}\{\mathcal{U}_{\textrm{sat}};f;\mathcal{V}\}\Longleftrightarrow(\{ f(U)\}_{U\in\mathcal{U}_{\textrm{sat}}}=\mathcal{V})(\mathcal{U}_{\textrm{sat}}=\{ f^{-}(V)\}_{V\in\mathcal{V}}),\label{Eqn: INI}$$
which reduces to the following for a homeomorphism $f$ that satisfies both $\textrm{sat}(A)=A$ for $A\subseteq X$ and $_{f}B=B$ for $B\subseteq Y$ $$U\textrm{ and }V\in\textrm{HOM}\{\mathcal{U};f;\mathcal{V}\}\Longleftrightarrow(\mathcal{U}=\{ f^{-1}(V)\}_{V\in\mathcal{V}})(\{ f(U)\}_{U\in\mathcal{U}}=\mathcal{V})\label{Eqn: HOM}$$
and compares with $$\begin{gathered}
U\textrm{ and }V\in\textrm{OC}\{\mathcal{U};f;\mathcal{V}\}\Longleftrightarrow(\textrm{sat}(U)\in\mathcal{U}\!:\{ f(U)\}_{U\in\mathcal{U}}=\mathcal{V}_{\textrm{comp}})\wedge\\
\wedge(\textrm{comp}(V)\in\mathcal{V}\!:\{ f^{-}(V)\}_{V\in\mathcal{V}}=\mathcal{U}_{\textrm{sat}})\label{Eqn: OC}\end{gathered}$$
for an open-continuous $f$.
The following is a slightly more general form of the restriction on the inclusion that is needed for image continuity to behave well for subspaces of $Y$.
**Theorem A2.3.** *Let* $q\!:(X,\mathcal{U})\rightarrow(Y,\textrm{FT}\{\mathcal{U};q\})$ *be an image continuous* *function. For a subspace* $B$ of $(Y,\textrm{FT}\{\mathcal{U};q\})$,$$\textrm{FT}\{\textrm{IT}\{ j;\mathcal{U}\};q_{<}\}=\textrm{IT}\{ i;\textrm{FT}\{\mathcal{U};q\}\}$$
*where* $q_{<}\!:(q^{-}(B),\textrm{IT}\{ j;\mathcal{U}\})\rightarrow(B,\textrm{FT}\{\textrm{IT}\{ j;\mathcal{U}\};q_{<}\})$, *if either $q$ is an* *open map or $B$ is an open set of* $Y$.$\qquad\square$
In summary we have the useful result that an open preimage continuous function is image continuous and an open image continuous function is preimage continuous, where the second assertion follows on neglecting non-saturated open sets in $X$; this is permitted in as far as the generation of the final topology is concerned, as these sets produce the same images as their saturations. Hence *an image continuous function* $q\!:X\rightarrow Y$ *is preimage continuous iff every open set in $X$ is saturated with respect to $q$,* and *a preimage continuous function* $e\!:X\rightarrow Y$ *is image continuous iff the $e$-image of every open set of $X$ is open in $Y$.*
**A3. More on Topological Spaces**
This Appendix — which completes the review of those concepts of topological spaces begun in Tutorial4 that are needed for a proper understanding of this work — begins with the following summary of the different possibilities in the distribution of $\textrm{Der}(A)$ and $\textrm{Bdy}(A)$ between sets $A\subseteq X$ and its complement $X-A$, and follows it up with a few other important topological concepts that have been used, explicitly or otherwise, in this work.
**Definition A3.1.** ***Separation, Connected Space*.** *A* *separation* *(disconnection)* *of $X$ is a pair of mutually disjoint nonempty open (and therefore closed) subsets $H_{1}$ and $H_{2}$ such that $X=H_{1}\cup H_{2}$* *A space $X$ is said to be* *connected* *if it has no separation, that is if it cannot be partitioned into two open or two closed nonempty subsets. $X$ is* *separated (disconnected)* *if it is not connected.$\qquad\square$*
It follows from the definition, that for a disconnected space $X$ the following are equivalent statements.
\(a) There exist a pair of disjoint nonempty open subsets of $X$ that cover $X$.
\(b) There exist a pair of disjoint nonempty closed subsets of $X$ that cover $X.$
\(c) There exist a pair of disjoint nonempty clopen subsets of $X$ that cover $X.$
\(d) There exists a nonempty, proper, clopen subset of $X$.
By a *connected subset* is meant a subset of $X$ that is connected *when provided with its relative topology making it a subspace of $X$.* Thus any connected subset of a topological space must necessarily be contained in any clopen set that might intersect it: if $C$ and $H$ are respectively connected and clopen subsets of $X$ such that $C\bigcap H\neq\emptyset$, then $C\subset H$ because $C\bigcap H$ is a nonempty clopen set in $C$ which must contain $C$ because $C$ is connected.
For testing whether a subset of a topological space is connected, the following relativized form of (a)$-$(d) is often useful.
**Lemma A3.1.** *A subset $A$ of $X$ is disconnected iff there are disjoint open sets $U$ and $V$ of $X$ satisfying* $${\textstyle U\bigcap A\neq\emptyset\neq V\bigcap A\textrm{ such that }A\subseteq U\bigcup V,\;\textrm{with }U\bigcap V\bigcap A=\emptyset}\label{Eqn: SubDisconnect1}$$ *or there are disjoint closed sets $E$ and $F$ of $X$ satisfying* $${\textstyle E\bigcap A\neq\emptyset\neq F\bigcap A\textrm{ such that }A\subseteq E\bigcup F,\;\textrm{with }E\bigcap F\bigcap A=\emptyset.}\label{Eqn: SybDisconnect2}$$ *Thus $A$ is disconnected iff there are disjoint clopen subsets in the relative topology of $A$ that cover $A$.$\qquad\square$*
**Lemma A3.2.** *If $A$ is a subspace of $X$, a* *separation of* *$A$ is a pair of disjoint nonempty subsets $H_{1}$ and $H_{2}$ of $A$ whose union is $A$ neither of which contains a cluster point of the other. $A$ is connected iff there is no separation of $A.$* *$\qquad\square$*
**Proof.** Let $H_{1}$ and $H_{2}$ be a separation of $A$ so that they are clopen subsets of $A$ whose union is $A$. As $H_{1}$ is a closed subset of $A$ it follows that $H_{1}=\textrm{Cl}_{X}(H_{1})\bigcap A$, where $\textrm{Cl}_{X}(H_{1})\bigcap A$ is the closure of $H_{1}$ in $A$; hence $\textrm{Cl}_{X}(H_{1})\bigcap H_{2}=\emptyset$. But as the closure of a subset is the union of the set and its adherents, an empty intersection signifies that $H_{2}$ cannot contain any of the cluster points of $H_{1}$. A similar argument shows that $H_{1}$ does not contain any adherent of $H_{2}$.
Conversely suppose that neither $H_{1}$ nor $H_{2}$ contain an adherent of the other: $\textrm{Cl}_{X}(H_{1})\bigcap H_{2}=\emptyset$ and $\textrm{Cl}_{X}(H_{2})\bigcap H_{1}=\emptyset$. Hence $\textrm{Cl}_{X}(H_{1})\bigcap A=H_{1}$ and $\textrm{Cl}_{X}(H_{2})\bigcap A=H_{2}$ so that both $H_{1}$ and $H_{2}$ are closed in $A.$ But since $H_{1}=A-H_{2}$ and $H_{2}=A-H_{1}$, they must also be open in the relative topology of $A$. *$\qquad\blacksquare$*
Following are some useful properties of connected spaces.
(c1) The closure of any connected subspace of a space is connected. In general, every $B$ satisfying $$A\subseteq B\subseteq\textrm{Cl}(A)$$ is connected. Thus any subset of $X$ formed from $A$ by adjoining to it some or all of its adherents is connected so that *a topological space with a dense connected subset is connected.*
(c2) The union of any class of connected subspaces of $X$ with nonempty intersection is a connected subspace of $X$.
(c3) A topological space is connected iff there is a covering of the space consisting of connected sets with nonempty intersection. Connectedness is a topological property: Any space homeomorphic to a connected space is itself connected.
(c4) If $H_{1}$ and $H_{2}$ is a separation of $X$ and $A$ is any connected subset $A$ of $X$, then either $A\subseteq H_{1}$ or $A\subseteq H_{2}$*.*
While the real line $\mathbb{R}$ is connected, a subspace of $\mathbb{R}$ is connected iff it is an interval in $\mathbb{R}$.
The important concept of total disconnectedness introduced below needs the following
**Definition A3.2.** ***Component*.** *A* *component $C^{*}$* *of a space $X$ is a maximally* (with respect to inclusion) *connected subset of $X$.* *$\qquad\square$*
Thus a component is a connected subspace which is not properly contained in any larger connected subspace of $X$. The maximal element need not be unique as there can be more than one component of a given space and a “maximal” criterion rather than “maximum” is used as the component need not contain every connected subsets of $X$; it simply must not be contained in any other connected subset of $X$. Components can be constructively defined as follows: Let $x\in X$ be any point. Consider the collection of all connected subsets of $X$ to which $x$ belongs Since $\{ x\}$ is one such set, the collection is nonempty. As the intersection of the collection is nonempty, its union is a nonempty connected set $C$. This the largest connected set containing $x$ and is therefore a component containing $x$ and we have
(C1) Let $x\in X$. The unique component of $X$ containing $x$ is the union of all the connected subsets of $X$ that contain *$x$.* Conversely **any nonempty connected subset $A$ of $X$ **is contained in that unique component of $X$ to which each of the points of $A$ belong*.* Hence *a* *topological space is connected iff it is the unique component of itself.*
(C2) Each component $C^{*}$ of $X$ is a closed set of $X$: By property (c1) above, $\textrm{Cl}(C^{*})$ is also connected and from $C^{*}\subseteq\textrm{Cl}(C^{*})$ it follows that $C^{*}=\textrm{Cl}(C^{*})$. Components need not be open sets of $X$: an example of this is the space of rationals $\mathbb{Q}$ in reals in which the components are the individual points which cannot be open in $\mathbb{R}$; see Example (2) below.
(C3) Components of $X$ are equivalence classes of $(X,\sim)$ with $x\sim y$ iff they are in the same component: while reflexivity and symmetry are obvious enough, transitivity follows because if $x,y\in C_{1}$ and $y,z\in C_{2}$ with $C_{1}$, $C_{2}$ connected subsets of $X$, then $x$ and $z$ are in the set $C_{1}\bigcup C_{2}$ which is connected by property c(2) above as they have the point $y$ in common. Components are connected disjoint subsets of $X$ whose union is $X$ (that is they form a partition of $X$ with each point of $X$ contained in exactly one component of $X$) such that any connected subset of $X$ can be contained in only one of them. Because a connected subspace cannot contain in it any clopen subset of $X$, it follows that *every clopen connected subspace must be a component of $X$.*
Even when a space is disconnected, it is always possible to decompose it into pairwise disjoint connected subsets. If $X$ is a discrete space this is the only way in which $X$ may be decomposed into connected pieces. If $X$ is not discrete, there may be other ways of doing this. For example, the space $$X=\{ x\in\mathbb{R}\!:(0\leq x\leq1)\vee(2<x<3)\}$$
has the following three distinct decomposition into connected subsets:
$$\begin{array}{rcl}
{\displaystyle X} & = & [0,1/2)\bigcup[1/2,1]\bigcup(2,7/3]\bigcup(7/3,3)\\
X & = & \{0\}\bigcup{\displaystyle \left(\bigcup_{n=1}^{\infty}\left(\frac{1}{n+1},\frac{1}{n}\right]\right)}\bigcup(2,3)\\
X & = & [0,1]\bigcup(2,3).\end{array}$$
Intuition tells us that only in the third of these decompositions have we really broken up $X$ into its connected pieces. What distinguishes the third from the other two is that neither of the pieces $[0,1]$ or $(2,3)$ can be enlarged into bigger connected subsets of $X$.
As connected spaces, the empty set and the singleton are considered to be *degenerate* and any connected subspace with more than one point is *nondegenerate.* At the opposite extreme of the largest possible component of a space $X$ which is $X$ itself, are the singletons $\{ x\}$ for every $x\in X$. This leads to the extremely important notion of a
**Definition A3.3.** ***Totally disconnected space.*** *A space $X$ is* *totally disconnected* *if every pair of distinct points in it can be separated by a disconnection of $X$.$\qquad\square$*
$X$ is totally disconnected iff the components in $X$ are single points with the only nonempty connected subsets of $X$ being the one-point sets: If $x\neq y\in A\subseteq X$ are distinct points of a subset $A$ of $X$ then $A=(A\bigcap H_{1})\bigcup(A\bigcap H_{2})$, where $X=H_{1}\bigcup H_{2}$ with $x\in H_{1}$ and $y\in H_{2}$ is a disconnection of $X$ (it is possible to choose $H_{1}$ and $H_{2}$ in this manner because $X$ is assumed to be totally disconnected), is a separation of $A$ that demonstrates that any subspace of a totally disconnected space with more than one point is disconnected.
A totally disconnected space has interesting physically appealing separation properties in terms of the (separated) Hausdorff spaces; here a topological space $X$ is *Hausdorff, or $T_{2}$,* iff each two distinct points of $X$ can be *separated* by disjoint neighbourhoods, so that for every $x\neq y\in X$, there are neighbourhoods $M\in\mathcal{N}_{x}$ and $N\in\mathcal{N}_{y}$ such that $M\bigcap N=\emptyset$. This means that for any two distinct points $x\neq y\in X$, it is impossible to find points that are arbitrarily close to both of them. Among the properties of Hausdorff spaces, the following need to be mentioned.
(H1) $X$ is Hausdorff iff for each $x\in X$ and any point $y\neq x$, there is a neighbourhood $N$ of $x$ such that $y\not\in\textrm{Cl}(N)$. This leads to the significant result that for any $x\in X$ the closed singleton $$\{ x\}=\bigcap_{N\in\mathcal{N}_{x}}\textrm{Cl}(N)$$ *is the intersection of the closures of any local base at that point,* which in the language of nets and filters (Appendix A1) means that a net in a Hausdorff space cannot converge to more than one point in the space and the adherent set $\textrm{adh}(\mathcal{N}_{x})$ of the neighbourhood filter at $x$ is the singleton $\{ x\}$.
(H2) Since each singleton is a closed set, each finite set in a Hausdorff space is also closed in $X$. Unlike a cofinite space, however, there can clearly be infinite closed sets in a Hausdorff space.
(H3) Any point $x$ in a Hausdorff space $X$ is a cluster point of $A\subseteq X$ iff every neighbourhood of $x$ contains infinitely many points of $A$, a fact that has led to our mental conditioning of the points of a (Cauchy) sequence piling up in neighbourhoods of the limit. Thus suppose for the sake of argument that although some neighbourhood of $x$ contains only a finite number of points, $x$ is nonetheless a cluster point of $A$. Then there is an open neighbourhood $U$ of $x$ such that $U\bigcap(A-\{ x\})=\{ x_{1},\cdots,x_{n}\}$ is a finite closed set of $X$ not containing $x$, and $U\bigcap(X-\{ x_{1},\cdots,x_{n}\})$ being the intersection of two open sets, is an open neighbourhood of $x$ not intersecting $A-\{ x\}$ implying thereby that $x\not\in\textrm{Der}(A)$; infact $U\bigcap(X-\{ x_{1},\cdots,x_{n}\})$ is simply $\{ x\}$ if $x\in A$ or belongs to $\textrm{Bdy}_{X-A}(A)$ when $x\in X-A$. Conversely if every neighbourhood of a point of $X$ intersects $A$ in infinitely many points, that point must belong to $\textrm{Der}(A)$ by definition.
Weaker separation axioms than Hausdorffness are those of $T_{0}$, respectively $T_{1}$, spaces in which for every pair of distinct points *at least one,* respectively *each one,* has some neighbourhood not containing the other; the following table is a listing of the separation properties of some useful spaces.
Space $T_{0}$ $T_{1}$ $T_{2}$
------------------------- -------------- -------------- --------------
Discrete $\checkmark$ $\checkmark$ $\checkmark$
Indiscrete $\times$ $\times$ $\times$
$\mathbb{R}$, standard $\checkmark$ $\checkmark$ $\checkmark$
left/right ray $\checkmark$ $\times$ $\times$
Infinite cofinite $\checkmark$ $\checkmark$ $\times$
Uncountable cocountable $\checkmark$ $\checkmark$ $\times$
$x$-inclusion/exclusion $\checkmark$ $\times$ $\times$
$A$-inclusion/exclusion $\times$ $\times$ $\times$
: \[Table: separation\][Separation properties of some useful spaces.]{}
It should be noted that that as none of the properties (H1)–(H3) need neighbourhoods of both the points simultaneously, it is sufficient for $X$ to be $T_{1}$ for the conclusions to remain valid.
From its definition it follows that any totally disconnected space is a Hausdorff space and is therefore both $T_{1}$ and $T_{0}$ spaces as well. However, if a Hausdorff space has a base of clopen sets then it is totally disconnected; this is so because if $x$ and $y$ are distinct points of $X$, then the assumed property of $x\in H\subseteq M$ for every $M\in\mathcal{N}_{x}$ and some clopen set $M$ yields $X=H\bigcup(X-H)$ as a disconnection of $X$ that separates $x$ and $y\in X-H$; note that the assumed Hausdorffness of $X$ allows $M$ to be chosen so as not to contain $y$.
**Example A3.1.** (1) Every indiscrete space is connected; every subset of an indiscrete space is connected. Hence if $X$ is empty or a singleton, it is connected. A discrete space is connected iff it is either empty or is a singleton; the only connected subsets in a discrete space are the degenerate ones. This is an extreme case of lack of connectedness, and a discrete space is the simplest example of a total disconnected space.
\(2) $\mathbb{Q}$, the set of rationals considered as a subspace of the real line, is (totally) disconnected because all rationals larger than a given irrational $r$ is a clopen set in $\mathbb{Q}$, and $${\textstyle \mathbb{Q}=((-\infty,r)\bigcap\mathbb{Q})\bigcup(\mathbb{Q}\bigcap(r,\infty))\qquad r\textrm{ is an irrational}}$$
is the union of two disjoint clopen sets in the relative topology of $\mathbb{Q}$. The sets *$(-\infty,r)\cap\mathbb{Q}$* and $\mathbb{Q}\cap(r,\infty)$ are clopen in $\mathbb{Q}$ because neither contains a cluster point of the other. Thus for example, any neighbourhood of the second must contain the irrational $r$ in order to be able to cut the first which means that any neighbourhood of a point in either of the relatively open sets cannot be wholly contained in the other. The only connected sets of $\mathbb{Q}$ are one point subsets consisting of the individual rationals. In fact, a connected piece of $\mathbb{Q}$, being a connected subset of $\mathbb{R}$, is an interval in $\mathbb{R}$, and a nonempty interval cannot be contained in $\mathbb{Q}$ unless it is a singleton. It needs to be noted that the individual points of the rational line are not (cl)open because any open subset of $\mathbb{R}$ that contains a rational must also contain others different from it. This example shows that a space need not be discrete for each of its points to be a component and thereby for the space to be totally disconnected.
In a similar fashion, the set of irrationals is (totally) disconnected because all the irrationals larger than a given rational is an example of a clopen set in $\mathbb{R}-\mathbb{Q}$.
\(3) The $p$-inclusion ($A$-inclusion) topology is connected; a subset in this topology is connected iff it is degenerate or contains $p$. For, a subset inherits the discrete topology if it does not contain $p$, and $p$-inclusion topology if it contains $p$.
\(4) The cofinite (cocountable) topology on an infinite (uncountable) space is connected; a subset in a cofinite (cocountable) space is connected iff it is degenerate or infinite (countable).
\(5) Removal of a single point may render a connected space disconnected and even totally disconnected. In the former case, the point removed is called a *cut point* and in the second, it is a *dispersion point.* Any real number is a cut point of $\mathbb{R}$ and it does not have any dispersion point only.
\(6) Let $X$ be a topological space. Considering components of $X$ as equivalence classes by the equivalence relation $\sim$ with $Q\!:X\rightarrow X/\sim$ denoting the quotient map, $X/\sim$ is totally disconnected: As $Q^{-}([x])$ is connected for each $[x]\in X/\sim$ in a component class of $X$, and as any open or closed subset $A\subseteq X/\sim$ is connected iff $Q^{-}(A)$ is open or closed, it must follow that $A$ can only be a singleton.$\qquad\blacksquare$
The next notion of compactness in topological spaces provides an insight of the role of nonempty adherent sets of filters that lead in a natural fashion to the concept of attractors in the dynamical systems theory that we take up next.
**Definition A3.4.** ***Compactness.*** *A topological space $X$ is* *compact* *iff every open cover of $X$ contains a finite subcover of $X$*. *$\qquad\square$*
This definition of compactness has an useful equivalent contrapositive reformulation: *For any given collection of open sets of $X$ if none of its finite subcollections cover $X$, then the entire collection also cannot cover $X$.* The following theorem is a statement of the fundamental property of compact spaces in terms of adherences of filters in such spaces, the proof of which uses this contrapositive characterization of compactness.
**Theorem A3.1.** *A topological space $X$ is compact iff each class of closed subsets of $X$ with finite intersection property has nonempty intersection.* *$\qquad\square$*
**Proof.** *Necessity.* Let $X$ be a compact space. Let $\mathcal{F}=\{ F_{\alpha}\}_{\alpha\in\mathbb{D}}$ be a collection of closed subsets of $X$ with finite FIP, and let $\mathcal{G}=\{ X-F_{\alpha}\}_{\alpha\in\mathbb{D}}$ be the corresponding open sets of $X$. If $\{ G_{i}\}_{i=1}^{N}$ is a nonempty finite subcollection from $\mathcal{G}$, then $\{ X-G_{i}\}_{i=1}^{N}$ is the corresponding nonempty finite subcollection of $\mathcal{F}$. Hence from the assumed finite intersection property of $\mathcal{F}$, it must be true that $$\begin{array}{ccl}
{\displaystyle X-\bigcup_{i=1}^{N}G_{i}} & = & {\displaystyle \bigcap_{i=1}^{N}(X-G_{i})}\qquad(\textrm{DeMorgan}'\textrm{s Law})\\
& \neq & \emptyset,\end{array}$$
so that no finite subcollection of $\mathcal{G}$ can cover $X$. Compactness of $X$ now implies that $\mathcal{G}$ too cannot cover $X$ and therefore $$\bigcap_{\alpha}F_{\alpha}=\bigcap_{\alpha}(X-G_{\alpha})=X-\bigcup_{\alpha}G_{\alpha}\neq\emptyset.$$ The proof of the converse is a simple exercise of reversing the arguments involving the two equations in the proof above.$\qquad\blacksquare$
Our interest in this theorem and its proof lies in the following corollary — *which essentially means that for every filter $\mathcal{F}$ on a compact space the adherent set* $\textrm{adh}(\mathcal{F})$ *is not empty —* from which follows that every net in a compact space must have a convergent subnet.
**Corollary.** *A space $X$ is compact iff for every class $\mathcal{A}=(A_{\alpha})$ of nonempty subsets of $X$ with* FIP*,* $\textrm{adh}(\mathcal{A})=\bigcap_{A_{\alpha}\in\mathcal{A}}\textrm{Cl}(A_{\alpha})\neq\emptyset$*.*$\qquad\square$
The proof of this result for nets given by the next theorem illustrates the general approach in such cases which is all that is basically needed in dealing with attractors of dynamical systems; compare Theorem A1.3.
**Theorem A3.2.** *A topological space $X$ is compact iff each net in $X$ adheres in $X$*.$\qquad\square$
**Proof.** *Necessity.* Let $X$ be a compact space, $\chi\!:\mathbb{D}\rightarrow X$ a net in $X$, and $\mathbb{R}_{\alpha}$ the residual of $\alpha$ in the directed set $\mathbb{D}$. For the filter-base $(_{\textrm{F}}\mathcal{B}_{\chi(\mathbb{R}_{\alpha})})_{\alpha\in\mathbb{D}}$ of nonempty, decreasing, nested subsets of $X$ associated with the net $\chi$, compactness of $X$ requires from $\bigcap_{\alpha\preceq\delta}\textrm{Cl}(\chi(\mathbb{R}_{\alpha})\supseteq\chi(\mathbb{R}_{\delta})\neq\emptyset$, that the uncountably intersecting subset $$\textrm{adh}(_{\textrm{F}}\mathcal{B}_{\chi}):=\bigcap_{\alpha\in\mathbb{D}}\textrm{Cl}(\chi(\mathbb{R}_{\alpha}))$$
of $X$ be non-empty. If $x\in\textrm{adh}(_{\textrm{F}}\mathcal{B}_{\chi})$ then because $x$ is in the closure of $\chi(\mathbb{R}_{\beta})$, it follows from Eq. (\[Eqn: Def: Closure\]) that $N\bigcap\chi(\mathbb{R}_{\beta})\neq\emptyset$[^33] for every $N\in\mathcal{N}_{x}$, $\beta\in\mathbb{D}$. Hence $\chi(\gamma)\in N$ for some $\gamma\succeq\beta$ so that $x\in\textrm{adh}(\chi)$; see Eq. (\[Eqn: adh net2\]).
*Sufficiency.* Let *$\chi$* be a net in $X$ that adheres at $x\in X$. From any class $\mathcal{F}$ of closed subsets of $X$ with FIP, construct as in the proof of Thm. A1.4, a decreasing nested sequence of closed subsets $C_{\beta}=\bigcap_{\alpha\preceq\beta\in\mathbb{D}}\{ F_{\alpha}\!:F_{\alpha}\in\mathcal{F}\}$ and consider the directed set $_{\mathbb{D}}C_{\beta}=\{(C_{\beta},\beta)\!:(\beta\in\mathbb{D})(x_{\beta}\in C_{\beta})\}$ with its natural direction (\[Eqn: DirectionIndexed\]) to define the net $\chi(C_{\beta},\beta)=x_{\beta}$ in $X$; see Def. A1.10. From the assumed adherence of $\chi$ at some $x\in X$, it follows that $N\bigcap F\neq\emptyset$ for every $N\in\mathcal{N}_{x}$ and $F\in\mathcal{F}$. Hence $x$ belongs to the closed set $F$ so that $x\in\textrm{adh}(\mathcal{F})$; see Eq. (\[Eqn: adh filter\]). Hence $X$ is compact.$\qquad\blacksquare$
Using Theorem A1.5 that specifies a definite criterion for the adherence of a net, this theorem reduces to the useful formulation that *a space is compact iff each net in it has some convergent subnet.* An important application is the following: Since every decreasing sequence $(F_{m})$ of nonempty sets has FIP (because $\bigcap_{m=1}^{M}F_{m}=F_{M}$ for every finite $M$), *every decreasing sequence of nonempty* closed *subsets* *of a compact spac*e *has nonempty intersection.* For a complete metric space this is known as the *Nested Set Theorem,* and for $[0,1]$ and other compact subspaces of $\mathbb{R}$ as the *Cantor Intersection Theorem.*[^34]
For subspaces $A$ of $X$, it is the relative topology that determines as usual compactness of $A$; however the following criterion renders this test in terms of the relative topology unnecessary and shows that the topology of $X$ itself is sufficient to determine compactness of subspaces: *A subspace $K$ of a topological space $X$ is compact iff each open cover of $K$ in $X$ contains a finite cover of $K$.*
A proper understanding of the distinction between compactness and closedness of subspaces — which often causes much confusion to the non-specialist — is expressed in the next two theorems. As a motivation for the first that establishes that not every subset of a compact space need be compact, mention may be made of the subset $(a,b)$ of the compact closed interval $[a,b]$ in $\mathbb{R}$.
**Theorem A3.3.** *A closed subset $F$ of a compact space $X$ is compact.* *$\qquad\square$*
**Proof.** Let $\mathcal{G}$ be an open cover of $F$ so that an open cover of $X$ is $\mathcal{G}\bigcup(X-F)$, which because of compactness of $X$ contains a finite subcover $\mathcal{U}$. Then $\mathcal{U}-(X-F)$ is a finite collection of $\mathcal{G}$ that covers $F$.*$\qquad\blacksquare$*
It is not true in general that a compact subset of a space is necessarily closed. For example, in an infinite set $X$ with the cofinite topology, let $F$ be an infinite subset of $X$ with $X-F$ also infinite. Then although $F$ is not closed in $X$, it is nevertheless compact because $X$ is compact. Indeed, let $\mathcal{G}$ be an open cover of $X$ and choose any nonempty $G_{0}\in\mathcal{G}$. If $G_{0}=X$ then $\{ G_{0}\}$ is the required finite cover of $X$. If this is not the case, then because $X-G_{0}=\{ x_{i}\}_{i=1}^{n}$ is a finite set, there is a $G_{i}\in\mathcal{G}$ with $x_{i}\in G_{i}$ for each $1\leq i\leq n$, and therefore $\{ G_{i}\}_{i=0}^{n}$ is the finite cover that demonstrates the compactness of the cofinite space $X$. Compactness of $F$ now follows because the subspace topology on $F$ is the induced cofinite topology from $X$. The distinguishing feature of this topology is that it, like the cocountable, is not Hausdorff: If $U$ and $V$ are any two nonempty open sets of $X$, then they cannot be disjoint as the complements of the open sets can only be finite and if $U\bigcap V$ were to be indeed empty, then $${\textstyle X=X-\emptyset={X-(U\bigcap V)=(X-U)\bigcup(X-V)}}$$
would be a finite set. An immediate fallout of this is that in an infinite cofinite space, a sequence $(x_{i})_{i\in\mathbb{N}}$ (and even a net) with $x_{i}\neq x_{j}$ for $i\neq j$ behaves in an extremely unusual way: *It converges,* as in the indiscrete space, *to* *every point of the space.* Indeed if $x\in X$, where $X$ is an infinite set provided with its cofinite topology, and $U$ is any neighbourhood of $x$, any infinite sequence $(x_{i})_{i\in\mathbb{N}}$ in $X$ must be eventually in $U$ because $X-U$ is finite, and ignoring of the initial set of its values lying in $X-U$ in no way alters the ultimate behaviour of the sequence (note that this implies that the filter induced on $X$ by the sequence agrees with its topology). Thus $x_{i}\rightarrow x$ for any $x\in X$ is a reflection of the fact that there are no small neighbourhoods of any point of $X$ with every neighbourhood being almost the whole of $X$, except for a null set consisting of only a finite number of points. This is in sharp contrast with Hausdorff spaces where, although every finite set is also closed, every point has arbitrarily small neighbourhoods that lead to unique limits of sequences. A corresponding result for cocountable spaces can be found in Example A1.2 Continued.
This example of the cofinite topology motivates the following “converse” of the previous theorem.
**Theorem A3.4.** *Every compact subspace of a Hausdorff space is closed.$\qquad\square$*
**Proof.** Let $K$ be a nonempty compact subset of $X$, Fig. \[Fig: cmpct\_clsd\], and let $x\in X-K$. Because of the separation of $X$, for every $y\in K$ there are disjoint open subsets $U_{y}$ and $V_{y}$ of $X$ with $y\in U_{y}$, and $x\in V_{y}$. Hence $\{ U_{y}\}_{y\in K}$ is an open cover for $K$, and from its compactness there is a finite subset $A$ of $K$ such that $K\subseteq\bigcup_{y\in A}U_{y}$ with $V=\bigcap_{y\in A}V_{y}$ an open neighbourhood of $x$; $V$ is open because each $V_{y}$ is a neighbourhood of $x$ and the intersection is over finitely many points $y$ of $A$. To prove that $K$ is closed in $X$ it is enough to show that $V$ is disjoint from $K$: If there is indeed some $z\in V\bigcap K$ then $z$ must be in some $U_{y}$ for $y\in A$. But as $z\in V$ it is also in $V_{y}$ which is impossible as $U_{y}$ and $V_{y}$ are to be disjoint. **This last part of the argument infact shows that *if $K$ is a compact subspace of a Hausdorff space $X$ and $x\notin K$, then there are disjoint open sets $U$ and $V$ of $X$ containing $x$ and $K$.$\qquad\blacksquare$*
The last two theorems may be combined to give the obviously important
**Corollary.** *In a compact Hausdorff space, closedness and compactness of its subsets are equivalent concepts.$\qquad\square$*
In the absence of Hausdorffness, it is not possible to conclude from the assumed compactness of the space that every point to which the net may converge actually belongs to the subspace.
**Definition A3.5.** *A subset $D$ of a topological space* *$(X,\mathcal{U})$* *is* *dense in $X$ if* $\textrm{Cl}(D)=X$*. Thus the closure of $D$ is the largest open subset of $X$, and every neighbourhood of any point of $X$ contains a point of $D$ not necessarily distinct from it; refer to the distinction between Eqs. (\[Eqn: Def: Closure\]) and (\[Eqn: Def: Derived\]).$\qquad\square$*
Loosely, $D$ is dense in $X$ iff every point of $X$ has points of $D$ arbitrarily close to it. A *self-dense* (*dense in itself*) set is a set without any isolated points; hence $A$ is self-dense iff $A\subseteq\textrm{Der}(A)$. A closed self-dense set is called a *perfect set* so that a closed set $A$ is perfect iff it has no isolated points. Accordingly **$$A\textrm{ is perfect}\Longleftrightarrow A=\textrm{Der}(A),$$
means that the closure of a set without any isolated points is a perfect set.
**Theorem A3.5.** *The following are equivalent statements.*
\(1) *$D$ is dense in $X$*.
\(2) *If $F$ is any closed set of $X$ with $D\subseteq F$, then $F=X$*; *thus the only closed superset of $D$ is $X$.*
\(3) *Every nonempty (basic) open set of $X$ cuts $D;$ thus the only open set disjoint from $D$ is the empty set $\emptyset$.*
\(4) *The exterior of $D$ is empty.$\qquad\square$*
**Proof.** (3) If $U$ indeed is a nonempty open set of $X$ with $U\bigcap D=\emptyset$, then $D\subseteq X-U\neq X$ leads to the contradiction $X=\textrm{Cl}(D)\subseteq\textrm{Cl}(X-U)=X-U\neq X$, which also incidentally proves (2). From (3) it follows that for any open set $U$ of $X$, $\textrm{Cl}(U)=\textrm{Cl}(U\bigcap D)$ because if $V$ is any open neighbourhood of $x\in\textrm{Cl}(U)$ then $V\bigcap U$ is a nonempty open set of $X$ that must cut $D$ so that $V\bigcap(U\bigcap D)\neq\emptyset$ implies $x\in\textrm{Cl}(U\bigcap D)$. Finally, $\textrm{Cl}(U\bigcap D)\subseteq\textrm{Cl}(U)$ completes the proof.$\qquad\blacksquare$
**Definition A3.6.** (a) *A set $A\subseteq X$ is said to be* *nowhere dense* *in* ***$X$ if* $\textrm{Int}(\textrm{Cl}(A))=\emptyset$ *and* *residual* *in* ***$X$ if* $\textrm{Int}(A)=\emptyset$*.$\qquad\square$*
$A$ is nowhere dense in $X$ iff $$\textrm{Bdy}(X-\textrm{Cl}(A))=\textrm{Bdy}(\textrm{Cl}(A))=\textrm{Cl}(A)$$
so that $${\textstyle \textrm{Cl}(X-\textrm{Cl}(A))={(X-\textrm{Cl}(A))\bigcup\textrm{Cl}(A)=X}}$$
from which it follows that $$A\textrm{ is nwd in }X\Longleftrightarrow X-\textrm{Cl}(A)\textrm{ is dense in }X$$
and $$A\textrm{ is residual in }X\Longleftrightarrow X-A\textrm{ is dense in }X.$$
Thus $A$ is nowhere dense iff $\textrm{Ext}(A):=X-\textrm{Cl}(A)$ **is dense in *$X$,* and in particular a closed set is nowhere dense in $X$ iff its complement is open dense in $X$ with open-denseness being complimentarily dual to closed-nowhere denseness. The rationals in reals is an example of a set that is residual but not nowhere dense. The following are readily verifiable properties of subsets of $X$.
\(1) A set $A\subseteq X$ is nowhere dense in $X$ iff it is contained in its own boundary, iff it is contained in the closure of the complement of its closure, that is $A\subseteq\textrm{Cl}(X-\textrm{Cl}(A))$. In particular a closed subset $A$ is nowhere dense in $X$ iff $A=\textrm{Bdy}(A)$, that is iff it contains no open set.
\(2) From $M\subseteq N\Rightarrow\textrm{Cl}(M)\subseteq\textrm{Cl}(N)$ it follows, with $M=X-\textrm{Cl}(A)$ and $N=X-A$, that a nowhere dense set is residual, but a residual set need not be nowhere dense unless it is also closed in $X$.
\(3) Since $\textrm{Cl}(\textrm{Cl}(A))=\textrm{Cl}(A)$, $\textrm{Cl}(A)$ is nowhere dense in $X$ iff $A$ is.
\(4) For any $A\subseteq X$, both $\textrm{Bdy}_{A}(X-A):=\textrm{Cl}(X-A)\bigcap A$ and $\textrm{Bdy}_{X-A}(A):=\textrm{Cl}(A)\bigcap(X-A)$ are residual sets and as Fig. \[Fig: DerSets\] shows $$\textrm{Bdy}_{X}(A)=\textrm{Bdy}_{X-A}(A)\bigcup\textrm{Bdy}_{A}(X-A)$$ is the union of these two residual sets. When $A$ is closed (or open) in $X$, its boundary consisting of the only component $\textrm{Bdy}_{A}(X-A)$ (or $\textrm{Bdy}_{X-A}(A)$) as shown by the second row (or column) of the figure, being a closed set of $X$ is also nowhere dense in $X$; infact *a closed nowhere dense set is always the boundary of some open set.* Otherwise, the boundary components of the two residual parts — as in the donor-donor, donor-neutral, neutral-donor and neutral-neutral cases — need not be individually closed in $X$ (although their union is) and their union is a residual set that need not be nowhere dense in $X$: the union of two nowhere dense sets is nowhere dense but the union of a residual and a nowhere dense set is a residual set. One way in which a two-component boundary can be nowhere dense is by having $\textrm{Bdy}_{A}(X-A)\supseteq\textrm{Der}(A)$ or $\textrm{Bdy}_{X-A}(A)\supseteq\textrm{Der}(X-A)$, so that it is effectively in one piece rather than in two, as shown in Fig. \[Fig: DerSets1\](b).
**Theorem A3.6.** *$A$ is nowhere dense in $X$ iff each non-empty open set of $X$ has a non-empty open subset disjoint from* Cl *$\qquad\square$*
**Proof.** If $U$ is a nonempty open set of $X$, then $U_{0}=U\cap\textrm{Ext}(A)\neq\emptyset$ as $\textrm{Ext}(A)$ is dense in $X$; $U_{0}$ is the open subset that is disjoint from $\textrm{Cl}(A)$. It clearly follows from this that each non-empty open set of $X$ has a non-empty open subset disjoint from a nowhere dense set $A$.$\qquad\blacksquare$
What this result (which follows just from the definition of nowhere dense sets) actually means is that no point in $\textrm{Bdy}_{X-A}(A)$ can be isolated in it.
**Corollary.** $A$ *is nowhere dense in $X$ iff* Cl$(A)$ *does not contain any nonempty open set of $X$* *any nonempty open set that contains $A$ also contains its closure.* *$\qquad\square$*
**Example A3.2.** Each finite subset of $\mathbb{R}^{n}$ is nowhere dense in $\mathbb{R}^{n}$; the set $\{1/n\}_{n=1}^{\infty}$ is nowhere dense in $\mathbb{R}$. The Cantor set $\mathcal{C}$ is nowhere dense in $[0,1]$ because every neighbourhood of any point in $\mathcal{C}$ must contain, by its very construction, a point with $1$ in its ternary representation. That the interior and the interior of the closure of a set are not necessarily the same is seen in the example of the rationals in reals: The set of rational numbers $\mathbb{Q}$ has empty interior because any neighbourhood of a rational number contains irrational numbers (so also is the case for irrational numbers) and $\mathbb{R}=\textrm{Int}(\textrm{Cl}(\mathbb{Q}))\supseteq\textrm{Int}(\mathbb{Q})=\emptyset$ justifies the notion of a nowhere dense set.$\qquad\blacksquare$
The following properties of $\mathcal{C}$ can be taken to define any subset of a topological space as a Cantor set; set-theoretically it should be clear from its classical middle-third construction that the Cantor set consists of all points of the closed interval $[0,1]$ whose infinite triadic (base 3) representation, expressed so as not to terminate with an infinite string of $1$’s, does not contain the digit $1$. Accordingly, any end-point of the infinite set of closed intervals whose intersection yields the Cantor set, is represented by a repeating string of either $0$ or $2$ while a non end-point has every other arbitrary collection of these two digits. Recalling that any number in $[0,1]$ is a rational iff its representation in any base is terminating or recurring — thus any decimal that neither repeats or terminates but consists of all possible sequences of all possible digits represents an irrational number — it follows that both rationals and irrationals belong to the Cantor set.
($\mathcal{C}1$) ***$\mathcal{C}$ is*** *******totally disconnected.*** If possible, let $\mathcal{C}$ have a component containing points $a$ and $b$ with $a<b$. Then $[a,b]\subseteq\mathcal{C}\Rightarrow[a,b]\subseteq C_{i}$ for all $i$. But this is impossible because we may choose $i$ large enough to have $3^{-i}<b-a$ so that $a$ and $b$ must belong to two different members of the pairwise disjoint closed $2^{i}$ subintervals each of length $3^{-i}$ that constitutes $C_{i}$. Hence $$[a,b]\textrm{ is not a subset of any }C_{i}\Longrightarrow[a,b]\textrm{ is not a subset of }\mathcal{C}.$$
($\mathcal{C}2$) ***$\mathcal{C}$ is perfect*** so that for any $x\in\mathcal{C}$ every neighbourhood of $x$ must contain some other point of $\mathcal{C}$. Supposing to the contrary that the singleton $\{ x\}$ is an open set of $\mathcal{C}$, there must be an $\varepsilon>0$ such that in the usual topology of $\mathbb{R}$$${\textstyle \{ x\}=\mathcal{C}\bigcap(x-\varepsilon,x+\varepsilon).}\label{Eqn: Cantor_Perfect}$$
Choose a positive integer $i$ large enough to satisfy $3^{-i}<\varepsilon$. Since $x$ is in every $C_{i}$, it must be in one of the $2^{i}$ pairwise disjoint closed intervals $[a,b]\subset(x-\varepsilon,x+\varepsilon)$ each of length $3^{-i}$ whose union is $C_{i}$. As $[a,b]$ is an interval, at least one of the endpoints of $[a,b]$ is different from $x$, and since an endpoint belongs to $\mathcal{C}$, $\mathcal{C}\cap(x-\varepsilon,x+\varepsilon)$ must also contain this point thereby violating Eq. (\[Eqn: Cantor\_Perfect\]).
($\mathcal{C}3$) ***$\mathcal{C}$ is nowhere dense*** because each neighbourhood of any point of $\mathcal{C}$ intersects $\textrm{Ext}(\mathcal{C})$; see Thm. A3.6.
($\mathcal{C}4$) ***$\mathcal{C}$ is compact*** because it is a closed subset contained in the compact subspace $[0,1]$ of $\mathbb{R}$, see Thm. A3.3. The compactness of $[0,1]$ follows from the Heine-Borel Theorem which states that any subset of the real line is compact iff it is both closed and bounded with respect to the Euclidean metric on $\mathbb{R}$.
Compare ($\mathcal{C}1$) and ($\mathcal{C}2$) with the essentially similar arguments of Example A3.1(2) for the subspace of rationals in $\mathbb{R}$.
**A4. Neutron Transport Theory**
This section introduces the reader to the basics of the *linear* neutron transport theory where graphical convergence approximations to the singular distributions, interpreted here as multifunctions, led to the present study of this work. The one-speed (that is mono-energetic) neutron transport equation in one dimension and plane geometry, is $$\mu\frac{\partial\Phi(x,\mu)}{\partial x}+\Phi(x,\mu)=\frac{c}{2}\int_{-1}^{1}\Phi(x,\mu^{\prime})d\mu^{\prime},\,0<c<1,\,-1\leq\mu\leq1\label{Eqn: NeutronTransport}$$
where $x$ is a non-dimensional physical space variable that denotes the location of the neutron moving in a direction $\theta=\cos^{-1}(\mu)$, $\Phi(x,\mu)$ is a neutron density distribution function such that $\Phi(x,\mu)dxd\mu$ is the expected number of neutrons in a distance $dx$ about the point $x$ moving at constant speed with their direction cosines of motion in $d\mu$ about $\mu$, and $c$ is a physical constant that will be taken to satisfy the restriction shown above. Case’s method starts by assuming the solution to be of the form $\Phi_{\nu}(x,\mu)=e^{-x/\mu}\phi(\mu,\nu)$ with a normalization integral constraint of $\int_{-1}^{1}\phi(\mu,\nu)d\mu=1$ to lead to the simple equation $$(\nu-\mu)\phi(\mu,\nu)=\frac{c\nu}{2}\label{Eqn: case_eigen}$$ for the unknown function $\phi(\nu,\mu)$. Case then suggested, see @Case1967, the non-simple complete solution of this equation to be $$\phi(\mu,\nu)=\frac{c\nu}{2}\mathcal{P}\frac{1}{\nu-\mu}+\lambda(v)\delta(v-\mu),\label{Eqn: singular_eigen}$$
where $\lambda(\nu)$ is the usual combination coefficient of the solutions of the homogeneous and non-homogeneous parts of a linear equation, $\mathcal{P}(\cdot)$ is a principal value and $\delta(x)$ the Dirac delta, to lead to the full-range $-1\leq\mu\leq1$ solution valid for $-\infty<x<\infty$ $$\Phi(x,\mu)=a(\nu_{0})e^{-x/\nu_{0}}\phi(\mu,\nu_{0})+a(-\nu_{0})e^{x/\nu_{0}}\phi(-\nu_{0},\mu)+\int_{-1}^{1}a(\nu)e^{-x/\nu}\phi(\mu,\nu)d\nu\label{Eqn: CaseSolution_FR}$$
of the one-speed neutron transport equation (\[Eqn: NeutronTransport\]). Here the real $\nu_{0}$ and $\nu$ satisfy respectively the integral constraints $$\frac{c\nu_{0}}{2}\ln\frac{\nu_{0}+1}{\nu_{0}-1}=1,\qquad\mid\nu_{0}\mid>1$$ $$\lambda(\nu)=1-\frac{c\nu}{2}\ln\frac{1+\nu}{1-\nu},\qquad\nu\in[-1,1],$$ with $$\phi(\mu,\nu_{0})=\frac{c\nu_{0}}{2}\frac{1}{\nu_{0}-\mu}$$
following from Eq. (\[Eqn: singular\_eigen\]).
It can be shown [@Case1967] that the eigenfunctions **$\phi(\nu,\mu)$** satisfy the full-range orthogonality condition $$\int_{-1}^{1}\mu\phi(\nu,\mu)\phi(\nu^{\prime},\mu)d\mu=N(\nu)\delta(\nu-\nu^{\prime}),$$
where the odd normalization constants $N$ are given by
$$\begin{array}{ccl}
{\displaystyle N(\pm\nu_{0})} & = & {\displaystyle \int_{-1}^{1}\mu\phi^{2}(\pm\nu_{0},\mu)d\mu}\qquad\textrm{for }\mid\nu_{0}\mid>1\\
& = & {\displaystyle \pm\frac{c\nu_{0}^{3}}{2}\left(\frac{c}{\nu_{0}^{2}-1}-\frac{1}{\nu_{0}^{2}}\right)},\end{array}$$
and$$N(\nu)=\nu\left(\lambda^{2}(\nu)+\left(\frac{\pi c\nu}{2}\right)^{2}\right)\qquad\textrm{for }\nu\in[-1,1].$$
With a source of particles $\psi(x_{0},\mu)$ located at $x=x_{0}$ in an infinite medium, Eq. (\[Eqn: CaseSolution\_FR\]) reduces to the boundary condition, with $\mu,\textrm{ }\nu\in[-1,1]$, $$\psi(x_{0},\mu)=a(\nu_{0})e^{-x_{0}/\nu_{0}}\phi(\mu,\nu_{0})+a(-\nu_{0})e^{x_{0}/\nu_{0}}\phi(-\nu_{0},\mu)+\int_{-1}^{1}a(\nu)e^{-x_{0}/\nu}\phi(\mu,\nu)d\nu\label{Eqn: BC_FR}$$
for the determination of the expansion coefficients $a(\pm\nu_{0}),\textrm{ }\{ a(\nu)\}_{\nu\in[-1,1]}$. Use of the above orthogonality integrals then lead to the complete solution of the problem to be $$a(\nu)=\frac{e^{x_{0}/\nu}}{N(\nu)}\int_{-1}^{1}\mu\psi(x_{0},\mu)\phi(\mu,\nu)d\mu,\qquad\nu=\pm\nu_{0}\textrm{ or }\nu\in[-1,1].$$
For example, in the infinite-medium Greens function problem with $x_{0}=0$ and $\psi(x_{0},\mu)=\delta(\mu-\mu_{0})/\mu$, the coefficients are $a(\pm\nu_{0})=\phi(\mu_{0},\pm\nu_{0})/N(\pm\nu_{0})$ when $\nu=\pm\nu_{0}$, and $a(\nu)=\phi(\mu_{0},\nu)/N(\nu)$ for $\nu\in[-1,1]$.
For a half-space $0\leq x<\infty$, the obvious reduction of Eq. (\[Eqn: CaseSolution\_FR\]) to
$$\Phi(x,\mu)=a(\nu_{0})e^{-x/\nu_{0}}\phi(\mu,\nu_{0})+\int_{0}^{1}a(\nu)e^{-x/\nu}\phi(\mu,\nu)d\nu\label{Eqn: CaseSolution_HR}$$
with boundary condition, $\mu,\textrm{ }\nu\in[0,1]$, $$\psi(x_{0},\mu)=a(\nu_{0})e^{-x_{0}/\nu_{0}}\phi(\mu,\nu_{0})+\int_{0}^{1}a(\nu)e^{-x_{0}/\nu}\phi(\mu,\nu)d\nu,\label{Eqn: BC_HR}$$
leads to an infinitely more difficult determination of the expansion coefficients due to the more involved nature of the orthogonality relations of the eigenfunctions in the half-interval $[0,1]$ that now reads for $\nu,\textrm{ }\nu^{\prime}\in[0,1]$ [@Case1967]
$$\begin{aligned}
\int_{0}^{1}W(\mu)\phi(\mu,\nu^{\prime})\phi(\mu,\nu)d\mu & = & \frac{W(\nu)N(\nu)}{\nu}\delta(\nu-\nu^{\prime})\nonumber \\
\int_{0}^{1}W(\mu)\phi(\mu,\nu_{0})\phi(\mu,\nu)d\mu & = & 0\nonumber \\
\int_{0}^{1}W(\mu)\phi(\mu,-\nu_{0})\phi(\mu,\nu)d\mu & = & c\nu\nu_{0}X(-\nu_{0})\phi(\nu,-\nu_{0})\nonumber \\
\int_{0}^{1}W(\mu)\phi(\mu,\pm\nu_{0})\phi(\mu,\nu_{0})d\mu & = & \mp\left(\frac{c\nu_{0}}{2}\right)^{2}X(\pm\nu_{0})\label{Eqn: HR Ortho}\\
\int_{0}^{1}W(\mu)\phi(\mu,\nu_{0})\phi(\mu,-\nu)d\mu & = & \frac{c^{2}\nu\nu_{0}}{4}X(-\nu)\nonumber \\
\int_{0}^{1}W(\mu)\phi(\mu,\nu^{\prime})\phi(\mu,-\nu)d\mu & = & \frac{c\nu^{\prime}}{2}(\nu_{0}+\nu)\phi(\nu^{\prime},-\nu)X(-\nu)\nonumber \end{aligned}$$
where the half-range weight function $W(\mu)$ is defined as
$$W(\mu)=\frac{c\mu}{2(1-c)(\nu_{0}+\mu)X(-\mu)}\label{Eqn: W(mu)}$$
in terms of the $X$-function $$X(-\mu)=\textrm{exp}-\left\{ \frac{c}{2}\int_{0}^{1}\frac{\nu}{N(\nu)}\left[1+\frac{c\nu^{2}}{1-\nu^{2}}\right]\ln(\nu+\mu)d\nu\right\} ,\qquad0\leq\mu\leq1,$$
that is conveniently obtained from a numerical solution of the nonlinear integral equation $$\Omega(-\mu)=1-\frac{c\mu}{2(1-c)}\int_{0}^{1}\frac{\nu_{0}^{2}(1-c)-\nu^{2}}{(\nu_{0}^{2}-\nu^{2})(\mu+\nu)\Omega(-\nu)}d\nu\label{Eqn: Omega(-mu)}$$
to yield $$X(-\mu)=\frac{\Omega(-\mu)}{\mu+\nu_{0}\sqrt{1-c}},$$
and the $X(\pm\nu_{0})$ satisfy $$X(\nu_{0})X(-\nu_{0})=\frac{\nu_{0}^{2}(1-c)-1}{2(1-c)v_{0}^{2}(\nu_{0}^{2}-1)}.$$ Two other useful relations involving the $W$-function are given by $\int_{0}^{1}W(\mu)\phi(\mu,\nu_{0})d\mu=c\nu_{0}/2$ and $\int_{0}^{1}W(\mu)\phi(\mu,\nu)d\mu=c\nu/2$.
The utility of these full and half range orthogonality relations lie in the fact that a suitable class of functions of the type that is involved here can always be expanded in terms of them, see @Case1967. An example of this for a full-range problem has been given above; we end this introduction to the generalized — traditionally known as singular in neutron transport theory — eigenfunction method with two examples of half-range orthogonality integrals to the half-space problems A and B of Sec. 5.
**Problem A: The Milne Problem.** In this case there is no incident flux of particles from outside the medium at $x=0$, but for large $x>0$ the neutron distribution inside the medium behaves like $e^{x/\nu_{0}}\phi(-\nu_{0},\mu)$. Hence the boundary condition (\[Eqn: BC\_HR\]) at $x=0$ reduces to $$-\phi(\mu,-\nu_{0})=a_{\textrm{A}}(\nu_{0})\phi(\mu,\nu_{0})+\int_{0}^{1}a_{\textrm{A}}(\nu)\phi(\mu,\nu)d\nu\qquad\mu\geq0.$$
Use of the fourth and third equations of Eq. (\[Eqn: HR Ortho\]) and the explicit relation Eq. (\[Eqn: W(mu)\]) for $W(\mu)$ gives respectively the coefficients $$\begin{aligned}
{\displaystyle a_{\textrm{A}}(\nu_{0})} & = & X(-\nu_{0})/X(v_{0})\nonumber \\
a_{\textrm{A}}(\nu) & = & -\frac{1}{N(\nu)}\textrm{ }c(1-c)\nu_{0}^{2}\nu X(-\nu_{0})X(-\nu)\label{Eqn: Milne_Coeff}\end{aligned}$$
The extrapolated end-point $z_{0}$ of Eq. (\[Eqn: extrapolated\]) is related to $a_{\textrm{A}}(\nu_{0})$ of the Milne problem by $a_{\textrm{A}}(\nu_{0})=-\exp(-2z_{0}/\nu_{0})$.
**Problem B: The Constant Source Problem.** Here ****the boundary condition at $x=0$ is $$1=a_{\textrm{B}}(\nu_{0})\phi(\mu,\nu_{0})+\int_{0}^{1}a_{\textrm{B}}(\nu)\phi(\mu,\nu)d\nu\qquad\mu\geq0$$
which leads, using the integral relations satisfied by $W$, to the expansion coefficients
$$\begin{aligned}
{\displaystyle a_{\textrm{B}}(\nu_{0})} & = & -2/c\nu_{0}X(v_{0})\label{Eqn: Constant_Coeff}\\
a_{\textrm{B}}(\nu) & = & \frac{1}{N(\nu)}\textrm{ }(1-c)\nu(\nu_{0}+\nu)X(-\nu)\nonumber \end{aligned}$$
where the $X(\pm\nu_{0})$ are related to Problem A as $$\begin{aligned}
X(\nu_{0}) & = & \frac{1}{\nu_{0}}\sqrt{\frac{\nu_{0}^{2}(1-c)-1}{2a_{\textrm{A}}(\nu_{0})(1-c)(\nu_{0}^{2}-1)}}\\
X(-\nu_{0}) & = & \frac{1}{\nu_{0}}\sqrt{\frac{a_{A}(\nu_{0})\left(\nu_{0}^{2}(1-c)-1\right)}{2(1-c)(\nu_{0}^{2}-1)}}.\end{aligned}$$
This brief introduction to the singular eigenfunction method should convince the reader of the great difficulties associated with half-space, half-range methods in particle transport theory; note that the $X$-functions in the coefficients above must be obtained from numerically computed tables. In contrast, full-range methods are more direct due to the simplicity of the weight function $\mu$, which suggests the full-range formulation of half-range problems presented in Sec. 5. Finally it should be mentioned that this singular eigenfunction method is based on the theory of singular integral equations.
**Acknowledgment**
It is a pleasure to thank the referees for recommending an enlarged Tutorial and Review revision of the original submission *Graphical Convergence, Chaos and Complexity*, **and the Editor Professor Leon O Chua for suggesting a pedagogically self-contained, jargonless no-page limit version accessible to a wider audience for the present form of the paper. Financial assistance during the initial stages of this work from the National Board for Higher Mathematics is also acknowledged.
[^1]: \[Foot: UNConf\]A partial listing of papers is as follows: *Chaos and Politics: Application of nonlinear dynamics to socio-political issues; Chaos in Society: Reflections on the impact of chaos theory on sociology; Chaos in neural networks; The impact of chaos on mathematics; The impact of chaos on physics; The impact of chaos on economic theory; The impact of chaos on engineering; The impact of chaos on biology; Dynamical disease:* and *The impact of nonlinear dynamics and chaos on cardiology and medicine.*
[^2]: \[Foot: ScienceMag\]The eight Viewpoint articles are titled: *Simple Lessons from Complexity; Complexity in Chemistry; Complexity in Biological Signaling Systems; Complexity and the Nervous System; Complexity, Pattern, and Evolutionary Trade-Offs in Animal Aggregation; Complexity in Natural Landform Patterns; Complexity and Climate* and *Complexity and the Economy*.
[^3]: [\[Foot: reln&graph\]We do not distinguish between a relation and its graph although technically they are different objects. Thus although a functional relation, strictly speaking, is the triple $(X,f,Y)$ written traditionally as $f\!:X\rightarrow Y$, we use it synonymously with the graph $f$ itself. Parenthetically, the word]{} *functional* [in this work is not necessarily employed for a scalar-valued function, but is used in a wider sense to distinguish between a function and an arbitrary relation (that is a multifunction). Formally, whereas an arbitrary relation from $X$ to $Y$ is a subset of $X\times Y$, a functional relation must satisfy an additional restriction that requires $y_{1}=y_{2}$ whenever $(x,y_{1})\in f$ and $(x,y_{2})\in f$. In this subset notation, $(x,y)\in f\Leftrightarrow y=f(x)$. ]{}
[^4]: [\[Foot: EquivRel\]An useful alternate way of expressing these properties for a relation $\mathscr{M}$ on $X$ are]{}
[$\quad$(ER2) $\mathscr{M}$ is symmetric iff $\mathscr{M}=\mathscr M^{-}$ ]{}
[$\quad$(ER3) $\mathscr{M}$ is transitive iff $\mathscr{M}\circ\mathscr{M}\subseteq\mathscr{M}$, ]{}
[with $\mathscr{M}$ an equivalence relation only if $\mathscr{M}\circ\mathscr{M}=\mathscr{M}$, where for $\mathscr{M}\subseteq X\times Y$ and $\mathscr{N}\subseteq Y\times Z$, the composition $\mathscr{N}\circ\mathscr{M}:=\{(x,z)\in X\times Z\!:(\exists y\in Y)\textrm{ }((x,y)\in\mathscr{M})\wedge((y,z)\in\mathscr{N})\}$]{}
[^5]: [\[Foot: family\]A function $\chi\!:\mathbb{D}\rightarrow X$ will be called a]{} *family* [in $X$ indexed by $\mathbb{D}$ when reference to the domain $\mathbb{D}$ is of interest, and a]{} *net* [when it is required to focus attention on its values in $X$.]{}
[^6]: [\[Foot: extension\]Observe that it is]{} *not* [being claimed that $f$ belongs to the same class as $(f_{k})$. This is the single most important cornerstone on which this paper is based: the need to “complete” spaces that are topologically “incomplete”. The classical high-school example of the related problem of having to enlarge, or extend, spaces that are not big enough is the solution space of algebraic equations with real coefficients like $x^{2}+1=0$. ]{}
[^7]: [\[Foot: support\]By definition, the support (or supporting interval) of $\varphi(x)\in\mathcal{C}_{0}^{\infty}[\alpha,\beta]$ is $[\alpha,\beta]$ if $\varphi$ and all its derivatives vanish for $x\leq\alpha$ and $x\geq\beta$. ]{}
[^8]: [\[Foot: integral\]Both Riemann and Lebesgue integrals can be formulated in terms of the so-called]{} *step functions* [$s(x)$, which are piecewise constant functions with values $(\sigma_{i})_{i=1}^{I}$on a finite number of bounded subintervals $(J_{i})_{i=1}^{I}$ (which may reduce to a point or may not contain one or both of the end-points) of a bounded or unbounded interval $J$, with integral $\int_{J}s(x)dx\overset{\textrm{def}}=\sum_{i=1}^{I}\sigma_{i}|J_{i}|$. While the Riemann integral of a bounded function $f(x)$ on a bounded interval $J$ is defined with respect to sequences of step functions $(s_{j})_{j=1}^{\infty}$ and $(t_{j})_{j=1}^{\infty}$ satisfying $s_{j}(x)\leq f(x)\leq t_{j}(x)$ on $J$ with $\int_{J}(s_{j}-t_{j})\rightarrow0$ as $j\rightarrow\infty$ as $R\int_{J}f(x)dx=\lim\int_{J}s_{j}(x)dx=\lim\int_{J}t_{j}(x)dx$, the less restrictive Lebesgue integral is defined for arbitrary functions $f$ over bounded or unbounded intervals $J$ in terms of Cauchy sequences of step functions $\int_{J}|s_{i}-s_{k}|\rightarrow0$, $i,k\rightarrow\infty$, converging to $f(x)$ as $$s_{j}(x)\rightarrow f(x)\textrm{ pointwise almost everywhere on }J,$$ ]{}
[to be $$\int_{J}f(x)dx\overset{\textrm{def}}=\lim_{j\rightarrow\infty}\int_{J}s_{j}(x)dx.$$ ]{}
[That the Lebesgue integral is more general (and therefore is the proper candidate for completion of function spaces) is illustrated by the example of the function defined over $[0,1]$ to be $0$ on the rationals and $1$ on the irrationals for which an application of the definitions verify that whereas the Riemann integral is undefined, the Lebesgue integral exists and has value $1$. The Riemann integral of a bounded function over a bounded interval exists and is equal to its Lebesgue integral. Because it involves a larger family of functions, all integrals in integral convergences are to be understood in the Lebesgue sense. ]{}
[^9]: [\[Foot: delta\]The observant reader cannot have failed to notice how mathematical ingenuity successfully transferred the “troubles” of $(\delta_{k})_{k=1}^{\infty}$ to the sufficiently differentiable benevolent receptor $\varphi$ so as to be able to work backward, via the resultant trouble free $(\delta_{k}^{(-m)})_{k=1}^{\infty}$, to the final object $\delta$. This necessarily hides the true character of $\delta$ to allow only a view of its integral manifestation on functions. This unfortunately is not general enough in the strongly nonlinear physical situations responsible for chaos, and is the main reason for constructing the multifunctional extension of function spaces that we use. ]{}
[^10]: [\[Foot: cont=3Dbound\]Recall that for a linear operator continuity and boundedness are equivalent concepts. ]{}
[^11]: [\[Foot: OrthoMatrix\]A real matrix $A$ is an orthogonal projector iff $A^{2}=A$ and $A=A^{\textrm{T}}$. ]{}
[^12]: [\[Foot: class\]In this sense, a]{} *class* [is a set of sets. ]{}
[^13]: [\[Foot: interval\]By definition, an interval $I$ in a totally ordered set $X$ is a subset of $X$ with the property $$(x_{1},x_{2}\in I)\wedge(x_{3}\in X\!:x_{1}\prec x_{3}\prec x_{2})\Longrightarrow x_{3}\in I$$ ]{}
[so that any element of $X$ lying between two elements of $I$ also belongs to $I$.]{}
[^14]: [\[Foot: entropy\]Although we do not pursue this point of view here, it is nonetheless tempting to speculate that the answer to the question]{} *“Why* [does the entropy of an isolated system increase?” may be found by exploiting this line of reasoning that seeks to explain the increase in terms of a visible component associated with the usual topology as against a different latent workplace topology that governs the dynamics of nature.]{}
[^15]: [\[Foot: subspace\]In a subspace $A$ of $X$, a subset $U_{A}$ of $A$ is open iff $U_{A}=A\bigcap U$ for some open set $U$ of $X$. The notion of subspace topology can be formalized with the help of the inclusion map $i\!:A\rightarrow(X,\mathcal{U})$ that puts every point of $A$ back to where it came from, thus $$\begin{array}{ccl}
\mathcal{U}_{A} & = & \{ U_{A}=A\bigcap U\!:U\in\mathcal{U}\}\\
& = & \{ i^{-}(U)\!:U\in\mathcal{U}\}.\end{array}$$ ]{}
[^16]: [\[Foot: assoc&embed\]A surjective function is an]{} *association* [iff it is image continuous and an injective function is an]{} *embedding* [iff it is preimage continuous. ]{}
[^17]: [\[Foot: 0=3Dphi\]If $y\notin\mathcal{R}(f)$ then $f^{-}(\{ y\}):=\emptyset$ which is true for any subset of $Y-\mathcal{R}(f)$. However from the set-theoretic definition of natural numbers that requires $0:=\emptyset$, $1=\{0\}$, $2=\{0,1\}$ to be defined recursively, it follows that $f^{-}(y)$ can be identified with $0$ whenever $y$ is not in the domain of $f^{-}$. Formally, the successor set $A^{+}=A\bigcup\{ A\}$ of $A$ can be used to write $0:=\emptyset$, $1=0^{+}=0\bigcup\{0\}$, $2=1^{+}=1\bigcup\{1\}=\{0\}\bigcup\{1\}$ $3=2^{+}=2\bigcup\{2\}=\{0\}\bigcup\{1\}\bigcup\{2\}$ etc. Then the set of natural numbers $\mathbb{N}$ is defined to be the intersection of all the successor sets, where a successor set $\mathcal{S}$ is any set that contains $\emptyset$ and $A^{+}$ whenever $A$ belongs to $\mathcal{S}$. Observe how in the successor notation, countable union of singleton integers recursively define the corresponding sum of integers. ]{}
[^18]: [See footnote \[Foot: 0=3Dphi\] for a justification of the definition when $b$ is not in $\mathcal{R}(a)$.]{}
[^19]: [\[Foot: subnet\]A subnet is the generalized uncountable equivalent of a subsequence; for the technical definition, see Appendix A1. ]{}
[^20]: [\[Foot: point\_inter\]Equation (\[Eqn: func\_bi\]) is essentially the intersection of the pointwise topologies (\[Eqn: point\]) due to $f$ and $f^{-}$. ]{}
[^21]: [\[Foot: strict reln\]If $\preceq$ is an order relation in $X$ then the]{} *strict relation $\prec$ in $X$* [corresponding to $\preceq$, given by $x\prec y\Leftrightarrow(x\preceq y)\wedge(x\neq y)$,]{} *is not an order relation* [because unlike $\preceq$, $\prec$ is not reflexive even though it is both transitive and asymmetric.]{} **
[^22]: [\[Foot: infinite\]This makes $T$, and hence $X$, inductively defined infinite sets. It should be realized that (ST3)]{} *does not mean* [that every member of $T$ is obtained from $g$, but only ensures that the immediate successor of any element of $T$ is also in $T.$ The infimum $_{\rightarrow}T$ of these towers satisfies the additional property of being totally ordered (and is therefore essentially a sequence or net) in $(X,\preceq)$ to which (ST2) can be applied. ]{}
[^23]: [\[Foot: Hausdorff\]Recall that this means that if there is a totally ordered chain $C$ in $(X,\preceq)$ that succeeds $C_{+}$, then $C$ must be $C_{+}$ so that no chain in $X$ can be strictly larger than $C_{+}$. The notation adopted here and below is the following: If $X=\{ x,y\}$ is a non-empty set, then $\mathcal{X}:=\mathcal{P}(X)=\{ A\!:A\subseteq X\}=\{\emptyset,\{ x\},\{ y\},\{ x,y\}\}$ is the set of subsets of $X$, and $\mathfrak{X}:=\mathcal{P}^{2}(X)=\{\mathcal{A}:\mathcal{A}\subseteq\mathcal{X}\}$, the set of all subsets of $\mathcal{X}$, consists of the $16$ elements $\emptyset$, $\{\emptyset\}$, $\{\{ x\}\}$, $\{\{ y\}\}$, $\{\{ x,y\}\}$, $\{\{\emptyset\},\{ x\}\}$, $\{\{\emptyset\},\{ y\}\}$, $\{\{\emptyset\},\{ x,y\}\}$, $\{\{ x\},\{ y\}\}$, $\{\{ x\},\{ x,y\}\}$, $\{\{ y\},\{ x,y\}\}$, $\{\{\emptyset\},\{ x\},\{ y\}\}$, $\{\{\emptyset\},\{ x\},\{ x,y\}\}$, $\{\{\emptyset\},\{ y\},\{ x,y\}\}$, $\{\{ x\},\{ y\},\{ x,y\}\}$, and $\mathcal{X}$: an element of $\mathcal{P}^{2}(X)$ is a subset of $\mathcal{P}(X)$, any element of which is a subset of $X$. Thus if $C=\{0,1,2\}$ is a chain in $(X=\{0,1,2\},\leq)$, then $\mathcal{C}=\{\{0\},\{0,1\},\{0,1,2\}\}\subseteq\mathcal{P}(X)$ and $\mathfrak{C}=\{\{\{0\}\},\{\{0\},\{0,1\}\},\{\{0\},\{0,1\},\{0,1,2\}\}\}\subseteq\mathcal{P}^{2}(X)$ represent chains in $(\mathcal{P}(X),\subseteq)$ and $(\mathcal{P}^{2}(X),\subseteq)$ respectively . ]{}
[^24]: [\[Foot: supremum\]A similar situation arises in the following more intuitive example. Although the subset $A=\{1/n\}_{n\in Z_{+}}$ of the interval $I=[-1,1]$ has no a smallest or minimal elements, it does have the infimum 0. Likewise, although $A$ is bounded below by any element of $[-1,0)$, it has no greatest lower bound in $[-1,0)\bigcup(0,1]$. ]{}
[^25]: [\[Foot: omega-limit\]How does this happen for $A=\{ f^{i}(x_{0})\}_{i\in\mathbb{N}}$ that is not the constant sequence $(x_{0})$ at a fixed point? As $i\in\mathbb{N}$ increases, points are added to $\{ x_{0},f(x_{0}),\cdots,f^{I}(x_{0})\}$ not, as would be the case in a normal sequence, as a piled-up Cauchy tail, but as points generally lying between those already present; recall a typical graph as of Fig. \[Fig: tent4\] for example.]{}
[^26]: \[Foot: gen\_eigen\][The technical definition of a generalized eigenvalue is as follows. Let $\mathcal{L}$ be a linear operator such that there exists in the domain of $\mathcal{L}$ a sequence of elements $(x_{n})$ with $\Vert x_{n}\Vert=1$ for all $n$. If $\lim_{n\rightarrow\infty}\Vert(\mathcal{L}-\lambda)x_{n}\Vert=0$ for some $\lambda\in\mathbb{C}$, then this $\lambda$ is a]{} *generalized eigenvalue* [of $\mathcal{L}$, the corresponding eigenfunction $x_{\infty}$ being a]{} *generalized eigenfunction.*
[^27]: \[Foot: cluster\]This is also known as a *cluster point*; we shall, however, use this new term exclusively in the sense of the elements of a derived set, see Definition 2.3.
[^28]: \[Foot: Filter\_conv\][The restatement $$\mathcal{F}\rightarrow x\Longleftrightarrow\mathcal{N}_{x}\subseteq\mathcal{F}\label{Eqn: Def: LimFilter}$$ of Eq. (\[Eqn: lim filter\]) that follows from (F3), and sometimes taken as the definition of convergence of a filter, is significant as it ties up the algebraic filter with the topological neighbourhood system to produce the filter theory of convergence in topological spaces. From the defining properties of $\mathcal{F}$ it follows that for each $x\in X$, $\mathcal{N}_{x}$ is the coarsest (that is smallest) filter on $X$ that converges to $x$.]{}
[^29]: \[Foot: adh\_seq\][In a first countable space, while the corresponding proof of the first part of the theorem for sequences is essentially the same as in the present case, the more direct proof of the converse illustrates how the convenience of nets and directed sets may require more general arguments. Thus if a sequence $(x_{i})_{i\in\mathbb{N}}$ has a subsequence $(x_{i_{k}})_{k\in\mathbb{N}}$ converging to $x$, then a more direct line of reasoning proceeds as follows. Since the subsequence converges to $x$, its tail $(x_{i_{k}})_{k\geq j}$ must be in every neighbourhood $N$ of $x$. But as the number of such terms is infinite whereas $\{ i_{k}\!:k<j\}$ is only finite, it is necessary that for any given $n\in\mathbb{N}$, cofinitely many elements of the sequence $(x_{i_{k}})_{i_{k}\geq n}$ be in $N$. Hence $x\in\textrm{adh}((x_{i})_{i\in\mathbb{N}})$. ]{}
[^30]: \[Foot: seq xxx\][This is uncountable because interchanging any two eventual terms of the sequence does not alter the sequence. ]{}
[^31]: [Note that $\{ x\}$ is a $1$-point set but $(x)$ is an uncountable sequence.]{}
[^32]: \[Foot: e&q\][We adopt the convention of denoting arbitrary preimage and image continuous functions by $e$ and $q$ respectively even though they are not be injective or surjective; recall that the embedding $e\!:X\supseteq A\rightarrow Y$ and the association $q\!:X\rightarrow f(X)$ are $1:1$ and onto respectively. ]{}
[^33]: \[Foot: fil-nbd\][This is of course a triviality if we identify each $\chi(\mathbb{R}_{\beta})$ (or $F$ in the proof of the converse that follows) with a neighbourhood $N$ of $X$ that generates a topology on $X$.]{}
[^34]: **Nested-set theorem.** *If $(E_{n})$ is a decreasing sequence of nonempty, closed, subsets of a complete metric space $(X,d)$ such that* [$\lim_{n\rightarrow\infty}\textrm{dia}(E_{n})=0$]{}*, then there is a unique point* [$$x\in\bigcap_{n=0}^{\infty}E_{n}.$$ The uniqueness arises because the limiting condition on the diameters of $E_{n}$ imply, from property (H1), that $(X,d)$ is a Hausdorff space. ]{}
| ArXiv |
---
abstract: 'Let $\Omega\subset \mathbb{C}^2$ be a bounded pseudoconvex complete Reinhardt domain with a smooth boundary. We study the behavior of analytic structure in the boundary of $\Omega$ and obtain a compactness result for Hankel operators on the Bergman space of $\Omega$.'
address: 'Bowling Green State University, Department of Mathematics and Statistics, Bowling Green, Ohio 43403 '
author:
- 'Timothy G. Clos'
bibliography:
- 'rrrefs.bib'
title: Hankel Operators on the Bergman spaces of Reinhardt Domains and Foliations of Analytic Disks
---
Introduction
============
Let $\Omega\subset \mathbb{C}^n$ for $n\geq 2$ be a bounded domain. We let $dV$ be the (normalized) Lebesgue volume measure on $\Omega$. Then $L^2(\Omega)$ is the space of measurable, square integrable functions on $\Omega$. Let $\mathcal{O}_{\Omega}$ be the collection of all holomorphic (analytic) functions on $\Omega$. Then the Bergman space $A^2(\Omega):=\mathcal{O}_{\Omega}\cap L^2(\Omega)$ is a closed subspace of $L^2(\Omega)$, a Hilbert space. Therefore, there exists an orthogonal projection $P:L^2(\Omega)\rightarrow A^2(\Omega)$ called the Bergman projection. Then the Hankel operator with symbol $\phi\in L^{\infty}(\Omega)$ is defined as $$H_{\phi}f:=(I-P)(\phi f)$$ where $I$ is the identity operator and $f\in A^2(\Omega)$.
Previous Work
=============
Compactness of Hankel operators on the Bergman spaces of bounded domains and its relationship between analytic structure in the boundary of these domains is an ongoing research topic. In one complex dimension, Axler in [@Axler86] completely characterizes compactness of Hankel operators with conjugate holomorphic, $L^2$ symbols. There, the emphasis is on whether the symbol belongs to the little Bloch space. This requires that the derivative of the complex conjugate of the symbol satisfy a growth condition near the boundary of the domain.\
The situation is different in several variables for conjugate holomorphic symbols. In [@clos], the author completely characterizes compactness of Hankel operator with conjugate holomorphic symbols on convex Reinhardt domains in $\mathbb{C}^n$ if the boundary contains a certain class of analytic disks. The proof relied on using the analytic structure in the boundary to show that a compact Hankel operator with a conjugate holomorphic symbol must be the zero operator, assuming certain conditions on the boundary of the domain. In particular, the symbol is identically constant if certain conditions are satisfied. An example of a domain where these conditions are satisfied is the polydisk in $\mathbb{C}^n$ (as seen in [@Le10] and [@clos]).\
In [@CelZey] the authors studied the compactness of Hankel operators with symbols continuous up to the closure of bounded pseudoconvex domains via compactness multipliers. They showed if $\phi\in C(\overline{\Omega})$ is a compactness multiplier then $H_{\phi}$ is compact on $A^2(\Omega)$. The authors of [@CelZey] approached the problem using the compactness estimate machinery developed in [@StraubeBook].\
Hankel operators with symbols continuous up to the closure of the domain is also studied in [@CuckovicSahutoglu09] and [@ClosSahut]. The paper [@CuckovicSahutoglu09] considered Hankel operators with symbols that are $C^1$-smooth up to the closure of bounded convex domains in $\mathbb{C}^2$. The paper [@ClosSahut] considered symbols that are continuous up to the closure of bounded convex Reinhardt domains in $\mathbb{C}^2$. Thus the regularity of the symbol was reduced at the expense of a smaller class of domains.\
Many of these results characterize the compactness of these operators by the behavior of the symbol along analytic structure in the domain. For bounded pseudoconvex domains in $\mathbb{C}^n$, compactness of the $\overline{\partial}$-Neumann operator implies the compactness of Hankel operators with symbols continuous up to the closure of the domain. See [@FuSt] and [@StraubeBook] for more information on compactness of the $\overline{\partial}$-Neumann operator. For example the ball in $\mathbb{C}^n$ has compact $\overline{\partial}$-Neumann operator and hence any Hankel operator with symbol continuous up the closure of the ball is compact on the Bergman space of the ball. The compactness of the $\overline{\partial}$-Neumann operator on the ball in $\mathbb{C}^n$ follows from the convexity of the domain and absence of analytic structure in the boundary of the domain. See [@StraubeBook].\
As shown in [@dbaressential], the existence of analytic structure in the boundary of bounded convex domains is an impediment to the compactness of the $\overline{\partial}$-Neumann operator. It is therefore natural to ask whether the Hankel operator with symbol continuous up to the closure of the domain can be compact if the $\overline{\partial} $-Neumann operator is not compact. As we shall see, the answer is yes. On the polydisk in $\mathbb{C}^n$, [@Le10] showed that the answer to this question is yes, despite the non-compactness of the $\overline{\partial}$-Neumann operator. For bounded convex domains in $\mathbb{C}^n$ for $n\geq 2$, relating the compactness of the Hankel operator with continuously differentiable symbols to the geometry of the boundary is well studied. See [@CuckovicSahutoglu09]. They give a more general characterization than [@Le10] for symbols that are $C^1$-smooth up to the closure of the domain. For symbols that are only continuous up to the closure of bounded convex Reinhardt domains in $\mathbb{C}^2$, there is a complete characterization in [@ClosSahut].\
The Main Result
===============
In this paper we investigate the compactness of Hankel operators on the Bergman spaces of smooth bounded pseudoconvex complete Reinhardt domains. These domains may not be convex as in [@ClosSahut] but are instead almost locally convexifiable. That is, for any $(p_1,p_2)\in b\Omega$ and if $(p_1,p_2)$ are away from the coordinate axes, then there exists $r>0$ so that $$B((p_1,p_2),r):=\{(z_1,z_2)\in \mathbb{C}^2: |z_1-p_1|^2+|z_2-p_2|^2<r^2\}$$ and a biholomorphism $T:B((p_1,p_2),r)\rightarrow \mathbb{C}^2$ so that $ B((p_1,p_2),r)\cap \Omega$ is a domain and $T(B((p_1,p_2),r)\cap \Omega)$ is convex. We will use this fact along with a result in [@CuckovicSahutoglu09] to localize the problem. We then analyze the geometry on analytic structure in the resulting convex domain. Then we perform the analysis on the boundary of this convex domain using the boundary geometry previously established to show the main result.\
\[thmmain\] Let $\Omega\subset\mathbb{C}^2$ be a bounded pseudoconvex complete Reinhardt domain with a smooth boundary. Then $\phi\in C(\overline{\Omega})$ so that $\phi\circ f$ is holomorphic for any holomorphic $f:\mathbb{D}\rightarrow b\Omega$ if and only if $H_{\phi}$ is compact on $A^2(\Omega)$.
We will assume $\phi\circ f$ is holomorphic for any holomorphic function $f:\mathbb{D}\rightarrow b\Omega$ and show that $H_{\phi}$ is compact on $A^2(\Omega)$, as the converse of this statement appears as a corollary in [@CCS].
Analytic structure in the boundary of pseudoconvex complete Reinhardt domains in $\mathbb{C}^2$
===============================================================================================
We fill first investigate the geometry of non-degenerate analytic disks in the boundary of Reinhardt domains. We define the following collection for any bounded domain $\Omega\subset \mathbb{C}^n$. $$\Gamma_{\Omega}:=\overline{\bigcup_{f\in A(\mathbb{D})\cap C(\overline{\mathbb{D}})\, , f \,\text{non-constant}}\{f(\mathbb{D}) |f:\mathbb{D}\rightarrow b\Omega\}}$$
Let $\Omega\subset \mathbb{C}^n$ for $n\geq 2$ be a domain. We say $\Gamma\subset b\Omega$ is an analytic disk if there exists $F:\mathbb{D}\rightarrow \mathbb{C}^n$ so that every component function of $F$ is holomorphic on $\mathbb{D}$ and continuous up to the boundary of $\mathbb{D}$ and $F(\mathbb{D})=\Gamma$.\
One observation is for any Reinhardt domain $\Omega\subset \mathbb{C}^n$, if $F(\mathbb{D})\subset b\Omega$ is an analytic disk where $F(\zeta):=(F_1(\zeta), F_2(\zeta),...,F_n(\zeta))$, then for any $(\theta_1,\theta_2,...,\theta_n)\in \mathbb{R}^n$, $G(\mathbb{D})\subset b\Omega$ is also an analytic disk where $$G(\zeta):=(e^{i\theta_1}F_1(\zeta), e^{i\theta_2}F_2(\zeta),...,e^{i\theta_n}F_n(\zeta)).$$
We say an analytic disk $f(\mathbb{D})$ where $f=(f_1,f_2,...,f_n)$ is trivial or degenerate if $f_j$ is identically constant for all $j\in \{1,2,...,n\}$. Otherwise, we say an analytic disk is non-trivial or non-degenerate.\
Let $\Omega\subset \mathbb{C}^2$ be a bounded pseudoconvex complete Reinhardt domain with a smooth boundary. If $g(\mathbb{D})\subset b\Omega$ is an analytic disk so that $\overline{g(\mathbb{D})}\cap \{z_2=0\}\neq \emptyset$ or $\overline{g(\mathbb{D})}\cap \{z_1=0\}\neq \emptyset$, then $g(\zeta)=(g_1(\zeta),0)$ or $g(\zeta)=(0,g_2(\zeta))$, respectively.
They are possibly infinitely many continuous families of non-trivial analytic disks in the boundary of bounded complete Reinhardt domains $\Omega$ in $\mathbb{C}^2$. Hence by compactness of the boundary of $\Omega$, there are subsets of $b\Omega$ that are accumulation sets of families of analytic disks. This next lemma gives us some insight on the structure of these accumulation sets.
\[disklim\] Suppose $\Omega\subset \mathbb{C}^2$ is a bounded complete Reinhardt domain and $\{\Gamma_j\}_{j\in \mathbb{N}}\subset b\Omega$ is a sequence of pairwise disjoint, continuous families of analytic disks so that $\Gamma_j\rightarrow \Gamma_0$ as $j\rightarrow \infty$, where $\Gamma_0=\{e^{i\theta}F(\mathbb{D}):\theta\in [0,2\pi]\}$. Then, there exists $c_1,c_2\in \mathbb{C}$ so that $F\equiv (c_1,c_2)$.
Let $\sigma$ be the Lebesgue measure on the boundary. Without loss of generality, we may assume $\Gamma_j$ are families of non-degenerate analytic disks and so we may assume $\sigma(\Gamma_j)>0$ for all $j\in \mathbb{N}$. If $\sigma(\Gamma_0)>0$, then we consider the sequence of indicator functions on $\Gamma_j$, called $\chi_{\Gamma_j}$. By assumption, $\chi_{\Gamma_j}\rightarrow \chi_{\Gamma_0}$ pointwise as $j\rightarrow \infty$. Hence an application of Lebesgue dominated convergence theorem shows that $\sigma(\Gamma_j)\rightarrow \sigma(\Gamma_0)$, and so $\sigma(\Gamma_j)\geq \delta>0$ for sufficiently large $j\in \mathbb{N}$. Since $\Gamma_j$ are pairwise disjoint and $\Omega$ is bounded, this is a contradiction. So $\sigma(\Gamma_0)=0$. Now assume $\Lambda_j(\zeta):=(f_j(\zeta),g_j(\zeta))$ where $f_j$, $g_j$ are holomorphic on $\mathbb{D}$ and continuous up to the boundary of $\mathbb{D}$. Furthermore, $$\Gamma_j=\{e^{i\theta}\Lambda_j:\theta\in [0,2\pi]\}.$$ Then, there exists $f,g$ so that $$\sup\{\text{dist}(((f_j(\zeta),g_j(\zeta)), f(\zeta),g(\zeta))):\zeta\in \overline{\mathbb{D}}\}\rightarrow 0$$ as $j\rightarrow \infty$. Therefore, one can show $f_j\rightarrow f$ and $g_j\rightarrow g$ uniformly on $\overline{\mathbb{D}}$ as $j\rightarrow \infty$. So $f$ and $g$ are holomorphic on $\mathbb{D}$ and continuous on $\overline{\mathbb{D}}$. To show $f$ and $g$ are constant it suffices to show they are constant on some open subset of $\mathbb{D}$. Assume $f$ is not identically constant. If $g$ is constant then by open mapping theorem, $F(\mathbb{D})$ is open in $\mathbb{C}\times \mathbb{R}$ and also $\sigma(F(\mathbb{D}))=0$, which cannot occur by the open mapping theorem. So, we assume both $F$ and $g$ are not identically constant, so the zeros of $f'$ and $g'$ have no accumulation point in $\mathbb{D}$. Then by a holomorphic change of coordinates, there exists an open simply connected set $D\subset \mathbb{D}$ so that $F(D)$ is biholomorphic to a subset $K$ of $\{z_1\in \mathbb{C}\}\times\{0\}$. Hence again by the open mapping theorem, $f$ is constant on $K$ since $K$ has measure zero and so $f$ is constant on $\overline{\mathbb{D}}$ by the identity principle.
\[lembiholo\] Let $\Omega\subset \mathbb{C}^2$ be a bounded pseudoconvex complete Reinhardt domain with a smooth boundary. Suppose $f:\mathbb{D}\rightarrow b\Omega$ and $g:\mathbb{D}\rightarrow b\Omega$ are holomorphic functions on $\mathbb{D}$ and continuous on $\overline{\mathbb{D}}$. Assume that $\overline{f(\mathbb{D})}\cap \overline{g(\mathbb{D})}\neq \emptyset$. Furthermore, assume $\overline{f(\mathbb{D})}\cap(\{z_1=0\}\cup \{z_2=0\})=\emptyset$ and $\overline{g(\mathbb{D})}\cap(\{z_1=0\}\cup \{z_2=0\})=\emptyset$. Then, $f(\mathbb{D})$ and $g(\mathbb{D})$ are biholomorphically equivalent to analytic disks contained in a unique complex line.
Let $\zeta_0\in \mathbb{D}$ and $\zeta_1\in \mathbb{D}$ be such that $f(\zeta_0)=g(\zeta_1)$. Without loss of generality, by composing with a biholomorphism of the unit disk that sends $\zeta_0$ to $\zeta_1$, we may assume $f(\zeta_0)=g(\zeta_0)$. Then, there exists $r>0$ and a biholomorphism $T:B(f(\zeta_0),r)\rightarrow \mathbb{C}^2$ so that $f^{-1}(B(f(\zeta_0),r)\cap f(\mathbb{D}))\subset \mathbb{D}$ and $g^{-1}(B(f(\zeta_0),r)\cap g(\mathbb{D}))\subset \mathbb{D}$ and $T(B(f(\zeta_0),r)\cap \Omega)$ is convex. Then, $A:=f^{-1}(B(f(\zeta_0),r)\cap f(\mathbb{D}))\cap g^{-1}(B(f(\zeta_0),r)\cap g(\mathbb{D}))$ is an open, non-empty, simply connected, and bounded. By the Riemann mapping theorem, there exists a biholomorphism $R:\mathbb{D}\rightarrow A$. Then, $T\circ f\circ R$ and $T\circ g\circ R$ are analytic disks in the boundary of a bounded convex domain. Hence they are contained in a complex line by [@CuckovicSahutoglu09 Lemma 2]. In fact, they are contained in the same complex line because both disks have closures with non-empty intersection and the domain has a smooth boundary. That is, if $L_{\alpha}:=\{(a_1\zeta+b_{\alpha},c_1\zeta+d_{\alpha}):\zeta\in \mathbb{C}\}$ and $L_{\beta}:=\{(a_2\zeta+b_{\beta},c_2\zeta+d_{\beta}):\zeta\in \mathbb{C}\}$ are one parameter continuous (continuously depending on the parameter) families of complex lines depending on parameters $\alpha$ and $\beta$ that locally foliate the boundary, with $(L_{\alpha_0}\cap L_{\beta_0})\cap b\Omega\neq \emptyset$, then $a_1=a_2$. The argument uses the fact that boundary normal vectors must vary smoothly. Furthermore, one can conclude $L_{\alpha_0}=L_{\beta_0}$ since one can show $b_{\alpha_0}=b_{\beta_0}$ and $d_{\alpha_0}=d_{\beta_0}$.
\[propconvex\] Let $\Omega\subset \mathbb{C}^2$ be a smooth bounded convex domain. Let $\{\Gamma_j\}_{j\in \mathbb{N}}$ be a collection of analytic disks in $b\Omega$ so that $$\nabla:=\overline{\bigcup_{j\in \mathbb{N}}\Gamma_j}$$ is connected. Then there exists a convex set $S$ and a non-constant holomorphic function $F:\mathbb{D}\rightarrow b\Omega$ so that $F$ is continuous up to $\overline{\mathbb{D}}$, $F(\mathbb{D})=S$ and $\nabla\subset \overline{S}$.
By Lemma \[lembiholo\], there exists a complex line $L=\mathbb{C}\times \{0\}$ so that $\nabla\subset L$ and by convexity of the domain, $L\cap \Omega=\emptyset$. Then the convex hull of $\nabla$, called $\mathcal{H}(\nabla)$, is contained in $L\cap\overline{\Omega}$. Since $\nabla$ contains a non-trivial analytic disk, the interior of $\mathcal{H}(\nabla)$ is non-empty. We denote this non-empty interior as $I$. Assume $\overline{I}\neq \mathcal{H}(\nabla)$. Let $z_0\in \mathcal{H}(\nabla)\setminus \overline{I}$. Then there is a positive Euclidean distance from $z_0 $ to $\overline{I}$. Let $\mathcal{L}$ denote the collection of all line segments from $z_0$ to $bI$, called $K$. Then $K$ has non-empty interior, which contradicts the convexity of $\mathcal{H}(\nabla)$. Therefore, $I$ is a non-empty simply connected bounded open set in $\mathbb{C}$, so there is biholomorphism from $\mathbb{D}$ to $I$ that extends continuously to $\overline{\mathbb{D}}$ by smoothness of the boundary of $\Omega$.
Then Lemma \[lembiholo\] implies that any disk in the boundary of a bounded pseudoconvex complete Reinhardt domain in $\Omega\subset \mathbb{C}^2$ is contained in a continuous family of analytic disks, called $\Gamma$. Furthermore, this continuous family can be represented as $$\Gamma=\{(e^{i\theta}F_1(\zeta), e^{i\theta}F_2(\zeta)):\theta\in [0,2\pi]\land \zeta\in \mathbb{D}\}$$ since $b\Omega$ is three (real) dimensional and $\Gamma$ locally foliates $b\Omega$.
Locally Convexifiable Reinhardt domains in $\mathbb{C}^2$
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\[almost\] Let $\Omega\subset\mathbb{C}^n$ be a bounded pseudoconvex complete Reinhardt domain. Then, $\Omega$ is almost locally convexifiable. That is, for every $(p_1,p_2,...,p_n)\in b\Omega\setminus(\{z_1=0\}\cup \{z_2=0\}\cup...\cup\{z_n=0\})$ there exists $r>0$ and there exists a biholomorphism $L$ on $B((p_1,p_2,...,p_n),r)$ so that $L(B((p_1,p_2,...,p_n),r)\cap \Omega)$ is convex.
Our understanding of analytic structure in the boundary of bounded convex domains is a crucial part of the proof the Theorem \[thmmain\]. The following proposition is proven in [@CuckovicSahutoglu09].
\[prophull\] Let $\Omega\subset \mathbb{C}^n$ be a bounded convex domain. Let $F:\overline{\mathbb{D}}\rightarrow b\Omega$ be a non-constant holomorphic map. Then the convex hull of $F(\mathbb{D})$ is an affine analytic variety.
We note there are no analytic disks in the boundary of $B((p_1,p_2,...,p_n),r)$ because of convexity and the fact that Property (P) (see [@Cat]) is satisfied on the boundary.
We define the following directional derivatives. We assume $\phi\in C(\overline{\Omega)}$. Let $\vec{U}=(u_1,u_2)$ be a unit complex tangential vector at $p:=(p_1,p_2)\in b\Omega$. Then if they exist as pointwise limits, $$\partial_b^{\vec{U},p}\phi:=\lim_{t\rightarrow 0}\frac{\phi(p_1+tu_1,p_2+tu_2)-\phi(p_1,p_2)}{t}$$ and $$\overline{\partial}_b^{\vec{U},p}\phi:=\lim_{t\rightarrow 0}\frac{\phi(\overline{p_1+tu_1},\overline{p_2+tu_2})-\phi(\overline{p_1},\overline{p_2})}{t}$$ The following lemma uses these directional derivatives to characterize when a continuous function $\phi$ is holomorphic ’along’ analytic disks in the boundary of the domain.
\[directional\] Let $\Omega\subset \mathbb{C}^2$ be a bounded pseudoconvex complete Reinhardt domain with a smooth boundary. Suppose $\phi\in C(\overline{\Omega})$. Then $\phi\circ g$ is holomorphic for any non-constant holomorphic $g:\mathbb{D}\rightarrow b\Omega$ if and only if for every $p\in g(\mathbb{D})$ and $\vec{U}$ tangent to $b\Omega$ at $p$, $$\partial_b^{\vec{U},p}\phi$$ exists as a pointwise limit and $$\overline{\partial}_b^{\vec{U},p}\phi=0$$
Suppose $\phi\circ f$ is holomorphic for any $f:\mathbb{D}\rightarrow b\Omega$ holomorphic. The first case to consider is if $\overline{f(\mathbb{D})}$ intersects either coordinate axis.
If $\overline{f(\mathbb{D})}$ intersects either coordinate axis, then by smoothness of $b\Omega$, $f(\mathbb{D})$ is contained in an affine analytic variety and is either vertical or horizontal. That is, $f(\mathbb{D})$ is contained in the biholomorphic image of $\mathbb{D}$. And so one can show $\partial_b^{\vec{U},p}\phi$ exists and $\overline{\partial}_b^{\vec{U},p}\phi=0$.\
That is, we may assume $f:=(f_1,f_2):\mathbb{D}\rightarrow b\Omega$ is holomorphic and neither $f_1$ nor $f_2$ is identically constant. This implies $\overline{f(\mathbb{D})}$ is away from either coordinate axis. Then $f(\mathbb{D})$ is contained in a family of analytic disks $\Gamma$ which foliate the boundary near $f(\mathbb{D})$. Let $p\in f(\mathbb{D})$. By Lemma \[almost\] and Proposition \[prophull\], there exists a biholomorphism $T:B(p,r)\rightarrow \mathbb{C}^2$ so that $T(f(\mathbb{D}))\subset \mathbb{C}\times \{\alpha\}$ for some $\alpha\in [-1,1]$. Furthermore, we may assume $T\circ f:=g$ where $g=(g_1,\alpha)$ and $g_1:\mathbb{D}\rightarrow \mathbb{C}$ is a biholomorphism with a continuous extension to the unit circle. We may assume $g_1$ is a biholomophism by Proposition \[propconvex\].
Let $\phi\circ T^{-1}=\widetilde{\phi}$. We will first show the tangential directional derivative $\partial_b^{\vec{U},p}\widetilde{\phi}$ and the conjugate tangential directional derivative $\overline{\partial}_b^{\vec{U},p}\widetilde{\phi}$ exists on $T(\Gamma)\subset \{(z_1,\alpha): z_1\in \mathbb{C}\land \alpha\in [-1,1]\}$ and $\overline{\partial}_b^{\vec{U},p}\widetilde{\phi}=0$ on $T(\Gamma)$ if and only if $\widetilde{\phi}\circ g$ is holomorphic for any holomorphic $g$ so that $g(\mathbb{D})\subset T(\Gamma)$. First we suppose $\widetilde{\phi}\circ g$ is holomorphic and $g(\mathbb{D})\subset T(\Gamma)$. Then we consider a unit vector $\vec{U}=(u,0)$ so that $\vec{U}$ is tangent to $g(\mathbb{D})$. We may consider the restriction of $\phi$ to $\overline{T(\Gamma)}$ to be a function of $(z_1,\overline{z_1},\alpha)$. That is, $$\phi|_{\overline{T(\Gamma)}}=\phi(z_1,\overline{z_1},\alpha).$$ Then for $({p_1},\alpha)\in g(\mathbb{D})$ we chose $t_0\in \mathbb{R}\setminus\{0\}$ so that for all $t$, $|t_0|>|t|>0$ we have $({p_1+tu},\alpha)\in g(\mathbb{D})$. Then using the fact that $\widetilde{\phi}\circ g$ is holomorphic, we have
$$\begin{aligned}
&\frac{\widetilde{\phi}(p_1,\overline{p_1+tu},\alpha)-\widetilde{\phi}(p_1,\overline{p_1},\alpha)}{t}\\
=&\frac{\widetilde{\phi}(g_1\circ g_1^{-1}(p_1),\overline{g_1}\circ \overline{g_1^{-1}}(\overline{p_1+tu}),\alpha)-\widetilde{\phi}(g_1\circ g_1^{-1}(p_1),\overline{g_1}\circ \overline{g_1^{-1}}(\overline{p_1}),\alpha)}{t}\\
\rightarrow &\frac{\partial(\phi\circ g\circ g_1^{-1})}{\partial \overline{z_1}}=0\\\end{aligned}$$
as $t\rightarrow 0$ and at $(p_1,\alpha)\in g(\mathbb{D})$. By a similar argument, it can be shown that $$\partial_b^{\vec{U},p}\widetilde{\phi}:=\lim_{t\rightarrow 0}\frac{\widetilde{\phi}({p_1+tu},\alpha)-\widetilde{\phi}({p_1},\alpha)}{t}$$ exists and is finite on $T(\Gamma)$.
Next we assume $$\overline{\partial}_b^{\vec{U},p}\widetilde{\phi}:=\lim_{t\rightarrow 0}\frac{\widetilde{\phi}(\overline{p_1+tu},\alpha)-\widetilde{\phi}(\overline{p_1},\alpha)}{t}=0$$ on $T(\Gamma)$ and $$\partial_b^{\vec{U},p}\widetilde{\phi}:=\lim_{t\rightarrow 0}\frac{\widetilde{\phi}({p_1+tu},\alpha)-\widetilde{\phi}({p_1},\alpha)}{t}$$ exists and is finite on $T(\Gamma)$.
Then $$\frac{\partial (\widetilde{\phi}\circ g)(\zeta)}{\partial \overline{\zeta}}=\partial_b^{\vec{U},p}\widetilde{\phi}\frac{\partial g}{\partial\overline{\zeta}}+\overline{\partial}_b^{\vec{U},p}\widetilde{\phi}\frac{\partial \overline{g}}{\partial\overline{\zeta}}=0$$
so by composing $\widetilde{\phi}$ with $T$, we have that $\phi\circ f$ is holomorphic.
\[approx\] Let $\Omega\subset \mathbb{C}^2$ be a bounded pseudoconvex complete Reinhardt domain with a smooth boundary. Suppose $\phi\in C(\overline{\Omega})$ is such that $\phi\circ f$ is holomorphic for any holomorphic $f:\mathbb{D}\rightarrow b\Omega$. Let $\Gamma\subset b\Omega$ be a continuous family of non-trivial analytic disks so that $\overline{\Gamma}$ is disjoint from the closure of any other non-trivial family of analytic disks in $b\Omega$. Then there exists $\{\psi_n\}_{n\in \mathbb{N}}\subset C^{\infty}(\overline{\Omega})$ so that the following holds.
1. $\phi_n\rightarrow \phi$ uniformly on $\overline{\Gamma}$ as $n\rightarrow \infty$.
2. $\phi_n\circ f$ is holomorphic for any holomorphic $f$ so that $f(\mathbb{D})\subset \Gamma$
Let $\nabla\subset b\Omega$ be a non-degenerate analytic disk so that $f(\mathbb{D})=\nabla$ where $f=(f_1,f_2)$ is holomorphic and continuous up to $\overline{\mathbb{D}}$. Furthermore, assume $\nabla$ is away from the coordinate axes. By Lemma \[lembiholo\] and Proposition \[prophull\], there is a local holomorphic change of coordinates $T$ so that $T(\nabla)$ is contained in an affine analytic variety. By Proposition \[propconvex\], we may assume $T(\nabla)$ is convex and $\overline{T(\nabla)}\subset \overline{T(\Gamma)}$ where $\Gamma$ is the continuous family of disks containing $f(\mathbb{D})$ and away from the closure of any other non-degenerate analytic disk. Then the restriction $\phi|_{\Gamma}=\phi(z_1,\alpha)$ where $z_1\in (T(U\cap b\Omega))\subset\{(z_1,z_2)\in \mathbb{C}^2:z_2=0\}$ and $\alpha\in [-1,1]$. Without loss of generality, extend $\phi$ as a continuous function on $\mathbb{C}^2$. As notation, $\mathbb{D}_{\frac{1}{n}}:=\{z\in \mathbb{C}:|z|<\frac{1}{n}\}$.
We let $\chi\in C^{\infty}_0(\mathbb{D})$ so that $0\leq \chi\leq 1$, $\chi$ is radially symmetric, and $\int_{\mathbb{C}}\chi=1$.\
Similarly, we let $\widetilde{\chi}\in C^{\infty}_0(-1,1)$, $0\leq \widetilde{\chi}\leq 1$, and radially symmetric so that $\int_{\mathbb{R}}\widetilde{\chi}=1$.\
Then we define the smooth mollifier $\{\chi_n\}_{n\in \mathbb{N}}\subset C^{\infty}_0(\mathbb{D}_{\frac{1}{n}}\times \left(-\frac{1}{n},\frac{1}{n}\right))$ as $$\chi_n(z_1,\alpha):=n^3\chi(nz_1)\widetilde{\chi}(n\alpha).$$
Then, there exists a holomorphic change of coordinates $H:V\rightarrow \mathbb{C}^2$ so that $T(\Gamma)\subset V$ and $H(T(\Gamma))=\mathbb{D}_s\times (-1,1)$ for some fixed radius $s>0$. For every $n\in \mathbb{N}$, chose $0<r_n<1$ so that $$-1<r_n(\alpha-\beta)<1$$ and $$|r_n(z_1-\lambda)|<s$$ for every $(z_1,\alpha)\in \mathbb{D}_s\times (-1,1)$ and for all
$$(\lambda,\beta)\in \mathbb{D}_{\frac{1}{n}}\times \left(-\frac{1}{n},\frac{1}{n}\right).$$
Then we define the convolution of $\phi\circ T^{-1}$ with $\{\chi_n\}$ in the following manner. $$\psi_n(z_1,\alpha):=\int_{\mathbb{C}\times \mathbb{R}}\phi\circ T^{-1}(r_n(z_1-\lambda),r_n(\alpha-\beta))\chi_n(\lambda,\beta)dA(\lambda)d\beta.$$
Let us extend $\psi_n$ trivially to $\mathbb{C}^2$ and denote this trivial extension as $\psi_n$, abusing the notation.
Now, we have everything we need to show $\psi_n\circ g$ are holomorphic for $g:\mathbb{D}\rightarrow T(\Gamma)$ holomorphic.
Using Lemma \[directional\], for every $n\in \mathbb{N}$, $$\lim_{t\rightarrow 0}\frac{\phi\circ T^{-1}(r_n(\overline{z_1+tu}-\lambda),r_n(\alpha-\beta))-\phi\circ T^{-1}(r_n(\overline{z_1}-\lambda),r_n(\alpha-\beta))}{t}=0$$
pointwise and
$$\lim_{t\rightarrow 0}\frac{\phi\circ T^{-1}(r_n({z_1+tu}-\lambda),r_n(\alpha-\beta))-\phi\circ T^{-1}(r_n({z_1}-\lambda),r_n(\alpha-\beta))}{t}$$ exists and is finite for every $(\lambda,\beta)\in \mathbb{D}_{\frac{1}{n}}\times (-\frac{1}{n},\frac{1}{n})$. Therefore, using the fact that $\chi_n$ are compactly supported and using the Lebesgue dominated convergence theorem, we have that $$\overline{\partial}_b^{\vec{U},p}\psi_n=0$$ for any unit vector $\vec{U}$ tangent to $T(\Gamma)$ for all $p\in T(\Gamma)$, and for all $n\in \mathbb{N}$. Furthermore, $${\partial}_b^{\vec{U},p}\psi_n$$ exists for any unit vector $\vec{U}$ tangent to $T(\Gamma)$, $p\in T(\Gamma)$, and $n\in \mathbb{N}$. Therefore by Lemma \[directional\], $\psi_n$ are holomorphic along analytic disks in $T(\Gamma)$.
Furthermore, it can be shown that $\psi_n\circ T\rightarrow \phi$ uniformly on $\overline{\Gamma}$ as $n\rightarrow \infty$. Now if $\Gamma$ intersects the coordinate axes, then the analytic disks are horizontal or vertical by smoothness of $b\Omega$. So, we perform the convolution procedure as in [@ClosSahut] without using a holomorphic change of coordinates.
For a linear operator $T:G\rightarrow H$ between Hilbert spaces, we define the essential norm as $$\|T\|_e:=\inf\{\|T-K\|\, ,K:G\rightarrow H\,\text{compact}\}$$
\[enorm\] Let $\Omega\subset \mathbb{C}^n$ be a bounded convex domain. Suppose $\Gamma_{\Omega}\neq \emptyset$ is defined as above. Assume $\{\phi_n\}_{n\in \mathbb{N}}\subset C(\overline{\Omega})$ so that $\phi_n\rightarrow 0$ uniformly on $\Gamma_{\Omega}$ as $n\rightarrow \infty$. Then, $\lim_{n\rightarrow \infty}\|H_{\phi_n}\|_e=0$
The next proposition is similar to the theorem in [@CuckovicSahutoglu09], with one major difference, namely they assumed smoothness of the boundary. Here, we assume the boundary is piecewise smooth.
\[onefamily\] Let $\Omega\subset \mathbb{C}^2$ be a bounded convex domain so that the boundary of $\Omega$ contains no analytic disks except for one continuous family, called $\Gamma_{\Omega}$. Let $\phi\in C^{\infty}(\overline{\Omega})$ so that $\phi\circ f$ is holomorphic for any holomorphic $f:\mathbb{D}\rightarrow b\Omega$. Then, $H_{\phi}$ is compact on $A^2(\Omega)$.
Without loss of generality, we may assume $$\Gamma_{\Omega}\subset \{(z_1,\alpha):z_1\in \mathbb{C}\, , \alpha\in (-1,1)\}.$$ Assuming $\phi\circ f$ is holomorphic for any $f:\mathbb{D}\rightarrow b\Omega$, one can show that the tangential directional derivative $\overline{\partial}_b \phi$ exists along $\Gamma_{\Omega}$. Furthermore $\frac{\partial\phi}{\partial\overline{z_1}}=0$ on $\Gamma_{\Omega}$. We wish to construct smooth function $\psi\in C^{\infty}(\overline{\Omega})$ so that $\psi \equiv \phi$ on $\Gamma_{\Omega}$ and $\overline{\partial}(\psi)=0$ on $\Gamma_{\Omega}$. To do this, we will use the idea of a defining function. There exists a smooth function $\rho\in C^{\infty}(\mathbb{C}^2)$ so that $\rho\equiv 0$ on $\overline{\{(z_1,\alpha):z_1\in \mathbb{C}\, , \alpha\in (-1,1)\}}$ and $|\nabla \rho|>0$ on $\overline{\{(z_1,\alpha):z_1\in \mathbb{C}\, , \alpha\in (-1,1)\}}$. Furthermore, by scaling the tangential and normal vector fields on $\overline{\{(z_1,\alpha):z_1\in \mathbb{C}\, , \alpha\in (-1,1)\}}$, we may assume $$\frac{\partial\rho}{\partial\overline{z_1}}|_{\overline{\{(z_1,\alpha):z_1\in \mathbb{C}\, , \alpha\in (-1,1)\}}}=0$$ and $$\frac{\partial\rho}{\partial\overline{z_2}}|_{\overline{\{(z_1,\alpha):z_1\in \mathbb{C}\, , \alpha\in (-1,1)\}}}=1.$$ Now we define $$\psi:=\phi-\rho\left(\frac{\partial\phi}{\partial\overline{z_2}}\right).$$ Then $\overline{\partial}\psi=0$ on $\Gamma_{\Omega}$ and also $\psi=\phi$ on $\Gamma_{\Omega}$. Then by Proposition \[enorm\], $\|H_{\phi-\psi}\|_e=0$ and so $H_{\phi-\psi}$ is compact on $A^2(\Omega)$. To show $H_{\psi}$ is compact we use the fact that $\overline{\partial}\psi=0$ on $\Gamma_{\Omega}$ together with the same argument seen in [@CuckovicSahutoglu09] that shows $H_{\widetilde{\beta}}$ is compact if $\overline{\partial}\widetilde{\beta}=0$ on $\Gamma_{\Omega}$. Therefore we conclude $H_{\phi}$ is compact.
Proof of Theorem \[thmmain\]
============================
The idea is to use the following result which will allow us to localize the problem.
\[local\] Let $\Omega\subset \mathbb{C}^n$ for $n\geq 2$ be a bounded pseudoconvex domain and $\phi\in L^{\infty}(\Omega)$. If for every $p\in b\Omega$ there exists an open neighbourhood $U$ of $p$ such that $U\cap \Omega$ is a domain and $$H^{U\cap \Omega}_{R_{U\cap\Omega}(\phi)}R_{U\cap\Omega}$$ is compact on $A^2(\Omega)$, then $H^{\Omega}_{\phi}$ is compact on $A^2(\Omega)$.
We will also use the following lemma appearing in [@CuckovicSahutoglu09].
\[bi\] Let $\Omega_1$ and $\Omega_2$ be bounded pseudoconvex subsets of $\mathbb{C}^n$. Suppose $\phi\in C^{\infty}(\overline{\Omega_1})$ so that $H_{\phi}$ is compact on $A^2(\Omega_1)$. Let $T:\Omega_2\rightarrow \Omega_1$ be a biholomorphism with a smooth extension to the boundary. Then $H_{\phi\circ T}$ is compact on $A^2( \Omega_1)$.
As we shall see, this collection of all non-constant analytic disks in $b\Omega$ will play a crucial role in our understanding of the compactness of Hankel operators on various domains in $\mathbb{C}^n$ for $n\geq 2$. There are several cases to consider depending on where $p\in b\Omega$ is located.
1. $p\in \Gamma_{\Omega}\subset b\Omega$ but away from the coordinate axes.
2. $p\in b\Omega\setminus \Gamma_{\Omega}$.
3. $p\in \{z_1=0\}\cup \{z_2=0\}$
We will first consider the case where $p$ is away from $\Gamma_{\Omega}$. We let $p:=(p_1,p_2)\in b\Omega$ and assume $p\in b\Omega\setminus \Gamma_{\Omega}$. So there exists an $r>0$ sufficiently small so that the ball $b(B(p,r)\cap \Omega)$ contains no analytic disks. Furthermore, there exists a biholomorphism $T:B(p,r)\rightarrow \mathbb{C}^2$ so that $T(B(p,r)\cap \Omega)$ is a convex domain. Therefore, since any analytic disk in $bT(B(p,r)\cap \Omega)$ must be the image (under $T$) of a disk in $b(B(p,r)\cap\Omega)$, there are no analytic disks in $bT(B(p,r)\cap \Omega)$. By convexity and compactness of the $\overline{\partial}$-Neumann operator, the Hankel operator $$H_{\phi\circ T^{-1}}^{T(B(p,r)\cap \Omega)}$$ is compact on $A^2(T(B(p,r)\cap \Omega))$. And so this proves $H^{U\cap \Omega}_{R_{U\cap\Omega}(\phi)}$ is compact on $A^2(U\cap\Omega)$ where $U:=B(p,r)$.\
If $p\in (\{z_1=0\}\cup \{z_2=0\})\cap b\Omega$, then by smoothness of the domain, either $p$ is contained in an analytic disk, $p$ is a limit point of a sequence of analytic disks, or $p$ is contained in part of the boundary satisfying property (P). If $p\in b\Omega$ is contained in a non-degenerate analytic disk, then locally the analytic disks are horizontal or vertical, by smoothness of the domain. Without loss of generality, assume the family of analytic disk is vertical. So, using the argument in [@ClosSahut], we can approximate the continuous symbol $\phi$ uniformly on $\Gamma_{U\cap\Omega}$ for some ball $U$ centered at $p$ with a sequence of smooth functions $\psi_n$ so that $\psi_n$ is holomorphic along any analytic disk contained in $b(U\cap\Omega)$. As in [@ClosSahut], we use [@CuckovicSahutoglu09] and the uniform approximation on $\Gamma_{U\cap\Omega}$ to conclude that $H^{U\cap\Omega}_{\phi|_{U\cap\Omega}}$ is compact on $A^2(U\cap\Omega)$.\
Note that if $p\in b\Omega$ is contained in part of the boundary satisfying property (P) (see [@Cat]), then the local $\overline{\partial}$-Neumann operator $N_1^{U\cap\Omega}$ is compact since there exists a convex neighbourhood $U$ of $p$ so that $U\cap\Omega$ is convex, and so $H^{U\cap\Omega}_{\phi|_{U\cap\Omega}}$ is compact on $A^2(U\cap\Omega)$.\
Lastly, if $p\in b\Omega\setminus (\{z_2=0\}\cup \{z_1=0\})$ and $p\in \Gamma_{\Omega}$. We will first assume $p$ is contained in a limit set of a discrete sequence of families of analytic disks. We may assume discreteness due to Lemma \[lembiholo\], Proposition \[propconvex\], and smoothness of the boundary of $\Omega$. Then by Lemma \[disklim\], this limit set exactly equals $\{p\}$. We will first assume $p$ is not contained in the closure of a single non-trivial analytic disk.
Let $U:=B(p,r)$ chosen so that $U\cap \Omega$ is a domain and $T(U\cap\Omega)$ is convex for some biholomorphism $T:U\rightarrow \mathbb{C}^2$. Denote this discrete collection of continuous families of analytic disks as $\{\Gamma_j\}_{j\in \mathbb{N}}\subset b(U\cap\Omega)$. Furthermore, we may assume $$\Gamma_{T(U\cap\Omega)}=\bigcup_{j\in \mathbb{N}}\Gamma_j.$$
Then $\{T(\Gamma_j)\}_{j\in \mathbb{N}}$ is a discrete collection of families of affine analytic disks. Then for each $j\in \mathbb{N}$ there exists open pairwise disjoint neighborhoods $V_j$ with a strongly pseudoconvex boundary so that $T(\Gamma_j)\subset V_j$. Let $\rho_j$ be smooth cutoff functions so that $\rho_j\equiv 1$ on a neighborhood of $T(\Gamma_j)$ and $\rho_j$ are compactly supported in $V_j$. Define $$\widetilde{\phi}_j:=\rho_j (\phi\circ T^{-1}-\phi(p_1,p_2)).$$ We wish to show $H_{ \widetilde{\phi}_j}$ are compact on $A^2(T(U\cap\Omega))$ for all $j\in \mathbb{N}$. By Lemma \[onefamily\] and Proposition \[approx\], we approximate $\phi\circ T^{-1}-\phi(p_1,p_2)$ with a sequence $\{\psi^j_n\}_{n\in \mathbb{N}}\subset C^{\infty}(\mathbb{C}^2)$ so that $\psi^j_n\rightarrow \phi\circ T^{-1}-\phi(p_1,p_2)$ uniformly on $\overline{T(\Gamma_j)}$ as $n\rightarrow \infty$ and $\psi^j_n$ are holomorphic along $T(\Gamma_j)$. Then, $\rho_j\psi^j_n$ are holomorphic along any analytic disk in $bT(U\cap\Omega)$ for all $j,n\in \mathbb{N}$ and $\rho_j\psi^j_n\in C^{\infty}(\mathbb{C}^2)$. Fix $j, n\in \mathbb{N}$. Then, there exists a function $\delta_{j,n}\in C^{\infty}(\mathbb{C}^2)$ so that
1. $\overline{\partial}\delta_{j,n}=0$ on $\Gamma_{T(U\cap\Omega)}$.
2. $\delta_{j,n}=\rho_j\psi^j_n$ on $\Gamma_{T(U\cap\Omega)}$.
Therefore by an argument similar to the proof of Proposition \[onefamily\], $H^{T(U\cap\Omega)}_{\delta_{j,n}}$, $H^{T(U\cap\Omega)}_{\rho_j\psi^j_n-\delta_{j,n}}$, and therefore $H^{T(U\cap\Omega)}_{\rho_j\psi^j_n}$ are compact on $A^2(T(U\cap\Omega))$ for all $j,n\in \mathbb{N}$.\
Furthermore, $$\rho_j\psi^j_n\rightarrow \widetilde{\phi}_j$$ uniformly on $\Gamma_{T(U\cap\Omega)}$ as $n\rightarrow \infty$. Then by convexity of $T(U\cap\Omega)$ and Proposition \[enorm\], $H_{\widetilde{\phi}_j}$ are compact on $A^2(T(U\cap\Omega))$ for all $j\in \mathbb{N}$. One can show that $$\alpha_N:=\sum_{j=1}^N\widetilde{\phi}_j$$ converges uniformly to $\phi\circ T^{-1}-\phi(p_1,p_2)$ on $\Gamma_{T(U\cap\Omega)}$ as $N\rightarrow \infty$. Also, $H_{\alpha_N}$ are compact on $A^2(T(U\cap\Omega))$ for all $N\in \mathbb{N}$ as the finite sum of compact operators. Furthermore, $\alpha_N\in C^{\infty}(\overline{T(U\cap\Omega)})$ for all $N$. Then by Lemma \[bi\], $H_{\alpha_N\circ T}$ are compact on $A^2(U\cap\Omega)$ for all $N$ and so $H^{U\cap\Omega}_{\phi|_{U\cap\Omega}}$ is compact on $A^2(U\cap \Omega)$.
So, we have the following. For all $p:=(p_1,p_2)\in b\Omega$ there exists $r>0$ so that $B(p,r)\cap\Omega$ is a domain and $$H^{B(p,r)\cap\Omega}_{\phi|_{U\cap\Omega}}$$ is compact on $A^2(B(p,r)\cap \Omega)$. Then by composing with the restriction operator $R:A^2(\Omega)\rightarrow A^2(B(p,r)\cap\Omega)$, we have that $$H^{B(p,r)\cap\Omega}_{\phi|_{U\cap\Omega}}R$$ is compact on $A^2(\Omega)$. Then by Proposition \[local\], $H_{\phi}$ is compact on $A^2(\Omega)$.
Next, we assume there exists a non-trivial analytic disk $\Gamma_0\in bT(U\cap\Omega)$ so that $p\in \overline{\Gamma_0}$ and $\{p\}$ is the limit set of $\{\Gamma_j\}_{j\geq 1}$. Then we can represent $$\Gamma_{U\cap\Omega}=\bigcup_{j\geq 0,\,\theta\in [0,2\pi]}\{e^{i\theta}\Gamma_j\}.$$ For $0<r<1$ we define $$\Gamma_r:=\bigcup_{f(\mathbb{D})\subset \Gamma_{U\cap\Omega},\,\theta\in [0,2\pi]}\{e^{i\theta}f(r\mathbb{D})\}$$ By convolving $\phi$ with a mollifier in $[0,2\pi]$, there exists $\{\tau_n\}_{n\in \mathbb{N}}\subset C(\overline{\Omega})$ so that $\tau_n\rightarrow \phi$ uniformly on $\overline{\Gamma_r}$ as $n\rightarrow \infty$, and for every $(z_1,z_2)\in \Gamma_r$ and $\vec{T}$ complex tangent to $bU\cap\Omega$ at $(z_1,z_2)$ the directional derivative of $\tau_n$ in the direction of $\vec{T}$ at $(z_1,z_2)$ exists. Furthermore, by the smoothness of $\tau_n$ in the $\theta$ variable, the directional derivative in the complex normal direction at $(z_1,z_2)$ also exists. Thus $\tau_n$ satisfies the compatibility condition for the Whitney extension theorem. See [@stein] and [@mal] for more information on the Whitney extension theorem. Therefore, there exits $\widetilde{\tau}_n\in C^1(\overline{\Omega})$ so that $\widetilde{\tau}_n\equiv \tau_n$ on $\Gamma_r$ and both tangential and normal directional derivatives of $\widetilde{\tau}_n$ agree with $\tau_n$. That is, $\tau_n\circ f$ are holomorphic on $\mathbb{D}$ for any $n\in \mathbb{N}$ and $f(\mathbb{D})\subset \Gamma_{U\cap\Omega}$. Thus $H_{\tau_n}$ is compact on $A^2(\Omega)$ by [@CuckovicSahutoglu09] and Proposition \[enorm\]. And so using Proposition \[enorm\] again and letting $r\rightarrow 1^-$, we conclude $H_{\phi|_{U\cap\Omega}}R_{U\cap\Omega}$ is compact on $A^2(\Omega)$. And so by Proposition \[local\], $H_{\phi}$ is compact on $A^2(\Omega)$.
| ArXiv |
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abstract: 'We clarify certain important issues relevant for the geometric interpretation of a large class of $N = 2$ superconformal theories. By fully exploiting the phase structure of these theories (discovered in earlier works) we are able to clearly identify their geometric content. One application is to present a simple and natural resolution to the question of what constitutes the mirror of a rigid Calabi-Yau manifold. We also discuss some other models with unusual phase diagrams that highlight some subtle features regarding the geometric content of conformal theories.'
author:
- |
Paul S. Aspinwall and Brian R. Greene\
F.R. Newman Lab. of Nuclear Studies,\
Cornell University,\
Ithaca, NY 14853\
title: |
On the Geometric Interpretation of\
$N$ = 2 Superconformal Theories\
---
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Introduction and Summary {#s:intro}
========================
One of the most intriguing problems in string theory is to understand how space-time emerges naturally. Since the vacuum configuration for a critical string is given by a conformal field theory a question which arises in this context is the following. Given a conformal field theory, can one construct some corresponding geometrical interpretation? In this paper we will discuss this question for particularly troublesome conformal field theories. It is worthwhile to emphasize at the outset that in general when a conformal theory does have a geometrical interpretation it may not be unique. A perusal of even simple systems such as conformal theories with central charge $c = 1$ makes this clear. For instance, in this moduli space it is known that a string on the group manifold $SU(2)$ is equivalent to a string on a circle of radius $\sqrt{\alpha^\prime}$. Both target spaces have an equal right to be declared [*the*]{} geometrical interpretation of the conformal field theory. Similarly a circle of radius $R$ is equivalent to a circle of radius $\alpha^\prime/R$. Mirror symmetry, in which strings propagating on distinct Calabi-Yau spaces give identical physical models, is another substantial arena in which geometrical interpretations are not unique. These ambiguities are a reflection of the rich structure of quantum geometry; they arise because of the extended nature of the string.
When there are multiple geometric interpretations of a given model, there is no reason why one should be forced to choose between the possibilities. Rather, one can exploit the geometric ambiguity as some interesting physical questions are more easily answered from one interpretation rather than another.
In this paper we shall focus our investigation into the geometric content of certain of $N = 2$ conformal theories using the framework established in [@W:phase; @AGM:I; @AGM:II]. This approach has the virtue of giving us a physical and mathematical understanding of [*global*]{} properties of the moduli space of these theories as well as of the theories themselves. It also gives us the proper arena for understanding the global implications of mirror symmetry. We will apply this approach to study some theories whose geometrical content has been quite puzzling. For some of these theories, previous papers have proposed possible geometrical interpretations [@Drk:Z; @Schg:gen; @Set:sup]. We will see that when phrased in the language of [@W:phase; @AGM:I; @AGM:II], the previous puzzles are seen to disappear and the geometric status of these theories becomes apparent. Following our remarks above, there need not be one unique interpretation of a given model; however, we do feel that the approach provided here is especially enlightening and economical. We will also see that the less natural constructions of [@Drk:Z; @Schg:gen; @Set:sup] can give misleading results for properties of the corresponding physical model.
We now recall some important background material which will naturally lead us to a summary of the problems we address and the solutions we offer.
Our understanding of the geometric content of $N = 2, c = 3d$ superconformal theories has undergone impressive growth and revision over the last few years. The initial picture which emerged from numerous studies is schematically given in figure \[fig:1\]a. We have an abstract $N = 2, c = 3d$ conformal field theory moduli space that is geometrically interpretable in terms of complex structure and Kähler structure deformations of an associated Calabi-Yau manifold of $d$ complex dimensions and a fixed topological type. The space of Kähler forms naturally exists as a bounded domain (the complexification of the “Kähler cone”) which we denote as a cube. The moduli space of complex structures does not have this form and is more usually compactified to form a compact space. Observables in each of the conformal theories in the moduli space are related to geometrical constructs on the corresponding Calabi-Yau space, the latter being taken as the target space of a nonlinear sigma model.
This picture was extended to that given in figure \[fig:1\]b after the discovery of mirror symmetry. Two Calabi-Yau spaces $X$ and $Y$ constitute a mirror pair if they yield isomorphic conformal theories when taken as the target space for a two-dimensional supersymmetric nonlinear , with the explicit isomorphism being a change in sign of the left moving $U(1)$ charges of all fields. Geometrically this implies that the Hodge numbers $h^{1,1}(X)$ and $h^{d-1,1}(X)$ are related to those of $Y$ by $h^{1,1}(X) = h^{d-1,1}(Y)$ and $h^{d-1,1}(X) = h^{1,1}(Y)$. Since the cohomology groups $H^{1,1}$ and $H^{d-1,1}$ correspond to Kähler and complex structure deformations, respectively, we see that the underlying conformal field theory moduli space has the two geometrical interpretations given in the figure. This immediately led to a problem since, as mentioned above, the geometric form of the moduli spaces of Kähler forms and complex structures appeared to be quite different.
This was resolved by the works of [@W:phase; @AGM:I; @AGM:II] to that shown in figure \[fig:1\]c. Here we see that the appropriate interpretation of the conformal field theory moduli space has required that the Kähler moduli space of $X$ be replaced by its “enlarged Kähler moduli space” (and similarly for $Y$). The latter contains numerous regions in addition to the Kähler cone of the topological manifold $X$. For instance, it typically contains regions corresponding to the Kähler cones of Calabi-Yau spaces related to $X$ by the birational operation of flopping a rational curve, regions corresponding to the moduli space of singular blow-downs of $X$ and its birational partners, and regions interpretable in terms of the parameter space of (gauged or ungauged) Landau-Ginzburg models fibered over various compact spaces. The complex structure moduli space can also be equipped with a phase structure [@AGM:sd] — as must happen to preserve mirror symmetry. We note that from the point of view the phase regions in the complex structure moduli space have a less pronounced physical interpretation. This is because in analyzing the we use perturbation theory in Kähler modes (which fix the size of the Calabi-Yau) and hence this approximation method is not mirror symmetric. However, the phase structure in the complex structure moduli space of $X$ [*is*]{} the phase structure in the enlarged Kähler moduli space of $Y$ and it is the latter interpretation where this phase structure is most manifest. For the purposes of this paper we may ignore the phase structure in the complex structure part of the moduli space and for this reason we have put parentheses around this in \[fig:1\]c.
The results of the present paper all stem directly from a careful study of the phase diagrams of figure \[fig:1\]c. We shall review the quantitative construction of these phase spaces in section \[s:ph\]; for now we will content ourselves with the schematic description given and summarize our results with a similar level of informality.
There are numerous ways of constructing $N = 2$ superconformal theories with $c = 3d$. Some constructions, such as the Calabi-Yau s described above, are manifestly geometric in character. Other constructions do not begin with a geometrical target space and hence their geometrical content, if any, can only be assessed after more detailed study. More generally and pragmatically, given an abstract conformal field theory in some presentation, how do we determine if it has a geometrical interpretation? We will not seek to answer this question in generality, but rather will focus attention on those theories for which we can construct the phase diagram illustrated in figure \[fig:1\]c. For theories of this sort, as we shall review, toric geometry supplies us with a geometric description of each theory. We hasten to emphasize, though, that Calabi-Yau s are but one kind of corresponding geometry. We will see, for instance, that Landau-Ginzburg orbifolds can be associated with noncompact, generally singular, configuration spaces. From our brief discussion here and also from [@W:phase; @AGM:I; @AGM:II] one might think that any theory with a phase diagram such as that in figure \[fig:1\]c, has regions interpretable in terms of Calabi-Yau s. After all, our progression from figures \[fig:1\]a through \[fig:1\]c has centered around Kähler cones of Calabi-Yau spaces. This conclusion, as we shall see in detail in section \[s:appl\], is false and comes to bear on a number of issues, including that of the generality of mirror symmetry. Namely, there are Calabi-Yau manifolds that are rigid, i.e. that have trivial $H^{d-1,1}$. The mirror to such a space, therefore, should have $h^{1,1} = 0$. This is troublesome, though, because Calabi-Yau spaces are Kähler and hence have at least one nontrivial element in $H^{1,1}$. With the above discussion, and explicit calculation in section \[s:appl\], the resolution to this puzzle becomes clear: the enlarged Kähler moduli space for the theory mirror to the one associated with the rigid space $X$ [*does not contain a region interpretable in terms of a Calabi-Yau* ]{}. In fact, the enlarged Kähler moduli space, in contrast to the generic case illustrated in figure \[fig:1\]c, is zero-dimensional and consists of a single point. By direct analysis, we show that the corresponding theory is a Landau-Ginzburg orbifold - not a Calabi-Yau - and hence it is perfectly consistent for the theory to lack a Kähler modulus. We note at the outset that possible resolutions to the question of the identity of mirrors to rigid Calabi-Yau spaces have been previously presented in [@Drk:Z; @Schg:gen; @Set:sup]. These authors have invoked unexpected additional structures such as non-Calabi-Yau spaces of dimension greater than $d$ and supermanifolds in an attempt to resolve this issue. Contrary to these works, we see here that absolutely no additional structure is required. Rather, rigid Calabi-Yau manifolds fit perfectly into the general framework introduced in [@W:phase; @AGM:I; @AGM:II].
In addition to applying our analysis to the case of rigid Calabi-Yau manifolds and their mirrors, we also study two other interesting phenomena. First, we present an example of a theory with nonzero dimensional enlarged Kähler moduli space that does not contain a geometric region thus showing that the mere existence of a would-be Kähler form does not guarantee a Calabi-Yau interpretation. Second, we briefly discuss an example (first pointed out in [@W:phase]) whose enlarged Kähler moduli space has a phase whose target space has the desired dimension but is not of the Calabi-Yau type.
Phase Diagrams: Supersymmetric Gauge Theory and Toric Geometry {#s:ph}
==============================================================
The moduli space of $N = 2$ superconformal theories is most naturally interpretable in terms of a collection of regions within which the theory assumes a particular phase. Amongst the possibilities are smooth and singular geometric Calabi-Yau phases, gauged and ungauged Landau-Ginzburg phases, as well as orbifolds and hybrids thereof. This is the burden of figure \[fig:1\]c.
The existence and quantitative construction of these phase diagrams has been approached from two distinct vantage points in the works of [@W:phase] and [@AGM:II]. In fact, a point which is not as fully appreciated as it might be is that these two approaches, although phrased in different languages, are [*isomorphic*]{}. Different questions, though, are often more easily answered from one of the two formalisms and hence it is important to fully understand both approaches and their precise relationship. It is the purpose of the present section to explain these issues. We note that the material in this section is implicit in [@W:phase] and [@AGM:II]; for our purposes we need to make the relation explicit.
In brief, both [@W:phase] and [@AGM:II] build constrained $N = 2$ supersymmetric quantum field theories. In the physical approach of [@W:phase] these constraints are phrased in terms of [*symplectic*]{} quotients. In the mathematical approach of [@AGM:II] these constraints are phrased in terms of [*holomorphic*]{} quotients. The well-known equivalence [@Kirwan:; @Ness:] of these two approaches then implies that each constructs the same theory and hence also the same phase diagrams. The proper language for establishing these statements is that of [*toric geometry*]{} for which the reader can find a primer in [@AGM:II]. In the following we will try to convey the main points with a minimum of unnecessary technical detail.
Complex projective space may be considered to be the prototypical toric variety. One constructs $\P^n$ by taking the $n+1$ homogeneous coordinates, $x_i$, spanning $\C^{n+1}$, removing the origin $x_i=0$ and modding out by the $\C^*$-action $x_i\to\lambda x_i$, $\lambda\neq0$. A toric variety is simply a generalization of this concept with perhaps more than one $\C^*$-action and a possibly more complicated point set removed prior to the modding out process.
The most natural way of building a $N$=2 with a complex projective target space appears to be in terms of a $U(1)$-gauged field theory [@ADL:CPn]. In this construction, one begins with the homogeneous coordinates, $x_i$, denoting chiral superfields, each with the same $U(1)$ charge, $Q_i$, (which we may take to be 1). The classical vacuum of such a theory may be determined by finding the minimum of the classical potential energy. Solving the algebraic equations for the auxiliary D-component of the gauge multiplet and including the result in the scalar potential yields the familiar contribution $$( |x_1|^2 + |x_2|^2 +\ldots+|x_{n+1}|^2 - r )^2, \label{eq:ve1}$$ where $r$, a real number, is the coefficient of the familiar Fayet-Illiopoulos D-term. We take $r$ to be positive here to avoid naïvely breaking supersymmetry (see section 3.2 of [@W:phase] for a discussion on negative values of $r$). Minimizing the energy forces us to require that (\[eq:ve1\]) should vanish. This immediately removes the origin $x_i=0$ from consideration. It also forces the $x_i$ to lie on the sphere $S^{2n+1}$. We may now divide out by the $U(1)$ (i.e., $S^1$) action to form $S^{2n+1}/S^1\cong\P^n$. The process of dividing by $\C^*$, in the usual formulation of $\P^n$ may be viewed to having taken place in two stages. First we fix an $\R_+$ degree of freedom by imposing the vanishing of (\[eq:ve1\]) and then we divide out by $S^1$. The equivalence of these two constructions then follows from the fact that $\C^*\cong \R_+\times S^1$. Dividing by the former is a simple example of a holomorphic quotient; dividing by the latter is a simple example of a symplectic quotient. We have just seen, therefore, the essential reason why these two are equivalent. Let us now discuss how Witten generalized upon this quantum field theory approach of generating symplectic quotients. We will then discuss their equivalent holomorphic quotient description as in [@AGM:II].
Witten [@W:phase] extended the above model to describe not a complex projective space but the “canonical” line bundle of complex projective space (see, for example, [@GH:alg] for the precise definition of this bundle). Let us reserve $n$ to denote the dimension of the toric variety in question so now we are looking at a line bundle over $\P^{n-1}$ and the variable $x_{n+1}$ will now be treated differently to the others. This space is then built from $\C^{n+1}$ by removing the point $x_1=x_2=\ldots=x_n=0$ and modding out by the action $$\eqalign{
x_i&\to \lambda x_1,\quad i=1\ldots n,\cr
x_{n+1}&\to\lambda^{-n}x_{n+1}.\cr} \label{eq:Cs1}$$ To produce this from the gauged point of view we put $Q_i=1$ for $i=1\ldots n$ and $Q_{n+1}=-n$. The vanishing of the classical potential now implies $$|x_1|^2 + |x_2|^2 +\ldots+|x_n|^2 -n|x_{n+1}|^2 = r. \label{eq:ve2}$$ We see that there are classical vacuum solutions for $r$ for either sign. If $r>0$, we thus recover the required target space as in the case of the projective space. If however $r<0$, we find that $x_{n+1}\neq0$ and we have no condition on $x_1,\ldots, x_n$. Let us consider this space more closely.
Removing the point set $x_{n+1}=0$ from $\C^{n+1}$ and dividing by the action (\[eq:Cs1\]) produces another toric variety. $x_{n+1}$ may be fixed by choosing a value for $\lambda^n$ leaving the $n$th roots of unity to act on the space spanned by $x_1,\ldots,x_n$. Thus the toric variety is $\C^n/\Z_n$. Therefore we see that the geometry of the target space can change discontinuously as we vary $r$. This theory is said to have two [*phases*]{} where the relevant toric variety is either the canonical line bundle of $\P^{n-1}$ or $\C^n/\Z_n$.
The construction of [@W:phase] doesn’t quite stop here. One may introduce a $U(1)$-invariant superpotential, $W$, i.e., a $\C^*$-invariant polynomial over the $x_i$’s. Minimizing the classical potential now also implies that we are at a critical point of $W$.
Our toric “ambient” space will always turn out to be non-compact. This however will contain compact subspaces which may also be considered as toric varieties themselves. Clearly $\P^{n-1}$ is a toric subspace of the canonical line bundle over $\P^{n-1}$. The only compact toric subspace of $\C^n/\Z_n$ is the point at the origin. Assuming that $W$ is suitably generic, the effect of including the superpotential term is to force the classical vacuum to be equal to, or contained in some compact toric subspace of the ambient space.
In our example, a suitable $W$ is $(x_1^n+x_2^n+\ldots+x_n^n)x_{n+1}$. In the canonical line bundle over $\P^{n-1}$ case, the critical point set of $W$ consists of the hypersurface $x_1^n+x_2^n+\ldots+x_n^n=0$ in $\P^{n-1}$. This is a compact $(n-2)$-fold. This is thus named, the [**]{} phase. In the $\C^n/\Z_n$ case, the origin is the critical point set of $W$. Thus our classical vacuum is simply one point. The effective superpotential of this theory however allows for massless fluctuations around this point given by a Landau-Ginzburg superpotential $x_1^n+x_2^n+\ldots+x_n^n$. This is thus the [**]{} phase. Note that all fluctuations around the vacuum in the phase are massive.
Let us fix some notation.[^1] We will call the ambient non-compact toric space $\Vbig$. This contains a maximal compact toric subset $\Vlit$ (which may be reducible). Within $\Vlit$ we have the classical vacuum of the quantum field theory which we denote $X$.
A simple generalization of the above construction is to consider a weighted projective space for $\Vlit$. Clearly this may be achieved by giving different charges to $x_1,\ldots,x_n$. Following the above formalism we would again obtain two phases depending on whether $r$ was less than or greater than zero. When we look at the associated conformal field theory it turns out that this does not capture the full moduli space, i.e., $h^{1,1}>1$ for many of these theories. It is not hard to generalize the present description to include at least some of these other degrees of freedom. For each such independent direction in the moduli space we are able to access in this formalism, we introduce a $U(1)$ gauge factor and a corresponding parameter $r_l$. Thus, the total gauge group is $G=U(1)^{s}$ where $s$ is the dimension of this subspace of the moduli space . The chiral fields will in general be charged under all of the $U(1)$ factors, and hence we write $Q_i^{(l)}$ to denote the charge of the $i^{th}$ chiral superfield under $U(1)_{(l)}$. The superpotential $W$ must now be a $G$-invariant combination of the chiral superfields.
It turns out that the language of toric geometry is precisely suited for determining all of the data needed for building such a model. Namely, in the case of $s = 1$ (or more generally, $s$ is the number of distinct toric factors making up the ambient space) it is straightforward to figure out appropriate charges so that minimization of the scalar potential yields the desired model. When $s$ is not of this form, the problem requires a more systematic treatment; this is precisely what the formalism of toric geometry supplies. Furthermore, for these more general cases, it proves increasingly difficult to determine the phase diagram of the model by studying the minimum of the scalar potential for various values of the $r_1,...,r_{s}$. The formalism of toric geometry, as described in [@AGM:II], supplies us with a far more efficient means of determining the phase structure, as well. Hence, let us now recast the above formulation directly in terms of toric geometry.
The homogeneous coordinates (in the sense of [@Cox:]) $x_1,\ldots, x_N$ form a natural representation of the group $(\C^*)^N$. Let us form a toric variety by removing some point set and dividing the resultant space by $(\C^*)^{N-n}$. Clearly the space formed, $\Vbig$, is acted upon non-trivially by $(\C^*)^n$. Let us introduce $\zeta_j$, $j=1,\ldots,n$, as the natural representation of this $(\C^*)^n$-action. That is, the $\zeta_j$ provide coordinates on a dense open subset of $\Vbig$. This follows since $\Vbig$ may be regarded itself as a compactification of $(\C^*)^n$. Let us relate these new “affine” coordinates to the homogeneous coordinates by $$\zeta_j = \prod_{i=1}^N x_i^{\alpha_{ij}}, \label{eq:aff}$$ where $\alpha_{ij}\in\Z$. We may represent the $N\times n$ matrix, $\alpha_{ij}$, by a collection of $N$ points, which we denote $\cA$, living in an $n$-dimensional real space where $\alpha_{ij}$ is the $j$th coordinate of the $i$th point. Let us demand that $\cA$ is such that there exists an $n$-dimensional lattice ${\bf N}$ within this same space (which we denote ${\bf N}_{\R}={\bf N}\otimes_{\Z}\R$) such that $$\cA = {\bf N} \cap (\hbox{Convex hull of $\cA$}\,).
\label{eq:Acvx}$$ The notation $\alpha_i$ will denote the position vector of the $i$th point of $\cA$ in ${\bf N}$.
Consider now the charges of the homogeneous coordinates under the $(\C^*)^{N-n}$ by which we modded out. Denote these $Q_i^{(l)}$ where $i=1,\ldots,N$ and $l=1,\ldots,N-n$. The obvious short exact sequence $$1\to(\C^*)^n\to(\C^*)^N\to(\C^*)^{N-n}\to1,$$ induces, $$\sum_{i=1}^N Q^{(l)}_i\alpha_{ij} = 0,\quad\forall l,j.
\label{eq:krnl}$$
Thus, we see that the charges $Q^{(l)}_i$ are simply the [*kernel of the transpose of the matrix whose elements are*]{} $\alpha_{ij}$. The reader should check that in the simple case, say, of projective space discussed earlier, that the charge assignment posited can in fact be derived in this manner.
Now define ${\bf M}$ as the dual lattice to ${\bf N}$. Let us demand that there is an element $\mu\in{\bf M}$ such that $$\langle\mu,\alpha_i\rangle=1,\quad\forall i. \label{eq:hypln}$$ This condition is similar to stating that $\Vbig$ be a space with vanishing canonical class, $K$, (or zero first Chern class). Actually $\Vbig$ need not be smooth so be need to be more careful about our language. The correct term from algebraic geometry is that $\Vbig$ is [*Gorenstein*]{} (see, for example, [@Reid:mm]). Applying (\[eq:hypln\]) to (\[eq:krnl\]) tells us that $$\sum_{i=1}^N Q^{(l)}_i =0,\quad\forall l. \label{eq:Q0}$$ This appears as an important condition in [@W:phase] ensuring freedom from anomalies in certain chiral currents which should be present if there is an infrared limit with $N$=2 superconformal invariance. It is curious to note that (\[eq:Q0\]) is not sufficient to guarantee (\[eq:hypln\]). We may have $\langle\mu,\alpha_i\rangle=k$ for example, for some integer $k$. $\Vbig$ would then be $\Q$-Gorenstein which is roughly saying that $kK=0$ but $K$ may be a non-trivial torsion element. The effect of this in terms of the two dimension quantum field theory has not been studied.
This point set $\cA$ gives us all the information we require to build $\Vbig$ except which point set should be removed from $(\C^*)^N$ before performing the quotient. This is performed in toric geometry by building a fan, $\Delta$. A fan is a collection of tesselating cones in ${\bf N}_\R$ with apexes at the origin. The intersection of this fan with the hyperplane containing $\cA$ will be a set of tesselating polytopes. The convex hull of this set of polytopes must be the convex hull of $\cA$ and the vertices of the polytopes must be elements of $\cA$. Thus each cone, $\sigma$, in $\Delta$ is “generated” by a subset of $\cA$. We say $\alpha_i\in\sigma$ if $\alpha_i$ is one of the generators, i.e., $\alpha_i$ lies at a vertex of the intersection of $\sigma$ with the hyperplane in ${\bf N}_\R$ containing $\cA$. The point set $F_\Delta$ removed from $\C^N$ prior to quotienting is then specified by $$\bigcap_{\sigma\in\Delta} \Bigl\{ x\in\C^N;
\prod_{{\alpha_i\in\cA,} \atop {\alpha_i\not\in\sigma}}\!\!x_i=0 \Bigr\},
\label{eq:Fset}$$ where $x$ is the point with coordinates $x_i$.
The fact that different fans may be associated with the point-set $\cA$ gives rise to the phase structure. We need only consider the case where all the $\sigma$’s are simplicial based cones, i.e., we induce a simplicial decomposition of triangulation of $\cA$. To each such fan (satisfying in addition a certain “convexity” property, see [@AGM:II] for more details) we associate a phase. Other fans consistent with $\cA$ not satisfying these conditions may always be considered as models on the boundary between two or more phases. The parameters, $r$, in the linear approach give us an identical fan structure. This is best understood from examining figure 11 of [@AGM:II]. The $r$ parameters, in essence, fix the heights of the points in this figure and hence following the discussion of section of 3.8 of [@AGM:II] their values determine a triangulation of the point set $\cA$. From a physical point of view we can group together those values for the $r$ parameters which yield the same phase for the model. In this way we partition the space of all possible $r$’s into a phase diagram. This phase diagram is the “secondary fan” for the moduli space as discussed in [@AGM:II].
We now have a dictionary between [@W:phase] and the toric approach: [*Specifying generic values of [“$r$”]{} parameters is equivalent to specifying a triangulation of $\cA$. The non-vanishing conditions on the fields $x_i$ specified by minimizing the $D$-term part of the classical potential is equivalent to removing the point set $F_\Delta$ given by*]{} (\[eq:Fset\]).
Note that requiring $\cA$ to be “complete” in the sense of (\[eq:Acvx\]) is not necessary in the analysis of [@W:phase]. By imposing this condition we gain access to the largest subspace of the moduli space we can reach by this toric method.
One point in the dictionary between [@W:phase] and the toric approach which we have not spelled out explicitly as yet is how we determine the superpotential $W$ from the toric data. This is straightforward as we now describe. Let us $G$ to denote the group $(\C^*)^{N-n}$. $W$ is a $G$-invariant polynomial in the chiral superfields. From (\[eq:krnl\]) we see that any monomial of the form $$\prod_{i=1}^N x_i^{\langle\alpha_i, v\rangle} \label{eq:prod}$$ for a fixed but arbitrary vector $v$ is $G$-invariant. However, we want all terms in $W$ to not only be $G$-invariant but also to have nonnegative integral exponents. Towards this end we are naturally led to introduce the cone $\Upsilon$ in ${\bf M}_\R$, dual to $\Sigma$ which is the cone over the convex hull of $\cA$ in ${\bf N}_\R$, defined by $$\Upsilon = \left\{ s\in{\bf M}_\R; \langle s,t\rangle\geq0, \forall
t\in\Sigma \right\}.$$ The integral lattice points in $\Upsilon$, when substituted for the vector $v$ in (\[eq:prod\]), will then generate $G$-invariant monomials with nonnegative exponents. To systematize this, we now define $\cBp\subset \Upsilon$ by $$\cBp = {\bf M} \cap \Upsilon,$$ the integral lattice points contained in the dual cone. Any point in $\cBp$, if substituted for the vector $v$ in (\[eq:prod\]), yields a $G$-invariant nonnegative exponent monomial. Finally, we note that we would like $W$ to be a suitably “quasihomogeneous” polynomial of lowest nontrivial degree in the $x_i$. This will remove any “irrelevant” terms in the superpotential [@VW:] and may be achieved as follows. Let the monomials in this reduced superpotentials be labeled by elements of $\cB \subset \cBp$. Following [@BB:mir] let us put one last condition on $\cA$, namely that when we derive the point set $\cB$ exists a vector $\nu\in{\bf N}$ such that $$\langle \beta_v,\nu\rangle = 1, \quad\forall\beta_v\in\cB,$$ and that the vectors given by the elements of $\cB$ (or a subset of $\cB$) generate $\Upsilon$. We also impose the condition on $\cB$ paralleling our discussion for the point set $\cA$. Namely, we can say $$\cB = {\bf M} \cap (\hbox{Convex hull of $\cB$}\,),$$ with the elements of $\cB$ at the vertices of this convex hull generating $\Upsilon$. We denote by $M$ the number of points in $\cB$ so that $v=1,\ldots,M$. The superpotential $W$ is then constructed according to $$W = \sum_{v=1}^M a_v w_v,$$ for $a_v\in\C$ with $$w_v = \prod_{i=1}^N x_i^{\langle\beta_v,\alpha_i\rangle}.
\label{eq:mon}$$
We may note at this point that mirror symmetry is conjectured to exchange the sets $\{{\bf M},\mu,M,\cA\,\}\leftrightarrow\{{\bf
N},\nu,N,\cB\}$. This may be regarded as a generalization of the “monomial-divisor mirror map” of [@AGM:mdmm].[^2] The mirror pairs of [@GP:orb] (which is established at the conformal field theory level) are a subset of this general construction and the examples in sections \[ss:Z\] and \[ss:h1\] are in this subset. Thus statement concerning mirror symmetry with regards to these examples may be regarded as definitely true. Also note that our analysis of the phases of the moduli space does not depend on the mirror map and thus does not depend on this mirror conjecture.
Now let us try to calculate the central charge $3d$ of the conformal field theory associated to this model. We may apply the same reasoning as was used in [@VW:] to determine this. Firstly we have $N$ chiral superfields each of which contributes $+1$ to $d$. This may be taken to correspond to the string propagating in $\C^N$. We also have $N-n$ vector superfields which we take to contribute $-1$ to $d$ since each removes one complex dimension from the target space. Thus, so far we have $d=n$. However, the string is further confined by the superpotential $W$ and we expect this to reduce the value of $d$ as we now show.
Consider now rescaling by an element of $(\C^*)^N$, $$x_i\to\lambda^{\omega_i}x_i.$$ The monomial $w_v$ then scales to $\lambda^\chi w_v$ where $$\chi=\sum_{i=1}^N \langle\beta_v,\alpha_i\rangle\omega_i.$$ Consider choosing the weights $\omega_i$ such that $$\sum_{i=1}^N \omega_i\alpha_i = \nu. \label{eq:wcond}$$ Then all the monomials transform $w_v\to\lambda w_v$ and thus, declaring $a_v$ to be invariant, we have $W\to\lambda W$. Taking the inner product of (\[eq:wcond\]) with $\mu$ gives $$\sum_{i=1}^N\omega_i = \langle \mu,\nu\rangle.$$ It was shown in [@VW:] that the effect of the superpotential is to contribute $-2\sum\omega_i$ to $d$. Thus we have $$d=n-2\langle \mu,\nu\rangle, \label{eq:d}$$ in agreement with the conjecture in [@BB:mir].[^3]
For the cases considered in [@AGM:II] based upon the construction of [@Bat:m] we had $\langle\mu,\nu\rangle=1$. This then is a generalization. It should be noted that this more generalized picture could have been deduced directly by applying the toric language to Witten’s formulation of [@W:phase] although historically it was first written in the form of [@Boris:m] where it was used specifically for conjecturing the mirror map for complete intersections in toric varieties.
To summarize so far, all the data we require to build an abelian gauged linear of the form studied in [@W:phase] is the matrix $\alpha_{ij}$. To provide a consistent model for a conformal field theory we demand that this matrix be compatible with $\mu$ and $\nu$ and be consistent with the existence of ${\bf N}$ in the form of (\[eq:Acvx\]). Once we have this information we may apply the technology of [@W:phase; @AGM:II] to determine the geometry of the various phases in the moduli space of Kähler forms. This is most easily determined in terms of triangulations of the point set $\cA$.
There is one more piece of information we will need before moving on to some examples concerning orbifolding. The toric variety $\Vbig$ is acted upon by $(\C^*)^n$. It is simple in toric geometry to describe the orbifold of $\Vbig$ by a discrete subgroup of this $(\C^*)^n$. Consider the affine coordinates introduced by (\[eq:aff\]). Let us consider the element, $g\in(\C^*)^n$ which acts by $$g:(\zeta_1,\zeta_2,\ldots,\zeta_n)\mapsto(e^{2\pi ig_1}\zeta_1,
e^{2\pi ig_2}\zeta_2,\ldots,e^{2\pi ig_n}\zeta_n) \label{eq:iden}$$ where $0\leq g_j<1$. We can see (for more details consult [@Reid:yp]) that dividing $\Vbig$ by the group generated by $g$ is equivalent to replacing the lattice ${\bf N}$ by a lattice generated by ${\bf
N}$ and the vector $(g_1,g_2,\ldots,g_n)\in{\bf N}_\R$. The reason for this is that lattice points $p$ in ${\bf N}$ represent one (complex) parameter group actions on the toric variety $$p:(\zeta_1,\zeta_2,\ldots,\zeta_n)\mapsto(\lambda^{p_1}\zeta_1,
\lambda^{p_2}\zeta_2,\ldots,\lambda^{p_n}\zeta_n).$$ For points $p$ whose components are non-integral, such a map is only well defined if certain global identifications are made on the $(\zeta_1,\zeta_2,\ldots,\zeta_n)$. In particular, one directly sees that taking $p$ to be $(g_1,g_2,\ldots,g_n)$ requires the desired identification of (\[eq:iden\]).
Applications {#s:appl}
============
Let us now illustrate the general method of the previous section by applying it to various examples. The possibilities offered by this formulation appear to be very rich but we select here a few key examples to emphasize points relevant to our discussion.
The Hypersurface Case {#ss:hyp}
---------------------
Suppose that $\langle \mu,\nu\rangle=1$. In this case it is easy to see that $\nu\in\cA$ and that this point lies properly in the interior of the convex hull of $\cA$ (since $\langle\beta_v,\nu\rangle$ is strictly positive). One possible triangulation of the point set $\cA$ thus consists of drawing lines from $\nu$ to each point on the vertices of the convex hull and filling this skeleton in with a suitable set of simplices to form a triangulation. The resultant set forms a complete fan of dimension $n-1$ with center $\nu$. This fan $\delta$ corresponds to a compact $(n-1)$-dimensional toric sub-variety $\Vlit$ of $\Vbig$. Let us denote by $p$ the homogeneous coordinate corresponding to $\nu$. We see from (\[eq:mon\]) that every term in the superpotential appears linearly in $p$. Thus we may write $W=pG$ where $G$ is a function of the $N-1$ homogeneous coordinates describing $\Vlit$. Thus the condition $\partial W/\partial p=0$ implies $G=0$ — i.e., we are on a hypersurface within $\Vlit$. For a generic $G$, the other derivatives of $W$ imply that $p=0$.
The target space, $X$, is now a hypersurface within $\Vlit$ which itself has dimension $n-1$. $X$ is thus of dimension $n-2$. The equation (\[eq:d\]) tells us that $d=n-2$. In fact $X$ is an anticanonical divisor of $\Vlit$ [@Bat:m] and is thus a space of $d$ dimensions. Note that $X$ may not be smooth but these singularities can often be removed by further refinements of the fan $\delta$. Actually, in the case $d\leq3$ the singularities may always be removed in this way.[^4]
For the case $\langle \mu,\nu\rangle=1$ we therefore always have a “” phase. That is, some limit in the moduli space where we may go to build some non-linear of the conformal field theory (although in case $d>3$ we may have to include considerations such as terminal orbifold singularities in our model). The case considered here is basically of the type studied in [@Bat:m; @AGM:II] as shown in [@BB:mir]. It also includes the example of the model with the phase in $\C^n/\Z_n$ and the hypersurface in $\P^{n-1}$ discussed above.
The Mirror of the Z-orbifold {#ss:Z}
----------------------------
We now turn to the issue of rigid Calabi-Yau spaces and their mirrors. For concreteness we focus on the Z-orbifold of [@CHSW:]. Recall that this is the torus of six real dimensions divided by a diagonal $\Z_3$ action. It has 36 (1,1)-forms (9 from the original torus and 27 associated with blow up modes) and no (2,1)-forms. It is therefore rigid. Using the construction of [@GP:orb], it was shown in [@AL:geom] how to construct the Z-orbifold in terms of an orbifold of a Gepner model [@Gep:]. To phrase this more carefully allowing for the phase structure, one builds a conformal field theory as an orbifold of a Gepner model which may be deformed via marginal operators to a theory corresponding to a whose target space is the blown-up Z-orbifold. It was also shown how to build a conformal field theory giving the mirror of the above theory, also as a orbifold of the Gepner model.
The Gepner model itself is believed to be equivalent to an orbifold of a theory. In the case under consideration (the ${\bf 1}^9$ model) the configuration space of this orbifold theory is $\C^9/\Z_3$. The space $\C^9$ is a toric variety with $n=9$ described simply by the fan consisting of one cone, $\sigma$, isomorphic to the positive quadrant of $\R^9$. That is, $\cA$ consists of the points $(1,0,0,0,0,0,0,0,0),(0,1,0,0,0,0,0,0,0),
\ldots,(0,0,0,0,0,0,0,0,1)$. The required $\Z_3$ quotient is performed by adding the generator $$g_1=(\ff13,\ff13,\ff13,\ff13,\ff13,\ff13,\ff13,\ff13,\ff13),
\label{eq:g1Z}$$ to the integral lattice of $\R^9$. It was shown in [@AL:geom] that the mirror of the Z-orbifold was obtained by dividing by a further $\Z_3$ (i.e., taking a $\Z_3$ orbifold of the Gepner model) given by the vector $$g_2=(0,0,0,\ff13,\ff13,\ff13,\ff23,\ff23,\ff23),$$ to give the required ${\bf N}$-lattice. We may apply a $Gl(9,\R)$ transformation to ${\bf N}_\R$ to rotate ${\bf N}$ back into the standard integral lattice. This will act on $\sigma$ so that it is no longer the positive quadrant. One choice of transformation leaves $\sigma$ generated by $$\eqalignsq{
\alpha_1&=(3,0,0,1,1,1,-1,-1,-3)\cr
\alpha_2&=(0,1,0,0,0,0,0,0,0)\cr
\alpha_3&=(0,0,1,0,0,0,0,0,0)\cr
\alpha_4&=(0,0,0,1,0,0,0,0,0)\cr
\alpha_5&=(0,0,0,0,1,0,0,0,0)\cr
\alpha_6&=(0,0,0,0,0,1,0,0,0)\cr
\alpha_7&=(0,0,0,0,0,0,1,0,0)\cr
\alpha_8&=(0,0,0,0,0,0,0,1,0)\cr
\alpha_9&=(0,-1,-1,-2,-2,-2,0,0,3).\cr} \label{eq:AZ}$$ These 9 points lie in the hyperplane defined by $\mu=(3,1,1,1,1,1,1,1,3)$. It is a simple matter to show that the dual cone gives $\nu=(1,0,0,0,0,0,0,0,0)$ so that $\cA$ has the required properties.
The important property of this model stems from the fact that the points $\alpha_1,\ldots,\alpha_9$ form the vertices of a simplex with [*no*]{} interior points lying on the lattice ${\bf N}$. That is, the set $\cA$ consists only of those points listed in (\[eq:AZ\]). Thus the only triangulation of $\cA$ consists of this simplex! This model has $\Vbig\cong\C^9/(\Z_3\times\Z_3)$ with superpotential $$W = a_1x_1^3+a_2x_2^3+\ldots+a_9x_9^3+a_{10}x_1x_2x_3+\ldots. \label{eq:WZ}$$ The critical point of $W$ is the origin. Thus we have an orbifold of a theory as expected. Since there is no other triangulation of $\cA$ there is no other phase and, in particular, [*no phase*]{}. Since $\langle \mu,\nu\rangle=3$ we are not in conflict with the section \[ss:hyp\]. As expected we see that $d=3$ in agreement with the fact that this theory is the mirror of a smooth threefold (i.e., the blow-up of the Z-orbifold).
Thus, by properly understanding the full content of mirror symmetry — as a symmetry between the moduli spaces of $N = 2$ superconformal theories — we see that there is no puzzle regarding the mirror of a rigid Calabi-Yau manifold. The mirror description simply does not have a Calabi-Yau phase and hence the absence of a Kähler form causes no conflict.
It is important to realize that we have all the information we need to study this model without recourse to finding some other effective target space geometry. In particular, deformations of complex structure are achieved by deforming the $a_v$ parameters in (\[eq:WZ\]) in the usual way and one may then use mirror symmetry to study the moduli space of Kähler forms of the Z-orbifold as was done in [@Drk:Z].[^5]
The lack of a phase appears due to the existence of [*terminal*]{} singularities in algebraic geometry as we now discuss (see also [@Reid:yp] for a more thorough account). In section \[s:ph\] we discussed the case of a Landau-Ginzburg theory in $\C^n/\Z_n$. In this case, the $\Z_n$ symmetry is generated by $(\ff1n,\ff1n,\ldots)$. This singularity may be blown-up to give the canonical line bundle over $\P^{n-1}$. This smooth space has trivial canonical class. Thus the singularity $\C^n/\Z_n$ may be blown up without adding something non-trivial into the canonical class. Such a blow-up mode is always visible in the associated conformal field theory as a truly marginal operator since it may be regarded as a deformation of the Kähler form.
A terminal singularity is a singularity which cannot be resolved (or even partially resolved) without adding something non-trivial to the canonical class. The singularity $\C^9/\Z_3$ generated by $g_1$ of (\[eq:g1Z\]) is precisely such a singularity. As such, from the conformal field theory point of view, it is “stuck”. This agrees with the fact that the Gepner model contains no marginal operators corresponding to deformations of the Kähler form.
One can go ahead and blow-up the $\C^9/\Z_3$ singularity if one really desires some smooth manifold. There is no unique prescription for this but one may, for example, form the space $\O_{\P^8}(-3)$. This is a line bundle over $\P^8$ with $K<0$ (i.e., $c_1>0$). The homogeneous coordinates of this projective space may be given by the coordinates of the original $\C^9$. The superpotential of the Landau-Ginzburg theory is cubic in these fields and so one might try to associate this model to the cubic hypersurface in $\P^8$. This is the essence of the construction of [@Drk:Z; @Schg:gen]. Note that in the language of this paper, we no longer satisfy (\[eq:Q0\]) and so our field theory is expected to have undesirable properties in the infrared limit.
When we try to describe the mirror of the Z-orbifold, the situation becomes even worse. The second $\Z_3$ quotient given by $g_2$ induces further terminal quotient singularities on $\P^8$ which require considerably more to be added to the canonical class. We hope the reader sees that this procedure of forcing a smooth geometrical interpretation when terminal singularities appear is completely unnatural when written in terms of the underlying conformal field theory and it is unnecessary when one adopts the phase picture. The mirror of the Z-orbifold need only be described as an orbifold of a Landau-Ginzburg theory in $\C^9$.
We should add that the construction of [@Set:sup] should be expected to overcome the renormalization group flow problem inherent in the above hypersurface in $\P^8$ of [@Drk:Z; @Schg:gen]. In the construction of [@Set:sup] one adds ghost fields to reduce the effective dimension of the target space back down to that of $d$. Assuming this is the case, this target space with ghosts can be proposed as a good geometric interpretation of the conformal field theory. It should be pointed out however that such geometric interpretations are probably highly ambiguous. That is, one conformal field theory can be given many interpretations. This occurs in [@Set:sup] where constructions of K3 conformal field theories are given in terms of a 4 complex dimensional space with ghosts whereas the complete moduli space is already understood completely in terms of K3 surfaces [@AM:K3p]. In fact, it is probable that any geometric model may be blown-up to give $K<0$ and then nonzero contributions the the $\beta$-function be cancelled by adding suitable extra fields. Since there are an infinite number of such blow-ups for any model there is the possibility of ascribing an infinity of geometric interpretations of this form.
Finally note that it might be possible to associate some geometry with the case discussed in this section by considering orbifolds with discrete torsion [@Berg:dt]. Since we do not understand precisely how to relate quotient singularities with discrete torsion to classical singularities we will not discuss this interpretation here.
A Case with $h^{1,1}=1$ {#ss:h1}
-----------------------
The above example may be considered rather trivial in that our phase space was zero dimensional, i.e., consisted of only one point. Let us now give a less trivial example which still has no phase.
Consider dividing $\C^9$ by the group $\Z_4\times\Z_4$ groups generated by $$\eqalign{g_1&=(\ff14,\ff14,\ff12,\ff14,\ff14,\ff12,\ff14,\ff14,\ff12)\cr
g_2&=(\ff14,\ff34,0,0,0,0,0,0,0).\cr}
\label{eq:gZ4}$$ Using the arguments of [@GP:orb; @AL:geom] one may show that the theory in this space is the mirror of the orbifold $T^3/(\Z_4\times\Z_2)$ where $T$ is a complex torus, the $\Z_4$ group is generated by $(z_1,z_2,z_3)\mapsto(iz_1,-iz_2,z_3)$ and $\Z_2$ by $(z_1,z_2,z_3)\mapsto(z_1,-z_2,-z_3)$, where $z_i$ are the complex coordinates on the tori. The $(2,1)$-form $dz_1\wedge dz_2\wedge d\bar
z_3$ is invariant under this group. Indeed $h^{2,1}$ for this orbifold is equal to 1. Thus we expect the case in question to have $h^{1,1}=1$.
The point set $\cA$ corresponding to such a space is given by $n=9$ and $$\eqalignsq{
\alpha_1&=(4,-3,0,0,0,0,0,0,0)\cr
\alpha_2&=(0,1,0,0,0,0,0,0,0)\cr
\alpha_3&=(0,0,1,0,0,0,0,0,0)\cr
\alpha_4&=(0,0,0,1,0,0,0,0,0)\cr
\alpha_5&=(0,0,0,0,1,0,0,0,0)\cr
\alpha_6&=(0,0,0,0,0,1,0,0,0)\cr
\alpha_7&=(0,0,0,0,0,0,1,0,0)\cr
\alpha_8&=(-4,2,-2,-1,-1,-2,-1,4,-2)\cr
\alpha_9&=(0,0,0,0,0,0,0,0,1)\cr
\alpha_{10}&=(3,-2,0,0,0,0,0,0,0)\cr
\alpha_{11}&=(2,-1,0,0,0,0,0,0,0)\cr
\alpha_{12}&=(1,0,0,0,0,0,0,0,0),\cr} \label{eq:Ao}$$ with $\mu=(1,1,1,1,1,1,1,3,1)$ and $\nu=(0,0,0,0,0,0,0,1,0)$. Therefore this theory has $d=3$ again.
The points $\alpha_1,\ldots,\alpha_9$ form a simplex with $\alpha_{10},\alpha_{11},\alpha_{12}$ positioned along the edge joining $\alpha_1$ and $\alpha_2$. Thus all the interesting part of the point set as regards triangulations is contained in this line $\alpha_1\alpha_2$: $$\setlength{\unitlength}{0.007in}\begin{picture}(330,30)(155,635)
\thinlines
\put(400,660){\circle{10}}
\put(160,660){\circle*{10}}
\put(480,660){\circle*{10}}
\put(320,660){\circle{10}}
\put(240,660){\circle{10}}
\put(160,660){\line( 1, 0){320}}
\put(315,635){\makebox(0,0)[lb]{$\alpha_{11}$}}
\put(155,635){\makebox(0,0)[lb]{$\alpha_1$}}
\put(235,635){\makebox(0,0)[lb]{$\alpha_{10}$}}
\put(395,635){\makebox(0,0)[lb]{$\alpha_{12}$}}
\put(475,635){\makebox(0,0)[lb]{$\alpha_2$}}
\end{picture}$$ The points $\alpha_{10},\alpha_{11},\alpha_{12}$ may, or may not be included in the triangulation (and are hence shown as circles rather than dots).
If none of the points $\alpha_{10},\alpha_{11},\alpha_{12}$ are included in the triangulation, we have one simplex with vertices $\alpha_1,\ldots,\alpha_9$ and the associated toric variety is $\C^9/(\Z_4\times\Z_4)$ as expected. If all these points are included in the triangulation we have 4 simplices. The resulting space is a partial resolution of the $\C^9/(\Z_4\times\Z_4)$ space. The exceptional divisor introduced is a “plumb product” of three $\P^1$ spaces. Each of the points $\alpha_{10},\alpha_{11},\alpha_{12}$ may be taken to correspond to one of these $\P^1$ components. This is shown in figure \[fig:Z4b\]. The black dot on the left hand side shows the isolated singularity. On the right hand side the singularity (which is now terminal) covers the whole exceptional divisor. Clearly, other triangulations represent intermediate steps in this blow-up.
Let us now analyze the critical point set of $W$. Finding $\cB$ we determine from (\[eq:mon\]) that $$W = a_1x_1^4x_{10}^3x_{11}^2x_{12}+a_2x_2^4x_{10}x_{11}^2x_{12}^3+
a_3x_3^2+a_4x_4^4+a_5x_5^4+a_6x_6^2+a_7x_7^4+a_8x_8^4+a_9x_9^2+
\ldots \label{eq:WZ4}$$ In total there are 87 points in $\cB$ but we need only consider the above terms with nonzero $a_1,\ldots,a_9$ for a sufficiently generic $W$.
Consider the maximal triangulation. This includes all three points $\alpha_{10},\alpha_{11},\alpha_{12}$. Since $N=12$ we need to remove the set $F_\Delta$ given by (\[eq:Fset\]) from $\C_{12}$. This amounts in imposing $x_2x_{11}x_{12}\neq0$ or $x_1x_2x_{12}\neq0$ or $x_1x_2x_{10}\neq0$ or $x_1x_{10}x_{11}\neq0$. We also wish to impose $\partial W/\partial x_i=0$ for $i=1,\ldots,12$. It is straight-forward to show that these conditions require $$\eqalign{x_3=x_4=x_5=x_6=x_7&=x_8=x_9=x_{11}=0\cr
x_1&\neq0\cr
x_2&\neq0\cr}$$ and that $x_{10}$ and $x_{12}$ cannot both be zero simultaneously. As $N-n=3$ we have three $\C^*$ actions to divide this subspace of $\C^{12}$ by. Two may be used to fix $x_1$ and $x_2$ to specific values. The other $\C^*$ may be used to turn $x_{10}$ and $x_{12}$ into homogeneous coordinates parametrizing $\P^1$. The vacuum is thus $\P^1$. One may also determine the superpotential in this vacuum to show that we have a theory fibered over this $\P^1$ to obtain the familiar hybrid-type models of [@W:phase; @AGM:II]. One may also show that the fiber has a $\Z_4$-quotient singularity at the zero section.
In terms of the ambient toric variety $V_\Delta$, what we have just described in the previous paragraph is the $\P^1$ that appears in the middle of the chain of three $\P^1$’s on the right in figure \[fig:Z4b\]. Thus although $\Vbig$ appears to have three degrees of freedom for the Kähler form — giving the three independent sizes of the three $\P^1$’s, only one makes it down to $X$, the critical point set of $W$. Therefore $X$ only has one Kähler-type deformation. Sometimes additional modes appear in the fibre for these hybrid models but in this case the fibre contains no twist fields with the correct charges to be considered a (1,1)-form. We will therefore assert that $h^{1,1}(X)=1$. Thus we are in agreement with the assertions concerning the mirror space at the start of this section.
Analyzing the other possible triangulations we find that we reproduce one of the two phases we know about — either the orbifold in $\C^9/(\Z_4\times\Z_4)$ or the hybrid model over $\P^1$. The points $\alpha_{10}$ and $\alpha_{12}$ may be ignored when considering $X$. Thus we have constructed a model with a non-trivial phase diagram — there are 2 phases — but neither is a space.
In general there is a homomorphism: $$\kappa:H^{1,1}(V_\Delta)\to H^{1,1}(X).
\label{eq:kappa}$$ In general however $\kappa$ is neither injective nor surjective. The example in this section shows a failure of injectivity since $h^{1,1}(V_\Delta)=3$ and $h^{1,1}(X)=1$. In the more simple case of $\langle\mu,\nu\rangle=1$ it was shown in [@AGM:mdmm] that the kernel of $\kappa$ was described by points in the interior of co-dimension one faces of the convex hull of $\cA$. In the case described in this section we see that such a simple criterion cannot be used — all the points $\alpha_{10},\alpha_{11},\alpha_{12}$ lie in a co-dimension 7 face and yet $\alpha_{10}$ and $\alpha_{12}$ contribute to the kernel and $\alpha_{11}$ survives through to $H^{1,1}(X)$. At this point in we know of no simple method of determining the image of $\kappa$ except to explicitly calculate the critical point set of $W$ on a case by case basis.
Let us conclude this section by discussing the short-comings of analyzing this model in terms of the “generalized manifolds” of [@Schg:gen]. The $\Z_4$ singularity in $\C^9$ generated by $g_1$ of (\[eq:gZ4\]) may be partially resolved by a line bundle over the weighted projective space $\P^8_{\{1,1,2,1,1,2,1,1,2\}}$. The resultant space has $K<0$. The “generalized manifold”, $R$, would be identified as the hypersurface in this weighted projective space given by the vanishing of (\[eq:WZ4\]) with $x_1,\ldots,x_9$ taken to be the quasi-homogeneous coordinates and $x_{10}=x_{11}=x_{12}=1$. The $\Z_4$-action of $g_2$ acts on $R$ to induce $\Z_4$-quotient singularities over some subspace of codimension two. These latter singularities are not terminal and may be resolved without adding anything further to $K$. In fact, resolving these latter singularities may be achieved by introducing the points $\alpha_{10},\alpha_{11},\alpha_{12}$ into the toric fan.
It is easy to see that something similar will happen in general. That is, any $K=0$ toric resolutions we may perform in $\Vbig$ may also be performed after blowing up any terminal singularities in $\Vbig$. It follows that the points in the interior of the convex hull of $\cA$ may be counted by analyzing singularities which may be locally resolved with $K=0$ in $R$. (Note that this is a rather inefficient way of proceeding in our picture — one may as well just analyze the singularities in $\Vbig$ without any destroying the $K=0$ condition.) This observation sheds light on a conjecture in [@Schg:fno] that $h^{1,1}(X)$ could be determined by counting the contribution to $h^{1,1}$ of any resolutions of singularities within the $R$. We see now that this will count $h^{1,1}(V_\Delta)$ which is, in general, not equal to $h^{1,1}(X)$. Thus this conjecture is false. In the example above, counting this way would imply that $h^{1,1}(X)=3$.
With regards to determining $h^{1,1}(X)$, it appears hard to save the construction of [@Set:sup] from a similar fate. The problem is that the divisors associated with $\alpha_{10}$, $\alpha_{11}$ and $\alpha_{12}$ appear on equal footing in $R$. Thus unless some unsymmetric rules are devised for resolving canonical singularities in superspace one cannot obtain the correct answer $h^{1,1}(X)=1$.
A Case with $X=\hbox{\bigbbbfont P}^3$ {#ss:P3}
--------------------------------------
One might be led to suspect the following to be the general picture for the geometric interpretation of an $N$=2 superconformal field theory. Either $X$ is a space and the string is free to move within $X$ and there are no massless modes normal to $X$, or $X$ is a space of dimension $<d$ in which the string is free to move and there are massless modes governed by some superpotential normal to $X$ inside some bigger ambient space $\Vbig$ containing $X$. We now show an example (which also appeared in [@W:phase]) which is an exception to this.
Consider the following point set for $\cA\,$: $$\eqalignsq{
\alpha_1&=(1,0,0,0,0,0,0,1,0,0,0)\cr
\alpha_2&=(0,1,0,0,0,0,0,1,0,0,0)\cr
\alpha_3&=(0,0,1,0,0,0,0,0,1,0,0)\cr
\alpha_4&=(0,0,0,1,0,0,0,0,1,0,0)\cr
\alpha_5&=(0,0,0,0,1,0,0,0,0,1,0)\cr
\alpha_6&=(0,0,0,0,0,1,0,0,0,1,0)\cr
\alpha_7&=(0,0,0,0,0,0,1,0,0,0,1)\cr
\alpha_8&=(-1,-1,-1,-1,-1,-1,-1,0,0,0,1)\cr
\alpha_9&=(0,0,0,0,0,0,0,1,0,0,0)\cr
\alpha_{10}&=(0,0,0,0,0,0,0,0,1,0,0)\cr
\alpha_{11}&=(0,0,0,0,0,0,0,0,0,1,0)\cr
\alpha_{12}&=(0,0,0,0,0,0,0,0,0,0,1).\cr}
\label{eq:AP3}$$ Thus $\mu=(0,0,0,0,0,0,0,1,1,1,1)$. One can also determine $\cB$ with a little effort and find $\nu=(0,0,0,0,0,0,0,1,1,1,1)$. Thus $d=3$ again. The superpotential $W$ may be written $$W = x_9G_1 + x_{10}G_2 + x_{11}G_3 + x_{12}G_4,$$ where the $G_k$ are generic homogeneous polynomials of total degree two in $x_1,\ldots,x_8$.
There are two triangulations of the point set $\cA$. The first consists of taking 8 simplices each of which has $\alpha_9,\ldots,\alpha_{12}$ as 4 of its vertices with the other 7 vertices taken from the set $\{\alpha_1,\ldots,\alpha_8\}$. In terms of $F_\Delta$ this amounts to removing the point $x_1=\ldots=x_8=0$ from consideration. Restricting to the critical point set of $W$ forces $x_9=\ldots=x_{12}=0$ and $G_1=\ldots=G_4=0$. Dividing out by the single required $\C^*$-action forms $\P^7$ with homogeneous coordinates $x_1,\ldots,x_8$. Thus $X$ is the intersection of 4 quadric equations $G_k=0$ in $\P^7$. This is a known space dating back to [@CHSW:].
The other triangulation consists of 4 simplices with 8 vertices given by $\alpha_1,\ldots,\alpha_8$ with the other 3 taken from the set $\{\alpha_9,\ldots,\alpha_{12}\}$. This amounts to removing $x_9=\ldots=x_{12}=0$ from consideration. Restricting to the critical point set of $W$ forces $x_1=\ldots=x_8=0$. Now the $\C^*$-action may be used to form $\P^3$ with homogeneous coordinates $x_9,\ldots,x_{12}$. Thus $X$ in this phase is $\P^3$.
Our phase diagram consists of two phases — both of which have the dimension of $X$ equal to $d$. One phase is a manifold with $h^{1,1}=1$ and $h^{2,1}=65$ which we understand. The other phase is $\P^3$. The reader might be alarmed at the appearance of the latter since $\P^3$ is not a space and lacks a nonvanishing holomorphic 3-form for example. The resolution is as follows. Although we have correctly identified the vacuum of the field theory as $X$ we have to be a little careful in declaring it to be the effective target space of a conformal field theory. Let us consider the variables $\alpha_1,\ldots,\alpha_8$ which we forced to zero. The superpotential is quadratic in these variables so we certainly haven’t missed any massless degrees of freedom (which would only add to our troubles by increasing $d$ anyway). The point is that there is actually a $\Z_2$-quotient singularity coming from the identification of homogeneous coordinates in $\Vbig$ to affine ones. Thus we have a fibration of a orbifold theory over $\P^3$ which may appear trivial in that the superpotential is quadratic but we may expect twist fields is add to our spectrum. In particular we expect to have an analog to a Calabi-Yau $H^{3,0}$ mode (i.e. a field of charge (3,0) under the $U(1)_L \times U(1)_R$ of the superconformal algebra) coming from such twisted sectors. Of course, such a mode cannot be given a literal geometric interpretation in terms of a (3,0)-form.
It is interesting also to ask how literally we can take this $\P^3$ to be a target space for the conformal field theory. To find the actual size of truly conformally invariant target space one needs to solve the Picard-Fuchs system as described in [@AGM:sd]. We will not present the details here since they are rather lengthy but we may quickly summarize as follows. One solves equations (42) of [@AGM:sd] where the “$\beta$” vector of this system is set equal to $-\nu$. (One could then count rational curves on this space if one so desired.) The complexified Kähler form $B+iJ$ of the phase can then be analytically continued into $\P^3$ phase (which is most easily done by the method of [@me:min-d]). Taking $z$ to be the local coordinate on the moduli space where $z=0$ corresponds to the limit point in the $\P^3$ phase we obtain $$B+iJ=-\ff12 -\frac{3\pi i}{2\log(z)} +O(\log(z)^{-2}).$$ Thus $J\geq 0$ in the region near the limit point $|z|\ll1$. The effective size of target space is very small as $|z|\to0$. In other words, the effect of integrating out the massive modes in the linear of [@W:phase] has caused an infinite renormalization of the “$r$” parameter (unlike what is believed to happen for the phase).
To summarize we see that the phase picture can produce phases with dimension equal to $d$ which do not correspond to non-linear s. To understand these phases more completely will require a better understanding of the hybrid models.
Conclusions {#s:conc}
===========
Geometrical methods have proven themselves to be a powerful conceptual and calculational tool in understanding the physical content of certain conformal theories and their associated string models. As such, it is a worthwhile task to gain as complete an understanding as possible of the geometrical status of conformal field theories, especially for the case of $N = 2$ worldsheet supersymmetry relevant for spacetime supersymmetric string models. The phase structure of such $N = 2$ models, as found in [@W:phase; @AGM:II], goes a long way towards capturing the full geometric content of these theories, and, in particular, certainly provides the correct framework for discussing mirror symmetry. In this paper we have used this phase structure analysis to address certain previously puzzling issues regarding the geometrical content of certain theories. In particular, at first sight mirror symmetry seems to come upon the puzzle regarding the identity of the mirror of a rigid manifold. We have seen, though, that this appears to be a puzzle only because the question itself is not phrased in the correct context. That is, mirror symmetry tells us that certain [*a priori*]{} distinct pairs of families of conformal theories actually are composed of isomorphic members. When the phase structure of each family in such a pair contains a Calabi-Yau region, then these Calabi-Yau’s form a mirror pair. However, in certain cases, at least one of the families does [*not*]{} have a Calabi-Yau region. In such cases mirror symmetry will simply not yield a mirror pair of Calabi-Yau manifolds. A family which has a rigid Calabi-Yau phase, as we have seen, provides one such example — the mirror family does not have a Calabi-Yau region. Thus the absence of a Kähler form for the mirror is not an issue. The mirror moduli space has no Calabi-Yau phase and hence does not require a Kähler form. The previous puzzle disappears, therefore, when the question is phrased in the correct context.
Beyond the rigid case, we have also seen that even when there is a conformal mode that can play the part of a Kähler form, there need not be a Calabi-Yau phase on which it can realize this potential. So, whereas in the previous problem we resolved the issue of a “Calabi-Yau in search of a Kähler form” here we have “a Kähler form in search of a Calabi-Yau”. We have established that there are examples in which there simply is no Calabi-Yau to be found.
It is worth noting that the map $\kappa$ in (\[eq:kappa\]) is neither injective or surjective. We saw the effect on this map not being injective in section \[ss:h1\]. The failure of surjectivity shows that some (1,1)-forms on $X$ do not come from the toric ambient space $\Vbig$. In the case of models of the form discussed in section \[ss:hyp\] it is still possible to count the number $h^{1,1}(X)$ because of the properties of hypersurfaces [@Bat:m; @AGM:mdmm]. In the cases considered here however we deal with more general complete intersections. The problem of counting $h^{1,1}(X)$ in this context is the mirror of the problem of counting $h^{2,1}$ for the mirror model. The analogue of the fact that $\kappa$ is not an isomorphism is the fact that deforming the polynomial giving the superpotential is not the same as deformation the complex structure. It would be interesting to see if methods along the lines of [@GH:poly] could be applied in this context to determine the Hodge numbers of $X$.
An interesting question, to which we do not know the answer, is whether there are examples in which [*neither*]{} family in a mirror pair has a Calabi-Yau region. Such an example would establish that there are $N = 2, c = 3d$ conformal theories which are not interpretable in terms of Calabi-Yau compactifications (or analytic continuations thereof). To answer this question is difficult because of the failure of the surjectivity of $\kappa$. One may find mirror pairs of orbifolds of the Gepner model, ${\bf 1}^9$, for which neither has an obvious interpretation. An example with $h^{1,1}$ and $h^{2,1}$ equal to 4 and 40 was mentioned in [@AL:geom]. This is a good candidate for a situation where neither of the mirror partners have a phase (despite the assertions of [@AL:geom]). Unfortunately if $X$ is the model with $h^{1,1}=4$ then the image of $\kappa$ is trivial, i.e., none of the (1,1)-forms come from $\Vbig$. Because of this the methods of this paper cannot be used to draw any conclusions regarding the lack of phase for this example.
Acknowledgements {#acknowledgements .unnumbered}
================
It is a pleasure to thank D. Morrison and R. Plesser for useful conversations. The work of P.S.A. is supported by a grant from the National Science Foundation. The work of B.R.G. is supported by a National Young Investigator Award, the Alfred P. Sloan Foundation and the National Science Foundation.
Note Added {#note-added .unnumbered}
==========
The mirror of the example of section \[ss:P3\] was studied in [@lots:per] where it was discovered that there is an extra $\Z_2$ symmetry in the moduli space. This should act on the moduli space in section \[ss:P3\] to identify the two phases with each other. This may be viewed as a new kind of $R\leftrightarrow1/R$ symmetry.
[10]{}
E. Witten, , Nucl. Phys. [**B403**]{} (1993) 159–222.
P. S. Aspinwall, B. R. Greene, and D. R. Morrison, , Phys. Lett. [**303B**]{} (1993) 249–259.
P. S. Aspinwall, B. R. Greene, and D. R. Morrison, , Nucl. Phys. [**B416**]{} (1994) 414–480.
P. Candelas, E. Derrick, and L. Parkes, , Nucl.Phys. [**B407**]{} (1993) 115–154.
R. Schimmrigk, , Phys. Rev. Lett. [**70**]{} (1993) 3688–3691.
S. Sethi, , Nucl. Phys. [**B430**]{} (1994) 31–50.
P. S. Aspinwall, B. R. Greene, and D. R. Morrison, , Nucl. Phys. [**B420**]{} (1994) 184–242.
F. C. Kirwan, , Number 31 in Mathematical Notes, Princeton University Press, Princeton, 1984.
L. Ness, , Amer. J. Math. [**106**]{} (1984) 1281–3129.
A. D’Adda, P. D. Vecchia, and M. L[ü]{}scher, , Nucl. Phys. [**B152**]{} (1979) 125–144.
P. Griffiths and J. Harris, , Wiley-Interscience, 1978.
D. A. Cox, , Amherst 1992 preprint, alg-geom/9210008, to appear in J. Algebraic Geom.
M. Reid, , in S. Iitaka, editor, “Algebraic Varieties and Analytic Varieties”, volume 1 of Adv. Studies in Pure Math., pages 131–180, Kinokuniya, Tokyo, 1983.
C. Vafa and N. Warner, , Phys. Lett [**218B**]{} (1989) 51–58.
V. V. Batyrev and L. A. Borisov, , Essen and Michigan 1994 preprint, alg-geom/9402002.
P. S. Aspinwall, B. R. Greene, and D. R. Morrison, , Internat. Math. Res. Notices [**1993**]{} 319–338.
S. Katz and D. Morrison, , to appear.
B. R. Greene and M. R. Plesser, , Nucl. Phys. [**B338**]{} (1990) 15–37.
V. V. Batyrev, , J. Alg. Geom. [**3**]{} (1994) 493–535.
L. Borisov, , Michigan 1993 preprint, alg-geom/9310001.
M. Reid, , Proc. Symp. Pure Math. [**46**]{} (1987) 345–414.
P. Candelas, G. Horowitz, A. Strominger, and E. Witten, , Nucl. Phys. [**B258**]{} (1985) 46–74.
P. S. Aspinwall and C. A. L[ü]{}tken, , Nucl. Phys. [**B353**]{} (1991) 427–461.
D. Gepner, , Phys. Lett. [**199B**]{} (1987) 380–388.
P. S. Aspinwall and D. R. Morrison, , Duke and IAS 1994 preprint DUK-TH-94-68, IASSNS-HEP-94/23, hep-th/9404151, to appear in “Essays on Mirror Manifolds 2”.
P. Berglund, , Phys. Lett. [**319B**]{} (1993) 117–124.
R. Schimmrigk, , Bonn 1994 preprint BONN-TH-94-007, hep-th/9405086.
P. S. Aspinwall, , Nucl. Phys. [**B431**]{} (1994) 78–96.
P. Green and T. H[ü]{}bsch, , Commun. Math. Phys. [**113**]{} (1987) 505–528.
P. Berglund et al., , Nucl. Phys. [**B419**]{} (1994) 352–403.
[^1]: This notation is not entirely consistent with [@AGM:II]. For example the $\Delta$ of this paper is the $\Delta^+$ of [@AGM:II].
[^2]: This has also been studied independently by S. Katz and D. Morrison [@KM:mir].
[^3]: Except that there appears to be a typographical error in conjecture (2.17) of [@BB:mir].
[^4]: This is because Gorenstein singularities can only be terminal in more than 3 dimensions [@Reid:mm].
[^5]: Note that the periods deduced in [@Drk:Z] can be determined from the analysis of the Picard-Fuchs equation as we mention briefly later.
| ArXiv |
---
abstract: 'The growing amount of intermittent renewables in power generation creates challenges for real-time matching of supply and demand in the power grid. Emerging ancillary power markets provide new incentives to consumers (e.g., electrical vehicles, data centers, and others) to perform demand response to help stabilize the electricity grid. A promising class of potential demand response providers includes energy storage systems (ESSs). This paper evaluates the benefits of using various types of novel ESS technologies for a variety of emerging smart grid demand response programs, such as regulation services reserves (RSRs), contingency reserves, and peak shaving. We model, formulate and solve optimization problems to maximize the net profit of ESSs in providing each demand response. Our solution selects the optimal power and energy capacities of the ESS, determines the optimal reserve value to provide as well as the ESS real-time operational policy for program participation. Our results highlight that applying ultra-capacitors and flywheels in RSR has the potential to be up to 30 times more profitable than using common battery technologies such as LI and LA batteries for peak shaving.'
author:
-
title: |
Optimizing Energy Storage Participation\
in Emerging Power Markets
---
Acknowledgment {#acknowledgment .unnumbered}
==============
This paper is supported by the NSF Grant 1464388.
| ArXiv |
---
abstract: 'I present the results of first principles calculations of the electronic structure and magnetic interactions for the recently discovered superconductor YFe$_2$Ge$_2$ and use them to identify the nature of superconductivity and quantum criticality in this compound. I find that the Fe $3d$ derived states near the Fermi level show a rich structure with the presence of both linearly dispersive and heavy bands. The Fermi surface exhibits nesting between hole and electron sheets that manifests as a peak in the susceptibility at $(1/2,1/2)$. I propose that the superconductivity in this compound is mediated by antiferromagnetic spin fluctuations associated with this peak resulting in a $s_\pm$ state similar to the previously discovered iron-based superconductors. I also find that various magnetic orderings are almost degenerate in energy, which indicates that the proximity to quantum criticality is due to competing magnetic interactions.'
author:
- Alaska Subedi
title: 'Unconventional sign-changing superconductivity near quantum criticality in YFe$_2$Ge$_2$'
---
Unconventional superconductivity and quantum criticality are two of the most intriguing phenomena observed in physics. The underlying mechanisms and the properties exhibited by the systems in which these two phenomena occur has not been fully elucidated because unconventional superconductors and materials at quantum critical point are so rare. The dearth of realizable examples has also held back the study of the relationship and interplay between unconventional superconductivity and quantum criticality, if there are any.
Therefore, the recent report of non-Fermi liquid behavior and superconductivity in YFe$_2$Ge$_2$ by Zou *et al.* is of great interest despite a low superconducting $T_c$ of $\sim$1.8 K.[@zou13] This material is also interesting because it shares some important features with the previously discovered iron-based high-temperature superconductors. Like the other iron-based superconductors, its structural motif is a square plane of Fe that is tetrahedrally coordinated, in this case, by Ge. This Fe$_2$Ge$_2$ layer is stacked along the $z$ axis with an alternating layer of Y ions. The resulting body-centered tetragonal structure ($I4/mmm$) of this compound is the same as that of the ‘122’ family of the iron-based superconductors. The nearest neighbor Fe–Ge and Fe–Fe distances of 2.393 and 2.801 Å, respectively, in this compound[@vent96] are similar to the Fe–As and Fe–Fe distances of 2.403 and 2.802 Å, respectively, found in BaFe$_2$As$_2$.[@rott08] This raises the possibility that the direct Fe–Fe hopping is important to the physics of this material, which is the case for the previously discovered iron-based superconductors.[@sing08a]
Furthermore, Zou *et al.* report that the superconductivity in this compound exists in the vicinity of a quantum critical point that is possibly associated with antiferromagnetic spin fluctuations.[@zou13] A related isoelectronic compound LuFe$_2$Ge$_2$ that occurs in the same crystal structure exhibits antiferromagnetic spin density wave order below 9 K,[@avil04; @fers06] and the magnetic transition is continuously suppressed in Lu$_{1-x}$Y$_x$Fe$_2$Ge$_2$ series as Y content is increased, with the quantum critical point lying near the composition Lu$_{0.81}$Y$_{0.19}$Fe$_2$Ge$_2$.[@ran11] The proximity of YFe$_2$Ge$_2$ to quantum criticality is observed in the non-Fermi liquid behavior of the specific-heat capacity and resistivity. Zou *et al.* find that the unusually high Sommerfeld coefficient with a value of $C/T \simeq 90$ mJ/mol K$^2$ at 2 K further increases to a value of $\sim$100 mJ/mol K$^2$ as the temperature is lowered, although the experimental data is not detailed enough to distinguish between a logarithmic and a square root increase. They also find that the resistivity shows a behavior $\rho \propto T^{3/2}$ up to a temperature of 10 K.
In this paper, I use the results of first principles calculations to discuss the interplay between superconductivity and quantum criticality in YFe$_2$Ge$_2$ in terms of its electronic structure and competing magnetic interactions. I find that the fermiology in this compound is dominated by Fe $3d$ states with the presence of both heavy and linearly dispersive bands near the Fermi level. The Fermi surface consists of five sheets. There is an open tetragonal electron cylinder around $X = (1/2, 1/2, 0)$. A large three dimensional closed sheet that is shaped like a shell of a clam is situated around $Z =
(0, 0, 1/2) = (1, 0, 0)$. This sheet encloses a cylindrical and two almost spherical hole sheets. The tetragonal cylinder sheet around $X$ nests with the spherical and the cylindrical sheets around $Z$, which manifests as a peak at $(1/2,1/2)$ in the bare susceptiblity. I propose that the superconductivity in this compound is mediated by antiferromagnetic spin fluctuations associated with this peak, and the resulting superconductivity has a sign-changing $s_\pm$ symmetry with opposite signs on the nested sheets around $X$ and $Z$. This superconductivity is similar to the one proposed for previously discovered iron-based superconductors.[@mazi08; @kuro08] Furthermore, I find that there are competing magnetic interactions in this compound, and the quantum criticality is due to the fluctuations associated with these magnetic interactions.
The results presented here were obtained within the local density approximation (LDA) using the general full-potential linearized augmented planewave method as implemented in the WIEN2k software package.[@wien2k] Muffin-tin radii of 2.4, 2.2, and 2.1 a.u. for Y, Fe, and Ge, respectively, were used. A $24 \times 24 \times 24$ $k$-point grid was used to perform the Brillouin zone integration in the self-consistent calculations. An equivalently sized or larger grid was used for supercell calcualtions. Some magnetic calculations were also checked with the ELK software package.[@elk] I used the experimental parameters ($a$ = 3.9617 and $c$ = 10.421 Å),[@vent96] but employed the internal coordinate for Ge ${z_{\textrm{Ge}}}$ = 0.3661 obtained via non-spin-polarized energy minimization. The calculated value for ${z_{\textrm{Ge}}}$ is different from the experimentally determined value of ${z_{\textrm{Ge}}}$ = 0.3789. The difference in the Ge height between the calculated and experimental structures is 0.13 Å, which is larger than the typical LDA error in predicting the crystal structure. Such a discrepancy is also found in the iron-based superconductors.[@sing08a] This may suggest that YFe$_2$Ge$_2$ shares some of the underlying physics with the previously discovered iron-based superconductors.
![ Top: LDA non-spin-polarized band structure of YFe$_2$Ge$_2$. Bottom: A blow-up of the band structure around Fermi level. The long $\Gamma$–$Z$ direction is from $(0,0,0)$ to $(1,0,0)$ and the short one is from $(0,0,0)$ to $(0, 0, 1/2)$. The $X$ point is $(1/2,1/2,0)$. The stacking of the Brillouin zone is such that $(1,0,0) = (0,0,1/2)$. See Fig. 1 of Ref. for a particularly illuminating illustration of the reciprocal-space structure. []{data-label="fig:yfg-bnd"}](yfg-bnd-l.ps "fig:"){width="\columnwidth"}\
![ Top: LDA non-spin-polarized band structure of YFe$_2$Ge$_2$. Bottom: A blow-up of the band structure around Fermi level. The long $\Gamma$–$Z$ direction is from $(0,0,0)$ to $(1,0,0)$ and the short one is from $(0,0,0)$ to $(0, 0, 1/2)$. The $X$ point is $(1/2,1/2,0)$. The stacking of the Brillouin zone is such that $(1,0,0) = (0,0,1/2)$. See Fig. 1 of Ref. for a particularly illuminating illustration of the reciprocal-space structure. []{data-label="fig:yfg-bnd"}](yfg-bnd-s.ps "fig:"){width="\columnwidth"}
![ (Color online) Electronic density of states non-spin-polarized of YFe$_2$Ge$_2$ and projections on to the LAPW spheres per formula unit both spin basis. []{data-label="fig:yfg-dos"}](yfg-dos.eps){width="\columnwidth"}
The non-spin-polarized LDA band structure and density of states (DOS) are shown in Figs. \[fig:yfg-bnd\] and \[fig:yfg-dos\], respectively. The lowest band that starts out from $\Gamma$ at $-$5.2 eV relative to the Fermi energy has Ge $4p_z$ character. There is only one band with Ge $4p_z$ character below Fermi level, and there is another band with this character above the Fermi level. This indicates that the Ge ions make covalent bonds along the $c$ axis, which is not surprising given the short Ge–Ge distance in that direction. The four bands between $-$1.2 and $-$4.8 eV that start out from $\Gamma$ at $-$1.5 and $-$2.6 eV have Ge $4p_x$ and $4p_y$ character. Rest of the bands below the Fermi level have mostly Fe $3d$ character. Similar to the other iron-based superconductors,[@sing08a] there is no gap-like structure among the Fe $3d$ bands splitting them into a lower lying $e_g$ and higher lying $t_{2g}$ states. This shows that Fe–Ge covalency is minimal and direct Fe–Fe interactions dominate. Almost all of the Fe $4s$ and Y $4d$ and $5s$ character lie above the Fermi level. This indicates a nominal occupation of Fe $3d^{6.5}$, although the actual occupancy will be different because there is some covalency of Fe $3d$ states with Y $4d$ and Ge $4p$ states.
The electronic states near the Fermi level come from Fe $3d$ derived bands and show a rich structure. The electronic DOS at the Fermi level is $N(E_F) = 4.50$ eV$^{-1}$ on a per formula unit both spin basis corresponding to a calculated Sommerfeld coefficient of 10.63 mJ/mol K$^2$. The Fermi level lies at the bottom of a valley with a large peak due to bands of mostly $d_{xz}$ and $d_{yz}$ characters on the left and a small peak due to a band of mostly $d_{xy}$ character on the right. (The local coordinate system of the Fe site is rotated by 45$^\circ$ in the $xy$ plane with respect to the global cartesian axes such that the Fe $d_{x^2-y^2}$ orbital points away from the Ge $p_x$ and $p_y$ orbitals.) There is a pair of linearly dispersive band with mostly $d_{xz}$ and $d_{yz}$ as well as noticeable Ge $p_z$ characters either side of $Z$. If they are not gapped in the superconducting state, they will provide the system with a massless excitation. In addition to this pair of linearly dispersive bands, there is also a very flat band near the Fermi level along $X$–$\Gamma$. This band has an electron-like nature around $X$ and crosses the Fermi level close to it. Along the $X$–$\Gamma$ direction, it reaches a maximum at 0.08 eV above the Fermi level, turns back down coming within 0.01 eV of touching the Fermi level, and again moves away from the Fermi level. It may be possible to access these band critical points that have vanishing quasiparticle velocities via small perturbations due to impurities, doping, or changes in structural parameters. The role of such band critical points in quantum criticality has been emphasized recently,[@neal11] and similar physics may be relevant in this system.
![(Color online) Top: LDA Fermi surface of YFe$_2$Ge$_2$. Bottom: The Fermi surface without the large sheet. The shading is by velocity.[]{data-label="fig:yfg-fs"}](yfg-fs1v2 "fig:"){width="0.8\columnwidth"} ![(Color online) Top: LDA Fermi surface of YFe$_2$Ge$_2$. Bottom: The Fermi surface without the large sheet. The shading is by velocity.[]{data-label="fig:yfg-fs"}](yfg-fs2v2 "fig:"){width="0.8\columnwidth"}
The Fermi surface of this compound is shown in Fig. \[fig:yfg-fs\]. There is an open very two dimensional tetragonal electron cylinder around $X$. This has mostly $d_{xz}$ and $d_{yz}$ character. There are four closed sheets around $Z$. One of them is a large three dimensional sheet with the shape like the shell of a clam with $d_{xz}$, $d_{yz}$, $d_{xy}$, and $d_{z^2}$ characters. There are two almost spherical hole sheets. These have mostly $d_{xz}$ and $d_{yz}$ characters, with the smaller one also containing noticeable Ge $p_z$ character. These two spherical sheets are enclosed by a closed cylindrical hole sheet that has mostly $d_{xy}$ character.
The cylindrical and larger spherical sheets centered around $Z$ touch at isolated points. Otherwise, the Fermi surface is comprised of disconnected sheets. If one considers the $\Gamma$–$Z$–$\Gamma$ path along the $k_z$ direction, there is a series of box-shaped cylindrical hole sheet that encloses the two spherical sheets. Although there are no sections around $\Gamma$, these sheets around $Z$ enclose almost two-third of the $\Gamma$–$Z$–$\Gamma$ path. Therefore, there is likely to be substantial nesting between the sheets around $Z$ and $X$ that will lead to a peak in the susceptibility at the wave vector $(1/2,1/2)$.
I have calculated the Lindhard susceptibility $$\chi_0(q,\omega) = \sum_{k,m,n} |M_{k,k+q}^{m,n}|^2
\frac{f(\epsilon_k^m) - f(\epsilon_{k+q}^n)}{\epsilon_k^m -
\epsilon_{k+q}^n - \omega - \imath \delta}$$ at $\omega \to 0$ and $\delta \to 0$, where $\epsilon_k^m$ is the energy of a band $m$ at wave vector $k$ and $f$ is the Fermi distribution function. $M$ is the matrix element, which is set to unity. The real part of the susceptibility is shown in Fig. \[fig:yfg-suscep\], and it shows peaks at $\Gamma$, $Z$, and $X$ with the peak at $X$ having the highest magnitude. Note, however, that the cylinders around $Z$ and $X$ have different characters, which should reduce the peak $X$ and make it broader as well. The peak at $\Gamma$ is equal to the DOS $N(E_F)$. The peak at $Z$ reflect the nesting along the flat sections of the sheets along $(0,0,1/2)$ direction, while the peak at $X$ is due to the nesting between the hole cylinder and spheres centered around $Z$ and the electron cylinder centered around $X$.
The bare Lindhard susceptibility is further enhanced due to the RPA interaction, and its real part is related to magnetism and superconductivity. It is found experimentally that pure YFe$_2$Ge$_2$ does not order magnetically down to a temperature of 2 K although it shows non-Fermi liquid behavior in the transport and heat capacity measurements that is likely due to proximity to a magnetic quantum critical point.[@zou13] As the temperature is lowered further, superconductivity manifests in the resistivity measurements at $T_c^\rho$ = 1.8 K and DC magnetization at $T_c^{\textrm{mag}}$ = 1.5 K. This superconductivity can be due spin fluctuations associated with the peak in the susceptibility.[@berk66; @fay80] The pairing interaction has the form $$V(q=k-k') = - \frac{I^2(q) \chi_0(q)}{1 - I^2(q) \chi_0^2(q)}$$ in the singlet channel and is repulsive. (In the triplet channel, the interaction is attractive and also includes an angular factor.) Here $I(q)$ is the Stoner parameter which microscopically derives from Coulomb repulsion between electrons.
![The real part of bare susceptibility calculated with the matrix element set to unity.[]{data-label="fig:yfg-suscep"}](yfg-suscep-color){width="0.6\columnwidth"}
In the present case, the structure of the calculated susceptibility leads to the off-diagonal component of the interaction matrix to have a large negative value $-\lambda$ for the pairing between the hole sheets at $Z$ and electron cylinder at $X$ in the singlet channel. The diagonal component of the interaction matrix $\lambda_d$ pairing interactions on the hole and electron sheets are small and ferromagnetic. (For simplicity, I have assumed that the density of states are same for the hole and electron sections.) The eigenvector corresponding to the largest eigenvalue of this interaction matrix has opposite signs between the hole sheets around $Z$ and electron cylinder around $X$, and this is consistent with a singlet $s_\pm$ superconductivity with a wave vector $(1/2,1/2)$. This superconductivity is similar to the previously discovered iron-based superconductors.[@mazi08; @kuro08]
The proposed superconducvity in YFe$_2$Ge$_2$ and the previously discovered iron-based superconductor is similar, but the $T_c$ = 1.8 K for YFe$_2$Ge$_2$ is much smaller than those reported for other iron-based superconductors. One reason for this may be the smaller nesting in this compound leading to a smaller peak in susceptibility. The hole cylinder around $Z$ has mostly $d_{xy}$ character whereas the hole spheres around $Z$ and the electron cylinder around $X$ have mostly $d_{xz}$ and $d_{yz}$ character. These factors should lead to a slightly smaller and broader peak at $X$. I note, however, that nesting in the other iron-based superconductors is also not perfect[@mazi08] and the band characters between the nested sheets also vary.[@kuro08]
Another reason for the smaller $T_c$ in YFe$_2$Ge$_2$ may be due to the existence of competing magnetic fluctuations associated with the proximity to quantum criticality. The DOS from non-spin-polarized calculation is $N(E_F)$ = 1.125 eV$^{-1}$ per spin per Fe, which puts this material on the verge of a ferromagnetic instability according to the Stoner criterion. Ferromagnetism is pair-breaking for the singlet pairing and will suppress the $T_c$ in this compound. Furthermore, there is a peak in the susceptibility at $Z$ as well. The presence of additional antiferromagnetic interactions might reduce the phase space available for the spin fluctuation associated with the pairing channel and may be pair-breaking as well.
Energy (meV/Fe) Moment ($\mu_B$/Fe)
----------------- ----------------- ---------------------
NSP 0 0
FM $-$6.29 0.59
AFM (0,0,1/2) $-$11.63 0.64
SDW (1/2,1/2,0) $-$6.52 0.72
: \[tab:mag\] The relative energies of various magnetic orderings and the moments within the Fe spheres. These are almost degenerate, indicating the proximity to quantum criticality is due to competing magnetic interactions.
I performed magnetic calculations with various orderings on $(1 \times
1 \times 2)$ and $(\sqrt{2} \times \sqrt{2} \times 2)$ supercells to check the strength of competing magnetic interactions. The relative energies and the Fe moments are summarized in Table \[tab:mag\]. I was able to stabilize various magnetic configurations, and their energies are close to that of the non-spin-polarized configuration. However, I was not able to stabilize the checkerboard antiferromagnetic order in the $ab$ plane. When the magnetic order is stabilized, the magnitude of the Fe moment is less than 1 $\mu_{B}$, and the magnitudes vary between different orderings. This indicates that the magnetism is of itinerant nature. It is worthwhile to note that LDA calculations overestimate the magnetism in this compound as it does not exhibit any magnetic order experimentally. This disagreement between LDA and experiment is different from that for the Mott insulating compounds where LDA in general underestimates the magnetism.
Although this compound does not magnetically order experimentally, it nonetheless shows proximity to magnetism. It is found that partial substitution of Y by isovalent Lu causes the system to order antiferromagnetically, with 81% Lu substitution being the critical composition.[@ran11] At substitution values below the critical composition, the system shows non-Fermi liquid behavior in the heat capacity and transport measurements.[@zou13] The unusually high Sommerfeld coefficient of $\sim$90 mJ/mol K$^2$ at 2 K further increases as the temperature is lowered and the resistivity varies as $\rho \propto T^{3/2}$ up to around 10 K. This non-Fermi liquid behavior and the large renormalization of the magnetic moments may happen because there is a large phase for competing magnetic tendencies in this compound. This is due to the fluctuation-dissipation theorem, which relates the fluctuation of the moment to the energy and momentum integrated imaginary part of the susceptibility.[@mori85; @solo95; @ishi98; @agua04; @lars04] If the quantum criticality is due to competing magnetic interactions, the inelastic neutron scattering experiments, which measures the imaginary part of the susceptibility, would exhibit the structure related to the competing interactions. Therefore, even though this compound does not show magnetic ordering, it would be useful to perform such experiments and compare with the results presented here.
In any case, I indeed find that various magnetic orderings and the non-spin-polarized configuration are close in energy (see Table \[tab:mag\]). The energy of the lowest magnetic configuration is only 11.6 meV/Fe lower than the non-spin-polarized one, and the energies of the different magnetic orderings are within 6 meV/Fe of each other. As a comparison, the difference in energy between the non-magnetic configuration and the most stable magnetic ordering in BaFe$_2$As$_2$ is 92 meV/Fe, and the energy of the magnetic ordering closest to the most stable one is higher by 51 meV.[@sing08b] Signatures of quantum criticality has been reported for BaFe$_2$As$_2$ and related compounds.[@ning09; @jian09; @kasa10] YFe$_2$Ge$_2$ should show pronounced effects of proximity to quantum criticality as the competition between magnetic interactions is even stronger.
In summary, I have discussed the superconductivity and quantum criticality in YFe$_2$Ge$_2$ in terms of its electronic structure and competing magnetic interactions. The electronic states near the Fermi level are derived from Fe $3d$ bands and show a rich structure with the presence of both linearly dispersive and heavy bands. The Fermi surface consists of five sheets. There is an open rectangular electron cylinder around $X$. A big sheet shaped like the shell of a clam encloses a hole cylinder and two hole spheres around $Z$. There is a peak in the bare susceptibility at $(1/2,1/2)$ due to nesting between the hole sheets around $Z$ and the electron cylinder around $X$. I propose that the superconductivity in YFe$_2$Ge$_2$ is due to antiferromagnetic spin fluctuations associated with this peak. The resulting superconducting state has a $s_\pm$ state similar to that of previously discovered iron-based superconductors. I also find that different magnetic configurations are close in energy, which suggests the presence of competing magnetic interactions that are responsible for the proximity to quantum criticality observed in this compound.
I am grateful to Antoine Georges for helpful comments and suggestions. This work was partially supported by a grant from Agence Nationale de la Recherche (PNICTIDES).
Y. Zou, Z. Feng, P. W. Logg, J. Chen, G. I. Lampronti, and F. M. Grosche, arXiv e-print, 1311:0247.
G. Venturini, and B. Malaman, J. Alloys Compounds [**235**]{}, 201 (1996).
M. Rotter, M. Tegel, D. Johrendt, I. Schellenberg, W. Hermes, and R. Pöttgen, Phys. Rev. B [**78**]{}, 020503(R) (2008).
D. J. Singh, and M.-H. Du, Phys. Rev. Lett. [ **100**]{}, 237003 (2008).
M. Avila, S. Bud’ko, and P. Canfield, J. Magn. Magn. Mater. [**270**]{}, 51 (2004).
J. Ferstl, H. Rosner, and C. Geibel, Physica B: Condens. Matter [**378-380**]{}, 744 (2006).
S. Ran, S. L. Bud’ko, and P. C. Canfield, Philosophical Magazine [**91**]{}, 4388 (2011).
I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du, Phys. Rev. Lett. [**101**]{}, 057003 (2008).
K. Kuroki, S. Onari, R. Arita, H. Usui, Y. Tanaka, H. Kotani, H. Aoki, Phys. Rev. Lett. [**101**]{}, 087004 (2008).
P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, and J. Luitz, “WIEN2k, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties” (K. Schwarz, Tech. Univ. Wien, Austria) (2001).
http://elk.sourceforge.net.
B. P. Neal, E. R. Ylvisaker, and W. E. Pickett, Phys. Rev. B [**84**]{}, 085133 (2011).
J. T. Park, D. S. Inosov, A. Yaresko, S. Graser, *et al.*, Phys. Rev. B [**82**]{}, 134503 (2010).
N. F. Berk and J. R. Schrieffer, Phys. Rev. Lett. [**17**]{}, 433 (1966).
D. Fay and J. Appel, Phys. Rev. B [**22**]{}, 3173 (1970).
T. Moriya, *Spin Fluctuations in Itinerant Electron Magnetism* (Springer, Berlin, 1985).
A. Z. Solontsov and D. Wagner, Phys. Rev. B [**51**]{}, 12410 (1995).
A. Ishigaki and T. Moriya, J. Phys. Soc. Jpn. [ **67**]{}, 3924 (1998).
A. Aguayo, I. I. Mazin, and D. J. Singh, Phys. Rev. Lett. [**92**]{}, 147201 (2004).
P. Larson, I. I. Mazin, and D. J. Singh, Phys. Rev. B [**69**]{}, 064429 (2004).
D. J. Singh, Phys. Rev. B [**78**]{}, 094511 (2008).
F. Ning, K. Ahilan, T. Imai, A. S. Sefat, R. Jin, M. A. McGuire, B. C. Sales, D. Mandrus, J. Phys. Soc. Jpn. [**78**]{}, 013711 (2009).
S. Jiang, H. Xing, G. Xuan, C. Wang, Z. Ren, C. Feng, J. Dai, Z. Xu, G. Cao, J. Phys. Condens. Matter [**21**]{}, 382203 (2009).
S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, Y. Matsuda, Phys. Rev. B [**81**]{}, 184519 (2010).
| ArXiv |
---
abstract: 'Semi-supervised and unsupervised systems provide operators with invaluable support and can tremendously reduce the operators’ load. In the light of the necessity to process large volumes of video data and provide autonomous decisions, this work proposes new learning algorithms for activity analysis in video. The activities and behaviours are described by a dynamic topic model. Two novel learning algorithms based on the expectation maximisation approach and variational Bayes inference are proposed. Theoretical derivations of the posterior estimates of model parameters are given. The designed learning algorithms are compared with the Gibbs sampling inference scheme introduced earlier in the literature. A detailed comparison of the learning algorithms is presented on real video data. We also propose an anomaly localisation procedure, elegantly embedded in the topic modeling framework. It is shown that the developed learning algorithms can achieve $95\%$ success rate. The proposed framework can be applied to a number of areas, including transportation systems, security and surveillance.'
author:
- 'Olga Isupova, Danil Kuzin, and Lyudmila Mihaylova, [^1]'
bibliography:
- 'TNNLS-2016-P-6800-R1-Biblist.bib'
title: Learning Methods for Dynamic Topic Modeling in Automated Behaviour Analysis
---
behaviour analysis, unsupervised learning, learning dynamic topic models, variational Bayesian approach, expectation maximisation, video analytics
Introduction
============
analysis is an important area in intelligent video surveillance, where abnormal behaviour detection is a difficult problem. One of the challenges in this field is informality of the problem formulation. Due to the broad scope of applications and desired objectives there is no unique way, in which normal or abnormal behaviour can be described. In general, the objective is to detect unusual events and inform in due course a human operator about them.
This paper considers a probabilistic framework for anomaly detection, where less probable events are labelled as abnormal. We propose two learning algorithms and an anomaly localisation procedure for spatial detection of abnormal behaviours.
Related work
------------
There is a wealth of methods for abnormal behaviour detection, for example, pattern-based methods [@Raghavendra2011; @Yen13; @Ouivirach13]. These methods extract explicit patterns from data and use them as behaviour templates for decision making. In [@Raghavendra2011] the sum of the visual features of a reference frame is treated as a normal behaviour template. Another common approach for representing normal templates is using clusters of visual features [@Yen13; @Ouivirach13]. Visual features can range from raw intensity values of pixels to complex features that exploit the data nature [@Zhou2016].
In the testing stage new observations are compared with the extracted patterns. The comparison is based on some similarity measure between observations, e.g., the Jensen-Shannon divergence in [@Su2014] or the $Z$-score value in [@Yen13; @Ouivirach13]. If the distance between the new observation and any of the normal patterns is larger than a threshold, then the observation is classified as abnormal.
Abnormal behaviour detection can be considered as a classification problem. It is difficult in advance to collect and label all kind of abnormalities. Therefore, only one class label can be expected and one-class classifiers are applied to abnormal behaviour detection: e.g., a one-class Support Vector Machine [@Cheng2013], a support vector data description algorithm [@Liu2010], a neural network approach [@Maddalena2013], a level set method [@Osher1988] for normal data boundary determination [@Ding2015].
Another class of methods rely on the estimation of probability distributions of the visual data. These estimated distributions are then used in the decision making process. Different kinds of probability estimation algorithms are proposed in the literature, e.g., based on non-parametric sample histograms [@Adam2008], Gaussian distribution modelling [@Basharat2008]. Spatio-temporal motion data dependency is modelled as a coupled Hidden Markov Model in [@Kratz2009]. Auto-regressive process modelling based on self-organised maps is proposed in [@Brighenti2011].
An efficient approach is to seek for feature sets that tend to appear together. These feature sets form typical activities or behaviours in the scene. Topic modeling [@Hofmann99; @Blei03LDA] is an approach to find such kinds of statistical regularities in a form of probability distributions. The approach can be applied for abnormal behaviour detection, e.g., [@Mehran09; @Li2008; @Varadarajan2009]. A number of variations of the conventional topic models for abnormal behaviour detection have been recently proposed: clustering of activity distributions [@Wang09], modelling temporal dependencies among activities [@Hospedales2011], a continuous model for an object velocity [@Jeong14].
Within the probabilistic modelling approach [@Jeong14; @Li2008; @Mehran09; @Basharat2008; @Wang09; @Kratz2009] the decision about abnormality is mainly made by computing likelihood of a new observation. The comparison of the different abnormality measures based on the likelihood estimation is provided in [@Varadarajan2009].
Topic modeling is originally developed for text mining [@Hofmann99; @Blei03LDA]. It aims to find latent variables called *“topics”* given the collection of unlabelled text *documents* consisted of *words*. In probabilistic topic modeling documents are represented as a mixture of topics, where each topic is assumed to be a distribution over words.
There are two main types of topic models: Probabilistic Latent Semantic Analysis (PLSA) [@Hofmann99] and Latent Dirichlet Allocation (LDA) [@Blei03LDA]. The former considers the problem from the frequentist perspective while the later studies it within the Bayesian approach. The main learning techniques proposed for these models include maximum likelihood estimation via the Expectation-Maximisation (EM) algorithm [@Hofmann99], variational Bayes inference [@Blei03LDA], Gibbs sampling [@Griffiths2004] and Maximum a Posteriori (MAP) estimation [@Chien2008].
Contributions
-------------
In this paper inspired by ideas from [@Hospedales2011] we propose an unsupervised learning framework based on a Markov Clustering Topic Model for behaviour analysis and anomaly detection. It groups possible topic mixtures of visual documents and forms a Markov chain for the groups.
The key contributions of this work consist in developing new learning algorithms, namely MAP estimation using the EM-algorithm and variational Bayes inference for the Markov Clustering Topic Model (MCTM), and in proposing an anomaly localisation procedure that follows concepts of probabilistic topic modeling. We derive the likelihood expressions as a normality measure of newly observed data. The developed learning algorithms are compared with the Gibbs sampling scheme proposed in [@Hospedales2011]. A comprehensive analysis of the algorithms is presented over real video sequences. The empirical results show that the proposed methods provide more accurate results than the Gibbs sampling scheme in terms of anomaly detection performance.
Our preliminary results with the EM-algorithm for behaviour analysis are published in [@Isupova2015]. In contrast to [@Isupova2015] we now consider a fully Bayesian framework, where we propose the EM-algorithm for MAP estimates rather than the maximum likelihood ones. We also propose here a novel learning algorithm based on variational Bayes inference and a novel anomaly localisation procedure. The experiments are performed on more challenging datasets in comparison to [@Isupova2015].
The rest of the paper is organised as follows. Section \[sec:features\] describes the overall structure of visual documents and visual words. Section \[sec:model\] introduces the dynamic topic model. The new learning algorithms are presented in Section \[sec:inference\], where the proposed MAP estimation via the EM-algorithm and variational Bayes algorithm are introduced first and then the Gibbs sampling scheme is reviewed. The methods are given with a detailed discussion about their similarities and differences. The anomaly detection procedure is presented in Section \[sec:abnormality\]. The learning algorithms are evaluated with real data in Section \[sec:experiments\] and Section \[sec:conlusion\] concludes the paper.
Video analytics within the topic modeling framework {#sec:features}
===================================================
Video analytics tasks can be formulated within the framework of topics modeling. This requires a definition of visual documents and visual words, e.g., as in [@Wang09; @Hospedales2011]. The whole video sequence is divided into non-overlapping short clips. These clips are treated as visual documents. Each frame is divided next into grid cells of pixels. Motion detection is applied to each of the cells. The cells where motion is detected are called moving cells. For each of the moving cells the motion direction is determined. This direction is further quantised into four dominant ones - up, left, down, right (see Figure \[fig:feature\_extraction\]). The position of the moving cell and the quantised direction of its motion form a visual word.
Each of the visual documents is then represented as a sequence of visual words’ identifiers, where identifiers are obtained by some ordering of a set of unique words. This discrete representation of the input data can be processed by topic modeling methods.
The Markov Clustering Topic Model for behavioural analysis {#sec:model}
==========================================================
Motivation
----------
In topic modeling there are two main kinds of distributions — the distributions over words, which correspond to topics, and the distributions over topics, which characterise the documents. The relationship between documents and words is then represented via latent low-dimensional entities called topics. Having only an unlabelled collection of documents, topic modeling methods restore a hidden structure of data, i.e., the distributions over words and the distributions over topics.
Consider a set of distributions over topics and a topic distribution for each document is chosen from this set. If the cardinality of the set of distributions over topics is less than the number of documents, then documents are clustered into groups such that documents have the same topic distribution within a group. A unique distribution over topics is called a *behaviour* in this work. Therefore, each document corresponds to one behaviour. In topic modeling a document is fully described by a corresponding distribution over topics, which means in this case a document is fully described by a corresponding behaviour.
There are a number of applications where we can observe documents clustered into groups with the same distribution over topics. Let us consider some examples from video analytics where a visual word corresponds to a motion within a tiny cell. As topics represent words that statistically often appear together, in video analytics applications topics define some motion patterns in local areas.
Let us consider a road junction regulated by traffic lights. A general motion on the junction is the same with the same traffic light regime. Therefore, the documents associated to the same traffic light regimes have the same distributions over topics, i.e., they correspond to the same behaviours.
Another example is a video stream generated by a video surveillance camera from a train station. Here it is also possible to distinguish several types of general motion within the camera scene: getting off and on a train and waiting for it. These types of motion correspond to behaviours, where the different visual documents showing different instances of the same behaviour have very similar motion structures, i.e., the same topic distribution.
Each action in real life lasts for some time, e.g., a traffic light regime stays the same and people get on and off a train for several seconds. Moreover, often these different types of motion or behaviours follow a cycle and their changes occur in some order. These insights motivate to model a sequence of behaviours as a Markov chain, so that the behaviours remain the same during some documents and change in a predefined order. The model that has these described properties is called a Markov Clustering Topic Model (MCTM) in [@Hospedales2011]. The next section formally formulates the model.
Model formulation
-----------------
This section starts from the introduction of the main notations used through the paper. Denote by $\mathcal{X}$ the vocabulary of all visual words, by $\mathcal{Y}$ the set of all topics and by $\mathcal{Z}$ the set of all behaviours, $x$, $y$ and $z$ are used for elements from these sets, respectively. When an additional element of a set is required it is denoted with a prime, e.g., $z'$ is another element from $\mathcal{Z}$.
Let $\mathbf{x}_t = \{x_{i, t}\}_{i = 1}^{N_t}$ be a set of words for the document $t$, where $N_t$ is the length of the document $t$. Let $\mathbf{x}_{1:T_{tr}} = \{\mathbf{x}_t\}_{t = 1}^{T_{tr}}$ denote a set of all words for the whole dataset, where $T_{tr}$ is the number of documents in the dataset. Similarly, denote by $\mathbf{y}_t = \{y_{i, t}\}_{i = 1}^{N_t}$ and $\mathbf{y}_{1:T_{tr}} = \{\mathbf{y}_t\}_{t = 1}^{T_{tr}}$ a set of topics for the document $t$ and a set of all topics for the whole dataset, respectively. Let $\mathbf{z}_{1:T_{tr}} = \{z_t\}_{t = 1}^{T_{tr}}$ be a set of all behaviours for all documents.
Note that $x$, $y$ and $z$ without subscript denote possible values for a word, topic and behaviour from $\mathcal{X}$, $\mathcal{Y}$ and $\mathcal{Z}$, respectively, while the symbols with subscript denote word, topic and behaviour assignments in particular places in a dataset.
Here, $\boldsymbol{\Phi}$ is a matrix corresponding to the distributions over words given the topics, $\boldsymbol{\Theta}$ is a matrix corresponding to the distributions over topics given behaviours. For a Markov chain of behaviours a vector $\boldsymbol{\pi}$ for a behaviour distribution for the first document and a matrix $\boldsymbol{\Xi}$ for transition probability distributions between the behaviours are introduced: $$\begin{aligned}
\boldsymbol{\Phi} &= \{\phi_{x, y}\}_{x \in \mathcal{X}, y \in \mathcal{Y}}, &\phi_{x, y} &= p(x | y), &\boldsymbol{\phi}_y &= \{\phi_{x, y}\}_{x \in \mathcal{X}};\\
\boldsymbol{\Theta} &= \{\theta_{y, z}\}_{y \in \mathcal{Y}, z \in \mathcal{Z}}, &\theta_{y, z} &= p(y | z), &\boldsymbol{\theta_z} &= \{\theta_{y, z}\}_{y \in \mathcal{Y}};\\
\boldsymbol{\pi} &= \{\pi_z\}_{z \in \mathcal{Z}}, &\pi_z &= p(z); \\
\boldsymbol{\Xi} &= \{\xi_{z', z}\}_{z' \in \mathcal{Z}, z \in \mathcal{Z}}, &\xi_{z', z} &= p(z' | z), &\boldsymbol{\xi}_z &= \{\xi_{z', z}\}_{z' \in \mathcal{Z}},\end{aligned}$$ where the matrices $\boldsymbol{\Phi}$, $\boldsymbol{\Theta}$ and $\boldsymbol{\Xi}$ and the vector $\boldsymbol{\pi}$ are formed as follows. An element of a matrix on the $i$-th row and $j$-th column is a probability of the $i$-th element given the $j$-th one, e.g., $\phi_{x, y}$ is a probability of the word $x$ in the topic $y$. The columns of the matrices are then distributions for corresponding elements, e.g., $\boldsymbol\theta_z$ is a distribution over topics for the behaviour $z$. Elements of the vector $\boldsymbol\pi$ are probabilities of behaviours to be chosen by the first document. All these distributions are categorical.
The introduced distributions form a set $$\boldsymbol{\Omega} = \{\boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\pi}, \boldsymbol{\Xi}\}$$ of model parameters and they are estimated during a learning procedure.
Prior distributions are imposed to all the parameters. Conjugate Dirichlet distributions are used: $$\begin{aligned}
\boldsymbol{\phi}_y &\sim Dir(\boldsymbol{\phi}_y | \boldsymbol{\beta}), &\forall y \in \mathcal{Y};\\
\boldsymbol{\theta}_z &\sim Dir(\boldsymbol{\theta}_z | \boldsymbol{\alpha}), &\forall z \in \mathcal{Z};\\
\boldsymbol{\pi} &\sim Dir(\boldsymbol{\pi} | \boldsymbol{\eta}); \\
\boldsymbol{\xi}_z &\sim Dir(\boldsymbol{\xi}_z | \boldsymbol{\gamma}), &\forall z \in \mathcal{Z},\end{aligned}$$ where $Dir(\cdot)$ is a Dirichlet distribution and $\boldsymbol{\beta}$, $\boldsymbol{\alpha}$, $\boldsymbol{\eta}$ and $\boldsymbol{\gamma}$ are the corresponding hyperparameters. As topics and behaviours are not known a priori and will be specified via the learning procedure, it is impossible to distinguish two topics or two behaviours in advance. This is the reason why all the prior distributions are the same for all topics and all behaviours.
![Graphical representation of the Markov Clustering Topic Model.[]{data-label="fig:graph_model"}](TNNLS-2016-P-6800-R1-MCTM_graph_model)
The generative process for the model is as follows. All the parameters are drawn from the corresponding prior Dirichlet distributions. At each time moment $t$ a behaviour $z_t$ is chosen first for a visual document. The behaviour is sampled using the matrix $\boldsymbol{\Xi}$ according to the behaviour chosen for the previous document. For the first document the behaviour is sampled using the vector $\boldsymbol\pi$.
Once the behaviour is selected, the procedure of choosing visual words repeats for the number of times equal to the length of the current document $N_t$. The procedure consists of two steps — sampling a topic $y_{i, t}$ using the matrix $\boldsymbol\Theta$ according to the chosen behaviour $z_t$ followed by sampling a word $x_{i, t}$ using the matrix $\boldsymbol\Phi$ according to the chosen topic $y_{i, t}$ for each token $i \in \{1, \dotsc, N_t\}$, where a token is a particular place inside a document where a word is assigned. The generative process is summarised in Algorithm \[alg:generative\_em\]. The graphical model, showing the relationships between the variables, can be found in Figure \[fig:graph\_model\].
The number of clips – $T_{tr}$, the length of each clip – $N_t$ $\forall t = \{1, \dotsc, T_{tr}\}$, the hyperparameters – $\boldsymbol{\beta}$, $\boldsymbol{\alpha}$, $\boldsymbol{\eta}$, $\boldsymbol{\gamma}$; The dataset $\textbf{x}_{1:T_{tr}} = \{x_{1, 1}, \dotsc, x_{i, t}, \dotsc, x_{N_{T_{tr}}, T_{tr}}\}$; draw a word distribution for the topic $y$: $$\boldsymbol\phi_y \sim Dir(\boldsymbol\phi_y | \boldsymbol\beta);$$ draw a topic distribution for behaviour $z$: $$\boldsymbol\theta_z \sim Dir(\boldsymbol\theta_z | \boldsymbol\alpha);$$ draw a transition distribution for behaviour $z$: $$\boldsymbol\xi_z \sim Dir(\boldsymbol\xi_z | \boldsymbol\gamma);$$ draw a behaviour probability distribution for the initial document $$\boldsymbol\pi \sim Dir(\boldsymbol\phi | \boldsymbol\eta);$$
draw a behaviour for the document from the initial distribution: $z_t \sim Cat(z_t | \boldsymbol{\pi})$; draw a behaviour for the document based on the behaviour of the previous document: $z_t \sim Cat(z_t | \boldsymbol{\xi}_{z_{t-1}})$; \[alg\_step:gen\_behaviour\] draw a topic for the token $i$ based on the chosen behaviour: $y_{i, t} \sim Cat(y_{i, t} | \boldsymbol{\theta}_{z_t}$); \[alg\_step:gen\_topic\] draw a visual word for the token $i$ based on the chosen topic: $x_{i, t} \sim Cat(x_{i, t} | \boldsymbol{\phi}_{y_{i, t}})$; \[alg\_step:gen\_word\]
The full likelihood of the observed variables $\mathbf{x}_{1:T_{tr}}$, the hidden variables $\mathbf{y}_{1:T_{tr}}$ and $\mathbf{z}_{1:T_{tr}}$ and the set of parameters $\boldsymbol\Omega$ can be written then as follows:
$$\begin{aligned}
&p(\mathbf{x}_{1:T_{tr}}, \mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}, \boldsymbol\Omega | \boldsymbol\beta, \boldsymbol\alpha, \boldsymbol\eta, \boldsymbol\gamma) = \nonumber\\
& \underbrace{p(\boldsymbol\pi | \boldsymbol\eta) \, p(\boldsymbol\Xi | \boldsymbol\gamma) \, p(\boldsymbol\Theta | \boldsymbol\alpha) \, p(\boldsymbol\Phi | \boldsymbol\beta)}_{\text{Priors}} \times \nonumber\\
\label{eq:full_likelihood}
& \underbrace{p(z_1 | \boldsymbol\pi) \left[\prod\limits_{t = 2}^{T_{tr}} p(z_t | z_{t-1}, \boldsymbol\Xi) \right] \prod\limits_{t = 1}^{T_{tr}} \prod\limits_{i = 1}^{N_t} p(x_{i, t} | y_{i, t}, \boldsymbol\Phi) p(y_{i, t} | z_t, \boldsymbol\Theta)}_{\text{Likelihood}}\end{aligned}$$
In [@Hospedales2011] Gibbs sampling is implemented for parameters learning in the MCTM. We propose two new learning algorithms: based on an EM-algorithm for the MAP estimates of the parameters and based on variational Bayes inference to estimate posterior distributions of the parameters. We introduce the proposed learning algorithms below and briefly review the Gibbs sampling scheme.
Parameters learning {#sec:inference}
===================
Learning: EM-algorithm scheme {#sec:em}
-----------------------------
We propose a learning algorithm for MAP estimates of the parameters based on the Expectation-Maximisation algorithm [@Dempster77]. The algorithm consists of repeating E and M-steps. Conventionally, the EM-algorithm is applied to get maximum likelihood estimates. In that case the M-step is: $$\mathcal{Q}(\boldsymbol\Omega, \boldsymbol\Omega^{\text{old}}) \longrightarrow \max\limits_{\boldsymbol\Omega},$$ where $\boldsymbol\Omega^{\text{old}}$ denotes the set of parameters obtained at the previous iteration and $\mathcal{Q}(\boldsymbol\Omega, \boldsymbol\Omega^{\text{old}})$ is the expected logarithm of the full likelihood function of the observed and hidden variables: $$\begin{gathered}
\mathcal{Q}(\boldsymbol\Omega, \boldsymbol\Omega^{\text{old}}) = \\
\mathbb{E}_{p(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}} | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}})} \log p(\mathbf{x}_{1:T_{tr}}, \mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}} | \boldsymbol\Omega).\end{gathered}$$ The subscript of the expectation sign means the distribution, with respect to which the expectation is calculated. During the E-step the posterior distribution of the hidden variables is estimated given the current estimates of the parameters.
In this paper the EM-algorithm is applied to get MAP estimates instead of traditional maximum likelihood ones. The M-step is modified in this case as: $$\label{eq:M_step_functional}
\mathcal{Q}(\boldsymbol\Omega, \boldsymbol\Omega^{\text{old}}) + \log p(\boldsymbol\Omega | \boldsymbol\beta, \boldsymbol\alpha, \boldsymbol\eta, \boldsymbol\gamma) \longrightarrow \max\limits_{\boldsymbol\Omega},$$ where $p(\boldsymbol\Omega | \boldsymbol\beta, \boldsymbol\alpha, \boldsymbol\eta, \boldsymbol\gamma)$ is the prior distribution of the parameters.
As the hidden variables are discrete, the expectation converts to a sum of all possible values for the whole set of the hidden variables $\{\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}\}$. The substitution of the likelihood expression from (\[eq:full\_likelihood\]) into (\[eq:M\_step\_functional\]) allows to marginalise some hidden variables from the sum. The remaining distributions that are required for computing the $\mathcal{Q}$-function are as follows:
- $p(z_1 = z | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}})$ — the posterior distribution of a behaviour for the first document;
- $p(z_t = z', z_{t-1} = z | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}})$ — the posterior distribution of two behaviours for successive documents;
- $p(y_{i, t} = y | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}})$ — the posterior distribution of a topic assignment for a given token;
- $p(y_{i, t} = y, z_t = z | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}})$ — the joint posterior distribution of a topic and behaviour assignments for a given token.
With the fixed current values for these posterior distributions the estimates of the parameters that maximise the required functional of the M-step (\[eq:M\_step\_functional\]) can be computed as: $$\begin{aligned}
\label{eq:M:phi}
\widehat{\phi}_{x, y}^{\, \text{EM}} &= \dfrac{\left(\beta_x + \hat{n}_{x, y}^{\,\text{EM}} - 1\right)_+}{\sum\limits_{x' \in \mathcal{X}} \left(\beta_{x'} + \hat{n}_{x', y}^{\,\text{EM}} - 1\right)_{+}}, &\forall x \in \mathcal{X}, y \in \mathcal{Y};\\
\label{eq:M:theta}
\widehat{\theta}_{y, z}^{\, \text{EM}} &= \dfrac{\left(\alpha_y + \hat{n}_{y, z}^{\,\text{EM}} - 1\right)_+}{\sum\limits_{y' \in \mathcal{Y}} \left(\alpha_{y'} + \hat{n}_{y', z}^{\,\text{EM}} - 1\right)_+}, &\forall y \in \mathcal{Y}, z \in \mathcal{Z};\\
\label{eq:M_psi_k,l}
\widehat{\xi}_{z', z}^{\, \text{EM}} &= \dfrac{\left(\gamma_{z'} + \hat{n}_{z', z}^{\, \text{EM}} - 1\right)_+}{\sum\limits_{\check{z} \in \mathcal{Z}} \left(\gamma_{\check{z}} + \hat{n}_{\check{z}, z}^{\, \text{EM}} - 1\right)_{+}}, &\forall z', z \in \mathcal{Z};\\
\label{eq:M:pi}
\widehat{\pi}_z^{\, \text{EM}} &= \dfrac{\left(\eta_z + \hat{n}_{z}^{\, \text{EM}} - 1\right)_+}{\sum\limits_{z' \in \mathcal{Z}} \left(\eta_{z'} + \hat{n}_{z'}^{\, \text{EM}} - 1\right)_+}, &\forall z \in \mathcal{Z},\end{aligned}$$ where $(a)_+ \stackrel{\text{def}}{=} \max(a, 0)$ [@Vorontsov2014ARTMArticle]; $\beta_x$, $\alpha_y$ and $\gamma_{z'}$ are the elements of the hyperparameter vectors $\boldsymbol\beta$, $\boldsymbol\alpha$ and $\boldsymbol\gamma$, respectively; and $\hat{n}_{x, y}^{\,\text{EM}} = \sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} p(y_{i, t} = y | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}}) \mathbb{I}(x_{i, t} = x)$ is the expected number of times, when the word $x$ is associated to the topic $y$, where $\mathbb{I}(\cdot)$ is the indicator function; $\hat{n}_{y, z}^{\,\text{EM}} = \sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} p(y_{i,t} = y, z_t = z | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}})$ is the expected number of times, when the topic $y$ is associated to the behaviour $z$; $\hat{n}_{z}^{\, \text{EM}} = p(z_1 = z | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}})$ is the “expected number of times”, when the behaviour $z$ is associated to the first document, in this case the “expected number” is just a probability, the notation is used for the similarity with the rest of the parameters; $\hat{n}_{z', z}^{\, \text{EM}} = \sum\limits_{t = 2}^{T_{tr}} p(z_t = z', z_{t - 1} = z| \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}})$ is the expected number of times, when the behaviour $z$ is followed by the behaviour $z'$.
During the E-step with the fixed current estimates of the parameters $\boldsymbol\Omega^{\text{old}}$, the updated values for the posterior distributions of the hidden variables should be computed. The derivation of the updated formulae for these distributions is similar to the Baum-Welch forward-backward algorithm [@Murphy2012], where the EM-algorithm is applied to the maximum likelihood estimates for a Hidden Markov Model (HMM). This similarity appears because the generative model can be viewed as extension of a HMM.
For effective computation of the required posterior distributions the additional variables $\acute{\alpha}_z(t)$ and $\acute{\beta}_z(t)$ are introduced. A dynamic programming technique is applied for computation of these variables. Having the updated values for $\acute{\alpha}_z(t)$ and $\acute{\beta}_z(t)$ one can update the required posterior distributions of the hidden variables. The E-step is then formulated as follows (for simplification of notation the superscript “old” for the parameters variables is omitted inside the formulae):
$$\begin{aligned}
\label{eq:E:alpha}
&\begin{cases}
\begin{aligned}
&\acute{\alpha}_z(t) = \prod\limits_{i = 1}^{N_t} \sum\limits_{y \in \mathcal{Y}} \phi_{x_{i, t}, y} \, \theta_{y, z} \times\\
&\quad\sum\limits_{z'\in \mathcal{Z}} \acute{\alpha}_{z'}(t-1) \xi_{z, \tilde{z}}, \,\text{if}\, t \geq 2;
\end{aligned}\\
\acute{\alpha}_{z}(1) = \pi_z \prod\limits_{i = 1}^{N_1} \sum\limits_{y\in \mathcal{Y}} \phi_{x_{i, 1}, y} \, \theta_{y, z};
\end{cases}\\
\label{eq:E:beta}
&\begin{cases}
\begin{aligned}
&\acute{\beta}_{z}(t) = \sum\limits_{z' \in \mathcal{Z}} \acute{\beta}_{z'}(t+1) \xi_{z', z} \times \\
&\quad \prod\limits_{i = 1}^{N_{t+1}} \sum\limits_{y \in \mathcal{Y}} \phi_{x_{i, t+1}, y} \, \theta_{y, z'} , \,\text{if}\, t < T_{tr};
\end{aligned}\\
\acute{\beta}_{z}(T_{tr}) = 1;
\end{cases}\\
\label{eq:E:normalisation_const}
&K = \sum\limits_{z \in \mathcal{Z}} \acute{\alpha}_{z}(1) \acute{\beta}_{z}(1);\\
\label{eq:E:z_t}
&p(z_1 | \textbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}}) = \dfrac{\acute{\alpha}_{z_1}(1) \acute{\beta}_{z_1}(1)}{K}; \\
\label{eq:E:z_t, z_t-1}
&\begin{aligned}
&p(z_t, z_{t-1} | \textbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}}) = \dfrac{\acute{\alpha}_{z_{t-1}}(t-1) \acute{\beta}_{z_t}(t) \xi_{z_t, z_{t-1}}}{K} \times\\
&\quad \prod\limits_{i = 1}^{N_t} \sum\limits_{y \in \mathcal{Y}} \phi_{x_{i, t}, y} \theta_{y, z_t};
\end{aligned}\\
\label{eq:E:y_i,t, z_t}
&\begin{cases}
\begin{aligned}
&p(y_{i, t}, z_t | \textbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}})
= \dfrac{\phi_{x_{i, t}, y_{i, t}} \theta_{y_{i, t}, z_t} \acute{\beta}_{z_t}(t)}{K} \times\\
&\quad\sum\limits_{z' \in \mathcal{Z}} \acute{\alpha}_{z'}(t-1) \xi_{z_t, z'} \prod\limits_{\substack{j = 1 \\ j \neq i}}^{N_t} \sum\limits_{y' \in \mathcal{Y}} {\phi_{x_{j, t}, y'} \theta_{y', z_t}},\,\text{if}\, t \geq 2;
\end{aligned} \\
\begin{aligned}
&p(y_{i, 1}, z_1 | \textbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}})
= \dfrac{\phi_{x_{i, 1}, y_{i, 1}} \theta_{y_{i, 1}, z_1} \acute{\beta}_{z_1}(1)}{K} \times\\
&\quad \pi_{z_1} \prod\limits_{\substack{j = 1 \\ j \neq i}}^{N_1} \sum\limits_{y' \in \mathcal{Y}} {\phi_{x_{j, 1}, y'} \theta_{y', z_1}};
\end{aligned}
\end{cases} \\
\label{eq:E:y_i,t}
&\begin{aligned}
p(y_{i, t} | \textbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}}) &= \sum\limits_{z \in \mathcal{Z}} p(y_{i, t}, z | \textbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}}),
\end{aligned}\end{aligned}$$
where $K$ is a normalisation constant for all the posterior distributions of the hidden variables.
Starting with some random initialisation of the parameter estimates, the EM-algorithm iterates the E and M-steps until convergence. The obtained estimates of the parameters are used for further analysis.
Learning: Variational Bayes scheme {#sec:vb}
----------------------------------
We also propose a learning algorithm based on the variational Bayes (VB) approach [@Jordan1999] to find approximated posterior distributions for both the hidden variables and the parameters.
In the VB inference scheme the true posterior distribution, in this case the distribution of the parameters and the hidden variables $p(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}, \boldsymbol\Omega | \mathbf{x}_{1:T_{tr}}, \boldsymbol\eta, \boldsymbol\gamma, \boldsymbol\alpha, \boldsymbol\beta)$, is approximated with a factorised distribution — $q(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}, \boldsymbol\Omega)$. The approximation is made to minimise the Kullback-Leibler divergence between the factorised distribution and true one. We factorise the distribution in order to separate the hidden variables and the parameters: $$\begin{gathered}
\label{eq:vb_factorisation}
\hat{q}(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}, \boldsymbol\Omega) = \hat{q}(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}) \hat{q}(\boldsymbol\Omega) \stackrel{\text{def}}{=} \\
\operatorname*{argmin}\mathrm{KL} \left(q(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}) q(\boldsymbol\Omega) || \right.\\
\left.p(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}, \boldsymbol\Omega | \mathbf{x}_{1:T_{tr}}, \boldsymbol\eta, \boldsymbol\gamma, \boldsymbol\alpha, \boldsymbol\beta)\right),\end{gathered}$$ where $\mathrm{KL}$ denotes the Kullback-Leibler divergence. The minimisation of the Kullback-Leibler divergence is equivalent to the maximisation of the evidence lower bound (ELBO). The maximisation is done by coordinate ascent [@Jordan1999].
During the update of the parameters the approximated distribution $q(\boldsymbol\Omega)$ is further factorised: $$\label{eq:q_param_factorisation}
q(\boldsymbol\Omega) = q(\boldsymbol\pi) q(\boldsymbol\Xi) q(\boldsymbol\Theta) q(\boldsymbol\Phi).$$ Note that this factorisation of approximated parameter distributions is a corollary of our model and not an assumption.
The iterative process of updating the approximated distributions of the parameters and the hidden variables can be formulated as an EM-like algorithm, where during the E-step the approximated distributions of the hidden variables are updated and during the M-step the approximated distributions of the parameters are updated.
The M-like step is as follows: $$\begin{aligned}
\label{eq:VB:beta}
&\begin{cases}
q(\boldsymbol\Phi) = \prod\limits_{y \in \mathcal{Y}} Dir\left(\boldsymbol\phi_y; \tilde{\boldsymbol\beta}_y\right),\\
\tilde{\beta}_{x, y} = \beta_x + \hat{n}_{x, y}^{\, \text{VB}}, &\forall x \in \mathcal{X}, y \in \mathcal{Y};
\end{cases}\\
&\begin{cases}
q(\boldsymbol\Theta) = \prod\limits_{z \in \mathcal{Z}} Dir(\boldsymbol\theta_z; \tilde{\boldsymbol\alpha}_z),\\
\tilde{\alpha}_{y, z} = \alpha_y + \hat{n}_{y, z}^{\, \text{VB}}, &\forall y \in \mathcal{Y}, z \in \mathcal{Z};
\end{cases}\\
&\begin{cases}
q(\boldsymbol\pi) = Dir(\boldsymbol\pi; \tilde{\boldsymbol\eta}),\\
\tilde{\eta}_z = \eta_z + \hat{n}_z^{\,\text{VB}}, &\forall z \in \mathcal{Z};
\end{cases}\\
\label{eq:VB:gamma}
&\begin{cases}
q(\boldsymbol\Xi) = \prod\limits_{z \in \mathcal{Z}} Dir(\boldsymbol\xi_{z}; \tilde{\boldsymbol\gamma}_z),\\
\tilde{\gamma}_{z', z} = \gamma_{z'} + \hat{n}_{z', z}^{\, \text{VB}}, &\forall z', z \in \mathcal{Z},
\end{cases}\end{aligned}$$ where $\tilde{\boldsymbol\beta}_y$, $\tilde{\boldsymbol\alpha}_z$, $\tilde{\boldsymbol\eta}$ and $\tilde{\boldsymbol\gamma}_z$ are updated hyperparameters of the corresponding posterior Dirichlet distributions; and $\hat{n}_{x, y}^{\, \text{VB}} = \sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} \mathbb{I}(x_{i, t} = x) q(y_{i, t} = y)$ is the expected number of times, when the word $x$ is associated with the topic $y$. Here and below the expected number is computed with respect to the approximated posterior distributions of the hidden variables; $\hat{n}_{y, z}^{\, \text{VB}} = \sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} q(y_{i, t} = y, z_t = z)$ is the expected number of times, when the topic $y$ is associated with the behaviour $z$; $\hat{n}_z^{\,\text{VB}} = q(z_1 = z)$ is the “expected number” of times, when the behaviour $z$ is associated to the first document; $\hat{n}_{z', z}^{\, \text{VB}} = \sum\limits_{t = 2}^{T_{tr}} q(z_t = z', z_{t-1} = z)$ is the expected number of times, when the behaviour $z$ is followed by the behaviour $z'$.
The following additional variables are introduced for the E-like step: $$\begin{aligned}
\label{eq:VB:introduced_pi}
\tilde{\pi}_z &= \exp\left(\psi\left(\tilde{\eta}_z\right) - \psi\left(\sum\limits_{z' \in \mathcal{Z}} \tilde{\eta}_{z'}\right)\right)\\
\tilde{\xi}_{\tilde{z}, z} &= \exp\left(\psi\left(\tilde{\gamma}_{\tilde{z}, z}\right) - \psi\left(\sum\limits_{z' \in \mathcal{Z}} \tilde{\gamma}_{z', z}\right)\right);\\
\tilde{\phi}_{x, y} &= \exp\left(\psi\left(\tilde{\beta}_{x, y} \right) - \psi\left(\sum\limits_{x' \in \mathcal{X}} \tilde{\beta}_{x', y} \right)\right);\\
\label{eq:VB:introduced_theta}
\tilde{\theta}_{y, z} &= \exp\left(\psi\left(\tilde{\alpha}_{y, z}\right) - \psi\left(\sum\limits_{y' \in \mathcal{Y}}\tilde{\alpha}_{y', z}\right)\right),\end{aligned}$$ where $\psi(\cdot)$ is the digamma function.
Using these additional notations, the E-like step is formulated the same as the E-step of the EM-algorithm, replacing everywhere the estimates of the parameters with the corresponding tilde introduced notation and true posterior distributions of the hidden variables with the corresponding approximated ones in (\[eq:E:alpha\]) – (\[eq:E:y\_i,t\]).
The point estimates of the parameters can be obtained by expected values of the posterior approximated distributions. An expected value for a Dirichlet distribution (a posterior distribution for all the parameters) is a normalised vector of hyperparameters. Using the expressions for the hyperparameters from (\[eq:VB:beta\]) – (\[eq:VB:gamma\]), the final parameters estimates can be obtained by: $$\begin{aligned}
\label{eq:VB:phi}
\widehat{\phi}_{x, y}^{\,\text{VB}} &= \dfrac{\beta_x + \hat{n}_{x, y}^{\, \text{VB}}}{\sum\limits_{x' \in \mathcal{X}} \left(\beta_{x'} + \hat{n}_{x', y}^{\, \text{VB}}\right)}, &\forall x \in \mathcal{X}, y \in \mathcal{Y};\\
\label{eq:VB:theta}
\widehat{\theta}_{y, z}^{\,\text{VB}} &= \dfrac{\alpha_y + \hat{n}_{y, z}^{\, \text{VB}}}{\sum\limits_{y' \in \mathcal{Y}} \left(\alpha_{y'} + \hat{n}_{y', z}^{\, \text{VB}}\right)}, &\forall y \in \mathcal{Y}, z \in \mathcal{Z};\\
\label{eq:VB:xi}
\widehat{\xi}_{z', z}^{\,\text{VB}} &= \dfrac{\gamma_{z'} + \hat{n}_{z', z}^{\, \text{VB}}}{\sum\limits_{\check{z} \in \mathcal{Z}} \left(\gamma_{\check{z}} + \hat{n}_{\check{z}, z}^{\, \text{VB}}\right)}, &\forall z', z \in \mathcal{Z};\\
\label{eq:VB:pi}
\widehat{\pi}_{z}^{\,\text{VB}} &= \dfrac{\eta_z + \hat{n}_z^{\,\text{VB}}}{\sum\limits_{z' \in \mathcal{Z}} \left(\eta_{z'} + \hat{n}_{z'}^{\,\text{VB}}\right)}, &\forall z \in \mathcal{Z}.\end{aligned}$$
Learning: Gibbs sampling algorithm {#sec:gibbs}
----------------------------------
In [@Hospedales2011] the collapsed version of Gibbs sampling (GS) is used for parameter learning in the MCTM. The Markov chain is built to sample only the hidden variables $y_{i, t}$ and $z_t$, while the parameters $\boldsymbol{\Phi}$, $\boldsymbol{\Theta}$ and $\boldsymbol{\Xi}$ are integrated out (note that the distribution for the initial behaviour choice $\boldsymbol\pi$ is not considered in [@Hospedales2011]).
During the burn-in stage the hidden topic and behaviour assignments to each token in the dataset are drawn from the conditional distributions given all the remaining variables. Following the Markov Chain Monte Carlo framework it would draw samples from the posterior distribution $p(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}} | \mathbf{x}_{1:T_{tr}}, \boldsymbol\beta, \boldsymbol\alpha, \boldsymbol\eta, \boldsymbol\gamma)$. From the whole sample for $\{\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}\}$ the parameters can be estimated by [@Griffiths2004]: $$\begin{aligned}
\label{eq:GS:phi}
\widehat{\phi}_{x, y}^{\,\text{GS}} &= \dfrac{\hat{n}_{x, y}^{\,\text{GS}} + \beta_x}{\sum\limits_{x' \in \mathcal{X}} \left(\hat{n}_{x', y}^{\,\text{GS}} + \beta_{x'}\right)}, &\forall x \in \mathcal{X}, y \in \mathcal{Y};\\
\widehat{\theta}_{y, z}^{\,\text{GS}} &= \dfrac{\hat{n}_{y, z}^{\,\text{GS}} + \alpha_y}{\sum\limits_{y' \in \mathcal{Y}} \left(\hat{n}_{y', z}^{\,\text{GS}} + \alpha_{y'} \right)}, &\forall y \in \mathcal{Y}, z \in \mathcal{Z};\\
\label{eq:GS:xi}
\widehat{\xi}_{z', z}^{\,\text{GS}} &= \dfrac{\hat{n}_{z', z}^{\,\text{GS}} + \gamma_{z'}}{\sum\limits_{\check{z} \in \mathcal{Z}} \left(\hat{n}_{\check{z}, z}^{\,\text{GS}} + \gamma_{\check{z}} \right)}, &\forall z', z \in \mathcal{Z},\end{aligned}$$ where $\hat{n}_{x, y}^{\,\text{GS}}$ is the count for the number of times when the word $x$ is associated with the topic $y$, $\hat{n}_{y, z}^{\,\text{GS}}$ is the count for the topic $y$ and the behaviour $z$ pair, $\hat{n}_{z', z}^{\,\text{GS}}$ is the count for the number of times when the behaviour $z$ is followed by the behaviour $z'$.
Similarities and differences of the learning algorithms {#sec:comparison}
-------------------------------------------------------
The point parameter estimates for all three learning algorithms (\[eq:M:phi\]) – (\[eq:M:pi\]), (\[eq:VB:phi\]) – (\[eq:VB:pi\]) and (\[eq:GS:phi\]) – (\[eq:GS:xi\]) have a similar form. The EM-algorithm estimates differ up to the hyperparameters reassignment — adding one to all the hyperparameters in the VB or GS algorithms ends up with the same final equations for the parameters estimates in the EM-algorithm. We explore this in the experimental part. This “-1” term in the EM-algorithm formulae (\[eq:M:phi\]) – (\[eq:M\_psi\_k,l\]) occurs because it uses modes of the posterior distributions while the point estimates obtained by the VB and GS algorithms are means of the corresponding posterior distributions. For a Dirichlet distribution, which is a posterior distribution for all the parameters, mode and mean expressions differ by this “-1” term.
The main differences of the methods consist in the ways the counts $n_{x, y}$, $n_{y, z}$ and $n_{z', z}$ are estimated. In the GS algorithm they are calculated by a single sample from the posterior distribution of the hidden variables $p(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}} | \mathbf{x}_{1:T_{tr}}, \boldsymbol\beta, \boldsymbol\alpha, \boldsymbol\gamma)$. In the EM-algorithm the counts are computed as expected numbers of the corresponding events with respect to the posterior distributions of the hidden variables. In the VB algorithm the counts are computed in the same way as in the EM-algorithm up to replacing the true posterior distributions with the approximated ones.
Our observations for the dynamic topic model confirm the comparison results for the vanilla PLSA and LDA models provided in [@Asuncion2009].
Anomaly detection {#sec:abnormality}
=================
This paper presents on-line anomaly detection with the MCTM in video streams. The decision making procedure is divided into two stages. At a learning stage the parameters are estimated using $T_{tr}$ visual documents by one of the learning algorithms, presented in Section \[sec:inference\]. After that during a testing stage a decision about abnormality of new upcoming testing documents is made comparing a marginal likelihood of each document with a threshold. The likelihood is computed using the parameters obtained during the learning stage. The threshold is a parameter of the method and can be set empirically, for example, to label 2% of the testing data as abnormal. This paper presents a comparison of the algorithms (Section \[sec:experiments\]) using the measure independent of threshold value selection.
We also propose an anomaly localisation procedure during the testing stage for those visual documents that are labelled as abnormal. This procedure is designed to provide spatial information about anomalies, while documents labelled as abnormal provide temporal detection. The following sections introduce both the anomaly detection procedure on a document level and the anomaly localisation procedure within a video frame.
Abnormal documents detection
----------------------------
The marginal likelihood of a new visual document $\mathbf{x}_{t+1}$ given all the previous data $\mathbf{x}_{1:t}$ can be used as a normality measure of the document [@Hospedales2011]: $$\begin{gathered}
\label{eq:online_likelihood_integral}
p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}) = \\
\iiint p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}, \boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi}) p(\boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi} | \mathbf{x}_{1:t}) \mathrm{d}\boldsymbol\Phi \mathrm{d}\boldsymbol\Theta \mathrm{d} \boldsymbol{\Xi}.\end{gathered}$$
If the likelihood value is small it means that the current document cannot be fitted to the learnt behaviours and topics, which represent typical motion patterns. Therefore, this is an indication for an abnormal event in this document. The decision about abnormality of a document is then made by comparing the marginal likelihood of the document with the threshold.
In real world applications it is essential to detect anomalies as soon as possible. Hence an approximation of the integral in (\[eq:online\_likelihood\_integral\]) is used for efficient computation. The first approximation is based on the assumption that the training dataset is representative for parameter learning, which means that the posterior probability of the parameters would not change if there is more observed data: $$p(\boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi} | \mathbf{x}_{1:t}) \approx p(\boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi} | \mathbf{x}_{1:Tr}) \quad \forall t \geq T_{tr}.$$
The marginal likelihood can be then approximated as $$\begin{gathered}
\label{eq:online_likelihood_integral_train}
\iiint p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}, \boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi}) p(\boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi} | \mathbf{x}_{1:t}) \mathrm{d}\boldsymbol\Phi \mathrm{d}\boldsymbol\Theta \mathrm{d} \boldsymbol{\Xi} \approx \\
\iiint p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}, \boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi}) p(\boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi} | \mathbf{x}_{1:T_{tr}}) \mathrm{d}\boldsymbol\Phi \mathrm{d}\boldsymbol\Theta \mathrm{d} \boldsymbol{\Xi}.\end{gathered}$$
Depending on the algorithm used for learning the integral in (\[eq:online\_likelihood\_integral\_train\]) can be further approximated in different ways. We consider two types of approximation.
### Plug-in approximation
The point estimates of the parameters can be plug-in in the integral (\[eq:online\_likelihood\_integral\_train\]) for approximation: $$\begin{gathered}
\label{eq:online_likelihood_plugin}
\iiint p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}, \boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi}) p(\boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi} | \mathbf{x}_{1:Tr}) \mathrm{d}\boldsymbol\Phi \mathrm{d}\boldsymbol\Theta \mathrm{d} \boldsymbol{\Xi} \approx\\
\iiint p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}, \boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi}) \delta_{\hat{\boldsymbol{\Phi}}}(\boldsymbol{\Phi}) \delta_{\hat{\boldsymbol{\Theta}}}(\boldsymbol{\Theta}), \delta_{\hat{\boldsymbol{\Xi}}}(\boldsymbol{\Xi}) \mathrm{d}\boldsymbol\Phi \mathrm{d}\boldsymbol\Theta \mathrm{d} \boldsymbol{\Xi} = \\
p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}, \hat{\boldsymbol{\Phi}}, \hat{\boldsymbol{\Theta}}, \hat{\boldsymbol{\Xi}}),\end{gathered}$$ where $\delta_{a}(\cdot)$ is the delta-function with the centre in $a$; $\hat{\boldsymbol{\Phi}}$, $\hat{\boldsymbol{\Theta}}$, $\hat{\boldsymbol{\Xi}}$ are point estimates of the parameters, which can be computed by any of the considered learning algorithms using (\[eq:M:phi\]) – (\[eq:M\_psi\_k,l\]), (\[eq:VB:phi\]) – (\[eq:VB:xi\]) or (\[eq:GS:phi\]) – (\[eq:GS:xi\]).
The product and sum rules, the conditional independence equations from the generative model are then applied and the final formula for the plug-in approximation is as follows: $$\begin{gathered}
\label{eq:online_likelihood_final}
p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}) \approx p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}, \hat{\boldsymbol{\Phi}}, \hat{\boldsymbol{\Theta}}, \hat{\boldsymbol{\Xi}}) =\\
\sum\limits_{z_t}\sum\limits_{z_{t+1}} \left[ p(\mathbf{x}_{t+1} | z_{t+1}, \hat{\boldsymbol{\Phi}}, \hat{\boldsymbol{\Theta}}) \times \right.\\
\left. p(z_{t+1} | z_t, \hat{\boldsymbol{\Xi}}) p(z_t | \mathbf{x}_{1:t}, \hat{\boldsymbol{\Phi}}, \hat{\boldsymbol{\Theta}}, \hat{\boldsymbol{\Xi}})\right],\end{gathered}$$ where the predictive probability of the behaviour for the current document, given the observed data up to the current document, can be computed via the recursive formula: $$\begin{gathered}
\label{eq:predictive_behaviour}
p(z_{t} | \mathbf{x}_{1:t}, \hat{\boldsymbol{\Phi}}, \hat{\boldsymbol{\Theta}}, \hat{\boldsymbol{\Xi}}) = \\
\sum_{z_{t-1}} \dfrac{p(\mathbf{x}_{t} | z_{t}, \hat{\boldsymbol{\Phi}}, \hat{\boldsymbol{\Theta}}) p(z_{t} | z_{t-1}, \hat{\boldsymbol{\Xi}}) p(z_{t-1} | \mathbf{x}_{1:t-1}, \hat{\boldsymbol{\Phi}}, \hat{\boldsymbol{\Theta}}, \hat{\boldsymbol{\Xi}})}{p(\mathbf{x}_{t} | \mathbf{x}_{1:t-1}, \hat{\boldsymbol{\Phi}}, \hat{\boldsymbol{\Theta}}, \hat{\boldsymbol{\Xi}})}.\end{gathered}$$
The point estimates can be computed for all three learning algorithms, therefore a normality measure based on the plug-in approximation of the marginal likelihood is applicable for all of them.
### Monte Carlo approximation
If samples $\{\boldsymbol{\Phi}^{s}, \boldsymbol{\Theta}^{s}, \boldsymbol{\Xi}^{s}\}$ from the posterior distribution $p(\boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi} | \mathbf{x}_{1:T_{tr}})$ of the parameters can be obtained, the integral (\[eq:online\_likelihood\_integral\_train\]) is further approximated by the Monte Carlo method: $$\begin{gathered}
\iiint p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}, \boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi}) p(\boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi} | \mathbf{x}_{1:T_{tr}}) \mathrm{d}\boldsymbol\Phi \mathrm{d}\boldsymbol\Theta \mathrm{d} \boldsymbol{\Xi} \approx\\
\dfrac{1}{S} \sum\limits_{s = 1}^{S} p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}, \boldsymbol{\Phi}^{s}, \boldsymbol{\Theta}^{s}, \boldsymbol{\Xi}^{s}),\end{gathered}$$ where $S$ is the number of samples. These samples can be obtained (i) from the approximated posterior distributions $q(\boldsymbol{\Phi})$, $q(\boldsymbol{\Theta})$, and $q(\boldsymbol{\Xi})$ of the parameters, computed by the VB learning algorithm, or (ii) from the independent samples of the GS scheme. For the conditional likelihood $p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}, \boldsymbol{\Phi}^{s}, \boldsymbol{\Theta}^{s}, \boldsymbol{\Xi}^{s})$ the formula (\[eq:online\_likelihood\_final\]) is valid.
Note that for the approximated posterior distribution of the parameters, i.e., the output of the VB learning algorithm, the integral (\[eq:online\_likelihood\_integral\_train\]) can be resolved analytically, but it would be computationally infeasible. This is the reason why the Monte Carlo approximation is used in this case.
Finally, in order to compare documents of different lengths the normalised likelihood is used as a normality measure $s$: $$s(\mathbf{x}_{t+1}) = \dfrac{1}{N_{t+1}} p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}).$$
Localisation of anomalies {#sec:localisation}
-------------------------
The topic modeling approach allows to compute a likelihood function not only of the whole document but of an individual word within the document too. Recall that the visual word contains the information about a location in the frame. We propose to use the location information from the least probable words (e.g., 10 words with the least likelihood values) to localise anomalies in the frame. Note, we do not require anything additional to a topic model, e.g., modelling regional information explicitly as in [@Haines2010] or comparing a test document with training ones as in [@Pathak2015]. Instead, the proposed anomaly localisation procedure is general and can be applied in any topic modeling based method, where spatial information is encoded to visual words.
The marginal likelihood of a word can be computed in a similar way to the likelihood of the whole document. For the point estimates of the parameters and plug-in approximation of the integral it is: $$p(x_{i, t+1} | \mathbf{x}_{1:t}) \approx
p(x_{i, t+1} | \mathbf{x}_{1:t}, \hat{\boldsymbol{\Phi}}, \hat{\boldsymbol{\Theta}}, \hat{\boldsymbol{\Xi}}).$$ For the samples from the posterior distributions of the parameters and the Monte Carlo integral approximation it is: $$p(x_{i, t+1} | \mathbf{x}_{1:t}) \approx
\dfrac{1}{S} \sum\limits_{s = 1}^{S} p(x_{i, t+1} | \mathbf{x}_{1:t}, \boldsymbol{\Phi}^{s}, \boldsymbol{\Theta}^{s}, \boldsymbol{\Xi}^{s}).$$
Performance validation {#sec:experiments}
======================
We compare the two proposed learning algorithms, based on EM and VB, with the GS algorithm, proposed in [@Hospedales2011], on two real datasets.
Setup
-----
The performance of the algorithms is compared on the QMUL street intersection data [@Hospedales2011] and Idiap traffic junction data [@Varadarajan2009]. Both datasets are $45$-minutes video sequences, captured busy traffic road junctions, where we use a $5$-minute video sequence as a training dataset and others as a testing one. The documents that have less than $20$ visual words are discarded from consideration. In practice these documents can be classified to be normal by default as there is no enough information to make a decision. The frame size for both datasets is $288 \times 360$. Sample frames are presented in Figure \[fig:sample\_frames\].
The size of grid cells is set to $8 \times 8$ pixels for spatial quantisation of the local motion for visual word determination. Non-overlapping clips with a one second length are treated as visual documents.
We also study the influence of the hyperparameters on the learning algorithms. In all the experiments we use the symmetric hyperparameters: $\boldsymbol\alpha = \{\alpha, \dotsc, \alpha\}$, $\boldsymbol\beta = \{\beta, \dotsc, \beta\}$, $\boldsymbol\gamma = \{\gamma, \dotsc, \gamma\}$ and $\boldsymbol\eta = \{\eta, \dotsc, \eta\}$. The three groups of the hyperparameters settings are compared: $\{\alpha = 1, \beta = 1, \gamma = 1, \eta = 1\}$ (referred as “prior type 1”), $\{\alpha = 8 , \beta = 0.05, \gamma = 1, \eta = 1\}$ (“prior type H”) and $\{\alpha = 9, \beta = 1.05, \gamma = 2, \eta = 2\}$ (“prior type H+1”). Note that the first group corresponds to the case when in the EM-algorithm learning scheme the prior components are cancelled out, i.e., the MAP estimates in this case are equal to the maximum likelihood ones. The equations for the point estimates in the EM learning algorithm with the prior type H+1 of the hyperparameters settings are equal to the equations for the point estimates in the VB and GS learning algorithms with the prior type H of the settings. The corresponding Dirichlet distributions with all used parameters are presented in Figure \[fig:dirichlet\_pdf\].
Note that parameter learning is an ill-posed problem in topic modeling [@Vorontsov2014ARTMArticle]. This means there is no unique solution for parameter estimates. We use $20$ Monte Carlo runs for all the learning algorithms with different random initialisations resulting with different solutions. The mean results among these runs are presented below for comparison.
All three algorithms are run with three different groups of hyperparameters settings. The number of topics and behaviours is set to $8$ and $4$, respectively, for the QMUL dataset, $10$ and $3$ are used for the corresponding values for the Idiap dataset. The EM and VB algorithms are run for $100$ iterations. The GS algorithm is run for $500$ burn-in iterations and independent samples are taken with a $100$ iterations delay after the burn-in period.
Performance measure
-------------------
Anomaly detection performance of the algorithms depends on threshold selection. To make a fair comparison of the different learning algorithms we use a performance measure, which is independent of threshold selection.
In binary classification the following measures [@Murphy2012] are used: $\text{TP}$ — true positive, a number of documents, which are correctly detected as positive (abnormal in our case); $\text{TN}$ — true negative, a number of documents, which are correctly detected as negative (normal in our case); $\text{FP}$ — false positive, a number of documents, which are incorrectly detected as positive, when they are negative; $\text{FN}$ — false negative, a number of documents, which are incorrectly detected as negative, when they are positive; $\text{precision} = \dfrac{\text{TP}}{\text{TP} + \text{FP}}$ — a fraction of correct detections among all documents labelled as abnormal by an algorithm; $\text{recall} = \dfrac{\text{TP}}{\text{TP} + \text{FN}}$ — a fraction of correct detections among all truly abnormal documents.
The area under the precision-recall curve is used as a performance measure in this paper. This measure is more informative for detection of rare events than the popular area under the receiver operating characteristic (ROC) curve [@Murphy2012].
Parameter learning
------------------
We visualise the learnt behaviours for the qualitative assessment of the proposed framework (Figures \[fig:qmul\_behaviours\] and \[fig:idiap\_behaviours\]). For illustrative purposes we consider one run of the EM learning algorithm with the prior type H+1 of the hyperparameters settings.
The behaviours learnt for the QMUL data are shown in Figure \[fig:qmul\_behaviours\] (for visualisation words representing $50\%$ of probability mass of a behaviour are used). One can notice that the algorithm correctly recognises the motion patterns in the data. The general motion of the scene follows a cycle: a vertical traffic flow (the first behaviour in Figure \[fig:qmul\_behav\_1\]), when cars move downward and upward on the road; left and right turns (the fourth behaviour in Figure \[fig:qmul\_behav\_4\]): some cars moving on the “vertical” road turn to the perpendicular road at the end of the vertical traffic flow; a left traffic flow (the second behaviour in Figure \[fig:qmul\_behav\_2\]), when cars move from right to left on the “horizontal” road; and a right traffic flow (the third behaviour in Figure \[fig:qmul\_behav\_3\]), when cars move from left to right on the “horizontal” road. Note that the ordering numbers of behaviours correspond to their internal representation in the algorithm. The transition probability matrix $\boldsymbol{\Xi}$ is used to recognise the correct behaviours order in the data.
Figure \[fig:idiap\_behaviours\] presents the behaviours learnt for the Idiap data. In this case the learnt behaviours have also a clear semantic meaning. The scene motion follows a cycle: a pedestrian flow (the first behaviour in Figure \[fig:idiap\_behav\_1\]), when cars stop in front of the stop line and pedestrians cross the road; a downward traffic flow (the third behaviour in Figure \[fig:idiap\_behav\_3\]), when cars move downward along the road; an upward traffic flow (the second behaviour in Figure \[fig:idiap\_behav\_2\]), when cars from left and right sides move upward on the road.
Anomaly detection {#anomaly-detection}
-----------------
In this section the anomaly detection performance achieved by all three learning algorithms is compared. The datasets contain the number of abnormal events, such as jaywalking, car moving on the opposite lane, disruption of the traffic flow (see examples in Figure \[fig:sample\_abnormalities\]).
For the EM learning algorithm the plug-in approximation of the marginal likelihood is used for anomaly detection. For both the VB and GS learning algorithms both the plug-in and Monte Carlo approximations of the likelihood are used. Note that for the GS algorithm samples are obtained during the learning stage, $5$ and $100$ independent samples are taken. For the VB learning algorithm samples are obtained after the learning stage from the posterior distributions, parameters of which are learnt. This means that the number of samples that are used for anomaly detection does not influence on the computational cost of learning. We test the Monte Carlo approximation of the marginal likelihood with $5$ and $100$ samples for the VB learning algorithm.
As a result, we have $21$ methods to compare: obtained by three learning algorithms, three different groups of hyperparameters settings, one type of marginal likelihood approximation for the EM learning algorithm, two types of marginal likelihood approximation for the VB and GS learning algorithms, where two Monte Carlo approximations are used with $5$ and $100$ samples. The list of methods references can be found in Table \[tab:methods\_references\].
Note that we achieve a very fast decision making performance in our framework. Indeed, anomaly detection is made for approximately $0.0044$ sec per visual document by the plug-in approximation of the marginal likelihood, for $0.0177$ sec per document by the Monte Carlo approximation with $5$ samples and for $0.3331$ sec per document by the Monte Carlo approximation with $100$ samples[^2].
Reference Learning algorithm Hyper-parameters settings Marginal likelihood approximation Number of posterior samples
--------------- -------------------- --------------------------- ----------------------------------- -----------------------------
EM 1 p EM type 1 Plug-in —
EM H p EM type H Plug-in —
EM H+1 p EM type H+1 Plug-in —
VB 1 p VB type 1 Plug-in —
VB 1 mc 5 VB type 1 Monte Carlo 5
VB 1 mc 100 VB type 1 Monte Carlo 100
VB H p VB type H Plug-in —
VB H mc 5 VB type H Monte Carlo 5
VB H mc 100 VB type H Monte Carlo 100
VB H+1 p VB type H+1 Plug-in —
VB H+1 mc 5 VB type H+1 Monte Carlo 5
VB H+1 mc 100 VB type H+1 Monte Carlo 100
GS 1 p GS type 1 Plug-in —
GS 1 mc 5 GS type 1 Monte Carlo 5
GS 1 mc 100 GS type 1 Monte Carlo 100
GS H p GS type H Plug-in —
GS H mc 5 GS type H Monte Carlo 5
GS H mc 100 GS type H Monte Carlo 100
GS H+1 p GS type H+1 Plug-in —
GS H+1 mc 5 GS type H+1 Monte Carlo 5
GS H+1 mc 100 GS type H+1 Monte Carlo 100
: Methods references[]{data-label="tab:methods_references"}
The mean areas under precision-recall curves for anomaly detection for all $21$ compared methods can be found in Figure \[fig:mean\_results\]. Below we examine the results with respect to hyperparameters sensitivity, an influence of the likelihood approximation on the final performance, we also compare the learning algorithms and discuss anomaly localisation results.
### Hyperparameters sensitivity
This section presents sensitivity analysis of the anomaly detection methods with respect to changes of the hyperparameters.
The analysis of the mean areas under curves (Figure \[fig:mean\_results\]) suggests that the hyperparameters almost do not influence on the results of the EM learning algorithm, while there is a significant dependence between hyperparameters changes and results of the VB and GS learning algorithms. These conclusions are confirmed by examination of the individual runs of the algorithms. For example, Figure \[fig:hyperparam\_sensitivity\] presents the precision-recall curves for all $20$ runs with different initialisations of $4$ methods for the QMUL data: the VB learning algorithm using the plug-in approximation of the marginal likelihood with the prior types 1 and H of the hyperparameters settings and the EM learning algorithm with the same prior groups of the hyperparameters settings. One can notice that the variance of the curves for the VB learning algorithm with the prior type 1 is larger than the corresponding variance with the prior type H, while the similar variances for the EM learning algorithm are very close to each other.
Note that the results of the EM learning algorithm with the prior type 1 do not significantly differ from the results with the other priors, despite of the fact that the prior type 1 actually cancels out the prior influence on the parameters estimates and equates the MAP and maximum likelihood estimates. We can conclude that the choice of the hyperparameters settings is not a problem for the EM learning algorithm and we can even simplify the derivations considering only the maximum likelihood estimates without the prior influence.
The VB and GS learning algorithms require a proper choice of the hyperparameters settings as they can significantly change the anomaly detection performance. This choice can be performed empirically or with the type II maximum likelihood approach [@Murphy2012].
### Marginal likelihood approximation influence
In this section the influence of the type of the marginal likelihood approximation on the anomaly detection results is studied.
The average results for both datasets (Figure \[fig:mean\_results\]) demonstrate that the type of the marginal likelihood approximation does not influence remarkably on anomaly detection performance. As the plug-in approximation requires less computational resources both in terms of time and memory (as there is no need to sample and store posterior samples and average among them) this type of approximation is recommended to be used for anomaly detection in the proposed framework.
### Learning algorithms comparison
This section compares the anomaly detection performance obtained by three learning algorithms.
The best results in terms of a mean area under a precision-recall curve are obtained by the EM learning algorithm, the worst results are obtained by the GS learning algorithm (Figure \[fig:mean\_results\] and Table \[tab:mean\_area\]). In Table \[tab:mean\_area\] for each learning algorithm the group of hyperparameters settings and the type of marginal likelihood approximation is chosen to have the maximum of the mean area under curves, where a mean is taken over independent runs of the same method and maximum is taken among different settings for the same learning algorithm.
Dataset EM VB GS
--------- -------- -------- --------
QMUL 0.3166 0.3155 0.2970
Idiap 0.3759 0.3729 0.3673
: Mean area under precision-recall curves[]{data-label="tab:mean_area"}
Figure \[fig:best\_worst\_curves\] presents the best and the worst precision-recall curves (in terms of the area under them) for the individual runs of the learning algorithms. The figure shows that among the individual runs the EM learning algorithm also demonstrates the most accurate results. Although, the minimum area under the precision-recall curve for the EM learning algorithm is less than the area under the corresponding curve for the VB algorithm. It means that the variance among the individual curves for the EM learning algorithm is larger in comparison with the VB learning algorithm.
The variance of the precision-recall curves for both VB and GS learning algorithms is relatively small. However, the VB learning algorithm has the curves higher than the curves obtained by the GS learning algorithm. It can be confirmed by examination of the best and worst precision-recall curves (Figure \[fig:best\_worst\_curves\]) and the mean values of the area under curves (Figure \[fig:mean\_results\] and Table \[tab:mean\_area\]).
We also present the results of classification accuracy, i.e., the fraction of the correctly classified documents, for anomaly detection, which can be achieved with some fixed threshold. The best classification accuracy for the EM learning algorithm in both datasets can be found in Table \[tab:accuracy\].
Dataset Accuracy
--------- ----------
QMUL 0.9544
Idiap 0.8891
: Best classification accuracy for the EM learning algorithm[]{data-label="tab:accuracy"}
### Anomaly localisation
We apply the proposed method for anomaly localisation, presented in Section \[sec:localisation\], and get promising results. We demonstrate the localisation results for the EM learning algorithm with the prior type H+1 on both datasets in Figure \[fig:abnormality\_localisation\]. The red rectangle is manually set to locate the abnormal events within the frame, the arrows correspond to the visual words with the smallest marginal likelihood computed by the algorithm. It can be seen that the abnormal events correctly localised by the proposed method.
For quantitative evaluation we analyse $10$ abnormal events ($5$ from each dataset). For each clip for a given number $N_{\text{top}}$ of the least probable words, we measure the recall: $\text{recall} = \dfrac{\text{TP}}{N_{\text{an}}}$, where $N_{\text{an}}$ is the maximum possible number of abnormal words among $N_{\text{top}}$, i.e., $N_{\text{an}} = N_{\text{top}}$ if $N_{\text{top}} \leq N_{\text{total an}}$, where $N_{\text{total an}}$ is the total number of abnormal words, and $N_{\text{an}} = N_{\text{total an}}$ if $N_{\text{top}} > N_{\text{total an}}$. Figure \[fig:quantitative\_localisation\] presents the mean results for all events. One can notice, for example, that when the localisation procedure can possibly detect $45\%$ of the total number of abnormal words, it correctly finds $\approx 90\%$ of them.
![Recall results of the proposed anomaly localisation procedure[]{data-label="fig:quantitative_localisation"}](TNNLS-2016-P-6800-R1-localisation_quantitative){width="0.9\columnwidth"}
Conclusions {#sec:conlusion}
===========
This paper presents two learning algorithms for the dynamic topic model for behaviour analysis in video: the EM-algorithm is developed for the MAP estimates of the model parameters and a variational Bayes inference algorithm is developed for calculating the posterior distributions of them. A detailed comparison of these proposed learning algorithms with the Gibbs sampling based algorithm developed in [@Hospedales2011] is presented. The differences and the similarities of the theoretical aspects for all three learning algorithms are well emphasised. The empirical comparison is performed for abnormal behaviour detection using two unlabelled real video datasets. Both proposed learning algorithms demonstrate more accurate results than the algorithm proposed in [@Hospedales2011] in terms of anomaly detection performance.
The EM learning algorithm demonstrates the best results in terms of the mean values of the performance measure, obtained by the independent runs of the algorithm with different random initialisations. Although, it is noticed that the variance among the precision-recall curves of the individual runs is relatively high. The variational Bayes learning algorithm shows the smaller variance among the precision-recall curves than the EM-algorithm. The results show that the VB algorithm answers are more robust to different initialisation values. However, it is shown that the results of the algorithm are significantly influenced by the choice of the hyperparameters. The hyperparameters require additional tuning before the algorithm can be applied to data. Note that the results of the EM learning algorithm only slightly depend on the choice of the hyperparameters settings. Moreover, the hyperparameters can be even set in such a way as the EM algorithm is applied to obtain the maximum likelihood estimates instead of the maximum a posteriori ones. Both proposed learning algorithms — EM and VB — provide more accurate results in comparison to the Gibbs sampling based algorithm.
We also demonstrate that consideration of marginal likelihoods of visual words rather than visual documents can provide satisfactory results about locations of anomalies within a frame. In our best knowledge the proposed localisation procedure is the first general approach in probabilistic topic modeling that requires only presence of spatial information encoded in visual words.
EM-algorithm derivations
========================
This Appendix presents the details of the proposed EM learning algorithm derivation. The objective function in the EM-algorithm is: $$\begin{aligned}
&\mathcal{Q}(\boldsymbol\Omega, \boldsymbol\Omega^{\text{old}}) + \log p(\boldsymbol\Omega | \boldsymbol\beta, \boldsymbol\alpha, \boldsymbol\eta, \boldsymbol\gamma) = \nonumber\\
&\sum_{\mathbf{y}_{1:T_{tr}}} \sum_{\mathbf{z}_{1:T_{tr}}} \left( p(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}} | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old}) \times \right.\nonumber\\
&\left. \log{p(\mathbf{x}_{1:T_{tr}}, \mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}} | \boldsymbol\Omega, \boldsymbol\alpha, \boldsymbol\beta, \boldsymbol\gamma, \boldsymbol\eta)} \vphantom{p(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}} | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old})} \right) + \nonumber\\
&+ \log p(\boldsymbol\Omega | \boldsymbol\beta, \boldsymbol\alpha, \boldsymbol\eta, \boldsymbol\gamma) = \nonumber\\
&= Const + \sum_{z_1 \in \mathcal{Z}} \left( \log{\pi_{z_1}} \, p(z_1 | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old}) \right) + \nonumber\\
&\sum_{t = 2}^{T_{tr}} \sum_{z_t \in \mathcal{Z}} \sum_{z_{t-1} \in \mathcal{Z}} \left( \log{\xi_{z_t, z_{t-1}}} \, p(z_t, z_{t-1} | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old}) \right) + \nonumber\\
&\sum_{t = 1}^{T_{tr}} \sum_{i = 1}^{N_t} \sum_{y_{i, t} \in \mathcal{Y}} \left( \log{\phi_{x_{i, t}, y_{i, t}}} \, p(y_{i, t} | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old}) \right) + \nonumber\\
&+ \sum_{t = 1}^{T_{tr}} \sum_{i = 1}^{N_t} \sum_{z_t \in \mathcal{Z}} \sum_{y_{i, t} \in \mathcal{Y}} \left( \log{\theta_{y_{i, t}, z_t}}\, p(y_{i, t}, z_t | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old}) \right) + \nonumber\\
&\sum\limits_{z \in \mathcal{Z}} (\eta_z - 1) \log\pi_{z} + \sum\limits_{z \in \mathcal{Z}}\sum\limits_{z' \in \mathcal{Z}} (\gamma_z - 1) \log\xi_{z, z'} + \nonumber\\
\label{eq:em_maximised_function}
&\sum\limits_{z \in \mathcal{Z}} \sum\limits_{y \in \mathcal{Y}} (\alpha_y - 1) \log\theta_{y, z} + \sum\limits_{y \in \mathcal{Y}} \sum\limits_{x \in \mathcal{X}} (\beta_x - 1) \log\phi_{x, y}\end{aligned}$$
On the M-step the function (\[eq:em\_maximised\_function\]) is maximised with respect to the parameters $\boldsymbol\Omega$ with fixed values for $p(z_1 | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old})$, $p(z_t, z_{t-1} | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old})$, $p(y_{i, t} | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old})$, $p(y_{i, t}, z_t | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old})$. The optimisation problem can be solved separately for each parameter, which leads to the equations (\[eq:M:phi\]) – (\[eq:M\_psi\_k,l\]).
On the E-step for the efficient implementation the forward-backward steps are developed for the auxiliary variables $\acute{\alpha}_z(t)$ and $\acute{\beta}_z(t)$: $$\begin{gathered}
\label{eq:alpha_def}
\acute{\alpha}_z(t) \stackrel{\text{def}}{=} p(\mathbf{x}_1, \dotsc, \mathbf{x}_t, z_t = z | \boldsymbol\Omega^{Old}) = \\
\sum\limits_{\mathbf{z}_{1:t-1}} \pi^{Old}_{z_1} \left[ \prod_{\acute{t} = 2}^{t-1} \xi^{Old}_{z_{\acute{t}}, z_{\acute{t}-1}} \right] \left[\prod_{\acute{t} = 1}^{t-1} \prod_{\vphantom{\acute{t}} i = 1}^{N_{\acute{t}}} \sum\limits_{\vphantom{\acute{t}} y \in \mathbf{Y}} \phi^{Old}_{x_{i, \acute{t}}, y} \theta^{Old}_{y, z_{\acute{t}}}\right] \times \\
\xi^{Old}_{z_t = k, z_{t-1}} \prod\limits_{i = 1}^{N_t} \sum\limits_{y \in \mathcal{Y}} \phi^{Old}_{x_{i, t}, y} \theta^{Old}_{y, z_t = z}.\end{gathered}$$ Reorganisation of the terms in (\[eq:alpha\_def\]) leads to the recursive expressions (\[eq:E:alpha\]).
Similarly for $\acute{\beta}_z(t)$: $$\begin{gathered}
\label{eq:beta_def}
\acute{\beta}_k(t) \stackrel{\text{def}}{=} p(\mathbf{x}_{t+1}, \dotsc, \mathbf{x}_{T_{tr}} | z_t = z, \boldsymbol\Omega^{Old}) = \\
\sum\limits_{\mathbf{z}_{t+1 : T_{tr}}} \xi^{Old}_{z_{t+1}, z_t = z} \left[\prod\limits_{\acute{t} = t+2}^{T_{tr}} \xi^{Old}_{z_{\acute{t}}, z_{\acute{t}-1}} \right] \prod\limits_{\acute{t} = t+1}^{T_{tr}} \prod\limits_{\vphantom{\acute{t}} i = 1}^{N_{\acute{t}}} \sum\limits_{\vphantom{\acute{t}} y \in \mathcal{Y}} \phi^{Old}_{x_{i, \acute{t}}, y} \theta^{Old}_{y, z_{\acute{t}}}.\end{gathered}$$ The recursive formula (\[eq:E:beta\]) is obtained by interchanging the terms in (\[eq:beta\_def\]).
The required posterior of the hidden variables terms $p(z_1 | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old})$, $p(z_t, z_{t-1} | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old})$, $p(y_{i, t} | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old})$, $p(y_{i, t}, z_t | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old})$ are then expressed via the axillary variables $\acute{\alpha}_z(t)$ and $\acute{\beta}_z(t)$, which leads to (\[eq:E:z\_t\]) – (\[eq:E:y\_i,t\]).
VB algorithm derivations
========================
This Appendix presents the details of the proposed variational Bayes inference derivation. We have separated the parameters and the hidden variables. Let us consider the update formula of the variational Bayes inference scheme [@Murphy2012] for the parameters: $$\begin{aligned}
\label{eq:q_param_full}
&\log q(\boldsymbol\Omega) = Const + \nonumber\\
&\mathbb{E}_{q(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}})} \log p(\mathbf{x}_{1:T_{tr}}, \mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}, \boldsymbol\Omega | \boldsymbol\eta, \boldsymbol\gamma, \boldsymbol\alpha, \boldsymbol\beta)= \nonumber\\
&Const + \mathbb{E}_{q(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}})} \left( \sum\limits_{z \in \mathcal{Z}} \left(\eta_z - 1 \right) \log \pi_z + \right.\nonumber\\
&\sum\limits_{z \in \mathcal{Z}} \sum\limits_{\tilde{z} \in \mathcal{Z}} \left(\gamma_{\tilde{z}} - 1\right) \log \xi_{\tilde{z}, z} + \sum\limits_{z \in \mathcal{Z}} \sum\limits_{y \in \mathcal{Y}} \left(\alpha_y - 1\right) \log \theta_{y, z} + \nonumber\\
&\sum\limits_{y \in \mathcal{Y}} \sum\limits_{x \in \mathcal{X}} \left(\beta_x - 1 \right) \log \phi_{x, y} + \sum\limits_{z \in \mathcal{Z}} \mathbb{I}(z_1 = z) \log \pi_z + \nonumber\\
&\sum\limits_{t = 2}^{T_{tr}} \sum\limits_{z \in \mathcal{Z}} \sum\limits_{\tilde{z} \in \mathcal{Z}} \mathbb{I}(z_t = \tilde{z}) \mathbb{I}(z_{t-1} = z) \log\xi_{\tilde{z}, z} + \nonumber\\
&\sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} \sum\limits_{y \in \mathcal{Y}} \mathbb{I}\left(y_{i, t} = y\right) \log \phi_{x_{i, t}, y} + \nonumber\\
&\left. \sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} \sum\limits_{z \in \mathcal{Z}} \sum\limits_{y \in \mathcal{Y}} \mathbb{I}(y_{i, t} = y) \mathbb{I}(z_t = z) \log \theta_{y, z} \right)\end{aligned}$$
One can notice that $\log q(\boldsymbol\Omega)$ is further factorised as in (\[eq:q\_param\_factorisation\]). Now each factorisation term can be considered independently. Derivations of the equations (\[eq:VB:beta\]) – (\[eq:VB:gamma\]) are very similar to each other. We provide the derivation only of the term $q(\boldsymbol{\Phi})$: $$\begin{aligned}
&\log q(\boldsymbol\Phi) = Const + \nonumber\\
&\mathbb{E}_{q(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}})} \left(\vphantom{\sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} \sum\limits_{y \in \mathcal{Y}}}\sum\limits_{y \in \mathcal{Y}} \sum\limits_{x \in \mathcal{X}} \left(\beta_x - 1 \right) \log \phi_{x, y} + \right. \nonumber\\
&\left.\sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} \sum\limits_{y \in \mathcal{Y}} \mathbb{I}\left( y_{i, t} = y \right) \log \phi_{x_{i, t}, y} \right) = \nonumber\\
&Const + \sum\limits_{y \in \mathcal{Y}} \sum\limits_{x \in \mathcal{X}} \left(\beta_x - 1 \right) \log \phi_{x, y} + \nonumber\\
&\sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} \sum\limits_{y \in \mathcal{Y}} \log \phi_{x_{i, t}, y} \underbrace{\mathbb{E}_{q(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}})}\left( \mathbb{I}\left(y_{i, t} = y\right)\right)}_{q(y_{i, t} = y)} = \nonumber\\
&Const + \nonumber\\
\label{eq:log_q_phi}
&\sum\limits_{y \in \mathcal{Y}} \sum\limits_{x \in \mathcal{X}} \log \phi_{x, y} \left( \beta_x - 1 + \sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} \mathbb{I}(x_{i, t} = x) q(y_{i,t} = y) \right)\end{aligned}$$ It can be noticed from (\[eq:log\_q\_phi\]) that the distribution of $\boldsymbol\Phi$ is a product of the Dirichlet distributions (\[eq:VB:beta\]).
The update formula in the variational Bayes inference scheme for the hidden variables is as follows: $$\begin{aligned}
&\log q(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}) = Const + \nonumber\\
&\mathbb{E}_{q(\boldsymbol\pi)q(\boldsymbol\Xi)q(\boldsymbol\Theta)q(\boldsymbol\Phi)} \log p(\mathbf{x}_{1:T_{tr}}, \mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}, \boldsymbol\Omega | \boldsymbol\eta, \boldsymbol\gamma, \boldsymbol\alpha, \boldsymbol\beta) = \nonumber\\
&Const + \sum\limits_{z \in \mathcal{Z}} \mathbb{I}\left(z_1 = z\right) \mathbb{E}_{q(\boldsymbol\pi)} \log \pi_z + \nonumber\\
&\sum\limits_{t = 2}^{T_{tr}} \sum\limits_{z \in \mathcal{Z}} \sum\limits_{\tilde{z} \in \mathcal{Z}} \mathbb{I}\left(z_t = \tilde{z}\right) \mathbb{I}\left(z_{t-1} = z\right) \mathbb{E}_{q(\boldsymbol\Xi)} \log \xi_{\tilde{z}, z} + \nonumber\\
&\sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} \sum\limits_{y \in \mathcal{Y}} \mathbb{I}\left(y_{i, t} = y\right) \mathbb{E}_{q(\boldsymbol\Phi)} \log \phi_{x_{i, t}, y} + \nonumber\\
\label{eq:log_q_yz_beginning}
&\sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} \sum\limits_{z \in \mathcal{Z}} \sum\limits_{y \in \mathcal{Y}} \mathbb{I}\left(y_{i, t} = y \right) \mathbb{I} \left(z_t = z\right) \mathbb{E}_{q(\boldsymbol\Theta)} \log \theta_{y, z}\end{aligned}$$ We know from the parameters update (\[eq:VB:beta\]) – (\[eq:VB:gamma\]) that their distributions are Dirichlet. Therefore, $\mathbb{E}_{q(\boldsymbol\pi)}\log \pi_z = \psi\left(\tilde{\eta}_z\right) - \psi\left(\sum_{z' \in \mathcal{Z}} \tilde{\eta}_{z'}\right)$ and similarly for all the other expected value expressions.
Using the introduced notations (\[eq:VB:introduced\_pi\]) – (\[eq:VB:introduced\_theta\]) the update formula (\[eq:log\_q\_yz\_beginning\]) for the hidden variables can be then expressed as: $$\begin{aligned}
&\log q(\mathbf{y}_{1:{T_{tr}}}, \mathbf{z}_{1:T_{tr}}) = Const +
\sum\limits_{z \in \mathcal{Z}} \mathbb{I}\left(z_1 = z\right) \log \tilde{\pi}_z + \nonumber\\
&\sum\limits_{t = 2}^{T_{tr}} \sum\limits_{z \in \mathcal{Z}} \sum\limits_{\tilde{z} \in \mathcal{Z}} \mathbb{I}\left(z_t = \tilde{z}\right) \mathbb{I}\left(z_{t-1} = z\right) \log \tilde{\xi}_{\tilde{z}, z} + \nonumber\\
&\sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} \sum\limits_{y \in \mathcal{Y}} \mathbb{I}\left(y_{i, t} = y\right) \log\tilde{\phi}_{x_{i, t}, y} + \nonumber\\
&\sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} \sum\limits_{z \in \mathcal{Z}} \sum\limits_{y \in \mathcal{Z}} \mathbb{I} \left(y_{i, t} = y\right) \mathbb{I}\left(z_t = z\right) \log\tilde{\theta}_{y, z}\end{aligned}$$
The approximated distribution of the hidden variables is then: $$\begin{gathered}
\label{eq:q_yz}
q(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}) = \\
\dfrac{1}{\tilde{K}} \tilde{\pi}_{z_1} \left[\prod\limits_{t = 2}^{T_{tr}} \tilde{\xi}_{z_t, z_{t-1}}\right] \prod\limits_{t = 1}^{T_{tr}} \prod\limits_{i = 1}^{N_t} \tilde{\phi}_{x_{i, t}, y_{i, t}} \tilde{\theta}_{y_{i, t}, z_t},\end{gathered}$$ where $\tilde{K}$ is a normalisation constant. Note that the expression of the true posterior distribution of the hidden variables is the same up to replacing the true parameters variables with the corresponding tilde variables: $$\begin{gathered}
p(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}} | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega) = \\
\dfrac{1}{K} \pi_{z_1} \left[\prod\limits_{t = 2}^{T_{tr}} \xi_{z_t, z_{t-1}} \right] \prod\limits_{t = 1}^{T_{tr}} \prod\limits_{i = 1}^{N_t} \phi_{x_{i, t}, y_{i, t}} \theta_{y_{i, t}, z}\end{gathered}$$ Therefore, to compute the required expressions of the hidden variables $q(z_1 = z)$, $q(z_{t-1} = z, z_t = z')$, $q(y_{i, t} = y, z_t = z)$ and $q(y_{i, t} = y)$ one can use the same forward-backward procedure and update formula as in the E-step of the EM-algorithm replacing all the parameter variables with the corresponding introduced tilde variables.
Acknowledgments {#acknowledgments .unnumbered}
===============
Olga Isupova and Lyudmila Mihaylova would like to thank the support from the EC Seventh Framework Programme \[FP7 2013-2017\] TRAcking in compleX sensor systems (TRAX) Grant agreement no.: 607400. Lyudmila Mihaylova also acknowledges the UK Engineering and Physical Sciences Research Council (EPSRC) for the support via the Bayesian Tracking and Reasoning over Time (BTaRoT) grant EP/K021516/1.
[Olga Isupova]{} is a PhD student at the Department of Automatic Control and Systems Engineering at the University of Sheffield and an Early Stage Researcher in the FP7 Programme TRAX. She received the Specialist (eq. to M.Sc.) degree in Applied Mathematics and Computer Science, 2012, from Lomonosov Moscow State University, Moscow, Russia. Her research is on machine learning, Bayesian nonparametrics, anomaly detection.
[Danil Kuzin]{} is a PhD student at the Department of Automatic Control and Systems Engineering at the University of Sheffield and an engineer at the Rinicom, Ltd. He received the Specialist degree in Applied Mathematics and Computer Science, 2012, from Lomonosov Moscow State University, Moscow, Russia. His research is mainly in sparse modelling for video. His other research interests include nonparametric Bayes and deep reinforcement learning.
\[[{width="1in" height="1.25in"}]{}\] [Lyudmila Mihaylova]{} (M’98, SM’2008) is Professor of Signal Processing and Control at the Department of Automatic Control and Systems Engineering at the University of Sheffield, United Kingdom. Her research is in the areas of machine learning and autonomous systems with various applications such as navigation, surveillance and sensor network systems. She has given a number of talks and tutorials, including the plenary talk for the IEEE Sensor Data Fusion 2015 (Germany), invited talks University of California, Los Angeles, IPAMI Traffic Workshop 2016 (USA), IET ICWMMN 2013 in Beijing, China. Prof. Mihaylova is an Associate Editor of the IEEE Transactions on Aerospace and Electronic Systems and of the Elsevier Signal Processing Journal. She was elected in March 2016 as a president of the International Society of Information Fusion (ISIF). She is on the board of Directors of ISIF and a Senior IEEE member. She was the general co-chair IET Data Fusion $\&$ Target Tracking 2014 and 2012 Conferences, Program co-chair for the 19th International Conference on Information Fusion, 2016, academic chair of Fusion 2010 conference.
[^1]: O.Isupova, D.Kuzin, L.Mihaylova are with the Department of Automatic Control and Systems Engineering, The University of Sheffield, Sheffield, UK e-mail: [email protected], [email protected], [email protected]
[^2]: The computational time is provided for a laptop computer with i7-4702HQ CPU with 2.20GHz, 16 GB RAM using Matlab R2015a implementation.
| ArXiv |
---
abstract: 'We review the stabilization of the radion in the Randall–Sundrum model through the Casimir energy due to a bulk conformally coupled field. We also show some exact self–consistent solutions taking into account the backreaction that this energy induces on the geometry.'
address: |
IFAE, Departament de F[í]{}sica, Universitat Aut[ò]{}noma de Barcelona,\
08193 Bellaterra $($Barcelona$)$, Spain
author:
- 'Oriol Pujol[à]{}s\'
title: 'Effective potential in Brane-World scenarios'
---
epsf
**Abstract**
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$ UAB-FT-504
Introduction
============
Recently, it has been suggested that theories with extra dimensions may provide a solution to the hierarchy problem [@gia; @RS1]. The idea is to introduce a $d$-dimensional internal space of large physical volume ${\cal V}$, so that the the effective lower dimensional Planck mass $m_{pl}\sim {\cal V}^{1/2} M^{(d+2)/2}$ is much larger than $M \sim TeV$- the true fundamental scale of the theory. In the original scenarios, only gravity was allowed to propagate in the higher dimensional bulk, whereas all other matter fields were confined to live on a lower dimensional brane. Randall and Sundrum [@RS1] (RS) introduced a particularly attractive model where the gravitational field created by the branes is taken into account. Their background solution consists of two parallel flat branes, one with positive tension and another one with negative tension embedded in a a five-dimensional Anti-de Sitter (AdS) bulk. In this model, the hierarchy problem is solved if the distance between branes is about $37$ times the AdS radius and we live on the negative tension brane. More recently, scenarios where additional fields propagate in the bulk have been considered [@alex1; @alex2; @alex3; @bagger].
In principle, the distance between branes is a massless degree of freedom, the radion field $\phi$. However, in order to make the theory compatible with observations this radion must be stabilized [@gw1; @gw2; @gt; @cgr; @tm]. Clearly, all fields which propagate in the bulk will give Casimir-type contributions to the vacuum energy, and it seems natural to investigate whether these could provide the stabilizing force which is needed. Here, we shall calculate the radion one loop effective potential $V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}(\phi)$ due to conformally coupled bulk scalar fields, although the result shares many features with other massless bulk fields, such as the graviton, which is addressed in [@gpt]. As we shall see, this effective potential has a rather non-trivial behaviour, which generically develops a local extremum. Depending on the detailed matter content, the extremum could be a maximum or a minimum, where the radion could sit. For the purposes of illustration, here we shall concentrate on the background geometry discussed by Randall and Sundrum, although our methods are also applicable to other geometries, such as the one introduced by Ovrut [*et al.*]{} in the context of eleven dimensional supergravity with one large extra dimension [@ovrut]. This report is based on a work done in collaboration with Jaume Garriga and Takahiro Tanaka [@gpt].
Related calculations of the Casimir interaction amongst branes have been presented in an interesting paper by Fabinger and Hořava [@FH]. In the concluding section we shall comment on the differences between their results and ours.
The Randall-Sundrum model and the radion field
==============================================
To be definite, we shall focus attention on the brane-world model introduced by Randall and Sundrum [@RS1]. In this model the metric in the bulk is anti-de Sitter space (AdS), whose (Euclidean) line element is given by $$ds^2=a^2(z)\eta_{ab}dx^{a}dx^{b}=
a^2(z)\left[dz^2 +d{\bf x}^2\right]
=dy^2+a^2(z)d{\bf x}^2.
\label{rsmetric}$$ Here $a(z)=\ell/z$, where $\ell$ is the AdS radius. The branes are placed at arbitrary locations which we shall denote by $z_+$ and $z_-$, where the positive and negative signs refer to the positive and negative tension branes respectively ($z_+ < z_-$). The “canonically normalized” radion modulus $\phi$ - whose kinetic term contribution to the dimensionally reduced action on the positive tension brane is given by $${1\over 2}\int d^4 x \sqrt{g_+}\, g^{\mu\nu}_+\partial_{\mu}\phi
\,\partial_{\nu}\phi,
\label{kin}$$ is related to the proper distance $d= \Delta y$ between both branes in the following way [@gw1] $$\phi=(3M^3\ell/4\pi)^{1/2} e^{- d/\ell}.$$ Here, $M \sim TeV$ is the fundamental five-dimensional Planck mass. It is usually assumed that $\ell \sim M^{-1}$ . Let us introduce the dimensionless radion $$\lambda \equiv \left({4\pi \over 3M^3\ell}\right)^{1/2} {\phi} =
{z_+ \over z_-} = e^{-d/\ell},$$ which will also be refered to as [*the hierarchy*]{}. The effective four-dimensional Planck mass $m_{pl}$ from the point of view of the negative tension brane is given by $m_{pl}^2 = M^3 \ell
(\lambda^{-2} - 1)$. With $d\sim 37 \ell$, $\lambda$ is the small number responsible for the discrepancy between $m_{pl}$ and $M$.
At the classical level, the radion is massless. However, as we shall see, bulk fields give rise to a Casimir energy which depends on the interbrane separation. This induces an effective potential $V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}(\phi)$ which by convention we take to be the energy density per unit physical volume on the positive tension brane, as a function of $\phi$. This potential must be added to the kinetic term (\[kin\]) in order to obtain the effective action for the radion: $$S_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}[\phi]
=\int d^4x\, a_+^4 \left[{1\over 2}g_+^{\mu\nu}\partial_{\mu}\phi\,
\partial_{\nu}\phi +
V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}(\lambda(\phi))
\right].
\label{effect}$$ In the following Section, we calculate the contributions to $V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}$ from conformally invariant bulk fields.
Massless scalar bulk fields
===========================
The effective potential induced by scalar fields with arbitrary coupling to the curvature or bulk mass and boundary mass can be addressed. It reduces to a similar calculation to minimal the coupling massless field case, which is sovled in [@gpt], and correponds to bulk gravitons. However, for the sake of simplicity, we shall only consider below the contribution to $V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}(\phi)$ from conformally coupled massless bulk fields. Technically, this is much simpler than finding the contribution from bulk gravitons and the problem of backreaction of the Casimir energy onto the background can be taken into consideration in this case. Here we are considering generalizations of the original RS proposal [@alex1; @alex2; @alex3] which allow several fields other than the graviton only (contributing as a minimally coupled scalar field).
A conformally coupled scalar $\chi$ obeys the equation of motion $$-\Box_g \chi + {D-2 \over 4 (D-1)}\ R\ \chi =0,
\label{confin}$$ $$\Box^{(0)} \hat\chi =0.
\label{fse}$$ Here $\Box^{(0)}$ is the [*flat space*]{} d’Alembertian. It is customary to impose $Z_2$ symmetry on the bulk fields, with some parity given. If we choose even parity for $\hat\chi$, this results in Neumann boundary conditions $$\partial_{z}\hat\chi = 0,$$ at $z_+$ and $z_-$. The eigenvalues of the d’Alembertian subject to these conditions are given by $$\label{flateigenvalues}
\lambda^2_{n,k}=\left({n \pi \over L}\right)^2+k^2,$$ where $n$ is a positive integer, $L=z_{-}-z_+$ is the coordinate distance between both branes and $k$ is the coordinate momentum parallel to the branes. [^1]
Similarly, we could consider the case of massless fermions in the RS background. The Dirac equation,[^2] $$\gamma^{n}e^a_{\ n}\nabla_a\,\psi=0.$$ is conformally invariant [@bida], and the conformally rescaled components of the fermion obey the flat space equation (\[fse\]) with Neumann boundary conditions. Thus, the spectrum (\[flateigenvalues\]) is also valid for massless fermions.
Flat Spacetime
---------------
Let us now consider the Casimir energy density in the conformally related flat space problem. We shall first look at the effective potential per unit area on the brane, ${\cal A}$. For bosons, this is given $$V^b_0 = {1\over 2 {\cal A}} {\rm Tr}\ {\rm\ln} (-\bar\Box^{(0)}/\mu^2).$$ Here $\mu$ is an arbitrary renormalization scale. Using zeta function regularization (see e.g. [@ramond]), it is straightforward to show that $$V^b_0 (L)= {(-1)^{\eta-1} \over (4\pi)^{\eta} \eta!}
\left({\pi\over L}\right)^{D-1} \zeta'_R(1-D).
\label{vboson}$$ Here $\eta=(D-1)/2$, and $\zeta_R$ is the standard Riemann’s zeta function. The contribution of a massless fermion is given by the same expression but with opposite sign: $$V^{f}_0(L) = - V_0^b(L).
\label{vfermion}$$ The expectation value of the energy momentum tensor is traceless in flat space for conformally invariant fields. Moreover, because of the symmetries of our background, it must have the form [@bida] $$\langle T^z_{\ z}\rangle_{flat}= (D-1) \rho_0(z),\quad
\langle T^i_{\ j}\rangle_{flat}= -{\rho_0(z)}\ \delta^i_{\ j}.$$ By the conservation of energy-momentum, $\rho_0$ must be a constant, given by $$\rho_0^{b,f} = {V_0^{b,f} \over 2 L} = \mp {A \over 2 L^D},$$ where the minus and plus signs refer to bosons and fermions respectively and we have introduced $$A\equiv{(-1)^{\eta} \over (4\pi)^{\eta} \eta!}
\pi^{D-1} \zeta'_R(1-D) > 0.$$ This result [@adpq; @dpq], which is a simple generalization to codimension-1 branes embedded in higher dimensional spacetimes of the usual Casimir energy calculation, and it reproduces the same kind of behaviour: the effective potential depends on the interbrane distance monotonously. So, depending on $D$ and the field’s spin, it induces an atractive or repulsive force, describing correspondingly the collapse or the indefinite separation of the branes, just as happened in the Appelquist and Chodos calculation [@ac]. In this case, then, the stabilization of the interbrane distance cannot be due to quantum fluctuations of fields propagationg into the bulk.
AdS Spacetime
--------------
Now, let us consider the curved space case. Since the bulk dimension is odd, there is no conformal anomaly [@bida] and the energy momentum tensor is traceless in the curved case too.[^3] This tensor is related to the flat space one by (see e.g. [@bida]) $$<T^{\mu}_{\ \nu}>_g = a^{-D} <T^{\mu}_{\ \nu}>_{flat}.$$ Hence, the energy density is given by $$\rho = a^{-D} \rho_0.
\label{dilute}$$ The effective potential per unit physical volume on the positive tension brane is thus given by $$V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}(\lambda) =
2\ a_+^{1-D} \int a^D(z) \rho\, dz =
\mp \ell^{1-D}{A \lambda^{D-1} \over (1-\lambda)^{D-1}}.
\label{ve1}$$ Note that the background solution $a(z)=\ell/z$ has only been used in the very last step.
The previous expression for the effective potential takes into account the casimir energy of the bulk, but it is not complete because in general the effective potential receives additional contributions from both branes. We can always add to $V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}$ terms which correspond to finite renormalization of the tension on both branes. These are proportional to $\lambda^0$ and $\lambda^{D-1}$. The coefficients in front of these two powers of $\lambda$ cannot be determined from our calculation and can only be fixed by imposing suitable renormalization conditions which relate them to observables. Adding those terms and particularizing to the case of $D=5$, we have $$V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}(\lambda) = \mp \ell^{-4}\left[{A\lambda^4 \over (1-\lambda)^4} +
\alpha+\beta\lambda^4\right],
\label{confveff}$$ where $A\approx 2.46 \cdot 10^{-3}$. The values $\alpha$ and $\beta$ can be obtained from the observed value of the “hierarchy”, $\lambda_{obs}$, and the observed value of the effective four-dimensional cosmological constant, which we take to be zero. Thus, we take as our renormalization conditions $$V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}(\lambda_{obs})
={dV_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}\over d\lambda}(\lambda_{obs})=0.
\label{renc}$$ If there are other bulk fields, such as the graviton, which give additional classical or quantum mechanical contributions to the radion potential, then those should be included in $V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}$. From the renormalization conditions (\[renc\]) the unknown coefficients $\alpha$ and $\beta$ can be found, and then the mass of the radion is calculable. In Fig. \[fig1\] we plot (\[confveff\]) for a fermionic field and a chosen value of $\lambda_{obs}$.
=10 cm\
From (\[renc\]), we have $$\beta = - A (1-\lambda_{obs})^{-5},\quad \alpha= -\beta
\lambda_{obs}^5.
\label{consts}$$ These values correspond to changes $\delta \sigma_{\pm}$ on the positive and negative brane tensions which are related by the equation $$\delta\sigma_+ = -\lambda^5_{obs}\ \delta\sigma_-.
\label{reltensions}$$ As we shall see below, Eq. (\[reltensions\]) is just what is needed in order to have a static solution according to the five dimensional equations of motion, once the casimir energy is included.
We can now calculate the mass of the radion field $m_\phi^{(-)}$ from the point of view of the negative tension brane. For $\lambda_{obs}\ll 1$ we have: $$m^{2\ (-)}_\phi =
\lambda_{obs}^{-2}\ m^{2\ (+)}_\phi = \lambda_{obs}^{-2}\
{d^2 V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}\over
d\phi^2}\approx \mp
\lambda_{obs} \left({5\pi^3 \zeta'_R(-4)\over 6 M^3 l^5}\right).
\label{massconf}$$ The contribution to the radion mass squared is negative for bosons and positive for fermions. Thus, depending on the matter content of the bulk, it is clear that the radion may be stabilized due to this effect.
Note, however, that if the “observed” interbrane separation is large, then the induced mass is small. So if we try to solve the hierarchy problem geometrically with a large internal volume, then $\lambda_{obs}$ is of order $TeV/m_{pl}$ and the mass (\[massconf\]) is much smaller than the $TeV$ scale. Such a light radion would seem to be in conflict with observations. In this case we must accept the existence of another stabilization mechanism (perhaps classical or nonperturbative) contributing a large mass to the radion. Of course, another possibility is to have $\lambda_{obs}$ of order one, with $M$ and $\ell$ of order $m_{pl}$, in which case the radion mass (\[massconf\]) would be very large, but then we must look for a different solution to the hierarchy problem.
Casimir Energy Backreatcion
----------------------------
Due to conformal invariance, it is straightforward to take into account the backreaction of the Casimir energy on the geometry. First of all, we note that the metric (\[rsmetric\]) is analogous to a Friedmann-Robertson-Walker metric, where the nontrivial direction is space-like instead of timelike. The dependence of $a$ on the transverse direction can be found from the Friedmann equation $$\left({a'\over a}\right)^2 = {16\pi G_5 \over 3} \rho - {\Lambda \over 6}.
\label{friedmann}$$ Here a prime indicates derivative with respect to the proper coordinate $y$ \[see Eq. (\[rsmetric\])\], and $\Lambda<0$ is the background cosmological constant. Combined with (\[dilute\]), which relates the energy density $\rho$ to the scale factor $a$, Eq. (\[friedmann\]) becomes a first order ordinary differential equation for $a$. We should also take into account the matching conditions at the boundaries $$\left({a'\over a}\right)_{\pm}={\mp 8\pi G_5 \over 6} \sigma_{\pm}.
\label{matching}$$ A static solution of Eqs. (\[friedmann\]) and (\[matching\]) can be found by a suitable adjustment of the brane tensions. Indeed, since the branes are flat, the value of the scale factor on the positive tension brane is conventional and we may take $a_+=1$. Now, the tension $\sigma_+$ can be chosen quite arbitrarily. Once this is done, Eq. (\[matching\]) determines the derivative $a'_+$, and Eq. (\[friedmann\]) determines the value of $\rho_0$. In turn, $\rho_0$ determines the co-moving interbrane distance $L$, and hence the location of the second brane. Finally, integrating (\[friedmann\]) up to the second brane, the tension $\sigma_-$ must be adjusted so that the matching condition (\[matching\]) is satisfied. Thus, as with other stabilization scenarios, a single fine-tuning is needed in order to obtain a vanishing four-dimensional cosmological constant.
This is in fact the dynamics underlying our choice of renormalization conditions (\[renc\]) which we used in order to determine $\alpha$ and $\beta$. Indeed, let us write $\sigma_+=\sigma_0 + \delta\sigma_+$ and $\sigma_- =-\sigma_0 +\delta\sigma_-$, where $\sigma_0=(3 / 4\pi \ell G_5)$ is the absolute value of the tension of the branes in the zeroth order background solution. Elliminating $a'/a$ from (\[matching\]) and (\[friedmann\]), we easily recover the relation (\[reltensions\]), which had previously been obtained by extremizing the effective potential and imposing zero effective four-dimensional cosmological constant (here, $\delta\sigma_{\pm}$ is treated as a small parameter, so that extremization of the effective action coincides with extremization of the effective potential on the background solution.) In that picture, the necessity of a single fine tuning is seen as follows. The tension on one of the walls can be chosen quite arbitrarily. For instance, we may freely pick a value for $\beta$, which renormalizes the tension of the brane located at $z_-$. Once this is given, the value of the interbrane distance $\lambda_{obs}$ is fixed by the first of Eqs. (\[consts\]). Then, the value of $\alpha$, which renormalizes the tension of the brane at $z_+$, must be fine-tuned to satisfy the second of Eqs. (5.8).
Eqs. (\[friedmann\]) and (\[matching\]) can of course be solved nonperturbatively. We may consider, for instance, the situation where there is no background cosmological constant ($\Lambda=0$). In this case we easily obtain $$a^3(z)={6\pi G A \over (z_- -z_+)^5}(C-z)^2
={{3\over 4}\pi^3 \zeta'_R(-4) G_5 }{(z_0-z)^2\over(z_- -z_+)^5},$$ where the brane tensions are given by $$2\pi G \sigma_{\pm}=\pm (C-z_{\pm})^{-1}$$ and $C$ is a constant. This is a self–consistent solution where the warp in the extra dimension is entirely due to the Casimir energy.
Of course, the conformal interbrane distance $(z_--z_+)$ is different from the physical $d$, although they are related. For instance, imposing $a(z_+)=1$, which we can rewrite as $$6\pi A G_5 = \left({z_--z_+\over z_0-z_+}\right)^2 (z_--z_+)^3$$ and we get the relation $$d=(z_--z_+)\left[{3\over5}\sqrt{(z_--z_+)^3 \over 6\pi G_5 A}\left(
1-\biggl(1-\sqrt{6\pi G_5 A\over(z_--z_+)^3} \;\biggr)^{5/3}\right)\right].$$ Here we can see that the case of negligible Casimir energy, ${6\pi G_5 A/(z_--z_+)^3} \ll 1$, indeed corresponds to the flat case, in which the conformal and the physical distances coincide.
We can also integrate Eq. (\[friedmann\]) in the general case [@tesina], and get $$a(y)=\left({16\pi A M^3 \over{-\Lambda
(z_--z_+)^5}}\right)^{1/5}\sinh^{2/5}\left({5\over2}\sqrt{-\Lambda/6}\;(y_0-y)\right),$$ with brane tensions given by $$\sigma_{\pm}=\pm
{3\over{4\pi}}{\sqrt{-\Lambda/6}\over
G_5}\coth\left({5\over2}\sqrt{-\Lambda/6}\;(y_0-y_\pm)\right).$$ Here we are assuming $\Lambda<0$, and $y_0$ is an integration constant. Moreover we can explicitely check how this reduces to RS solution in the limit of small Casimir energy compared to the cosmological constant, [*i.e.*]{}, when $${16\pi G_5 \over 3} \rho_0 \ll {\Lambda \over 6}.$$ Again fixing $a(z_+)=1$ we find $$y_0={2\over5}\sqrt{-6\over\Lambda}\; {\rm arcsinh}\left(\left({32\pi
\rho_0\over-\Lambda M^3}\right)^{-1/2}\right) >> 1$$ since $(32\pi \rho_0/(-\Lambda M^3)) << 1 $, so that we can write the warp factor as a power series in the parameter $(32\pi \rho_0/(-\Lambda M^3))^{1/5} \ll 1$: $$a(y)\approx e^{-\sqrt{-\Lambda/6}\;y}\left( 1 - {1\over5}\left({128\pi
\rho_0 \over -\Lambda M^3}\right)^{2/5} e^{2\sqrt{-\Lambda/6}\;y} +\dots
\right).$$
Conclusions and discussion
==========================
We have shown that in brane-world scenarios with a warped extra dimension, it is in principle possible to stabilize the radion $\phi$ through the Casimir force induced by bulk fields. Specifically, conformally invariant fields induce an effective potential of the form (\[confveff\]) as measured from the positive tension brane. From the point of view of the negative tension brane, this corresponds to an energy density per unit physical volume of the order $$V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}^{-}\sim m_{pl}^4
\left[{A\lambda^4 \over (1-\lambda)^4}+\alpha+\beta\lambda^4\right],$$ where $A$ is a calculable number (of order $10^{-3}$ per degree of freedom), and $\lambda \sim \phi/(M^3 \ell)^{1/2}$ is the dimensionless radion. Here $M$ is the higher-dimensional Planck mass, and $\ell$ is the AdS radius, which are both assumed to be of the same order, whereas $m_{pl}$ is the lower-dimensional Planck mass. In the absence of any fine-tuning, the potential will have an extremum at $\lambda \sim 1$, where the radion may be stabilized (at a mass of order $m_{pl}$). However, this stabilization scenario without fine-tuning would not explain the hierarchy between $m_{pl}$ and the $TeV$.
A hierarchy can be generated by adjusting $\beta$ according to (\[consts\]), with $\lambda_{obs}\sim (TeV/m_{pl}) \sim
10^{-16}$ (of course one must also adjust $\alpha$ in order to have vanishing four-dimensional cosmological constant). But with these adjustement, the mass of the radion would be very small, of order $$m^{2\ (-)}_{\phi} \sim\lambda_{obs}\ M^{-3} \ell^{-5}
\sim \lambda_{obs} (TeV)^2.
\label{smallmass}$$ Therefore, in order to make the model compatible with observations, an alternative mechanism must be invoked in order to stabilize the radion, giving it a mass of order $TeV$.
Goldberger and Wise [@gw1; @gw2], for instance, introduced a field $v$ with suitable classical potential terms in the bulk and on the branes. In this model, the potential terms on the branes are chosen so that the v.e.v. of the field in the positive tension brane $v_+$ is different from the v.e.v. on the negative tension brane $v_-$. Thus, there is a competition between the potential energy of the scalar field in the bulk and the gradient which is necessary to go from $v_+$ to $v_-$. The radion sits at the value where the sum of gradient and potential energies is minimized. This mechanism is perhaps somewhat [*ad hoc*]{}, but it has the virtue that a large hierarchy and an acceptable radion mass can be achieved without much fine tuning. It is reassuring that in this case the Casimir contributions, given by (\[smallmass\]), would be very small and would not spoil the model.
The graviton contribution to the radion effective potential can be computed as well. Each polarization of the gravitons contribute as minimally coupled massles bulk scalar field [@tamaproof], so since gravitons are not conformally invariant, the calculation is considerably more involved, and a suitable method has been developed for this purpose [@gpt]. The result is that gravitons contribute a negative term to the radion mass squared, but this term is even smaller than (\[smallmass\]), by an extra power of $\lambda_{obs}$. More over this method works also in AdS space for scalar fields of any kind (massive, nonminimally coupled …).
In an interesting recent paper [@FH], Fabinger and Hořrava have considered the Casimir force in a brane-world scenario similar to the one discussed here, where the internal space is topologically $S^1/Z_2$. In their case, however, the gravitational field of the branes is ignored and the extra dimension is not warped. As a result, their effective potential is monotonic and stabilization does not occur (at least in the regime where the one loop calculation is reliable, just like in the original Kaluza-Klein compactification on a circle [@ac]). The question of gravitational backreaction of the Casimir energy onto the background geometry is also discussed in [@FH]. Again, since the gravitational field of the branes is not considered, they do not find static solutions. This is in contrast with our case, where static solutions can be found by suitable adjustment of the brane tensions.
Finally, it should be pointed out that the treatment of backreaction (here and in [@FH]) applies to conformally invariant fields but not to gravitons. Gravitons are similar to minimally coupled scalar fields, for which it is well known that the Casimir energy density diverges near the boundaries [@bida]. Therefore, a physical cut-off related to the brane effective width seems to be needed so that the energy density remains finite everywhere. Presumably, our conclusions will be unchanged provided that this cut-off length is small compared with the interbrane separation, but further investigation of this issue would be interesting.
It seems also interesting to clarify whether the same stabilization mechanism works in other kind of warped compactified brane world models, such as some coming form M-theory [@ovrut]. In this case the bulk instead of a slice of AdS (which is maximally symmetric), consists of a power-law warp factor, and consequently a less symmetric space. This complicates the calculation since, for instance, there are two 4-d massless moduli fields (apart from the 4-d gravitons) to stabilize.
After the work reported here [@gpt; @tesina] was complete, Ref. [@muko] appeared with some overlapping results, and also [@bmno; @noz] with related topics.
Acknowledgements
================
I devote special thanks to Jaume Garriga and Takahiro Tanaka for the entire realization of this work. I would like to thank Enric Verdaguer and Edgar Gunzig for his kind hospitallity in the Peyresq-5 Meeting. I acknowledge support from CICYT, under grant AEN99-0766, and from grant 1998FI 00198.
[99]{} N. Arkani-Hamed, S. Dimopoulos, G. Dvali, “The Hierarchy Problem and New Dimensions at a Millimeter,” Phys.Lett. [**B429**]{}, 263 (1998) \[hep-ph/9803315\]; “Phenomenology, Astrophysics and Cosmology of Theories with Submillimeter Dimensions and TeV Scale Quantum Gravity,” Phys.Rev. [**D59**]{} 086004 (1999) \[hep-ph/9807344\]; “New Dimensions at a Millimeter to a Fermi and Superstrigs at a TeV,” I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys.Lett. [**B436**]{}, 257 (1998) \[hep-ph/9804398\]. L. Randall and R. Sundrum, “A Large Mass Hierarchy from a Small Extra Dimension,” Phys. Rev. Lett.[**83**]{}, 3370 (1999) \[hep-ph/9905221\]. T. Gherghetta and A. Pomarol, “Bulk fields and supersymmetry in a slice of AdS,” hep-ph/0003129. A. Pomarol, “Gauge bosons in a five-dimensional theory with localized gravity,” hep-ph/9911294. A. Pomarol, “Grand unified theories without the desert,” hep-ph/0005293. R. Altendorfer, J. Bagger and D. Nemeschansky, “Supersymmetric Randall-Sundrum scenario,” hep-th/0003117. W. D. Goldberger and M. B. Wise, “Modulus stabilization with bulk fields,” Phys. Rev. Lett. [**83**]{}, 4922 (1999) \[hep-ph/9907447\]. W. D. Goldberger and M. B. Wise, “Phenomenology of a stabilized modulus,” hep-ph/9911457. J. Garriga and T. Tanaka, “Gravity in the Randall-Sundrum Brane World,” Phys. Rev. Lett. [**84**]{}, 2778 (1999) \[hep-th/9911055\]. C. Charmousis, R. Gregory, V. Rubakov “Wave function of the radion in a brane world”, hep-th/9912160. T. Tanaka and X. Montes, “Gravity in the brane-world for two-branes model with stabilized modulus,” hep-th/0001092. J. Garriga, O. Pujol[à]{}s and T. Tanaka, “Radion effective potential in the brane-world,” \[hep-th/0004109\]; A. Lukas, B. A. Ovrut, K. S. Stelle and D. Waldram, “The universe as a domain wall,” Phys. Rev. [**D59**]{}, 086001 (1999) \[hep-th/9803235\]. M. Fabinger and P. Hořava, “Casimir effect between world-branes in heterotic M-theory,” hep-th/0002073. N. D. Birrell and P. C. Davies, “Quantum Fields In Curved Space,” [*Cambridge, Uk: Univ. Pr.*]{} (1982) 340p. P. Ramond, “Field Theory: A Modern Primer,” [*(Frontiers in Physics, 74), Redwood City, USA: Addison-Wesley*]{} (1989) 329p. I. Antoniadis, S. Dimopoulos, A. Pomarol and M. Quiros, “Soft masses in theories with supersymmetry breaking by TeV-compactification,” Nucl. Phys. [**B544**]{}, 503 (1999) \[hep-ph/9810410\]. A. Delgado, A. Pomarol and M. Quiros, “Supersymmetry and electroweak breaking from extra dimensions at the TeV-scale,” Phys. Rev. [**D60**]{}, 095008 (1999) \[hep-ph/9812489\]. T. Appelquist and A. Chodos, “Quantum Effects in Kaluza-Klein Theories,” Phys. Rev. Lett. [**50**]{}, 141 (1983); “The Quantum Dynamics of Kaluza-Klein Theories,” Phys. Rev. [**D28**]{}, 772 (1983). K. Kirsten, “The a(5) heat kernel coefficient on a manifold with boundary,” Class. Quant. Grav. [**15**]{}, L5 (1998) \[hep-th/9708081\]. D.V. Vassilevich, “Vector fields on a disk with mixed boundary conditions,” J. Math. Phys. [**36**]{}, 3174 (1995). O. Pujol[à]{}s, [*Master Thesis*]{}, unpublished. T. Tanaka, in preparation. S. Mukohyama “Quantum effects, brane tension and large hierarchy in the brane-world, ” Phys. Rev. [**D63**]{}, 044008 (2001) I. Brevik, K. A. Milton, S. Nojiri and S. D. Odintsov, “Quantum (in)stability of a brane-world AdS(5) universe at nonzero temperature,” hep-th/0010205. S. Nojiri, S. D. Odintsov and S. Zerbini, “Bulk versus boundary (gravitational Casimir) effects in quantum creation of inflationary brane world universe,” Class. Quant. Grav. [**17**]{}, 4855 (2000) \[hep-th/0006115\].
[^1]: If we considered an odd parity field, then we would impose Dirichlet boundary conditions, $\hat\chi(z_-)=\hat\chi(z_+)=0$, and the set of eigenvalues would be the same except for the zero mode, which only the even field has.
[^2]: Here, $e^a_{\ n}$ is the f[ü]{}nfbein, $n,m,\dots$ are flat indices, $a,b,\dots$ are “world” indices, and $\gamma^n$ are the Dirac matrices. The covariant derivative can be expressed in terms of the spin connection $\omega_{an m}$ as $\nabla_a=\partial_a+{1\over 2} \omega_{anm} \Sigma^{nm}$, where $\Sigma^{nm}={1\over 4}[\gamma^n,\gamma^m]$ are the generators of the Lorentz transformations in spin $1/2$ representation.
[^3]: One can see that for the conformally coupled case and for the spacetime we are considering the anomaly vanishes even on the branes. Since the conformal anomally is given by the Seeley-de Witt coefficient $a_{5/2}$, and this is a conformal invariant quantity, in the conformally coupled case $a_{5/2}$ is the same as the flat spacetime related problem, which is zero in this case because the branes are flat. We can also use the expressions found for $a_{5/2}$ presented in Ref. [@klaus; @vass] to show how in this case it vanishes.
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